HIGHER-ORDER SLIDING MODE TECHNIQUES FOR FAULT DIAGNOSIS

Transcription

1 Ph.D. in Electronic and Computer Engineering Dept. of Electrical and Electronic Engineering University of Cagliari HIGHER-ORDER SLIDING MODE TECHNIQUES FOR FAULT DIAGNOSIS Nicola Orani Advisor: Elio Usai Curriculum: ING-INF/04 Automatica XXII Cycle March 2010

2

3 Ph.D. in Electronic and Computer Engineering Dept. of Electrical and Electronic Engineering University of Cagliari HIGHER-ORDER SLIDING MODE TECHNIQUES FOR FAULT DIAGNOSIS Nicola Orani Advisor: Elio Usai Curriculum: ING-INF/04 Automatica XXII Cycle March 2010

4

5 Dedicated to my family

6

7 Acknowledgments Special thanks to my supervisor Elio Usai. Also, i would like to thank Alessandro Pisano and Giorgio Bartolini for their indispensable collaboration. Finally, special thanks to my wife Valentina and to my children Federica and Gabriele, both born during my PhD studies, that supported, tolerated and motivated me during these three years. Tanks again. i

8

9 Contents 1 FDI-Generalities Nomenclature Introduction Fault diagnosis Modelling of faulty systems (Process) Model-Based Fault Detection Methods Residual generation Parameter Estimation Techniques Parity equations techniques State estimation and observer based techniques Basic fault detection observer-based schemes Unknown Input Observer for Fault Detection FDI schemes based on UIO and output observers Structural fault detectability, isolability and identifiability Structural fault detectability Structural fault isolability Structural fault identifiability Sliding Mode Observers SMO for Linear Systems SMO for linear systems partially driven by unknown inputs A classical approach HOSMO to weaken the 1-relative degree condition Non-linear approaches to Sliding Mode Observers design Systems in the companion form Triangular systems Quasi-continuous HOSM observers Algebraic observers Sliding Modes for FDI SMO for faults reconstruction SMO for reconstructing of the input fault signals SMO for reconstructing of the output fault signals Filtration-free fault reconstruction via full order HOSMO iii

10 iv CONTENTS Actuator faults reconstruction Filtration-free fault reconstruction via reduced order HOSMO Fault observer design Actuator fault reconstruction Simulation results Discrete state reconstruction via HOSMO Introduction Problem formulation Assumptions Comments on the considered class of systems Discrete state observer design Observer input design Discrete state reconstruction Application results Three-tank system case study Mathematical model Simulation results Fault reconstruction via HOSM Discrete state estimation via HOSM Experimental results System identification Fault reconstruction via HOSM Discrete state estimation via HOSM Concluding remarks 123 Bibliography 125

11 List of Figures 1.1 The fault topology in the controlled system The monitored system and fault topology Time-dependency of faults : a)abrupt; b)incipient; c)intermittent Linear input output model and faults Basic model faults: additive(a); multiplicative (b) General scheme for analytical fault-detection and diagnosis method Residual generation general structure Residual generation via output estimator Model structure for parameter estimation with equation error Model structure for parameter estimation with output error Scheme for Output error via parity equation method The referred system model Scheme for equation error via parity equation method Parity equation method for a MIMO state-space model; Differentiator filter Process and state observer Multivariable process with disturbance v(t), w(t) and faults f u (t), f y (t) Process and output observer The UIO Structure Bank of estimators for output residual generation (DOS) The GOS scheme for FDI of system inputs Utkin Observer The state variable and the state observation error The actual fault and the reconstructed signal fault Configuration of the considered three-tank system System input-output Shape of tank Shape of tank Shape of tank Fault topology System inputs, outputs and disturbances Test 1. Faults and disturbance reconstruction performance Test 2. Reconstructed faults and disturbance in presence of noisy measures Test 1. δ i (t) vs. δ i (t). From top to bottom: i = 1,2, v

12 vi LIST OF FIGURES 6.11 Test 1. δ i (t) vs. ˆδ i (t). From top to bottom: i = 1,2, Test 1. Actual σ(t) and reconstructed ˆσ(t) Test 2. σ(t) (solid) and ˆσ(t) (dashed) varying observer gain Test 3. Actual δ i (t) vs. δ i (t) discrete inputs Test 3. Actual σ(t) vs. ˆσ(t) discrete states The experimental setup Principle of parameters identification Test 1- Reference signals Test 1- Measured signals Test 2- Reference signals Test 2- Measured signals Test 1- Reconstructed faults Test 2- Reconstructed faults Actual δ 1 (t) and non-thresholded reconstructed δ 1 (t) discrete inputs δ 1 (t) (solid) and ˆδ 1 (t) (dashed) varying observer gain δ i (t) vs ˆδ i (t). From top to bottom: i = 1,2, Actual σ(t) and reconstructed ˆσ(t) discrete state

13 List of Tables 1.1 Fault Signature Parameters of the Three Tank System vii

14

15 Summary The field of research inherent this thesis is that of "Sliding Modes (SM) techniques for Model Based Fault Diagnosis and Identification (FDI) for dynamic systems". In general this topic is of great importance and interest in industry and engineering, mainly for economic, security and reliability reasons, in recent decades, there was a significant development in science and a wide range of theoretical approaches have been formulated and are therefore available in literature. Many contributions to FDI are based on the principle of analytical redundancy, which is essentially based on using a model of the system under observation in order to estimate the evolution of the corresponding real variables. If a fault occurs it causes a discrepancy or "residual" between the actual behavior and the model, which is then used to identify faults. A very important property that FDI systems should have is the rejection of false alarms that could occur in the presence of model mismatch or measurement noises and therefore discern between residues generated by faults and those due to disturbances. To this purpose it is important to use "robust techniques" to overcome the problems due to measurement noises and uncertainties. In general sliding modes theory has good robustness properties and it can be used to deal with both structural and unmatched uncertainties, so that the application of sliding mode techniques to robust FDI offers good potential. First of all a careful analysis of the literature was made to assess the state of the art about "model based Fault Diagnosis and Identification", with particular attention on available Sliding Modes techniques for Model Based FDI. We proceeded first to examine the techniques currently available in the literature and at the same time, deepen and improve some methodologies SM Observer-Based for Nonlinear Uncertain Systems. It must be highlighted that dissimilar to initial approaches aimed to design SM observers like residual generators, for which in the presence of a fault the sliding motion was destroyed, the underlying philosophy of the mayor promising techniques available in literature provide for the design of SMOs in which the sliding motion is maintained even in the presence of faults, which are then detected and identified by analyzing the so-called equivalent output injection. In this regard, it must be highlighted that, the major contributions of my research activities involving FDI concerns the use of HOSMO (Higher Order Sliding Mode Observers). The use of continuous time output injection signals, in order to circumvents some limitations resulting from the use of a conventional first order SM observers, provide for finite time reconstruction, theoretically exact, of additive actuator faults for nonlinear uncertain systems. Furthermore, based on the assumption that it is always possible to associate a specific nonlinear dynamic to a system affected by specific faults, an HOSMO for the identification of the actual dynamic, and therefore of the fault signature, 1

16 2 LIST OF TABLES for nonlinear uncertain switched system has been proposed. The studied techniques have been extensively verified by using a the three tank water process experimental set up. It is worth to remark that the problems studied in the present thesis are related to the topics of the ongoing FP7 project PRODI - Power plants Robustification based On fault Detection and Isolation algorithms" which supported part of the work made during my PhD course.

17 Chapter 1 FDI-Generalities 1.1 Nomenclature By going through the literature, one recognizes immediately that the terminology in the field of the FDI (Fault Detection and Identification) is not consistent. This makes it difficult to understand the goals of the contributions and to compare the different approaches. For example, which are the differences between faults, failures, malfunctioning and errors? Or the differences between fault (or failure) detection, isolation, identification and diagnosis? Or supervision and monitoring? Or supervisory functions and supervisory control? The SAFEPROCESS Technical Committee discussed this matter, and tried to find commonly accepted definitions. Below, some definitions are suggested, based on the discussions within the Committee. States and Signals Fault An unpermitted deviation of at least one characteristic property or parameter of the system from the acceptable, usual or standard condition. Failure A permanent interruption of a system s ability to perform a required function under specified operating conditions. Malfunction An intermittent irregularity in the fulfillment of a system s desired function. Error A deviation between a measured or computed value of an output variable and its true or theoretically correct one. Disturbance An unknown and uncontrolled input acting on a system. Perturbation An input acting on a system, which results in a temporary departure from the current state. Residual A fault indicator, based on a deviation between measurements and model-equationbased computations. Symptom A change of an observable quantity from normal behavior. 3

18 4 CHAPTER 1. FDI-GENERALITIES Functions Fault detection Determination of faults present in a system and the time of detection. Fault isolation Determination of the kind, location and time of detection of a fault. It follows fault detection. Fault identification Determination of the size and time-variant behavior of a fault. It follows fault isolation. Fault diagnosis Determination of the kind, size, location and time of detection of a fault. It follows fault detection. It includes fault detection and identification. Monitoring A continuous real-time task to determining the conditions of a physical system, by recording information, recognizing and indicating anomalies in the behavior. Supervision Monitoring a physical and taking appropriate actions to maintain the operation in the case of fault. Protection Means by which a potentially dangerous behavior of the system is suppressed if possible, or means by which the consequences of a dangerous behaviour are avoided. Models Quantitative model Use of static and dynamic relations among system variables and parameters in order to describe a system s behaviour in quantitative mathematical terms. Qualitative model Use of static and dynamic relations among system variables in order to describe a system s behaviour in qualitative terms such as causalities and IF- THEN rules. Diagnostic model A set of static or dynamic relations which link specific input variables, the symptoms, to specific output variables, the faults. Analytical redundancy Use of more (not necessarily identical) ways to determine a variable, where one way uses a mathematical process model in analytical form. System properties Reliability Ability of a system to perform a required function under stated conditions, within a given scope, during a given period of time. Measure: MTBF = Mean Time Between Failures. Safety Ability of a system not to cause danger to persons or equipment or the environment. Availability Probability that a system or equipment will operate satisfactorily and effectively at any point of time. Measure: A=MTBF/(MTBF+MTTR) where MTTR = Mean Time To Repair.

19 1.2. INTRODUCTION 5 Dependability A form of availability that has the property of always being available when required. It is the degree to which a system is operable and capable of performing its required function at any randomly chosen time during its specified operating time. Time dependency of faults Abrupt fault Fault modeled as stepwise function. It represents bias in the monitored signal. Incipient fault Fault modeled by using ramp signals. It represents drift of the monitored signal. Intermittent fault Combination of impulses with different amplitudes. Fault terminology Additive fault Influences a variable by an addition of the fault itself. They may represent, e.g., offsets of sensors. Multiplicative fault Are represented by the product of a variable with the fault itself. They can appear as parameter changes within a process. 1.2 Introduction For the improvement of reliability, safety and efficiency advanced methods of supervision, fault-detection and fault diagnosis become increasingly important for many technical processes. This holds especially for safety related to power plants and chemical plants. In active FTC (Fault Tolerant Control), FDI plays a vital role to provide information on faults/failures in the system and to enable appropriate reconfiguration to take place. Therefore the main function of FDI is to detect a fault or failure and to find its location so that corrective action can be made to eliminate or minimize the effect on the overall system performance. The classical approaches are limit or trend checking of some measurable output variables based on hardware or physical redundancy methods which use multiple sensors, actuators, components to measure and control a particular variable. Typically, a voting technique is applied to the hardware redundant system to decide if a fault has occurred and its location among all the redundant system components. The major problems encountered with hardware redundancy are the extra equipment and maintenance cost, as well as the additional space required to accommodate the equipment. Furthermore they do not give a deeper insight into the process behavior and usually do not allow for an effective prior fault diagnosis. In view of the conflict between reliability and the cost of adding more hardware, it is possible to use the dissimilar measured values together to cross-compare each other, rather than replicating each hardware individually. This is the meaning of analytical or functional redundancy. In the analytical redundancy scheme, the resulting difference generated from the comparison of different variables is called a residual or symptom signal. In brief the residual should be zero when the system is in normal operation and should be different from zero when a fault has occurred. This property of the residual is used to determine whether or not faults have occurred. Consistency checking in analytical redundancy is normally achieved through a comparison

20 6 CHAPTER 1. FDI-GENERALITIES between measured signals and their estimated values. The estimation is generated by a mathematical model of the considered plant. The comparison is done using the residual quantities which are computed as differences between the measured signals and the corresponding signals generated by the mathematical model. Classical approaches, consisting in limit or trend checking of some measurable output variables, do not give a deeper insight into the process since they are only able to react after a relatively large change of a feature occurs, i.e., after either a large sudden fault or a long-lasting gradually increasing fault. In addition, an in-depth fault diagnosis is usually not possible. Therefore advanced methods of supervision and fault diagnosis are needed which satisfy the following requirements: 1. Early detection of small faults with abrupt or incipient time behavior; 2. Diagnosis of faults in the actuator, process components or sensors; 3. Detection of faults in closed loops; 4. Supervision of processes in transient states. To this purpose, model-based methods of FDI were developed by using input and output signals together with dynamic process models. Various methods are proposed in the literature. In the following section the main tasks of fault detection and fault diagnosis are introduced, some basic problems and methods in supervision, fault detection and fault diagnosis are considered. Later model-based fault-detection methods are analyzed, which allow a deep insight into the process behavior. 1.3 Fault diagnosis The task of fault diagnosis consists of determining the type, size and location of the fault as well as its time of detection based on the observed analytical and heuristic symptoms. If no further knowledge on fault symptom causalities is available, classification methods can be applied which allow for a mapping of symptom vectors into fault vectors. To this end, methods like statistical and geometrical classification or neural nets and fuzzy clustering can be used. Note that geometrical analysis, whilst simple to implement, has a few drawbacks. The most serious is that, in the presence of noise, input variations and change of operating point of the monitored process, false alarms are possible. If a-priori knowledge of fault-symptom causalities is available, e.g. in the form of causal networks, diagnostic reasoning strategies can be applied. Forward and backward chaining, with Boolean algebra for binary facts and with approximate reasoning for probabilistic or possibilistic facts, are examples. Finally a fault decision indicates the type, size and location of the most possible fault, as well as its time of detection. In the following we focalized our attention on the techniques based on the dynamic mathematical process model of the system under supervision, i.e. the model-based faultdetection methods are considered, which allow a deep insight the process behavior.

21 1.3. FAULT DIAGNOSIS Modelling of faulty systems The first step in FDI model-based approach consists of providing a mathematical description of the system under investigation which shows all the possible faulty conditions, as well. The detailed scheme for FDI techniques here presented is depicted by Figure 1.4. The main components are the Plant under investigation, the Actuators and Sensors, which can be further sub-divided as input and output sensors, and finally the controller. In the following, the system working conditions will be monitored by means of its input u(t) and output y(t) measurements and signals from the controller u R which are supposed completely available for FDI purposes. It is worth noting that also the behavior of any controller that drives the system can be taken into consideration for FDI purpose. Concerning the occurrence of malfunctions, the location of faults and their modelling, the system under diagnosis can be separated into the following different parts which can be affected by faults: Actuators Process or system components Input sensors Output sensors Controller Typical examples of such faults are: structural defects, such as cracks, ruptures, fractures, leaks, and loose parts; defects in the gears, and aging effects; faults in sensors, such as scaling errors, hysteresis, drift, dead zones, shortcuts, and contact failures; abnormal parameter variations; external obstacles, such as collisions and clogging of outflows. This fault scenario can be summarized by the Figure (1.1). Figure (1.1) also shows the situation where the controller can be affected by faults, since the monitored process consists of a closed-loop system. However, because of technological reasons (e.g., the control action is performed by a digital computer), when the actuator is considered as a part or a component of the whole controller device, the former can be treated as subsystem where faults are likelier to occur whilst the latter remains free from faults. Under these assumptions, as depicted in Figure (1.2) when system is considered in view of fault location, since input and output measurements are supposed completely available for FDI purposes, the controller behavior in the design of a fault diagnosis scheme can be neglected as well as the interconnection between the control system and the process. A fault is defined as an unpermitted deviation of at least one characteristic property of a variable from an acceptable behavior. Therefore, the fault is a state that may lead to a malfunction or failure of the system. The time dependency of faults can be distinguished, as shown in Fig (1.3), abrupt fault (stepwise), incipient fault (driftlike), intermittent fault. Now lumped-parameter processes are considered, which operate in open loop. The static behavior (steady states) is frequently expressed by a non-linear characteristic. By considering small signal deviations around an operating point (Y 00 /U 00 ) the input/output behavior of a SISO process can frequently be described by ordinary linear differential equations

22 8 CHAPTER 1. FDI-GENERALITIES Figure 1.1: The fault topology in the controlled system Figure 1.2: The monitored system and fault topology y(t) + a 1 y (1) (t) + + a n y (n) (t) = b 0 u(t) + b 1 u (1) (t) + + b m u (m) (t) (1.1) where y i represent the i-th time derivative and y(t) = Y (t) Y 00 ;u(t) = U (t) U 00. The corresponding transfer function becomes, through Laplace transformation: G P (s) = G yu (s) = y(s) u(s) = b 0 + b 1 s + + b m s m 1 + a 1 s + + a n s n (1.2) In Fig. (1.4) has been denoted how different faults can affect the nominal system, where input signal faults f u and output signal faults f y, are additive faults and a i, b i represent change in model parameters. Must be highlight that, regard to the process models, as we ll see with more detail, the faults can be further classified. Additive faults appear, e.g., as offsets of sensors, whereas multiplicative faults are parameter changes within a process. According to Fig. (1.5) addi-

23 1.3. FAULT DIAGNOSIS 9 Figure 1.3: Time-dependency of faults : a)abrupt; b)incipient; c)intermittent Figure 1.4: Linear input output model and faults tive faults influence a variable Y by an addition of the fault f, and multiplicative faults by the product of another variable U with f. Figure 1.5: Basic model faults: additive(a); multiplicative (b) Under the hypothesis of linearity, process dynamics can be described by the following continuous time, time-invariant, linear dynamic system (LTI) in the state-space form { ẋ = Ax(t) + Bu(t) y = C x(t) + Du(t) (1.3) where x R n, u R p, y R m and with the corresponding transfer function matrix G yu (s) = y(s) u(s) = C (si A) 1 B + D (1.4) There exist a number of ways to model faults, among them the extension of model 1.4 to y(s) = G yu (s)u(s) +G y f (s)f (s) (1.5)

24 10 CHAPTER 1. FDI-GENERALITIES is a widely used one, where f R q is an unknown vector that represents all possible faults and will be zero in the fault-free case. In this work f is assumed to be deterministic. Suppose that a minimal state-space realization of 1.5 is given by { ẋ = Ax(t) + Bu(t) + E f (t) y = C x(t) + Du(t) + F f (t) (1.6) with known matrices E,F. Then we have { G yu (s) = y(s) u(s) = C (si A) 1 B + D G y f (s) = y(s) f (s) = C (si A) 1 E + F (1.7) It becomes evident that the fault distribution matrices E,F indicate where a fault occur and its influence on the system components. A shown in figure Fig. (1.2) we divide the fault in three categories: sensor faults f s : these faults directly act on the process measurement actuator faults f a : these faults cause changes in the actuator response process faults f p : they are used to indicated malfunctions within the process It s straightforward to note that sensor faults can be modeled by setting F = I, i.e, y = C x(t)+du(t)+ f S (t), while actuators faults by setting E = B, F = D, i.e, ẋ = Ax(t)+Bu(t)+ B f A (t); y = C x(t) + Du(t) + D f A (t). For systems affected by actuator, sensor and process faults, we define f = f A f p f S, E = [B E P 0], F = [D F P I ] (1.8) for some E P and F P and apply 1.6 to represent the system dynamics. Due to the way how they affect the system dynamic, the faults modeled in 1.6 are called additive faults. It s very important to note that the occurrence of an additive fault will not affect the system stability, independently from the fact that a feedback control loop is integrated into the system under observation or not. Typical additive faults met in practice are, for instance, sensors and actuators offsets, described by a constant, or drift in sensors. The former can be described by a constant, while the latter by a ramp. In practice, malfunction in the process or in the sensors and actuators often cause changes in the model parameters. They are called multiplicative faults and generally modeled in term of parameter change. They can be described by extending 1.6 to { ẋ = (A + A)x(t) + (B + B)u(t) y = (C + C )x(t) + (D + D)u(t) (1.9) where A, B, C, D represent the multiplicative faults in the plant, actuators and sensors, respectively. It is assumed that A = l A i=1 A i θ Ai B = l B i=1 B i θ Bi C = l C i=1 C i θ Ci D = l D i=1 D (1.10) i θ Di

25 1.4. (PROCESS) MODEL-BASED FAULT DETECTION METHODS 11 where A i,b i,c i,d i are matrices of known and of appropriate dimensions, and θ Ai,θ Bi, θ Ci,θ Di are unknown time functions. Multiplicative faults are characterized by their possible direct influence on the system stability. This fact is evident for the faults described by A. It must be highlighted that also multiplicative faults could be modeled as additive faults, in such case the new fault vector is a function of the state and input variable of the system and thus will affect the system stability. In any case, the major focus of this work will be on the diagnosis of additive faults. Considering additive and multiplicative faults, and taking into account that environmental disturbance, uncertainty or a mismatch between the nominal plant and the damaged plant as well as measurement and process noises are often modeled as unknown input vectors, we ll denote them by d,η,ν and integrate them into input-output state space models 1.9 and 1.6 as follows { ẋ = Āx(t) + Bu(t) + E f + E d d + η y = C x(t) + Du(t) + F f + F d d + ν (1.11) where Ā = A+ A, B = B + B, C = C + C, D = D + D, d R K D represent a deterministic unknown input vector, and η R n,ν R m represents a steady stochastic process which is assumed to be, if no additional remarks is made, a white, normal distributed noise vector with zero mean. 1.4 (Process) Model-Based Fault Detection Methods Different approaches to FDI using mathematical models have been developed in the last 30 years, see, e.g., [41], [42],[43],[46],[63],[52],[58],[62],[61],[68],... According to the nomenclature explained in previous paragraph, model-based FDI can be defined as the detection, isolation and identification of faults The task consists of the diagnosis of faults in the processes, actuators and sensors by using the dependencies between different measurable signals, i.e., using the priori information on the process. These dependencies are expressed by mathematical process models. Figure 1.6 shows the basic structure of model-based FDI. Based on measured input signals U and output signals Y, the detection methods generate residuals r, parameter estimates or state estimates ˆx, which are called features. By comparison with the normal features (nominal values), changes of features are detected, leading to analytical symptoms s. In FDI model-based techniques "symptoms" and "residual" terms are treated as synonymous, and in the following we shall refer to residual term. In Fig. (1.6) two main blocks can be recognized: 1. Residual generation: First this block generates feature signals (parameters, state estimation or directly residual if a parity equation method is used to detect the feature) using available inputs and outputs from the monitored system. This feature of fault, in conjunction of normal behavior, should indicate that a fault has occurred. The residual should normally be zero or close to zero under no fault condition, whilst distinguishably different from zero when a fault occurs.

26 12 CHAPTER 1. FDI-GENERALITIES Figure 1.6: General scheme for analytical fault-detection and diagnosis method 2. Residual evaluation: This block examines symptoms for the likelihood of faults and a decision rule is then applied to determine if any faults have occurred. The residual evaluation block may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residuals. On the other hand, it may consist of statistical methods, e.g., generalized likelihood ratio testing or sequential probability ratio testing Most contributions in the field of quantitative model-based FDI focus on the residual generation problem, since the decision-making problem can be considered relatively straightforward if residuals are well-designed. The generation of residual (i.e. symptoms) is the main issue in model-based fault diagnosis. A variety of methods are available in the literature for residual generation and this section presents briefly some of the most common methods. The following basic process model-based fault detection schemes will be considered and summarized: 1. Observers-based approach (Output Observers, State Observers, estimators, filters); 2. Parity equations; 3. Identification and parameter estimation An important aspect of these methods is the kind of fault to be detected. As noted in the following, one can distinguish between additive faults, which influence the variables of the process by a summation, and multiplicative faults, which are products of the process variables. The basic methods show different results, depending on these types of faults. If only output signals y(t) can be measured, signal based methods can be applied, e.g. vibrations can be detected, which are related to rotating machinery or electrical circuits. Typical signal model-based methods of fault detection are:

27 1.5. RESIDUAL GENERATION Bandpass Filters; 2. Spectral analysis (FFT); 3. Maximum-entropy estimation; The characteristic quantities or features generated by fault detection methods show stochastic behavior with mean values and variances. Deviations from the normal behavior must then be detected by methods of change detection like: 1. Mean and variance estimation; 2. Likelihood-ratio test; 3. Run-sum test; As previously observed each basic method offers different potentiality in term of capability of fault detection. This is due to different aspects that in following shall be considered. 1.5 Residual generation The most frequently used FDI methods exploit the a priori knowledge of the characteristics of certain signals. As an example, the spectrum, the dynamic range of the signal and its variations may be checked. However, the necessity of a priori information concerning the monitored signals and the dependence of the signal characteristics on unknown working conditions of the system under diagnosis are the main drawbacks of such a class of methods. The most significant contribution in modern model-based approaches is the introduction of the symptom or residual signals, which depend on faults and are independent of system operating states. They represent the inconsistency between the actual system measurements and the corresponding signals of the mathematical model. The residual generator block can be interpreted as illustrated in Figure (1.7) [45]. Figure 1.7: Residual generation general structure

28 14 CHAPTER 1. FDI-GENERALITIES In the above structure, the auxiliary redundant signal z(t) is generated by the function (possibly dynamical) W 1 (u( ), y( )) and, together with the measurement y(t), the symptom signal r (t) is computed by means of W 2 (z( ), y( )). When a fault occurs in the plant, the residual r (t) will be different from zero. The simplest residual generator is depicted in Fig (1.8) and it is obtained when the system W 1 is a model z(t) = W 1 (u( )) of the plant or it is an input-output description for the actual process obtained from system identification procedure (e.g., an Auto Regressive exogenous (ARX) model). In the former case, the measurement y(t) is not required in W 1 because it is a system simulator. The signal z(t) represents the simulated output and the residual is computed as r (t) = y(t) z(t). Since it is an open-loop system, the process simulation may become unstable. Figure 1.8: Residual generation via output estimator An extension to the model-based residual generation is to replace W 1 (u( )) by W 1 (u( ), y( )), i.e. an output estimator fed by both system input and output. In such a case, function W 1 generates an estimation of a linear function of the output W 1 (u( ), y( )) = M y(t), whilst function W 2 can be defined as W 2 (z( ), y( )) = W (z(t) M y(t)), W being a weighting matrix. Concluding, no matter which type of method is used, the residual generation process is nothing but a linear mapping whose inputs consist of process inputs and outputs. According to the definition, ideally, the residual signal r(t) has to be designed to become zero for fault-free case and different from zero in case of failures. This means that r (t) = 0 if and only if f (t) = 0 (1.12) However, modelling errors and disturbances are inevitable. In general, both faults and uncertainty affect the residual, and discrimination between these two effects is difficult. Aspect inherent to the robustness of residuals has to be considerate. To overcome false alarm the simplest and most widely used way to fault detection, after generating the residual, is achieved by directly comparing residual signal r (t) or a residual function J(r (t)) with a fixed threshold ε or a threshold function ε(t) as follows { J(r (t)) ε(t) f or f (t) = 0 J(r (t)) ε(t) f or f (t) 0 (1.13) where f (t) is the general fault vector. If the residual exceeds the threshold, a fault may be occurred. This test works well especially with fixed thresholds if the process operates

29 1.5. RESIDUAL GENERATION 15 approximately in a steady state and it reacts after relatively large feature, i.e. after either a large sudden or a long-lasting gradually increasing fault. Clearly the residual signal should be near zero for the fault-free case, with consequent small thresholds to improve the sensitivity of fault detection, and should increase significantly when a fault appears in the system. On the other hand, adaptive thresholds ε(t) can be exploited which depend on plant operating conditions, for example when ε(t) is expressed as a function of plant inputs. The generation of symptoms is the main issue in model-based fault diagnosis. A variety of methods are available in the literature for residual generation and this section presents briefly some of the most common methods. Most of the residual generation techniques are based on both continuous and discrete system models [46], [63] Parameter Estimation Techniques In most practical cases, the process parameters are not known at all, or they are not known exactly enough. Then, they can be determined by means of parameter estimation methods, measuring input and output signals, u(t) and y(t), if the basic structure of the model is known [63]. This approach is based on the assumption that the faults are reflected in the physical system parameters and the basic idea is that the parameters of the actual process are estimated on-line using well-known parameter estimations methods. The results are thus compared with the parameters of the reference model, obtained initially under faultfree assumptions. Any discrepancy can indicate that a fault may have occurred. Now we compare two different approaches for modelling the input-output behavior of the monitored system: 1. Equation Error Methods 2. Output Error Methods Equation Error Methods The process model written in following form G P (s) = G yu (s) = y(s) u(s) = b 0 + b 1 s + + b m s m 1 + a 1 s + + a n s n (1.14) can be rewritten in vector form as y(t) = Ψ T (t)θ (1.15) where Θ = [a 1... a n,b 1...b m ] T is the parameter vector and Ψ T (t) is Ψ T (t) = [ y (1) (t)... y (n) (t) + u(t)... + u (m) (t)] (1.16) According to Figure (1.9), for parameter estimation, the equation error e(t) is introduced e(t) = y(t) Ψ T (t)θ (1.17) or, assuming the transfer function of the process in the following form G P (s) = G yu (s) = y(s) u(s) = B(s) A(s) (1.18)

30 16 CHAPTER 1. FDI-GENERALITIES the equation error, via La Place transform, becomes e(t) = L 1 ( ˆB(s)u(s) Â(s)y(s)) (1.19) in which Â(s) and ˆB(s) correspond to the estimate of A(s) an B(s). The least-square (LS) estimate ˆΘ = [Ψ T Ψ] 1 Ψ T y (1.20) is obtained if the minimization of the sum of least-squares is computed { J(Θ) = Σt e 2 (t) = e T e d J(s)) dθ) = 0 (1.21) Figure 1.9: Model structure for parameter estimation with equation error The equivalent procedure in z-domain is straightforward to obtain and will be omitted. The least-squares estimated can be also expressed in recursive form (RLS) [46] with respect to the estimates at the instant k, with k = t T 0 = 1,2,3,... ˆΘ(k + 1) = ˆΘ(k) + Υ(K )[Υ(K + 1) Ψ T (K + 1) ˆΘ(k)] (1.22) where { Υ(K ) = 1 P(k)Ψ(k + 1) Ψ T (K +1)P(k)Ψ(K +1) P(k + 1) = [I Υ(K )Ψ T (K + 1)]P(k) (1.23) To improve estimates, filtering methods can be also exploited. In particular, when measurements are affected by noise, a Kalman filter can be used for the parameter estimation. Output Error Methods Instead of the equation error computed previously, the output error e(t) = y(t) ŷ(θ, t) (1.24)

31 1.5. RESIDUAL GENERATION 17 where can also be used as depicted in Fig.(1.10). ŷ(θ, s) = ˆB(s) u(s) (1.25) Â(s) Figure 1.10: Model structure for parameter estimation with output error Unfortunately, direct calculation of the parameter estimate Θ is not possible, because e(t) is non-linear in the parameters. Therefore, the problem of minimization of the sum of least-squares has to be done by numerical optimization methods. The computational effort is then much larger and on-line real-time application is in general impossible. However, relatively precise parameter estimates may be obtained. If a fault within the process changes one or several parameters by Θ, the output signal changes for small deviations according to y(t) = Ψ T (t) Θ(t) + Ψ T (t)θ(t) + Ψ T (t) Θ(t) (1.26) and the parameter estimator indicates a change Θ(t). Generally, the process parameters Θ depend on physical process coefficients p (like stiffness, damping factor, resistance,...) Θ(t) = f (p) (1.27) via non-linear algebraic equations. If the inversion of the relationship exists, changes p of the process coefficients can be calculated. These changes in the coefficients are in many cases directly related to faults. Thus, although the knowledge of p facilitates the fault diagnosis problem, it is not necessary for fault detection only. Parameter estimation can also be applied to non-linear static process models Parity equations techniques The basic idea of the parity relations approach is to provide a proper check of the parity (consistency) of the measurements acquired from the monitored system. In the early development of fault diagnosis, the parity vector (relation) approach was applied to static or

32 18 CHAPTER 1. FDI-GENERALITIES parallel redundancy schemes which may be obtained directly from measurements (hardware redundancy) or from analytical relations (analytical redundancy). A survey of these methods can be found in [67],[47]. In the case of hardware redundancy, two methods can be exploited to obtain redundant relations. The first requires the use of several sensors having identical or similar functions to measure the same variable. The second approach consists of dissimilar sensors to measure different variables but with their outputs being relative to each other. Even if these techniques have been successfully applied for fault diagnosis, the attention of this section is focused on analytical forms of redundancy. A straightforward model-based method of fault detection is to take a model in parallel to the process B(s) A(s) The methodology is depicted in Fig. 1.11, thereby an output error vector e(t) is obtained e(t) = L 1[( B(s) A(s) ˆB(s) ) ] u(s) Â(s) ˆB(s) and run it Â(s) (1.28) Figure 1.11: Scheme for Output error via parity equation method It worth noting that the model parameter and structure of the monitored process have to be known a priori. Let consider the situation depicted in Fig Figure 1.12: The referred system model For the system in Fig under assumption of exact agreement between process and model, the output error assume the following form e(t) = L 1( B(s) ) A(s) f u(s) + f y (s) (1.29)

33 1.5. RESIDUAL GENERATION 19 According to Fig. 1.13, another possibility is to generate a polynomial error ē(s) = Â(s)y(s) ˆB(s)u(s) (1.30) = B(s)f u (s) + A(s)f y (s) (1.31) Figure 1.13: Scheme for equation error via parity equation method In both cases, different time responses are obtained for an additive input or output fault. Moreover, the error vector e(s) corresponds to the output error of parameter estimation method computed by e(t) = y(t) ŷ(θ, t). On the other hand, e(t) concerns the equation error, of the corresponding parameter estimation method, in the form e(t) = y(t) Ψ T (t)θ. The equations relative to e(s) and ē(s) generate residuals, and can therefore used to implement and design the residual generation system in order to meet fault detection and isolation specifications, and are called parity equations [48]. under the assumptions of fault occurrence and of exact agreement between process and model: G M (s) = G P (s) Therefore in the following can be assume i.e. B(s) A(s) = ˆB(s) Â(s) (1.32) e(t) ē(t) r (t) (1.33) However, it must be highlighted that, within the parity equations, the model parameters are assumed to be known and constant, whereas the parameter estimations can vary the parameters of Â(s) and ˆB(s) in order to minimize the residuals. Moreover, for the amplification of specific characteristics of the parity vector r (s) the residuals can be filtered according to matrix G f (s) to compute the filtered residual vector r f (s) with accordance of r f (s) = G f (s)r (s) (1.34) r f (s) can therefore be used to implement and design the residual generation system, in order to meet fault detection and isolation specification, as well. However, for SISO processes only one residual can be generated and it is obviously not easy to distinguish between different faults. On the other hand, more freedom in the design of parity equations can be obtained, for SISO processes, when intermediate signals can be measured, or

34 20 CHAPTER 1. FDI-GENERALITIES for MIMO systems. As an extension of the parity equation method, the parity relation concept presented here can be generalized [49], [51], [50] and then extended to state-space descriptions, as shown in [58] for discrete-time models. The redundancy relations for a continuous-time model are now specified mathematically as follow. Given the system { ẋ = Ax(t) + Bu(t) y = C x(t) (1.35) by substituting the second of Equations in the first one and differential several times, the following system is obtained y(t) ẏ(t) ÿ(t). = C C A C A 2 that can be rewrite in compact form as. x(t) C B 0 0 C AB C B u(t) u(t) ü(t). (1.36) Y f (t) = T x(t) +QU f (f ) (1.37) In order to remove the non-measurable states x(t), and to obtain a parity vector useful for FDI, equation above is multiplied by W, such that W T = 0 (1.38) This leads to residuals as shown in Fig r (t) = W Y f (t) W QU f (t) (1.39) Figure 1.14: Parity equation method for a MIMO state-space model; Differentiator filter The filtered input and output vectors U f and Y f are obtained by digital state variable filters for order n 3. The design of the matrix W gives some freedom to generate a structured set of residuals. One possibility is to select the elements of W such that one measured variable has no impact on a specific residual. Then, this residual remains small in the case of an additive fault on this variable, and the other residuals increase [50], [52].

35 1.5. RESIDUAL GENERATION 21 Finally, because of the previous results, it is clear that some correspondence exists between parity relation and observer-based methods. This aspect was firstly pointed out in [54] and later was demonstrated by [66]. The problem was re-examined in detail by Chen and Patton [50] and the equivalence under different conditions and in different meanings was discussed. It was shown that the parity relation approach is equivalent to the use of a dead-beat observer. This implies that the parity relation scheme provides less design flexibility when compared with methods which are based on observers without any restriction. More recently, a comparison between observer-based and parity space techniques was proposed [57]. Both the methods were first explored for SISO systems and then they were extended to the comparison of MIMO systems. The comparison was performed using linear discrete-time models. In particular, considering MIMO systems described by estimated input-output discretetime forms (e.g., ARX or Auto Regressive Moving Average exogenous (ARMAX) models) of parity equations leads to a representation in which parameters redundancy can not be avoided. To overcome this drawback Delmaire et al. proposed in [57] to use observers designed from identified canonical state-space forms [56]. Moreover, in the case of parameters redundancy, multiple identification of some parameters may occur, leading to inconsistent estimations which might produce inconsistent FDI decisions [57]. This fact states again the FDI capabilities of the observer-based methods with respect to parity relation schemes.

36 22 CHAPTER 1. FDI-GENERALITIES 1.6 State estimation and observer based techniques The most common model-based approach makes use of observers [66].[59]to generate diagnostic signals - residuals. In the framework of FDI, faults are detected by setting a (fixed or variable) threshold on each residual signal. A number of residuals can be designed, each having special sensitivity to individual faults occurring in different locations in the system. The subsequent analysis of each residual, once a threshold is exceeded, then leads to fault isolation [66]. Therefore, the essential issue in model-based FDI is the generation of residuals. Model-based FDI is built upon a number of idealized assumptions, one of which is that the mathematical model used is a faithful replica of the plant dynamics. This is, of course, not possible in practice, as an accurate and complete mathematical description of a process is never available. Sometimes the mathematical structure of the dynamic system is not fully known. For other applications, the parameters of the system may not be fully known, or may only be known over a limited range of the plant s operation. There is therefore always a "model-reality mismatch" between the plant dynamics and the model used for FDI. The basic idea behind the observer or filter-based techniques is to reconstruct the state of the system by using either Luenberger observers (in a deterministic setting) or Kalman filters (in a stochastic noisy environment) and, by means of the reconstructed state, to compute an "expected" output which can be compared with the measured output therefore building the so-called "output estimation error". Such output estimation error (or its weighted value) can be therefore used as residual. It is worth noting that when an observer is exploited for FDI purpose, the estimation of the outputs is necessary, whilst the estimation of the state vector is usually not needed [52]. Moreover, the advantage of using the observer is the flexibility in the selection of its gains which leads to a rich variety of FDI schemes [60]-[61] In order to obtain the structure of a (generalized) observer, the continuous-time, timeinvariant, linear dynamic model for the process under consideration in a state-space form is considered { ẋ = Ax(t) + Bu(t) y = C x(t) (1.40) Assuming that all matrices A, B and C are perfectly known, an observer is used to reconstruct the system variables based on the measured inputs and outputs u(u) and y(t). { ˆx(t) = A ˆx(t) + Bu(t) + He(t) e(t) = y(t) C ˆx(t) The observer scheme described by previous equation is depicted in Fig For the state estimation error e x (t), it follows that { ex (t) = x(t) ˆx(t) ė x (t) = (A HC )e x (t) The state error e x (t) vanishes asymptotically (1.41) (1.42) lim e x(t) = 0 (1.43) t if the observer is stable, which can be achieved by proper design of the observer feedback matrix H. Let the process be influenced by disturbance and faults as depicted in Fig. 1.16

37 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 23 Figure 1.15: Process and state observer Figure 1.16: Multivariable process with disturbance v(t), w(t) and faults f u (t), f y (t) it is described by the following system { ẋ(t) = Ax(t) + Bu(t) +Qν(t) + L1 f (t) y(t) = C x(t) + Rw(t) + L 2 f (t) (1.44) where v(t) and w(t) represent the non-measurable disturbance vector at the input and at the output respectively, and the entries of vector f (t) = [f u (t) T f y (t) T ] R k correspond to specific faults acting on the system through fault distribution matrices L 1 and L 2, and they can represent actuator, process, input and output sensor additive faults. For the state estimation error, under assumption that disturbance are neglected, the following equation hold and the output error becomes ė x (t) = (A HC )e x (t) + L 1 f (t) HL 2 f (t) (1.45)

38 24 CHAPTER 1. FDI-GENERALITIES e(t) = Ce x (t) + L 2 f (t) (1.46) The vector f (t) represent additive faults because they influence e(t) and x(t) by a summation. When sudden and permanent faults f(t) occur, the state estimation error will deviate from zero, e x (t) as well as e(t) show dynamic behavior which are different for L 1 f (t) and L 2 f (t). Both e x (t) and e(t) can be taken as residuals. In particular, the residual e(t) is the basis for different fault detection methods based on output estimation. For the generation of residual with special properties, the design of the observer feedback matrix H is of interest [52]-[53]. Limiting conditions are the stability and the sensitivity against disturbances v(t) and w(t). If the signals are affected by noise, the Kalman filter must be used instead of classical observers. If faults appear as changes A or B or C of the parameters, the process behavior becomes { ẋ(t) = (A + A)x(t) + (B + B)u(t) (1.47) y(t) = (C + C )x(t) while the state e x (t) and the output estimation e(t) errors { ėx (t) = (A HC )e x (t) + ( A H C )x(t) + Bu(t) y(t) = Ce x (t) + C x(t) (1.48) The changes A, B and C are then multiplicative faults [62]-[63]. In this case, the changes in the residuals depend on the parameter changes, as well as input and state variable changes. Hence, the influence of parameter changes on the residuals is not as straightforward as in the case of the additive faults f(t) Basic fault detection observer-based schemes The following observer-based fault detection schemes and configurations are briefly summarized and recalled [66],[62],[52],[63]. 1. Dedicated observers for MIMO process 2. Fault detection filters for MIMO process 3. Output observers Dedicated observers for MIMO process Observer excited by one output: one observer is driven by one sensor output. The other outputs are reconstructed and compared with measured outputs y(t). This allows the detection of single output sensor faults. Bank of observers, excited by all outputs: several observers are designed for a definite fault signal and detected by hypothesis test. Bank of observers, excited by single outputs: several observers for single sensors outputs are used. The estimated outputs are compared with the measured outputs y(t). This allows for the detection of multiple sensor fault (DOS, Dedicated Observer Scheme).

39 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 25 Bank of observers, excited by all outputs and all input except one: as before, but each observer is excited by all inputs except one sensor input, which is supervised (GOS, Generalized Observer Scheme). Fault detection filters for MIMO process The feedback matrix H of the state observer is chosen so that the residual vectors caused by different fault sources have well distinct direction in the residual space. With such "directional" residual vectors, the fault isolation problem consists of determining which of the known fault signature directions the residual vector lies the closest to. The original form of the "Failure detection filter" was proposed by Beard [64] and Jones [65] to generate directional residual vectors. Many more straightforward methods have followed, including methods to achieve "robust fault detection filter" [61]. The fault (or failure) detection is a class of Luenberger observers with a specially designed feedback gain matrix. It allows output estimation errors having directional characteristics associated with some known fault directions, to be obtained. These fault detection methods mostly require several measurable output signals and make use of internal analytical redundancy of multivariable systems. Recently it was proposed to improve their robustness with respect to process parameter changes and unknown input signals v(t) and w(t) [50],[61]. This can be reached, for example, through filtering the output error of the observer by r (t) = W e(t) together with a special design of the observer feedback matrix H (see Fig. 1.15). Output observers Another possibility is the use of output observers (or UIO, see next section) in the reconstruction of the output signals, if the estimate of the state variable ˆx(t) is not of primary interest. In this context, it is worthy to mention the paper by Chen, Patton and Zhang [61] concerning the design of output observers for robust FDI using eigenstructure assignment method. Through a linear transformation the state-space representation of the observer becomes and the residual is determinate by The state estimation error z(t) = T x(t) (1.49) ẑ(t) = F ẑ(t) + Ju(t) +G y(t) (1.50) r (t) = W z ẑ(t) +W y y(t) (1.51) e(t) = ẑ(t) T x(t) (1.52) and the residuals r (t) are then designed, such that they are independent of the process states x(t), the known input u(t) and the unknown inputs v(t) and w(t), as depicted in Figure In this way the residuals are dependent only on fault signals f (t).

40 26 CHAPTER 1. FDI-GENERALITIES Figure 1.17: Process and output observer Unknown Input Observer for Fault Detection As the complexity of a dynamic system increases, the harder is the task of modeling the system and its disturbances, and one can speak of an "uncertain" system, for which there is an uncertainty of knowledge of the system s structure, its parameters and the disturbance effect. There are therefore robustness problems in FDI with respect to modelling errors and disturbances. The goal of robust FDI is to discriminate between the fault effects and the effects of uncertain signals and perturbations. The robustness problem in FDI is thus defined as the maximization of the detectability of faults, together with the minimization of the effect of modelling errors and disturbances on the FDI procedure. The ultimate goal of robustness is to provide rapid and reliable detection and isolation of system faults when the plant under control is disturbed, and when the mathematical model upon which the diagnosis is based cannot faithfully reproduce the full dynamic operation of the plant. There are many approaches for residual generation [66]-[68]. The most common one uses observers. The basic idea behind the observer or filter-based approach is to estimate the outputs of the system from the measurements (or a subset of measurements) by using either Luenberger observer(s) in a deterministic setting or Kalman filter(s) in a stochastic setting. Then, the (weighted) output estimation error (or innovations in the stochastic case), is used as a residual. It is worth noting that for FDI purposes, only the output estimation is required, the estimation of the state vector is not strictly necessary. The most important task in model-based FDI is the generation of residuals which are independent of disturbances. The method is based on disturbance decoupling principle. In this approach, uncertain factors in system modeling or identification are considered to act by means of an "unknown input", the disturbance, on a linear system model. The disturbance vector is unknown but its distribution matrix is usually assumed known. Based on the disturbance distribution matrix obtained by modelling or identification procedure, the unknown input can be de-coupled from the residual. The principle of the Unknown Input Observer (UIO) is to make the state (or output) estimation error decoupled from the unknown inputs. Since the residual is a weighted estimation error, it may be de-coupled from each disturbance. The first step in the disturbance

41 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 27 de-coupled residual generation consists of designing an UIO. Very important contributions to this subject can be found in [52]. UIO or other disturbance de-coupling based residual generation approaches require that the unknown input distribution matrix must be known a priori. The actual unknown input itself does not need to be known. When uncertainties are caused by modelling errors, linearization errors, parameter variations, etc, such a disturbance de-coupling approach cannot be directly applied because the distribution matrix E is normally unknown. To solve this problem, which is of paramount importance in real industrial system applications, various techniques to identify the disturbance distribution matrix are available in literature. The problem of designing an observer for unknown inputs has been studied for nearly two decades and many approaches for the design of both full-order and reduced-order UIO have been proposed (geometric and algebraic methods, singular value decomposition and matrix inversion techniques, linear transformation algorithms)[52]. It is worth noting that for fault detection purpose also the faults can be considered as an unknown input acting on the system. Differently from other type of disturbance acting on systems, for which residuals must be insensitive, residual signals have to be sensitive to faults themselves. In the following we will see a procedure for fault detection using UIO scheme. In this paragraph, a full-order UIO structure is considered and a mathematical method for designing UIO is presented. The necessary and sufficient conditions for this observer to exist are also recalled. These conditions are easy to verify and the design procedure is easy to implement. Let consider a continuous time, time-invariant, linear dynamic systems with an additive unknown disturbance term in the following form: { ẋ = Ax(t) + Bu(t) + E d(t) y = C x(t) (1.53) Where, x(t) R n is the state vector, y(t) R m is the output vector, u(t) R r is the known input vector and d(t) R q the unknown input vector. A,B,E,C are known matrices with appropriate dimensions. It must be highlight that the unknown term E d(t) can be used to describe an additive disturbance, different kinds of modelling uncertainties (noise, unmodelled non-linear terms, time-varying dynamics, etc.) as well as fault terms. The unknown input term may also appear in the output equation, i.e., y(t) = C x(t) + E y d(t) (1.54) but this case is not considered because the term E d(t) can be nullified by using a transformation of the output signal y(t) [52]. Sometimes for systems described by Equation 1.53, there is a term relating the control input u(t) in the output equation, i.e., y(t) = C x(t) + Du(t) (1.55) however, the term Du(t) is omitted in the following since this does not affect the generality of the discussion on the observer design. Definition 1. An observer is defined as an Unknown Input Observer for the system described by 1.53, if its state estimation error vector e x (t) approaches zero asymptotically, regardless of the presence of the unknown input term. In other words an UIO is a robust observer in which the state estimation errors become insensitive to disturbance.

42 28 CHAPTER 1. FDI-GENERALITIES The full-order UIO, for the system 1.53, has the following mathematical form [61] { ż(t) = F z(t) + T Bu(t) + K y(t) ˆx(t) = z(t) + H y(t) (1.56) Where z(t) R n is the state of the UIO, ˆx(t) the estimated state vector x(t) whilst F,T, H and K are matrices to be designed to achieve the unknown input de-coupling. The observer structure is depicted in Figure Figure 1.18: The UIO Structure The state estimation error (e x (t) = x(t) ˆx(t)) obtained by the UIO 1.56 applied to the system 1.53 is described by the equation: ė x (t) = [A HC A K 1 C ]e x (t) [F (A HC A K 1 C )]z(t) [K 2 + (A HC A K 1 C )H]y(t) [T (I HC )]Bu(t) (HC I )E d(t) (1.57) where K = K 1 + K 2. If the following relations hold: then the state estimation error becomes equal to: (HC I )E = 0 I HC = T A HC A K 1 C = F F H = K 2 (1.58) ė x (t) = Fe x (t) (1.59) This means that, if all the eigenvalues of F are stable, e x (t) will approach zero asymptotically, i.e., ˆx(t) x(t). Hence, according to the Definition 1, the observer described

43 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 29 by Equations 1.56 is an UIO for the system The design of this UIO consists of solving matrix equalities stated in eq and making all eigenvalues of the system matrix F with negative real part. A special solution for the matrix H under conditions 1.58 is given in [52]: H = E(C E) + (1.60) where ( ) + represent the pseudo-inverse matrix notation. The following Theorem states the existence conditions for the UIO. Theorem.1. Necessary and sufficient conditions for the existence of an UIO 1.56 for the system defined by Equation 1.53 are [52]: 1. r ank(c E) = r ank(e) 2. (A 1,C ) is a detectable pair, where A 1 = A E(C E) + C A It is worth noting that the number of independent row of the matrix C must not be less than the number of the independent columns of the matrix E to satisfy condition 1 in Theorem 1. It means that the maximum number of disturbances which can be decoupled cannot be larger than the number of the independent measurements. Moreover, without unknown inputs in the system, by setting T = I, H = 0 and E = 0, the observer 1.56 will be a simple Luenberger observer. In this situation, condition 2 in Theorem 1 clearly holds true and such condition is the detectability of pair (A,C ). UIO design procedure It can be seen how K 1 is a free matrix of parameters in the design of an UIO. After K 1 is computed, in order to stabilize the dynamic system matrix F, other parameter matrices in the UIO can be computed by the relation K = K 1 + K 2 and conditions Some design freedom left in the choice of K 1 may be exploited to make the diagnostic residual has directional characteristics. In this work, as we ll see later, because the input-output link of the Multiple-Input Multiple-Output (MIMO) system under investigation is obtained by means of the identification of a collection of Multiple-Input Single-Output (MISO) models, this further degree of freedom will not be used in the residual design. Under these assumptions, if the pair (A 1,C ) is observable, in order to stabilize the system matrix F = A 1 K 1 C, the pole placement routine available in the Control System Toolbox for MATLAB can be used. If (A 1,C ) is not observable, an observable canonical decomposition should be applied to the pair. If (A 1,C )is detectable, the matrix F can be stabilized FDI schemes based on UIO and output observers Considering the system in 1.53 and explaining the dependency from fault signals { ẋ = Ax(t) + Bu(t) + B fu (t) y = C x(t) + f y (t) (1.61) The vectors faults f u (t) = [f u1 (t),..., f ur (t)] T and f y (t) = [f y1 (t)... f ym (t)] T represent actuator and sensor faults respectively, and either assuming values different from

44 30 CHAPTER 1. FDI-GENERALITIES zero only in the presence of faults. Usually these signals are described by step and ramp functions representing abrupt (bias) and incipient faults (drift). { u(t) = u (t) + n u y(t) = y (t) + n y (1.62) It is worth noting that actual input and output measurement can be corrupted with signal noise, usually assumed as white, zero mean, uncorrelated Gaussian signal. To uniquely isolate a fault concerning one of the output sensors, f y (t), under the hypothesis that inputs are fault-free (f u (t) = 0) a bank of classical dynamic Luenberger observers, in a deterministic setting or Kalman Filter (KF) in a stochastic setting, is used, according to Figure Figure 1.19: Bank of estimators for output residual generation (DOS) This observer configuration represents the Dedicated Observer Scheme (DOS) [69]. The number of these observers (estimators) is equal to the number m of system outputs, and each device is driven by a single output and all the inputs of the system. In this case a fault on the i th output affects only the residual function of the output observer or filter driven by the i th output. To uniquely isolate a fault concerning one of the system inputs, f ui (t), under the assumption that outputs are fault-free, (f y (t) = 0) a bank of UIO or UIKF is used (Fig. 1.20). Such a solution is known as the Generalized Observer Scheme (GOS) [66]. The number of these observers is equal to the number r of control inputs. The i-th observer is driven by all but the i-th input and all outputs of the system and generates a residual function which is sensitive to all but the i-th input fault f ui (t), i-th unknown input. In this way the detection of single input measurement faults is possible, since a fault on the i-th input affects all the residual functions except that of the device which is insensitive to the i-th unknown input. In order to summarize the isolation capabilities of the schemes presented, the table below shows the "fault signatures" for the case of a single fault in each input-output signal. The residuals which are affected by the input and output faults are described by an entry 1 in the corresponding table entry, while an entry 0 means that the input or output

45 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 31 Figure 1.20: The GOS scheme for FDI of system inputs Table 1.1: Fault Signature u 1 u 2 u r y 1 y 2 y m r U IO r U IO r U IO r O r O r Om fault does not a affect the corresponding residual. Note how multiple faults in the system outputs can be isolated since a fault on the i-th output signal affects only the residual function r Oi of the output observer driven by the i-th output, but all the UIO or UIKF residual functions r U IOi. On the other hand, multiple faults on the inputs channel cannot be isolated by means of this simplified technique since all the residual functions are sensitive to faults regarding different inputs. However, to overcoming this limitations, dedicated UIO for multiple fault input channel can be designed. DOS With reference to Figure 1.19, in order to diagnose a fault on the i th system output when the measurement noises are negligible (n u = 0,n y = 0) and f u (t) = 0, the model of the i th observer (i = 1,...,m) has the form x i (t) = A i x i (t) + B i u(t) + K i (y i (t) C i x i (t)) (1.63)

46 32 CHAPTER 1. FDI-GENERALITIES where x i (t) is the observer state vector and the triple (A i,b i,c i ) is a minimal state space representation (completely observable) of the link among the inputs of the process and its i th output y i (t). Such a triple can be obtained by means of a MISO identified model. The entries of K i must be designed in order to assign stable, and suitably chosen, eigenvalues to the matrix (A i K i C i ). In this situation and in the absence of faults, i.e., f y (t) = 0, it can be verified that for the i th output residual r i (t) the following relation holds lim r i (t) = lim (y i (t) C i x i (t)) = 0 (1.64) t t and the rate of convergence depends on the position of the eigenvalues of the (A i K i C i ) matrix in the complex left half plane. In the presence of a fault (step or ramp signal) on the i th process output only the i th output residual reaches a value different from zero and this situation leads to a complete failure diagnosis. GOS With reference to the device for the FDI of the input channels, depicted in Figure 1.20, the structure of the i th UIO (i = 1,...,r ) for residual generation [52], under the assumptions (n u = 0,n y = 0) and f y (t) = 0, is the following { żi (t) = (T i A K i C )z i (t) + J i u(t) + S i y(t) r i (t) = L 1 i x i (t) + L 2 i y(t) (1.65) where z i (t) R n denotes the observer state vector, r i (t) R m is the residual vector and F i, J i,s i,l i 1 and Li 2 are matrices to be designed with appropriate dimensions. Let T i be a linear transformation of the state x(t) of the system and define the state estimation error as e x i (t) = z i (t) T i x(t) (1.66) On the imposition (n u = 0,n y = 0) and f y (t) = 0 it can be shown that the dynamics of the state estimation error become e x i (t) = F i e x i (t) + (F i T i T i A + S i C )x(t) + (J i T i B)u(t) T i B f u (t) (1.67) whilst the residual vector is given by It can be seen that if the following holds r i (t) = L 1 i e x i (t) + (L 1 i T i + L 2 i C )x(t) (1.68) Equation 1.67 and 1.68 becomes F i T i T i A + S i C = 0 J i = 0 L 1 i T i + L 2 i C = 0 (1.69) { ex i (t) = F i e x i (t) + T i B f u (t) r i (t) = L 1 i e x i (t) (1.70)

47 1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 33 If the linear transformation T i is chosen as [70] T i = I n B i (C B i ) + C (1.71) where B i is the i th column on B matrix and K i is selected such that F i = T i A K i C is asymptotically stable, then, the solutions of Equation 1.69 are obtained as F i = T i A + K i C S i = K i + F i B i (C B i ) + J i = T i B L 1 i = C L 2 i = [I m (C B i )(C B i ) + ] (1.72) The selection of the B i matrix in Equations 1.71 and 1.72 sets to zero the i th column of the J i matrix. That is, the estimation error and then the residual of the i th UIO become independent of the i t h system input. Under the hypothesis of observability of the system 1.61 and in the absence of input faults (f u (t) = 0), it can be seen that the i th residual vector reaches zero as t approaches infinity and the rate of convergence depends on the position of the eigenvalues of F i matrix. In the presence of a fault on the i th input, the i th residual reaches zero asymptotically while the residuals of the r 1 remaining observers are sensitive to the fault signal. This situation leads to the possibility of unique detection and isolation of all process input faults. The design of this UIO requires the knowledge of a minimal form model (A,B,C ) for the system Such a triple can be computed by using a realization procedure from a MIMO identified model. Residual Robustness The model-based FDI uses a mathematical model for the system. As discussed in earlier sections, the main and most challenging task of model-based FDI is the generation of residuals in which outputs and inputs of the system are processed to generate a fault indicator signal (residual). Ideally, this signal should be near to zero for the fault-free case, and should increase significantly when a fault appears in the system. The better the model represents the system, the better will be the reliability and performance in FDI. However, modelling errors and disturbances are inevitable, and hence there is a need to develop robust FDI algorithms. A robust FDI system is sensitive only to faults, even in the presence of a model reality mismatch. To achieve robustness in FDI, the residual should be insensitive to uncertainty, whilst sensitive to faults, and therefore robust ([60], [66], [61]). The residual that has this property can then be used to detect and isolate faults reliably. In general, both faults and uncertainty affect the residual, and discrimination between these two effects is difficult. The effects of disturbances act as a source of false alarms which must be minimized. The ideal case is to make the residual itself become de-coupled from disturbances (robust residual generation). This is the principle of a robust residual generator which can be achieved by minimizing the effect of disturbances on residuals. In particular, we have to consider that the basis of fault diagnosis technique, under consideration, is the use of mathematical models. Hence, the model should have a certain accuracy. In order to make a diagnosis algorithm robust against modelling uncertainty, we should also have some knowledge about modelling uncertainty. Otherwise, if an FDI algorithm can be made robust without a priori knowledge of the modelling, a model would clearly not be required in the first place. The information of modelling uncertainty is normally represented by assumptions on uncertainty. These assumptions should be easy to handle

48 34 CHAPTER 1. FDI-GENERALITIES by the robust design in a systematic manner, otherwise it does not provide assistance for robust design. The disturbance representation of uncertainty can be handled by the unknown input observer or the eigenstructure assignment. However, this assumption is not realistic, i.e., the distribution matrix cannot always be obtained directly in practice. In real situation, we can obtain some descriptions about uncertainty, for example, parameters of the system are within a certain bound. However, these descriptions are not easy to handle in designing robust FDI algorithms.

49 Chapter 2 Structural fault detectability, isolability and identifiability The concept of structural fault detectability, isolability and identifiability are introduced to describe the structural property of a system from the FDI point of view. Generally speaking one have to distinguish those properties from the performance of a generic FDI scheme, therefore such structural properties will be introduced without any reference to the specified FDI scheme. Study on structural fault detectability, isolability and identifiability plays a central role in the structural analysis for the construction of a technical process and for the design of an FDI scheme. In following, according to the notation in [42], we shall introduce the concepts of structural fault detectability, isolability and identifiability, study their checking criteria and illustrate the major results. In our analysis we ll only consider additive faults and, according to 1.6, we ll refer to the following faulty MIMO system { ẋ(t) = Ax(t) + Bu(t) + E f (t) y(t) = C x + Du(t) + F f (t) (2.1) where f is the fault vector f = [f 1,..., f q ] T. 2.1 Structural fault detectability In the literature, one can find a number of definition of fault detectability, introduced under different aspect. In order to give a definition valid to introduce also the concepts of isolability and identifiability, we first specify our intension of introducing the concept of structural fault detectability. First structural fault detectability should be understood as a structural property of the system under consideration, which describes how a fault affects the system behavior. It should be expressed independent of the system input variables, disturbances as well as model uncertainties. Secondly structural fault detectability should indicate if a fault would cause changes in the system output. Finally, the structural fault detectability should be independent of the type and the size of the fault under consideration. Bearing these in mind, we adopt an intuitive definition of fault detectability 35

50 36 CHAPTER 2. STRUCTURAL FAULT DETECTABILITY, ISOLABILITY AND IDENTIFIABILITY which says: A fault is detectable if its occurrence, independent of its size and type, would cause a change in the nominal behavior of the system outputs. Definition 1 Given system 2.1, a fault f i is structurally detectable if for some u result y fi d f i 0 i = 1,..., q (2.2) f i =0 Therefore a fault become detectable if its occurrence, independently of its size and type, lead to change in the system outputs at least at some time instant and for some system input. Theorem 1 Given system 2.1, a fault f i is structurally detectable iff C (si A) 1 E i + F i 0 (2.3) where E i,f i denoting the i-th column of matrices E,F respectively. It must be highlight that an additive fault is structurally detectable as far as the transfer function from the fault to the system output is non zero. In the following the transfer matrix G fi = C (si A) 1 E i + F i (2.4) will be called fault transfer matrix. In this work, the rank of a transfer matrix is understood as the so-called normal rank, i.e. maximum rank, if no additional specification is given. It must be highlighted that the detection of additive faults can be realized independently of the system inputs [42]. 2.2 Structural fault isolability We say that a group of faults are isolable if any simultaneous occurrence of these faults would lead to a change in the system output. For the sake of simplicity, if we consider only two different detectable faults f i, f j,i j, we say that the faults are isolable if the changes in the system output caused by these two simultaneous faults are distinguishable. Definition 2 Given system 2.1. The faults in a fault vector f are isolable, when y d f 0 (2.5) f f =0 In general case, we say that a group of faults are isolable if any simultaneous occurrence of these faults would lead to a change in the system output. It must be highlighted the similarity between the isolability of additive faults and the so-called input observability which is widely used for the purpose of input reconstruction. Consider the system { ẋ(t) = Ax(t) + E f (t) y(t) = C x(t) + F f (t) (2.6) in which, without loss of generality, we consider the fault vector f as the system input. The input f(t) of system 2.6 is said observable if y(t) = 0 for t > 0 implies f (t) = 0 for t > 0 provided that x(0) = 0 [44].

51 2.2. STRUCTURAL FAULT ISOLABILITY 37 Except the assumption of initial condition x(0) = 0, the physical meanings of the isolability of additive faults and input observability are equivalent. With the aid of the concept of fault transfer matrices 2.4, we now derive existence conditions for the structural fault isolability. Theorem 2 Given system 2.1, than f (t) with fault transfer matrix is structurally isolable iff G f (s) = [G f1 (s)...g fq (s)] (2.7) r ank(g f (s)) = q r ank(g fi (s)) (2.8) i=1 Due to the its straightforwardness, the proof given in [42] will be omitted. Corollary 1 Given the system 2.1 and assume that f i, i = 1,...,l q, are additive faults. Then, these l faults with fault matrix G f (s), are isolable iff r ank(g f (s)) = l (2.9) This result reveals that, to isolate l different faults, we need at least an l-dimensional subspace in the measurement space spanned by the fault transfer matrix. Considering that r ank(g f (s)) mi n{m,l}, where m is the number of sensors, we have the following claim which is very useful for the practical application. Claim. The additive faults are isolable only if the number of the faults is not larger than the number of the sensors. The condition 2.9 can be equivalently expressed in terms of the matrices of the state space description. Indeed, denoting the minimal state space realization of G f (s) by G f (s) = C (si A) 1 E + F (2.10) the following corollary hold Corollary 2 Given the system 2.1 and assume that f i (i = 1,...,l q m) are additive faults. Then these l faults are isolable iff [ A si E r ank C F ] = n + l (2.11) Noting that the relationship between fault isolability (for additive faults) and input observability, as previously discussed consequently to Definition 2, can be further proved by the rank condition in 2.11, since this is the condition for which the input f (t) of system 2.6 is observable with knowledge of initial condition x(0) [44]. To the aim of find out alternative conditions for checking condition 2.9 or 2.11, difficult to use in practical context, now we ll derive some sufficient conditions, on the assumption that m l, for the fault isolability. Let us consider a generic additive fault vector ξ = [ξ 1,ξ 2,...,ξ l ] T, acting on the system 2.6, to which correspond the fault transfer matrix G f (s) = C (si A) 1 E ξ + F ξ. In the following, for the sake of simplicity, we ll refer to matrix

52 38 CHAPTER 2. STRUCTURAL FAULT DETECTABILITY, ISOLABILITY AND IDENTIFIABILITY E ξ and F ξ without subscripts. Initially, to simplify our study, we first consider F = 0. It follows from Cayley-Hamilton Theorem that G f (s) = C (si A) 1 E = 1 n Φ(s) C ( S i s n 1 )E = 1 n Φ(s) C ( α i (s)a i 1 )E (2.12) i=1 i=1 Φ(s) = det(si A) = s n + a 1 s n 1 + a 2 s n a n 1 s + a n S i = S i 1 A + a i 1 I, i = 2,,n; S 1 = I α 1 (s) = s n 1 + a 1 s n a n 1 ; ;α n 1 (s) = s + α 1 (s), α n (s) = 1 which can be rewritten into C (si A) 1 E = 1 Φ(s) [α 1I α 2 I α n I ] C E C AE. C A n 1 E (2.13) (2.14) Thus r ank C E C AE. C A n 1 E = l (2.15) builds a necessary condition for the fault isolability. Now if for some j {1,,n} r ank(c A j 1 E) = l (2.16) where l is the dimension of fault vector, then 2.14 can be rewritten into C (si A) 1 E = 1 Φ(s) [α j I + n i=1,i j α i (s)q i ]C A j 1 E (2.17) where Q i R m m (i = 1,,n, i j ) are some matrices. Considering that r ank(α j I + n i=1,i j α i (s)q i ) = m l, r ank(c A j 1 E) = l (2.18) we finally have r ank(c (si A) 1 E) = l. This proves the following theorem. Theorem 3 Given C (si A) 1 E as defined in 2.12 with m l and satisfying Assume that for some j {1,,n}, r ank(c A j 1 E) = l. Then r ank(c (si A) 1 E) = l (2.19) In the framework of linear system theory, C A i E, with i = 0,1,..., are called Markov matrices. The theorem above provide us a sufficient condition for checking the isolability of additive faults by means of Markov matrices.

53 2.3. STRUCTURAL FAULT IDENTIFIABILITY 39 In a similar manner, like the proof of the previous theorem, we are able to prove the following theorem that gives an alternative sufficient condition for the fault isolability. Theorem 4 Given C (si A) 1 E. Let Γ i = C S i E, i = 1,,n and assume that for some j {1,,n} r ank(γ j ) = l (2.20) then the r ank(c (si A) 1 E) = l. The procedure to extend the founded condition to the more general case of system 2.1, for which F 0, has been given in [42]. In particular it result that condition 2.15 can be equivalently written as F C E C AE. C A n 1 E while conditions 2.16 and 2.20 respectively as and { r ank(c A j 1 E) = l (2.21) r ank(f ) = l,i f j = 1 r ank(c A j 2 E) = l,i f j {2,,n + 1} } (2.22) r ank(γ j ) = l, j {0,,n},Γ 0 = F, 0. 0 Γ j = [a n I a n 1 I a 1 I I ] F, j {1,,n} C E. C A j 1 E (2.23) 2.3 Structural fault identifiability Roughly speaking, the concept of structural fault identifiability is understood as a characterization of system structure that is essential to reconstruct faults from the system outputs. From the mathematical point of view, fault identifiability characterize the mapping from the system output to the faults under consideration. If this mapping is unique, then the faults are identifiable. Usually, we intend to express this mapping in terms of the model from the faults to the system output, then the structural fault identifiability is equivalent to the model invertibility. Motivated by this fact, the concept of structural fault identifiability will be introduce in terms of fault transfer matrices. Definition 3 Given system 2.1 and let

54 40 CHAPTER 2. STRUCTURAL FAULT DETECTABILITY, ISOLABILITY AND IDENTIFIABILITY G f (s) = [G f1 (s)...g fl (s)] (2.24) be the fault transfer matrix of fault vector f (t) = [f 1 (t) f l (t)] T. The vector f (t) is called structurally identifiable if G f (s) is invertible and its inverse is stable and causal [42]. Note that the requirements on the stability and causality of the inverse of G f (s) is an expression for the realizability of inverting G f (s). It s evident that without these two requirements, the structural fault identifiability would be equivalent to the structural fault isolability. In another word, the structural fault isolability is a necessary condition for the fault to be identifiable. In the work of Hou and Patton [44] has been considered the problem of input reconstruction from the viewpoint of input observability. In particular has been shown that the input observability is a necessary and sufficient condition for the existence of an estimator for reconstructing inputs. To understand better the structural conditions behind these concepts it s important to introduce the concept of invariant zero, since it has also important implications for the arguments addressed in the following chapter. Consider the system (2.1) and assume that the initial state is given by x(0). Taking Laplace transformation of the system representation yields [ A si E C F The polynomial system matrix ][ X (s) f (s) [ A si E P(s) = C F ] [ X (0 ) = Y (s) ] ] (2.25) (2.26) is sometimes referred to as Roosenbrock s matrix. A necessary and sufficient condition for an "input" f (t) = f (0)e zt (2.27) to yield rectilinear motion in the state space of the form x(t) = x(0)e zt (2.28) such that the output of the system is identically zero for all time is that z, x(0) and f (0) satisfy [ ] x(0) P(z) = 0 (2.29) f (0) This result a set of complex frequencies z which are associated with specific directions x(0) and f (0) in the state and input spaces for which the output of the system is zero. These elements are called invariant zeros. Practically, in the case of square system, i.e. system with equal number of inputs and outputs, in order for equation (2.29) to have a nonzero solution for x(0) and u(0), det(p(z)) must be zero. In such cases invariant zeros are presents and the condition (2.11) is not true.

55 2.3. STRUCTURAL FAULT IDENTIFIABILITY 41 A similar discussion may also be extended in terms of input observability [44]. In fact, setting y(t) = 0 in (2.25), we obtain [ A si E C F ][ X (s) f (s) ] [ X (0 ) = 0 ] (2.30) whatever x(0) and therefore x(t) are, system (2.30) has a unique solution for f as f = 0 iff matrix P(s) has no invariant zeros. However if the the Roosenbrock s matrix P(s), related to the system (2.6), has invariant zeros C it results that the "input" ( for our case the fault) is detectable, where the input f (t) is said to be detectable if y(0) = 0 for t 0 implies f (t) 0 as t 0. In these cases, as shown in the next chapter, the fault reconstruction problem can still be solved.

56

57 Chapter 3 Sliding Mode Observers The concept of sliding mode control has been extended to the problem of state estimation by an observer, for linear systems, uncertain linear systems and nonlinear systems. Using the same design principle as for variable structure control, the observer trajectories are constrained to evolve after a finite time on a suitable sliding manifold by the use of an injection signal designed according a SM control algorithm. Subsequently the sliding motion provides an estimate, asymptotically or in finite time, of the system states. It worth noting that the sliding manifold is usually given by the difference between the observer and the system outputs, therefore in such cases we refer to the control signal as output injection signal. This chapter present an overview of both linear and nonlinear sliding mode observer paradigms. Many of the concepts in this chapter are closely based on the book by Edwards and Spurgeon [16] 3.1 SMO for Linear Systems We consider initially the linear system described by { ẋ = Ax(t) + Bu(t) y = C x(t) (3.1) where A R nxn,b R nxm,c R pxn and p m. Assume that the matrices B and C are of full rank and the pair (A,C ) is observable. Since a sliding motion on the error output space is going to be enforced, it is convenient to introduce a coordinate transformation so that the outputs appear as the last p components of the states. One possibility is to consider the non-singular transformation x T c x as T c = [ Nc T C ] (3.2) where N c R nx(n p) and the columns span the null space of C. This transformation is non-singular, and with respect to this new coordinate system, the distribution matrices of the similar system are 43

58 44 CHAPTER 3. SLIDING MODE OBSERVERS [ Ã = T c AT 1 A11 A 12 c = A 21 A 22 ] ; B = T c B = then the nominal system (3.1) can be rewritten as [ B1 B 2 ] ; C = C T 1 c = [0 I p ] (3.3) where { ẋ1 (t) = A 11 x 1 (t) + A 12 y(t) + B 1 u(t) ẏ(t) = A 21 x 1 (t) + A 22 y(t) + B 2 u(t) T c x = [ ] x1 n p y p The observer proposed by Utkin [1]-[2] has the form { ˆx 1 (t) = A 11 ˆx 1 (t) + A 12 ŷ(t) + B 1 u(t) + Lν ŷ(t) = A 21 ˆx 1 (t) + A 22 ŷ(t) + B 2 u(t) ν (3.4) (3.5) (3.6) where ( ˆx 1, ŷ) represent the state estimates for x 1 and y, L R (n p)xp is a constant feedback gain matrix and the discontinuous vector ν, of appropriate dimension, is define componentwise by ν i = M sg n(ŷ i y i ) (3.7) where M R +. If the errors between the estimates and the true states are written as e 1 = ˆx 1 x 1 and e y = ŷ y, then from equations (3.4) and (3.6) the following error dynamical system is obtained { ė1 (t) = A 11 e 1 (t) + A 12 e y (t) + Lν (3.8) ė y (t) = A 21 e 1 (t) + A 22 e y (t) ν that in compact form can be rewritten as follows [ ] L ė = Ãe(t) + Γν wher e Γ = I p (3.9) Since the pair (A,C ) is observable, the pair (A 11, A 21 ) is also observable. As a consequence, L can be chosen to make the spectrum of A 11 + L A 21 lie in C. Define a further change of coordinates, dependent on L, by it results [ ẽ1 ẽ = ẽ y [ In p L T = 0 I p ] [ e1 (t) + Le y (t) = Te = e y (t) ] ] (3.10) (3.11) and system in compact form (3.9), with respect to the new coordinates, can be rewrite as From (3.11), system (3.12) can be rewritten as ẽ = T Ãe(t) + T Γν (3.12) { ẽ 1 (t) = Ã 11 ẽ 1 (t) + Ã 12 e y (t) ė y (t) = A 21 ẽ 1 (t) + Ã 22 e y (t) ν (3.13)

59 3.1. SMO FOR LINEAR SYSTEMS 45 where à 11 = A 11 + L A 21, à 12 = A 12 + L A 22 à 11 L and à 22 = A 22 A 21 L. It follows from (3.13) that in the domain Ω = {(ẽ 1 (t),e y ) : A 21 ẽ 1 (t) λ max(ã 22 + à T 22 ) e y < M η} (3.14) where η < M is some small positive scalar, the reachability condition e T y ė y < η e y (3.15) is satisfied. Consequently, an ideal sliding motion will take place on the surface S o = {(ẽ 1,e y ) : e y = 0} (3.16) It follows that after some finite time t s, for all subsequent time, e y = 0 and ė y = 0 (in mean value, i.e., in the Filippov sense). Therefore from equation (3.13) result ẽ 1 (t) = à 11 ẽ 1 (t) (3.17) which, by choice of L, represents a stable system and so ẽ 1 0, i.e., e 1 0 and consequently ˆx 1 x 1 asymptotically. Equation (3.17) presents the reduced order sliding mode error dynamics. Example Consider now the problem of designing a sliding mode observer for the system in (3.1) described by [ 0 1 A = 2 0 ] [ 0 B = 1 ] C = [ 1 1 ] (3.18) which is observable since (r ank([c ;C A]) = 2) and represent a simple harmonic oscillator. For simplicity assume u = 0. Define a nonsingurar matrix [ 1 0 T c = 1 1 ] (3.19) and the change of coordinates according to (3.10)-(3.5) [ C = C T 1 c = [0 1]; à = T c AT c = 3 1 ] ; B = T c B = [ ] 0 1 (3.20) The system is now in the form given in equation (3.4). An appropriate choice of gain in the observer given in (3.6) is L = 0.5 which results in an error system governed by à 11. The simulation results which follows were obtained setting the gain of the discontinuous output injection term M = 1 and the following initial conditions: [x 1 (0), y(0)] = [1 0], [ ˆx 1 (0), ŷ(0)] = [0 0].

60 46 CHAPTER 3. SLIDING MODE OBSERVERS Figure 3.1: Utkin Observer

61 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS SMO for linear systems partially driven by unknown inputs The problem of designing an observer for a multivariable linear system partially driven by unknown inputs is of great interest. Such a problem arises in systems subject to disturbances or with inaccessible/unmeasurable inputs and in many applications such as fault detection and isolation, parameter identification and cryptography. This problem has been studied extensively for the last two decades. Consider initially the uncertain dynamical system { ẋ = Ax(t) + Bu(t) + E d(t, y,u) y = C x(t) (3.21) where A R nxn,b R nxm,c R pxn and E R nxq with p q. Assume that the matrices B, C and D are full rank and the function ξ(t, y,u) is unknown but bounded, so that d(t, y,u) r 1 u + α(t, y) (3.22) where r 1 is a known scalar and α : R + R p R + is a known function A classical approach The problem of estimating the states of the uncertain system given in (3.21) was approached by W al cot t and Ż ak (1987) [17]. Such a strategy, although intuitively appealing, necessitates the use of algebraic manipulation tools to effectively solve an associated constrained Lyapunov problem for systems of reasonable order. Edwards and Spurgeon (1998) [16] propose an observer strategy, similar in style to that in [17], which circumvents the use of symbolic manipulation and offers an explicit design algorithm. This approach will be outlined here. Let (A,E,C ) represent the linear part of the uncertain system which represents the propagation of the uncertainty through the output. Define an observer for the uncertain system (3.21) of the form ż(t) = Az(t) + Bu(t) G l Ce(t) +G n ν (3.23) where e=z-x and ν is discontinuous about the hyperplane S 0 = {e R n : e y = Ce = 0} (3.24) and G l,g n R n p are gain matrices whose precise structure is to be determinate. demonstrated in [16] the following proposition holds: As Proposition 1 A sliding mode observer of the form (3.23) which rejects the uncertainty class in (3.21) exist if and only if the nominal linear system, defined by the triple (A,E,C), satisfies: rank(ce)=q any invariant zeros of (A,E,C) must lie in C

62 48 CHAPTER 3. SLIDING MODE OBSERVERS For a square system, where p = q, it should be noted that the above two conditions fundamentally require the triple (A,E,C) to be relative degree one and minimum phase. Note also that these system theoretic conditions depend upon a specific selection of uncertainty channel. In the following a canonical form will be introduced, as presented in [16], since it s useful to explore the pertinent characteristics of SMO for linear, uncertain systems. Lemma 1: Let the triple (A,E,C ) represent a linear system with p > q and suppose r ank(c D) = q. Then a change of coordinates x T 0 x exist [16] so that the triple (Ā,Ē, C ) with respect to the new coordinates has the following structure: (a) The system matrix can be written as 1 Ā = T 0 AT 0 A 11 A 12 A 211 A 212 A 22 (3.25) where A 11 R (n p) (n p), A 211 R (p q) (n p) and when partitioned have the structure A 11 = [ A 0 11 A A 0 22 ] and A 211 = [0 A 0 21 ] (3.26) where A 0 11 Rr r and A 0 21 R(p q) (n p r ) for some r 0 and the pair (A 0 22, A0 21 ) is completely observable. Furthermore, the eigenvalues of A 0 11 are the invariant zeros of (A,E,C). (b) The disturbance distribution matrix has the form [ Ē = T 0 E = 0Ē2 where E 2 R q q is nonsingular. (c) The output distribution matrix has the form: ] [ 0 and Ē 2 = E 2 ] p q q (3.27) C = C T 0 1 = [ 0 T ] (3.28) where T R pxp is orthogonal. By referring to this canonical form, the necessary and sufficient existence conditions for the existence of the observer (3.23), that provides quadratic stability of the estimate error system despite the presence of bounded matched uncertainty, can be obtained as formally proved in [16]. For completeness, an outline of the key arguments is presented here in order to emphasize the key structure of a SMO as in (proof of necessity) Let G l and G n in (3.23) be appropriate gain matrices so that A 0 = A G l C is stable (3.29)

63 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS 49 and assume first that an ideal sliding mode insensitive to the uncertainty exist on the hyperplane in the error space given by S 0 in (3.24). Defining the observation error in such way e = z x, and considering systems (3.21) and (3.23), the error system satisfies: ė(t) = A 0 e(t) E d(t, y,u) +G n ν (3.30) Suppose at time T s the observation error lie on hyperplane S 0 = {e R n : Ce = 0}, i.e., an ideal sliding motion take place. Mathematically this can be expressed as Ce(t) = 0 and C ė(t) = 0 for all t > T s. Substituting for ė(t) from (3.30) gives C ė(t) = C A 0 e(t) C E d(t, y,u) +CG n ν = 0 (3.31) Definition 1: If det(cg n ) 0, the equivalent control associated with the nominal system (3.30), written as ν eq, is defined to be the unique solution to the algebraic equation (3.31), namely ν eq = (CG n ) 1 (C E d(t, y,u) C A 0 e(t)) (3.32) It must be highlight that the equivalent control is directly dependent from the uncertain term, as expected. The ideal sliding motion is given by substituting the expression of equivalent control (3.32) into equation (3.30) which result in a free motion, i.e., a motion independent of the control action and given by ė(t) = (I G n (CG n ) 1 C )A 0 e(t) (I G n (CG n ) 1 C )E d(t, y,u) (3.33) To be insensitive to the uncertainty it follows that (I G n (CG n ) 1 C )E = 0, or equivalently E = G n (CG n ) 1 C E (3.34) Since by assumption, for original uncertain linear system in (3.21), r ank(e) = q, it follows immediately that from equation (3.34) rank(ce)=q, further it means that we can assume, without loss of generality, that the system (A,E,C ) is in canonical form given by the change of coordinates stated in Lemma 1. If the nonlinear gain matrix is partitioned so that G n = [ G1 G 2 ] n p p (3.35) then, from (3.28) and (3.35), result CG n = TG 2 and so det(g 2 ) 0. From equation (3.33) and using standard arguments, it follows that the poles of the (linear) reduced-order motion, given by ė(t) = (I G n (CG n ) 1 C )A 0 e(t), are λ((a 0 ) 11 G 1 G 1 2 (A 0) 21 ) (3.36) where (A 0 ) 11 and (A 0 ) 21 represent the top left and bottom left sub-block of the closedloop matrix A 0 partitioned in a compatible way to the canonical form. By definition of the matrix A 0, given in (3.29), result: (A 0 ) 11 = A 11 (G 1 C ) 11, where (G 1 C ) 11 represents the top left sub-block of the square matrix (G 1 C ). However, it s straightforward to verify that (G 1 C ) 11 = 0 for all G l R n p and so (A 0 ) 11 = A 11. Similarly it can be shown that (A 0 ) 21 = A 21 and consequently λ((a 0 ) 11 G 1 G 1 2 (A 0) 21 ) = λ(a 11 G 1 G 1 2 A 21) (3.37)

64 50 CHAPTER 3. SLIDING MODE OBSERVERS From equation (3.34) it follows that G 1 G2 1 Ē 2 = 0 which, after considering the structure of Ē 2, implies G 1 G2 1 = [Ḡ 0], where Ḡ R (n p) (p q) and therefore from the definition of A 21 it follows that A 11 G 1 G2 1 A 21 = A 11 Ḡ A 211. By construction the pair (A 11, A 211 ) is such that {zer os o f (A,E,C )} = λ(a 0 11 ) λ(a 11 Ḡ A 211 ) f or al l Ḡ R (n p) (n p) (3.38) And therefore for a stable sliding motion is necessary that any invariant zeros of (A,E,C ) must lie in C. (proof of sufficiency) Conversely, let (A,E,C ) represent the system and suppose r ank(c E) = q and any invariant zeros lie in C. It s assume that the system is already in the canonical form required to facilitate sliding mode observer design where the matrix A 0 11 is stable. As a consequence there exists a matrix L R (n p) (p q) such that A 11 +L A 211 is stable. Define a non-singular transformation as [ ] In p L T L = (3.39) 0 T where L = [L 0 (n p) q ] (3.40) After changing coordinates with respect to T L, the new output distribution matrix becomes Ẽ = C T 1 L = [0 I p ] (3.41) and from (3.27)-(3.40) result that L E 2 = [ L 0 ][ 0 E 2 and so the uncertainty distribution matrix is given by [ L E 2 Ẽ = T L E = T E 2 ] [ = ] = 0 (3.42) 0 T E 2 Finally, if à = T L AT 1, it can be shown by direct evaluation that which is stable by choice of L. The system triple (Ã, D, C ) is now in the form L ] (3.43) à 11 = A 11 + L A 211 (3.44) { ẋ1 = A 11 x 1 (t) + A 12 y(t) + B 1 u(t) ẏ = A 21 x 1 (t) + A 22 y(t) + B 2 u(t) + E 2 d(t, y,u) (3.45) where x 1 R (n p), y R p and the matrix A 11 has stable eigenvalues.

65 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS 51 Define the the corresponding observer by { ˆx 1 = A 11 ˆx 1 (t) + A 12 ŷ(t) + B 1 u(t) A 12 e y (t) ŷ = A 21 ˆx 1 (t) + A 22 ŷ(t) + B 2 u(t) (A 22 A S 22 )e y (t) + ν (3.46) where A S 22 is a stable design matrix and e y = ŷ y. Let P 2 R p p be symmetric positive Lyapunov matrix for A S 22, then the discontinuous vector ν is defined by ν = ρ(t, y,u) E 2 P 2e y P 2 e y where the scalar function ρ : R + R p R m R + satisfies (3.47) ρ(t, y,u) r 1 u + α(t, y) + γ 0 (3.48) and γ 0 is a positive scalar. If the state estimation error e 1 = ˆx 1 x 1, then it s straightforward to show { ė1 (t) = A 11 e 1 (t) ė y (t) = A 21 e 1 (t) + A S 22 e y (t) + ν E 2 d (3.49) Define Q 1 R (n p) (n p) and Q 2 R (p p) as symmetric positive definite design matrices and define P 2 R (p p) as the unique symmetric positive definite solution to the Lyapunov equation P 2 A S 22 + (AS 22 )T P 2 = Q 2 (3.50) Using the computed value of P 2 define ˆQ = A T 21 P 2Q 1 2 P 2 A 21 +Q 1 (3.51) Note that ˆQ = ˆQ T > 0 and let P 1 R (n p) (n p) be the unique symmetric positive definite solution to the Lyapunov equation Taking the quadratic form P 1 A 11 + A T 11 P 1 = ˆQ (3.52) V (e 1,e y ) = e T 1 P 1e 1 + e T y P 2e y (3.53) as a candidate Lyapunov function, and considering the derivative along the system trajectory it result V (e 1,e y ) < 0 [16] for (e 1,e y ) 0, i.e., the error system is quadratically stable. Therefore considering the hyperplane given in (3.24) then it result that An ideal Sliding motion takes place on (3.24) in finite time. This finite time convergence property is a key advantage of sliding mode observer schemes. Many other observer paradigms guarantee only asymptotical properties. If ˆx(t) represents the state estimate for x and e = ˆx x, then the robust observer can conveniently be written as ˆx(t) = A ˆx(t) + Bu(t) G l Ce(t) +G n ν (3.54) where the linear gain [ G l = T0 1 A 12 A 22 A S 22 ]. (3.55)

66 52 CHAPTER 3. SLIDING MODE OBSERVERS and the non-linear gain [ G n = E 2 T0 1 0 I P ] (3.56) and ν = ρ(t, y,u) P 2Ce P 2 Ce (3.57) A key development in this formulation of the sliding mode observer design framework is that there is no requirement for (A,C ) to be observable. This is straightforward to demonstrated as a SMO can be developed for a system that is unobservable as long as the nominal triple (A,E,C ) has r ank(c E) = q and any invariant zeros of (A,E,C ) lie in C. In many sense, the problem of sliding mode observer design for systems which can be assumed to possess a core linear triple is solved. Clearly, when transmission zeros are present in the representation, these will appear in the poles of the sliding mode dynamics and only asymptotically stable error convergence will be possible [9].

67 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS HOSMO to weaken the 1-relative degree condition The issue of broadening the class of systems so that the r ank(c E) = q and/or the stability condition on the invariant zeros of (A,E,C ) is of ongoing interest. Floquet et. al. [24] show that the relative degree condition can be weakened if a classical sliding mode observer is combined with SM exact differentiators to essentially generate additional independent output signals from the available measurements. It s proved that the transmission zeros of the nominal triple (A,E,C ) still contribute to the dynamics of the error signal in the sliding mode and thus the condition that any invariant zeros of (A,E,C ) must lie in C still remains. We will refer to the linear time-invariant system subject to unknown inputs or disturbances in 3.21, where the output takes the following expression y = [y 1... y p ] T = C x, y i = C i x (3.58) As for the previous case d(t, y,u) R q stands for the bounded, unknown inputs. It is further supposed that q p. Basing on Proposition 1, if r ank(c E) = r ank(e) = q, i.e. observer matching condition hold, there exist a linear change of coordinates that puts the original system into the canonical form given in 3.45 for which there exist an observer of the form ˆx(t) = A ˆx(t) + Bu(t) +G l (y C ˆx) +G n ν (3.59) where G l and G n are design gains and ν is an injection signal which depends on the output estimation error in such a way that a sliding motion in the state estimation error space is induced in finite time, and therefore the state estimation error e = x ˆx is asymptotically stable and independent of the unknown signal d during the sliding motion. Here the aim is to extend the previous result so that a sliding mode observer can be designed for (3.58) when the standard matching condition is not satisfied, more in particular, when rank(ce)<q. To this end, introduce the notation of relative degree µ j N +, 1 j p of the system with respect to the output y j, that is to say the number of times the output y j must be differentiated in order to have the unknown input d explicitly appear. Thus, µ j is defined as follows: C j A k E = 0 f or al l k < µ j 1 C j A µ j 1 E 0 (3.60) without loss of generality, it is assumed that µ 1... µ p. The following assumptions are made: the invariant zeros of {A,E,C} lie in C there exists a full rank matrix C a = C 1. C 1 A µ α 1 1. C p. C p A µ αp 1 (3.61)

68 54 CHAPTER 3. SLIDING MODE OBSERVERS where the integers 1 µ αi µ i are such that r ank(c a E) = r ank(e) = q and the µ αi are chosen such that P i=1 µ α i is minimal. Before describing the observer scheme proposed in [24] we introduce the following lemma that in the cited article is also demonstrated. Lemma 2: the invariant zeros of the triples {A,E,C } and {A,E,C a } are identical The previous lemma, on which practically the main idea of the method under investigation stands, says that the invariant zeros of the triple {A,E,C } and the newly created triple with additional (derivative) outputs {A,E,C a } are identical. Consequently, if the original system is minimum phase the new triple {A,E,C a } is both minimum phase and relative degree one and hence a "classical" observer of the form given in (3.59) can be designed for {A,E,C a }. The observer design is based in two step. First a sliding mode observer is designed for a system described by the following triangular observable form: ξ 1 = ξ 2 + b1 T u ξ 2 = ξ 3 + b2 T u. ξ l 1 = ξ l + b T l 1 u ξ l = b l+1 T θ + b T l u y ξ = ξ 1 (3.62) where ξ = [ξ 1 ξ l ] T R l, (l>1) is the state vector, y ξ R is the output, u R m is the known input vector and θ R q stands for some unknown inputs. The vectors b i, with i = 1,...,l, of appropriate dimension. Assume that the system is bounded input bounded state in finite time and that the signal output y ξ, the unknown input θ and θ are bounded. Most of the sliding mode observer designs for (3.62) are based on a step-by-step procedure using successive filtered values of the so-called equivalent output injections obtained from recursive first-order sliding mode observers (see e.g. [3]). However, the approximation of the equivalent injections by low pass filters at each step will typically introduce delays that lead to inaccurate estimates or to instability for high-order systems. To overcome this problem it s possible to replace the discontinuous first-order sliding mode output injection by a continuous second-order sliding mode one. The observer is built as follows: ˆξ 1 = ν( ξ 1 ˆξ 1 ) + b1 T u ˆξ 2 = E 1 ν( ξ 2 ˆξ 2 ) + b2 T u. ˆξ l 1 = E l 2 ν( ξ l 1 ˆξ l 1 ) + b T l 1 u (3.63) ˆξ l = E l 1 ν( ξ l ˆξ l ) + b T l u y ξ = ξ 1 where ξ 1 := y ξ and

69 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS 55 ξ j := ν( ξ j 1 ˆξ j 1 ), 2 j l (3.64) where the continuous output error injection ν( ) is given by the so-called super twisting algorithm [14, 15]: ν(s) = ϕ(s) + λ s s 1 2 si g n(s) ϕ(s) = α s si g n(s) λ s,α s > 0 For i = 1,...,l 1, the scalar functions E i are defined as (3.65) E i = 1 i f ξ j ˆξ j ɛ f or all j i el se E i = 0 (3.66) Denoting ξ = ξ ˆξ, the error dynamic is given by ξ 1 = ξ 2 ν(y ξ ˆξ 1 ) ξ 2 = ξ 3 E 1 ν( ξ 2 ˆξ 2 ). ξ l 1 = ξ l E l 2 ν( ξ l 1 ˆξ l 1 ) ξ l = b l+1 T θ E l 1 ν( ξ l ˆξ l ) (3.67) As argued in [3] the sliding manifolds are reached one by one sequentially, from i = 1 to i = l. At each step, a sub-dynamic of dimension one is obtained and consequently no peaking phenomena appear. It can be verified that with a suitable choice of gains λ s and α s, a sliding mode is attained in finite time on the manifold ξ 1 = ξ 2,...,= ξ L = 0 and the following equivalent output injection is obtained ν( ξ l ˆξ l ) = b T l+1 θ (3.68) Note that the step-by-step observer achieves finite time recovery of the state components. As second step, in order to estimate the state of system (3.58), the following SMO has been proposed [24] ẑ(t) = Az(t) + Bu(t) +G l (y a C a z) +G n ν(y a C a z) (3.69) where the auxiliary output y a is defined by y a = y 1 ν(ỹ 1 y 1 1 ). ν(ỹ µ α y µ α ). y p. ν(ỹ µ αp 1 p y µ αp 1 p ) (3.70)

70 56 CHAPTER 3. SLIDING MODE OBSERVERS where the constituent signals in (3.70) are given from the step-by-step observer: ẏ 1 i = ν(ỹ 1 i y 1 i ) +C i Bu ẏ 2 i = E 1ν(ỹ 2 i y 2 i ) +C i ABu (3.71). ẏ µ α i 1 = E i µαi 2ν(ỹ µ α i 1 y µ α i 1 ) +C i i i A µ α i 2 Bu i for 1 i p, with ỹ 1 i := y i ỹ j j 1 := ν(ỹ y j 1 ), 2 j µ i i i αi 1 (3.72) where the injection operator ν( ) is defined by (3.65). The discontinuous output injection ν c in (3.69) could be defined by ν c (y a C a z) = ρ P 2(y a C a z) P 2 (y a C a z) (3.73) where ρ is a positive constant larger than the upper bound of d. The definition of the symmetric positive matrix P 2 has been given in Let us define the state estimation error e = x z and the augmented output estimation error e y = C a x ȳ, with where e y = [e1 1,...,eµ α i 1 1,...,ep 1,...,eµ αp 1 p ] T ȳ = [y1 1,..., yµ α i 1 1,..., yp 1,..., yµ αp 1 (3.74) p ] T ė 1 i = C i Ax ν(y i y 1 i ) ė 2 i = C i A 2 x E 1 ν(ỹ 2 i y 2 i ). ė µ α i 1 i = C i A µ α 1 i x E µαi 2ν(ỹ µ α i 1 y µ α i 1 ) i i (3.75) for i i p. Thus, choosing suitable output injections ν, the following relations hold after a finite time T: ν(y i y 1 i ) = C i Ax ν(ỹ 2 i y 2 i ) = C i A 2 x (3.76). ν(ỹ µ α i 1 y α i 1 ) = C i i i A 2 x for i i p. This means that y a = C a x Then it is straightforward to show that ė = Ae + Dw G l (y a C a z) G n ν c (y a C a z) (3.77) Thus, for all t > T, the error dynamics (3.77) are given by ė = (A G l C a )e + E d G n ν c (C a e) (3.78) Since by construction r ank(c a E) = r ank(e) and by assumption the invariant zeros of the triple (A,E,C a ) lie in the left half plane, the design methodology presented in 3.2.1

71 3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS 57 can be applied so that e = 0 is an asymptotically stable equilibrium point of (3.78) and the dynamics are independent of d once a sliding motion on the sliding manifold {e : s = C a e = 0} has been attained. In addition, the method here presented enables an estimation of the unknown inputs. Define (v c ) eq as the equivalent output error injection required to maintain the sliding motion in 3.78, during the sliding motion Since e 0 and using (3.78) ṡ = C a ė = C a (A G l C a )e +C a E d C a G n v c (C a e) = 0 (3.79) C a G n (v c ) eq C a E d (3.80) As C a E is full rank, an approximation ˆ d of d can be obtained from v c eq by ˆ d = ((C a E) T C a E) 1 (C a E) T C a G n v c eq (3.81)

72 58 CHAPTER 3. SLIDING MODE OBSERVERS 3.3 Non-linear approaches to Sliding Mode Observers design In this sections the problem of designing observers for state estimation of non-linear systems using sliding mode will be addressed [9]. Early contribution in this area were developed independently by Walcott and Zak ([10]-[11]) and Slotine et al. [12], where the latter team considered a more extended class of system. Probably the next major breakthrough in the development of SMO for non-linear systems appears in the article by Drakunov and Utkin [2] where the equivalent injection concepts was introduced for observer design. Barbot et al. [3] developed a sliding mode observer for a non-linear system in a triangular input form. Such systems were originally considered in the work of Drakunov and Utkin [2] and are important because it is possible to develop an observer without using input derivatives. It must be highlight that sliding mode observers, are usually designed under the assumptions that the nominal uncertain system can be put into a triangular observable form, where the uncertain term act only on the last dynamics. This assumption is usually known as the observability matching condition for analogy to the well-known matching condition for a sliding mode controller to be insensitive to matched perturbations. In [25] is considered the case of a step-by-step SMO for autonomous nonlinear systems with unknown inputs, based on the hierarchical application of the supertwisting algorithm in such a way similar to that presented in In such approach the system under observation is assumed already in triangular observable form. In this section will be presented an approach to finite time HOSMO [31] that does not require the system to be reduced to any normal form, which can be difficult to achieve in the presence of model uncertainties. An important implications of the former observer is the possibility of reconstruct the uncertain term affecting the uncertain nonlinear system. Another method very useful when the problem of the input reconstruction is considered, is associated to the the concept of algebraic observability. The problem of the input reconstruction for non-linear time invariant dynamic system, making use on the concept of algebraic observability and sliding mode differentiators, has been approached successfully in the work of Cannas et al. [36].

73 3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN Systems in the companion form As in Slotine et al.[12], let us consider a non-linear system in companion form x (n) = f (x, t) (3.82) where f (x, t) is a non-linear, uncertain function of the system state and x 1 is the single measurement available. Define a corresponding sliding mode observer ˆx 1 = α 1 e 1 + ˆx 2 k 1 sg n(e 1 ) ˆx 2 = α 2 e 1 + ˆx 3 k 1 sg n(e 1 ) ˆx n = α n e 1 + f ˆ k n sg n(e 1 ) (3.83) where e 1 = ˆx 1 x 1, f ˆ is an estimate of f (x, t) and the constant αi are chosen as for a classical Luenbergher observer to ensure asymptotical error decay of a corresponding linearized system representation, where k i = 0. The corresponding error dynamics are given by ė 1 = α 1 e 1 + e 2 k 1 sg n(e 1 ) ė 2 = α 2 e 1 + e 3 k 1 sg n(e 1 ) (3.84) ė n = α n e 1 + f k n sg n(e 1 ) where f = ˆ f f is assumed bounded and The sliding condition (d/d t)(e 1 ) 2 < 0 is satisfied in the region k (n) f (3.85) e 2 k 1 + α 1 e 1 i f e 1 > 0 e 2 k 1 + α 1 e 1 i f e 1 < 0 (3.86) From first equation of (3.84), when a sliding mode is attained on e 1 = 0 it follows that in Filippov sense the following hold and therefore e 2 k 1 sg n(e 1 ) = 0 ė 2 = e 3 k 2 k 1 e 2 ė n = f k n k 1 e 2 The values α i are thus seen to only affect the dynamic performance prior to the reaching of this region, which is often called the sliding patch, and the dynamics on the patch are determined by k 2 k λi n 1 k 3 k = 0 (3.87) k n k

74 60 CHAPTER 3. SLIDING MODE OBSERVERS Assuming k n is selected as a constant ratio with k 1 and that the poles defining the dynamics on the patch are critically damped i.e. are real and equal to some constant value γ, then Slotine et al. [12] show that e (i ) 2 (2γ)i k 1 i = 0,...,n 2 (3.88) from which the precision of the state estimates can be determined. As well as defining the concept of the sliding patch, the contributions of Slotine et al. [12] discussed the effect of measurement noise on sliding mode observer. It was demonstrated that, as would be expected, the system does not attain a sliding mode in the presence of noise, but effectively remains within a region of the sliding patch which is determined by the bound on the noise. Moreover, it was demonstrated that the average dynamics can be modified by selection of the k i which in turn can tailor the contribution of the noise to the state estimates.

75 3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN Triangular systems As in [3], let us consider a non-linear system in the triangular input form ξ 1 = ξ 2 + ḡ 1 (ξ 1,u) ξ 2 = ξ 3 + ḡ 2 (ξ 1,ξ 2,u) ξ n 1 = ξ n + ḡ n 1 (ξ 1,ξ 2,...,ξ n 1,u) ξ n = f n (ξ 1,ξ 2,...,ξ n ) + ḡ n (ξ 1,ξ 2,...,ξ n,u) (3.89) where y = ξ 1, the terms ḡ i ( ) are assumed known, ḡ n (,0) = 0 for i = 1,...,n, and the system is assumed bounded input bounded state (BIBS) in finite time. As in Drakunov and Utkin [2], Barbot et al. [3] define the following SMO ˆξ 1 = ˆξ 2 + ḡ 1 (ξ 1,u) + λ 1 si g n(ξ 1 ˆξ 1 ) where ˆξ 2 = ˆξ 3 + ḡ 2 (ξ 1, ξ 2,u) + λ 2 si g n( ξ 2 ˆξ 2 ) ˆξ n 1 = ˆξ n + ḡ n 1 (ξ 1, ξ 2,..., ξ n 1,u) + λ n 1 si g n( ξ n 1 ˆξ n 1 ) ˆξ n = f n (ξ 1, ξ 2,..., ξ n ) + ḡ n (ξ 1, ξ 2,..., ξ n,u) + λ n si g n( ξ n ˆξ n ) (3.90) ξ i = ˆξ i + λ i 1 si g n(ξ i 1 ˆξ i 1 ) (3.91) for i = 2,...,n 1 and the si g n( ) function is computed using filtered version of argument and the anti-peaking methodology of Khalil [13] is employed. Effectively the observation error information is not used in the implementation before the corresponding sliding manifold is reached. The manifolds are reached sequentially one by one and ξ i ˆξ i converges to zero if the error ξ j ˆξ j, with j < i, have already converged to zero. The motivation for the above, which is effectively a sequential consideration of a series of first-order dynamics, is easily seen by forming the error dynamics for e i = ξ i ˆξ i : ė 1 = e 2 λ 1 si g n(ξ 1 ˆξ 1 ) ė 2 = e 3 + ḡ 2 (ξ 1,ξ 2,u) ḡ 2 (ξ 1, ξ 2,u) λ 2 si g n( ξ 2 ˆξ 2 ) ė n 1 = e n + ḡ n 1 (ξ 1,ξ 2,...,ξ n 1,u) ḡ n 1 (ξ 1, ξ 2,..., ξ n 1,u) λ n 1 si g n( ξ n 1 ˆξ n 1 ) ė n = f n (ξ 1,ξ 2,...,ξ n 1,u) f n (ξ 1, ξ 2,..., ξ n,u) +ḡ n (ξ 1,ξ 2,...,ξ n,u) ḡ n (ξ 1, ξ 2,..., ξ n,u) λ n si g n( ξ n ˆξ n ) (3.92) It can be verified that for sufficiently large λ 1, a sliding mode is attained on e 1 = 0 in finite time and it follows that e 2 = λ 1 si g n(ξ 1 ˆξ 1 ) (3.93) which with 3.91 yields ξ 2 = ξ 2. The observation error dynamics become

76 62 CHAPTER 3. SLIDING MODE OBSERVERS ė 1 = 0 ė 2 = e 3 λ 2 si g n(e 2 ) ė n 1 = e n + ḡ n 1 (ξ 1,ξ 2,...,ξ n 1,u) ḡ n 1 (ξ 1,ξ 2,..., ξ n 1,u) λ n 1 si g n( ξ n 1 ˆξ n 1 ) ė n = f n (ξ 1,ξ 2,...,ξ n 1,u) f n (ξ 1, ξ 2,..., ξ n,u) +ḡ n (ξ 1,ξ 2,...,ξ n,u) ḡ n (ξ 1, ξ 2,..., ξ n,u) λ n si g n( ξ n ˆξ n ) (3.94) Proceeding as before it can be shown that for sufficiently large λ 2 a sliding mode is then attained on e 2 in finite time and it follows that e 3 = λ 2 si g n(ξ 2 ˆξ 2 ) (3.95) which yields ξ 3 = ξ 3. Applying the same methodology up to the e n dynamic produce ė 1 = 0 ė 2 = 0 ė n 1 = 0 ė n = λ n si g n(e n ) (3.96) and it follows trivially that a sliding mode is finally attained on e n = 0 in finite time. This finite time property of sliding mode observer is very attractive and has led to the development of a number of application specific sliding mode observers.

77 3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN Quasi-continuous HOSM observers The referred nonlinear uncertain system is the following [31] ẋ = f (x) + f (x) y = h(x) (3.97) with the state vector x X R n and the scalar output y Y R. The vector fields f (x) : X R n and h(x) : X Y represent the known nominal part of the system dynamics, while f (x) : X R n is assumed to be uncertain. The aim is that of designing a finite-time converging observer for the above system. The following observer structure has been considered [31] ˆx = f ( ˆx) + g ( ˆx)u ŷ = h( ˆx) (3.98) with observed state vector ˆx R n and observed output variable ŷ R. The design of the vector field g ( ) and the corrective term u R are the main topics of the following discussions. In particular, under a certain observability condition that is going to be specified, we shall select the vector function g ( ) in such a way that the observer output has full relative degree n with respect to the observer input u. To the aim of introducing some important concepts let us introduce the operator d q(z) applied to a generic scalar function q with vector argument z defined on an open set Ω R n, q(z) : R n R. We denote d q(z) = q(z) z Now, let us define the matrix M(z) = [ q(z) = z 1, q(z),..., q(z) ] z 2 z n dh(z) dl f (z) h(z). dl n 2 f (z) h(z) dl n 1 f (z) h(z) (3.99) where L f (z) h(z) is the so-called Lie derivative of h(z) along f (z) and is defined as L f (z) h(z) = h(z) f (z) and the kth derivative of h(z) along f (z) is defined as L k z f (z) h(z) = L k 1 f (z) h(z) z f (z). The following assumption is assumed to hold Assumption 1 The matrix M(z) is nonsingular for every possible value of z. The vector g ( ˆx) in (3.98) will be designed according to equation from which, from assumption 1, descend M( ˆx)g ( ˆx) = [0,0,...,1] T (3.100) g ( ˆx) = M 1 ( ˆx) [0,0,...,1] T (3.101)

78 64 CHAPTER 3. SLIDING MODE OBSERVERS i.e., g ( ˆx) is the last column of matrix M 1 ( ˆx). In light of (3.101) the observer input/output dynamics is d d t ŷ ŷ. ŷ (n 1) L f ( ˆx) h( ˆx) = L 2 h( ˆx) f ( ˆx). + L (n) h( ˆx) f ( ˆx) u (3.102) which means that the observer output ŷ has full relative degree n with respect to the observer input u. Let us define the n-dimensional output error vector ε, containing the output error e y = ŷ y R and its first n 1 derivatives: ε = ε 1 ε 2. ε n = e y ė y. e (n 1) y. (3.103) and the state observation error e = ˆx x = e 1 e 2. e n. It must be highlight that the observer output error possesses the same relative degree n with respect to u. In [31] has been demonstrated that, under assumption 1, the observer (3.98), (3.101) can reconstruct the state of the nominal system (3.97) with f (x) = 0, i.e., the system is perfectly known, exactly and in finite time, provided that the observer input u is selected in such a way that the vector ε is steered to zero in finite time. In other words it means that ε = 0 e = 0 (3.104) It must be highlight that it is true for any uniformly observable system (see [32]), i.e., for systems which are observable independently of the inputs. Taking into account the uncertain term f (x), not modeled in the observer structure, an additional, necessary and sufficient, condition is needed to guaranty the preservation of implication Such condition, referring to the system (3.97), consists in the following requirement L f (x) h(x) L f (x) L f (x) h(x). = L f (x) L n 2 f (x) h(x) (3.105)

79 3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN 65 It means that the relative degree of the disturbance with respect the output, and therefore also with respect the output error e y, must be equal to n. In order to design a robust state observer, for the system (3.97), of the form (3.98), (3.101), under the condition 3.105, a controller which stabilizes in finite time the observation error ε must be designed. The observation output error dynamics takes the following Brunovsky canonical form where ε 1 = ε 2 ε 2 = ε 3... ε n = Φ(e, ˆx) + u (3.106) Φ(e, ˆx) = L n f ( ˆx) h( ˆx) Ln f ( ˆx e) h( ˆx e) L f ( ˆx e) L n 1 f ( ˆx e) h( ˆx e) (3.107) for which the following assumption is meet Φ(e, ˆx) < Γ (3.108) In [31], the quasi-continuous arbitrary-order sliding mode controller [33, 34], based on the so-called arbitrary-order" sliding-mode approach [33], was suggested in order to stabilize (3.106)-(3.108) in finite time. For completeness, the cited algorithm is reported in following. Let i = 1,...,n 1 and denote ϕ 0,n = e y, N 0,n = e y (3.109) Ψ 0,n = ϕ 0,n /N 0,n = sign e y, (3.110) ϕ i,n = e (i ) y + β (n i )/(n i+1) i N i 1,n Ψ i 1,n, (3.111) (n i )/(n i+1) N i,n = e (i y ) i N, i 1,n (3.112) Ψ i,n = ϕ i,n /N i,n (3.113) where β 1,..., β n 1 are positive numbers. The quasi-continuous n-sliding controller is u = αψ n 1,n (e y, ė y,..., e (n 1) y ). (3.114) It was shown in [33, 34] that provided that the tuning parameters β 1,..., β n 1,α are chosen sufficiently large in the given order then the control law defined by (3.109)-(3.114) stabilizes the system (3.106)-(3.108) in finite time. Note that control defined by (3.109)-(3.114) is globally bounded ( u α) and continuous everywhere but the origin of the n-dimensional error space, from which the name of the controller was derived. Following are reported example of second and third-order quasi-continuous controllers: u = α ė y + e y 1/2 signe y ė y + e y 1/2, u = α ë y +2( ė y + e y 2/3 ) 1/2 (ė y + e y 2/3 signe y ) ë y +2( ė y + e y 2/3 ) 1/2 (3.115) The n-th order quasi-continuous controller requires the availability of the successive derivatives of the output estimation error up to the order n 1. In order to reconstruct such derivatives exactly and in finite time, the well known Arbitrary-Order slidingmode differentiator by A. Levant [35] can be used.

80 66 CHAPTER 3. SLIDING MODE OBSERVERS The n-th order differentiator can be expressed in the following non-recursive form ż 0 = v 0 = z 1 κ 0 z 0 e y (t) n n+1 si g n(z 0 e y (t)), ż 1 = v 1 = z 2 κ 1 z 1 v 0 n 1 n si g n(z 1 v 0 ),... ż i = v i = z i κ i z i v i 1 n i n i+1 si g n(z i v i 1 ),... ż n = κ n si g n(z n v n 1 ) (3.116) for suitable positive constant coefficients κ i to be chosen recursively large in the given order ([35]). Under the assumption that the n 1th derivative of e y is Lipshitz, i.e., a real positive a constant C exists such that e (n) y C, the following equalities are true in the absence of measurement noise after a finite time transient process: z i e (i ) y (t) = 0 i = 0,...,n (3.117) Clearly, for the considered closed-loop system (3.106)-(3.108) the C constant exists and it is is overestimated by C = Γ + α. The separation and robustness results relevant to the combined use of the above differentiator and any n-sliding homogenous controller were discussed in [35]. It was demonstrated by Levant [35] that non-idealities like measurement noise and finite frequency commutation cause a bounded error in the estimated derivatives and, as a result, a bounded loss of accuracy for the controller that uses the "noisy" derivative estimates [33], [34], [35]. An important implications of the presented observer is the possibility of reconstruct the uncertain term f (x) of system In the work of Davila et al. [31] a special instance of the general uncertain dynamics f (x) represented by some bounded unknown input term v(x, t) R premultiplied by a known distribution matrix D of appropriate dimension, has been considered, and a technique aimed to reconstruct the exact unknown input v(x, t) has been proposed.

81 3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN Algebraic observers Let us introduce some concepts about the algebraic observability and the input reconstruction by differential algebra [37]. Let we refer to the following non-linear system ẋ = f (x,u(t)) y = h(x) (3.118) where x is the n-dimensional state variable vector, u is a scalar input, y a scalar output, and f and h, are algebraic function vectors in x. We assume that the state is not directly available and only the scalar output is measured. The goal to be attained is the reconstruction of the system input u by using only the information contained in y. An algebraic observer for the input u of system is a polynomial of the form P = p(u, y, y (1),..., y (m) ). (3.119) with m n. In such cases the input of system can be reconstructed if a smooth function g exists such that u = g (y, y (1),..., y (m) ). The algebraic observability property can be easily tested within the differential algebra context by resorting to the concept of "Characteristic Set" associated to the dynamic equations. In order to define the Characteristic Set, we need to introduce some concepts inherent to the Differential Algebra. Remark 1 If one chooses the ranking of the variables and their derivatives u < u (1) < u (2) < < y < x 1 < x 2 < < y (1) < x 1 (1) < x 2 (1) < < y (n) < x n (2) < x 2 (n) < (3.120) for a system of the form 3.118, the Characteristic Set exhibit n +1 differential polynomials, that is: an Input/Output (I/O) relation that is a differential polynomial in u, y and their derivatives, and is denoted by k(u, y); n differential polynomials, triangular with respect to the state components, and denoted by the n-dimensional vector K (u, x, y). Property 1 The input is algebraically observable if at least one of the following equivalent relations is verified: 1. derivatives of the input u do not appear in the Characteristic Set; 2. k(u, y) is of order n in the output y. Proposition 1 A necessary condition for the finite time global reconstruction of the input of system (1) is that the I /O relation k(u, y) is an algebraic observer of the input of the type P = p(u, y, y (1),..., y (m) )

82 68 CHAPTER 3. SLIDING MODE OBSERVERS It means that if the I/O relation is not an algebraic observer, it contains some derivatives of the input, hence the input is the solution of a differential equation whose initial conditions are unknown, therefore if no further information is available the input admits infinite solutions. Conversely, if the I/O relation is an algebraic observer, the number of solutions is finite. Proposition 2 The input function can be globally recovered in a finite time iff it appears in the I/O relation with order zero and degree one. Therefore, considering the reference dynamic system described by Eq and assuming that the input is algebraically observable, there exists an algebraic I /O relation between the input u and the output y with its first m n derivatives, i.e., ξ(y, y (1), y (2),, y (n),u) = 0 (3.121) For a globally recoverable input, Eq can be written as u = g (y, y (1), y (2),, y (n) ) (3.122) Thus, in the case of global reconstruction, an m-order differentiator, realized for instance with the Arbitrary-Order sliding-mode differentiator by A. Levant [35] in 3.116, allows one to determine the input to the system uniquely by starting from the output measures and independently of the initial conditions. Conversely, in the case of local reconstruction, the state of the system and the input signal are determined uniquely, provided that the initial conditions of the reconstructor are close to the initial conditions of the system [24] and [36].

83 Chapter 4 Sliding Modes for FDI It is well known that the core element of model-based fault detection in control systems is the generation of residual signals which act as indicators of faults. The residual signals are generated through comparison between measurement estimates and real measured quantities. For the design of residual generators, various approaches have been discussed in the literature. In particular, the basic idea behind the use of the observer for fault detection is to estimate the outputs of the system from the measurements by using some type of observer, and then construct the residual by a properly weighted output estimate error. The residual is then examined for the likelihood of faults by using a fixed or adaptive threshold. When, in the observer-based approaches, a full order observer is used in residual generator design, the main design procedure becomes an equivalent state feedback control problem because of the dual relation between the state feedback control and the full order observer design. Based on this idea, some well-established approaches for state feedback control can be readily applied to robust fault detection using full order unknown input observers. Because of the existence of system complexities such as nonlinearities, disturbances, and uncertainties in a typical complex control system, fault diagnosis for such dynamical systems still pose a number of challenging problems. Amongst various uncertainties, unknown inputs are one type of uncertainty that has received considerable attention. To deal with the unknown inputs, robust approaches are often employed. Furthermore when the system under consideration is subject to unknown disturbance or unknown inputs, to achieve effective fault detection, the effect of the disturbance has to be de-coupled from the residual signal to avoid false alarms in detection. This problem is known in the literature as robust fault detection or fault detection using unknown input observers. Sliding mode techniques have good robustness and are completely insensitive to so-called matched uncertainty [16]-[7]. It has been shown that sliding mode techniques can also be used to deal with both structural and unmatched uncertainty [18], therefore the application of sliding mode techniques for robust FDI offers good potential. A sliding mode observer was used for FDI in the early nineties [21] where Sreedhar, Fernandez and Masada consider a model-based sliding mode observer approach although in their design procedure it is assumed that the states of the system are available. A different approach is adopted by Hermans and Zarrop [22], who attempt to design an observer in such a way that in the presence of a fault the sliding motion is destroyed. More recently 69

84 70 CHAPTER 4. SLIDING MODES FOR FDI Edwards et al.(2000) [4] proposed an approach based on the concept of equivalent output injection in which the resulting reconstruction signal can approximate the actuator faults to any required accuracy (this is called precise fault reconstruction). In [4] the authors consider the practical situation when the system states are not available. The observer is designed to maintain a sliding motion even in the presence of faults which are detected and identified by analyzing the so-called equivalent output injection. The novelty lies in the manipulation of the equivalent output injection signal to explicitly reconstruct fault signals rather than detect the presence of a fault through a residual signal (This may be allied to the equivalent control signal which appears in the analysis of sliding mode). Fault reconstruction is a powerful alternative to the detection of a fault via the use of a residual signal as long as the location of the fault effect on the system is known. The residual approach is more suited to the combined problem of fault detection and fault isolation, when the structure of the fault influence on the system is not perfectly known. A bank of dissimilar (but redundant) residual signals can then be used to infer the location of the fault in the system. On the other hand, the fault estimation approach is a direct way of providing fault information which, when compared with other fault estimation signals (from the same system), can be used to isolate all faults. The fault estimation method also provides a direct estimate of the size and severity of the fault, which can be important in many applications. Later, it was extended by Tan and Edwards (2002) [19] where sensor faults were considered. However, uncertainty was not considered in these early papers. It is well-known that the observer-based approach is very dependent on the system model. In practice a precise and accurate model for a real system is often not available due to unknown exogenous disturbances and/or time-varying parameters (component aging). Modelling uncertainty can cause false and missed alarms, which may make the FDI system useless. Hence, it is very important to consider robustness when implementing FDI schemes. A FDI scheme for a class of linear systems with uncertainty was proposed by Tan and Edwards (2003) [20] which focused on minimizing the L2 gain between the uncertainty and the fault reconstruction signal by using linear matrix inequalities (LMI). A robust fault detection method for nonlinear systems with disturbances was considered in Floquet et. al [23] where strict geometric conditions are exploited to the aim of design a residual generator when the faults cannot be decoupled from the disturbance inputs. It should be emphasized that precise fault reconstruction is very challenging for nonlinear systems especially in the presence of uncertainty. When uncertainty is considered, all the results concerning sliding mode observer-based fault reconstruction only provide an estimate of the fault signal. It is a valuable meaningful task to establish an approach for fault reconstruction in nonlinear systems, or to find conditions under which precise fault reconstruction is feasible. Moreover, since FDI is required to take place on-line in real engineering systems, this requires the reconstruction fault signal to be based only on the available measured information. Recently Yan and Edwards (2007) [5] extends previous result, Edwards et al. (2000) [4], for a class of nonlinear uncertain systems where the uncertainty is allowed to have a nonlinear bound. A sufficient condition based on LMIs is presented for the existence and stability of a robust sliding mode observer. Then, fault estimation and fault reconstruction methods are presented using the equivalent output injection approach. It is shown that, under certain geometric conditions associated to the uncertainty structure matrix and the fault distribution matrix, precise fault reconstruction is available for a class of nonlinear systems by exploit-

85 ing the features of the sliding motion and the structure of the uncertainty. The proposed reconstruction signal converges to the fault with arbitrary accuracy even in the presence of uncertainty. If the geometric condition does not hold, then a strategy is presented to estimate the fault signal, and the estimation error depends on the bounds on the uncertainty. The proposed fault estimation/reconstruction signals are only based on the available plant input/output information and can be calculated on-line. It must be highlight that the main limitation of the fault diagnosis schemes, that make use of conventional first order SMOs, is that the relative degrees from the inputs and/or the unknown inputs to the outputs must be one. Furthermore for the fault reconstruction signals, obtained by processing the so called "equivalent output error injection", the use of low pass filters is needed and the observer guaranties the exact reconstruction of fault only in Filippov sense. Because many physical systems such as satellite control systems, and mechanical systems can not satisfy this condition, new fault diagnosis beyond using conventional first order SMOs are needed. One promising strategy is to use the recently developed high order sliding mode techniques such as high order sliding mode observers and differentiators. Based on high order sliding modes, arbitrary-order exact robust differentiators have been studied in the literature. The proposed differentiators can provide exact estimation for the derivatives of a signal of any order if there is no measurement noise. When noise is present, the estimation errors of the derivatives will be small if the magnitude of the noise is small. These properties make high order sliding mode differentiators appealing in fault diagnosis. Although high order sliding mode observers and differentiators have appealing properties that could be used in fault diagnosis, their great potential has not been well recognized in the fault diagnosis community and there are very few results in this direction. 71

86 72 CHAPTER 4. SLIDING MODES FOR FDI 4.1 SMO for faults reconstruction As previously discuss, fault reconstruction is a powerful alternative to the detection of a fault via the use of a residual signal as long as the location of the fault effect on the system is known. Based on the approach described in 3.2.1, where an observer for linear system is designed to maintain a sliding motion even in the presence of matched unknown input, in this section will be presented a method [16] to detected and identified such unknown inputs by analyzing the so-called equivalent output injection. More precisely the manipulation of the equivalent output injection signal will be used to explicitly reconstruct fault signals rather than detect the presence of a fault through a residual signal. In this section we consider a nominal linear system subject to certain faults described by { ẋ = Ax(t) + Bu(t) + E fi (t) (4.1) y = C x(t) + f o (t) where A R nxn,b R nxm,c R pxn and E R nxq with q p < n, and matrices B,C and D are of full rank. It must be highlight that in 4.1 we denote with E the fault distribution matrix in order to apply directly the results presented in The functions f i (t) and f o (t) are deemed to represent actuator and sensor faults, respectively, and are assumed to be bounded. It s further assumed that the states of the system are unknown, i.e. p < n and only the signals u(t) and y(t) are available. As one can see the fault signal f i (t) takes the place of the disturbance term d(t, y,u) of the uncertain dynamical system considered in 3.21, furthermore. The objective is to synthesize an observer to generate a state estimate ˆx(t) and output estimate ŷ = C ˆx such that a sliding mode is established in which the output error e y (t) = ŷ(t) y(t) (4.2) is forced to zero in finite time. The particular observer structure described in section will be considered, namely ż(t) = Az(t) + Bu(t) G l e y (t) +G n ν (4.3) where G l and G n are defined in (3.55) and (3.56) respectively. The discontinuous output injection term ν, designed according the works [4]-[16] in which a first order SM control law has been proposed, takes the following form ν = ρ(t, y,u) P 2e y P 2 e y (4.4) where the upperbounding function ρ(t, y, u) and the symmetric positive design matrix P 2 are introduced in 3.48 and 3.50, respectively. It will be shown that, provided a sliding motion can be attained, estimates of f i (t) and f o (t) can be computed from approximating the so-called equivalent output injection signal required to maintain the sliding motion.

87 4.1. SMO FOR FAULTS RECONSTRUCTION SMO for reconstructing of the input fault signals Consider first the case when only actuator faults are presents, i.e., f o = 0. Assume that an observer has been designed as in and that a sliding motion has been established so that e y = ė y = 0 (in mean value). Therefore, in appropriate coordinates, the equation (3.49) becomes 0 = A 21 e 1 (t) E 2 f i (t) + ν eq (4.5) where ν eq is the equivalent control that represent the average behavior of the discontinuous component ν and represents the effort necessary to maintain the motion on the sliding surface. From (3.49) - (3.53) it result that the error system is quadratically stable and therefore, after a finite time, it result that ν eq E 2 f i (t) (4.6) A commonly used approach to reconstruct the equivalent injection is by the use of a low pass filter [7]. Since r ank(e 2 ) = q it follows from (4.6) that f i (t) (E T 2 E 2) 1 E T 2 ν eq (4.7) The key point is that the signal on the right-hand side of the equation above can be computed on-line and depends only on the output estimation error e y, thus the fault f i can be approximated to any degree of accuracy.

88 74 CHAPTER 4. SLIDING MODES FOR FDI SMO for reconstructing of the output fault signals Now consider the case when only sensor faults are presents, i.e., f i = 0. In such situation since y(t) = C x(t) + f o (t) it follows that e y (t) = Ce(t) f o (t) (4.8) therefore, assuming the referred system 4.1 in a the normal form of 3.45 and considering the particular observer structure given in 3.46, it follows a state estimation error of the form { ė1 (t) = A 11 e 1 (t) + A 12 f o (t) ė y (t) = A 21 e 1 (t) + A S 22 e y (t) + A 22 f o (t) f o (t) + ν Note that f o (t) and f o (t) appear as output disturbances and thus ρ( ) in equation (4.4) must be chosen to be sufficiently large to maintain sliding in the presence of these disturbances. Provided a sliding motion can be attained, making the assumption ė y = 0, the following holds 0 = A 21 e 1 (t) + A 22 f o (t) f o (t) + ν eq (4.10) Thus for slowly varying faults, i.e. fo (t) 0, provided the dynamics of the sliding motion are sufficiently fast, from the first component of the error dynamic 4.9, and assuming ė 1 = 0, we obtain e 1 = A 1 11 A 12 f o (t). Substituting such expression into the second component of the error dynamic and imposing e y = ė y = 0, the following holds (4.9) ν eq (A 22 A 21 A 1 11 A 12)f o (t) (4.11) As in the previous paragraph, the equivalent output injection ν eq can be calculated from (4.4) and consequently if (A 22 A 21 A 1 11 A 12) is non-singular, the fault signal can be obtained from equation (4.11). Note that from Schur expansion det(a) = det(a 11 )det(a 22 A 21 A 1 11 A 12) (4.12) and thus (A 22 A 21 A 1 11 A 12) is nonsingular if and only if det(a) 0. However even if (A 22 A 21 A 1 11 A 12) is singular, inference can still be made about certain fault channels depending on the precise nature of the rank deficiency.

89 4.2. FILTRATION-FREE FAULT RECONSTRUCTION VIA FULL ORDER HOSMO Filtration-free fault reconstruction via full order HOSMO Based on the same concepts, but applied in a nonlinear framework, in this section is presented an observer for a class of nonlinear uncertain systems, which contains a corrective term designed on a second-order sliding mode control algorithm featuring global convergence properties. Such observer allow the reconstruction in finite time of the actuator faults performed directly from the continuous observer output injection signal without require any filtration. Let us consider the class of uncertain nonlinear systems ([5],[82]) described by ẋ = Ax +G(x,u) + EΨ(x,u, t) + D f (y,u, t) (4.13) y = C x (4.14) where x R n, u R m and y R p are the state, input and output vectors, respectively. A R nxm, E R nxr, D R nxq and C R pxn (q p < n) are known constant matrices with D and C having both full rank. The known nonlinear vector field G(x,u) is assumed to be Lipschitz, the unknown nonlinear term Ψ(x,u, t) R r represent all model uncertainties and disturbance affecting the system, and the unknown functions f (y,u, t) R q represents the actuators faults to be identified. Some assumption are met about the uncertain terms affecting system (4.13)-(4.14). Let known function τ(t),ξ(t),τ d (t),ξ d (t), possibly depending on the system inputs/outputs, exist such that f (y,u, t) τ(t), Ψ(x,u, t) ξ(t) (4.15) f (y,u, t) τ d (t), Ψ(x,u, t) ξ d (t) (4.16) It is assumed that the known control input vector u(t) contains smooth functions. In particular it is assumed to know a positive function U d (y, t) such that The following assumption is also met: u(t) U d (y, t) (4.17) r ank(c [E D ]) = r ank([e D ]) p (4.18) Under assumption (4.18) it was shown [5] that it can be found a linear transformation, with matrix T 0, such that, in the new coordinates system, the system equations become ẋ 1 = A 1 x 1 + A 2 x 2 +G 1 (x,u) (4.19) ẋ 2 = A 3 x 1 + A 4 x 2 +G 2 (x,u) + E 2 Ψ(x,u, t) + D 2 f (y,u, t) (4.20) y = C 2 x 2 (4.21) where x = [x 1, x 2 ] T with x 1 R n p, G 1 (x,u) and G 2 (x,u) are known nonlinear vector fields. The triple (A,[E D],C ) assume therefore the following form ( [ A1 A 2 A 3 A 4 ], [ 0(n p) (r +q) E 2 D 2 ], [ 0p (n p) C 2 ] ) (4.22)

90 76 CHAPTER 4. SLIDING MODES FOR FDI where A 1 R (n p) (n p), and C 2 R p p is non singular. If, furthermore, r ank([e D ]) = q q, then the transformation matrix can be selected in such a way that E 2 and D 2 have the following structure [ 0(p q) r E 2 = E 22 ] [ 0(p q) q,d 2 = D 22 where E 22 R q r, furthermore the matrix D 22 R q q is of full rank (see [4] and [5]). ] (4.23) Under the assumption that the invariant zeros of the triple (4.22) lie in C, there exist a matrix L R (n p) p of the form L = [ L 1 0 (n p) q ] (4.24) with L 1 R (n p) (p q) such that A 1 + L A 3 is Hurwitz [5]. Let us consider system (4.19)-(4.21), and introduce a new coordinate transformation z = T x where [ ] I(n p) L T =: (4.25) 0 I p and L is given by (4.24). Then, in the new coordinates the transformed dynamics become ż 1 = M 0 z 1 + M 1 z 2 + [I n p L]G(T 1 z,u) (4.26) ż 2 = A 3 z 1 + M 2 z 2 +G 2 (T 1 z,u) + E 2 Ψ(T 1 z,u, t) + D 2 f (y,u, t) (4.27) y = C 2 z 2 (4.28) where z = [z 1 z 2 ] T, with z 1 R n p and z 2 R p, and M 0 = A 1 + L A 3 M 1 = A 2 + L A 4 (A 1 + L A 3 )L M 2 = A 4 A 3 L As said before, Matrix M 0 is Hurwitz, it ll have important consequences on the next developments. For the system (4.26)-(4.28), consider a dynamical observer of the following form ẑ 1 = M 0 ẑ 1 + M 1 C 2 1 y + [I n p L]G(T 1 ẑ,u) (4.29) ẑ 2 = A 3 ẑ 1 + M 2 ẑ 2 +G 2 (T 1 ẑ,u) K (y C 2 ẑ) + w (4.30) ŷ = C 2 ẑ 2 (4.31) where ẑ = [ẑ 1 C 2 1 y] T, ŷ is the observer output, and w is an appropriate control law to be designed. Let e 1 = z 1 ẑ 1 and e y = y ŷ then the error dynamics becomes ė 1 = M 0 e 1 + [I n p L](G(T 1 z,u) G(T 1 ẑ,u)) (4.32) ė y = C 2 A 3 e 1 +C 2 M y e y +C 2 (G 2 (T 1 z,u) G 2 (T 1 ẑ,u)) +C 2 E 2 Ψ(T 1 z,u, t) +C 2 D 2 f (y,u, t) C 2 w (4.33)

91 4.2. FILTRATION-FREE FAULT RECONSTRUCTION VIA FULL ORDER HOSMO 77 where M y = (A 4 A 3 L)C K and K R p p is a gain matrix selected to guarantee the stability of the linear part of the state estimation error dynamics, i.e., in order to make the following matrix an Hurwitz one [ M0 0 C 2 M 2 C 2 M y ] (4.34) Proposition 1: Consider system (4.32). If the following LMI is feasible Ā T P T + P Ā + 1 ɛ P P T + ɛl G 2 I n p + αp < 0 (4.35) P := P[I n p L], Ā := [ A1 A 3 ] (4.36) with P > 0, ɛ and α are positive constants, L G is the Lipschitz constant of G(x,u) with respect to x, and L has the structure in (4.24), then e 1 tends asymptotically to zero fulfilling the following inequality during the transient where M := λ max (P)/λ mi n (P). Proof: see [5]. e 1 (t) e 1max (t) M e 1 (0) exp{ α t/2} (4.37) Remark 1: The LMI (4.35) follows from a Lyapunov approach with candidate function V = e 1 T Pe 1, and, in particular, by imposing V αv. Proposition 1 explain that vector e 1 tends to zero asymptotically. It means that in order to reconstruct the system state implementing the equation (4.29) would be enough, but, in order to reconstruct the fault vector, additional dynamics are required. Then, the main problem here is to design the observer control w in such a way that vectors e y and ė y tends exactly to zero in finite time. In [5] a solution was suggested based on standard first order sliding mode control. Such an approach opens the way to achieve the fault reconstruction via using the equivalent control principle (i.e., via low pass filtering) and it is therefore an approximate method [7],[4],[5]. Here we propose a different approach based on second-order sliding modes that enable us to reconstruct the fault without any filtration, therefore leading to an exact solution. Consider the following well-defined transformation: e y = C 2 1 e y (4.38) and write the expression for the second derivative of e y were G y is compactly defined as follows e y = A 3 ė 1 + M y C 2 e y +Ġ y + E 2 Ψ + D 2 f ẇ (4.39) G y = (G 2 (T 1 z,u) G 2 (T 1 ẑ,u)) (4.40)

92 78 CHAPTER 4. SLIDING MODES FOR FDI as Considering (4.32)-(4.33) together with the transformation (4.38) yields to rewrite (4.39) e y = (A 3 M 0 + M y C 2 A 3 )e 1 + (M y C 2 ) 2 e y + ([I n p L] + M y C 2 )G y +M y C 2 E 2 Ψ + M y C 2 D 2 f +Ġ y + E 2 Ψ + D 2 f My C 2 w ẇ = ϕ(e 1,e y,u, w, t) ẇ (4.41) with implicit definition of the drift term ϕ( ). The equation (4.41) defines p scalar subsystems having the following form { γ1i = γ 2i, i = 1..p γ 2i = ϕ i (, t) + v i (4.42) where γ 1i and ϕ i (, t) are the i-th entry, for i = 1..p, of vectors e y and ϕ(, t), respectively, and v i = ẇ i. The problem is to find a set of discontinuous controls laws v i stabilizing the uncertain SISO systems (5.15) in finite time. To solve this problem the second-order sliding mode control approach appears to be particularly appropriate because of systems (5.15) have relative degree two. The boundedness properties of the drift term ϕ i plays a crucial role. In the standard literature on 2-SMC it was often assumed the existence of a constant upperbound Φ i, known a-priori, such that ϕ i Φ i. Here we refer to a recently proposed "Global" version [30, 8] of the suboptimal algorithm which can work under the more general assumption ϕ i (, t) Φ i (t) (4.43) where Φ i (t) is a (possibly time-depending) function given in real time. solution is sumarized in the following Theorem 1. The proposed Theorem 1: Consider system (5.15), whose uncertain dynamics satisfies (4.43). Apply the control law [ Φ i (t) + χ ] sign ( γ 1i (t) γ 1i (0) ) 0 t t Mi j v i (t) = [ Φ i (t) + χ ] sign (γ 1i (t Mi j )) t Mi j < t t ci j (4.44) where [ Φi (t) + Π + χ ] sign (γ 1i (t Mi j )) t ci < t t Mi,j +1 Π 1 3 η2 (4.45) χ and η are positive arbitrary constants t Mi j (j = 1,2,...) is the sequence of time instants at which γ 2i (t) = 0, and t ci j is the first time instant subsequent t Mi j at which one of the

93 4.2. FILTRATION-FREE FAULT RECONSTRUCTION VIA FULL ORDER HOSMO 79 following relationships is verified γ 1i (t ci j ) = 1 2 γ 1(t Mi j ) γ 2i (t ci j ) = η γ 1i (t Mi j ) (4.46) where γ 2i (t), which represents an instantaneous upperbound of γ 2i (t), is defined as γ 2i (t Mi j ) = 0 j = 1,2,... γ 2i (t) = 2(Φ i (t) + χ) t Mi j t t ci j 0 t ci j < t < t Mi,j +1 (4.47) Then, the global finite-time attainment of conditions is provided. Proof of Theorem 1: See [30, 8]. γ 1i = γ 2i = 0 (4.48) Remark 2: The parameter η, which is free to be chosen according to the desired transient specifications, defines the prescribed upperbound for the modulus of γ 2i during the transient process. Note that the smaller η is, the slower the convergence process. Furthermore, it must be highlight that, the proposed algorithm requires the sequence of the values of γ 1i at the time instants at which γ 2i is zero. The corresponding time instants can be detected, with an arbitrarily-small delay, by existing peak-detector devices. In previous works [6], the sub-optimal algorithm has been shown to be robust against the approximate detection of the singular points, and the same considerations still apply to the actual case. Let Φ i (t) be positive functions fulfilling (4.43). Define Φ(t) = col(φ 1 (t),φ 2 (t),...,φ p (t)) (4.49) Let us derive an expression for the norm of the upperbounding vector Φ(t) that is required for the synthesis of the control law. By (4.41) we can rewrite the drift term ϕ(, t) = col(ϕ 1 (, t),ϕ 2 (, t),...,ϕ p (, t)) as ϕ(, t) = Me 1 + (M y C 2 ) 2 e y + ([I n p L] + M y C 2 )G y + + M y C 2 E 2 Ψ + M y C 2 D 2 f +Ġ y + E 2 Ψ + D 2 f My C 2 w (4.50) where M A 3 M 0 + M y C 2 A 3. It is worth noting that whilst e y is known and measurable, e 1 is unknown, and, in order to evaluate explicitly an upperbound to (4.50), it must be overestimated. To this end, the following proposition is useful.

94 80 CHAPTER 4. SLIDING MODES FOR FDI From (4.24) and (4.25), it can be concluded that in light of which it follows from (4.40) that x ˆx T 1 z T 1 ẑ = e 1 (4.51) G y = G 2 (T 1 z,u) G 2 (T 1 ẑ,u) L G e 1 (4.52) Furthermore we are interested in finding an upperbound of Ġ y, so expliciting the derivative of Ġ y : Ġ y (x, ˆx,u) = G 2 x (x,u)ẋ G 2 ˆx ( ˆx,u) ˆx + G 2 u (x,u) u G 2 ( ˆx,u) u (4.53) u Adding and subtracting the term G 2 x (x,u) ˆx, and making some manipulations, yields the following inequality Ġ y L G2x Rė 1 + LĠ2x Re 1 ˆx + LĠ2u Re 1 u (4.54) where R is an appropriate matrix, and L G2x,LĠ2x,LĠ2u represent the Lipschitz constants with respect to x, uniformly for u, of G 2, G 2 x and G 2 u, respectively. Eq. (4.54) together with (4.32) implies that Ġ y { R L G2x ( M 0 + L G [I n p L] ) + R LĠ2x ˆx + LĠ2u K u } e 1 (4.55) that can be rewritten in compact form as Ġ y Ġ ymax e 1 (4.56) Now we have all the ingredients to evaluate the upper bounding function (4.49). By (4.50), and considering the initial assumptions (4.15) (4.17) together with (4.56), (4.52) and (4.37), it can be found a scalar function Φ (t) such that Φ (t) Φ i (t) i which has the following structure Φ (t) = ( M +Ġ ymax + Γ 1 )e 1max (t) + (M y C 2 ) 2 e y Γ 2 ξ(t) + Γ 3 τ(t) + E 2 ξ d (t) + D 2 τ d (t) + M y C 2 w (4.57) where the positive constants Γ i (i = 1,2,3) are defined as Γ 1 = ([I n p L] + M y C 2 ) L G Γ 2 = M y C 2 E 2 Γ 3 = M y C 2 D 2 (4.58) The upperbound defined in (4.57)-(4.58) is available in real time and can be used in all the control laws in Theorem 1 (in other words, the conservative approximation Φ i (t) = Φ (t) could be made, for the simplicity sake, when implementing the proposed observer).

95 4.2. FILTRATION-FREE FAULT RECONSTRUCTION VIA FULL ORDER HOSMO Actuator faults reconstruction We address in this section the problem of reconstructing the vector f (y,u, t) representing the actuator faults. This procedure is conventionally approached by resorting to the equivalent control method [4]-[5] and, more precisely, by exploiting the possibility of achieving on-line an estimate of the equivalent control by means of low pass filtering. In this contest the use of second order sliding mode control, that provides a continuous observer control signal w steering to zero both e y and ė y in finite time, allows us to reconstruct the fault without filtration. Assume that Im(E 22 ) Im(D 22 ) = {0}, where the matrices E 22 and D 22 are those given in (4.23). Then [5] there exist a nonsingular matrix W R q q such that [ H1 0 W [E 22 D 22 ] = 0 H 2 ] (4.59) where H 1 R ( q q) r and H 2 R (q q). If (4.59) holds, the effect of the faults can be separated" from the effect of the uncertainty, thus permitting the precise reconstruction of the fault vector. From (4.33) and (4.38), imposing the conditions e y = ė y = 0, it yields that A 3 e 1 +G y (z, ẑ,u, t) + E 2 Ψ(T 1 z,u, t) + D 2 f (y,u, t) w = 0 (4.60) From the latter, taking into account (4.52), (4.37) and (4.23) it derives that E 22 Ψ(T 1 z,u, t) + D 22 f (y,u, t) w q = 0 (4.61) where w q denotes the last q elements of vector w. By (4.59), [ Ψ(T 1 z,u, t) W w q = W [E 22 D 22 ] f (y,u, t) ] [ H1 Ψ(T 1 z,u, t) = H 2 f (y,u, t) ] (4.62) Let W q denote the last q rows of W ; then it follows the following formula for reconstructing the fault vector f ˆ = H 1 2 W q w q (4.63) Remark 3: If Im(E 22 ) Im(D 22 ) {0} the exact reconstruction of the faults is no longer possible and the application of the given fault reconstruction formula gives rise to an estimation error overestimated as follows [5] f ˆ (t) f (y,u, t) D + 2 E 2 ξ(x,u, t)+d 2 L G e 1 where D + 2 is the left pseudo-inverse of D 2 (which exists since D 2 is full column rank). Simulative and experimental analysis of the proposed full order observer scheme is widely given in chapter 6, "Application Results".

96 82 CHAPTER 4. SLIDING MODES FOR FDI 4.3 Filtration-free fault reconstruction via reduced order HOSMO In this section, by dispensing with the request of estimating the state vector, in other words by only aiming to the reconstruction of the faults, a reduced order observer will be designed. Also for such case, as in previous section, the output injection of the proposed observer is built by means of a second-order sliding mode control algorithm, featuring global convergence properties, that permit the reconstruction of the actuator fault without requiring any filtration. Let us consider the same class of uncertain nonlinear systems of the former case (4.13) (4.14), except for the nonlinear vector G( ), for which, in this discussion, it may depend only from the output vector y and the control inputu. The same assumption (4.15) (4.18) are meet, furthermore we assume that the system is BIBS and that a controller, acting on the system, guarantees the following constraints x 1 Γ( y, u ) (4.64) where Γ : (R + R + ) R + is a positive scalar function. It worth noting that, under the rank condition (4.18) it can be found a linear transformation, such that the reference system in the new coordinates become ẋ 1 = A 1 x 1 + A 2 x 2 +G 1 (y,u) (4.65) ẋ 2 = A 3 x 1 + A 4 x 2 +G 2 (y,u) + E 2 Ψ(x,u, t) + D 2 f (y,u, t) (4.66) y = C 2 x 2 (4.67) where x = [x 1, x 2 ] T with x 1 R n p, G 1 (y,u) and G 2 (y,u) are known nonlinear vector fields, and C 2 R pxp is non singular. As one can see, in (4.66) only the disturbance vector Ψ(x,u, t) is dependent from the whole state vector x Fault observer design The aim is to design a dynamical system allowing us to reconstruct the actuator faults in finite time by dispensing with the request of estimating the whole state vector. Consider system (4.65)-(4.67) and rewrite (4.66) as follows by embedding the state vector component x 1 into the new, larger, disturbance" vector ζ where ẋ 2 = A 4 x 2 +G 2 (y,u) + Pζ(x,u, t) + D 2 f (y,u, t) (4.68) y = C 2 x 2 (4.69) P = [A 3 E 2 ] P R p (n p+r ) (4.70) [ ] x 1 ζ(x,u, t) = ζ R (n p+r ) 1 (4.71) Ψ(x,u, t)

97 4.3. FILTRATION-FREE FAULT RECONSTRUCTION VIA REDUCED ORDER HOSMO 83 Consider the following reduced-order observer ˆx 2 = A 4 x 2 +G 2 (y,u) + w (4.72) ŷ = C 2 ˆx 2 (4.73) where ŷ is the observer output and w is an appropriate control law. The aim is to reconstruct the actuator fault vector f (y,u, t) exactly and in finite time. Let e y = y ŷ, and consider the following transformation: from (4.68), (4.72) and (4.74) it results The second derivative of e y is e y = C 2 1 e y x 2 ˆx 2 (4.74) ė y = Pζ(x,u, t) + D 2 f (y,u, t) w (4.75) ë y = P ζ(x,u, t) + D 2 f (y,u, t) ẇ (4.76) = ϕ(, t) ẇ (4.77) with implicit definition of the drift term vector field ϕ( ). The problem of stabilizing e y and ė y, exactly and in finite time, is formally equivalent to that considered for the system 4.41 and therefore can be solved by using the same approach. Let Φ i (t) be positive functions fulfilling (4.43). Define Φ (t) as a positive scalar function such that Φ (t) Φ i (t) i = 1,2,..., p (4.78) Let us derive an expression for Φ(t) which is required for the synthesis of the control law. By (4.77) we can rewrite the drift term ϕ(, t) = col(ϕ 1 (, t),ϕ 2 (, t),...,ϕ p (, t)) as ϕ(, t) = P ζ(x,u, t) + D 2 f (y,u, t) (4.79) First, from the expression for ζ given in (4.71), it follows that ζ(x,u, t) = [ ẋ 1 Ψ(x,u, t) ] (4.80) so that, from (4.16) ζ(x,u, t) ẋ 1 + ξ d (t) (4.81) Now considering the system equation (4.65), in conjunction with assumption (4.64), it follows that ẋ 1 A 1 x 1 + A 2 x 2 + G 1 (y,u) A 1 Γ( y, u ) + K (y,u) Γ d ( y, u ) (4.82) with implicit definition of the positive scalar function Γ d

98 84 CHAPTER 4. SLIDING MODES FOR FDI Now we have all ingredients to evaluate the upper bounding function Φ (t), according to (4.78), as follows Φ (t) = P [Γ d ( y, u ) + ξ d (t)] + D 2 τ d (t) (4.83) The upperbound defined in (4.83) can be used in all the control laws in Theorem 1 (in other words, the conservative approximation Φ i (t) = Φ (t) could be made, for the simplicity sake, when implementing the proposed observer) Actuator fault reconstruction Now we address the problem of reconstructing the vector f (y,u, t) representing the actuator faults. As for the full order observer case, 4.2.1, the use of second order sliding mode control, that provides a continuous observer control signal w steering to zero both e y and ė y exactly and in finite time, allows us to reconstruct the fault without filtration. Assume that Im(P) Im(D 2 ) = {0}, then [5] there exist a nonsingular matrix W R p p such that [ H1 0 W [P D 2 ] = 0 H 2 ] (4.84) where H 2 R (q q). If (4.84) holds, the effect of the faults can be separated" from the effect of the uncertainty, thus permitting the precise reconstruction of the fault vector [5]. From (4.75), imposing the condition e y = ė y = 0, the following holds after a finite time Pζ(x,u, t) + D 2 f (y,u, t) w = 0 (4.85) Bringing w into the left hand side and multiplying both sides for W, it results [ ζ(x,u, t) W w = W [P D 2 ] f (y,u, t) ] (4.86) Let W 2 denote the last q rows of W, then it follows the following formula for reconstructing the fault vector f ˆ = H 1 2 W 2 w (4.87) Remark 4: It must be highlight that, unlike the approaches in [4]-[5] where ė y = 0 is true only in Filippov sense, here the condition e y = ė y = 0 is attained in finite time, and, since the control signal w is already continuous, no filtering of discontinuous signal is necessary anymore.

99 4.3. FILTRATION-FREE FAULT RECONSTRUCTION VIA REDUCED ORDER HOSMO Simulation results The effectiveness of the suggested observation and fault reconstruction scheme is tested on a simple example. Consider a system in the general form (4.13)-(4.14) with n = 4, m = 1, p = 3 and the following matrices and vector fields: A = G(y,u) =, C = y u si n(y 2 ) 3u u As one can see the nonlinear vector G( ) depend only from the output vector. Introduce a linear transformation with matrix T 0 = The matrices of the transformed dynamic system become [ ] A1 A = A 3 A si n(y G(y,u) = col(g 1,G 2 ) = 2 ) + y 2 2 2u u y u 0 [ ] D = = D 2 1 C 2 = [ ] E = = E and Ψ(x,u, t) = x si n 2 x 3. The P in (4.70) takes the following form P =

100 86 CHAPTER 4. SLIDING MODES FOR FDI Let us design the observer according to (4.72)-(4.73). For the system under analysis, it result that Im(P) Im(D 2 ) = {0}, so there exist a matrix W R 3x3, as in (4.84), that allows to decouple uncertainties from faults thereby permitting to reconstruct the fault signals from w in accordance with the formula (4.87). A suitable choice for decoupling matrix conduce to W 2 = [ ] H 2 = 3.5 The reconstruction of fault is made via (4.87). A fault signal, shown in the top plot of Figure 2, has been introduced in input channel. Figure 4.1: The state variable and the state observation error. The initial conditions of the state variables are x 1 (0) = x 2 (0) = x 3 (0) = x 4 (0) = 1.0, while the initial conditions of the observed states are ˆx 2 (0) = ˆx 3 (0) = ˆx 4 (0) = 0.0. Fig.4.1 shows the actual state variables, and the state observation errors. Fig.4.2 shows the actual and reconstructed fault signal and the reconstruction error. The finite time convergence of the estimate, and the high accuracy of the estimate, are both apparent from the analysis of Fig.4.2.

101 4.3. FILTRATION-FREE FAULT RECONSTRUCTION VIA REDUCED ORDER HOSMO 87 Figure 4.2: The actual fault and the reconstructed signal fault.

102

103 Chapter 5 Discrete state reconstruction via HOSMO 5.1 Introduction As explained in Chapter 1 the knowledge of a model for the system under investigation, the better possible accurate, is the main point in model based FDI systems, either for techniques based on residual generation or not. In particular it is well-known that the observer-based approaches are very dependent on the system model since modelling uncertainty can cause false and missed alarms, which may make the FDI system useless. It must be highlighted that many process, in real applications, show nonlinear dynamic behavior, especially if wide areas of operation are involved during the functioning. It s a common practice, in several engineering fields, to refer to non-linear processes that operate at different regimes as distinct models each associated to an admissible operating condition. Based on these assumptions we focalize our attention into those cases in which the system under diagnosis is characterized by different nonlinear dynamics, varying according to an unknown logic. In such a case the use of a prefixed FDI scheme based on a unique model could led to a residual strongly dependent from model mismatches. Assuming indeed to know the active nonlinear model and its parameter, and also to be able of designing an appropriate FDI scheme for each single nonlinear model, the overall scheme will be obviously more accurate and the false alarm rate will strongly decrease. Furthermore, it could be possible to associate a specific model for the faulty system. In this case, the detection of the actual dynamics leads to fault detection, directly. In this respect a crucial issue in model based FDI is the capability to identifying the actual nonlinear dynamic of systems that could be described appropriately by a switched system. A switched system has a discrete dynamics represented by a finite state machine that evolves according to the occurrence of discrete events. To each discrete state (or "mode") a continuous dynamics is associated. In the last decade the control community has devoted a great deal of attention to the study of switched systems [71],[72],[73]. Therefore, a problem of great interest is the reconstruction of the discrete-state through the observation of measurable system outputs. The techniques developed in this framework can be applied to several problems where the discrete events are not observable. 89

104 90 CHAPTER 5. DISCRETE STATE RECONSTRUCTION VIA HOSMO In the framework of discrete event systems, several approaches have been proposed to estimate the discrete state [74],[75]. In a more general hybrid context, the discrete state estimation has been discussed in [76],[77]. In this chapter we investigate the problem of the discrete state reconstruction for switched systems building on the idea that a general class of switched systems can be modeled by nonlinear systems with an affine boolean input representing the system discrete state. Objective of the present work [81] is to reconstruct such a boolean input despite bounded uncertainties affecting the system dynamics. To this aim we propose a 2-SM based technique, that exhibit remarkable properties of robustness against uncertainties and disturbances [7], for reconstructing the unmeasurable quantities affecting the system.

105 5.2. PROBLEM FORMULATION Problem formulation In this paper we examine the class of switched systems that can be represented in the form: ẋ(t) = G(x,u, t) + D(x,u, t)δ(σ(t)) + ε(x, t) (5.1) where x(t) X R n is the continuous state, u(t) U R p is the input to the system, G(x,u, t) R n and D(x,u, t) R n L are known vector fields, and ε(x, t) R n is an uncertain term representing model mismatches and/or external disturbances. The piecewise-constant integer function σ(t) {0,...,k 1} represents the unknown discrete state of system (5.1). Vector δ(σ(t)) {0,1} L contains boolean elements. It maps the discrete state σ(t) into an L-dimensional boolean vector which encodes" the actual discrete state. Model (5.1) can represent switched systems with, at most, k 2 L different dynamics. The problem tackled in this paper is the reconstruction of the discrete state σ(t) Assumptions We now specify the assumptions which are met about the considered class of systems (5.1). The continuous state x(t) is supposed to be fully measurable, and G(x,u, t), D(x,u, t) are supposed to be known. The dimension L of vector δ(t) must not exceed the dimension of the continuous state: L n (5.2) The boolean vector δ(t) is not available due to the uncertainty in the discrete state. The discrete state σ(t) can be uniquely recovered from the boolean vector δ(t), and viceversa. Let the time evolutions of the continuous state x and exogenous input u variables be a-priori confined in the compact domains X and U. Assumption with respect the smoothness and norm-bounded are made with respect the full-rank matrix D(x, u, t) and the unmeasurable, state dependent, uncertainty term ε(x, t) D(x,u, t) D 0 Ḋ(x,u, t) D 1 (5.3) [D T (x,u, t)d(x,u, t)] 1 D T (x,u, t) D 2 (5.4) ε(x, t) ε 0 ε(x, t) ε 1 (5.5) Note that no boundedness or smoothing requirements are met on the vector field G(x,u, t) Comments on the considered class of systems We now show that the considered class of switched systems (5.1) is general enough to represent the following continuous-time switched nonlinear dynamics ẋ(t) = G σ(t) (x,u, t) + ε(x, t) (5.6)

106 92 CHAPTER 5. DISCRETE STATE RECONSTRUCTION VIA HOSMO where σ(t) {0,...,k 1} is the discrete state. A simple systematic procedure to put system (5.6) into the form (5.1) is now given. Define the matrices D(x,u, t) and G(x,u, t) as follows (omitting the dependence of its entries from x, u and t) D(x,u, t) = [ G 1 G 0 G 2 G 0 ] G k 1 G 0 G(x,u, t) = G 0 (x,u, t) (5.7) and let δ(σ(t)) = [δ 1 (σ(t)),δ 2 (σ(t)),...,δ L (σ(t))] T (5.8) where { 1 if j = σ(t) j = 1,2,...,L δ j (σ(t)) = 0 otherwise (5.9) One can readily verify that system (5.6) is equivalent to (5.1),(5.7)-(5.9), whose main characteristics is that of being affine in the boolean vector δ(σ(t)) defining the current mode of operation. 5.3 Discrete state observer design The proposed discrete state estimator (which assumes the knowledge of the continuous system state) takes the following form: ż = G(x,u, t) + w(t) (5.10) where z represents the observer state and w is an observer input to be designed. Let e = x z be the observer error variable. Then, from (5.1) and (5.10), the observer error dynamics is ė = D(x,u, t)δ(t) + ɛ(x, t) w(t) (5.11) Observer input design Our objective is to design an observer control vector w = [w 1, w 2,..., w n ] T guaranteeing the finite-time convergence to zero of e and ė. A second-order sliding modes based approach, that enables us to reconstruct the discrete state, will be proposed. Must be highlight that, dissimilar to other works [5] within a distinct framework related to a fault diagnosis, such an approach, theoretically exact, exhibits a solution converging in finite time. Consider the second time derivative of the error variable e ë = Ḋ(x,u, t)δ(t) + D d δ(t) ε(x, t) ẇ(t) (5.12) d t which can be rewritten in compact form as follows: ë = ϕ(x,u, t) ẇ(t) (5.13) The uncertain drift term" ϕ( ) = [ϕ 1 ( ),ϕ 2 ( ),...,ϕ n ( )] T takes the following form

107 5.3. DISCRETE STATE OBSERVER DESIGN 93 ϕ(x,u, t) = Ḋ(x,u, t)δ(t) + D d δ(t) ε(x, t) (5.14) d t which depend from δ(t) that must be reconstructed. Denote v(t) = [v 1, v 2,..., v n ] T ẇ(t), y i,1 = e i and y i,2 = ė i, where e i and ė i represent the i -th entry of vectors e and ė. Then it is possible to rewrite system (5.13) in terms of n de-coupled single input subsystems having the following form { ẏi,1 = y i,2, i = 1,2,..,n (5.15) ẏ i,2 = ϕ i (x,u, t) + v i The problem is to find a set of control inputs v i stabilizing the uncertain SISO systems (5.15) in finite time. To solve this problem, the second-order sliding mode control approach [15],[29] appears to be particularly appropriate because of systems (5.15) have relative degree two with respect to the inputs v i s which are treated as auxiliary control variables. The control task is complicated by the two issues: (a) variables y i,2 (i = 1,2,...,n) are not measurable, and (b) the drift terms ϕ i (x,u, t) are uncertain. Let the mode switching sequence of the hybrid dynamics have a dwell time t d. This means that t i+1 t i t d, for i 0. The main idea is to use the discontinuous controller in [26]. Under the condition that a constant upperbound Φ i to the drift term magnitude can be computed ϕ i (x,u, t) Φ i t (5.16) such controller is able to stabilize the uncertain SISO systems (5.15) in a finite time t << t d starting from any initial conditions (y i,1 (0), y i,2 (0)) norm bounded by arbitrarily large constants. Denote as t i (i = 1,2,...) switching instants at which the active dynamics is commuting. The discrete state σ(t), and then also the vector δ(t), are piecewise constant during the time intervals T i = (t i, t i+1 ) between two adjacent mode switchings. The effect of the impulsive term d d t δ(t) at the switching instants t i is a jump in the (e,ė) state trajectories of system (5.13), and in particular, from (5.12), it result ė = D(x,u, t) δ(t) D 0 n, since the norm of the n-dimensional boolean vector δ will never exceed the value n. So, considering the single i-th decoupled subsystem (5.15), at the first switching instant t 1 the point (y i,1 (t 1 ), y i,2 (t 1 )) will leave the origin according to (0, y i,2 (t 1 + )) with y i,2 (t + 1 ) D 0 (5.17) After a new transient whose duration can be made less than t the system will be steered back to the origin. Thus, at any time t [t 1 + t, t 2 ) and any i = 1,..,n, the conditions y i,1 (t) = y i,2 (t) = 0, i.e., e(t) = ė(t) = 0 will be satisfied. The reasoning is repeated over all the successive switching intervals. The key point is the capability of the robust controller presented in [26], that will be specified in the sequel, of steering to zero the SISO systems (5.15)arbitrarily fast during the time interval between two adjacent mode switchings. Along any interval T i (t i 1, t i ), (i = 1,2,...), t 0 0, the drift term (5.14) can be upper bounded in the form ϕ(x,u, t) Φ D 1 n + ε1, t (t i 1, t i ) (5.18) The existence of such a constant upper bound allows for designing a robust controller featuring the desired finite time convergence properties. Next theorem outlines the main

108 94 CHAPTER 5. DISCRETE STATE RECONSTRUCTION VIA HOSMO result by introducing the so-called Suboptimal second-order sliding mode control algorithm [26, 27], together with the tuning rules that allow to obtain an arbitrarily fast convergence, which is a basic requirement of the present problem. Theorem 1: Consider system (5.15) and the control law v i (t) = V M si g n ( y i,1 (t) 1 2 y i,1(ξ i,j ) ) ξ i,j t < ξ i,j +1 j = 1,2,... (5.19) where ξ i,0 = t 0 0, ξ i,j is the sequence of time instants at which y i,2 (t) = 0, and V M = Γ Φ (5.20) Denote the sequence of the switching instants as t h, h = 1,2,... Then there is Γ such that, for any Γ Γ, the following conditions are provided. y i,1 (t) = y i,2 (t) = 0, t (t h + t, t h+1 ) (5.21) where i = 1,2,...,n and with t being arbitrarily small. Proof of Theorem 1. The proof exploits the basic convergence properties of the suboptimal second-order sliding mode control algorithm [27]. Let us consider the single i th decoupled subsystem (5.15) and denote with t 1 the first switching instant. At t = t 1, when the y i,1 and y i,2 variables are assumed both zero, on the basis of (5.17), we can infer that y i,2 undergoes a jump that, for the worst case, lead the system states (y i,1 (t + 1 ), y i,2(t + 1 )) in the point (0,D 0 ). Starting from here, after a the time t M1 D 0 D 0 2 Φ(Γ 1), the extremal point (,0) is reached. The suboptimal control strategy, with the additional constraint 2 Φ(Γ 1) Γ > 2, causes the generation of a sequence of states, with coordinates (y i,1 M1,0), featuring the following contraction property: y i,1 M1+j α y 1 i,1m1, j = 1,2,..., α = [0,1) (5.22) Γ 1 Such sequence converge to the origin at finite time t that can be made arbitrarily small opportunely tuning the gain parameter Γ. By imposing the following condition t < t d, descend D 0 Φ(Γ 1) + D Φ(Γ 1) 2Γ Φ (Γ 1) 1 Φ(Γ + 1) 1 1 Γ 1 < t d (5.23) From (5.23) it follows the rule for selecting Γ in such a way that the convergence time t fulfills the inequality t < t d. An admissible interval for Γ exist due to the fact that the left and side, i.e. t, converge to zero for Γ. On the basis of previous observations, denoting with Γ the value of Γ such that t = t d, choosing the tuning parameter Γ ]Γ, ), the the condition t < t d is guarantees.

109 5.3. DISCRETE STATE OBSERVER DESIGN Discrete state reconstruction It was shown in the previous Section that there is t such that, in every inter-switching" interval T i (t i 1, t i ) the next conditions hold From the definition (5.11) of ė, its zeroing implies that e = ė = 0, t [t i 1 + t, t i ) (5.24) D(x,u, t)δ(t) + ɛ(x, t) w(t) = 0 (5.25) Notice that the observer input w(t) is obtained integrating the discontinuous signal v(t), whose sign switches at very high (theoretically infinite) frequency (Zeno behaviour), then w(t) is a continuous signal. By neglecting the uncertainty ɛ(x, t) in (5.25) it yields naturally the following reconstruction formula that defines a non-thresholded estimate of the boolean vector δ. δ(t) = [D T (x,u, t)d(x,u, t)] 1 D T (x,u, t)w(t) (5.26) The non-thresholded estimate is not robust against the uncertainty ɛ(x, t). By (5.25), the estimation error δ(t) δ(t) will be such that δ(t) δ(t) [D T (x,u, t)d(x,u, t)] 1 D T (x,u, t) ɛ (5.27) It can be fruitfully exploited the boolean nature of the vector δ(t) by introducing a thresholding that rounds the value of δ(t) to the closest integer value between 0 and 1. It yields the thresholded estimate ˆδ(t) defined according to { 1 δi (t) > 0.5 ˆδ i (t) = 0 δi (t) 0.5 (5.28) where δ i (t) and ˆδ i (t) are the i-th entries of vectors δ(t) and ˆδ(t) respectively. The thresholded estimate results to be robust against any error ( δ i (t) δ i (t)) less, in magnitude, than 0.5. Thus it can be explicitly given a bound to the maximal tolerated magnitude for the uncertainty term. From the requirement that δ(t) δ(t) 0.5 it yields by (5.27), (5.4), (5.5) the following maximal acceptable bound for the norm of the uncertainty term ɛ(x, t) ε D 2 (5.29) The fulfillment of (5.29) guarantees the insensitivity of the estimate ˆδ against the uncertainty, namely the condition ˆδ(t) = δ(t), t [t i 1 + t, t i ), i = 1,2,... (5.30) Lemma 1 Under the condition that the norm of the uncertain term ɛ(x, t) fulfills the restriction (5.29), the proposed estimation procedure given by (5.26), (5.28) provides for

110 96 CHAPTER 5. DISCRETE STATE RECONSTRUCTION VIA HOSMO the exact reconstruction of the boolean vector δ in the time intervals t [t i 1 + t, t i ), according to (5.30) Remark 1 The requirement of providing the observer convergence within the arbitrarily small transient time t << t d would correspond, in the linear context, to locate the eigenvalues of the error dynamics far away from the origin. Generally, this strongly deteriorates the robustness against the measurement noise of the resulting linear high gain" observer. It can be argued, due to the analysis made in [28, 29], that the magnification of the noise in the considered 2-SMC observer could be less severe than in the linear observer counterpart. This topic will be addressed in more detail in next research activities.

111 Chapter 6 Application results 6.1 Three-tank system case study As one of popular experimental systems in control laboratories, the three-tank water process is regarded as a valuable setup for investigating, theoretically and experimentally, nonlinear multivariable feedback control as well as fault diagnosis schemes ([38],[39],[40]). The multi-tank system that we shall consider, see in Fig.6.1, is composed of three tanks of different shape disposed vertically, so that the potential energy of water allows the water to flow from one tank to another in a lower position. A variable-velocity pump that supplies the upper tank1, and three automatic (electrical) regulation valves RV 1,RV 2 and RV 3 allow for regulating the outflow from each tank. A reflux tank is present below the lower tank3. Furthermore, three manual valves MV 1, MV 2 and MV 3 are also present besides the electrically controlled ones. The three tanks are equipped with piezo-resistive pressure transducers (P Z 1,P Z 2,P Z 3 ) which permit to measure the water levels H1, H2 and H3. As depicted in figure 6.2, we select, as the modifiable control inputs, the water inflow to the tank 1 (provided by the variable-speed DC pump) and the opening of the valves RV 1, RV 2 and RV 3. The input variables U (t),u 1 (t), U 2 (t) and U 3 (t) (whose exact meaning shall be explained later on) control the DC pump and the regulation valves RV 1, RV 2 and RV 3 respectively. Concerning the FDI aspects, we ll investigate the effectiveness of the FDI proposed scheme, discussed in Chapter 4, against additive faults f p, f 1, f 2 and f 3 acting on the pump and the three regulation valves RV 1, RV 2 and RV 3 respectively Mathematical model The flow balance equations leads to the following system model V 1 = q C 1 H1 (6.1) V 2 = C 1 H1 C 2 H2 (6.2) V 3 = C 2 H2 C 3 H3 (6.3) 97

112 98 CHAPTER 6. APPLICATION RESULTS Figure 6.1: Configuration of the considered three-tank system Figure 6.2: System input-output where V 1,V 2,V 3 represent the actual volume of water contained in the three tanks, q is the in-flow to the upper tank provided by the pump, H 1, H 2, H 3 are the nonnegative water levels, and C 1,C 2,C 3 are the outflow coefficients of the valves that can be adjusted. The time derivatives of the actual volumes of water depend on the time derivatives of the water levels inside the tanks according to the simple relationships V i = β i (H i )Ḣ i, i = 1,2,3 (6.4) where β i,i = 1,2,3 represent the cross sectional area of tank i at the level height H i.

113 6.1. THREE-TANK SYSTEM CASE STUDY 99 Figure 6.3: Shape of tank 1 Figure 6.4: Shape of tank 2 Figure 6.5: Shape of tank 3 An expression for cross-sectional areas is now given in explicit form. Taking into account the shapes of the tanks in Fig , which are derived from the experimental setup provided by Inteco [78], the following relationships hold, where a,b,c,w,r,h max are constant parameters reported in the Table 1. β 1 = aw (6.5) β 2 (H 2 ) = cw + bw(h 2 /H max ) (6.6) β 3 (H 3 ) = w R 2 (R H 3 ) 2 (6.7) Clearly, all the above functions are strictly positive. It yields the simple model

114 100 CHAPTER 6. APPLICATION RESULTS Ḣ 1 = (q C 1 H1 )β 1 1 (6.8) Ḣ 2 = (C 1 H1 C 2 H2 )β 1 2 (H 2) (6.9) Ḣ 3 = (C 2 H2 C 3 H3 )β 1 3 (H 3) (6.10) The coefficients C 1, C 2, C 3 actually depends on the valves opening. Obviously when the generic valve RV i is in fully closed position, the corresponding coefficient C i is equal to zero. Vice versa if the valve is in fully opened condition C i = C. Therefore seems i appropriate to represent such outflow coefficients by the notation C i = C i (t) = C i U i (t), where U i (t) [0,1] is time-varying and represents the relative actual valve opening. We assume that U (t), U 1 (t), U 2 (t) and U 3 (t), the input references for the system actuators, are user-modifiable and therefore known. We also denote the adjustable water inflow generated by the DC pump as It results the following model q = C U (t) U (t) [0,1] (6.11) Ḣ 1 = (C U (t) C 1 U 1 (t) H 1 )β 1 1 (6.12) Ḣ 2 = (C 1 U 1 (t) H 1 C 2 U 2 (t) H 2 )β 1 2 (H 2) (6.13) Ḣ 3 = (C 2 U 2 (t) H 2 C 3 U 3 (t) H 3 )β 1 2 (H 2) (6.14) In the following additive actuator faults will be induced both manually, making use of the manual values MV 1, MV 2, MV 3 depicted in Fig. 6.1, or numerically by additive terms corrupting the adjustable reference signals according to Fig.6.6. Figure 6.6: Fault topology

115 6.2. SIMULATION RESULTS Simulation results The effectiveness of the suggested fault reconstruction scheme and of the discrete state identification is studied preliminarily by making some simulative analysis. The parameter values used for simulating tests have been evaluated through a procedure of system identification. The dimensional parameters a, w,b,c,r have been directly measured and an identification procedure, described later, has been applied to evaluate appropriate values for C, C 1, C 2,C 3. The obtained values are Table 6.1: Parameters of the Three Tank System Parameter Value Unit C 4 m C C s 5 m3 s 5 m3 s 5 m3 s C a m w m b m c 0.1 m R m H i max 0.35 m

116 102 CHAPTER 6. APPLICATION RESULTS Fault reconstruction via HOSM In this paragraph we address the problem of detecting and exactly identifying certain faults and disturbances acting on a three tank system [80]. For this purpose we exploit a robust model-based technique based on 2-order SMO, and more precisely, on the manipulation of the equivalent output injection signal to explicitly reconstruct fault signals. Based on the particular structure of the system under investigation, the method presented in 4.2, for a class of nonlinear uncertain systems, seems particularly suitable. It is straightforward noting that the proposed scheme enables us to reconstruct certain additive faults and external disturbances without any filtering, therefore leading to a solution converging in finite time and theoretically exact. As depicted in figure 6.7, we select, as the modifiable control inputs, the water inflow to the Tank 1 (provided by the variable-speed DC pump) and the opening of the valves RV 1 and RV 2. Considered faults are associated to a malfunctioning in the control valves RV 1 and RV 2. The outflow through the valve RV 3 is regarded as a disturbance term. This term, to be treated as an external disturbance, will be reconstructed by the proposed observer. Concerning the FDI aspects, it is our aim to detect and reconstruct additive faults f 1 and f 2 on the regulation valves RV 1 and RV 2. Pumping systems are often redundant in practice, so that there is a limited possibility of pump fault. Figure 6.7: System inputs, outputs and disturbances In the following we denote, according to Eq. (6.3), the overall outflow from the lower tank 3 in compact form as ψ( ) = C 3 H3 (6.15) we also denote the adjustable water inflow generated by the DC pump as q(t) = C U (t) U (t) [0,1] (6.16)

117 6.2. SIMULATION RESULTS 103 According to (6.12) (6.14), the following model results Ḣ 1 = (C U (t) C 1 U 1 (t) H 1 )β 1 1 (6.17) Ḣ 2 = (C 1 U 1 (t) H 1 C 2 U 2 (t) H 2 )β 1 2 (H 2) (6.18) Ḣ 3 = (C 2 U 2 (t) H 2 ψ( ))β 1 3 (H 3) (6.19) As previously said, we assume that actuator faults (such as permanent or intermittent biases, or gain degradation) and component faults (such as leakage in the tanks and clogs in the pipes) can occur in the valves RV 1 and RV 2. Our aim is to realize an Observer-Based FDI scheme guaranteeing the precise reconstruction of the (possibly simultaneous) additive actuator faults f 1 and f 2 that can be occur in control valves RV 1 and RV 2, and of the disturbance term ψ(t). It worth noting that an estimate of the flow disturbance ψ(t) would be very useful for level control purpose. The actuator fault signals can be modeled by additive terms corrupting the adjustable valve signals U 1 (t) and U 2 (t). Thus, by including such possible fault signals in the model one can rewrite system (6.17)-(6.19) according the general form (4.13)-(4.14). ẋ = G(x,u) + D(x)f (x,u, t) + E(x)Ψ(x, t) (6.20) y = C x (6.21) where x = [H 1, H 2, H 3 ] T, u = [U,U 1,U 2 ] T, C = I 3 being the identity matrix, and G(x,u) = G (x)u with G (x) = D(x) = E(x) = 0 0 C β 1 C 1 H 1 β 1 0 C 0 1 H 1 β 2 (H 2 ) C 2 H 2 β 2 (H 2 ) C H 2 β 3 (H 3 ) 1 β 3 (H 3 ) C 1 H 1 β 1 0 C 1 H 1 β 2 (H 2 C 2 H 2 ) β 2 (H 2 ) C 0 2 H 2 β 3 (H 3 ) (6.22) (6.23), f (x,u, t) = [f 1, f 2 ] T (6.24) Ψ(x, t) = ψ( ) (6.25) The nonlinear vector fields G (x), D(x), E(x) are assumed to be known, while vectors f (x,u, t) and Ψ(x, t) represent the considered actuators faults and the external disturbance, respectively. As one can see, the measurements actually represent the complete state vector. Nevertheless, the design of an appropriate observer is necessary to identify and reconstruct signal faults and disturbance term. Assuming that an appropriate closed-loop control system has been designed, capable of guaranteeing that the tanks never become empties (i.e., H i > 0,i = 1,2,3), the follows hold H i x i 0; f or i = 1,2,3 no lost of full rank for E(x) and D(x) (6.26)

118 104 CHAPTER 6. APPLICATION RESULTS Observing the structure of (6.20) - (6.25), the system can be reviewed as or equivalently in the following compact form ẏ = C [G(x,u) + D(x)f (x,u, y) + E(x)Ψ(x, t)] (6.27) ẏ = C {G(x,u) + [D(x) E(x)]τ(x,u, t)} (6.28) with implicit definition of the uncertainty term and disturbance distribution matrix [ f (x,u, t) τ(x,u, t) = Ψ(x, t) ] ; [D(x) E(x)] = C 1 H 1 β C 1 H 1 β 2 (H 2 C 2 H 2 ) β 2 (H 2 0 ) 0 C 2 H 2 β 3 (H 3 ) 1/β 3 (H 3 ) (6.29) Under the full column-rank condition (6.26) for the matrices E(x) and D(x), the following assumption is globally met: r ank(c [E(x) D(x) ]) = r ank([e(x) D(x) ]) p 3 (6.30) Therefore, from (6.27)-(6.30), we can infer that the propagation of the overall uncertainty vector term τ(x,u, t) through the output is governed by a transfer function characterized by a relative degree one, i.e., all the uncertain element of τ(x,u, t) acting on the first derivative of the correspondent signal output. Furthermore, in view of applying the method presented in 4.2, we should verify that all invariant zeros of the triple (A,[E D], C) lie in C. The check, from which follows that no invariant zeros are present, is trivial and will be omitted. Making the assumption the non linear term G(x,u) is Lipschitz with respect to x uniformly for u (that belong to an admissible control set) and that the unknown nonlinear term τ(x,u, t) and its first derivative is norm bounded, with known bounds, the assumptions stated in 4.2 and 4.3 are met, so the proposed observer, making use of a 2SM based output injection control low, can be used to reconstruct in finite time the vector signal τ(x,u, t). In the following it will be assumed that the system in (6.20) (6.21) is under feedback control and the signals u(t) are (smooth) functions of the states x(t). In the absence of faults it is assumed that the controller has been well designed so that x(t) is close to its required operating point. If a fault occurs it is assumed that x(t) lies in a bounded compact set for at least a finite time t f > 0, starting from the onset of the fault, which allows time for detection to take place. The proposed observer has the following form: ˆx = G( ˆx, t) + ν (6.31) ŷ = C ˆx (6.32) From (6.28) and ( ), and considering that C = I, descend the following observation error dynamics: ė y = ẏ ŷ = ẋ ˆx = [G(x, t) G( ˆx, t)] + [D(x) E(x)]τ(x,u, t) ν (6.33)

119 6.2. SIMULATION RESULTS 105 where ν is the continuous control vector whose realization should be capable of guaranteeing the required global and finite-time convergence to zero of e y and ė y. The approach based on second-order sliding modes, presented in 4.2, has been used in order to enable us to reconstruct the fault in finite time without any filtration, therefore leading to an exact solution. Two simulative tests has been performed with sampling time T s = 0.001s. In the first test (Test 1) noise-free measurements are used. In the second test (Test 2), a band-limited additive white noise is taken into account. Fault and disturbance signals of different shape and magnitude have been considered. Note that a fault of magnitude of 0.5 represent an error which is the 50% of the overall valve opening run. In the figures (6.8) and (6.9) the results of the two simulation tests are shown. A precise reconstruction of the faults and disturbances is achieved in both tests. Figure 6.8: Test 1. Faults and disturbance reconstruction performance

120 106 CHAPTER 6. APPLICATION RESULTS Figure 6.9: Test 2. Reconstructed faults and disturbance in presence of noisy measures

121 6.2. SIMULATION RESULTS Discrete state estimation via HOSM Let us show that the three-tank water process can be modeled as a switched affine system [81] according to the general formulation in Eq The vertical multi-tank system that we shall consider has three on-off valves V 1 sw, V 2 sw, V 3 sw that determine whether an outflow from each Tank exists or not. The on-off state of the three valves define the 8 possible operating mode of the considered system. Referring to the schematic representation in Fig. 6.1, signal q(t), the inflow to the upper tank, represent a measurable input to the system, the binary signals U 1,U 2 and U 3 are the unknown states of the on-off valves, and H 1, H 2 and H 3 are the water levels which represent the continuous state of the three-tank system. It is our aim to present a scheme, based on the approach described in chapter 5, for reconstructing the states of the three on-off valves by assuming the knowledge of the water levels and of the input inflow q(t) to the upper tank. Let us consider the system formulation in ( ) where the three boolean variables U 1 (t),u 2 (t),u 3 (t) represent the status of the three valves. Due to the on-off behavior of the valves, the C i (t) coefficients (i = 1,2,3) can assume two values only according to { 0 when valve V isw is OFF C i (t) = when valve V i sw is ON C i (6.34) Collecting into a boolean vector the discrete states of the on-off valves as follows δ(t) = [U 1 (t),u 2 (t),u 3 (t)] T {0,1} 3 (6.35) it is straightforward to put the model (6.17)-(6.19) in the form (5.1) with x = [H 1, H 2, H 3 ] T, u = q(t) and D(x,u, t) = G(x,u, t) = C 1 H 1 q(t) β β C 1 H 1 β 2 (H 2 C 2 H 2 ) β 2 (H 2 0 ) C 0 2 H 2 β 3 (H 3 ) C 3 H 3 β 3 (H 3 ) (6.36) (6.37) According to the notation introduced in chapter 5, it must be highlighted that in our case a particular instance for (6.35) represents one of the possible k = 8 discrete states σ(t) in which the three-tank system could be found. In the derived three tank system the dimension L of vector δ(t) is L = 3 which does not exceed the dimension n = 3 of the continuous state, as required in assumption (5.2). The assumptions (5.3) on the matrix D(x,u, t) are trivially satisfied since the water levels H 1 (t), H 2 (t), H 3 (t) remain strictly positive during the observation process. Furthermore the assumption (5.4) requires that the square matrix D(x,u, t) is nonsingular. Since C 1 C 2 C 3 detd(x,u, t) = H1 H 2 H 3 (6.38) β 1 (H 1 )β 2 (H 2 )β 3 (H 3 ) again the assumption (5.4) is fulfilled if none of the water levels become zero during the observation process. Assuming that an appropriate closed-loop supervisory system has

122 108 CHAPTER 6. APPLICATION RESULTS been designed, capable of guaranteeing that H i (t) H > 0, i = 1,2,3, the proposed observer can provide for the reconstruction of the binary signal vector δ(t). i An additive error term ɛ(x, t) may take into account possible discrepancies between the actual and nominal system model as well as possible external disturbances. It has been stated, in the Lemma 1 of 5.3.2, that the discrete state can be still reconstructed exactly provided that the norm of ɛ(x, t) is sufficiently small. It is worth noting that the discrete state σ(t) {0,1,,7} can be reconstructed from the thresholded estimates ˆδ 1 (t), ˆδ 2 (t), ˆδ 3 (t) by means of the following expression ˆσ(t) = ˆδ 1 (t) ˆδ 2 (t) ˆδ 3 (t) 2 0 (6.39) The effectiveness of the suggested discrete state observer, presented in 5.3, is now studied by means of some simulative analysis conducted on the three tank model (6.12)- (6.14). The inflow input q(t) = C U (t) and the binary state δ(t) have been selected in such a way that the tanks never become empty, that would cause the loss of observability for the system. The parameter values used in the simulations, evaluated by means of an identification procedure, are reported in the Table 1. Euler integration method with the fixed sampling time T s = 0.001s has been used. A disturbance vector with elements of the form ε i (x, t) = 0.1( H 1 (t) + H 2 (t) + H 3 (t) )si n(ωt), i = 1,2,3 (6.40) is considered, and a band-limited additive white noise is added to the level measurements H 1, H 2, H 3. The binary signal inputs U 1 (t),u 2 (t),u 3 (t) defining the discrete state of the system have been selected as shown in the plot of the next figure 6.10 (the same profile for all the simulation tests has been used). It can be noted that a dwell time of 0.5s has been used. In the first test (Test 1), the disturbance vector ε(x, t) and the measurement noise are not included. The plots in the figure 6.10 show the actual δ i (t) values together with the non-thresholded reconstructed ones δ i (t). Figure 6.11 makes the same comparison by considering the thresholded reconstructed values ˆδ i (t). Figure 6.12 shows the actual and reconstructed discrete states σ(t) and ˆσ(t). It can be seen that the suggested method provides a prompt identification of the active mode. In "Test 2" it is shown that by increasing the V M observer parameter it can be achieved an arbitrarily fast identification of the current mode after the mode switchings. To this end three different values of V M have been considered, and a zoom across some switching instant is made in the Fig The differences in the transient duration confirm the expected performance. In the last test (Test 3), disturbances and measurement noise are considered. Figure 6.14 shows that signals δ 1i (t) are corrupted by the noise as compared with the Test 1. But, since the resulting errors are less than 0.5, the successive thresholding removes the errors and recovers the correct discrete state estimates according to Lemma 1 (see figure 6.15)

123 6.2. SIMULATION RESULTS Actual and reconstructed (non-thresholded) binary states Time (sec) Figure 6.10: Test 1. δ i (t) vs. δ i (t). From top to bottom: i = 1,2, Actual and reconstructed (thresholded) bynary states Time (sec) Figure 6.11: Test 1. δ i (t) vs. ˆδ i (t). From top to bottom: i = 1,2,3.

124 110 CHAPTER 6. APPLICATION RESULTS 8 Actual and reconstructed discrete state Time (sec) Figure 6.12: Test 1. Actual σ(t) and reconstructed ˆσ(t) 4 Actual and reconstructed discrete state 3 2 V M =0.1 V M =1.0 1 V M = Time (sec) Figure 6.13: Test 2. σ(t) (solid) and ˆσ(t) (dashed) varying observer gain

125 6.2. SIMULATION RESULTS Actual and Reconstructed (non thresholded) binary states Time (sec) Figure 6.14: Test 3. Actual δ i (t) vs. δ i (t) discrete inputs 8 Actual and reconstructed discrete states Time (sec) Figure 6.15: Test 3. Actual σ(t) vs. ˆσ(t) discrete states

126 112 CHAPTER 6. APPLICATION RESULTS 6.3 Experimental results Experimental results using the three-tank laboratory-size setup apparatus by Inteco [78] are presented and commented in this section. A picture of the experimental setup is shown in the figure Figure 6.16: The experimental setup The multi-tank system is interfaced with an external PC-based data acquisition and control system. The development of, both, the controller and observer systems is made in the MATLAB/Simulink environment, and the associated executable code is automatically generated by the RTW/RTWI rapid prototyping environment. The water level in the tanks are measured with piezo resistive pressure transducer and acquired by a DAC multipurpose I/O board. There are four control signals sent out from the DAC board to the multi-tank system: the three valve control signals and the DC pump control signal. The sampling time is set to 0.01 s. The closed-loop control system is made up of anti wind-up PI controllers. The identification of the plant, sensor and actuator parameters, is carried out in order to minimize the discrepancies between the real process and its mathematical model System identification A parameter identification procedure has been carried out for: i. Level sensors characteristic curves. ii. Dc pump actuator characteristic curves. iii. Proportional valve actuator characteristic curves.