System Identification And Fault Detection Of Complex Systems

Transcription

1 University of Central Florida Electronic Theses and Dissertations Doctoral Dissertation (Open Access) System Identification And Fault Detection Of Complex Systems 2006 Dapeng Luo University of Central Florida Find similar works at: University of Central Florida Libraries Part of the Mechanical Engineering Commons STARS Citation Luo, Dapeng, "System Identification And Fault Detection Of Complex Systems" (2006). Electronic Theses and Dissertations This Doctoral Dissertation (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of STARS. For more information, please contact

2 System Identification and Fault Detection of Complex Systems by Dapeng Luo B.S. Beijing Institute of Technology, 1993 M.S. Florida Atlantic University, 2003 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mechanical, Materials and Aerospace Engineering in the College of Engineering and Computer Science at the University of Central Florida Orlando, Florida Fall Term 2005 Major Professor: Alexander Leonessa

3 c 2005 by Dapeng Luo ii

4 Abstract The proposed research is devoted to devising system identification and fault detection approaches and algorithms for a system characterized by nonlinear dynamics. Mathematical models of dynamical systems and fault models are built based on observed data from systems. In particular, we will focus on statistical subspace instrumental variable methods which allow the consideration of an appealing mathematical model in many control applications consisting of a nonlinear feedback system with nonlinearities at both inputs and outputs. Different solutions within the proposed framework are presented to solve the system identification and fault detection problems. Specifically, Augmented Subspace Instrumental Variable Identification (ASIVID) approaches are proposed to identify the closed-loop nonlinear Hammerstein systems. Then fast approaches are presented to determine the system order. Hard-over failures are detected by order determination approaches when failures manifest themselves as rank deficiencies of the dynamical systems. Geometric interpretations of subspace tracking theorems are presented in this dissertation in order to propose a fault tolerance strategy. Possible fields of application considered in this research include manufacturing systems, autonomous vehicle systems, space systems and burgeoning bio-mechanical systems. iii

5 To my parents, Shigui Luo, Xiuyan He To my wife, Jie Chen iv

6 Acknowledgments I wish to express my most sincere thanks and heartfelt gratitude to my advisor Dr. Alexander Leonessa for his patience, guidance, motivation and support throughout my research and during each stage of my Ph.D program. I would like to thank other members of my Supervisory Committee, Dr. Johnson, Dr. Nicholson, Dr. Jayaram, and Dr. Choudhury, for accepting to be on my committee. I wish to thank the people of MMAE Department who helped and supported me in the research. I would like to extend my special thanks to Dr. Nashed (Professor and Chair, Department of Mathematics), Dr. Ren (Professor, Department of Mathematics) and Dr. You (Professor, Department of Statistics and Actuarial Science) for the fruitful discussions with them. I would also like to express my thanks to my friends, Yan Tang and Yannick Morel, for their helps and motivations in my dissertation. v

7 TABLE OF CONTENTS LIST OF TABLES xii LIST OF FIGURES xiv CHAPTER 1 INTRODUCTION System Identification Models The Nonlinear System Identification Procedure Subspace Identification Literature Review of Subspace Identification Fault Detection and Tolerance Dynamic Model of Fault Detection Fault Detection Approaches Faulty Modes Literature Review of Fault Detection for Complex Systems Proposed Work Problem Statement vi

8 1.3.2 Major Contributions Summary CHAPTER 2 PRELIMINARIES CHAPTER 3 SINGLE-INPUT SINGLE-OUTPUT OPEN LOOP HAM- MERSTEIN SYSTEM IDENTIFICATION Introduction Problem Formulation Identification Scheme Design Triangular Series Approximation Trigonometric Series Approximation Numerical Simulations Triangular Functions Approximation Trigonometric Functions Approximation Nonlinear Function Identification Test Application of Modeling the Dynamics of a Marine Thruster Mission Description Experimental Setup, Modeling and Validation Conclusion vii

9 CHAPTER 4 MULTIPLE-INPUT MULTIPLE-OUTPUT CLOSED LOOP HAMMERSTEIN SYSTEM IDENTIFICATION Introduction Problem Formulation Identification Scheme Design Analysis of A Deterministic Identification Scheme Subspace Instrumental Variable Identification Algorithm Consideration of Numerical Implementation Mission Description Augmented SIVID Algorithm Numerical Simulations and Experiments Orthonormal Series Approximation and Prediction Quality Measurements Example 1: System with Knowledge of Nonlinearities Example 2: System without Knowledge of Nonlinearities Investigation of Computational Efficiency Conclusion CHAPTER 5 NONLINEAR IDENTIFICATION EXPERIMENTAL DE- SIGN Introduction viii

10 5.2 Description of the Reaction Wheel Pendulum Linear Subspace Identification Scheme (LSID) Experiments on Prediction Model Recovery Choosing the Prediction Model Nonlinear Identifications with Scaled Multi-frequency Signals Conclusion CHAPTER 6 FAST ORDER ESTIMATION FOR SUBSPACE METH- ODS WITHOUT PRIOR ESTIMATED NOISE MODELS Introduction Problem Formulation Order Estimation Order Determination Methods Threshold Estimation Numerical Simulations Simulation Example: Deterministic MIMO Process Simulation Example: Combined Deterministic-stochastic Process Conclusions CHAPTER 7 HARD-OVER FAILURE DETECTION Introduction ix

11 7.2 Problem Formulation Main Result Simulation Hard-over Sensor Failure Detection Hard-over Actuator Failure Detection Conclusion CHAPTER 8 SIGNAL DETECTION AND GEOMETRIC INTERPRE- TATION OF SUBSPACE TRACKING THEOREMS Introduction Subspace Tracking Theorems Problem Formulation A Brief Review of The Subspace Tracking Theorems Proposed in [Yan95] Main Results Conclusions CHAPTER 9 CONCLUSIONS AND FUTURE WORK APPENDIX A PROOFS IN CHAPTER A.1 Proof of Theorem A.2 Proof of Theorem x

12 APPENDIX B PROOFS IN CHAPTER B.1 Proof of Theorem B.2 Proof of Theorem APPENDIX C PROOFS IN CHAPTER C.1 Proof of Theorem C.2 Proof of Theorem LIST OF REFERENCES xi

13 LIST OF TABLES 3.1 Parameter values for the trigonometric expansion with p = Van der Pol Oscillator in subsection 4.5.3: Comparison of average computing efficiency between the ASIVID algorithm and the SIVID algorithm with p, q = 4 and α = Experiment 1. Prediction errors (in percentage) e(k), k = 1, 2, with different system orders from ˆn = 1 to ˆn = 8 by using the LSID and the ASIVID algorithms, respectively Experiment 2. Prediction errors (in percentage) with different system orders from ˆn = 1 to ˆn = 8 by using the LSID and the ASIVID algorithms, respectively Deterministic case in subsection 6.4.1: system order determination by using the approximated distribution function Eq. (6.11) Deterministic case in subsection 6.4.1: system order determination by using the maximum ratio method Eq. (6.4) Combined Deterministic-stochastic case in subsection 6.4.2: system order determination by using the approximated distribution function Eq. (6.11). 127 xii

14 7.1 Order determination without hard-over failures Hard-over Sensor Failure Detection Hard-over Actuator Failure Detection xiii

15 LIST OF FIGURES 1.1 Linear Model Hammerstein Model (left) and Wiener Model (right) The system identification loop A complex system involving a failure-monitor Interpretation of the orthogonal projection in a 2-dimensional space Hammerstein model with ARX linear subsystem Triangular basis function Triangular approximation: predicted output error Triangular approximation: nonlinearity Triangular approximation: identification error of â and ˆb Trigonometric approximation: predicted output error Trigonometric approximation: nonlinearity Trigonometric approximation: identification error of â and ˆb Identified coefficients comparison SQUID AUV xiv

16 3.11 Marine thruster: predicted output error with triangular approximation Marine thruster: nonlinearity with triangular approximation Marine thruster: â and ˆb with triangular approximation Marine thruster: predicted output error with trigonometric approximation Marine thruster: nonlinearity with trigonometric approximation Marine thruster: â and ˆb with trigonometric approximation Marine thruster: identified coefficients comparison MIMO Hammerstein model with nonlinear feedback Input-Output Signals of Nonlinear Example Simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms (noise free case) Simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms (N=6000) (with noise) Simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms (N=7000) (with noise) Input-Output Signals of Van der Pol Oscillator Van der Pol Oscillator in subsection 4.5.3: simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms with p, q = xv

17 4.8 Van der Pol Oscillator in subsection 4.5.3: simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms with p, q = Van der Pol Oscillator in subsection 4.5.3: simulation output compared with the predicted output by using ASIVID (...) and SID( ) algorithms with N = 6000, p, q = 4 and α = Van der Pol Oscillator in subsection 4.5.3: Comparison of average computing efficiency between the ASIVID algorithm and the SIVID algorithm with p, q = 4 and α = Schematic diagram of the reaction wheel pendulum Block diagram of the prediction model for the reaction wheel pendulum Experiment 1. The sinusoidal input and the according outputs of reaction wheel pendulum Experiment 1. Difference between the true and predicted outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = Chirp signal (CHIRP) experiment Experiment 2. Difference between the true and predicted outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = Band Limit White Noise Signal (BLWN) xvi

18 5.8 Uniform Random Number (URN) Experiment 3, non-scaled signals. Difference between the true and predicted outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = CHIRP Signal: the frequency response of output measurements Experiment 4, BLWN Signal. Difference between the predicted and true outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = Experiment 4, CHIRP Signal. Difference between the predicted and true outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = Experiment 4, URN Signal. Difference between the predicted and true outputs by using the ASIVID algorithm (dash line) and the LSID algorithm (solid line), respectively, when ˆn = Deterministic case in subsection 6.4.1: input-output data Deterministic case in subsection 6.4.1: the tendency of the identified expected value of threshold by using the approximated distribution function Eq. (6.11) Combined Deterministic-stochastic case in subsection 6.4.2: the tendency of the identified expected value of threshold by using the approximated distribution function Eq. (6.11) xvii

19 CHAPTER 1 INTRODUCTION 1.1 System Identification System identification deals with the problem of building mathematical models of dynamical systems based on observed data. It has been decades since system identification algorithms were developed as powerful techniques to analyze system structures and approximate real systems behavior in specific operational ranges. Since dynamical systems are abundant in our lives, the techniques of system identification have many applications, such as, modeling, prediction, signal detection, fault diagnosis, etc [WI01, Lju99a, Zhu98, HC01, SB01]. Existing identification algorithms are not bulletproof methodologies that can be used without any interaction from the user. A few reasons for this includes: An appropriate model structure must be found. This can be a difficult problem, in particular if the dynamics of the system is nonlinear. There are certainly no perfect data in real life. The fact that the recorded data are disturbed by noise must be taken into consideration. 1

20 The process may vary with time, which can cause problems if an attempt is made to describe it with a time-invariant model. It may be difficult or impossible to measure some variables/signals that are of central importance for the model. The essential ingredients of a system identification problem include model selection, system analysis, experimental design, criteria of best fit, algorithm development and model validation. In the remainder of this section, we will discuss these topics in details Models Models of systems are of fundamental importance in virtually all disciplines. In engineering, models can be useful for system analysis, control, prediction and simulation. Advanced techniques for the controller design, optimization, supervision, fault detection, and diagnosis are also based on system models. Since the quality of the model determines the quality of the final solution, modeling is essential in developing the whole system. When we intend to identify a system, we need some pre-knowledge of how its components relate to each other. Generally speaking, we shall call such an assumed relationship among observed signals the model of the system. Obviously, models may be phrased in varying degrees of mathematical formula. The intended use will determine the degree of sophistication that is required to make the model purposeful. 2

21 Linear and Nonlinear Models It goes without saying that there is an infinite collection of mathematical models. In this dissertation, we have restricted ourselves to discrete-time state space models. This might seem like a highly restricted class of models; however, many industrial processes can be described very accurately by this type of models. Moreover, the number of control system design tools based on this type of models is almost without bound. For these reasons, this model class is a very interesting one. w(k) v(k) u(k) + y(k) G(q 1 ) Figure 1.1: Linear Model A linear model is shown in Figure 1.1 and its state-space model is given by [Lju99b] ẋ(k) = Ax(k) + Bu(k) + w(k), x(0) = x 0, k = 1, 2,..., (1.1) y(k) = Cx(k) + Du(k) + v(k), (1.2) where x(k) R n, u(k) R m and y(k) R l, k = 1, 2,..., are the state, input and output vectors, respectively. The matrices A R n n, B R n m, C R l n and D R l m represent linear dynamics. The signal v(k) R l, k = 0, 1,..., represents the additive measurement noise. Nonlinear systems have more complicated structures than the linear systems. System identification on nonlinear systems which have known structure but unknown parameters is referred to semi-physical modeling [Lju99b]. Wiener and Hammerstein Models are 3

22 celebrated examples of this method [Lju99b], as shown in Figure 1.2. The state-space w(k) v(k) w(k) v(k) u(k) f( ) + G(q 1 y(k) ) + G(q 1 ) u(k) z(k) + y(k) f( ) + Figure 1.2: Hammerstein Model (left) and Wiener Model (right) model of Hammerstein system is given by ẋ(k) = Ax(k) + Bf(u(k)) + w(k), x(0) = x 0, k = 1, 2,..., (1.3) y(k) = Cx(k) + Df(u(k)) + v(k), (1.4) and that of Wiener system is ẋ(k) = Ax(k) + Bf(u(k)) + w(k), x(0) = x 0, k = 1, 2,..., (1.5) z(k) = Cx(k) + Df(u(k)), (1.6) y(k) = f(z(k)) + v(k). (1.7) Black-box and Gray-box Models A system model is a description of (part of) its behavior, served for a certain purpose. The model need not to be a true, accurate and complete description of the system, nor need the user have to believe so, to serve its purpose [Lju99b]. Since engineers are not really interested in an exact model as such, but more in the potential engineering applications of models [OM96b], the modeling approximation techniques (i.e., wavelets decomposition, kernel estimators, B splines, fuzzy models, neural networks, etc. [H 95]) are widely used in 4

23 engineering applications when accurate models are unavailable or too complicated to be applied. Therefore engineers will typically use system identification techniques to build their models. Two different philosophies may guide the choice of parameterized model sets, which are Black-box model structures. The goal is to chose model structure from a model set that is flexible to accommodate a variety of systems, without looking into their internal structures [Lju99b, H 95]. Gray-box model structures. The goal is to incorporate physical insight into the model set, with a certain number of adjustable parameters actually unknown about the system [H 95]. Compared to models obtained from physical insight, black-box models are relatively easy to obtain and use and, even more importantly, these models are simple enough to make model-based control system design tractable from a mathematical and practical point of view. However, black-box models have a limited validity, limited working range, and in some cases have no direct physical meaning [H 95]. In control system design problems, it is essential developing a mathematical model of the system to be controlled. Therefore, it is always wise first to try and utilize physical insight to characterize possible structures which will be reflected by the model [Lju99b]. The models obtained by system identification have the following properties, in contrast to models based solely on mathematical modeling (i.e., physical insight) [SS89], 5

24 They have limited validity (they are valid for a certain working point, a certain type of input, a certain process, etc.). They give little physical insight, since in most cases the parameters of the model have no direct physical meaning. The parameters are used only as tools to give a good description of the system s overall behavior. They are relatively easy to construct and use The Nonlinear System Identification Procedure Modeling and identification of nonlinear systems is a challenging task because nonlinear processes are unique due to the fact that they do not share many properties. A major goal of research in this field is to extend the capability of nonlinear modeling to describe a wide class of systems. This section examines the major steps that have to be performed for a successful system identification. Figure 1.3 illustrates the order in which the following steps have to carried out [Nel00]. Model Inputs/Outputs: The input/output data are recorded during a specifically designed identification experiment, where the user may determine which signals to measure and when to measure them and may also choose the input signals. If the process under consideration cannot be actively excited, a training data set still has to be designed by selecting a data set from the gathered measurements that 6

25 start model inputs/outputs model architecture dynamics representation model order model structure model parameters model validation OK:accept not OK:revise Figure 1.3: The system identification loop is as representative as possible. Chapter 5 discussed issues on input choices, range selection and signal scaling. Model Architecture: This step contains the most subjectively influenced decision in the identification process. The following factors are important when choosing appropriate model architecture: intended use of the model, data length, quality and availability, constraints of time frame, offline or online identification, etc. 7

26 Dynamics Representation: This step is mainly determined by the prior knowledge of the studied systems and the intended use of the model. Currently, the advantages and drawbacks of different nonlinear dynamical representations are still under investigation. Thus, subjective factors such as the user s previous experience with a specific approach and preferences for certain purposes play an important role. Nonlinear Hammerstein models are discussed in Chapter 3 and 4 and applied to represent the dynamics of some mechatronic systems. Model Order: This step is typically carried out by a combination of prior knowledge and trial-and-error approaches. Compared with linear system identification, when dealing with nonlinear systems, the users have to consider nonlinear approximation quality. The reason for this is that the neglected approximation errors may manifest them as extra dynamics in the identified models resulting in an increase of model order. Most often high order is not desired due to the bias/variance dilemma in addition to the unnecessarily increased model complexity. Detailed discussions on model order estimation are taken in Chapter 5 and 6. Model Structure: This step is often carried out when a reduction of model complexity is required. Generally speaking, the model complexity is related to the number of parameters that the the model possesses. The objective of this step is to simplify the model while keeping the capability of capturing the process behaviors. Order-reduction is one of the often met problems in this field. Model Parameters: This step is usually carried out by the application of linear and nonlinear optimization techniques. Nonlinear optimization typically requires 8

27 some user interaction. If the systems to be identified are closed-loop, correlations between inputs and noises have to be considered. Pre-filtering and instrumental variable techniques are often used to solve this problem. Chapter 4 presents a detailed discussion on this topic. Model Validation: After having arrived at a particular model, it then remains to test whether this model is good enough, that is whether it is valid for its purpose. Such tests are known as model validation. The assessment of model quality is typically based on how the models perform when they attempt to reproduce the measured data. The validation tests involve various procedures to assess how the model relates to observed data, to prior knowledge, and to its intended use. Deficient model behavior in these aspects makes us reject or revise the model, while good performance will develop a certain confidence in the model. A model can never be accepted as a final and true description of the system. Rather, it can at best be regarded as a good enough description of certain aspects that are of particular interest to us Subspace Identification The large interest in subspace identification (SID) methods for system identification is motivated by the need for useful engineering tools to model linear multivariable dynamical systems using experimental data [MV89,Ver93], and it is fair to say that they are perhaps the most efficient and accurate methods available today for multivariable time series 9

28 identification [CP04]. An overview of theoretical and numerical issues can be found in [OM96b, PM00]. Many of the SID algorithms offer a consistent estimate of state-space models, i.e., the state space matrices A, B, C, and D, for multivariable dynamic systems with proper selection of the system order. A typical SID contains two steps: (1) identification of the extended observability matrix Γ α and a block triangular Toeplitz matrix H α ; and (2) calculation of {A, B, C, D} from the identified observability matrix and the Toeplitz matrix. The SID algorithms do not suffer from the problems caused by a priori parametrization and nonlinear optimizations. Moreover, SID has a better numerical reliability and a modest computational complexity compared with the prediction error method (PEM), particularly when the number of outputs and states is large. While most SID approaches appear as numerical algorithms, statistical properties such as consistency have recently been explored [DS95, JW98, DS99, HS99]. The SID algorithms are attractive not only because of their numerical simplicity and stability, but also for the state space form that is very convenient for estimation, filtering, prediction, and control Literature Review of Subspace Identification Several basic subspace-based approaches have been proposed for solving system identification problems: MOESP (Multivariable Output Error State space) [Ver93], N4SID (Numerical Subspace State-Space System IDentification) [MV89,OM94,OM96b] and CVA (Canonical Variate Analysis) [Lar90]. The main feature of the MOESP class of techniques 10

29 was the determination of an extended observability matrix of the deterministic part of the model (1.1). The main feature of the N4SID class of techniques was the determination of the state sequence of the linear time invariant system to be identified, or of an observer to reconstruct its state sequence, via the intersection of the row spaces of the Hankel-like matrices constructed from past and future input-output data. In the CVA method, the canonical states were first determined and the state space models were than simply determined by regression. Viberg gave an overview of SID methods and classified them into realization-based or direct types, and also pointed out the different ways to get system matrices via estimated states or extended observability matrix [Vib95]. Van Overschee and De Moor gave a unifying theorem based on lower order approximation of an oblique projection [OM95]. Here different methods were viewed as different choices of row and column weighting matrices for the reduced rank oblique projection. The basic structure and idea of their theorem was based on trying to cast these methods into the N4SID algorithm. It focused on the algorithms instead of concepts and ideas behind of these methods. In [JW98], the consistency of a large class of methods for estimating the extended observability matrix was analyzed. When systems had only measurement noise, persistence of excitation conditions on the input signal were given in this paper to guarantee consistent estimates. For systems with process noise, the author showed that a persistence of excitation condition on the input is not sufficient. It was also shown that this problem could be eliminated if stronger conditions on the input signal were imposed. 11

30 In [CP04], the authors provided an elementary error analysis of several well-known subspace methods when system inputs were ill-conditioned. The motivation of this paper was that experimental evidence showed that the performance of standard subspace algorithms from the literature might be surprisingly poor in certain experimental conditions, and the authors argued that the poor behavior might be attributed to a form of ill-conditioned system inputs. In this paper, two main possible causes of ill-conditioning were analyzed. The first had to do with near collinearity of the state and future input subspaces. The second had to do with the dynamical structure of the input signal and might roughly be attributed to the lack of excitation. The authors in [SH00b] investigated techniques for data compression of the inputoutput data, needed for the preliminary processing. Standard QR factorization and the Cholesky factorization of the correlation matrices exploiting the block Hankel structure were analyzed. Structure exploiting algorithms and dedicated linear algebra tools were presented. In [CV97], the authors applied Instrumental Variable (IV) techniques and SID methods to solve the Errors-in-Variables (EIV) problems. In this paper, noise processes were allowed to be correlated with each other. The authors presented a solution to the problem of identifying discrete time linear time-invariant state space models of finite order in a statistically consistent way. In [RV00], the authors applied and evaluated SID (MOESP) and other commercial softwares to identify the dynamic model of a robot manipulator. Results using the different software tools have been similar (but not identical) in accuracy and predictive power. 12

31 However, the identification of a state-space model combined with a friction model could provide more efficient means to modeling robotics, since the singular values analysis in SID methods suggested a lower model order than the other methods. In [LG02] the authors discussed the multipatch measurements setup, used to study moving sensors in vibration mechanics, for blind eigenstructure identification. The method consisted of merging the data first, and processing them globally using an output-only covariance-based subspace algorithm after a suitable normalization of the output covariances. This method could provide consistent estimates of the eigenstructure, even under nonstationary excitation. 1.2 Fault Detection and Tolerance Highly reliable control systems are necessary in many applications. For example, when considering some space missions, a system with hundreds of components is required to operate for a period of several years. Such systems must naturally employ highly sophisticated fault tolerant control systems with a redundant capacity to perform a given task [Mas86]. Another important example is the real time, reliable ship control system for marine vessels in order to guarantee the crew safety and mission success [Sim97]. The concept of fault detection filter was first introduced in [Bea] as part of the selfreorganizing control system. It is a type of Luenberger observer in that the failures have known and predictable characteristics when they are propagating in the dynamic systems [CS98]. It was first proposed by the author for linear, time-invariant, continuous systems, 13

32 and it was extended and expanded by the subsequent researchers to take care of time varying [CS98], nonlinear [QB96], and discrete systems [WW75] in various applications. The most important contributions are attributed to W. S. Willsky, H. L. Jones, R. V. Beard, P. J. Buxbaum, etc. for their original works [BH69, AC74, AC75, WJ76, WW75, Bea, Jon73]. Fault detection involved in this dissertation is to use subspace identification combined with order estimation methods in alarm detection. Specifically, an instrumental variable subspace identification method will be applied to obtain covariance matrices associated with system matrices. Then, order estimation approaches will be applied to these covariance matrices to determine order deficiency related to various faulty modes Dynamic Model of Fault Detection The relationship between the processing status and measurable signals can be represented by a mathematical model which exhibits a certain number of parameters referred to as structure parameters. The structure of a complex system involving a failure monitor is shown in Fig When a fault occurs, abnormal changes among these parameters can be identified by the detection filter, indicating structure changes caused by the fault. 14

33 ACTUATORS PLANT SENSORS COMMAND OUTPUT COMMAND COMMAND INPUT CONTROL LAW FILTERED SIGNALS FAILURE MONITOR Figure 1.4: A complex system involving a failure-monitor Fault Detection Approaches The objectives of system analysis are usually prediction and control of the system s behavior, where the achievement of such objectives is based on whether or not such an analysis interferes with the behavior of the system itself [AN88]. When an accurate prediction is available, desired performances can be obtained by applying proper control strategies to the system. To achieve this, prediction often requires a full utilization of all available knowledge, or experience, or information, about the system. When applying these principles to a fault detection problem, the objective is to monitor an operational system and its components, managing the fault tolerance schemes and system reconfiguration when a fault occurs, and ensuring that the system will keep working properly or that the corresponding mission is satisfactorily completed. For an autonomous system, inputs of sensor signals not only provide feedback information to the control system, but are also used in the fault monitoring subsystem. Once a fault is detected, measures like switching techniques or reconfiguration can then be taken 15

34 to correct the problem. Generally speaking, the process of fault detection can be divided into three main steps [Wil74]. 1) Alarm: The alarm task simply consists of making a binary decision, either something has gone wrong or everything is fine. In this process, information on current processing status is extracted from the signals measured using internal sensing. Many papers can be found in the literature on estimating the innovations of processes through Kalman filters [SS00, SS04, SH00a]. 2) Isolation: The isolation problem consists of determining the location of a failure, e.g. which sensor or actuator has failed [GS91], what type of arrhythmia of a heart has occurred [WW75], etc. In this process, the obtained features are compared with the predefined fault-free states, leading to the indication of a specific fault, if it occurs. This can be achieved by identifying the failure signature matrices which describe the propagation of various failures through the system and the detecting filter [WJ76]. 3) Estimation: The estimation problem involves the determination of the extent or severity of a failure. This can be achieved by identifying the parameters in the failure model. Worth noting is that a systematic method has been designed in [Bea, Jon73] to solve the isolation problem mentioned in Step 2, consisting of a fault detection filter applicable to a linear deterministic multi-variable dynamical system, able to decouple failures in 16

35 system outputs. Such a filter, through proper gain selecting schemes, manifests particular failure modes as estimation residues which remain in a fixed direction or a fixed plane Faulty Modes The design of fault detection systems involves several issues. The first is the tradeoff between the sensitivity of alarm detectors and the performance of optimal estimators. Recent hardware and software technological breakthroughs provide a solution to this problem, allowing the implementation of detector and estimator filters in parallel [Wil74]. In short, an optimal filter is designed for fault-free missions, but the pre-filtered signals are also provided to a fault detection filter for alarm detection purpose. As a fault is detected, the fault detection filter starts up the failure isolation process. Afterwards, correction measures can be taken to ensure the satisfactory completion of the designed mission. Another important issue we need to face is the different properties of faults, which affect the strategies chosen for fault tolerance and system reconfiguration. One possible fault is characterized by an abrupt change, for example a sudden gyro shift, which can be represented as an additive kronecker delta or step function to the system dynamics [AC74,AC75]. A difficult situation is the hard-over failure of an actuator or sensor, which can be represented by rank deficiency of system matrices [EW75]. In order to detect both of the above failures, the detection filter is required to have fast response to those high frequency triggering agents. Another often met fault is the performance degradation of the 17

36 system components, such as the slow shift of gyro bias due to temperature change, or the increasing noise variance that degrades the measurement performance. Mathematically, the above faults are often taken as the inputs of the faulty system models that researchers must be concerned with when dealing with fault detection problems. Supposing the system order is n, the following possible modifications of (1.1) corresponding to different faulty modes are used to represent faulty system dynamics. Dynamics and/or sensor jump/step [WJ76, WW75] x(k + 1) = A(k)x(k) + B(k)u(k) + ν 1 δ k+1,θ1, k = 1, 2,..., (1.8) z(k) = Cx(k) + Du(k) + ν 2 δ k,θ2, (1.9) where ν 1 R n and ν 2 R n are unknown vectors, θ 1 and θ 2 are the unknown times of failure, and δ is a Kronecker delta. Hard-over actuator failure [EW75] x(k + 1) = A(k)x(k) + [B + Mσ k+1,θ ]u(k), k = 1, 2,.... (1.10) Here we can take into account complete failure (or off ) of certain actuators, i.e. an off failure of the ith actuator can be modelled by choosing the matrix M to contain all zero except for the ith column, which is taken to be the negative of the ith column of B. Hard-over sensor failures [RW75] z(k) = Cx(k) + Du(k) + [Mx(k) + Su(k)]σ k,θ, k = 1, 2,.... (1.11) 18

37 Here the failures are functions of u and x, and a failure of the ith sensor is modeled by choosing the ith rows of M and S appropriately. Increased process noise failures [RW75] x(k + 1) = A(k)x(k) + Bu(k) + ξσ k+1,θ, k = 1, 2,..., (1.12) where ξ is the additional white process noise. Added sensor noise failures [EW75, RW75] z(k) = Cx(k) + Du(k) + ξσ k+1,θ, k = 1, 2,..., (1.13) where ξ is the same as that in (1.12). The hard-over failure detection will be presented in Chapter Literature Review of Fault Detection for Complex Systems A complex system can be thought of as multiple inter-dependent working subsystems individually subject to malfunction or failure which can cause the system to fail its mission with unacceptable results [Jay04]. It is therefore crucial to detect the malfunctioning of a faulty component as soon as possible, and to compensate by reconfiguring or substituting redundant elements so that the system continues to operate satisfactorily. If redundancy is not available, it could be necessary to decrease the expected performance of the system without jeopardizing its capability of successfully completing its mission. To this end, the 19

38 problem of health monitoring of complex systems involves two types of questions. First, the detection of failures or catastrophic events should be achieved. Second, the detection of smaller faults, namely sudden or gradual (incipient) modifications, which affect the process without causing stop, is also required to prevent the subsequent occurrence of more catastrophic events [RF89]. A. S. Willsky and his coauthors presented in [AC75] a multi-hypothesis method to detect shifts of gyros and accelerometers in a calibration/alignment problem. With this method, different failure directions, i.e. degrees of freedom, are hypothesized, and the conditional probability of the system states, given the measurement, are computed. It is assumed in this approach that the system noise has a fault-free distribution, that the probabilities of the system having fault-free process noise and that of including a faulty mode are known, and that the system noise has a high probability to be normal process noise and a small probability of including a burst in each of the failure directions. A limitation from a practical viewpoint is its highly computational burden due to the exponentially growing bank of filters required in the model fitting process. H. L. Jones presented in his Ph.D thesis [Jon73] a fault detection filter design methodology capable of detecting a wide variety of abrupt changes, i.e. thruster failures, both attitude and rate gyro uncertainties, and aileron deflection sensors uncertainties. The contribution is that the failures manifest themselves in the estimation residues in decoupled directions. This method is capable of a real-time failure detection and isolation in continuous, stochastic system with continuous measurements. It is assumed in this approach that an accurate model of the unfailed system dynamics is known and that the 20

39 system dynamics is linear and time invariant. In the presence of a failure, the failure detection filter works as a suboptimal state estimator. A. S. Willsky and his coauthors presented in [WJ76] a generalized likelihood ratio (GLR) method to develop an adaptive filter to detect the abrupt changes in linear dynamic systems. At the same time, a parallel Kalman-Bucy filter is implemented in the normal estimation mission such that the filters have high sensitivity to failures while keeping optimal performance under normal conditions. Then, the log-likelihood ratio of failure vs non-failure based on a statistical reference model is computed for a fault detection purpose. This approach is applied to study vehicle systems with components failure, small nonlinearities and under-estimated system states. In [CS98,DS03] the authors presented a novel approach, a game theoretic fault detection filter for the fault detection and isolation applicable to a class of dynamic systems in the presence of nuisance parameters, i.e. unknown inputs. The proposed method has been proved to be effective in bounding the transmission of all exogenous signals other than the fault to be detected, while embedding the nuisance into an unobservable, invariant subspace. This approach is used to detect an accelerometer fault in the presence of a wind gust disturbance and sensor noise, and a position sensor fault under the uncertainty of the mass rate input for a rocket, which is a time-varying problem. M. Luo and his coauthors presented in [M 02] a model free framework to detect sensor faults by applying pseudo power signatures (instantaneous energy distribution) of output measurements, which is computed by a Singular Value Decomposition (SVD) approach. The contribution of this method is its applications when a reliable system model is not 21

40 available, or input measurement is noise-contaminated. This method is applied to study the aircraft safety of a Boeing 747. I. Szaszi and his coauthors presented in [I 02] a subspace fault detection and isolation (FDI) approach for the linearized longitudinal dynamics of a Boeing 747 series 100/200. The fault detection filter is designed to be sensitive to elevator fault and pitch rate sensor fault. The contribution of this method is to transform the sensor failure to a pseudoactuator failure, and to use a geometric approach to obtain the detection subspace based on the mutually detectable property. S. Jayaram systematically developed a general methodology in [Jay04] for the near real-time autonomous health monitoring for flexible space structures, e.g. solar panels of a mini-satellite, and robust system reconfiguration when faulty actuators are detected. The main contributions of this work are shown in several aspects. A parallel extended Kalman filter processing algorithm is designed for near real time estimation and sensor fault detection, where robust control methodology is used for the near real-time actuator fault detection. A robust reconfiguration mechanism is then applied to realize a fault tolerant capability. Applications include integrating the proposed methodology into a structure health monitoring and control system, and software simulation of such a methodology applied to a flexible space structure. 22

41 1.3 Proposed Work Problem Statement Considering the research from the previous and present contributions from the various researchers and contributors, the statement of work for this dissertation is: 1. to design nonlinear experiments, including the choice of the model inputs, excitation signals, model architecture, etc; 2. to implement statistical analysis on processes and measurement noises; 3. to choose dynamic representations; 4. to design model identification approaches, including model transformations, numerical implementations, etc; 5. to design order estimation approaches; 6. to choose fault mode representations, and to design fault detection approaches; 7. to demonstrate the validity of the system identification, order estimation and fault detection approaches through examples and simulations Major Contributions The significance of the proposed research approach is to use the subspace detection/identification method for fault detection and fault tolerance in an integrated manner. 23

42 1. A Hammerstein system with nonlinear feedbacks contaminated by (colored) system and measurement noises will be used to model the system dynamics. This model can represent a larger class of mechanical and electrical systems than simplified linear models. Therefore, the predicted model will fit a larger operating region. Additionally, modifications will be considered to allow the basic system dynamics to include the faulty modes needed for fault detection purpose. 2. The research, focusing on nonlinear closed-loop Hammerstein systems, will implement system identification, order estimation and fault detection missions. (a) The ASIVID method will be allowed to estimate the closed-loop Hammerstein system model in a numerical efficient manner. (b) Fast order estimation approaches will be used to determine the system order without a priori knowledge of noise models with the advantages of both fast implementation and robustness. (c) In the presence of disturbances, hard-over failures will be detected by using a statistical system order determination method. 3. Optimal and suboptimal subspace trackings will be discussed in details, including a geometric method for optimal and sub-optimal subspace reconstruction, leading to a possible fault tolerance strategy by reformulating the faulty/degraded systems. 24

43 1.4 Summary The remainder of the dissertation is organized into seven chapters. Preliminary knowledge and some definitions used in this dissertation are presented in Chapter 2. Single-input single-output nonlinear Hammerstein system identification is presented in Chapter 3, and it is applied to model the dynamics of a marine thruster. Identification of closed-loop multi-input multi-output nonlinear Hammerstein systems is presented in Chapter 4. Issues on nonlinear identification experimental design are discussed in chapter 5 by presenting the problem of identifying the dynamics of a reaction wheel pendulum. A fast order identification algorithm is introduced in Chapter 6, and its application to hard-over failure detection is discussed in Chapter 7. Signal detections and a geometric interpretation of subspace tracking theorems are presented in Chapter 8, and their potential applications to constrained system identification and fault tolerance are discussed. Finally, Chapter 9 presents a brief conclusion and makes recommendations for future work on the subject. 25

44 CHAPTER 2 PRELIMINARIES The following notation is used in this dissertation. Let hat denote the estimated quantities, the superscript ( ) T the transposition operator, ( ) H the Hermitian transposition [SS90], 2 the Euclidean vector norm, 1 the l 1 vector norm, F the Frobenius matrix norm, I m the identity matrix with dimension m, tr( ) the trace operator, and diag(d 1,, d n ) a diagonal matrix consisting of the diagonal elements d i, i = 1,, n. Define N the set of integers, R the set of real numbers, C the set of complex numbers, R m the set of real m-component vectors, C m the set of complex m-component vectors, R m n the set of m n real matrices, and C m n the set of complex m n matrices. Suppose Z R m n, then R(Z) denotes the column space of Z, R(Z) the orthogonal complement of R(Z) [SS90], R(Z T ) the row space of Z, and N (Z) is the null space of Z. Denote Z R Z T (ZZ T ) the right inverse of Z, where ( ) denotes the Moore- Penrose pseudo-inverse of a matrix. Furthermore, the operator Π Z Z R Z = Z T (ZZ T ) Z projects the row space of a matrix onto R(Z T ), and the operator Π Z I Π Z projects the row space of a matrix onto the orthogonal complement of R(Z T ). Denote with X/ Y Z [XΠ Y ][ZΠ Y ] Z the oblique projection of the row space of X R i n along the 26

45 row space Y R j n on the row space of Z R m n. Denote X/Y the shorthand for the projection of the row space of the matrix X on the row space of the matrix Y. Definition 2.1 [SS90] Consider a matrix Z C m n with rank r, m n r. Then there are unitary matrices U and V such that the singular value decomposition (SVD) of Z is defined as Z = U Σ V H, where Σ + = diag(σ 1,, σ r ) and σ 1 σ r > 0 are singular values of Z. In particular, if Z C m m is Hermitian, the eigenvalue decomposition (EVD) of Z is defined as [GL89] Z = U Λ U H, where Λ + = diag(λ 1,, λ r ) and λ 1 λ r > 0 are the eigenvalues of Z. Note that the columns of the matrix U = [u 1,..., u m ] are the left singular vectors of Z, furthermore, the columns of the matrix U 1 [u 1,..., u r ] form an orthonormal basis for the column space of Z, while the columns of the matrix U 2 [u r+1,..., u m ] form an orthonormal basis for the orthogonal complement R(U 1 ) [SS90]. Definition 2.2 [SS90] Given a matrix Q X, where the columns of Q X form an orthonormal basis for a subspace X C n n, the matrix P X Q X Q H X, is called the orthogonal projection onto X. 27

46 Specifically, P X z indicates the operation that projects z C m orthogonally onto its components in X. The projection operator can be interpreted as shown in Figure 2.1. X y = P X z z x = P X z X Figure 2.1: Interpretation of the orthogonal projection in a 2-dimensional space. Remark 2.1 [SS90] Although the columns of Q X do not form a unique basis of X, P X = Q X Q H X is always the unique orthogonal projection operator onto X. Two different bases will provide different matrices Q X and Q X which have the property that Q X = Q X Ū where Ū is a unitary matrix. Remark 2.2 [SS90] If P X is an orthogonal projection operator onto X, P X is Hermitian (P H X = P X ), which implies that P X y = 0 for any y X, and idempotent ((P X ) i = P X, i = 1, 2,....), which implies that P X x = x for any x X. Lemma 2.1 Let P X be an orthogonal projection onto X. Consider matrices X, Y such that X = [x 1,, x n1 ] with x i X, i = 1,, n 1, and Y = [y 1,, y n2 ] with y i X, i = 1,, n 2. Then, P X X = X and P X Y = 0. Proof The proof is straightforward and follows from Remark

47 Denote E( ) the mathematical expectation [Rao73]. Given a sequence {z k }, k = 1, 2,..., the averaging operator Ē is defined as [MB97] Ē[z k ] lim 1 n 0 n 0 n 0 k=1 E[z k ]. (2.1) In order to simplify our notation in the following chapters, we introduce the following definitions. Definition 2.3 Consider a generic signal s(k), k = 1, 2,.... For a given index α > n, where α N and n is the system order, define the past and future stacked signal vectors as follows s p (k) [s T (k α),..., s T (k 1)] T, k =α+1,..., (2.2) s f (k) [s T (k),..., s T (k + α 1)] T, k =α+1,.... (2.3) The block Hankel matrices associated with s p (k) and s f (k) are defined as S p (k) [s p (k),..., s p (k + β 1)], k =α+1,..., (2.4) S f (k) [s f (k),..., s f (k + β 1)], k =α+1,..., (2.5) where β N represents the number of columns of block Hankel matrices. Denote S 1 α (k) S p (k), S α+1 2α (k) S f (k), S 1 2α (k) [ S p T (k) S α+2 2α (k) S f T (k) ] T, S + p (k) S 1 α+1 (k) and S f (k) 29

48 CHAPTER 3 SINGLE-INPUT SINGLE-OUTPUT OPEN LOOP HAMMERSTEIN SYSTEM IDENTIFICATION 3.1 Introduction In this chapter, a discrete-time nonlinear estimation algorithm is proposed to identify, from input/output data, the model of an open-loop SISO system consisting of both linear and nonlinear components. In particular, within a general framework, both triangular and trigonometric function expansions are considered to represent the nonlinear subsystem. The overall system is then expressed in a parametric form and regression analysis is used for recovering the relationship between input and output signals. Numerical examples are presented to verify the nonlinear identification algorithm, and the algorithm is finally used to recover the marine thruster dynamics using experimentally sampled input/output data. Here, we consider plants having the Hammerstein structure with its linear model represented by ARX as follows y(k) = a 1 y(k 1)... a na y(k n a ) + b 1 ū(k 1) b nb ū(k n b ) + e(k), 30

49 where ū(k) and y(k), k = 1, 2,..., are input and output sequences of the linear model, respectively, and a 1,..., a na and b 1,..., b nb are model parameters. 3.2 Problem Formulation e(k) 1 A(q 1 ) u(k) m(k) f( ) B(q 1 ) + y(k) A(q 1 ) + Figure 3.1: Hammerstein model with ARX linear subsystem We consider plants having the Hammerstein structure with ARX linear model shown in Figure 3.1 and described by y(k) = B(q 1 ) A(q 1 ) m(k) + 1 e(k), m(k) = f(u(k)), k = 0, 1,..., (3.1) A(q 1 ) where q 1 is the backward shift operator, and A(q 1 )=1+a 1 q 1 + +a na q n a, B(q 1 )=b 0 +b 1 q 1 + +b nb 1q n b+1. (3.2) The integers n a and n b 1, with n a n b, are the orders of A(q 1 ) and B(q 1 ), respectively. The signals u( ) and y( ), are the plant input and output, respectively. The function f( ), describing the nonlinear effects, is supposed to be memoryless within some given finite interval [u min, u max ] R. The signal m( ) is a nonavailable internal sequence related to the input only and e( ) is a bounded measurement disturbance which is supposed to be a zero-mean, white noise sequence. We have the following assumption on the system (3.1). 31

50 Assumption 3.1 A(q 1 ) and B(q 1 ) are coprime. Remark 3.1 Assumption 3.1 implies that G(q 1 ) is controllable. It also ensures the existence of a unique solution to the problem of identifying of B(q 1 ) and A(q 1 ) from the sampled input-output data [FR01]. In the next section, we will discuss a general identification scheme for Hammerstein systems. 3.3 Identification Scheme Design In this section we first introduce a general decomposition of the nonlinear function f( ) : R R as a finite sum of p + 1 terms (p N) [Lju99b], f(u) = p µ j η j (u) = µ T η(u), u [u min, u max ], (3.3) j=0 where µ = [µ 0 µ p ] T is a parameter vector, η(u) = [η 0 (u) η p (u)] T, and η j ( ), j = 0,..., p, represents the j th basis function. Many different expansions have been considered through the years and two simple but effective choices are exploited in the following subsections. One possible way to define the basis functions is as follows η j (u) = κ(α j (u γ j )), j = 0,..., p, u [u min, u max ], (3.4) 32

51 where α j, γ j R denote the dilation and translation parameters, respectively, and κ( ) : R R denotes a generator function which belongs to a large family of functions containing trigonometric functions, triangular functions, Gaussian functions, etc [Lju99b]. Next, we rewrite (3.1) as follows y(k) = (1 A(q 1 ))y(k) + B(q 1 )f(u(k)) + e(k), k = 1, 2,..., (3.5) Substituting (3.2) into (3.5), we obtain y(k)= a 1 y(k 1) a na y(k n a )+b 0 f(u(k))+ +b nb 1f(u(k n b +1))+e(k), which can be written in a parametric form as y(k) = θ T φ(k) + e(k), k = 1, 2,..., (3.6) where with [ a [ µ θ a θ b Rn a+n b (p+1), ] T a 1 a na R n a, θ b vec(µb T ) R nb(p+1), ] T [ ] T µ 0 µ p R p+1, b b 0 b nb R n b 1, and vec( ) is an operator which stacks the columns of a matrix into a vector. The vector φ(k) is defined as φ(k) φ y (k) φ η (k) Rn a+n b (p+1), k = 1, 2,..., 33

52 where φ y (k) y(k 1). y(k n a ) R na, φ η (k) η(u(k)). η(u(k n b + 1)) R nb(p+1). Next, consider input-output measurements pairs [u(k), y(k)], k = 1,..., l, where l > n a, and form the quadratic cost J(θ) = Y Φθ 2, where Y y(n a ). y(l) R l na+1, Φ φ T (n a ). φ T (l) R (l n a+1) (n a +n b (p+1)). The estimated value of the parameter vector θ R na+n b(p+1) is given by the following optimization problem ˆθ = argmin J(θ), θ R n a+n b (p+1) which we solve by applying the Least Squares Estimation (LSE) method and the Singular Value Decomposition (SVD) [PB01b, GB00, SS90], which provide the following identification procedure: Algorithm 3.1 Linear Regression Algorithm 1. Assuming that (Φ T Φ) 1 exists, we have [Lju99b] ˆθ = (Φ T Φ) 1 Φ T Y. (3.7) 34

53 2. Next, after extracting â and ˆθ b from ˆθ, we have [PB01b] ˆµˆb T = σ max (vec 1 (ˆθ b ))u 1 v T 1, (3.8) where σ max ( ) is the maximum singular value of vec 1 (ˆθ b ), u 1 is the first left eigenvector of vec 1 (ˆθ b ), and v 1 is the first right eigenvector of vec 1 (ˆθ b ). 3. It follows from (3.8) that ˆb and ˆµ are given by where β R is an arbitrary parameter. ˆµ = βσ max (vec 1 (ˆθ b ))u 1, (3.9) ˆb = 1 β v 1, (3.10) 4. The nonlinear function f( ) can be recovered from the estimate ˆf(u(k)) = ˆθ b φ η (k). (3.11) Consider the linear-in-parameter equation (3.6). An updating scheme of WLS (Weighted Least-Squares) Algorithm is listed below. The interested reader may refer to [GS84] for details. Algorithm 3.2 Recursive Linear Regression Algorithm 1. Given θ 0, set P (0) = P 0, where P 0 is a positive definite matrix, and weighting parameter a(t) The vector of parameters updates through ˆθ(t + 1) = ˆθ(t) + a(t)p (t 1)φ(t) 1 + a(t)φ T (t)p (t 1)φ(t) [y(t + 1) φt (t)ˆθ(t)]. (3.12) 35

54 3. The projection matrix updates through where P ( 1) = P 0. P (t) = P (t 1) a(t)p (t 1)φ(t)φT (t)p (t 1), (3.13) 1 + a(t)φ T (t)p (t 1)φ(t) 4. Then, steps 2-4 in Algorithm 3.1 should be applied following the estimation of θ. Remark 3.2 The parameter β R can be set to any nonzero value. The estimation of the actual value of β requires more information about the subsystems, which is unavailable in our case. However, different choices of β do not change the predicted output ŷ(t), defined as ŷ(k) = (1 Â(q 1 ))y(k) + ˆB(q 1 ) ˆf(u(k)), k = 1, 2, Triangular Series Approximation In this subsection we use the triangular basis functions illustrated in Figure 3.2 to linearly parameterize the nonlinear function f( ). For simplicity, the input interval [u min, u max ] is evenly divided into p N partitions separated by a set of points {u 0, u 1,..., u p }, such that u min = u 0 < u 1 < < u p = u max with u j u j 1. Take κ( ) as the unit triangular generator function, defined as 1 + x if 1 x < 0, κ(x) = 1 x if 0 x < 1, (3.14) 0 otherwise, 36

55 η j (u) 1 u u min = u 0 u 1 u j 1 u j u j+1 u p = umax Figure 3.2: Triangular basis function and take γ j = u 0 + j, α j = 1/, and µ j = f(u 0 + j ) with j = 0, 1,..., p. Then, from (3.3) (3.4) and (3.14) we obtain the triangular series expansion of any nonlinear function f( ) over the interval [u min, u max ] with η j (u) given by (u u j 1 )/ if u j 1 u < u j, η j (u) (u j+1 u)/ if u j u < u j+1, j = 0,..., p. (3.15) 0 otherwise Trigonometric Series Approximation If η j ( ), j = 0, 1,..., p, are orthonormal with respect to the positive weighting function ρ( ) : R R, we have b a 0 if i j, ρ(u)η i (u)η j (u)du = 1 if i = j. (3.16) In particular, if we choose the η j ( ) to be sin( ) and cos( ) functions, (3.3) becomes a truncated Fourier series expansion. Thus, the nonlinear function f( ), which is defined in 37

56 the interval [u min, u max ], has the following trigonometric sum expansion f(u) = c 0 + p j=1 c j cos jπ(u u m) L + p j=1 where u m = (u min + u max )/2 and L = (u max u min )/2. d j sin jπ(u u m), (3.17) L This result can be cast in the general framework (3.3) (3.4) by choosing κ(x) = {cos x, sin x} T, (3.18) and α j = jπ/l, γ j = u m, j = 0, 1,..., p. Then, from (3.3) (3.4) and (3.18) we obtain η(u) = [1, η T 1 (u),..., η T p (u)] T, µ = [c 0, c 1, d 1,..., c p, d p ] T, (3.19) where [ η j (u) = cos jπ(u u m), sin jπ(u u ] T m), j = 1, 2,..., p. (3.20) L L 3.4 Numerical Simulations The identification algorithm presented in the previous section is now tested using numerical examples. We consider the discrete-time system with Hammerstein structure, given by G(q 1 ) = q q q 2, and f(u) = u u 0.25, < u < 0.25, u u, 38

57 with sampling period T = 0.1 s. The chosen input signal was a zero-mean, white noise sequence uniformly distributed between 1 and 1. The measurement noise was a zeromean white noise Gaussian sequence. The resulting signal-to-noise ratio was SNR = 22, where SNR l Σ k=0 l Σ k=0 y 2 (k). v 2 (k) We suppose that the order of the linear dynamic subsystem is known. The system identification is then performed using the algorithm (3.12) (3.13) and (3.8) (3.11) with both triangular and trigonometric basis functions to approximate the nonlinear subsystem Triangular Functions Approximation Predicted output error Time [s] Figure 3.3: Triangular approximation: predicted output error 39

58 actual predicted m(u) u Figure 3.4: Triangular approximation: nonlinearity Identification error of a Identification error of a Time [s] Time [s] Identification error of b Time [s] Identification error of b Time [s] Figure 3.5: Triangular approximation: identification error of â and ˆb Figures show the simulation results when the nonlinearity f( ) is parameterized using the triangular basis functions (3.15) and p = 10. Figure 3.3 shows the predicted output error defined as e(k) y(k) ŷ(k), k = 0, 1,.... For clarity, only the last 8 seconds of the simulation are shown, which reveal that the predicted output matches the experimental data, no matter what the choice of β is. The quality of this match is 40

59 measured by the following quantity E N = N Σ k=1 (y(k) ŷ(k)) 2 N which represents the averaged predicted output error computed using the last N points. In our simulation E 200 = (the last 20 s). Figure 3.4 shows that the predicted nonlinearity closely fits the actual one. Finally, Figure 3.5 shows that, for β = 0.75, the prediction error of a 1, a 2, b 0 and b 1 are quite small, which guarantees that the predicted linear dynamics is very close to the actual one., Trigonometric Functions Approximation Figures show the simulation results when the nonlinearity f( ) is parameterized Predicted output error Time [s] Figure 3.6: Trigonometric approximation: predicted output error using the trigonometric basis functions (3.20) and p = 5. The results are very similar to 41

60 1 actual predicted 0.5 m(u) u Figure 3.7: Trigonometric approximation: nonlinearity Identification error of a Time [s] Identification error of b Time [s] Identification error of a Time [s] Identification error of b Time [s] Figure 3.8: Trigonometric approximation: identification error of â and ˆb those obtained in the previous case with the only difference that, in this case, E 200 = , proving that this parameterization provides better convergence characteristic than the previous one. 42

61 Note that in this example f( ) is an odd function, hence in (3.17), coefficients related to the cos( ) terms need to converge to zero as shown in Table 3.1, which presents the final identification results of c 0, c 1, d 1,..., c 5, d 5. Table 3.1: Parameter values for the trigonometric expansion with p = 5 c c d c d c d c d c d Nonlinear Function Identification Test Since the real value of β is unknown, the estimation of the nonlinearity is not unique. However, by comparing the identified coefficients corresponding to different parameterizations, the equivalence of the two approximations can be shown. First, we rewrite the dynamic equation (3.1) in the state space form x(k + 1) = Āx(k) + Bw(k), k = 0, 1,..., y(k) = Cx(k) + Dw(k) + v(k), (3.21) 43

62 where x(k) is the state vector and Ā, B, C, and D are the discrete dynamic matrices. Next, define ˆB T (k) = [ˆb 0,T (k), ˆb 1,T (k)] T and ŵ T (k) as the identified B and w(k) at the k th step, respectively, when using the triangular basis functions parameterization. Similarly, ˆB F (k) = [ˆb 0,F (k), ˆb 1,F (k)] T and ŵ F (k) are the identified B and w(k) at the k th step, respectively, when using the trigonometric basis functions parameterization. Note that different parameterizations do not affect the value of Bw(k), hence ˆBT (k)ŵ T (k) = ˆB F (k)ŵ F (k), k = 0, 1,..., which provides ˆb 0,F (k) ˆb 0,T (k) = ˆb 1,F(k) ˆb 1,T (k) = ŵ T(k), k = 0, 1,.... (3.22) ŵ F (k) To avoid problems with any of the denominators in (3.22) being close to zero, define x ε x 2 + ε 2, x R, where ε 0 is an arbitrary small number. Then, (3.22) can be replaced with ˆb 0,F (k) ε ˆb 0,T (k) ε = ˆb 1,F(k) ε ˆb 1,T (k) ε = ŵ T(k) ε ŵ F (k) ε, k = 0, 1,..., which collapses to (3.22) when ε = 0. Figure 3.9 shows ˆb 0,F (k) ε, ˆb 1,F(k) ε and ŵ T(k) ε ˆb 0,T (k) ε ˆb 1,T (k) ε ŵ F (k) ε for ε = 0.1. It can be observed that they are all bounded and close to each other. 44

63 b 0,F ε / b 0,T ε b 1,F ε / b 1,T ε w T ε / w F ε Time [s] Figure 3.9: Identified coefficients comparison 3.5 Application of Modeling the Dynamics of a Marine Thruster Mission Description Autonomous Underwater Vehicles (AUVs) are required for a variety of missions, which include mine classification, sea floor characterization, surf zone operations, etc. For those vehicles to be able to perform such missions, an adequate control framework should be developed to take into account the complicated issues related to high maneuverability. Implementation of classical control techniques for trajectory tracking control of an underwater vehicle is a challenging problem when considering the complexity of the nonlinear coupled differential equations used to describe the vehicle dynamics and uncertainties related to unknown inertia, damping, coriolis, or centripetal matrices. Furthermore, one of the main difficulties recognized in literature [SP01, PWY00, YCS90] is related to the in- 45

64 fluence of thruster dynamics on the AUV s behavior and their integration into the control system design. In order to solve these problems, sliding mode control, adaptive control, neural network based control, and fuzzy logic control have been developed in recent years where the system dynamics is assumed to be partially known (see for example [YCS90, TEE00, IFU00, LHL99]). A different approach which has not yet been carefully investigated consists in experimentally identifying the unknown dynamics and then using the identified model to design the control algorithm. This approach would solve some of the problems related to unknown parameters and unmodeled dynamics. Modeling of marine thrusters is a challenging problem due to the coupling of the dynamics of the brushless DC motor which drives the propeller and the hydrodynamic effects caused by the interaction between the propeller and the fluid [WY99a, WY99b, BWG00]. Important are also other nonlinear effects such as saturation constraints on the input signal [LHH01]. A considerable effort has been spent in recent years to derive reliable models for marine thrusters (see for example [BWG00]); however, researchers in this field have not yet been able to accurately model the behavior observed on real systems. There are no universal identification techniques for general nonlinear systems. All of them largely depends on a priori knowledge about the system, i.e., about its mathematical representation, or particular properties. The model representation described in this section has been chosen from intuition without a solid theoretical motivation due to the difficulties of mathematically describing the dynamics of brushless DC motor, especially 46

65 when Pulse Width Modulation (PWM) signal is used to drive the motor, since PWM signal is generated by a servo-controller board with its own internal controller for set-point regulation. Furthermore, the hydrodynamic effects caused by the interaction between the propeller and the fluid are very complex and although a considerable effort has been spent in recent years to derive a reliable model, researchers in the field have not yet been able to accurately model the behavior observed on real systems. In this section, we simplify the modeling problem by blindly proposing a Hammerstein model, which consists of a nonlinear memoryless subsystem followed by a linear dynamical subsystem, to represent the marine thruster dynamics, and then verifying our choice by using both triangular and trigonometric basis functions parameterizations Experimental Setup, Modeling and Validation For our experimental setup we used the Submersible Quiet Unmanned Intelligent Diver (SQUID) AUV, which is a 22-inch long AUV designed at FAU [Le 00]. The experiment was set up in the FAU s hydrodynamics lab, where the 7000 gallon wave tank with its 1.7 m 2 cross-section and a maximum flow speed of 0.54 m/s was a ideal facility to hold the SQUID AUV underwater (see Figure 3.10). The thrust was measured by a six degree-offreedom force sensor from ATI Automation. Suppose that the thruster dynamics is described by T (k) = G(q 1 )f(i m (k)) + v(k), k = 0, 1,..., 47

66 Aluminium structure Force sensor in its waterproof housing MicroAUV 166Mhz CPU QNX4.25 Ehernet cable: QNXnet & TCP/IP Vector thruster Figure 3.10: SQUID AUV with G(q 1 ) = b a 1 q 1 + a 2 q 2, where T (k) is the thrust, and i m (k) is the current to the servocontroller board. The coefficients a 1, a 2, and b 0, and the nonlinear function f( ) are unknown and need to be identified. 1 Predicted output error Time [s] Figure 3.11: Marine thruster: predicted output error with triangular approximation 48