A Sliding Mode Observer for State Estimation and Reconstruction of Unknown Inputs with Arbitrary Relative Degree: Application to Chemical Processes

Transcription

1 A Sliding Mode Observer for State Estimation and Reconstruction of Unknown Inputs with Arbitrary Relative Degree: Application to Chemical Processes Esteban López Aguirre Chemical Engineer Universidad Nacional de Colombia Facultad de Minas Departamento de Procesos y Energía Medellín, Colombia 2015

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3 A Sliding Mode Observer for State Estimation and Reconstruction of Unknown Inputs with Arbitrary Relative Degree: Application to Chemical Processes Un Observador por Modos Deslizantes para la Estimación de Estado y la Reconstrucción de Entradas Desconocidas con Grado Relativo Arbitrario: Aplicación a Procesos Químicos Esteban López Aguirre Chemical Engineer Thesis presented as partial requirement for the degree of: Master of Engineering - Chemical Engineering Advisor: Héctor Antonio Botero Castro, Ph.D. Grupo de Investigación en Procesos Dinámicos - KALMAN Universidad Nacional de Colombia Facultad de Minas Departamento de Procesos y Energía Medellín, Colombia 2015

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5 To Ligia Henao and Heroína Delgado: Two shining stars in the sky above, Two true examples of strength and love.

6 Acknowledgments Firstly, I would like to express my gratitude to my advisor, Professor Héctor Botero, for his constant support and guidance; without his valuable insight and his kind disposition to help me through his expertise, this work would not have been possible. Also, I am in great debt to my other mentors, Professors Lina Gómez and Hernán Álvarez, and my fellow researchers at the Dynamical Processes Research Group - KALMAN ; their encouragement, constructive comments, and assessment were fundamental in the development of this thesis. In addition, I wish to thank my dear friends Cristian, Lina, Jorge, Mateo, Julián, and Felipe for the gift of their company, which allowed me to clear my mind when necessary and to keep motivated. Finally, I am immensely grateful to my parents María Leticia and Marino and to my sister Carolina; they are the foundation upon which I have constructed all the achievements in my life, and their unconditional love was always a source of fortitude throughout the educational process that has led to the formulation of this thesis.

7 vii Abstract This work proposes a sliding mode observer for the estimation of the state variables and the reconstruction of the unknown inputs of nonlinear dynamical systems regardless of whether or not these systems satisfy relative-degree-related conditions. A chronological review of the evolution of sliding mode techniques for state estimation is presented and the two main approaches for the design of sliding mode observers are described. The mathematical tools, including a Lyapunov-like theorem for fixed-time attraction, a high order sliding mode differentiator, and an original method for the dynamical solution of time-varying algebraic equations, that are necessary to carry out this work are appropriately described. In addition, two approximate methods to assess if the proposed observer produces accurate estimates of the variables of a given system are presented. The use of the proposed estimator is illustrated by simulating its application to several chemical and biochemical systems, to which an extended Kalman filter is also applied for comparison purposes. This comparison leads to the conclusion that, in general, the devised sliding observer presents a significantly better performance. Finally, simulations involving situations of particular interest for the developed observer such as non-smoothness of the unknown inputs, presence of noise in the output, and coupling with a controller are presented. Keywords: Sliding mode observers, unknown input reconstruction, fixed-time stability, zero-finding methods.

8 viii Resumen Este trabajo propone un observador por modos deslizantes para la estimación de las variables de estado y la reconstrucción de las entradas desconocidas de sistemas dinámicos no lineales independientemente de si estos sistemas satisfacen o no condiciones relacionadas con el grado relativo. Se presenta una revisión cronológica de la evolución de las técnicas por modos deslizantes para la estimación de estado y se describen los dos enfoques principales para el diseño de observadores deslizantes. Las herramientas matemáticas, incluyendo un teorema tipo Lyapunov para la atracción en tiempo fijo, un diferenciador por modos deslizantes de alto orden y un método original para la solución dinámica de ecuaciones algebraicas variantes en el tiempo, que son necesarias para llevar a cabo este trabajo son descritas apropiadamente. Además, se presentan dos métodos aproximados para evaluar si el observador propuesto produce estimativos exactos de las variables de un sistema dado. El uso del estimador propuesto se ilustra mediante la simulación de su aplicación a varios procesos químicos y bioquímicos, a los cuales también les es aplicado un filtro de Kalman extendido con propósitos comparativos. Dicha comparación lleva a la conclusión de que, en general, el observador deslizante ideado presenta un desempeño significativamente superior. Finalmente, simulaciones que involucran situaciones de particular interés para el observador desarrollado tales como entradas desconocidas no suaves, presencia de ruido en la salida y acoplamiento con un controlador son presentadas. Palabras clave: Observadores por modos deslizantes, reconstrucción de entradas desconocidas, estabilidad en tiempo fijo, métodos para hallar raíces.

9 Contents List of Figures List of Tables List of Symbols and Abbreviations xii xiii xiv 1 Introduction Motivation Research Problem Objectives General objective Specific objectives Notation Publications Thesis Outline Sliding Modes in State Estimation A Historical Review Most Common Types of Sliding Mode Observers Luenberger-like sliding mode observer Sliding mode observer based on transformation to triangular form Chapter Summary Mathematical Tools Extended Kalman Filter Finite-Time Stability Sliding Mode Differentiator Sliding Mode Solver for Systems of Time-Varying Algebraic Equations Chapter Summary Proposed Sliding Mode Observer Description of The Proposed Observer Methods for Assessment of Convergence A method based on linearization ix

10 x Contents A method based on multiple simultaneous implementation Chapter Summary Simulations Batch Reactor Continuous Bioreactor Continuous Stirred Reactors in Series Simulations Under Adverse Circumstances Non-smooth unknown input Noisy output Application to Automatic Control Chapter Summary Conclusions and Future Work Conclusions Future Work References 78

11 List of Figures 3.1 Differentiation of φ(t) = sin(t) + cos(t) Suggested value of γ as a function of the initial conditions Solution of (3.24) Proposed observer scheme Value of the index (4.26) as a function of time during the application of the proposed SMO to system (4.27) State estimation and unknown input reconstruction for system (4.27) using the proposed SMO Value of the index (4.26) as a function of time during the application of the proposed SMO to system (4.29) State estimation and unknown input reconstruction for system (4.29) using the proposed SMO State estimation and unknown input reconstruction for system (4.31) using the proposed SMO with three different sets of initial conditions State estimation and unknown input reconstruction for system (4.36) using the proposed SMO with three different sets of initial conditions Illustration of a jacketed batch reactor State esatimation and unknown input reconstruction for a batch reactor Illustration of a continuous bioreactor Value of the index (4.26) as a function of time during the application of the proposed SMO to system (5.6) State esatimation and unknown input reconstruction for a bioreactor Illustration of two continuous stirred reactors in series Value of the index (4.26) as a function of time during the application of the proposed SMO to system (5.12) State esatimation and unknown input reconstruction for a system of two continuous stirred reactors in series State esatimation and unknown input reconstruction for a CSTR affected by a non-smooth unknown input Temperature inside a CSTR and its noisy measurement xi

12 xii List of Figures 5.11 State esatimation and unknown input reconstruction for a CSTR with noisy temperature measurements ITAE index for the state estimation of a CSTR with different standard deviations for the output noise Concentraction control, state esatimation, and unknown input reconstruction for a CSTR Control action for the concentraction control in a CSTR

13 List of Tables 5.1 Parameters for the batch reactor model ITAE index for the state estimation of a batch reactor Parameters for the bioreactor model ITAE index for the state estimation of a continuous bioreactor Parameters for the model of two continuous stirred reactor in series ITAE index for the state estimation of a system of two continuous stirred reactors in series ITAE index for the state estimation of a CSTR affected by a non-smooth unknown input ITAE index for the state estimation of a CSTR with noisy temperature measurements IAE index for the regulation error of several simulations of the CSTR system 69 xiii

14 List of Symbols and Abbreviations Latin Letters Symbol A C d f F g G h h j H Hj i I n I L L i f m M n p q P P 0 Description Linearized system matrix Linear or linearized output matrix Vector of unknown inputs Function that defines a dynamical system Function that defines a system of algebraic equations Input distribution matrix function Function of actual derivatives of the outputs Output function j-th component of the output function Function of theoretical derivatives of the outputs i-th time derivative of the j-th output as a function of the state and the time derivatives of the unknown inputs Identity matrix of size n Index for assessment of convergence Parameter in sliding mode differentiator Lie derivative operator of the i-th order along f Number of outputs Linear input distribution matrix Order of a dynamical system Number of unknown inputs Number of algebraic equations in a system Estimation error covariance matrix Initial condition of the estimation error covariance matrix xiv

15 xv Symbol P c P i q Q Q N r j rj, k R N t t 0 T T c T v v N V V 0 w ŵ w w N x X ˆx ˆx 0 y y j ŷ z Description IAE index for regulation error ITAE index for the estimation error of the i-th state variable Number of equations/variables in a system of algebraic equations Linearized error dynamics matrix for the degrees of freedom Process noise covariance matrix Relative degree of the j-th output Relative degree of the j-th output with respect to the k-th unknown input Measurement noise covariance matrix Time Initial time Settling-time function Upper bound on convergence time State transformation Vector of switching terms Measurement noise Lyapunov function candidate Initial condition of the Lyapunov function Vector of degrees of freedom Estimate of w Estimation error for w Process noise State vector Set of admissible states Estimate of the state vector Initial condition of the estimated state vector Output vector j-th output Estimate of the output vector Vector of unknown variables in a system of algebraic equations

16 xvi List of Symbols and Abbreviations Symbol z ẑ z 0 Z Description Solution to a system of algebraic equations Estimate of z Initial condition for the sliding mode algebraic solver Set of admissible values for unknown variables Greek Letters Symbol γ δ k λ i ν j ξ σ ζ i Description Parameter in fixed-time attraction theorem Highest derivative of the k-th unknown input considered in the estimation Tuning parameter in sliding mode differentiator Number of derivatives of the j-th output used in the state estimation Transformed state vector Sliding variable Estimate of the i-th derivative in the sliding mode differentiator Subindices Symbol eq Description Equivalent signal Abbreviations Abbreviation CSTR EKF IAE Meaning Continuous stirred tank reactor Extended Kalman filter Integral of absolute error

17 xvii Abbreviation ITAE MIMO PI SISO SMO Meaning Integral of time-weighted absolute error Multiple-inputs multiple-outputs Proportional-integral Single-input single-output Sliding mode observer

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19 1 Introduction 1.1 Motivation In the particular case of Control Theory applied to chemical processes, there are plenty of important variables that are very difficult to measure in real-time; some notable examples are concentrations inside chemical reactors, crystal size in crystallization operations, average molecular weight in polymerization reactions, moisture content in drying processes, and phase compositions in distillation towers. If these variables are not measured online, they cannot be monitored or controlled; because of this, the desired products may fail to satisfy the necessary quality specifications. Moreover, if these unsupervised variables get outside their safety ranges, damages and accidents may occur (Isermann, 2011). This problem may be approached by using conventional state estimators that assume the available system model is perfect and faithfully represents the real behavior of the process. Nonetheless, this assumption is rarely close to be valid in practice. In most cases, during the modeling process necessary simplifications are made in order to produce a model that is feasible and practical. In addition to this conscious simplifications, there often exist unidentified phenomena that are not taken into account while modeling and generate uncertainty. Another situation than hinders the correspondence of the model with reality is the presence of external disturbances, which are scenarios outside the foreseen operating conditions or changes in variables that affect the dynamical behavior of the process and cannot be measured. Finally, parametric changes, which are common in chemical processes due to variations of the properties of substances and materials over time, represent an additional difficulty for a model to predict the real evolution of a system. In the presence of situations like the ones described above, a software sensor based on a model that does not consider uncertainties usually delivers incorrect estimates for the state variables of the system; therefore, the development of state estimators that are robust against unknown inputs is of great interest. These unknown inputs are variables that affect the system (generally modifying the model through a structure that is assumed to be known) and may group different kinds of uncertainties. Besides, estimating the value of these unknown inputs is useful for fault detection, model assessment, parameter fitting, and identification of unknown dynamics. 1

20 2 1 Introduction Being aware of these facts and the advantages of sliding mode techniques, the scientific community has proposed and studied sliding mode observers with the ability to estimate the state variables of a dynamical system in the presence of uncertainties (Floquet and Barbot, 2007; Fridman et al., 2008; Martínez-Guerra et al., 2004; Walcott and Zak, 1986; Xiong and Saif, 2001). However, continuing to advance in the overcoming of the obstacles faced when handling unknown inputs in state estimation is of great academic and practical interest to Chemical Engineering and Control Theory. 1.2 Research Problem Information is certainly a valuable asset in the practice of engineering; it is fundamental for making decisions, generating reports, detecting possible problems, and developing solutions. Therefore, every engineer constantly faces the necessity of handling, finding, and processing information. Lately, this concept has acquired such a level of significance that the classic paradigm of industrial processes has been modified to include information processing in addition to matter and energy transformations (Álvarez et al., 2009). In particular, knowing the online values of the important variables in a process is highly desirable and, in many cases, indispensable for Chemical Engineering. This knowledge is used for monitoring (Botero and Álvarez, 2009; Martínez-Guerra et al., 2004; Dochain et al., 2009), fault detection and isolation (Floquet et al., 2004; Yan and Edwards, 2005; Nagesh and Edwards, 2014; Menon and Edwards, 2014; Ríos et al., 2014), and feedback control (Wang et al., 1997; Loza et al., 2013; Chalanga et al., 2014). In the best case scenario, all the variables of interest in an industrial process are measured online by means of very accurate physical sensors. Nonetheless, this ideal situation is usually precluded by several factors. In some cases, the cost of necessary sensors is prohibitive, while in other occasions the sensors for the properties of interest do not exist. It is also possible that the required measurement instruments exist but they are unreliable, are not suitable for the operating conditions, or provide delayed information (de Assis and Maciel-Filho, 2000; Oliveira et al., 2006; Mohd Ali et al., 2015). Although the impossibility to measure important variables is particularly prominent in chemical and biological processes, this obstacle is not exclusive to these fields but may affect any dynamical system. Consequently, Control Theory has approached this problem by implementing algorithms that seek to infer the behavior of the unknown variables based on the knowledge of the variables whose online values can be measured appropriately. These algorithms are called state estimators; furthermore, when the systems they are applied to are deterministic, they are termed state observers (Besançon, 2007; Botero, 2010). The union of the available physical sensors and the state estimation algorithm is called software sensor

21 1.2 Research Problem 3 (de Assis and Maciel-Filho, 2000). Some of the state observers reported in the literature are the Luenberger observer, the asymptotic observer, the adaptable observer, the high gain observer, and the sliding mode observer (SMO) (Botero, 2010). However, in many cases, these observers are based on models that are assumed to be perfect, which cause estimation errors when model uncertainties or disturbances are present. Among the aforementioned kinds of estimators, sliding mode observers have acquired great interest in the last decades due to their insensitivity to some uncertainties and their possibility to converge in a finite time interval (Spurgeon, 2008). Sliding mode applications to state estimation and automatic control are characterized by the use of discontinuous or non-smooth correction terms; although these terms are the key to the advantages of sliding mode techniques, they are also the source of their main disadvantage, which consists of small amplitude, high frequency oscillations in the response of the treated system known as chattering (Edwards and Spurgeon, 1994; Walcott and Zak, 1986). Current research efforts on this topic focus on chattering reduction and improvement of robustness (Ginoya et al., 2014; Utkin, 2013; Yang et al., 2013). Generally, the methods intended for robust state estimation based on sliding mode theory make use of uncertain models which are affected by unknown inputs through a known structure. The implemented observers are expected to be independent of these unknown inputs or to be able to estimate their values, thus achieving robustness against some uncertainties. However, there are considerable structural restrictions to the kinds of unknown inputs conventional sliding mode techniques can deal with. Some of the proposed siding mode observers for systems with unknown inputs require the satisfaction of the matching condition (Shtessel et al., 2014). This condition demands that the unknown inputs act upon the outputs in such a way that it is possible to match them to the output error injection term in order to cancel their effect on the estimation error dynamics. This condition is mathematically defined for linear systems and, in the singleinput single-output (SISO) case, it is equivalent to a requirement that asks for the relative degree of the unknown input with respect to the output to equal 1. Another tendency, focused on nonlinear systems, has developed sliding mode observers for dynamical systems in triangular observable form (Barbot et al., 1996; Sánchez-Torres et al., 2011a), which, in the absence of known inputs, corresponds to a cascade of integrators where the output is the first state variable. When an unknown input is considered, it can only affect the last differential equation (Floquet and Barbot, 2007). These methods are ideal for simple mechanical systems since they take this triangular form naturally; nevertheless, to

22 4 1 Introduction apply these ideas to other kinds of systems, it is necessary to employ a state transformation to put them into triangular observable form, which requires the relative degree to equal the order of the system. This imposes a restriction as strong as the matching condition; furthermore, if a state transformation is used, the estimation is not carried out explicitly for the original state variables and it is necessary to apply the inverse transformation to obtain them, something that is rarely an easy task. Although efforts to relax the previously described conditions have been carried out (Barbot et al., 2009; Floquet and Barbot, 2006; Fridman et al., 2008; Xiong and Saif, 2001), the resulting methods are based on state transformations that may be complex, difficult to find, and difficult to invert. 1.3 Objectives General objective To develop a sliding mode observer for deterministic nonlinear dynamical systems that, besides estimating the the value of the unmeasured state variables, permits to reconstruct their unknown inputs regardless of their relative degree or whether they satisfy or not the matching condition Specific objectives 1. To identify the most used techniques for the state estimation of dynamical systems. 2. To describe the advantages and disadvantages of the use of sliding mode techniques in state estimation. 3. To characterize the sliding mode observers that have been applied to uncertain dynamical systems in the literature. 4. To describe the mathematical tools useful in the formulation of the proposed technique. 5. To propose a nonlinear sliding mode observer for state estimation and reconstruction of unknown inputs. 6. To validate the performance of the proposed technique through the simulation of its application to chemical systems and the comparison with a conventional state estimator.

23 1.4 Notation Notation When reading this dissertation, the following items regarding notation should be considered: Vector and matrix variables are displayed using bold face type, although regular mathematical type is used for vector and tensor fields. The operator corresponds to the Euclidean norm. The symbol 0 represents either the zero vector or the zero matrix of dimensions that are easily determined from the context, and the symbol 0 m n corresponds to the zero matrix with a specified size of m n. Given any function f of the form f(z 1, z 2,..., z 3 ) that may be evaluated at the estimated variables ẑ 1, ẑ 2,..., ẑ 3, the notation for the Jacobian matrix f z i is reduced to z1 f ẑ =ẑ 1,z 2 =ẑ 2,...,z 3 =ẑ i for the sake of simplicity Publications The following article, based on preliminary results obtained in the process of devising this thesis, has been submitted to an international journal: López, E. and Botero, H. Finite-time sliding mode observer for uncertain nonlinear systems based on a tunable algebraic solver. International Journal of Robust and Nonlinear Control. (Pending submission of revised version). 1.6 Thesis Outline This thesis is structured as follows: Chapter 2 aims to provide a brief chronological review of state estimation through sliding mode techniques along with a general description of the most common sliding mode observers for systems with unknown inputs. The mathematical tools necessary to fulfill the objectives of this work, including an original sliding mode method for the solution of systems of time-varying algebraic equations, are presented in Chapter 3. Chapter 4 introduces the main contribution of this dissertation, which is a sliding mode observer for the state estimation and unknown input reconstruction of systems that do not necessarily satisfy the conditions usually requested by other observers in the literature. The application of the proposed observer to chemical and biochemical processes is simulated in Chapter 5, where a comparison with a conventional state estimation technique is also carried out. In Chapter 6 the conclusions of this work are presented and some recommendations for possible future work are provided.

24 2 Sliding Modes in State Estimation The purpose of this chapter is to provide a review of the the use of sliding mode techniques in state estimation, creating the background necessary to characterize the disadvantages of the current methods whose overcoming will eventually become the the contributions of this work. Section 2.1 consists of a brief chronological review of the evolution of sliding mode observers. Section 2.2 describes the two most common sliding mode techniques used to estimate the state of uncertain dynamical systems along with the restrictions they impose. 2.1 A Historical Review In the Mid-Twentieth Century, a control strategy that used variable structure systems was started to be investigated in the Soviet Union (Utkin, 1992). This strategy implemented control laws that were discontinuous with respect to the state variables so that controllers would switch among different structures. In the decade of 1960 a particular kind of variable structure control system, known as sliding mode control, emerged (Spurgeon, 2008). Sliding mode techniques for automatic control acquired great interest due its insensitivity to a certain class of uncertainties and disturbances (Utkin, 1977; Draženović, 1969). When such techniques are used, once the dynamical system has reached the region of the state space where the control law is discontinuous, the control signal switch among structures with a theoretically infinite frequency and forces the state trajectory to remain in this region, which is called the sliding manifold. This ability to confine the system to a particular subset of its state space provides significant advantages (Sabanovic et al., 2004), which have been widely exploited and continue to be researched (Shtessel et al., 2014). At first, the most studied application of sliding mode techniques was automatic control. However, in the decade of 1980 the pioneer work of Utkin (1981), Slotine et al. (1986, 1987), and Walcott and Zak (1986) paved the way for a promising research field by inquiring into the application of sliding modes to state estimation. These first explorations considered systems with a known linear part and possible nonlinearities or disturbances grouped as uncertainties (Edwards and Spurgeon, 1994); nevertheless, sliding mode observers did not take long to be applied to certain classes of nonlinear systems, even achieving convergence in finite time for all the estimates of the state variables (Drakunov, 1992; Barbot et al., 1996; Drakunov and Utkin, 1995). 6

25 2.2 Most Common Types of Sliding Mode Observers 7 With the passage of time, the study of sliding observers became popular among a part of the scientific community and, although some applications to chemical processes exist Wang et al. (1997); Martínez-Guerra et al. (2004), most researchers focused on mechanical and electromechanical systems (Kao and Moskwa, 1995; Chen and Moskwa, 1997; Bartolini et al., 2003; Aurora and Ferrara, 2007). The low interest in the application of these techniques to chemical systems was inherited from sliding mode control, which did not find a niche among systems of this type since their typical actuators would not be able to reproduce discontinuous control signals. Just like sliding mode controllers, sliding mode observers may be designed to be insensitive to certain types of unknown inputs so that they deliver the true values of the state variables even in the presence of some uncertainties or disturbances. Due to these robustness qualities, sliding mode observers were, and continue to be, used for fault detection and isolation (Edwards and Spurgeon, 2000; Edwards et al., 2000; Yan and Edwards, 2005; Nagesh and Edwards, 2012, 2014). Nonetheless, in order for sliding mode observers to possess this insensitivity to unknown inputs, the treated uncertain systems must routinely satisfy significantly restrictive requirements such as the observer matching condition (Floquet and Barbot, 2006) or relative degree conditions (Barbot et al., 2009). With the objective of bypassing these restrictions, Xiong and Saif (2001) and Fridman et al. (2008) proposed to use a state transformation defined by the Lie derivatives of the outputs that do not involve the unknown inputs and some other function that can be proven to exist, but that are not constructively given. In addition, Floquet and Barbot (2006) and Barbot et al. (2009) developed constructive algorithms that, in some cases, permit to find a state transformation that converts the system into a set of triangular blocks which are suitable for the application of previously designed observers intended for systems with this structure. However, the only way to find whether these algorithms work or not for a particular system is to follow them until the corresponding conclusion is reached at some step. 2.2 Most Common Types of Sliding Mode Observers In this section, the two main currents of sliding mode observer design are described. The first one adds discontinuous correction terms to the model of the system and requires the satisfaction of a matching condition to be able to be insensitive to the unknown inputs, while the second one is based on the transformation of the system into a series of triangular blocks and imposes conditions on the relative degree of the outputs in order for these blocks to be independent of the unknown inputs. Both kinds of observers are designed for a system of the form

26 8 2 Sliding Modes in State Estimation ẋ = f(x) + g(x) d y = h(x) (2.1) where x X R n is the state vector, y R m is the output vector, d R p is the vector of unknown inputs, f : X R p R n, g : R n R n p, and h : R n R m are sufficiently smooth functions, and X is the set of admissible states. The objective of the observers is to produce an estimated state vector ˆx that converges to the real state x of the system Luenberger-like sliding mode observer One of the first observers for deterministic systems was proposed by Luenberger (1964, 1966). Its working principle consists of adding output error injection terms to the model of the system evaluated at the state estimates. The most simple sliding mode observers mimic this principle but they use injection terms that are discontinuous functions of the output error which seek to drive the sliding variable σ = ŷ y to zero. Here, ŷ = h(ˆx) is the estimated output vector. The application of of this kind of observer to system (2.1) takes the following form: ˆx = f(ˆx) + K v (2.2) where K is a gain matrix which may be variable and v is a vector of switching terms. A possible choice for v is ρ 1 sign(ŷ 1 y 1 ) ρ 2 sign(ŷ 2 y 2 ) v =. ρ m sign(ŷ m y m ) (2.3) where ŷ j and y j are the j-th components of ŷ and y respectively and the gains ρ j are chosen so that the system satisfies some reachability condition that ensures sliding motion on the manifold σ = 0 is achieved (Shtessel et al., 2014). Sometimes, a linear output error injection term is also included in observer (2.2) with the purpose of enlarging the set of initial conditions for which convergence to the sliding manifold is guaranteed (Shtessel et al., 2014; Spurgeon, 2008); however, such linear feedback has no effect on the observer dynamics once sliding mode is attained and is not considered in this discussion for the sake of simplicity. Let x = ˆx x be the estimation error for the state vector. Then, subtracting (2.1) from (2.2) yields the following error dynamics:

27 2.2 Most Common Types of Sliding Mode Observers 9 x = f(ˆx) f(x) g(x) d + K v (2.4) Once the sliding manifold is reached, the action of v becomes equivalent to that of a certain signal v eq (Shtessel et al., 2014) and the derivative of the sliding variable σ = ẏ ŷ vanishes; then it follows that h ˆx ˆx h x ẋ = 0 (2.5) Consequently, the equivalent output error injection signal v eq must satisfy h ˆx (f(ˆx) + K v eq) h (f(x) + g(x) d) = 0 (2.6) x Assuming h/ ˆx is full rank for all ˆx X, an appropriate selection of the gain matrix K allows ( h/ ˆx) K to be invertible so that (2.6) can be solved for v eq to produce v eq = ( ) 1 [ ] h h ˆx K h (f(x) + g(x) d) x ˆx f(ˆx) (2.7) Substituting this expression for v eq into (2.4) and replacing f(ˆx) f(x) by f to save notation yields ( ) 1 [ ] h h x = f g(x) d + K ˆx K h (f(x) + g(x) d) x ˆx f(ˆx) (2.8) Equation (2.8) corresponds to the estimation error dynamics when the system is in sliding motion and it can be rearranged as ( ) 1 [ ] [ ( ) ] 1 h h x = f + K ˆx K h h f(x) x ˆx f(ˆx) I n K ˆx K h g(x) d (2.9) x where I n is the n n identity matrix. The conventional approach to robustness in sliding mode techniques requires the error dynamics to become independent of the uncertainties on account of the equivalent correction signal that acts while the system sustains sliding mode. In this case, for the error dynamics (2.9) to be insensitive to the unknown inputs, the following is necessary: From (2.10) it follows that [ ( ) ] 1 h I n K ˆx K h g(x) = 0 x X (2.10) x

28 10 2 Sliding Modes in State Estimation ( ) 1 ( ) h h g(x) = K ˆx K x g(x) x X (2.11) It is a well-known property of matrix product that if A, B, and P are matrices such that P = A B, then rank(p) rank(a) and rank(p) rank(b). According to this, if (2.11) is satisfied the following must be true: [ ] h rank [g(x)] rank x g(x) x X (2.12) Moreover, ( h/ x)g(x) is in turn the product of some matrix and g(x), which implies [ ] h rank x g(x) rank [g(x)] (2.13) From (2.12) and (2.13) it is obvious that a necessary condition for the estimation error dynamics of this kind of observer to be invariant with respect to the unknown inputs is [ ] h rank [g(x)] = rank x g(x) x X (2.14) Equation (2.14) constitutes the so-called matching condition for state estimation. Roughly, this condition requires the unknown inputs to act in the channels of the outputs so they can be matched by the discontinuous error injection terms and canceled by their equivalent signals. If an uncertain dynamical system meets this condition, its unknown inputs are said to be matched; on the other hand, if this requirement is not satisfied, the unknown inputs are called unmatched. In the SISO case, the matching condition is equivalent to requesting the relative degree of the system to equal one. Nevertheless, when there are multiple inputs and outputs (MIMO), if less than p outputs have a relative degree equal to one, this condition cannot be satisfied, while if at least p of the m outputs possess a relative degree equal to one, it may or may not be fulfilled. When system (2.1) is such that there are constant matrices, C R m n, and M R n p that satisfy y = h(x) = C x and g(x) = M, the matching condition takes the form rank (M) = rank (C M) (2.15) which is the most common form to express it since this technique is mainly applied to linear systems.

29 2.2 Most Common Types of Sliding Mode Observers 11 Although the matching condition is necessary to apply this kind of observer to uncertain systems, it is worth noting that its satisfaction does not guarantee convergence to the true state variables; even if the observer can be designed so that (2.10) is true, the rest of Equation (2.9) still has to constitute an asymptotically stable system for the error x to converge to zero Sliding mode observer based on transformation to triangular form More recent works have proposed sliding mode observers whose estimates converge to the actual state variables in finite time. These observers are usually based on a state transformation that permits to express the treated system as a set of subsystems in triangular form (Floquet et al., 2004; Floquet and Barbot, 2007; Shen et al., 2010). This paradigm is described in this subsection. Let the output and the unknown inputs distribution matrix of system (2.1) be such that y 1 h 1 (x) y 2 h 2 (x) y = = h(x) =.. h m (x) y m (2.16) g(x) = [g 1 (x), g 2 (x),..., g p (x)] (2.17) where g k (x) R n for all k {1, 2,..., p} and y j = h j (x) R for all j {1, 2,..., m}. Assume each output y j has a well-defined relative degree r j for all x X, that is, for all j {1, 2,..., m} L gk L i fh j (x) = 0 x X k {1, 2,..., p} i < r j 1 L gk L r j 1 f h j (x) 0 x X for at least one k {1, 2,..., p} (2.18) where L i f h j(x) is the i-th Lie derivative of h j (x) along f(x), defined by L 0 fh j (x) = h j (x) L i fh j (x) = Li 1 f h j f(x) x (2.19) These Lie derivatives are also used to define the following state transformation:

30 12 2 Sliding Modes in State Estimation ξ 1 T 1 (x) ξ 2 T 2 (x) ξ = = T (x) =.. T m (x) ξ m (2.20) where the ξ j = T j (x) are defined by ξ 1 j ξ ν j j ξ 2 ξ j = = T j (x) = j. h j (x) L f h j (x). L ν j 1 f h j (x) j {1, 2,..., m} (2.21) for some positive integers ν 1, ν 2,..., ν m such that ν 1 + ν ν m = n. It is desired that T (x) does not depend on d so that, in the new coordinates, the system takes the form of a series of m triangular blocks where the unknown input may only affect the last differential equation of each block. In that case, the j-th triangular block would be given by ξ j = y j = ξ 1 j ξ 1 j ξ 2 j. ξ ν j 1 j ξ ν j j ξ 3 = j. ξ ν j j L ν j f h j(t 1 (ξ)) + p k=1 L g k L ν j 1 f h j (T 1 (ξ)) d k ξ 2 j (2.22) For this to be possible, that is for T (x) to be independent of d, the following condition must be met: ν j r j j {1, 2,..., m} (2.23) This restriction favors high relative degrees, as opposed to the matching condition. In particular, for SISO systems, condition (2.23) asks for the relative degree to equal n. Notice that, for the same system, some sets of integers ν 1, ν 2,..., ν m may meet this specification and some others may not. However, there are systems for which no set ν 1, ν 2,..., ν m that satisfies (2.23) exist; the state variables of such systems cannot be accurately estimated through this

31 2.2 Most Common Types of Sliding Mode Observers 13 method. In addition to this relative degree restriction, it is clear that the transformation T must be invertible in X in order to recover estimates for x from estimates for ξ. Then, assuming (2.23) is fulfilled, the Jacobian matrix T / x is required to be such that ( ) T rank = n x X (2.24) x This actually corresponds to an observability condition, since T / x is clearly the nonlinear observability matrix. If conditions (2.23) and (2.24) are satisfied, the following sliding mode observer, based on the work by Drakunov (1992), can be applied to the transformed system: ˆξ j 1 kj 1 sign(ψj 1 ˆξ j 1 ) ˆξ j = ˆξ 2 kj = j. 2 sign(ψj 2 ˆξ j 2 ). ˆξ ν j k ν j j j sign(ψ ν j j ˆξ ν j j ) j {1, 2,..., m} (2.25) In this expression, ˆξ j = [ˆξ 1 j, ˆξ 2 j,..., ˆξ ν j j ]T is the estimate of ξ j, the k i j are scalar gains and the ψ i j are given by ψ 1 j = y j = ξ 1 j ψj i = {k i 1 j sign(ψ i 1 i 1 j ˆξ j )} eq i 2 (2.26) where the { } eq operator produces the equivalent signal (Shtessel et al., 2014) of the switching term given as its argument. If the necessary equivalent signals are ideally known and the k i j gains, which may be variable, are such that kj i > ξ i+1 j = L i f h j (x) i {1, 2,..., νj 1} j {1, 2,..., m} k ν j j > Lν j f h j(x) + p k=1 L gk L ν j 1 f (2.27) h j (x)) d k j {1, 2,..., m} then, through arguments similar to those by Drakunov (1992), it can be proved that ˆξ j converges to ξ j in finite time for all j {1, 2,..., m}. If ˆx and ˆξ correspond to the estimated state vectors of the original and the transformed systems respectively, doing

32 14 2 Sliding Modes in State Estimation ˆx = T 1 ( ˆξ) (2.28) makes ˆx converge to x in finite time. However, in practice, the equivalent signals are usually extracted through low-pass filters, which converge asymptotically. Although this precludes a rigorous finite time convergence of the whole observer scheme, the effect of this non-ideality is commonly negligible due to quick convergence of the low-pass filters. 2.3 Chapter Summary In this chapter, a historical and practical review of state estimation via sliding mode techniques has been provided. The major milestones in the evolution of sliding mode observers have been pointed out, indicating the fundamental advantages and disadvantages of the resulting methods. In addition, two observers that are representative of the main paradigms in sliding mode estimation have been described in more detail, formally identifying the restrictions they impose on the structure of the systems they can be applied to. One of this restrictions ask for the outputs to have a small relative degree, while the other one requires them to have a large relative degree. The next chapter introduces some mathematical tools that will be useful in achieving the central objectives of this work.

33 3 Mathematical Tools This chapter describes the mathematical tools necessary to devise the sliding mode observer this thesis intends to propose. Section 3.1 introduces the conventional state estimator that will be compared with the sliding mode observer to be developed. The main purpose of Section 3.2 is to provide a Lyapunov-like theorem for fixed-time stability. Section 3.3 describes a robust high order sliding mode differentiator. Finally, Section 3.4 presents an original sliding mode technique for the solution of systems of time-varying algebraic equations, which is one of the contributions of this work and will constitute a central part of the pursued observer. 3.1 Extended Kalman Filter The specific objectives of this work express that the sliding mode observer that will be proposed must be compared with a conventional state estimator. The selected conventional estimator is the extended Kalman filter (EKF), whose continuous version is described in this section. This algorithm was chosen because it is extensively used in academic literature and it may possess a certain degree of robustness through the use of fictitious process noise (Simon, 2006). Consider a stochastic dynamical system of the form ẋ = f(x, w N ) y = h(x) + v N (3.1) where x R n is the state vector, y R m is the output vector, f : X R p R n and h : R n R m are known functions, w N R p and v N R m are assumed to be Gaussian white noises with zero mean and with covariance matrices given by Q N R p p and R N R m m respectively. Let the variable matrices A R n n, C R m n, and L R n p be defined as A = f ˆx C = h ˆx 15 (3.2) (3.3)

34 16 3 Mathematical Tools L = f w N (3.4) x=ˆx Then the continuous extended Kalman filter for system (3.1) is given by (Simon, 2006) ˆx = f(ˆx, 0) + P C T R N 1 [y h(ˆx)] Ṗ = A P + P A T + L Q N L T P C T R N 1 C P (3.5) where P is the estimation error covariance matrix. The initial conditions ˆx(t 0 ) = ˆx 0 and P(t 0 ) = P 0 for estimator (3.5) should be chosen so that ˆx 0 is the expected value of x(t 0 ) and P 0 is the expected value of [x(t 0 ) ˆx 0 ][x(t 0 ) ˆx 0 ] T. 3.2 Finite-Time Stability One of the essential characteristics of sliding mode techniques is that the sliding manifold is reached in finite time. Because of this, Lyapunov-like theorems have been developed in order to evaluate finite-time stability (Haddad et al., 2008). These theorems are commonly applied as reachability conditions to determine whether a system enters sliding motion or not or to design it in a way that ensures it does (Shtessel et al., 2014). In this work, the stronger notion of fixed-time stability is of particular interest. This kind of stability implies the existence of a bound on the convergence time that is independent of the initial conditions of the system. This section focuses on the definitions of finite-time attractive and fixed-time attractive sets and on a theorem that provides sufficient conditions to verify the latter for a dynamical system of the form ẋ = f(x, t) (3.6) Definition 3.1. (Polyakov, 2012) Let S be a non-empty subset of R n. S is said to be globally finite-time attractive for system (3.6) if, for any initial condition x(t 0 ) = x 0 R n, the solution x(t) reaches S at some finite time t = T (x 0 ) and remains there for all t T (x 0 ), where T : R n [t 0, ) is the settling-time function. Definition 3.2. (Polyakov, 2012; Sánchez-Torres et al., 2015) Let S be a non-empty subset of R n. S is said to be globally fixed-time attractive for system (3.6) if it is globally finite-time attractive for the system with a settling-time function T : R n [t 0, ) that is bounded by a constant; that is, there exists a positive constant T max such that T (x 0 ) T max for all x 0 R n.

35 3.2 Finite-Time Stability 17 Theorem 3.1. (Sánchez-Torres et al., 2015) If there exists a continuous radially unbounded function V : R n R such that (i) V (x) 0 for all x R n (ii) V (x) = 0 if and only if x S (iii) For some T c > 0 and some γ (0, 1], its time derivative V = ( V/ x)f(x) satisfies V 1 γ T c exp (V γ ) V 1 γ x R n (3.7) Then, S is globally fixed-time attractive for (3.6) with T max = t 0 + T c. Proof. Equation (3.7) is easily solved through separation of variables; letting V (x(t 0 )) = V 0, its solution is V (x(t)) [ ( )] 1/γ t t0 ln + exp( V γ 0 ) (3.8) T c Assume τ is such that [ ( )] 1/γ τ t0 ln + exp( V γ 0 ) = 0 (3.9) T c Solving (3.9) for τ produces τ = t 0 + T c (1 exp( V 0 γ )) (3.10) Then, the right-hand side of (3.8) reaches zero at t = τ = t 0 + T c (1 exp( V 0 γ )). Therefore, there exists a time t c such that t 0 t c τ and V (x(t)) reaches zero exactly at t = t c. Since V only takes non-negative values, the expression [ exp (V γ ) V 1 γ ]/(γ T c ) can only take non-positive values. In consequence, once V (x(t)) reaches zero at t = t c, it remains zero thereafter. Equivalently, x(t) S for all t t c. In addition, given that V (x) is radially unbounded and (3.7) holds for all x R n, this is true for any initial condition x(t 0 ) = x 0. From the previous discussion, it is clear that S is globally finite-time attractive for (3.6) with a settling-time function given by T (x 0 ) = t 0 + T c (1 exp( V (x 0 ) γ )) (3.11) Furthermore, since 0 < exp( V 0 γ ) 1, T (x 0 ) is such that

36 18 3 Mathematical Tools T (x 0 ) t 0 + T c (3.12) Then, S is globally fixed-time attractive for (3.6) with a settling-time bound given by T max = t 0 + T c. This theorem, which was developed by Sánchez-Torres et al. (2015), presents a convenient condition for fixed-time stability if the parameter T c appears explicitly in system (3.6) and can be modified, since this provides a straightforward way to tune the system by setting a limit for its convergence time. 3.3 Sliding Mode Differentiator Real-time differentiation of a signal is a well-known problem that has found a satisfactory solution via sliding modes. There are two main factors that encouraged the development of differentiators within the scientific community that studies sliding mode techniques. In the first place, the implementation of higher order sliding mode control requires knowledge of the time derivatives of the sliding variable. Secondly, through transformation to a triangular form like the one described in Section 2.2, the state estimation problem is reduced to a differentiation problem. The main inconvenient of numerical differentiation is the difficulty to simultaneously achieve accuracy of the estimated derivatives and robustness in the presence of noise. However, the work by Levant (1998, 2003) overcomes this problem and introduces a high order sliding mode differentiator that delivers the exact derivatives under ideal conditions and possesses robustness with respect to noise and sampling. Such differentiation takes the following form (Levant, 2003): ζ 0 = v 0 v 0 = λ k L 1 k+1 ζ0 φ(t) k k+1 sign(ζ0 φ(t)) + ζ 1 ζ 1 = v 1 v 1 = λ k 1 L 1 k ζ1 v 0 k 1 k sign(ζ 1 v 0 ) + ζ 2. (3.13) ζ k 1 = v k 1 v k 1 = λ 1 L 1 2 ζk 1 v k sign(ζk 1 v k 2 ) + ζ k ζ k = λ 0 L sign(ζ k v k 1 )