Int. J. Automation and Control, Vol. 5, No. 2, 2011 119 On robust state estimation for linear systems with matched and unmatched uncertainties Boutheina Sfaihi* and Hichem Kallel Department of Physical Engineering and Instrumentation, National Institute of Applied Sciences and echnology, Centre Urbain Nord, B.P. N 676, UNIS CEDEX 1080, unisie E-mail: Boutheina.Sfaihi@isetr.rnu.tn E-mail: golden.k@gnet.tn *Corresponding author Abstract: his paper addresses the problem of robust observer design for linear dynamic systems with uncertain parametric state matri. wo class systems with non-bounded parametric uncertainties are considered for estimation: dynamic systems with matched conditions and dynamic systems with unmatched conditions. In this contet, a robust estimator with a second gain update output estimation error is developed to linear systems with matched uncertainties. he framework is generalised to linear systems with unmatched uncertainties. With lesser restrictive conditions, it is shown that the conceived estimator for a chosen nominal system is insensitive to non-bounded parameters variation and provides good performances. Analytical developments are detailed and adopted choices are justified. Upon satisfying some conditions, the convergence properties of the robust estimators are proved through Lyapunov method. Approaches validity are illustrated via detailed numerical eamples with etensive simulation results. Keywords: control; robust state estimation; linear systems; Lyapunov stability; matched uncertainties; unmatched uncertainties. Reference to this paper should be made as follows: Sfaihi, B. and Kallel, H. (2011) On robust state estimation for linear systems with matched and unmatched uncertainties, Int. J. Automation and Control, Vol. 5, No. 2, pp.119 133. Biographical notes: Boutheina Sfaihi received the Diploma in Engineer degree in 2004 and the Master degree in 2005, both in Automatic and Industrial Computing from the National Institute of Applied Sciences and echnology, INSA, unis, unisia. Currently, she is preparing the PhD in Automatic and Industrial Computing. Her research interests include robust control theory for linear and non-linear dynamic systems. Hichem Kallel received a PhD in Electrical Engineering from the Ohio State University in 1991, USA. Currently, he is a Professor in Electrical Engineering in the Department of Physics and Electrical Engineering at the National Institute of Applied Sciences and echnology, INSA, unis, unisia. His research interests concern non-linear systems control and the development of advanced control strategies and their applications in robotics. Copyright 2011 Inderscience Enterprises Ltd.
120 B. Sfaihi and H. Kallel 1 Introduction he theory of observers started with the work of Luenberger (1964, 1966, 1971) and according to Luenberger, any system driven by the output of the given system can serve as an observer for that system. One possible method for obtaining the state vector is to build a model of the given system, drive the model with the same inputs as the original system and use the state vector of the model as an approimation to the unknown state vector (Luenberger, 1966). his problem has been later generalised in various ways, and in relatively recent years there has been a great deal of research aiming at designing state observers for uncertain systems called robust observers. Essentially three ways of representing model uncertainties: noises, plant disturbances and modelling errors are in the literature and lead various approaches and results. In a stochastic way when dealing with noises, two techniques are essentially investigated in the field: H2 and H optimal filtering. Robust H2 filtering problems are communally developed in a Kalman filtering approach (Kalman and Bucy, 1961) where uncertain dynamic systems are subjected to white noise. he H2 filter is conceived in order to find filter parameters such that the worst case mean square estimation error is minimised. Petersen and McFarlane (1994), Xie et al. (1994), heodor and Shaked (1996) and Shaked and de Souza (1995) considered linear time-invariant systems with time-varying norm-bounded uncertainties in both the state and output matrices. de Souza and rofino (2000) considered a polytopic uncertainty model. Fu et al. (2001) and Dong and You (2006) considered in the finite horizon robust H2 filtering problem involving a norm-bounded uncertain block. Zhang et al. (2006) studied the Kalman filtering for continuous-time systems with delayed measurements. With robust H filtering technique, the filter is developed in order to minimise the worst case induced L2 gain from process noise to estimation error. An upper bound is derived and minimised using techniques based on Riccati equations or LMIs thereby. Xie and de Souza (1995), Xie et al. (1991) and Fu et al. (1992) considered systems with norm bounded unstructured uncertain dynamic systems and conceived H filter using a Riccati equation technique. Using LMI technique, Palhares and Peres (1998), Geromel (1999) and Gao et al. (2008) considered H filtering uncertain dynamic systems with polytopic parameter uncertainties. On the other hand, Li and Fu (1997) considered systems with integral quadratic constraints, and formulate the problem with matri inequalities. In a deterministic way, plant disturbances can be considered as unknown inputs. Many researches are developed in the field: Kudva et al. (1980) gave necessary conditions and provided a set of design formulas in generalised inverse form where Hou and Muller (1992), Darouach et al. (1994) and Yang and Wild (1988) gave a direct design procedure of full order observers. More recently, Xiong and Saif (2000) provided a new sliding mode observer for linear uncertain systems. In a development etension of unknown input observers, Xiong and Saif (2003) proposed two reduced-order input estimators built upon a state functional observer under less restrictive conditions than those of the previous work and rinh et al. (2008) introduced to a procedure for finding reduced-order scalar functional observers. Sundaram and Hadjicostis (2008) provided a characterisation of observers with delay. When dealing, in a deterministic contet, with robust estimation for uncertain systems with modelling errors, various formulations of the problem are developed. However, when considering state estimation as independent task of control there is no standard
On robust state estimation for linear systems 121 technique to solve the observation issue and several works developed this approach. In the 1970s, Jain (1975) considered the problem of guaranteed state estimation for uncertain systems over a finite time horizon using earlier methodologies developed in Bertsekas and Rhodes (1971) and Chang and Peng (1972). Akashi and Imai (1979) designed a robust observer in general multi-input multi-output discrete time linear systems, through the geometric approach developed by Wonham (1974). More recently, Savkin and Petersen (1994) considered robust state estimation for a class of uncertain systems involving structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint. In Gu and Poon (2001), the authors introduced an observer scheme derived by including an etra term and adopting the Lyapunov stability theorem with the algebraic Riccati equation solution, and in Sfaihi and Kallel (2009) a new technique of robust state estimation to linear systems with matched and non-bounded uncertainties is proposed. In this paper, we focus on robust estimation for the linear system with uncertain parametric state matri. A description of the system is presented in Section 2. In Section 3, we introduce to the robust observer scheme for systems with matched conditions. In Section 4, we develop a generalised robust observer technique to uncertain system with unmatched conditions. We prove the convergence proprieties of synthesised techniques through Lyapunov theory and in Section 5, we develop two illustrative eamples to validate the robust observer schemes. 2 Preliminaries and problem formulation hroughout this note, the following notations are adopted. n nm and denote the space of n-dimensional real vectors and m n real matrices, respectively. I n is the nn identity matri. A matri is said to be stable if its eigenvalues are all on the left-half open comple plane. denotes the Euclidean norm of a vector. Consider a class of linear uncertain parametric systems when the uncertainty is assumed to eist in the state matri. Dynamics system are given by t () Apt ( ) () But () yt () Ct () n m k where, u and y are, respectively, the state to be estimated, the control nn vector and the measured outputs. Ap ( ) with p P models the uncertainty in the nm kn system. B and C are all known constant matrices. he problem of designing a robust observer is to estimate system (1) states. Evidently, the solution of the problem depends on the type of assumptions made on uncertainty p() t. In this work, we consider an uncertainty non-bounded and unstructured in matched and unmatched conditions. Let us define the following observer ˆ() t Fˆ() t G u() t G y() t (2) 1 2 (1)
122 B. Sfaihi and H. Kallel nm where t ˆ( ) is the observer state vector, F is the state matri, G 1 is the nk input matri and G2 is the gain matri observer. Define the state estimation error as e() t t ˆ() t () (3) he state estimation error is governed by e () t ˆ () t () t (4) Our aim is to define the class of gain matrices F, G 1 and G 2 that guarantee the convergence of the estimation error e() t to zero for both observer schemes developed for system (1) with matched and unmatched uncertainties. he case when the matching condition is not satisfied is discussed in Section 4. We first assume that matching condition holds. nn 3 Robust observer with matched uncertainties Main result: we suppose that the uncertainty is in the range of output matri C as following A( p) A ( p) C (5) nn where A is a constant matri and ( p) is the matched uncertainty matri. Substituting (5) in (1) produces t () [ A( pct ) ] () But () yt () Ct () he following assumption is required for the design of observer (2). Assumption H1: For the uncertain part of the parametric system (6), we assume that where p0 ˆ ( ) p y p y p yˆ y 0 0 nk P is a nominal value of p and is a positive real scalar. Proposition 1: Let Assumption H1 holds and considers the dynamic robust observer (2) with F A p C G C (7) G1 0 2 B (8) 2 22 23 G G C G (9) (6)
On robust state estimation for linear systems 123 he estimation error system (3) is asymptotically stable, if and only if the conditions below hold: 2 1 he matri A( /2) I G C 0 or the pair 2 A I 2 n, C is observable 2 he gain matri n 22 1 G23 p0 p0 2 (11) nk kk where G22, G23, p 0 is a nominal value of p and is a positive definite scalar. Proof: Considering the parametric linear system (6) and the estimator (2), the dynamics of the state estimation error (4) can be described by the equations e () t Fˆ Gu 1 G2y (( A( p) C) Bu) A( p) CG2Ce (12) F A( p) CG C ˆ G B u 2 1 In order to obtain an asymptotic estimation error convergence, we deduce from relation (12) the epressions of matrices gain F and G 1, respectively as F A( p0) CG2C and G1 B. he fied value p 0 is a nominal value of p chosen arbitrarily. From (12), it follows that e AG2C e p0 Cˆ ( p) C (13) o prove observer (2) stability, let us consider the Lyapunov function candidate V e, t e ( t) e ( t) (14) he time derivatives of Vt () along the trajectories of system (13) satisfy Vt () 2 e() te () t (15) By substituting (13) into (15) yields Vt () 2e AGCe 2 e p Cˆ ( pc ) 2 0 2 0 2e AG C e 2 e p Cˆ ( p) C Using assumption (H1), we can write ˆ ( ) 0 0 e p C p C e p C e (10)
124 B. Sfaihi and H. Kallel Consequently, the time derivative of Vt () becomes Since, Vt () 2e AGCe2 e p Ce 2 0 2 0 0 0 2e p C e e e e C p p Ce it follows that, Vt () 2e AGCe 2 ee ec p p Ce 2 0 0 In order to reduce the conservatism of this inequality, we introduce a second update gain G 23, such that G 2 G 22 C G 23. We make the following choice to write the below developed epression 2 Vt () 2e A In G22C e 2 (16) e C p p 2G Ce 0 0 23 he derivative of Lyapunov function Vt () is negative for any matrices gain G 22 and G 23 2 verifying A( /2) In G22C 0 and G 23 (1 / 2) ( p 0 ) ( p 0 ). herefore, V ( e, t) is negative definite, the error observer (3) is stable, and consequently the robust observer (2) is stable. he proposed conditions (10) and (11) provide estimator gains G 22 and G 23 which are always stabilising. he particular choice of a relaed gain form G 2 allows more freedom and fleibility in the robust observer conception. On other hand, the particular parameter imposed by assumption H1, is a positive scalar non-negligible that as seen in H1 absorbs uncertainties variation, and consequently imposes to G 22 to be a high gain matri. hen, an admissible stable estimator can be given by (2), and the proof is completed. 4 Robust observer with unmatched uncertainties Main result: assume, now, a general case with linear unmatched uncertainty as following 0 Ap ( ) Ap Lp ( ) (17) where p0 P is a nominal parameter and L( p) is the uncertain matri. In this case, the uncertain system (1) becomes t () Ap0 L( p) t () But () (18) yt () Ct () Under following Assumption H2, the main contribution is stated in Proposition 2 providing stability conditions for the estimation error (3). Assumption H2: We assume that L ˆL( p) ˆ N nn
where On robust state estimation for linear systems 125 N L A p A p (19) 1 0 ( p0, p1) P are nominal parameters construction with p0 p1 and is a positive real scalar. Proposition 2: Let Assumption H2 holds and considers the dynamic robust observer (7) with F A p L G C (20) G1 0 N 2 B (21) he estimation error system (4) is asymptotically stable, if and only if the condition A p I G C (22) 0 n 2 0 or the pair ( Ap ( 0) In, C) is observable, is satisfied. Proof: he stability robustness condition will be analysed by considering the following error dynamics substituted from uncertain system (22) and robust state estimator (2) 0 2 0 2 ˆ 1 e () t A p L( p) G C e () t F A p L( p) G C () t G B u() t In order to structure the robust observer (2), let us consider the certain matrices L N, F and G 1 defined respectively by LN Ap ( 1) Ap ( 0), F Ap ( 0) LN GC 2 and G1 B. he state estimation error dynamics (4) become e () t A p L( p) G C e () t L L( p) () t which imply 0 2 N ˆ 0 2 N ˆ e () t A p G C e () t L () t L( p) () t (24) By considering the Lyapunov function candidate V( e, ) t e ( t) e( t) and relation (24) below, the time derivative Lyapunov function Vt () 2 e() te () t is written as 0 2 ˆ ˆ Vt () 2e Ap GCe 2 e L Lp ( ) N N 0 2 2e A p G C e 2 e L L( p) Considering now, Assumption H2, Vt () becomes maimised by the quadratic epression () 2 0 n 2 Vt e Ap I GCe (25) Following epression (25), one can ensure the asymptotic stability of the estimation error (3) by applying observer gain matri G 2 satisfying the condition Ap ( 0) In GC 2 0. Since Vt () is positive definite and Vt () is negative definite, it can be concluded that the (23)
126 B. Sfaihi and H. Kallel estimation error converges to zero asymptotically with time. herefore, the estimated state ˆ converges asymptotically to the true state, i.e. ˆ as t. As in first case, the particular parameter is a positive scalar non-negligible that absorbs uncertainties variation, and imposes to G 2 to be a high gain matri. 5 Numerical eamples In this section, two eamples are given to show the effectiveness of the proposed methods. he first eample, in illustration to the introduced robust observer for linear systems with matched uncertainties, and the second in illustration to the developed robust observer for linear systems with unmatched uncertainties. 5.1 Eample 1 Consider the uncertain system (1) with following data 2 p 1 1 p 1 1 0 1 Ap ( ) 1 8 0, B 0, C 0 1 0 14p 0 24p 1 where the uncertain parameter p() t verify following cases mentioned in able 1. (26) It is easy to show that the uncertain system (26) is a linear system with matched uncertainties (5), that can be considered in the form (6) with the system matrices 2 1 1 p 0 A 1 8 0, ( p) 0 0 1 0 2 4p 0 Observer synthesis: to proceed with the robust observer design, we define p0 10 and 12. We verify conditions (10) (11) by taking high gains matrices G 756.35 20,599 2 0 5.27 280.04, G 40 860 615 20,598 22 23 and hence, by applying relations (7) (9), the resulting robust observer (2) developed for the uncertain system with matched uncertainties (26) is given by 750.35 20,598 749.35 1 758.34 20,599 F 44.27 1,148 45.27, G 1 0, G 2 45.27 1,140 (27) 572 20,598 571 1 613 20,598 Simulations: to evaluate the designed observer, the observer (27) is implemented with initial conditions (0) ˆ (0) [0 0 0] and with control vector ut ( ) 106sin( t) 6sin (0.5 t) for t 10, ut ( ) 20 4sin(1.5 t) 12sin( t), for 10 t 20 and ut ( ) 8 8sin(0.8 t) 4sin( t) for t 20. he Plots a, b and c in Figures 1 and 2 show the real plant
On robust state estimation for linear systems 127 dynamics (26) and that predicted by the robust observer (27). As it can be seen, the observer (27) provides an accurate estimation of states vector despite the important parametric variation illustrated in Figures 1(e) and 2(e). he observer convergence is evidenced in Figures 1(d) and 2(d) by an estimation error converging to zero. A plot of norm evolution of H ( p0) Cˆ( p) C ( p0) C ˆ is provided in Figures 1(f) and 2(f) to verify the validity of assumption H1. able 1 Uncertainity variation in eample 1 Case 1 Case 2 p() t pt ( ) 2t10 for t10 Array of random numbers normally distributed with a mean value equal to 20 pt ( ) 30 for 10 t20 and a variance of 3 pt ( ) t50 for t20 Figure 1 Matched uncertainty- Case 1: (a) 1 and estimated 1 dynamics, (b) 2 and estimated 2 dynamics, (c) 3 and estimated 3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
128 B. Sfaihi and H. Kallel Figure 2 Matched uncertainty- Case 2: (a) 1 and estimated 1 dynamics, (b) 2 and estimated 2 dynamics, (c) 3 and estimated 3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours) 5.2 Eample 2 Let us now consider now the following continuous linear uncertain system modelled as 2 p 0 1 0.5 0 1 Ap ( ) 5, 0 p p p B, C 0 1 0 (28) 0 0 6 1 where the uncertain parameter p() t verify following cases mentioned in able 2.
On robust state estimation for linear systems 129 By choosing a nominal value parameter p0 20, above system can be considered in the structure (18) with 2 20 0 Ap0 20 20 100, 0 0 6 0 p 20 0 Lp ( ) p 20 p 20 5 p 100 0 0 0 Observer synthesis: by defining the construction parameters p 1 10 and 25 and applying relations (19) (22) of Proposition 2, the resulting robust nominal observer developed for the uncertain system (28) is described by following matrices F, G 1 and G 2 0.3508 2.1073 0.3506 1 4 F 10 0.0125 0.2588 0.0165, 1 0 G, 0.2197 2.0998 0.2191 1 0.3506 2.1083 4 G2 10 0.0115 0.2578 0.2197 2.0998 Simulations: the observer (30) is implemented with initial conditions (0) ˆ (0) [0 0 0] and with control vector ut ( ) 106sin( t) 6sin(0.5 t) for t 4, ut ( ) 204sin(1.5 t) 12sin( t), for 4 t 8 and ut () 88sin(0.8 t) 4sin( t) for t 8. he conceived nominal observer (30) shows a finite time convergence properties to real uncertain system (28) as shown in Figures 3 and 4, plots (a) (c), and strengthened by the estimation error in plot (d). In spite of the important time parameter variation, illustrated in Figures 3 and 4, plot (e), the nominal observer remains robustly stable and provides a high speed convergence to real states. In plot (f), we verify that the considered real scalar H LN ˆL( p) ˆ is negative or null during simulation time that validates the chosen assumption. able 2 Uncertainity variation in eample 2 Case 1 Case 2 p() t pt () 5t10for t4 Array of random numbers normally distributed with a mean value equal to 20 and a variance of pt ( ) 30 for t4 8 3 p() t 0.5t34for t 8 (29) (30)
130 B. Sfaihi and H. Kallel Figure 3 Unmatched uncertainty- Case 1: (a) 1 and estimated 1 dynamics, (b) 2 and estimated 2 dynamics, (c) 3 and estimated 3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
On robust state estimation for linear systems 131 Figure 4 Unmatched uncertainty- Case 2: (a) 1 and estimated 1 dynamics, (b) 2 and estimated 2 dynamics, (c) 3 and estimated 3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours) 6 Conclusion In this paper, we considered the problem of designing robust observer for a class of linear systems with uncertain state matri. Both of state estimation for systems with matched uncertainties and with unmatched uncertainties are solved. Robustness is achieved by introducing a high gain in the observer structure and it was proved that the nominal
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