On robust state estimation for linear systems with ... - Semantic Scholar
Int. J. Automation and Control, Vol. 5, No. 2, 2011
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 Technology, Centre Urbain Nord, B.P. N°676, TUNIS CEDEX 1080, Tunisie E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: This paper addresses the problem of robust observer design for linear dynamic systems with uncertain parametric state matrix. Two class systems with non-bounded parametric uncertainties are considered for estimation: dynamic systems with matched conditions and dynamic systems with unmatched conditions. In this context, a robust estimator with a second gain update output estimation error is developed to linear systems with matched uncertainties. The 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 examples with extensive 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 Technology, INSAT, Tunis, Tunisia. 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 Technology, INSAT, Tunis, Tunisia. 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.
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Introduction
The 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 approximation to the unknown state vector (Luenberger, 1966). This 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 Hf 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. The 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), Theodor 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 Trofino (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 Hf 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 Hf filter using a Riccati equation technique. Using LMI technique, Palhares and Peres (1998), Geromel (1999) and Gao et al. (2008) considered Hf 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 matrix 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 extension 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 Trinh 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 context, 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
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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 extra 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 matrix. 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 examples to validate the robust observer schemes.
2
Preliminaries and problem formulation
Throughout this note, the following notations are adopted. n and num denote the space of n-dimensional real vectors and m u n real matrices, respectively. I n is the n u n identity matrix. A matrix is said to be stable if its eigenvalues are all on the left-half open complex plane. || || denotes the Euclidean norm of a vector. Consider a class of linear uncertain parametric systems when the uncertainty is assumed to exist in the state matrix. Dynamics system are given by
x (t ) A( p) x(t ) Bu (t ) y (t ) Cx(t )
(1)
where x n , u m and y k are, respectively, the state to be estimated, the control vector and the measured outputs. A( p ) nun with p P models the uncertainty in the system. B num and C kun are all known constant matrices. The 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
xˆ (t )
Fxˆ (t ) G1u (t ) G2 y (t )
(2)
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where xˆ(t ) is the observer state vector, F nun is the state matrix, G1 num is the input matrix and G2 nuk is the gain matrix observer. Define the state estimation error as xˆ (t ) x(t )
ex (t )
(3)
The state estimation error is governed by xˆ (t ) x (t )
ex (t )
Our aim is to define the class of gain matrices F, G1 and G2 convergence of the estimation error ex (t ) to zero for both observer for system (1) with matched and unmatched uncertainties. The case condition is not satisfied is discussed in Section 4. We matching condition holds.
3
(4) that guarantee the schemes developed when the matching first assume that
Robust observer with matched uncertainties
Main result: we suppose that the uncertainty is in the range of output matrix C as following A( p)
A M ( p)C
(5)
where A nun is a constant matrix and M ( p ) nuk is the matched uncertainty matrix. Substituting (5) in (1) produces x (t ) [ A M ( p )C ] x(t ) Bu (t ) ® ¯ y (t ) Cx(t )
(6)
The following assumption is required for the design of observer (2). Assumption H1: For the uncertain part of the parametric system (6), we assume that
M p0 yˆ M ( p) y d O M p0 yˆ y where p0 P is a nominal value of p and O is a positive real scalar. Proposition 1: Let Assumption H1 holds and considers the dynamic robust observer (2) with F
A M p0 C G2C
(7)
G1
B
(8)
G2
G22 C T G23
(9)
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The estimation error system (3) is asymptotically stable, if and only if the conditions below hold: 1
The matrix A (O 2 / 2) I n G22C 0 or the pair
§ · O2 I n , C ¸¸ ¨¨ A 2 © ¹
(10)
is observable 2
The gain matrix 1 T G23 ! M p0 M p0 2
(11)
where G22 nuk , G23 k uk , p0 is a nominal value of p and O 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
Fxˆ G1u G2 y (( A M ( p)C ) x Bu ) A M ( p)C G2C ex F A M ( p )C G2 C xˆ G1 B u
ex (t )
(12)
In order to obtain an asymptotic estimation error convergence, we deduce from relation (12) the expressions of matrices gain F and G1, respectively as F A M ( p0 )C G2C and G1 B . The fixed value p0 is a nominal value of p chosen arbitrarily. From (12), it follows that ex
A G2C ex M p0 Cxˆ M ( p)Cx
(13)
To prove observer (2) stability, let us consider the Lyapunov function candidate V ex , t eTx (t )ex (t )
(14)
The time derivatives of V (t ) along the trajectories of system (13) satisfy 2eTx (t )ex (t )
V (t )
(15)
By substituting (13) into (15) yields
2eTx A G2C ex 2eTx M p0 Cxˆ M ( p)Cx
V (t ) d
2eTx
A G2C ex 2
eTx
M p0 Cxˆ M ( p)Cx
Using assumption (H1), we can write eTx M p0 Cxˆ M ( p)Cx d O eTx M p0 C ex
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Consequently, the time derivative of V (t ) becomes V (t ) d 2eTx A G2C ex 2O ex M p0 C ex Since, 2O eTx M p0 C ex O 2 eTx ex eTx C T M T p0 M p0 Cex it follows that, V (t ) d 2eTx A G2C ex O 2 eTx ex eTx C T M T p0 M p0 Cex In order to reduce the conservatism of this inequality, we introduce a second update gain G23, such that G2 G22 C T G23 . We make the following choice to write the below developed expression
§ · O2 V (t ) d 2eTx ¨¨ A I n G22 C ¸¸ ex 2 © ¹ eTx C T
M
T
(16)
p0 M p0 2G23 Cex
The derivative of Lyapunov function V (t ) is negative for any matrices gain G22 and G23 verifying A (O 2 / 2) I G C 0 and G ! (1 / 2)M T ( p )M ( p ) . Therefore, V (e , t ) is n
22
23
0
0
x
negative definite, the error observer (3) is stable, and consequently the robust observer (2) is stable. The proposed conditions (10) and (11) provide estimator gains G22 and G23 which are always stabilising. The particular choice of a relaxed gain form G2 allows more freedom and flexibility in the robust observer conception. On other hand, the particular parameter O imposed by assumption H1, is a positive scalar non-negligible that as seen in H1 absorbs uncertainties variation, and consequently imposes to G22 to be a high gain matrix. Then, 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 A( p)
A p0 L( p)
(17)
where p0 P is a nominal parameter and L ( p ) nun is the uncertain matrix. In this case, the uncertain system (1) becomes ª¬ A p0 L ( p) º¼ x(t ) Bu (t ) y (t ) Cx(t )
x (t )
(18)
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 LN xˆ L( p) x d O xˆ x
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where A p1 A p0
LN
(19)
( p0 , p1 ) P are nominal parameters construction with p0 z p1 and O is a positive real scalar.
Proposition 2: Let Assumption H2 holds and considers the dynamic robust observer (7) with F
A p0 LN G2C
(20)
G1
B
(21)
The estimation error system (4) is asymptotically stable, if and only if the condition A p0 O I n G2C 0
(22)
or the pair ( A( p0 ) O I n , C ) is observable, is satisfied. Proof: The stability robustness condition will be analysed by considering the following error dynamics substituted from uncertain system (22) and robust state estimator (2)
ex (t )
A p L ( p ) G C e (t ) F A p L( p) G C xˆ (t ) G B u (t ) 0
2
0
x
2
(23)
1
In order to structure the robust observer (2), let us consider the certain matrices LN, F and G1 defined respectively by LN A( p1 ) A( p0 ) , F A( p0 ) LN G2C and G1 B . The state estimation error dynamics (4) become ex (t )
A p0 L( p) G2C ex (t ) LN L( p) xˆ(t )
which imply ex (t )
A p0 G2C ex (t ) LN xˆ (t ) L( p) x(t )
(24)
By considering the Lyapunov function candidate V (ex , t ) eTx (t )ex (t ) and relation (24) below, the time derivative Lyapunov function V (t ) 2eTx (t )ex (t ) is written as V (t )
d 2eTx A p0 G2 C ex 2 eTx L N xˆ L ( p) x 2eTx A p0 G2C ex 2eTx LN xˆ L ( p) x
Considering now, Assumption H2, V (t ) becomes maximised by the quadratic expression V (t ) d 2eTx A p0 O I n G2C ex
(25)
Following expression (25), one can ensure the asymptotic stability of the estimation error (3) by applying observer gain matrix G2 satisfying the condition A( p0 ) O I n G2C 0 . Since V (t ) is positive definite and V (t ) is negative definite, it can be concluded that the
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estimation error converges to zero asymptotically with time. Therefore, the estimated state xˆ converges asymptotically to the true state x, i.e. xˆ o x as t o f . As in first case, the particular parameter O is a positive scalar non-negligible that absorbs uncertainties variation, and imposes to G2 to be a high gain matrix.
5
Numerical examples
In this section, two examples are given to show the effectiveness of the proposed methods. The first example, 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 Example 1 Consider the uncertain system (1) with following data A( p)
1 p º ª 2 p 1 « 1 8 0 »» , B « «¬ 1 4 p 0 2 4 p »¼
ª1 º « » « 0» , C «¬1 »¼
ª1 «0 ¬
0 1
1º 0 »¼
(26)
where the uncertain parameter p(t ) verify following cases mentioned in Table 1. 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 ª 2 1 1 º A «« 1 8 0 »» , M ( p) «¬ 1 0 2»¼
ª p «0 « «¬ 4 p
0 0 0
º » » »¼
Observer synthesis: to proceed with the robust observer design, we define p0 10 and O 12 . We verify conditions (10)–(11) by taking high gains matrices G22
ª 756.35 20,599 º « 5.27 280.04 »» , G23 « «¬ 615 20,598 »¼
0 º ª2 « 40 860 » ¬ ¼
and hence, by applying relations (7)–(9), the resulting robust observer (2) developed for the uncertain system with matched uncertainties (26) is given by F
ª 750.35 20,598 749.35º « 44.27 1,148 45.27 »» , G1 « «¬ 572 20,598 571 »¼
ª1 º «0 » , G « » 2 «¬1 »¼
ª 758.34 20,599 º « 45.27 1,140 » « » «¬ 613 20,598 »¼
(27)
Simulations: to evaluate the designed observer, the observer (27) is implemented with initial conditions x(0) xˆ (0) [0 0 0]T and with control vector u (t ) 10 6sin(S t ) 6sin (0.5S t ) for t 10 , u (t ) 20 4sin(1.5S t ) 12sin(S t ) , for 10 d t 20 and u (t ) 8 8sin(0.8S t ) 4sin(S t ) for t t 20 . The Plots a, b and c in Figures 1 and 2 show the real plant
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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). The 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 || I ( p0 )Cxˆ I ( p)Cx || O || I ( p0 )C |||| xˆ x || is provided in Figures 1(f) and 2(f) to verify the validity of assumption H1. Table 1
p(t )
Uncertainity variation in example 1 Case 1 p(t ) 2t 10 for t 10
p(t ) 30 for 10 d t 20 p(t ) t 50 for t t 20 Figure 1
Case 2 Array of random numbers normally distributed with a mean value equal to 20 and a variance of
3
Matched uncertainty- Case 1: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
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Figure 2
Matched uncertainty- Case 2: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
5.2 Example 2 Let us now consider now the following continuous linear uncertain system modelled as A( p)
0º ª 2 p « p p 5 p» , B « » «¬ 0 0 6 »¼
ª1 º «0 » , C « » «¬1 »¼
ª0.5 0 1º « 0 1 0» ¬ ¼
where the uncertain parameter p(t ) verify following cases mentioned in Table 2.
(28)
On robust state estimation for linear systems By choosing a nominal value parameter p0 the structure (18) with A p0
L( p )
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20, above system can be considered in
0 º ª 2 20 « 20 20 100» , « » «¬ 0 0 6 »¼ 0 º p 20 ª 0 « p 20 p 20 5 p 100 » « » «¬ 0 0 0 »¼
(29)
Observer synthesis: by defining the construction parameters p1 10 and O 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, G1 and G2 ª 0.3508 2.1073 0.3506 º F 10 «« 0.0125 0.2588 0.0165»» , G1 «¬ 0.2197 2.0998 0.2191»¼ ª 0.3506 2.1083º 4« G2 10 « 0.0115 0.2578»» «¬ 0.2197 2.0998»¼ 4
ª1 º «0 » , « » «¬1 »¼
(30)
Simulations: the observer (30) is implemented with initial conditions x(0) xˆ (0) [0 0 0]T and with control vector u (t ) 10 6sin(S t ) 6sin(0.5S t ) for t 4 , u (t ) 20 4sin(1.5S t ) 12sin(S t ) , for 4 d t 8 and u (t ) 8 8sin(0.8S t ) 4sin(S t ) for t t 8 . The 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 xˆ L( p) x || O || xˆ x || is negative or null during simulation time that validates the chosen assumption. Table 2
Uncertainity variation in example 2 Case 1
p(t )
p(t ) 5t 10 for t 4 p(t ) 30 for t d 4 8 p(t ) 0.5t 34 for t t 8
Case 2 Array of random numbers normally distributed with a mean value equal to 20 and a variance of
3
130 Figure 3
B. Sfaihi and H. Kallel Unmatched uncertainty- Case 1: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 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 Figure 4
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Unmatched uncertainty- Case 2: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
Conclusion
In this paper, we considered the problem of designing robust observer for a class of linear systems with uncertain state matrix. 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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observer is a robust estimator for the uncertain system. The observers convergence was demonstrated via Lyapunov method and sufficient conditions for asymptotic estimation convergence have been established. The main advantage of the proposed observers over previously proposed ones is that they can be designed under less restrictive existence conditions which has been demonstrated by two illustrative examples.
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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 Technology, Centre Urbain Nord, B.P. N°676, TUNIS CEDEX 1080, Tunisie E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: This paper addresses the problem of robust observer design for linear dynamic systems with uncertain parametric state matrix. Two class systems with non-bounded parametric uncertainties are considered for estimation: dynamic systems with matched conditions and dynamic systems with unmatched conditions. In this context, a robust estimator with a second gain update output estimation error is developed to linear systems with matched uncertainties. The 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 examples with extensive 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 Technology, INSAT, Tunis, Tunisia. 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 Technology, INSAT, Tunis, Tunisia. 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.
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Introduction
The 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 approximation to the unknown state vector (Luenberger, 1966). This 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 Hf 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. The 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), Theodor 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 Trofino (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 Hf 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 Hf filter using a Riccati equation technique. Using LMI technique, Palhares and Peres (1998), Geromel (1999) and Gao et al. (2008) considered Hf 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 matrix 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 extension 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 Trinh 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 context, 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
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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 extra 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 matrix. 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 examples to validate the robust observer schemes.
2
Preliminaries and problem formulation
Throughout this note, the following notations are adopted. n and num denote the space of n-dimensional real vectors and m u n real matrices, respectively. I n is the n u n identity matrix. A matrix is said to be stable if its eigenvalues are all on the left-half open complex plane. || || denotes the Euclidean norm of a vector. Consider a class of linear uncertain parametric systems when the uncertainty is assumed to exist in the state matrix. Dynamics system are given by
x (t ) A( p) x(t ) Bu (t ) y (t ) Cx(t )
(1)
where x n , u m and y k are, respectively, the state to be estimated, the control vector and the measured outputs. A( p ) nun with p P models the uncertainty in the system. B num and C kun are all known constant matrices. The 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
xˆ (t )
Fxˆ (t ) G1u (t ) G2 y (t )
(2)
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where xˆ(t ) is the observer state vector, F nun is the state matrix, G1 num is the input matrix and G2 nuk is the gain matrix observer. Define the state estimation error as xˆ (t ) x(t )
ex (t )
(3)
The state estimation error is governed by xˆ (t ) x (t )
ex (t )
Our aim is to define the class of gain matrices F, G1 and G2 convergence of the estimation error ex (t ) to zero for both observer for system (1) with matched and unmatched uncertainties. The case condition is not satisfied is discussed in Section 4. We matching condition holds.
3
(4) that guarantee the schemes developed when the matching first assume that
Robust observer with matched uncertainties
Main result: we suppose that the uncertainty is in the range of output matrix C as following A( p)
A M ( p)C
(5)
where A nun is a constant matrix and M ( p ) nuk is the matched uncertainty matrix. Substituting (5) in (1) produces x (t ) [ A M ( p )C ] x(t ) Bu (t ) ® ¯ y (t ) Cx(t )
(6)
The following assumption is required for the design of observer (2). Assumption H1: For the uncertain part of the parametric system (6), we assume that
M p0 yˆ M ( p) y d O M p0 yˆ y where p0 P is a nominal value of p and O is a positive real scalar. Proposition 1: Let Assumption H1 holds and considers the dynamic robust observer (2) with F
A M p0 C G2C
(7)
G1
B
(8)
G2
G22 C T G23
(9)
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The estimation error system (3) is asymptotically stable, if and only if the conditions below hold: 1
The matrix A (O 2 / 2) I n G22C 0 or the pair
§ · O2 I n , C ¸¸ ¨¨ A 2 © ¹
(10)
is observable 2
The gain matrix 1 T G23 ! M p0 M p0 2
(11)
where G22 nuk , G23 k uk , p0 is a nominal value of p and O 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
Fxˆ G1u G2 y (( A M ( p)C ) x Bu ) A M ( p)C G2C ex F A M ( p )C G2 C xˆ G1 B u
ex (t )
(12)
In order to obtain an asymptotic estimation error convergence, we deduce from relation (12) the expressions of matrices gain F and G1, respectively as F A M ( p0 )C G2C and G1 B . The fixed value p0 is a nominal value of p chosen arbitrarily. From (12), it follows that ex
A G2C ex M p0 Cxˆ M ( p)Cx
(13)
To prove observer (2) stability, let us consider the Lyapunov function candidate V ex , t eTx (t )ex (t )
(14)
The time derivatives of V (t ) along the trajectories of system (13) satisfy 2eTx (t )ex (t )
V (t )
(15)
By substituting (13) into (15) yields
2eTx A G2C ex 2eTx M p0 Cxˆ M ( p)Cx
V (t ) d
2eTx
A G2C ex 2
eTx
M p0 Cxˆ M ( p)Cx
Using assumption (H1), we can write eTx M p0 Cxˆ M ( p)Cx d O eTx M p0 C ex
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Consequently, the time derivative of V (t ) becomes V (t ) d 2eTx A G2C ex 2O ex M p0 C ex Since, 2O eTx M p0 C ex O 2 eTx ex eTx C T M T p0 M p0 Cex it follows that, V (t ) d 2eTx A G2C ex O 2 eTx ex eTx C T M T p0 M p0 Cex In order to reduce the conservatism of this inequality, we introduce a second update gain G23, such that G2 G22 C T G23 . We make the following choice to write the below developed expression
§ · O2 V (t ) d 2eTx ¨¨ A I n G22 C ¸¸ ex 2 © ¹ eTx C T
M
T
(16)
p0 M p0 2G23 Cex
The derivative of Lyapunov function V (t ) is negative for any matrices gain G22 and G23 verifying A (O 2 / 2) I G C 0 and G ! (1 / 2)M T ( p )M ( p ) . Therefore, V (e , t ) is n
22
23
0
0
x
negative definite, the error observer (3) is stable, and consequently the robust observer (2) is stable. The proposed conditions (10) and (11) provide estimator gains G22 and G23 which are always stabilising. The particular choice of a relaxed gain form G2 allows more freedom and flexibility in the robust observer conception. On other hand, the particular parameter O imposed by assumption H1, is a positive scalar non-negligible that as seen in H1 absorbs uncertainties variation, and consequently imposes to G22 to be a high gain matrix. Then, 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 A( p)
A p0 L( p)
(17)
where p0 P is a nominal parameter and L ( p ) nun is the uncertain matrix. In this case, the uncertain system (1) becomes ª¬ A p0 L ( p) º¼ x(t ) Bu (t ) y (t ) Cx(t )
x (t )
(18)
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 LN xˆ L( p) x d O xˆ x
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where A p1 A p0
LN
(19)
( p0 , p1 ) P are nominal parameters construction with p0 z p1 and O is a positive real scalar.
Proposition 2: Let Assumption H2 holds and considers the dynamic robust observer (7) with F
A p0 LN G2C
(20)
G1
B
(21)
The estimation error system (4) is asymptotically stable, if and only if the condition A p0 O I n G2C 0
(22)
or the pair ( A( p0 ) O I n , C ) is observable, is satisfied. Proof: The stability robustness condition will be analysed by considering the following error dynamics substituted from uncertain system (22) and robust state estimator (2)
ex (t )
A p L ( p ) G C e (t ) F A p L( p) G C xˆ (t ) G B u (t ) 0
2
0
x
2
(23)
1
In order to structure the robust observer (2), let us consider the certain matrices LN, F and G1 defined respectively by LN A( p1 ) A( p0 ) , F A( p0 ) LN G2C and G1 B . The state estimation error dynamics (4) become ex (t )
A p0 L( p) G2C ex (t ) LN L( p) xˆ(t )
which imply ex (t )
A p0 G2C ex (t ) LN xˆ (t ) L( p) x(t )
(24)
By considering the Lyapunov function candidate V (ex , t ) eTx (t )ex (t ) and relation (24) below, the time derivative Lyapunov function V (t ) 2eTx (t )ex (t ) is written as V (t )
d 2eTx A p0 G2 C ex 2 eTx L N xˆ L ( p) x 2eTx A p0 G2C ex 2eTx LN xˆ L ( p) x
Considering now, Assumption H2, V (t ) becomes maximised by the quadratic expression V (t ) d 2eTx A p0 O I n G2C ex
(25)
Following expression (25), one can ensure the asymptotic stability of the estimation error (3) by applying observer gain matrix G2 satisfying the condition A( p0 ) O I n G2C 0 . Since V (t ) is positive definite and V (t ) is negative definite, it can be concluded that the
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estimation error converges to zero asymptotically with time. Therefore, the estimated state xˆ converges asymptotically to the true state x, i.e. xˆ o x as t o f . As in first case, the particular parameter O is a positive scalar non-negligible that absorbs uncertainties variation, and imposes to G2 to be a high gain matrix.
5
Numerical examples
In this section, two examples are given to show the effectiveness of the proposed methods. The first example, 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 Example 1 Consider the uncertain system (1) with following data A( p)
1 p º ª 2 p 1 « 1 8 0 »» , B « «¬ 1 4 p 0 2 4 p »¼
ª1 º « » « 0» , C «¬1 »¼
ª1 «0 ¬
0 1
1º 0 »¼
(26)
where the uncertain parameter p(t ) verify following cases mentioned in Table 1. 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 ª 2 1 1 º A «« 1 8 0 »» , M ( p) «¬ 1 0 2»¼
ª p «0 « «¬ 4 p
0 0 0
º » » »¼
Observer synthesis: to proceed with the robust observer design, we define p0 10 and O 12 . We verify conditions (10)–(11) by taking high gains matrices G22
ª 756.35 20,599 º « 5.27 280.04 »» , G23 « «¬ 615 20,598 »¼
0 º ª2 « 40 860 » ¬ ¼
and hence, by applying relations (7)–(9), the resulting robust observer (2) developed for the uncertain system with matched uncertainties (26) is given by F
ª 750.35 20,598 749.35º « 44.27 1,148 45.27 »» , G1 « «¬ 572 20,598 571 »¼
ª1 º «0 » , G « » 2 «¬1 »¼
ª 758.34 20,599 º « 45.27 1,140 » « » «¬ 613 20,598 »¼
(27)
Simulations: to evaluate the designed observer, the observer (27) is implemented with initial conditions x(0) xˆ (0) [0 0 0]T and with control vector u (t ) 10 6sin(S t ) 6sin (0.5S t ) for t 10 , u (t ) 20 4sin(1.5S t ) 12sin(S t ) , for 10 d t 20 and u (t ) 8 8sin(0.8S t ) 4sin(S t ) for t t 20 . The Plots a, b and c in Figures 1 and 2 show the real plant
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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). The 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 || I ( p0 )Cxˆ I ( p)Cx || O || I ( p0 )C |||| xˆ x || is provided in Figures 1(f) and 2(f) to verify the validity of assumption H1. Table 1
p(t )
Uncertainity variation in example 1 Case 1 p(t ) 2t 10 for t 10
p(t ) 30 for 10 d t 20 p(t ) t 50 for t t 20 Figure 1
Case 2 Array of random numbers normally distributed with a mean value equal to 20 and a variance of
3
Matched uncertainty- Case 1: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
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Figure 2
Matched uncertainty- Case 2: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
5.2 Example 2 Let us now consider now the following continuous linear uncertain system modelled as A( p)
0º ª 2 p « p p 5 p» , B « » «¬ 0 0 6 »¼
ª1 º «0 » , C « » «¬1 »¼
ª0.5 0 1º « 0 1 0» ¬ ¼
where the uncertain parameter p(t ) verify following cases mentioned in Table 2.
(28)
On robust state estimation for linear systems By choosing a nominal value parameter p0 the structure (18) with A p0
L( p )
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20, above system can be considered in
0 º ª 2 20 « 20 20 100» , « » «¬ 0 0 6 »¼ 0 º p 20 ª 0 « p 20 p 20 5 p 100 » « » «¬ 0 0 0 »¼
(29)
Observer synthesis: by defining the construction parameters p1 10 and O 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, G1 and G2 ª 0.3508 2.1073 0.3506 º F 10 «« 0.0125 0.2588 0.0165»» , G1 «¬ 0.2197 2.0998 0.2191»¼ ª 0.3506 2.1083º 4« G2 10 « 0.0115 0.2578»» «¬ 0.2197 2.0998»¼ 4
ª1 º «0 » , « » «¬1 »¼
(30)
Simulations: the observer (30) is implemented with initial conditions x(0) xˆ (0) [0 0 0]T and with control vector u (t ) 10 6sin(S t ) 6sin(0.5S t ) for t 4 , u (t ) 20 4sin(1.5S t ) 12sin(S t ) , for 4 d t 8 and u (t ) 8 8sin(0.8S t ) 4sin(S t ) for t t 8 . The 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 xˆ L( p) x || O || xˆ x || is negative or null during simulation time that validates the chosen assumption. Table 2
Uncertainity variation in example 2 Case 1
p(t )
p(t ) 5t 10 for t 4 p(t ) 30 for t d 4 8 p(t ) 0.5t 34 for t t 8
Case 2 Array of random numbers normally distributed with a mean value equal to 20 and a variance of
3
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B. Sfaihi and H. Kallel Unmatched uncertainty- Case 1: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 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 Figure 4
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Unmatched uncertainty- Case 2: (a) x1 and estimated x1 dynamics, (b) x2 and estimated x2 dynamics, (c) x3 and estimated x3 dynamics, (d) estimation error e, (e) parameter p variation, (f) condition H test (see online version for colours)
Conclusion
In this paper, we considered the problem of designing robust observer for a class of linear systems with uncertain state matrix. 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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observer is a robust estimator for the uncertain system. The observers convergence was demonstrated via Lyapunov method and sufficient conditions for asymptotic estimation convergence have been established. The main advantage of the proposed observers over previously proposed ones is that they can be designed under less restrictive existence conditions which has been demonstrated by two illustrative examples.
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