Robust L2 − L∞ filter design with Parameter ...

15th international conference on Sciences and Techniques of Automatic control & computer engineering - STA'2014, Hammamet, Tunisia, December 21-23, 2014

STA'2014-PID3571- DFC

Robust L2 − L∞ filter design with Parameter dependent Lyapunov function approach for polytopic time-delay systems A.Hmamed1 , T.Zoulagh1 and A.EL Hajjaji2 Abstract— This paper presents new LMI conditions for full order robust L2 −L∞ filter design for continuous polytopic time-delay systems. This work is based on the PDLF approach to L2 − L∞ filtering. A new structure of slack variables is introduced to provide extra free dimensions in the solution space for L2 −L∞ optimization. A numerical example is presented to show the effectiveness of the proposed method.

Keywords: PDLF , L2 − L∞ filtering , Linear matrix inequalities (LMIs), polytopic uncertainty. I. Introduction The aim of L2 −L∞ filtering problem is to minimize the peak value of the estimation error for all possible bounded energy disturbances [1]. Thus, this problem is referred to as the energy-to-peak filtering problem [2]. In some literature, it is also called generalized H2 filtering problem [3]. Unlike the celebrated Kalman filtering, the L2 − L∞ filtering does not require any information on the stochastic properties of the exogenous inputs [1], [2]. Even in the case where the noise information is available, the L2 − L∞ filtering is more suitable than the Kalman filtering for such systems as hybrid systems, time-delay systems, systems with uncertainties, and so on. Therefore, much research effort has been made towards the L2 − L∞ filtering problem. Since the time-delay can degrade system performance and even destabilize the system, great effort has been made to investigate timedelay systems control and filtering problems [4][5][8]. Recently, the problem of L2 − L∞ filtering of continuous time-delay systems, has been investigated in [2], [6] via linear matrix inequality (LMI) approaches [7]. These approaches, however, are based on a common Lyapunov function (CLF) which is adequate for arbitrarily fast parameter-varying systems, the stability analysis and filter design conditions based on a CLF can be conservative since one CLF must hold for the entire uncertainty domain. To overcome this drawback, Zhang et al. [3] proposed a parameter-dependent Lyapunov function (PDLF) approach to robust L2 − L∞ filter design for continuous-time time-delay systems. *This work was not supported by any organization 1 LESSI, Department of Physics Faculty ences Dhar El Mehraz B.P. 1796 Fes-Atlas

of SciMorocco

hmamed− [email protected], [email protected] 2 Laboratoire Modélisation, Information et Systèmes, University of Picardie Jules Verne, 7, Rue moulin Neuf, 80000 Amiens, France

The so-called dilated inequality characterization[9], [10] is essential to obtain PDLF-based strict LMI conditions for robust filter design. By coupling the system matrices with a newly introduced slack matrix variable which then takes over the role of the Lyapunov function matrix, that provide extra free dimensions in the solution space for L2 − L∞ optimization . Since the filter gains are constructed from the matrix coupled with the system matrices, keeping the slack matrix variable parameter-independent enables one to obtain parameter-independent filter gains, which is the main reason for requiring suitable dilated inequality characterizations for PDLF-based robust filter design. For discrete-time systems, strict LMI conditions for robust filter design using PDLF approaches can be obtained straightforwardly from those using CLF approaches if the dilated inequality characterization of Oliveira et al.[9] is used; for example, refer to [14] for discrete-time non-delayed systems, and [15] for discrete time-delay systems. Unlike the discrete-time counterpart, PDLF-based strict LMI conditions for robust filter design are not readily obtained even for a simple continuous-time delay-free systems since the existing dilated inequality characterizations [10], [12], [13] are suitable only for some specific cases. So, more complicated continuous-time time-delay systems have not been well investigated for obtaining suitable dilated inequality characterizations. In this paper, we propose a strict LMI condition for robust L2 − L∞ filter design for continuous-time time-delay systems based on a PDLF approach. To facilitate the robust filter design using PDLFs, we propose a new dilated inequality characterization for L2 − L∞ performance analysis. Based on this characterization, we propose a sufficient condition for robust L2 − L∞ filter design, which consists of strict LMIs and thus can be solved via convex optimization. Thanks to the proposed dilated inequality characterization and an appropriate linearizing transformation, the condition does not have any nonlinear couplings between the system matrices and the Lyapunov function matrix, nor any systemdependent filter parametrization. These features are key ingredients for obtaining robust filters based on the PDLF approach. The proposed condition leads to less conservative results than previous conditions using CLF approaches. This is illustrated by a numerical example.

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III. Filter condition analysis for L2 − L∞ performance

II. Problem formulation Consider the uncertain continuous time-delay system described by:  ˙   x(t)

= A(α)x(t) + Ah (α)x(t − h) + B(α)w(t) = C(α)x(t) + D(α)w(t) = L(α)x(t) = φ(θ), θ ∈ [−h, 0]

y(t) z(t)   x(θ)

(1)

where x ∈ 0 ,and S(α) such that:  −S(α) − S(α)T ∗ ∗ ∗ ∗ Ξ1 Ξ2 ∗ ∗ ∗    < 0. (6)  Ξ3 0 −Q(α) ∗ ∗   Ξ4 0 0 −I ∗ S(α) 0 0 0 −P (α)

(2)

where Ψi = [Ai , Ahi , Bi , Ci , Di , Li ] ,i = 1, .., N , are constant matrices with appropriate dimensions and αi are time-invariant uncertainties. the robust filter of interest in this paper is given in the following form:  x ˆ˙ (t) zˆ(t)

In this section, we present a sufficient condition for the L2 − L∞ performance analysis of the filtering error system (4). The condition represented in this paper, is an extension of the ones represented in [9]-[13].



P (α) Ca (α)

∗ γ2I

∗ Λ2 Λ4 Λ7 Λ10

∗ ∗ Λ5 Λ8 Λ11

∗ ∗ ∗ Λ9 Λ12

∗ ∗  ∗   < 0. ∗ Λ13



(8)

 > 0.

(9)

(5)

is guaranteed under zero-initial conditions for all nonzero w(t) ∈ L2 [0, +∞) and prescribed γ > 0. over the entire polytope Ω.

where Λ1 = P (α) + Aa (α)T S(α) − R(α)T Λ2 = −P (α) + H T Q(α)H + Aa (α)T R(α) + R(α)T Aa (α) Λ3 = Aha (α)T S(α) − N (α)T Λ4 = Aha (α)T R(α) + N (α)T Aa (α)

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Λ5 = −Q(α) + Aha (α)T N (α) + N (α)T Aha (α) Λ6 = Ba (α)T S(α) − M (α)T Λ7 = Ba (α)T R(α) + M (α)T Aa (α) Λ8 = Ba (α)T N (α) + M (α)T Aha (α) Λ9 = −I + Ba (α)T M (α) + M (α)T Ba (α) Λ10 = R(α) + E(α)T Aa (α) Λ11 = N (α) + E(α)T Aha (α) Λ12 = M (α) + E(α)T Ba (α) Λ13 = E(α)T + E(α) − P (α)



P 1i  P T2i Li − K

Proof:Suppose that (8) and (9) hold.(8) can be written as:

0 ∗ T  P (α) −P (α) + H Q(α)H  0 0  0 0 0 0 ΦT χ+χT Φ < 0.



∗ ∗ −Q(α) 0 0

∗ ∗ ∗ −I 0

∗ ∗   + ∗  ∗ −P (α) (10)



  −I Aa (α) Aha (α) Ba(α) I where and  Φ = χ = S(α) R(α) N α) M (α) E(α) To complete the proof see ( Choi et al.[11])in the performance analysis section.  Remark1 : If we make R(α) = N (α) = M (α) = 0 we will find the results in ( Choi al.[11]).

IV. Robust L2 − L∞ performance filter design In the second section, we propose a sufficient condition for the existence of a robust L2 − L∞ filter over the entire polytope Ω. Our approach is based on a specific linearizing transformation through which we had a strict LMI condition for the filter design.

Theorem 2 The filtering error system (4) is asymptotically stable with an L2 − L∞ disturbance attenuation level γ (5), if there exist matrices W, Y, Z, N1i , M1i , F, G, K, P 1i > 0, P 2i , P 3i > 0 and Qi > 0,i = 1, ..., N , and λ1 , λ2 satisfying LMIs (11, 12) such that :   Θ11 Θ21 Θ31  Θ41 Θ51  Θ61  Θ71 Θ81

∗ Θ22 Θ32 Θ42 Θ52 Θ62 Θ72 Θ82

∗ ∗ Θ33 Θ43 Θ53 Θ63 Θ73 Θ83

∗ ∗ ∗ Θ44 Θ54 Θ64 Θ74 Θ84

∗ ∗ ∗ ∗ Θ55 Θ65 Θ75 Θ85

∗ ∗ ∗ ∗ ∗ Θ66 Θ76 Θ86

∗ ∗ ∗ ∗ ∗ ∗ Θ77 Θ87

∗ ∗  ∗   ∗  < 0. ∗    ∗  ∗ Θ88 (11)

∗ P 3i Li



∗ ∗  > 0. γ2I

(12)

where Θ11 = −Z − Z T Θ21 = −Z − W Θ22 = −Y − Y T Θ31 = ATi Z + P 1i − λ1 Z T Θ32 = ATi Y T + P 2i − λ1 Z T + GT + CiT F T Θ33 = −P 1i + Qi + λ1 (Z T Ai + ATi Z) T Θ41 = ATi Z + P 2i − λ1 W Θ42 = ATi Y T + P 3i − λ1 Y + CiT F T Θ43 = −P 2i + Qi + λ1 (Y Ai + ATi Z + G + F Ci ) Θ44 = −P 3i + Qi + λ1 (Y Ai + ATi Y T + F Ci + CiT F T ) T Θ51 = AThi Z − N1i T T T Θ52 = Ahi Y − N1i T T Θ53 = λ1 Ahi Z + N1i Ai T T T Θ54 = λ1 Ahi Y + N1i Ai T T Θ55 = −Qi + Ahi N1i + N1i Ahi T T Θ61 = Bi Z − M1i T Θ62 = BiT Y T − M1i + DiT F T T T Θ63 = λ1 Bi Z + M1i Ai T Θ64 = λ1 (BiT Y T + DiT F T ) + M1i Ai T T Θ65 = Bi N1i + M1i Ahi T Θ66 = −I + BiT M1i + M1i Bi T Θ71 = Z − λ2 Z Θ72 = W T − λ2 Z T Θ73 = λ1 Z + λ2 Z T Ai Θ74 = λ1 W T + λ2 Z T Ai Θ75 = λ2 Z T Ahi + N1i + W T − Y T Θ76 = λ2 Z T Bi + M1i Θ77 = −P 1i + λ2 (Z T + Z) Θ81 = Z − λ2 W Θ82 = Y T − λ2 Y Θ83 = λ1 Z + λ2 (Y Ai + G + F Ci ) Θ84 = λ1 Y T + λ2 (Y Ai + F Ci ) Θ85 = λ2 Y Ahi + N1i Θ86 = λ2 (Y Bi + F Di ) + M1i T Θ87 = −P 2i + λ2 (W + Z) Θ88 = −P 3i + λ2 (Y + Y T ) with the filter parameters are given by: Af = (W − Y )−1 G, Bf = (W − Y )−1 F ,Lf = K.

Remark2 : For given λ1 and λ2 (11, 12) are linear and the problem will be solved by using LMI Toolbox. Then, to find the optimal values of λ1 , λ2 in order to minimize the filtering error, we apply a numerical optimization algorithm, such as the program fminsearch in the optimization toolbox of Matlab [16].˙

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Proof:Suppose that (11) and (12) hold. Let’s partition S(α), R(α), N (α), M (α) and E(α) in(8) with new variables. Assuming that S(α), R(α)and E(α) are fixed matrices,i.e S(α) = S, R(α) = R and E(α) = E as: R(α) = λ1 S. E(α) =λ   2 S.   And N (α) = N1i 0 , M (α) = M1i 0 . Where:

T T T T N1i M1i N T T = N1i , M T T = M1i . to complete the proof see( Choi et al.[11]).  Figure 1 : The actual cost of the filter for various uncertain parameter |ρ|, |σ| for the first vertex.

V. Illustrative example To illustrate the effectiveness of our method, we compare the performance of our filter with an existing filters [11] via a numerical example. We use MATLAB LMI Control Toolbox [17] to solve the design problem. Considering a time-delay system in form (1) whose matrices are given by:     A(α) =

3+ρ −0.1 ,Ah = −5 0.2

0 −4



 B(α) =



C= 1

0 , −0.2

−0.4545 + 0.4ρ , 0.9090   2 , D = 1, L(α) = 3 + 2.95σ

Figure 2 : The actual cost of the filter for various uncertain parameter |ρ|, |σ| for the second vertex.



4 + 2.95σ .

where ρ and σ are constant but uncertain parameters. The desired filter is a sub-optimal L2 − L∞ that yields the least upper-bounded on the L2 − L∞ norm of filtering error system (4), that can be obtained by solving the convex optimization given in (5). We use the famous function ”fminsearch” to find λ1 ,λ2 which correspond to the local minimum obtained by solvers as ”mincx” in MATLAB LMI Control Toolbox.This command is used and the initial value of (λ1 , λ2 ) is chosen to be (0, 0) the minimum value of γ = 1.4241 in the first case is founded when (λ1 ,λ2 ) = (−0.0034, 0.0002).Our results are shown and compared with [11] in the following table: case1 case2 case3

|ρ|, |σ| |ρ| < 1, |σ| < 1 |ρ| < 2, |σ| < 2 |ρ| < 2.6, |σ| < 2.6

Therem 2 1.4241 3.7207 7.2344

Figure 3 : The actual cost of the filter for various uncertain parameter |ρ|, |σ| for the third vertex.

Choi al.[11] 1.4983 3.9681 9.5947

Table : Comparative results of Theorem 2 with [11]. we give the parameter filter and the figures corresponding to the first case of |ρ| and |σ| with (λ1 ,λ2 ) = (−0.0034, 0.0002):     Af =



1.2013 −3.6229

Lf = 2.5974

10.8085 −0.9399 Bf = −11.0066 1.3556

Figure 4 : The actual cost of the filter for various uncertain parameter |ρ|, |σ| for the fourth vertex.



4.4981 .

the actual performance of the resultant filter is given in the Fig.1,Fig.2,Fig.3 and Fig.4 that verify that the cost γ 2 for any admissible uncertainty is below the derived upper bound for the first case |ρ| < 1, |σ| < 1.

To illustrate the effectiveness of our method, we present in F ig.5 the relationship between the minimized γ and the maximal absolute value uncertainty σ in case |ρ| ≤ 1.

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Figure 5 : Comparison of performances when |ρ| < 1. The initial state are set to zero and the external disturbance input w(t) is chosen to be a rectangular signal whose value is 1 during the time from 5 to 6 s and 0 otherwise.The delay value is set to 2 s.The time responses of the estimation errors for the case1 are shown in the Fig.6.

Figure 6 : The estimation error of Theorem 2 and [11] for |ρ| < 1,|σ| < 1.

[3] W. Zhang, L. Yu, X. Jiang, Delay-dependent generalized H2 filtering for uncertain systems with multiple time-varying state delays, Signal Process.vol.87, pp.709-724,2007. [4] El-Kasri, C., Hmamed, A., TISSIR, E.H., Tadeo, F. (2012). Robust H1 Filtering for Uncertain Two-Dimensional Continuous Systems with Time-varying Delays. Multidimensional Systems Signal Process, no. DOI :10.1007/s11045-013-0242-7. [5] El-Kasri, C., Hmamed, A., Alvarez, T., Tadeo, F. (2012). Uncertain 2D Continuous Systems with State Delay : Filter Design using an H1 Polynomial Approach. International Journal of Computer Applications, 44(18), 13-21 [6] J. Xia, S. Xu, B. Song, Delay-dependent L2 − L∞ filter design for stochastic time-delay systems, Systems Control Lettvol.vol.56, pp. 579-587,2007. [7] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia,1994. [8] Hmamed, A., El-Kasri, C., Tissir, E.H., Alvarez, T., Tadeo, F. (2013). Robust H1 Filtering for Uncertain 2-D Continuous Systems with Delays. International Journal of Innovative Computing, Information and Control, 9(5), 2167-2183 [9] M.C. de Oliveira, J. Bernussou, J.C. Geromel, A new discretetime robust stability condition, Systems Control Lett.vol.37, pp. 261-265,1999. [10] H.D. Tuan, P. Apkarian, T.Q. Nguyen, Robust and reducedorder filtering: new LMI-based characterizations and methods, IEEETrans. Signal Process.vol.49, pp. 2975-2984,2001. [11] H-C.Choi, S.Han, J.H.Seo,Parameter-dependent Lyapunov function approach to robust L2 − L∞ filter design for uncertain time-delay systems,Signal Processingvol.vol. 91, pp. 323329,2011. [12] V.F. Montagner, R.C.L.F. Oliveira, P.L.D. Peres, Robust stability and H∞ performance of linear time-varying systems in polytopic domains, Int. J. Control.vol. 77, pp. 1343-1352, 2004. [13] Peaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J. (2000). A new robust D-stability condition for real convex polytopic uncertainty. Systems Control Letters.vol. 40, pp. 21-30, 2000. [14] J.C. Geromel, M.C. de Oliveira, J. Bernussou, Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions, SIAM J. Control Optim.vol. 41, pp. 700-711, 2002. [15] Y. He, G.-P. Liu, D. Rees, M. Wu, H∞ filtering for discretetime systems with time-varying delay, Signal Process. vol. 89, pp. 275-282, 2009. [16] Coleman, T.,Branch,M., Grace,.A.(1999).Optimization toolbox for use with matlab.Natick,MA:the MathWorks inc. [17] P.Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, MATLAB LMI Control Toolbox User’s Guide, Natick, The MathWorks, 1995.

Remark 3 : These figures also confirm the effectiveness of our method as the peak value of estimation error by the filter proposed is less than the existing one in ( Choi al.[11]).

VI. Conclusions This paper extends the existing results on robust L2 − L∞ filtering to present a new structure for strict LMIs and thus can be solved via convex optimization .Our new structure introduces free slack variables which guarantees to provide extra free dimensions in the solution space for the L2 − L∞ optimization.The effectiveness of the proposed method has been illustrated via a numerical example.

References [1] K.M. Grigoriadis, J.T. Watson Jr., Reduced-order H∞ and L2 − L∞ filtering via linear matrix inequalities, IEEE Trans. Aerosp. Electron.Syst.vol.33, pp. 1326-1338,1997. [2] H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems, Systems Control Lett.vol.55, pp. 101-111,2006.

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