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According to (4d), V (t k+1 ) = η(t k+1 ) T S(t k+1 )η(t k+1 ) = η(t − k+1 )T M(τ k , t k+1 ) T × ( S(t − k+1 ) + τ kC T C ) M(τ k , t k+1 )η(t − k+1 ) = V (t − k+1 ) − (2ρ − 1)τ ∥ k Cη(t − k+1 )∥ ∥ 2 +ρ 2 τk 2η(t− k+1 )CT CS −1 (t k+1 )C T Cη(t − k+1 ). Invoking Proposition 5, the following inequality holds: V (t k+1 ) ≤ V (t − k+1 ) − τ k × ∥ ∥ Cη(t − k+1 )∥ ∥ 2 . Using (13) and because (9) holds, ((2ρ − 1) − ρ2 τ k ‖C‖ 2 β 1 ) (15) V (t k+1 ) ≤ e −µτ k V (t k ). (16) Consequently, according to (12) and (16), for all t ≥ t 0 , V (t) ≤ e −µ(t−t0) V (t 0 ), ‖η(t)‖ ≤ √ β2 β 1 e − µ 2 (t−t0) ‖η(t 0 )‖ , the exponential stability of η = 0 for system (10)-(14) is proved. Using similar arguments than for variable η, in view of (3c) and (4c) and since function φ is continuous and u takes values in a compact set, it can be easily shown that there exists ¯λ ∈ R + such that ‖λ(t)‖ ≤ ¯λ, for t ≥ t 0 . On the other hand, according to (4d), ˜θ(t k+1 ) = ( I n − ∆(t k+1 )Cλ(t − k+1 )) ˜θ(tk ) −∆(t k+1 )Cη(t − k+1 ). (17) Attention is focused on the homogenous part of (17), which is typical of the normalized least mean square algorithm: ˜θ(t k+1 ) = ( I n − ∆(t k+1 )Cλ(t − k+1 )) ˜θ(tk ). (18) Using Theorem 2.8 in [1], since Hypothesis 2 holds, ˜θ = 0 is globally exponentially stable for system (18). The non-homogenous part of equation (17) vanishes exponentially fast, because of the exponential convergence of η to 0 and the boundedness of ∆. Therefore, applying Lemma 7, ˜θ = 0 is globally exponentially stable for system (17). Writing the observation error as e = η + λ˜θ, the exponential convergence of e to 0, along the solutions of (5a-b), can be deduced from the facts that ˜θ and η converge with an exponential rate to zero and that λ is bounded. ✷ Remark 10 Throughout the paper, no stability assumption is supposed to hold for A. In view of Theorem 9, even unstable matrices can be considered. Remark 11 The sampling interval must satisfy (9). In practice, after having numerically found admissible t 1 and α, the upper bound in (9) can be determined using parameters values. 4.4 Discussion on parameters choice The selection of the observer parameters µ, ρ, ǫ illustrate the logical compromise between convergence speed and bound on τ. As it can be seen through (3b), parameter µ (taken bigger that ξ in view of Theorem 9) acts on the observation gain in such a way that taking big values will increase the convergence speed of the observer at the price of a smaller admissible τ. Indeed, this will make the solution of (3b) goes faster to non-positive definite matrices and then decrease the value of β 1 (see Remark 6d). Concerning ρ, a quick analysis could conclude that ρ = 1 is the optimal value to be chosen in view of (9), but a bigger value will help η to converge faster to 0, again, under more severe conditions on τ (see (14) and the definition of M). On the other hand, parameter ε allows to modulate the convergence of the parameter estimate. 5 Illustrative example For illustrative purposes, consider a second-ordersystem modeled by, for k ≥ 0 and t ∈ [t k , t k+1 ), ẋ 1 = sin(u)x 2 , ẋ 2 = −0.1 sin(u)x 1 − 0.2x 2 + 0.1 cos(u)θ, y = x 1 , (19) where θ is unknown and u(t) = 2 − (cos(0.1t)) 2 . The sampling step is chosen constant at τ = 0.11. An observer of the form (3a-d)-(4a-d) is designed for system (19). The design parameters have been fixed at ρ = 1, ǫ = 1, µ = 2.1 and Σ = 1. Using numerical tools, the following values have been found α = 0.5, t 1 = 0.01, ξ = 2.0396, β 1 ≥ 0.1125, consequently, condition (9) is satisfied: τ ≤ min{0.1125, 0.21}. Therefore, since the input is a persistent input for system (19), Theorem 9 applies. The initial conditions are t 0 = 0, x 1 (0) = x 2 (0) = 50, ˆx 1 (0) = ˆx 2 (0) = ˆθ(0) = 0 and θ = 2. For comparative purpose, the Euler discretization of observer (11-15) in [6] has been simulated. Figure 1 shows that, contrary to the continuous-time observer, the state ˆx 2 given by the emulation does not converge asymptotically to x 2 but oscillates. As mentioned in the Introduction, this fact is not surprising since only semiglobal practical stability properties can be established when 5
using the emulation of a continuous-time observer [2]. Moreover, the behaviour of emulation states present a big overshoot by opposite to the results obtained by the observer (3a-d)-(4a-d). Obtained parameter estimates converge slowly asymptotically to the true value for the continuous-discrete observer, and again, practically and with large overshoots for the emulation. Although not presented here, simulation results show that the continuous-time observer still works efficiently for larger values of τ like τ = 3 whereas, for such a value, the emulation states explose. x1 100 0 −100 x1 − ˆx1 x2 − ˆx2 x2 − ˆx2 60 40 20 0 150 100 50 0 −50 4 2 0 t 0 2 4 6 8 10 12 14 16 18 20 t (3a-d)-(4a-d) emulation 0 2 4 6 8 10 12 14 16 18 20 t (3a-d)-(4a-d) (3a-d)-(4a-d) emulation 0 2 4 6 8 10 12 14 16 18 20 emulation −2 1200 1250 1300 1350 1400 1450 1500 t Figure 1. State x 1 and convergence of the state observation errors ˜θ ˜θ 10 5 0 −5 −10 −15 −20 0 10 20 30 40 50 60 70 80 90 100 2.1 2.05 2 1.95 1.9 1.85 t t x 1 (3a-d)-(4a-d) emulation 200 400 600 800 1000 1200 1400 Figure 2. Convergence of the parameter estimate 6 Conclusions The adaptive observation of a class of continuous MIMO systems with sampled measurements has been realized. Assuming the input acting on the system to satisfy some persistent excitation conditions and the sampling steps to respect given bounds, a class of global exponential observers has been developed. References [1] B.D.O. Anderson, R.R. Bitmead, C.R. Johnson, P.V. Kokotović, R.L. Kosut, I.M.Y. Mareels, L. Praly, and B.D. Riedle. Stability of Adaptive Systems: Passivity and Averaging Analysis. Signal Processing, Optimization and Control, MIT Press, Cambridge, U.S.A., 1986. [2] M. Arcak and D. Nešić. A framework for nonlinear sampleddata observer design via approximate discrete-time models and emulation. Automatica, 40:1931–1938, 2004. [3] C.-M. Astorga, N. Othman, S. Othman, H. Hammouri, and T.-F. McKenna. Nonlinear continuous-discrete observers: applications to emulsion polymerization reactors. Control Eng. Practice, 10:3–13, 2002. [4] G. Bastin and M. Gevers. Stable adaptive observers for nonlinear time varying systems. IEEE Trans. on Aut. Control, 33(7):650–658, 1988. [5] G. Besançon. Remarks on nonlinear adaptive observer design. Syst. & Control Lett., 41(7):271–280, 2000. [6] G. Besançon, J. De León-Morales, and O. Huerta-Guevara. On adaptive observers for state affine systems. Int. J. of Contr., 79(6):581–591, 2006. [7] E. Bıyık and M. Arcak. A hybrid redesign of newton observers in the absence of an exact discrete-time model. Systems & Control Letters, 55:429–436, 2006. [8] G. Bornard, F. Celle, and N. Couenne. Regularly persistent observers for bilinear systems. In Proc. of the 29th International Conference on Nonlinear Systems, New Trends in Nonlinear Systems Theory, volume 122, pages 130–140, Nantes, France, 1988. [9] F. Celle, J.P. Gauthier, D. Kazakos, and G. Sallet. Synthesis of nonlinear observers: a harmonic-analysis approach. Math. Syst. Theory, 22:291–322, 1989. [10] Y. M. Cho and R. Rajamani. A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans. on Aut. Control, 42:534–537, 1997. [11] A.M. Dabroom and H.K. Khalil. Output feedback sampleddata control of nonlinear systems using high-gain observers. IEEE Trans. on Aut. Control, 46(11):1712–1725, 2001. [12] F. Deza, E. Busvelle, J.P. Gauthier, and D. Rakotopora. High gain estimation for nonlinear systems. Sys. & Control Lett., 18:295–299, 1992. [13] H. Hammouri, P. Kabore, S. Othman, and J. Biston. Failure diagnosis and nonlinear observer. application to a hydraulic process. J. of the Franklin Institute, 339:455–478, 2002. [14] H. Hammouri and J. DeLeon Morales. Chapter: Topological properties of observer’s inputs. In Progress in Syst. and Control Theory, Birkhauser, Boston, U.S.A., 2002. [15] H. Hammouri, M. Nadri, and R. Mota. Constant gain observer for continuous-discrete time uniformly observable systems. In Proc. of the 45th IEEE Conf. on Dec. & Control, pages 6240–6244, San Diego, U.S.A., 2006. [16] G. Kresselmeier. Adaptive observers with exponential rate of convergence. IEEE Trans. on Aut. Control, AC-22:2–8, 1977. [17] G. Lüders and K.S. Narendra. An adaptive observer and identifier for a linear system. IEEE Trans. on Aut. Control, AC-18:496–499, 1973. [18] R. Marino and P. Tomei. Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans. on Aut. Control, 40:1300–1304, 1995. [19] R. Marino and P. Tomei. Nonlinear control design. ser. Information and system sciences, Prentice Hall, London, U.K., 1995. [20] P.E. Moraal and J.W. Grizzle. Observer design for nonlinear systems with discret-time measurements. IEEE Trans. on Aut. Control, 40(3):395–404, 1995. 6
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N° D’ORDRE 9655 THÈSE DE DOCTOR
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« Tout est déjà dans l’air il
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ix Remerciements Je remercie sincè
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Table des matières xi Table des ma
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Table des matières xiii 5.3.2 Obse
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Contributions 1 Contributions Les c
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Publications 5 Publications La list
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11 Première partie Contributions
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24 Chapitre 1. Introduction CTD DTD
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Annexes
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168 Annexe A. Rappels mathématique
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170 Annexe B. Rappels de stabilité
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172 Annexe B. Rappels de stabilité
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174 Annexe C. Preuve de la Proposit
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