Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi Sugeno model with unmeasurable premise variables

Transcription

1 Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi Sugeno model with unmeasurable premise variables Dalil Ichalal, Benoît Marx, José Ragot and Didier Maquin Centre de Recherche en Automatique de Nancy (CRAN) Nancy-University, CNRS 17th Mediterranean Conference on Control and Automation MED 09 June 24-26, 2009, Makedonia Palace, Thessaloniki, Greece Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

2 Motivations Objective Estimate simultanuously the state and unknown inputs of nonlinear systems described by a Takagi-Sugeno model using a Proportional Integral Observer (PIO) and Proportional Multiple Integral Observer (PMIO) Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

3 Motivations Objective Estimate simultanuously the state and unknown inputs of nonlinear systems described by a Takagi-Sugeno model using a Proportional Integral Observer (PIO) and Proportional Multiple Integral Observer (PMIO) Difficulties Unmeasurable premise variables Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

4 Outline 1 Takagi-Sugeno approach for modeling Takagi-Sugeno principle Takagi-Sugeno model 2 Proportional Integral Observer design 3 State and unknown input estimation 4 Numerical example 5 Conclusions Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

5 Outline 1 Takagi-Sugeno approach for modeling Takagi-Sugeno principle Takagi-Sugeno model 2 Proportional Integral Observer design 3 State and unknown input estimation 4 Numerical example 5 Conclusions Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

6 Outline 1 Takagi-Sugeno approach for modeling Takagi-Sugeno principle Takagi-Sugeno model 2 Proportional Integral Observer design 3 State and unknown input estimation 4 Numerical example 5 Conclusions Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

7 Outline 1 Takagi-Sugeno approach for modeling Takagi-Sugeno principle Takagi-Sugeno model 2 Proportional Integral Observer design 3 State and unknown input estimation 4 Numerical example 5 Conclusions Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

8 Outline 1 Takagi-Sugeno approach for modeling Takagi-Sugeno principle Takagi-Sugeno model 2 Proportional Integral Observer design 3 State and unknown input estimation 4 Numerical example 5 Conclusions Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

9 Takagi-Sugeno approach for modeling Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

10 Takagi-Sugeno principle Operating range decomposition in several local zones. A local model represent the behavior of the system in each zone. The overall behavior of the system is obtained by the aggregation of the sub-models with adequate weighting functions. ξ1(t) ξ1(t) zone 1 Operating zone 2 space zone 3 Nonlinear system ξ2(t) zone 4 ξ2(t) Multiple model representation Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

11 Takagi-Sugeno approach for modeling Interests of Takagi-Sugeno approach Simple structure for modeling complex nonlinear systems. The specific study of the nonlinearities is not required. Possible extension of the theoretical LTI tools for nonlinear systems. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

12 Takagi-Sugeno model The considered system is given by the following equations ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)) y(t) = r µ i (ξ(t))(c i x(t)+d i u(t)) r Interpolation mechanism : µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,...,r} The premise variable ξ(t) can be measurable (input u(t),output y(t),...) or unmeasurable (state x(t),...). Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

13 Takagi-Sugeno model The considered system is given by the following equations ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)) y(t) = r µ i (ξ(t))(c i x(t)+d i u(t)) r Interpolation mechanism : µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,...,r} The premise variable ξ(t) can be measurable (input u(t),output y(t),...) or unmeasurable (state x(t),...). Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

14 Takagi-Sugeno model The considered system is given by the following equations ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)) y(t) = r µ i (ξ(t))(c i x(t)+d i u(t)) r Interpolation mechanism : µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,...,r} The premise variable ξ(t) can be measurable (input u(t),output y(t),...) or unmeasurable (state x(t),...). Interests of T-S model with unmeasurable premise variables Exact representation of the model ẋ(t) = f(x(t), u(t)). Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

15 Takagi-Sugeno model The considered system is given by the following equations ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)) y(t) = r µ i (ξ(t))(c i x(t)+d i u(t)) r Interpolation mechanism : µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,...,r} The premise variable ξ(t) can be measurable (input u(t),output y(t),...) or unmeasurable (state x(t),...). Interests of T-S model with unmeasurable premise variables Exact representation of the model ẋ(t) = f(x(t), u(t)). Only one T-S model for FDI with observer banks. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

16 Takagi-Sugeno model The considered system is given by the following equations ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)) y(t) = r µ i (ξ(t))(c i x(t)+d i u(t)) r Interpolation mechanism : µ i (ξ(t)) = 1 and 0 µ i (ξ(t)) 1, t, i {1,...,r} The premise variable ξ(t) can be measurable (input u(t),output y(t),...) or unmeasurable (state x(t),...). Interests of T-S model with unmeasurable premise variables Exact representation of the model ẋ(t) = f(x(t), u(t)). Only one T-S model for FDI with observer banks. Exemple of application: security improvement in cryptography systems. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

17 Proportional Integral Observer design Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

18 Proportional Integral Observer design Let us consider the T-S system represented by ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

19 Proportional Integral Observer design Let us consider the T-S system represented by ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) Assumption 1 Assume that the following assumptions hold: A1. The system is stable A2. The signals u(t), d(t) and ω(t) are bounded. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

20 Proportional Integral Observer design Let us consider the T-S system represented by ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) Assumption 1 Assume that the following assumptions hold: A1. The system is stable A2. The signals u(t), d(t) and ω(t) are bounded. Assumption 2 The unknown inputs d(t) are assumed to be constant: A3. ḋ = 0 (This assumption will be relaxed in the PMIO design). Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

21 Proportional Integral Observer design Let us denote the estimated state by ˆx. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

22 Proportional Integral Observer design Let us denote the estimated state by ˆx. The system can be re-written as follows: where: ν = r ẋ = µ i (ˆx)(A i x + B i u + E i d + W i ω + ν) r (µ i (x) µ i (ˆx))(A i x + B i u + E i d + W i ω) Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

23 Proportional Integral Observer design Let us denote the estimated state by ˆx. The system can be re-written as follows: where: ν = r ẋ = µ i (ˆx)(A i x + B i u + E i d + W i ω + ν) r (µ i (x) µ i (ˆx))(A i x + B i u + E i d + W i ω) The assumption A3. allows to write the system in the following augmented form: ẋ a = r µ i (ˆx)(Ãi x a + B ) i u + Γ i ω y = Cx a + D ω where: [ Ai E Ã i = i 0 0 ] [ Bi, Bi = 0 ] [ I Wi, Γi = 0 0 C = [ C G ] [ ] [ ] x, D = 0 W, xa = d ] [ ] ν, ω = ω Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

24 Proportional Integral Observer design The proposed PI observer is given by the following equations: ˆx = r ( ) µ i (ˆx) A iˆx + B i u + E i ˆd + KPi (y ŷ) ŷ = Cˆx + Gˆd ˆd = r µ i (ˆx)K Ii (y ŷ) where ˆx and ˆd are the estimates of x and d. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

25 Proportional Integral Observer design The proposed PI observer is given by the following equations: ˆx = r ( ) µ i (ˆx) A iˆx + B i u + E i ˆd + KPi (y ŷ) ŷ = Cˆx + Gˆd ˆd = r µ i (ˆx)K Ii (y ŷ) where ˆx and ˆd are the estimates of x and d. A similar reasoning makes it possible to transform the proposed PI observer in the following augmented form: ˆx a = r µ i (ˆx)(Ãiˆx a + B i u + K ) i (y ŷ) ŷ = Cˆx a where: K i = [ KPi K Ii ] Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

26 Proportional Integral Observer design Augmented state estimation error e a (t) = x a (t) ˆx a (t) Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

27 Proportional Integral Observer design Augmented state estimation error e a (t) = x a (t) ˆx a (t) r ė a = µ i (ˆx) ((Ãi K i C)ea +( Γ i K ) i D) ω Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

28 Proportional Integral Observer design Augmented state estimation error e a (t) = x a (t) ˆx a (t) r ė a = µ i (ˆx) ((Ãi K i C)ea +( Γ i K ) i D) ω Objectives The goal is to determine the gain matrices K i such that: The system which generates the state and unknown inputs errors is stable. lim t e(t) = 0, pour ω 2 = 0 e(t) 2 < γ ω 2 pour ω 2 0 Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

29 Proportional Integral Observer design Lemma 1 [Tanaka and Wang 2001] Consider the continuous-time TS-system defined by: ẋ(t) = r µ i (x(t))(a i x(t)+b i u(t)) y(t) = Cx(t) The system is stable and verifies the L 2 -gain condition: y(t) 2 < γ u(t) 2 if there exists a symmetric positive definite matrix P such that the following LMIs are satisfied for i = 1,...,r: [ A T i P + PA i + C T ] C PB i Bi T P γ 2 < 0 I Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

30 Proportional Integral Observer design Theorem 1: PIO The PI observer for the system is determined by minimizing γ under the following LMI constraints in the variables P = P T > 0, M i and γ for i = 1,...,r: [ ÃT i P + PÃi M i C CT M T i + I P Γ i M i D Γ T i P D T M T i γi The gains of the observer are derived from: and the attenuation level is calculated by: K i = P 1 M i γ = γ ] < 0 Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

31 Proportional Multiple Integral Observer design Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

32 Proportional Multiple Integral Observer design Let us consider the T-S system represented by ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

33 Proportional Multiple Integral Observer design Let us consider the T-S system represented by ẋ(t) = r µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) Assumption 3 The unknown input is assumed to be a bounded time varying signal with null q th derivative: A4. d (q) (t) = 0 Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

34 Proportional Multiple Integral Observer design Consider the generalization of the proportional multiple-integral observer to T-S systems of the PMI observer: ˆx = r µ i (ˆx)(A iˆx + B i u + E i ˆd0 + K Pi (y ŷ)) ŷ = Cˆx + Gˆd ˆd 0 = r µ i (ˆx)K Ii 0(y ŷ)+ ˆd 1 ˆd 1 = r µ i (ˆx)K Ii 1(y ŷ)+ ˆd 2. ˆd q 2 = r µ i (ˆx)K q 2 Ii (y ŷ)+ ˆd q 1 ˆd q 1 = r µ i (ˆx)K q 1 Ii (y ŷ) where ˆd i, i = 1,2,...,(q 1) are the estimation of the (q 1) first derivatives of the unknown input d(t). Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

35 Proportional Multiple Integral Observer design The state and unknown inputs estimation errors are: e = x ˆx, e 0 = ḋ ˆd0,..., e q 1 = ḋq 1 ˆdq 1 Their dynamics are given in the following form: ė = r µ i (ˆx)((A i K Pi C)e+(Γ i K Pi W) ω +(Ei K Pi G)e 0 ) ė 0 = r µ i (ˆx)( K Ii 0Ce+ e 1 KIi 0 W ω KIi 0Ge 0) ė 1 = r µ i (ˆx)( K Ii 1Ce+ e 2 KIi 1 W ω KIi 1Ge 0). where: ė q 2 = r µ i (ˆx)( K 0 ė q 1 = r µ i (ˆx)( K q 1 Ii Ce+ e q 1 K q 2 Ii Ii Ce KIi 0 W ω K q 2 Ii Ge 0 ) W ω K q 1 Ii Ge 0 ) Γ i = [ I n W i ], W = [ 0 W ] Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

36 Proportional Multiple Integral Observer design In the augmented form, we have: [ e e0 ẽ = ] r µ i (ˆx)((Ãi K i C)ẽ+( Γ i K i W) ω) = Cẽ where: ẽ = e e 0 e 1. e q 2 e q , Ã i =... [ K i = KPi T KIi 0 T K 1T Ii A i E i I s I s K q 1T ] T Ii... K q 2T Ii C = [ C G ], Γ i = [ Γ T i ] T Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

37 Proportional Multiple Integral Observer design In the following, we are only interested with particular components e and e 0 of ẽ: [ ] e = Cẽ e0 where: [ C = I n 0 0 I s represents null matrix with appropriate dimensions. ] Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

38 Proportional Multiple Integral Observer design Theorem 2 The PMI observer for the system that minimizes the transfer from ω(t) to [e(t) T e 0 (t) T ] is obtained by finding the matrices P = P T > 0, M i and γ that minimize γ under the following LMI constraints for i = 1,...,r: [ ÃT i P + PÃi M i C CT Mi T + C T C P Γ i M i W Γ T i P W T Mi T γi The gains of the observer are derived from: and the attenuation level is calculated by: K i = P 1 M i γ = γ ] < 0 Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

39 Numerical example Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

40 Numerical example Consider a continuous-time T-S system defined by: A 1 = , A 2 = and d(t) = [d 1 (t) T d 2 (t) T ] T. B 1 = E 2 = , B 2 = 3 1 7, W 1 = W 2 = [ C = 1 0 1, E 1 = ] [ 5 0, G = ,, W = [ ], ], Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

41 Numerical example The weighting functions depend on the first component x 1 of the state vector x and are defined as follows: { µ1 (x) = 1 tanh(x 1) 2 (1) µ 2 (x) = 1 µ 1 (x) The weighting functions obtained without perturbations and unknown inputs are shown in figure 1. This figure shows that the system is clearly nonlinear since µ 1 and µ 2 are not constant functions µ 1 (t) µ 2 (t) Figure: Weighting functions µ 1 and µ 2 Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

42 Numerical example 3 2 original d 1 estimated d original d 2 estimated d Figure: Unknown input estimation with a PI observer Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

43 Numerical example 3 2 original d 1 estimated d original d 2 estimated d Figure: Unknown input estimation with a PMI observer Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

44 Numerical example 0.5 e 1 0 e 2 e Figure: State estimation error with a PI observer 0.5 e 1 0 e 2 e Figure: State estimation error with a PMI observer Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

45 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Perspectives Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

46 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Perspectives Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

47 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Comparison between the results obtained by PI and PMI observers Perspectives Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

48 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Comparison between the results obtained by PI and PMI observers Pole assignement in a specific region (see the paper). Perspectives Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

49 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Comparison between the results obtained by PI and PMI observers Pole assignement in a specific region (see the paper). Perspectives Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

50 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Comparison between the results obtained by PI and PMI observers Pole assignement in a specific region (see the paper). Perspectives Study and reduction of the conservatism when searching a common Lyapunov matrix P, which satisfies r LMIs. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

51 Conclusions and perspectives Conclusions PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno structure. Study of the case where the premise variables are unmeasurable Comparison between the results obtained by PI and PMI observers Pole assignement in a specific region (see the paper). Perspectives Study and reduction of the conservatism when searching a common Lyapunov matrix P, which satisfies r LMIs. Extension to fault tolerant control of nonlinear systems. Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29

52 Thank you for your attention! Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED / 29