Fault tolerant tracking control for continuous Takagi-Sugeno systems with time varying faults

Transcription

1 Fault tolerant tracking control for continuous Takagi-Sugeno systems with time varying faults Tahar Bouarar, Benoît Marx, Didier Maquin, José Ragot Centre de Recherche en Automatique de Nancy (CRAN) Nancy, France Mediterranean Conference on Control and Automation, June 2-23, 211 Aquis Corfu Holiday Palace, Corfu, Greece Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 1 / 39

2 Motivations Objective of diagnosis and fault tolerant control To detect and isolate the (actuator) fault and estimate its magnitude (diagnosis) To modify the control law to accomodate the fault Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 2 / 39

3 Motivations Objective of diagnosis and fault tolerant control To detect and isolate the (actuator) fault and estimate its magnitude (diagnosis) To modify the control law to accomodate the fault Difficulties Taking into account the system complexity in a large operating range Nonlinear behavior of the system The faults are time varying Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 2 / 39

4 Motivations Objective of diagnosis and fault tolerant control To detect and isolate the (actuator) fault and estimate its magnitude (diagnosis) To modify the control law to accomodate the fault Difficulties Taking into account the system complexity in a large operating range Nonlinear behavior of the system The faults are time varying Proposed strategy Takagi-Sugeno representation of nonlinear systems Observer-based fault tolerant control design Extension of the existing results on linear systems Consideration of an a priori model of the fault Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 2 / 39

5 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

6 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

7 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

8 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

9 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

10 Outline 1 Takagi-Sugeno approach for modeling 2 Observer and FTC law structures 3 A priori considered fault models 4 Controller design 5 Simulations results 6 Conclusions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

11 Takagi-Sugeno principle Operating range decomposition in several local zones. A local model represents the behavior of the system in a specific zone. The overall behavior of the system is obtained by the aggregation of the sub-models with adequate weighting functions. ξ 1 (t) ξ 1 (t) zone 1 Operating space zone 2 zone 3 Nonlinear system ξ 2 (t) zone 4 ξ 2 (t) Multiple Model representation Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 4 / 39

12 Takagi-Sugeno approach for modeling The main idea of Takagi-Sugeno approach Define local models M i, i = 1..r Define weighting functions µ i (ξ), µ i 1 Define an agregation procedure : M = µ i (ξ)m i Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 5 / 39

13 Takagi-Sugeno approach for modeling The main idea of Takagi-Sugeno approach Define local models M i, i = 1..r Define weighting functions µ i (ξ), µ i 1 Define an agregation procedure : M = µ i (ξ)m i 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. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 5 / 39

14 Takagi-Sugeno approach for modeling The main idea of Takagi-Sugeno approach Define local models M i, i = 1..r Define weighting functions µ i (ξ), µ i 1 Define an agregation procedure : M = µ i (ξ)m i 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. The difficulties How many local models? How to define the domain of influence of each local model? On what variables may depend the weighting functions µ i? Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 5 / 39

15 Takagi-Sugeno approach for modeling Obtaining a Takagi-Sugeno model Identification approach Choice of premise variables Choice of the number of modalities of each premise variables Choice of the structure of the local models Parameter identification Transformation of an a priori known nonlinear model Linearization around some well-chosen points Identification of the weighting function parameters to minimize the output error Nonlinear sector approach Rewriting of the model in a compact subspace of the state space ẋ(t)= f(x(t),u(t)) ẋ(t)= r µ i (ξ(t)) ( A i x(t)+b i u(t) ) i=1 y(t)=h(x(t),u(t)) y(t)= r µ i (ξ(t)) ( C i x(t)+d i u(t) ) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 6 / 39

16 Takagi-Sugeno approach for modeling Obtaining a Takagi-Sugeno model Identification approach Choice of premise variables Choice of the number of modalities of each premise variables Choice of the structure of the local models Parameter identification Transformation of an a priori known nonlinear model Linearization around some well-chosen points Identification of the weighting function parameters to minimize the output error Nonlinear sector approach Rewriting of the model in a compact subspace of the state space ẋ(t)= f(x(t),u(t)) ẋ(t)= r µ i (ξ(t)) ( A i x(t)+b i u(t) ) i=1 y(t)=h(x(t),u(t)) y(t)= r µ i (ξ(t)) ( C i x(t)+d i u(t) ) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 6 / 39

17 Takagi-Sugeno approach for modeling Obtaining a Takagi-Sugeno model Identification approach Choice of premise variables Choice of the number of modalities of each premise variables Choice of the structure of the local models Parameter identification Transformation of an a priori known nonlinear model Linearization around some well-chosen points Identification of the weighting function parameters to minimize the output error Nonlinear sector approach Rewriting of the model in a compact subspace of the state space ẋ(t)= f(x(t),u(t)) ẋ(t)= r µ i (ξ(t)) ( A i x(t)+b i u(t) ) i=1 y(t)=h(x(t),u(t)) y(t)= r µ i (ξ(t)) ( C i x(t)+d i u(t) ) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 6 / 39

18 Takagi-Sugeno approach for modeling Obtaining a Takagi-Sugeno model Identification approach Choice of premise variables Choice of the number of modalities of each premise variables Choice of the structure of the local models Parameter identification Transformation of an a priori known nonlinear model Linearization around some well-chosen points Identification of the weighting function parameters to minimize the output error Nonlinear sector approach Rewriting of the model in a compact subspace of the state space ẋ(t)= f(x(t),u(t)) ẋ(t)= r µ i (ξ(t)) ( A i x(t)+b i u(t) ) i=1 y(t)=h(x(t),u(t)) y(t)= r µ i (ξ(t)) ( C i x(t)+d i u(t) ) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 6 / 39

19 Takagi-Sugeno system Reference model ẋ(t)= r µ i (ξ(t))(a i x(t)+b i u(t)) i=1 y(t)= r µ i (ξ(t))(c i x(t)+d i u(t)) i=1 Interpolation mechanism r µ i (ξ(t))= 1 and µ i (ξ(t)) 1, t, i {1,...,r} i=1 The premise variable ξ(t) are measurable (like u(t), y(t)). Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 7 / 39

20 Takagi-Sugeno system Reference model ẋ(t)= r µ i (ξ(t))(a i x(t)+b i u(t)) i=1 y(t)= r µ i (ξ(t))(c i x(t)+d i u(t)) i=1 Interpolation mechanism r µ i (ξ(t))= 1 and µ i (ξ(t)) 1, t, i {1,...,r} i=1 The premise variable ξ(t) are measurable (like u(t), y(t)). The faulty system ẋ f (t)= r µ i (ξ(t))(a i x f (t)+b i u f (t)+g i f(t)) i=1 y f (t)= r µ i (ξ(t))(c i x f (t)+d i u f (t)+w i f(t)) i=1 f(t) represents the fault vector (to be detected and accomodated). Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 7 / 39

21 Fault tolerant control design Objective The objective is, in the one hand, to estimate the actuator fault f(t) and the state of the system x(t) (diagnosis) and, in the other hand, to reconfigure the control law allowing the convergence of x f (t) to x(t) (FTC). Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 8 / 39

22 Fault tolerant control design Objective The objective is, in the one hand, to estimate the actuator fault f(t) and the state of the system x(t) (diagnosis) and, in the other hand, to reconfigure the control law allowing the convergence of x f (t) to x(t) (FTC). Fault tolerant control scheme f(t) u(t) Fault Tolerant Controller Reference model x(t) + ˆx f (t) ˆf(t) Controller Observer u f (t) System y f (t) Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 8 / 39

23 Observer and FTC law structures Faulty system ẋ f (t)= r µ i (ξ(t))(a i x f (t)+b i u f (t)+g i f(t)) i=1 y f (t)= r µ i (ξ(t))(c i x f (t)+d i u f (t)+w i f(t)) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 9 / 39

24 Observer and FTC law structures Faulty system ẋ f (t)= r µ i (ξ(t))(a i x f (t)+b i u f (t)+g i f(t)) i=1 y f (t)= r µ i (ξ(t))(c i x f (t)+d i u f (t)+w i f(t)) i=1 PI Observer r ˆx f (t)= i=1 r ŷ f (t)= i=1 r ˆf (t)= µ i (ξ(t)) i=1 ( ) µ i (ξ(t)) A iˆx f (t)+b i u f (t)+g iˆf (t)+h 1 i (y f (t) ŷ f (t)) ) µ i (ξ(t)) (C iˆx f (t)+ D i u f (t)+w iˆf(t) ( ) Hi 2 (y f (t) ŷ f (t)) Hi 3 ˆf(t) Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 9 / 39

25 Observer and FTC law structures Faulty system ẋ f (t)= r µ i (ξ(t))(a i x f (t)+b i u f (t)+g i f(t)) i=1 y f (t)= r µ i (ξ(t))(c i x f (t)+d i u f (t)+w i f(t)) i=1 PI Observer r ˆx f (t)= i=1 r ŷ f (t)= i=1 r ˆf (t)= µ i (ξ(t)) i=1 ( ) µ i (ξ(t)) A iˆx f (t)+b i u f (t)+g iˆf (t)+h 1 i (y f (t) ŷ f (t)) ) µ i (ξ(t)) (C iˆx f (t)+ D i u f (t)+w iˆf(t) ( ) Hi 2 (y f (t) ŷ f (t)) Hi 3 ˆf(t) FTC law r ( u f (t)=u(t)+ µ i (ξ(t)) K i (x(t) ˆx f (t)) K fˆf ) i (t) i=1 Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 9 / 39

26 Considered faults Exponential faults f i (t)=e α it+β i, with α i,β i R,i = 1,...,q α i = α,i + α i α,i and α i representing respectively the nominal and the uncertain parts of α i Let us define : The uncertain part can be bounded as : α = diag(α 1,...,α q ) α = diag(α,1,...,α,q ) α = diag( α 1,..., α q ) ( α) T α ν where ν R q q is a known diagonal positive definite matrix. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 1 / 39

27 Controller design Estimation errors e p (t)=x(t) x f (t) : state tracking error e s (t)=x f (t) ˆx f (t) : state estimation error e d (t)=f (t) ˆf (t) : fault estimation error e y (t)=y f (t) ŷ f (t) : output error e u (t)=u(t) u f (t) : error between nominal and FTC law Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

28 Controller design Estimation errors e p (t)=x(t) x f (t) : state tracking error e s (t)=x f (t) ˆx f (t) : state estimation error e d (t)=f (t) ˆf (t) : fault estimation error e y (t)=y f (t) ŷ f (t) : output error e u (t)=u(t) u f (t) : error between nominal and FTC law Notation and hypothesis r X µ = µ(ξ(t))x i i=1 ḟ(t)=αf(t) Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

29 Controller design Estimation errors e p (t)=x(t) x f (t) : state tracking error e s (t)=x f (t) ˆx f (t) : state estimation error e d (t)=f (t) ˆf (t) : fault estimation error e y (t)=y f (t) ŷ f (t) : output error e u (t)=u(t) u f (t) : error between nominal and FTC law Notation and hypothesis r X µ = µ(ξ(t))x i i=1 ḟ(t)=αf(t) Dynamics of the estimation errors ė p (t)=a µ e p (t)+b µ e u (t) G µ f(t) ė s (t)=a µ e s (t)+g µ e d (t) H 1 µ ( e y(t) ) ė d (t)= H 2 µ e y (t) H 3 µ e d (t)+ α+ H 3 µ f (t) Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

30 Controller design Proposed approach Rather than replacing the expressions of e y (t) and e u (t) in the estimation errors, it is preferable to introduce them as static constraints (redundancy approach). Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

31 Controller design Proposed approach Rather than replacing the expressions of e y (t) and e u (t) in the estimation errors, it is preferable to introduce them as static constraints (redundancy approach). Introduction of virtual dynamics { ėy (t)= e y (t)+ C µ e s (t)+w µ e d (t) R p p ė u (t)= K f µ f(t)+k f µe d (t)+k µ e p (t)+ K µ e s (t)+ e u (t) R m m Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

32 Controller design Proposed approach Rather than replacing the expressions of e y (t) and e u (t) in the estimation errors, it is preferable to introduce them as static constraints (redundancy approach). Introduction of virtual dynamics { ėy (t)= e y (t)+ C µ e s (t)+w µ e d (t) R p p ė u (t)= K f µ f(t)+k f µe d (t)+k µ e p (t)+ K µ e s (t)+ e u (t) R m m Advantages It avoids the introduction of products between unknown matrices leading to BMI. The estimation error dynamics is described by a single sum rather than a double sum. It allows to consider nonlinear output equation of TS model without increasing the difficulties. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

33 Controller design Proposed approach Rather than replacing the expressions of e y (t) and e u (t) in the estimation errors, it is preferable to introduce them as static constraints (redundancy approach). Introduction of virtual dynamics { ėy (t)= e y (t)+ C µ e s (t)+w µ e d (t) R p p ė u (t)= K f µ f(t)+k f µe d (t)+k µ e p (t)+ K µ e s (t)+ e u (t) R m m Advantages It avoids the introduction of products between unknown matrices leading to BMI. The estimation error dynamics is described by a single sum rather than a double sum. It allows to consider nonlinear output equation of TS model without increasing the difficulties. Disdvantage The dynamics of the estimation errors is described by a descriptor system. However, this singular system is impulse-free. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

34 Controller design Estimation error system ) ẽ T (t) = (ep T (t) et s (t) et d (t) et y (t) et u (t) where E = diag ( I n I n I q p m ), Ã µ = E ẽ(t)=ãµ ẽ(t)+ B µ f(t) A µ B µ A µ G µ H 1 µ H 3 µ H 2 µ C µ W µ I K µ K µ Kµ f I B µ = G µ α+ H 3 µ K f µ Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

35 Controller design E ẽ(t)=ãµ ẽ(t)+ B µ f(t) Stability guarantee Quadratic Lyapunov function (tracking, state and fault estimation convergence) V ( e p (t),e s (t),e d (t) ) = ẽ T (t)epẽ(t) with EP = P T E Attenuation of the fault effect L 2 constraint where γ represents the attenuation level. t t ẽ T (τ)eẽ(τ)dτ γ 2 f T (τ)f(τ)dτ Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

36 Controller design Summary The tracking error e p (t), state e s (t) and fault e d (t) estimation errors must therefore satisfy the following inequality : ẽ T (t)epẽ(t)+ẽ T (t)ep ẽ(t)+ẽ T (t)eẽ(t) γ 2 f T (t)f(t)< This inequality is fulfilled if : ( ÃT µ P+ P T Ã µ + E B T µ P γ2 ) < Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

37 Controller design Summary The tracking error e p (t), state e s (t) and fault e d (t) estimation errors must therefore satisfy the following inequality : ẽ T (t)epẽ(t)+ẽ T (t)ep ẽ(t)+ẽ T (t)eẽ(t) γ 2 f T (t)f(t)< This inequality is fulfilled if : ( ÃT µ P+ P T Ã µ + E B T µ P γ2 ) < Structure of the Lyapunov matrix P 1 P 7 P = P 13 P 16 P 17 P 18 P 19 P 25 P 1 = P1 T, P 7 = P7 T, P 13 = P13 T, P 16, P 17, P 18, P 19 P 25 are free slack matrices with appropriate dimensions Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

38 Controller design Theorem 1 The system that describes the different estimation errors is stable and the L 2 -gain from the faults to the state tracking error, the state and fault estimation errors is bounded by γ, if there exists matrices P 1 = P1 T, P 7 = P7 T, P 13 = P13 T, P 16, P 17, P 18, P 19, P 25, F i, R i, S i, Q i and M i and positive scalars γ and τ such that the following LMI are verified, for i = 1,...,r : Υ 1,1 i Ci T P 16 Υ 2,2 i W T i P 16 Υ 3,2 i Υ 3,3 i P 16 Υ 4,2 i Υ 4,3 i Υ 4,4 Υ 5,1 i F i Q i Υ 5,5 G T i P 1 Υ 6,3 i Q T i (τ γ)i P 13 τνi < Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

39 Considered faults Polynomial faults f i (t)=λ i t+ δ i, with λ i,δ i R,i = 1,...,q As well as for exponential function, defining different diagonal matrices, λ = λ + λ, with λ verifying : ( λ) T λ ν where ν R q q is a known diagonal positive definite matrix. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

40 Controller design Fault estimation error ė d (t)= H 2 µ e y (t) H 3 µ e d (t)+h 3 µ f(t)+λ Estimation error system ẽ T (t) = ( ) ep T (t) es T (t) ed T (t) et y (t) eu T (t) where E = diag ( I n I n I q p m ), E ẽ(t)=ãµ ẽ(t)+ B µ f(t)+n à µ = A µ B µ A µ G µ H 1 µ H 3 µ H 2 µ C µ W µ I K µ K µ Kµ f I B µ = G µ H 3 µ K f µ N = λ Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

41 Simulation results Takagi-Sugeno model ẋ(t)= 2 µ i (ξ(u(t)))(a i x f (t)+b i u f (t)+g i f(t)) i=1 y(t)= 2 µ i (ξ(u(t)))(c i x f (t)+d i u f (t)+w i f (t)) i=1 with A 1 = B 2 = C 1 = , G 1 = 5 (.5.5.5, A 2 =, G 2 = ), C 2 = ( (.5, W 1 =.5 ), D 1 = µ 1 (u(t))= 1 tanh(.5 u(t)) 2 µ 2 (u(t))=1 µ 1 (u(t)). The nominal input signal is u(t) = sin(cos(.5t) t)., B 1 = ), W 2 = ( , ( 1.5 ), D 2 = ), (.8.5 ), Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

42 Simulation results Exponential fault Actual fault f(t)=e 1.15t 14 8s t 11s The controller and the observer are synthesized for α = 1 and α =.2. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 2 / 39

43 Simulation results Healthy system Faulty system Time (sec) FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

44 Simulation results.2 e s (t) f(t) Estimated f(t) u(t) u f (t) Time (sec) FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

45 Simulation results Exponential fault Actual fault f(t)=e 1.25t 14 8s t 11s The controller and the observer are synthesized for α = 1 and α =.3. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

46 Simulation results Time (sec) FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

47 Simulation results Time (sec) FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

48 Simulation results Polynomial fault Actual fault f(t)=.2t 1 9s t 12.5s The controller and the observer are synthesized for λ =.11 and λ =.25. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

49 Simulation results Healthy system states Faulty system states Time (sec) FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

50 Simulation results.5 e s (t) f(t) Estimated f(t) u(t) u f (t) Time (sec) FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

51 Simulation results Polynomial fault Actual fault f (t) 1 9s t 12.5s The controller and the observer are synthesized for λ =.11 and λ =.25. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

52 Simulation results Time FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED 11 3 / 39

53 Simulation results Time FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

54 Simulation results Polynomial fault Actual fault f (t)= sin(4t) 9s t 12.5s The controller and the observer are synthesized for λ =.11 and λ =.25. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

55 Simulation results Time FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

56 Simulation results Time FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

57 Simulation results Polynomial fault Actual fault f(t)=e 1.35t 14 8s t 11s The controller and the observer are synthesized for λ =.11 and λ =.25. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

58 Simulation results Time FIGURE: Reference model states vs. faulty system ones with FTC Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

59 Simulation results Time FIGURE: State estimation errors, fault and its estimation, nominal and FTC control inputs Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

60 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. Perspectives Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

61 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Perspectives Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

62 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Fault tolerant control with reference trajectory tracking Perspectives Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

63 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Fault tolerant control with reference trajectory tracking The problem of FTC design is expressed via an optimization problem subject to LMI (Linear Matrix Inequality) constraints. Perspectives Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

64 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Fault tolerant control with reference trajectory tracking The problem of FTC design is expressed via an optimization problem subject to LMI (Linear Matrix Inequality) constraints. Perspectives Study of the unmeasurable premise variable case (ξ(t) = x(t)). Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

65 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Fault tolerant control with reference trajectory tracking The problem of FTC design is expressed via an optimization problem subject to LMI (Linear Matrix Inequality) constraints. Perspectives Study of the unmeasurable premise variable case (ξ(t) = x(t)). Comparison with multiple integral observer approach Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

66 Conclusions and perspectives Conclusions Active fault tolerant control law for nonlinear systems represented by a Takagi-Sugeno structure. State and fault estimation are achieved simultaneously Fault tolerant control with reference trajectory tracking The problem of FTC design is expressed via an optimization problem subject to LMI (Linear Matrix Inequality) constraints. Perspectives Study of the unmeasurable premise variable case (ξ(t) = x(t)). Comparison with multiple integral observer approach Implementation of a bank of different controller each of them dedicated to a particular kind of fault and design of a switching control law depending on the measured performances. Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39

67 Get in touch Didier Maquin Professor in Automatic Control National Polytechnic Institute of Nancy High School of Electrical and Mechanical Engineering Research Center for Automatic Control More information? Personal : Research Lab : Didier Maquin (CRAN) Fault tolerant tracking control for TS systems MED / 39