1. Claudia Lizet Navarro Hernández PhD Student Supervisor: Professor S.P.Banks April 2004 Monash University Australia April 2004 The University of Sheffield Department of Automatic Control and Systems Engineering

2. 1.- Iteration Technique for Nonlinear Systems 2.- Design of Observers for Nonlinear Systems 3.- Fault Detection for Nonlinear Systems 4.- Summary and Conclusions 1 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

3. Having the nonlinear system where i=number of approximations, it can be shown that the solution of this sequence converges to the solution of the original nonlinear system if the Lipschitz condition is satisfied. and introducing the sequence of linear time varying equations: 2 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

4. 3 where is the solution of the original nonlinear system. - Tomas-Rodriguez, M., Banks, S., (2003) Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory, IMA Journal of Mathematical Control and Information, 20, 89- 103. is a Cauchy sequence and

5. Solution to Van der Pol oscillator and for the ith approximation, Solution and Approximations for the Van der Pol Oscillator 4 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

6. 5 -Optimal Control Theory (Banks & Dinesh, 2000) -Nonlinear delay systems (Banks, 2002) -Theory of chaos (Banks & McCaffrey, 1998) -Stability and spectral theory (Tomas-Rodriguez & Banks, 2003) -Design of Observers (Navarro Hernandez & Banks, 2003)

7. 1.- To represent a nonlinear system by a sequence of linear time-varying approximations 2.- To design an identity observer for linear time-varying systems 3.- To test the performance of the observer for the nonlinear system 6 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

8. Fig.1.1 State Reconstruction Process (Open-loop) Lineal invariant system: Auxiliary dynamical system: Mismatch Objective Non-linear system: Inaccessible system state x SYSTEM OBSERVER Reconstructed state 7 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

9. PROBLEM: Find state estimator Design proposed: “Design of a State Estimator for a Class of Time- Varying Multivariable Systems” (NGUYEN and LEE) Steps in design: 1.- Canonical transformation of the time-varying system 2.- Construction of a full order dynamical system 8 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

10. EXAMPLE: 1.- Canonical transformation a) Construction of the observability matrix b) Check for uniform observability c) Construction of an (n x n) matrix with rank n by eliminating the linearly dependent rows d) Construction of a transformation matrix 9 “Iteration technique for nonlinear systems and its applications to control theory” Monash University e) Transformation of the original system into an equivalent system

11. 2.- Construction of asymptotic estimator a) Choice of n stable eigenvalues for the state estimator b) Design of matrix such that is a constant matrix. c) Construction of the state estimator of form: d) Calculation of the estimate using the transformation matrix 10 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

12. Given a nonlinear system 1. Reduction to a sequence of linear time varying approximations 2. Design of observer at each time varying approximation 3. Test of observer at final approximation Proof as 11 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

13. EXAMPLES Fig. 1 State X1 and Estimate a) 12 “Iteration technique for nonlinear systems and its applications to control theory” Monash University Fig. 2 State X2 and Estimate

14. Fig. 4 Error of Estimates Fig. 3 State X3 and Estimate 13 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

15. b) Fig. 5 State X1 and Estimate Fig. 6 State X2 and Estimate 14 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

16. Fig. 7 State X3 and Estimate Fig. 8 Error of Estimates 15 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

17. 1.- To represent a nonlinear system by a sequence of linear time-varying approximations 2.- To design an unknown input observer for linear time-varying systems 3.- To apply the iteration technique to solve the nonlinear problem and test performance. 16 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

18. Lineal invariant system: Auxiliary dynamical system: Non-linear system: 17 “Iteration technique for nonlinear systems and its applications to control theory” Monash University Process Unknown Input Observer Measurements u fd Objectiveand or

19. 18 “Iteration technique for nonlinear systems and its applications to control theory” Monash University PROBLEM: Find linear observer and residual Where u control input v noise y measurement faults ( = target fault) w process noise faults directions Such that is primarly affected by the target fault and minimally by noises and nuissance faults

20. Given a nonlinear system 1. Reduction to a sequence of linear time varying approximations 2. Design of observer at each time varying approximation 3. Test of observer at final approximation in the presence of different target and nuissance faults 19 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

21. 20 “Iteration technique for nonlinear systems and its applications to control theory” Monash University - New method to study nonlinear systems by using known linear techniques - Nonlinear system replaced by a sequence of linear time-varying problems - The linear time-varying problem must have a solution