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Claudia Lizet Navarro Hernández PhD Student

The University of Sheffield Department of Automatic Control and Systems Engineering. Iteration Technique for Nonlinear Systems and its Application to Control Theory. Claudia Lizet Navarro Hernández PhD Student. Supervisor: Professor S.P.Banks. April 2004.

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Claudia Lizet Navarro Hernández PhD Student

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  1. The University of Sheffield Department of Automatic Control and Systems Engineering Iteration Technique for Nonlinear Systems and its Application to Control Theory Claudia Lizet Navarro Hernández PhD Student Supervisor: Professor S.P.Banks April 2004 Monash University Australia April 2004

  2. Contents 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. 1.- Iteration Technique Having the nonlinear system and introducing the sequence of linear time varying equations: 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. 2 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

  4. Proof is a Cauchy sequence and 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. 3 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

  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. Applications to Control Theory • 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) 5 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

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

  9. Observer for linear time-varying systems 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 e) Transformation of the original system into an equivalent system 9 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

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

  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. 6 State X2 and Estimate Fig. 5 State X1 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. 3.- Application to Fault Detection 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. f d Process Measurements u Unknown Input Observer Objective and or Unknown input observer for fault detection Lineal invariant system: Auxiliary dynamical system: Non-linear system: 17 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

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

  20. Fault detection observer for Nonlinear Systems 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. 4.- Summary and Conclusions - 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 20 “Iteration technique for nonlinear systems and its applications to control theory” Monash University

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