The University of Sheffield Department of Automatic Control

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

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

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. “Iteration technique for nonlinear systems and its applications to control theory” Monash University 2

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, 89103. “Iteration technique for nonlinear systems and its applications to control theory” Monash University 3

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

-Optimal Control Theory (Banks & Dinesh, 2000) -Nonlinear delay systems (Banks, 2002) -Theory of chaos (Banks & Mc. Caffrey, 1998) -Stability and spectral theory (Tomas-Rodriguez & Banks, 2003) -Design of Observers (Navarro Hernandez & Banks, 2003) “Iteration technique for nonlinear systems and its applications to control theory” Monash University 5

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 6

SYSTEM Inaccessible system state x OBSERVER Reconstructed state Fig. 1. 1 State Reconstruction Process (Open-loop) Objective Lineal invariant system: Auxiliary dynamical system: Mismatch Non-linear system: “Iteration technique for nonlinear systems and its applications to control theory” Monash University 7

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 8

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 9

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 10

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 11

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

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

b) Fig. 6 State X 2 and Estimate Fig. 5 State X 1 and Estimate “Iteration technique for nonlinear systems and its applications to control theory” Monash University 14

Fig. 7 State X 3 and Estimate Fig. 8 Error of Estimates “Iteration technique for nonlinear systems and its applications to control theory” Monash University 15

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. “Iteration technique for nonlinear systems and its applications to control theory” Monash University 16

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

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 18

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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 19

- 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 “Iteration technique for nonlinear systems and its applications to control theory” Monash University 20