1. 1 Adaptive Control Neural Networks 13(2000): Neural net based MRAC for a class of nonlinear plants M.S. Ahmed

2. 2 Introduction Controller parameters are needed to be modified against the change in the operating point Response of a nonlinear plant generally cannot be shaped to a desired pattern using a linear controller One of the main difficulties in designing the nonlinear controller is the lack of a general structure for it

3. 3 Researches on Adaptive Neural Control of Nonlinear Dynamical Systems Narendra and Parthasarathy (1990): dynamic back-propagation for identification and control employing MFNN Chen and Khalil (1992, 1995): use of MFNN in adaptive control of feedback linearizable minimum phase plants represented by an input-output model (local convergence) Jagannathan and Lewis (1996a,b): use of MFNN in adaptive control of feedback linearizable plants with all states accessible (convergence to a stable solution through the Lyapunov approach) Sanner and Stoline (1992): use of Gaussian RBF for the adaptive control of feedback linearizable continuous time plants (globally stable under mild assumptions) Rovithakis and Christodoulou (1994):adaptive control through two step procedure employing a dynamic neural netwrok (convergence to zero error) Polycarpuo (1996): a more general class of feedback linearizable plants Ahmed (1994, 1995), Ahmed and Anjum (1997): adaptive control of nonlinear plants of unknown structure (local convergence) Wang, Liu, Harris and Brown (1995): indirect adaptive control of similar plants employing neural networks

4. 4 Adaptive Control of a SISO Plant If the nonlinear function is first order differentiable:

5. 5 The Control Scheme

6. 6 Artificial Neural Networks Advantages of neural networks: 1.Describing nonlinear functions 2.Robustness 3.Parallel architecture 4.Fault tolerant In the classical nonlinear approximation methods, a function is frequently approximated by a set of continuous known basis functions. When the basis functions are not known, the following methods will be used: Memory based methods Potential function techniques ANN

7. 7 Regulation It is assumed that there exist a parameter matrix such that the functions of desired controller can be perfectly described by the basis vector for all possible values of operating point.

8. 8 Algorithm Analysis

9. 9 State Tracking It is assumed that there exist a parameter matrix such that the functions of controller can be perfectly described by the basis vector for all possible values of operating point.

10. 10 Algorithm Analysis Closed loop system:

11. 11 Adaptive Control of a MIMO Plant Extension of the above development to MIMO nonlinear plants depends on the existence of a pseudo-linear plant model that warrants existence of a suitable Lyapunov function.

12. 12 Adaptive Control of a MIMO Plant

13. 13 Regulation

14. 14 Algorithm Analysis Closed loop system: Closed loop sub system equation:

15. 15 State Tracking

16. 16 Algorithm Analysis

17. 17 Remarks Theorems 1-4 do not imply that the controller parameters in  will converge to those in  *. In the state tracking it is also possible to take the regressor vector as [e T r T ] T instead of [x T r T ] T. It was assumed that there exists a parameter set for the nonlinear controller that can drive the control system to zero error. When this assumption does not hold only bounded error can be secured.

18. 18 Simulation Studies Example 1: SISO plant

19. 19 Simulation Studies MLP based adaptive control:

20. 20 Simulation Studies Example 2: MIMO plant

21. 21 Simulation Studies

22. 22 Simulation Studies

23. 23 Simulation Studies

24. 24 Conclusion A globally stable neural net MRAC for a class of nonlinear systems Nonlinear canonical state variable description Time varying pseudo-linear state feedback control gain generated from the output of a ANN set Plant need not be feedback linearizable Regulation and tracking schemes are proposed Adaptation of the controller parameters based on Lyapunov function to ensure global convergence Extension to MIMO plants Simulation studies Drawback of the proposed controller: –In some cases a large number of adjustable controller parameters may be needed