1

Adaptive Control

Neural Networks 13(2000):

Neural net based MRACfor a class of nonlinear plants

M.S. Ahmed

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

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

Adaptive Control of a SISO Plant

If the nonlinear function is first order differentiable:

5

The Control Scheme

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

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

Algorithm Analysis

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

Algorithm AnalysisClosed loop system:

11

Adaptive Control of a MIMO PlantExtension 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

Adaptive Control of a MIMO Plant

13

Regulation

14

Algorithm AnalysisClosed loop system:

Closed loop sub system equation:

15

State Tracking

16

Algorithm Analysis

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 [eT rT]T instead of [xT rT]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

Simulation StudiesExample 1: SISO plant

19

Simulation StudiesMLP based adaptive control:

20

Simulation StudiesExample 2: MIMO plant

21

Simulation Studies

22

Simulation Studies

23

Simulation Studies

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