Performance oriented anti-windup for a class of neural network controlled systems

1. Performance oriented anti-windup for a class of neural network controlled systems SWAN 2006 - Automation and Robotics Research Institute, UTA G. Herrmann M. C. Turner and I. Postlethwaite Control and Instrumentation Research Group University of Leicester

2. Motivation • The plant: A linear plant with matched unknown non-linearities • The nominal control system: Linear Control with augmented NN-controller for disturbance rejection • Controller conditioning for anti-windup: • Preliminaries: Constrained multi-variable systems • Non-linear Controller Conditioning • Linear Controller Conditioning • An Example • Conclusions Anti-windup for a class of neural network controlled systems

3. ? Linear Controller + + + - Unknown Nonlinearity Adap- tation NN compen- sation Motivation Linear Plant Linear control performance in combination with NN-control – Examples of practical validation: G. Herrmann, S. S. Ge, and G. Guo, “Practical implementation of a neural network controller in a hard disk drive,” IEEE Transactions on Control Systems Technology, 2005. ——, “A neural network controller augmented to a high performance linear controller and its application to a HDD-track following servo system,” IFAC 2005 (under journal review). Anti-Windup (AW) Control - a possible approach to overcome controller saturation G. Grimm, J. Hatfield, I. Postlethwaite, A. R. Teel, M. C. Turner, and L. Zaccarian, “Antiwindup for stable linear systems with input saturation: An LMI based synthesis,” IEEE Trans. on Autom. Control, vol. 48, no. 9, pp. 1509–1525, 2003. Alternative for NN: W. Gao; R.R. Selmic, "Neural network control of a class of nonlinear systems with actuator saturation Neural Networks", IEEE Trans. on Neural Networks, Vol. 17, No. 1, 2006. NN-Control- Examples : S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. World Scientific, Singapore, 1998. Y. Kim and F.L. Lewis, High-Level Feedback Control with Neural Networks," World Scientific, Singapore, 1998. Anti-windup for a class of neural network controlled systems

4. - + Linear AW-Compen- sator Motivation: Principle of anti-windup compensation Linear Controller Linear Plant Anti-windup for a class of neural network controlled systems

5. The plant Stable, minimum-phase, strictly proper with matched nonlinear disturbance f(y) Anti-windup for a class of neural network controlled systems

6. The disturbance is continuous in y and bounded: so that it can be arbitrarily closely modelled by a neural network approach: - neural network basis function vector, - neural network modelling error - optimal (constant) weight matrix The plant Anti-windup for a class of neural network controlled systems

7. is assumed to be Hurwitz stable The linear controller component defines the closed loop steady state: and the controller error: The Nominal Controller – Linear Control Component d - exogenous demand signal Anti-windup for a class of neural network controlled systems

8. discontinuous sliding mode component - compensates for modeling error e estimate - compensates for non-linearity is a design parameter Estimation algorithm: is symmetric, positive definiteLearning Coefficient Matrix - Estimation error The Nominal Controller – Non-Linear Control Component Anti-windup for a class of neural network controlled systems

9. can asymptotically track thesignal yd so thatthe controller error: becomes zero. Theestimation error remainsbounded. The Nominal Controller Anti-windup for a class of neural network controlled systems

10. - + Linear AW-comp. + - + + NN compen- sation Unknown Nonlinearity Non- linear Algorithm Controller conditioning Linear Controller Linear Plant Adap- tation Anti-windup for a class of neural network controlled systems

11. Symmetric Multi-variable Saturation Function: The Deadzone - Counter-part of a Saturation Function: Controller conditioning - Preliminaries Multi-variable Saturation Function: Anti-windup for a class of neural network controlled systems

12. Linear Controller + + - + Unknown Nonlinearity Adap- tation Disturbance Limit NN compen- sation The controller amplitude is large enough to compensate for the unknown non-linearity. Permissible Range of Tracking Control System small design parameter Controller conditioning - Assumptions Linear Plant Saturation Limit: We do not assume that the transient behaviour has to satisfy this constraint. Anti-windup for a class of neural network controlled systems

13. is a small design dependent constant NN-control is used The NN-controller is cautiously disabled and replaced by a high gain controller. The NN-estimation algorithm is slowed down. Controller conditioning – Non-linear Control Element Anti-windup for a class of neural network controlled systems

14. with compensation signals compensation in practice 0 AW-compensator: to be designed Closed Loop: Note that The control limits are satisfied Controller conditioning – Linear Control Element Linear controller Anti-windup for a class of neural network controlled systems

15. z - + w NN compen- sation Non- linear Algorithm Controller conditioning – AW-Compensator Design Target Design target for linear AW-compensator: Minimize g for where is a designer chosen performance output Linear AW-comp. Linear AW-comp. - + d Linear Plant + Linear Controller - + y + + Unknown Nonlinearity Adap- tation This L2-gain optimization target ensures recovery of the nominal controller performance. Anti-windup for a class of neural network controlled systems

16. - + NN compen- sation Non- linear Algorithm Controller conditioning – AW-Compensator Design Target Design target for overall AW-compensator: The conditioned linear control uL term operating in connection with the constrained NN-controller uNL, will track asymptotically any permissible steady state. The NN-weight estimates will remain bounded. Linear AW-comp. - + d Linear Plant + Linear Controller - + y + + Unknown Nonlinearity Adap- tation Anti-windup for a class of neural network controlled systems

17. The nominal model used for linear controller design Other parameters: A Simulation Example Hsieh & Pan (2000) [12]: 6-th order model to include issues of static friction, i.e. the pre-sliding behaviour: [12] Hsieh & Pan (2000) Simulation for a direct drive DC-torque motor Assume both angle position x1 and angle velocity x2 are measurable Anti-windup for a class of neural network controlled systems

18. Nominal linear Controller: Nominal NN-Controller: Gaussian Radial Basis Function A Simulation Example Anti-windup for a class of neural network controlled systems

19. Saturation limit: Conditioning of NN-Controller: Linear AW-Compensator design: A Simulation Example Anti-windup for a class of neural network controlled systems

20. A Simulation Example Position signal Control signal Anti-windup for a class of neural network controlled systems

21. A Simulation Example Position signal Control signal Anti-windup for a class of neural network controlled systems

22. Conclusions • Development of a conditioning method for a linear controller & robust NN-controller combination: • Nominal NN-controller: Add-on to a linear controller for compensation of matched unknown non-linearities/disturbances • Linear controller conditioning: Specially structured AW-controller (considering former results) • NN-controller conditioning: The unknown non-linearity is bounded and can be counteracted by a variable structure component; once the NN-controller exceeds the bound. • Design target: • Retain asymptotic tracking for permissible demands and keep NN-estimates bounded • Optimization of linear AW-controller according to an L2-constraint • Simulation Result: Performance similar for un/conditioned controller Anti-windup for a class of neural network controlled systems