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

2. Anti-windup for a class of neural network controlled systems 2 1.Motivation 2.The plant: A linear plant with matched unknown non-linearities 3.The nominal control system: Linear Control with augmented NN- controller for disturbance rejection 4.Controller conditioning for anti-windup: –Preliminaries: Constrained multi-variable systems –Non-linear Controller Conditioning –Linear Controller Conditioning 5.An Example 6.Conclusions

3. Anti-windup for a class of neural network controlled systems 3 Unknown Nonlinearity + + Motivation Linear Plant Linear Controller + NN compen- sation - Adap- tation 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. ? 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). (AW) 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.

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

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

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

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

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

9. Anti-windup for a class of neural network controlled systems 9 The Nominal Controller can asymptotically track the signal y d so that the controller error: becomes zero. The estimation error remains bounded.

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

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

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

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

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

15. Anti-windup for a class of neural network controlled systems 15 Controller conditioning – AW-Compensator Design Target + NN compen- sation - Unknown Nonlinearity + + Linear Plant Linear Controller Adap- tation Linear AW-comp. + - Non- linear Algorithm + + - w z d y Linear AW-comp. where is a designer chosen performance output Design target for linear AW-compensator: Minimize  for This L 2 -gain optimization target ensures recovery of the nominal controller performance.

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

17. Anti-windup for a class of neural network controlled systems 17 A Simulation Example Simulation for a direct drive DC-torque motor [12] Hsieh & Pan (2000) Hsieh & Pan (2000) [12]: 6-th order model to include issues of static friction, i.e. the pre-sliding behaviour: The nominal model used for linear controller design Other parameters: Assume both angle position x 1 and angle velocity x 2 are measurable

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

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

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

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

22. Anti-windup for a class of neural network controlled systems 22 Conclusions I.Development of a conditioning method for a linear controller & robust NN-controller combination: 1.Nominal NN-controller: Add-on to a linear controller for compensation of matched unknown non-linearities/disturbances 2.Linear controller conditioning: Specially structured AW-controller (considering former results) 3.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. II.Design target: 1.Retain asymptotic tracking for permissible demands and keep NN-estimates bounded 2.Optimization of linear AW-controller according to an L 2 -constraint III.Simulation Result: Performance similar for un/conditioned controller