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 Automation and Robotics Research Institute, UTA

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

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, Y. Kim and F.L. Lewis, High-Level Feedback Control with Neural Networks," World Scientific, Singapore, ? 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, ——, “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, 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.

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

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 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:

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:

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

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.

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. + -

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:

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

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

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

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.

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

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

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

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

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

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

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