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

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Performance oriented anti windup for a class of neural network controlled systems l.jpg

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


Slide2 l.jpg

  • Motivation network controlled systems

  • 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


Motivation l.jpg

? network controlled systems

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


Motivation principle of anti windup compensation l.jpg

- network controlled systems

+

Linear

AW-Compen-

sator

Motivation: Principle of anti-windup compensation

Linear

Controller

Linear

Plant

Anti-windup for a class of neural network controlled systems


The plant l.jpg
The plant network controlled systems

Stable, minimum-phase, strictly proper with matched nonlinear disturbance f(y)

Anti-windup for a class of neural network controlled systems


The plant6 l.jpg

The disturbance is continuous in network controlled systemsy 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


The nominal controller linear control component l.jpg

is assumed to be Hurwitz stable network controlled systems

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


The nominal controller non linear control component l.jpg

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


The nominal controller l.jpg

can asymptotically track modeling error thesignal yd so thatthe controller error:

becomes zero.

Theestimation error

remainsbounded.

The Nominal Controller

Anti-windup for a class of neural network controlled systems


Controller conditioning l.jpg

- modeling error

+

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


Controller conditioning preliminaries l.jpg

Symmetric modeling error 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


Controller conditioning assumptions l.jpg

Linear modeling error

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


Controller conditioning non linear control element l.jpg

is a small design dependent constant modeling error

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


Controller conditioning linear control element l.jpg

with modeling error

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


Controller conditioning aw compensator design target l.jpg

z modeling error

-

+

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


Controller conditioning aw compensator design target16 l.jpg

- modeling error

+

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


A simulation example l.jpg

The nominal model used for linear controller design modeling error

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


A simulation example18 l.jpg

Nominal linear Controller: modeling error

Nominal NN-Controller:

Gaussian Radial

Basis Function

A Simulation Example

Anti-windup for a class of neural network controlled systems


A simulation example19 l.jpg

Saturation limit: modeling error

Conditioning of NN-Controller:

Linear AW-Compensator design:

A Simulation Example

Anti-windup for a class of neural network controlled systems


A simulation example20 l.jpg
A Simulation Example modeling error

Position signal

Control signal

Anti-windup for a class of neural network controlled systems


A simulation example21 l.jpg
A Simulation Example modeling error

Position signal

Control signal

Anti-windup for a class of neural network controlled systems


Conclusions l.jpg
Conclusions modeling error

  • 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


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