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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

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
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

motivation

?

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

-

+

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
The plant

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

Anti-windup for a class of neural network controlled systems

the plant6

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

the nominal controller linear control component

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

the nominal controller non linear control component

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

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

controller conditioning

-

+

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

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

controller conditioning assumptions

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

controller conditioning non linear control element

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

controller conditioning linear control element

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

controller conditioning aw compensator design target

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

controller conditioning aw compensator design target16

-

+

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

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

a simulation example18

Nominal linear Controller:

Nominal NN-Controller:

Gaussian Radial

Basis Function

A Simulation Example

Anti-windup for a class of neural network controlled systems

a simulation example19

Saturation limit:

Conditioning of NN-Controller:

Linear AW-Compensator design:

A Simulation Example

Anti-windup for a class of neural network controlled systems

a simulation example20
A Simulation Example

Position signal

Control signal

Anti-windup for a class of neural network controlled systems

a simulation example21
A Simulation Example

Position signal

Control signal

Anti-windup for a class of neural network controlled systems

conclusions
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

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