Control Based on Instantaneous Linearization

Eemeli Aroeemeli.aro@tkk.fi

16.11.2005

Structure

• Rationale• Instantaneous linearization• Controller implementation• Discussion

Rationale

• Linear control better understood than nonlinear control

• Many well-established linear design techniques exist

• Linearization only valid in a limited operating range

Extract a linear model from the current sample

Instantaneous linearization

Assume a deterministic neural network input-output model is available

with a regression vector

which is interpreted to define the state of the system

)()( tgty

Tmdtydtuntytyt )(),...,(),(),...,1()(

Instantaneous linearization

Then linearize g around (t=) to get an approximate model

where

)()()(~ iyityity

)()(

)()(

t

i itytg

a

)(~...)(~)(~...)1(~)( 01 mdtudtubntyatyaty n

)()(

)()(

t

i idtutg

b

)()()(~ iuituitu

Instantaneous linearization

Separating components of the current regression vector () into a bias term () :

where

)()()()()(1)(~ 11 tuqBqtyqAty d

)(...)1()()( 1 nyayayt n )(...)(0 mdubdub m

Instantaneous linearization

The coefficients {ai} and {bi} are collected into the polynomials

Thus the approximate model can be seen as a linear model affected by a constant disturbance ()

mm

nn

qbqbbqB

qaqaqA

...)(

...1)(1

101

11

1

Instantaneous linearization

For a multilayer perceptron network with one hidden layer on tanh units and a linear output,

01

01

)(tanh)(̂ WwtwWtyhn

jj

n

kkjkj

hn

jj

n

kkjkjij

i

wtwwWtty

10

1

2 )(tanh1)()(̂

Application to Control

Extractlinear model

Controldesign

Controller

System

Linearized model parameters

Input Output

ControllerParameters

Reference

Application to Control

• At each sample, extract a linear model from a neural network model of the system and design a linear controller

• Can be seen as a gain scheduling controller with an infinite schedule

Application to Control

• Structurally equivalent to an indirect self-tuning regulator, only difference is in how the linear model is extracted

• Control design based on certainty equivalence principle – the controller is designed assuming that the linear model perfectly describes the system

Application to Control

• Can implement any linear control design

• Need to compensate for bias term ()– e.g. by using integral action, which also

compensates for other constant disturbances

• Need to keep in mind narrow operating range of linearized model

Application to Control

• Pole placement design– Assuming a linearized deterministic

model

– The objective is to select the three polynomials R, S, and T so that the closed loop system will behave as

)()()()()(1)( 11 tuqBqtyqAty d

)()()(

)( 1

1

trqAqB

qtym

mdm

Application to Control

reference

System

1qΤ

1qS

1qR1

input outp

ut+

–

Application to Control

• Minimum variance design– For regulation, not trajectory following– Design the controller to minimize a

criterion J(t)– Generalized Minimum Variance

controller

where P, W and Q are rational transfer functions

tItuqQtrqWdtyqPtJ21211 )()()()()()()( E

Discussion

• Pro– Allows the use of linear design techniques– Reasonably simple implementation– Fast; linearization & design can be done

between samples– Allows control of systems with unstable

inverses (with approximate pole placement controller design without zero cancellation)

– Can be used to understand the dynamics of the system (poles, zeros, damping, natural frequency)

Discussion

• Con– Linearized model often valid only in a

narrow range– Can’t deal with hard nonlinearities– Requires understanding on linear control

theory

References

• M. Nørgaard, O. Ravn, N. K. Poulsen, and L. K. Hansen, "Neural Networks for Modelling and Control of Dynamic Systems," Springer-Verlag, London, 2000

• O. Ravn, "The NNCTRL Toolbox. Neural networks for control", Version 2, Technical University of Denmark, 2003, http://www.iau.dtu.dk/research/control/nnctrl.html