The Adaptive Deadbeat Controller

Copyright © [FAC Adaptive Systems in Control and Signal Processing, Glasgow, UK, 1989

THE ADAPTIVE DEADBEAT CONTROLLER A. Walsh Severn-Trent Laboratories, Coventry Laboratory, St. Martins Road, Finham , Coventry, UK

Abstract, A reduced computational adaptive controller is presented, suitable for time varying stable non minimwn phase systems. By linking a pole placement technique to a model reference system,

deadbeat

significant computational

response .

controller,

Keywords.

The

paper

reports

on

tuning of

predictive

theory

and

stability

chosen has a

criteria

of

such

a

Adaptive control ; control system synthesis; controllers; pole placement.

The coefficients of the model are calculated off-line (Cadzow. 1970) , using any linear plant

single

input ,

single

output

of

self tuner. control

using

a

n,

and

given

the

number

of

steps

to

providing N is greater or equal to

n. In practice , a range of model coefficients for a var lable choice of N can be stored in computer memory, its selection dependent on the

They are those based on theory

order

deadbeat N.

plant has received considerable attention , resulting in the emergence of two important

classes of

the

and example responses are included.

INTRODUCTION The self

savings can be made where the model

linear

quadratic control law proposed by Clarke (1975), and those based on pole placement adopted by Wellstead (1979).

order of the plant to be controlled and control signal amplitude constraints . To clarify the des.gn explanation of the controller. the model reference is split int o the model D(z)

Clarke's approach using input weighting allows a

within the text. the argwnents of a polynomial such as Ad (z - 1) are dropped and represented as Ad '

controller trade

off

properties

between

of

technique was

the later

regulation

closed

loop

and

servo

system.

The

and

model

plant.

For

convenience

improved by incorporating an

incremental predictor to overcome offset problems encountered due to the input weighting and prediction bias (Clarke, 1983). Wellstead (1979) used a design law, where the closed loop pole positions are assigned to specified locations. Whilst the regulator is robust and

Consider the following:

(a)

Desired response

can be appl led to non minimum phase systems, it is time consuming, requiring the solution of a

set of linear coefficients of computation

equations to obtain all the controller . To reduce

involved ,

a

technique

is

the the

proposed

involving the on-line synthesis of only the nwnerator coefficients for a D(z) controller (Cadzow. 1970). forming the basis of a single input. single output model following adaptive system . Nd and Dd are the coefficients of the D(z) controller. calculated off-line . for the plant Bd / Ad to achieve a desired Nth step deadbeat response . For deadbeat response. the poles of Ad cancel some zeros of Nd ' hence the open loop

THE ADAPTIVE D(z) CONTROLLER By forcing a controller / plant parameter combination to be equal to the model parameters of a deadbeat system. the controlled plant response becomes deadbeat. It can then be asswned that the adaptive controller is itself a deadbeat controller. Such a controller has the property that its zeros cancel the poles of the plant being controlled . Hence the poles of the adaptive controller become the open loop poles of the adapti ve system. To maintain the equality of the equation , these poles are equal to the fixed open loop poles of the model . On-line computation is therefore reduced to that of estimating the remaining zeros and gain of the adaptive controller at each iteration . to maintain a desired response in the presence of time varying plant parameters (Walsh. 1987) .

transfer function can be represented as:

(b)

Plant response

Bp(z-l) / Ap (z-l)

125

yP

t

A_ Walsh

126 For the plant response then:

= Nd

where

plant

the

at

equal

the

desired

Bd/Dd Ad

= Nd

Bd/Dd

coefficients

each

iteration

Bp

(2 )

and

using

a

parameter and hence

Ap and Bp coefficients are estimated at each and Nd' has been fixed, only N-n zeros, or N+l-ll coefficients of the adaptive O(z) need be resolved. The denominator of the controller is fixed for all time, and equal to the pole placement of the designed response.

Ap

are

recursive

least squares algorithm. The identification algor i thm is simple to implement, and thi s simplicity results in reduced computational time when compared to other techniques, such as the generalised least squares. It can be shown that although dynamic performance is degraded with poor good,

As

iteration,

Np Bp/Op Ap

estimated

to

response

estimate.. regulation remains the use of a recursive least

The design of a O(z) deadbeat controller assumes that all initial from start up,

conditions a number

are of

necessary

plant

adaptive

before

coefficient values. By

output

estimates the time

vectors

for

and

zero. However, iterations are

converge to this occurs,

the

controller their true the input.

adaptive

O(z)

will

squares can be justified for this reason.

contain non zero values,

By specifying the following:

being added to the plant response. To overcome this problem, the error between the desired and actual plant responses is fed directly back to the plant input. being summed with the controller output. This loop provides quick

i)

Zeros in Ap(z-l) .

Np(z - l)

all

cancel

poles

of

correction

to

a

plant

resulting

response

in

error,

an

offset

since

if

the error is fed back to the controller input. N ii)

That if (i) is true then Op(z-l)

sampling

Hence:

action will be applied to the plant. The complete adaptive controller is shown in fig 1.

intervals

occur

before

any

(3 )

but (4)

Therefore..

the

unknown

coelf icients

can

be

found from ( 5) (6 )

Hence,

the

complete

adaptive

controller

numerator coefficients are

(7)

Desired Transfe r Function

('

+

Plant parameter estimates

I

1--_ _ _

~--+--/ -1L-__PL_-\_':\_T~f------- j ..

Adapt i \-c Controller ' - - - - - - - - --

Fi~.

I

-- -- - -- - - - - - - - - - - - - - '

_.l.dapti\-e 0( : ) Con tr olle r

corrective

The Adaptive Deadbeat Controller SYSTEM ANALYSIS

65318 s (s2 • 195s • 13878)

It Cion be shown that the forward path of the control loop upon which stability of the control loop depends is: Kp Np Bp

I

Dp

Kf Kp Bp

and this reduces to: 1

.

Kf Kp Bp

I

Ap = 0

(11)

The time scaling of the simulation was slowed down by 1,000, and controlled using a sampling interval of 3.5 seconds. representing a simulated 3.5mS sampling rate. The example shows how the controller copes with noise generated due to AID and D/A conversion. and computational delay, in this case 29' of the sampling interval .

(8 )

•

Ap

127

(9 )

In the second example. Fig 3. a type 1 plant was simulated digitally. and represented by the

Which is the characteristic equation of the controlled plant with the feedback loop gain Kf'

difference equation .

or expressed in another form :

2.34Yt_1 1.86Yt_2' 0 . 52Yt_3 • 0 . 00039ut_1

Yt (10) Where Q equals Kf Kp'

• 0.0013ut_2 • 0.00032ut_3

The signifi c ..1fice of equation 10, lies in the fact that the adaptive stability is depende nt on the

characteristic

plant.

equation

The gain of

(12)

the

of

inner

the

loop ,

Here the plant gain varies continuously by l ' per iteration over sample intervals 150 to 250 . In both examples the controller and model were chosen to gi ve a 10 step deadbeat response ,

controlled

Kf'

can be

used to mo v e the r o ots of the characteristic equation to an operating point along its root locus trajectory, which ensures s tability and at the same time determining the rate of adaptive convergence. However, this is only possible if

using an inner loop galn o f 4.

part of the root l o cus lies within the unit circle, and all roots can be brought within the circle at the same time. Hence, naturally unstable plant, eg type 2 (DiStefano, 1967), always having some root of the characteristic

A pole placement, model following adaptive controller has been described. suitable for stable and non minimum phase plant. Computation has been reduced to that of plant identification and estimating the zeros and gain of the

equation

outside

the

unit

circle,

cannot

CONCLUSION

be

c ontroller .

controlled by the adaptive D(z) controller.

In

practice

plant

parameter

estimation may not be exac t and this can degrade the dynamic r esponse, although regulation rema ins

CONTROLLER SIMULATION

good.

The

controller

in

its

present

f o rm does not readily handle plant with a signifi c ant time delay present and further work is required in this area.

are presented. In the first, Two examples Fig 2, the controller, implemented on a Wang 2200VP computer. was applied to the analogue simulation of a time varying, type one plant, represented by the open loop transfer function.

Adap tive D(z ) Controll er Set poi nt Des ired r esponse Pl an t response

f! ,

I,·V til1'"'" rJ' ,I

I

I

1

,

,I'

,1 ,

'5~0

V Plan t gain doub l ed . ..

F ig. ~

Sys t em response for Examp le I .

_-

A. Walsh

128

Adaptive DCz) Crntroller 3

J

Setpoint Desired response Plant response

I

2

~I

!

t ,..,. I

:

.

I',

I:

,:

I:

1

~ +.--..LI_',,-,. ...L~.J..4--'+

Plant gain increased by 100% over 100 iterations

Fig 3 System response for Example 2 REFERENCES In F. F. Cadzow. J .A. and H.R. Martens (1970). Kuo(Ed). Discrete-Time and Computer Control Systems. Prentice Hall. Chap. 7. pp. 246-322. D.W. and P.J. Gawthrop (1975). Clarke. Self-tuning controller. Proc lEE. 122. No. 9. pp. 929-934. Clarke. D.W .• A.J.F. Hodgson. and P.S. Tuffs (1983). Offset problem and k-incremental predictors in self tuning control. Proc lEE. 130. Pt D. No. 5. pp. 217-225. DiStefano. J.J.. A.R. Stubberud. and I.J. Williams (1967). In Schaum's Outline Series. Feedback and Control Systems. McGraw-Hill. Chap, 11. pp. 187-236. Kuo. B.C. (1980). In M. E, Van Valkenburg (Ed). Digital Control Systems. Holt-Saunders. Chap. 3. pp. 78-162. Walsh. A. (1987). An adaptive controller with high

computational

efficiency,

PhD

thesis,

Dept. of Systems and Control. Coventry Polytechnic. Wellstead. P.E. (1979). Self-tuning pole/zero assignment regulators . Int. J. Control. 30. No. 1. pp. 1-26,