Robustness Properties of Smith Predictor Control

Copyright © IFAC Identification and System Parameter Estimation, Budapest, Hungary 1991

ROBUSTNESS PROPERTIES OF SMITH PREDICTOR CONTROL M. Habermayer Department of A ulOmation , Technical University of Budapest, Budapest, Goldmann Gyorgy ter 3, H-III1, Hungary

Abst-racL. The paper compares t-he t-racking and dist-urbance reject-ion propert-ies o~ mat-ched and mismat-ched Smit-h predict-or cont-rol. Then by det-ailed ~requency domain analysis it- enlight-ens how t-he accurat-ely modeled t-ime-delay. gain and delay-free process paramet-ers influence t-he performance of t-he mat-ched Smit-h cont-rol Keywords. Smit-h predict-or. adapt-ive robust-ness. frequency domain analysis

cont-rol.

est-imat-ion.

mismatch.

its disturbance rejection is not so good as its tracking ability and propose various met-hods to improve t-his drawback CWatanabe and co-wor·kers.1 983; Watanabe and Sa t.o. 1984; Chi en and co-wor k er s, 1985).

I NTRODUCTI ON For cant-rolling syst-ems with large t-ime-delay t-he Smit-h predict-or has been known as an effect·i ve cant-I" all er i ~ the plant parameters are known. In case of unknown constant or varying process parameters an adaptive Smith controller can be used which resul ts from t .he combination of an on-line identification method and t-he Smit-h predictor algorithm. When the process time-delay is also varying both the delay-free parameters and the time-delay at' the process must- be estimat-ed on-line CHabermayer and Keviczky. 1985a).

On the at-her hand Bahill has shown (1983) that the Smit-h control has removed the time-delay from the denominator of t-he closed-loop transfer funct-ion assuming per~ectmodel matching and dist-urbance effecting on the process input. but the system tracks the disturbance wit-h a time-delay. This latter study. however does not deal with either the disturbance rejection effecting on the process outputor the disturbance rejection of the mismatched Smith control.

The robust-ness propert-ies of an adapt-ive cant-roller depend on the control law and t-he parameter estimator. A "robust-" adaptive controller is obtained when a "robust" control law is combined with a "robust" parameter estimat-or . Therefore working out an adaptive controller it is pract-ical to begin with a robust control algorithm and then make it adaptive rat-her then to analyze the perfomance of a "high-performance "cant-roller '.. hi ch will almost- certainly be sensit-ive to parameter values (Goodwin. Hill and Palaniswami. 1985) .

This paper compares t-he tracking and disturbance rejection properties o~ the Smith control and shows that they can be handled in the same way for both matched and mismatched schemes. Then by ~requency domain analysis t-he paper enlightens how the inaccurately modelled time-delay in~luences the system per~ormance for various positive or negative mismatch degrees. In the end some Bode plots help to understand the per for mance modi t'yi ng effectof the inaccurate delay-~ree process model if the plant. is given by a first order time lag with time-delay.

Most of the publications dealing with the properties of the mismat-ched Smith predictor est-ablish t-hat this controller is a robust one, moreover for small positive or negative modelling errors the dynamic behaviour may be more favourable than for accurate mod@lling CIoannides and co-workers. 1979; Chotai and co-workers. 1984; Walton and Marshall. 1984; Habermayer and Keviczky. 1985b). The above conclusions. however. are not- generally valid and are mainly drawn from simulation result-s relating to part.icular examples and time-delay mismatches in most- cases.

TRACKING AND DISTURBANCE REJECTION PROPERTIES OF SMI TH CONTROL The structure of t-he Smith control scheme is given in Fig. 1. Here GCs) is the delay-free transfer function. T the time-delay o~ the process. GMCs) and T M

are the models

On t-he other hand est-ablishments obtained for the disturbance rejection of the Smith predictor control are contradictory. as well. Several papers state thata disadvantage of this controller is that-

o~

GCs) and

T.

are the reference. output and signals.

Yr

Y and

1

disturbance

Assuming that disturbance signal does not ef~ect on the process output the closed

1173

loop

~ransler

Y

ROBUSTNESS ANALYSIS OF THE MISMATCHED SMITH CONTROL IN THE FREQUENCY [)oMAIN

is as lollows:

I

(s) (s)

Y

func~ion

r

On the basis of Eqs (2 - 2) and (2 - 4) i~ is seen ~ha~ ~he misma~ch elfec~ 01 ~he Smith con~rol can be charac~erized by ~he ~ransler function

l(s)=O G (s)GCs)e -ST R

-ST

1 + G (s)G (s)(1- e R

-ST

+ G (s)GCs)e

M)

M

(2 -

By algebraic Y

1)

G

F

(s)

l(s) =0

R

G + G (s)G(s)[

G

R

ST

(s)

_M_ _

( 1-e

1)

which is delayed by

ST

(s)

Time - Delay

R

(2 -

2)

F

Here G (s) is Lhe Lransfer

M

01 Lhe

lunc~ion

F

feedback. For G (s)=GCs) and T

=T

~ha~

results in the scheme of the ma~ched con~rol. The structure belonging Eq.(2 - 2) is seen in Fig. 2 .

ycs)

Iy

l(s)

G

F

The closed loop transfer output dis~urbance is:

Smi~h

where

~o

lunction

T

lor

- e

-seT

+ e

is a

-ST

(3 -2)

number.

cons~an~

~o

~he

2) :

i

1

+

The ST

1 + G (s)G (s)(1 - e R

-

COS(WCT)

+

COS(WT)

+

j[sinCwcT) - sin(wT)J.

(3 -

3)

-ST

R

(3

absolut -

value

and

phase

angle

01

3):

M)

M

(2 -

i

= 1

eT and c

M

o

(s)

G (s)GCs)e

W (s)

(s) F

The frequency func~ion belonging transfer lunction given by Eq. (3 -

1

lit~le

Misma~ch

When c = 1 the accurately modelled S mith con~rol is obtained. For negative and positive time - delay mismatch cl. respect.ively.

W (s)

r

+

T.

M

modelling is G (s)=1 which

accura~e

M

signal

If only the time-delay is misma~ched in 1) ~hen the transfer function (3 G (s) = GCs) and

1+ G (s)GCs)G (s) R

as a Yl 01

opera~es

+ e

M)

G (s)GCs)e -ST

a

(3 -

G (s)

(2 - 4) and ~he system delaY- lree con~rol, outpu~

G (s)G(s)e -ST

Al~er

+ e

For perfect modelling G (s)=l, the time F delay is eliminated from ~he closed loop transfer lunc~ion G.cs) in Eqs (2 - 2) and

1

r

1

M

(s)

manipula~ion:

(s)

Y

-S T

G (s)

R

algebra 1

i~

3)

lollows:

(3 -

4)

(3

6)

and G (s)GCs)e

-ST

R

M

1

1

1

+ G (s)GCs)G (s) R

which has Lhe

si n( WCT) -si n( WT) 1 COS(WCT)+COS(WT) -

-ST

R

-

F

ST "') + e

1 + GR (S)GCS)[G (S)(l - e G (s)GCs)e

arc

'P (w)

-ST

G (s)

_G(s)e- ST

The above equa~ions show ~ha~ plot~ing ~he absolut value and phase angle of (3 4) and (3 - 6) against WT the only parame~er is ~he time-delay misma~ch degree c. In case of various c values Figs 4 - 11 show IGF(jw) I = f.cwT,c) and 'PF f 2 (wT,c)

i

F

realiza~ion

(2 -

4)

given in Fig.

3.

frequency Comparison 01 ~he schemes given in Fig. 2 and Fig. 3 shows ~hat ~he ~racking and ou~pu~ disturbance reJec~ion abili~y of ~he Smi~h control can be handled in ~he same way lor ma~ched and misma~ched cases,

charac~eris~ics

for T

1.

From the diagrams given in Figs 4-11 following conclusions can be drawn: The

~he

influence 01 ~he misma~ched on ~he s~ability 01 ~he Smi~h con~rol depends on the situa~ion of ~he cU~-Oll frequency of the matched ~ime-delay

~oo .

sys~em .

1174

1.1 In

low freque ncy domain ( if ~he of ~he proces s 4re hlgh in compa rison wi~h ~he proces s ~lm~ delay) ~he non acccura~el y modell ed ~ime-delay does no~ cause unstab le behavi our even for high misma~ch degree . 1 . 2 For in~ermedia~e !'reque ncies either improvemen~ or de~eriora~ion may occur in ~he sys~em perform ance. 1 . 3 If ~he cu~-off freque ncy of ~he perfec t.ly modell ed Sffil~h c o ntrol fal ls in ~he high freque ncy domaln th e system may be unstab le even f o r low time-d elay mismat ch . ~he

~ime

cons~an~s

2 The lower is ~he ~ime-delay ml s match degree ~he grea~er is ~he freque ncy domain in which ~he sys~em preser ves its s~abili~y . In case of adapti ve Snuth con~rol applyi ng on-lin e ~lme-aelay estima~ion ~his means : neglec t of the frac~ional ~ime-delay does not lead to uns~abl. behavi our genera lly .

freque ncy cl( > 2

~he

domain .

Smith

con~rol

Time

Cons~ant

Mismat ch

GL(Jw) .

In

~he previo us case t.he T of t.he freque ncy func~ion

in

~he

fac~or

la~~er

(

1

+ JwT'

case

T

~ransfer

( 3

L

substi~u~ing

G

(s)

K G

c

LW

J(

F

G (s) LW c J( -GCs) (1

(s )

J(

-ST -

e

W)

is a consta nt number

gain ffilsmat ch.

For cl(

=1

7)

the the

1 )

by

-

gl v1ng

8) ~he

the proces s gain

( 3

-

If only ~he gain is mismat c hed freque ncy functio n reduce s itsel f to

5.

The

9)

the

-

IGF(jw) I = g.cWT. c,.? agains t only varyin g parame ter is cl( ' Figs - 17 give ~he Bode plo~s of GFCjw) for -

l&.g when

c,,=

1

=

~he

+ COS(WT ) - jsinCw T).

=0.5

order

Becaus e of t.he limite d size of Lhis paper freque ncy charac~eris~ics of a misma~ched Smit.h contro l are no~ presen~ed here in case of a second order proces s w1~h ~he t.ransf er funcLi on o~ (3 12). Slmila r conclu sion can be drawn . howeve r. for the Lime constan~ T mismat. ch effecL as for t.he first. order proces s.

+

L

C3

f~r~t.

T

(Jw)

~he

12)

The negaLi ve time cons~ant misma~ch for c ~ 0 . 5 is also not. so danger ous from the T viewpo int of sLabil ity moreov er it oft.en resul~s in perform ance improve ment. . For C ,2 s~ability problem s may occur T especi ally if the cu~-off freque ncy of ~he exactl y modell ed Smith cont.ro l falls in ~he high freque ncy domain .

e

cl( GLWCJW) (l-COSC WCT)+ j sin(wcT ) )

Plot~ing

(3 -

Figs 18 - 25 demosL rate tha~ ~he misma~ch of t.he first. order lag t.ime cons~ant does not. lead t.o insLabili~y i~ 0 . 5 < c < 2.

- ST +

Cl

=

lS exactl y modell ed . The freque ncy functio n belong ing ~o (3 - 8) is G

dampin g

1n t.he freque n c y f unctio n of (3 9) 1n cas e of T 2 and T 1. Here ~he t.ime cons~ant model is : T c T where c W T T represen~s t.he mismat. ch degree of T . For T < 1 or T > 1 ~he diagram s are differen~ in some ex ~ e n t. Fi gs 24. -25 show ~he freque ncy charac terist. ics for T = 2, T =10 and c 0.5 and 2, respec tively . T

6)

-

( 3

where c

proces. s is

-

( 3

(s)

of ~he proces s model into ( 3 mismat ch effect is charac~erized t.ransf er funct1 0n G

~he

of

may be mismat .ched .

proces s g a ln By (3 - 6) and ~ransfer funct1 0n

.. LW

and

11)

J2(Tw

~he

K G

(s)

W

(3 -

1

functio n be

= K G (s)

1S

time

Figs 18 23 demons trat.e ~he mismat ch effect. of T if the delay- free part of t.he

the delay- t'ree

K

uns~able

1

Gain MiSmat ch

where

0 . 5 or

~

In the feedbaC K transf er functio n of (3-8) a first order or a second order lag is ~he most freque ntly used approxima~ion for cons~ant

posi~ive ~ime-delay mismat c h ~he size of ~he freque ncy domain in which the sys~em preser ves i~s s~abillty is a li~~le smalle r ~han for c
GCs)

c I( be

if ~he cut-of f freque ncy of the matche d system falls in the high freque ncy domain , but the possib ility of t.he unstab le behavi our is less if t.he gain mismat .ch degree is positiv e.

3 For

Le~

For may

M.isma~ch ot' the dampin g factor ( has an influe nce on the exactl y modell ed Smit.h conLro l perform ance mainly in t.he int.ermedia~e freque ncy domain and almost. never leads to instab le behavi our for bot.h over and undere stimat ion of (. Many times ~he inaccu rately modell ed dampin g factor may resulL consid ereble improv ement in ~he system perform ance in compa rison wit.h t.he exac~ly modell ed Smi~h con~ro l .

10)

wT 12

c I(

diagram s show that the small gain misma~ch does no~ result in unst.ab le behevi our even if the cut-of f freque ncy of t.he matche d system is in the

1175

CONCLUSIONS

~~TY' ~L-

The paper investigates robustness properties of the mismatched Smith predictor for both reference signa l tracking and disturbance rejection. First it is demonstra ted that the tracking and disturbance rejection abilities of this scheme can be handled in the same 'Way. Then by frequency domain analysis it is presented that the Smith control is robust enough for either small time - delay, gain, or delay - fr' ee parameter mismatch . Robustness of thi s contr'ol for small time-delay mismat c h ensures, for example, that neglect of the fractional time -delay in adaptive cont r ol does not lead to unstable behaviour. It i s also seen from the frequency diagrams that the higher is the mismatch degree the longer is the sampli ng di stance by 'Which the system preserves its stabilit y.

G

-ST ~

-ST

(s)

-"--(1-e

").eo

G (s)

Fig.

3

Output disturbance Smith control (y =0)

rejection

of

r

REFERENCES Bahi 11, A. T . Cl 983) . A si mpl e adapt.i ve SOli th predictor for controlling time-delay systems. IEEE Contr 01 Systems. Magazine, 16 - 22. Chot.ai ,A. ,Owens, D. H. ,Raya , A. ,and Wang, H. M. (1984). Design of Smi th control schemes for ti me '-del ay systems based on plant step dat a. International J . of Control ,40,297 - 316. Goodwin,G.C . , Salgado,M.E., Middleton,R .H. (1988) . Indirect adaptive control, an integrated approach. Amer' ican Control Conference, Atlant_a, 2440 - 2445. Habermayer,M. and Keviczky,L. C19S5a). I nvesti gati on of an adapti ve Smi t ·h controller by si mulation. 7th IFAC Conference on Digital Computer AP-Rlication to Process Control, Vienna, 413 - 417. Habermayer,M., and Keviczky,L. (1985b). Mismatch analysis of Smith predictor control in the time domain. IFAC/IFORS Conference on Control Science and Technoloq~~ Development, Bei jing , 961 - 972. Ioannides,A.C . ,Rogers,G . J., and Latham,V. (1979). Stability limits of a Smith controller in simple systems containing a time-delay . Int. J. of Control , 29, 557 - 563. Walton,K. and Marshall,J.E. (1984). Mismatch in a predictor control scheme some closed form solution. Int. J. of Control ,40, 403 - 419.

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Figs 10-11 Time-delay misma~ch e~~ecl o~ the Smith control

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1178