Internal Model Adaptive Control Systems

Copyright © IFAC Adaptive Systems in Control and Signa l Processing, Glasgow, U K, 1989

INTERNAL MODEL ADAPTIVE CONTROL SYSTEMS

J.

Guan, L. Wang and D. H. Owens

Dynamics and Control Division, Department of Mechanical and Process Engineering, UmveTSlty of Strathclyde, Glasgow, UK

Abstract: This paper gives a description of an mternal m:x:Iel adapt~ve control system. Sane performance robustness properties of non-adapt~ve 1nternal m:x:Iel control system against m:x:Ielling error and load disturbances are presented; . these properties may be irrp:>rtant for the adaptive control scheme. A stab~bty robustness condition is glven based on which a switchonl off algorithm is developed.

~!:!!!! .:. Internal M:x:Iel Control (IMC) Scheme; Adaptive Control; Robustness; Stab~lity.

Conslder

1. Im'RODUCTION

a discrete time plant m:x:Iel

kz--n(l-p,Z-')


CA(Z-I)-

'

.d:.tl.n~m

n(l-a,z-')

(2.ll

,

Assuming the controller design criterion L(y(t)-r)2+p(u(I)-C~I(I)r)

'-0

( 2.2) -1

Based on the analys1s similar to [3,41, a piecew~se stable adaptive system with slow-vary1ng parameters is a stable adapt~ve system, and the condition of robust stabllity can be checked online , a switch on /off control scheme is introduced into the IMC scheme.

where

y(iJ

~s

the output of plant

G (z

under

A

the step ~nput r, U(l) the control weightmg on the control energy.

signal, p

a

Lemna 2.1 [ 5 J : When p ... 0, the state feedback controller Wh1Ch minimize the crlterion (2.2) w1ll result 1n a closed-loop transfer funct10n of the form

Section 2 lS a revlew of IMC des~gn method and some advantages of IMC adaptive control system. In Section 3 the robust stabillty condition and one of its applications w~ll be presented.

(2.3 ) 2.

SISO

IM:: DESIGN Mr:rOOD ANLJ

ITS POTENTIAL MNA.VfN>ES ADAPTIVI: CONTROL

FOR

where

PI" IP,I~ 1 2.1 SISO Discrete Internal

~el

Control

Des~gn

p,-

[

f..1 p, I> 1

<2.4)

There are two maior reasons for the expos~t~on of the Il'X: deslgn method descnbed ln th~s subsection: (a) It will give a controller deslgn method for a stable plant and based on the des~gn further analvs~s ~s conducted; (b) We wlll be interested espeCially ~n the design criterlon wh~ch will be seen to have close link to the sensitivity of the closed-loop system.

The state-feedback controller lS fairlv flexible control strategy. If any output feedba~k control ler wh~ch achleves the sa/re closed loop transfer functwn as 1n (2.3), it may be regarded as an optll!lal controller. Hence (2.3) may be regarded as a different version of des~gn objectlve from the criter~on (2.2).

The design of an IM:: system can be d~vided into two stages: (1) Design of a perfect controller; (2) Design of a ill ter.

From (2.4) 1f I '\ l , Si B. . Therefore the plant zeros lns1de the un~te clrcie .z . =l ln the complex plane w~ll be cancelled out ~n eq.( 2.3 ), hence the

t

135

=

J. Guan , L.

136

Wang and D. H . Owens

best ach1evable closed-loop transfer function will be influenced by the plant zeros outaide the umte circle and the t1me delay element z-

L

Consider Fig.!.

,et)

a

conventional

control

scheme

(2.9)

in

d(t)

y

d

-~

±

yet) +

F1g.l

The Classic Feed-back Control

r(t )

y

Denobng U(z)_C -C(I+C C)" R(z) ,. •

(2.5)

The optimal design may be achieved by the requirerrent that

F1Q.

3(b) The IMC Stracture

F1g. J The Transformatlon Between the Classic and the II-l:: Conf1gurat10n The controller ln (2.6) may require large control energy since P.O in the design inl>lies that there are no constraints on control energy. A filter can be introduced in the case of constra1ned energy: . 12.6)

(I-a)lz" F(Z)-(

which

takes

all the non-ant1c1pat1ve (poles

not

-d

containing

z

and stable parts

-1

1nverse

G

of

the

plant

-1

(z);

G

A

1S therefore called

the

(2.10) be

(2.11)

T(z) - T o(z)F(z)

will be shown later that the requirement of performance (F(z)"U and of good robustness 1n the f11ter are conflicted; and the trade-offs can be eas11y ballanced owing to the siIrplicity of the 1nfluence of filter on the performance and robustness.

As

A

good

Since after the 1mplementation, the controller will be controlling the real plant G(z), i.e. as in Fig.2. d

r{t)

.')1

so that the best achievable performance will required to be

cO

1nvertible part of G (z).

1- az

~)9-...,---

y (t )

±

The fInal design of the controller systen wlll be of the form

of

the

C,(z) - C,o(z)F(z)- C~~(z)F(z)

II-l::

(2.12)

Remark : If we define a norm Fig.2

Implementation of system in Fig.l 11 x(l) 11-

By the transformat10n as in F1g. 3, an Internal M::ldel Control scheme is obtained 1n F1g.3. (b) clnd

The physical reahzation of the II-l:: system unstable G and G may not be stable, the

for II-l::

L>2(1)

(2.13)

.-,

denote Ilx(z)II-llx(t)11

A

system is useful only when G and G are stable. A

The factorization C •. (z)- To(z)

is

called the non-invertlble part of m:xlel

and C •• (z)- C~~(z)

lS call the lnvertible part. Obviously we have the relatlon

(2.71

G (z) A

where x(z) 1S the z-transforrnatlon of x(t), then the rrun~zat10n of (2.2) may be written into the form of

Miu\\YCz)-R(z)\\

(2.14)

,

where c lS controller.

the variable part

of

the

feedback

In an IMC system, (2.14) is equ1vallent to (2.8 )

Minll(C.(z)C,(z)-I)R(z)11

'.

(2.15)

Internal Model Adaptive Control Systems It is this property of the 1M2 system design that may make closed-loop system least sensitive to rrodelling error and load dlsturbances. It may partly contribute to the success of Ir-K: system ln the practice.

2.2 Properties of 1M2 System Coping with M::ldelling ~

Like in every feedback system, the function of the feedbac k line in the 1M2 scheme lS to cope with rrodelling error and dlsturbances. It may be analysed in terms of sensitivity reductlon when rrodelling error and load d1sturbance are small and in terms of performance robustness when model~ ling error and load disturbance are b1g. ~

The !~ scheme ~ be least sensitive to both small modelling errors and small load disturbances

use the definltions of sensitivity reduction simi lar to those in [6 J for the feedback system 1n Fig. 3(b) whlle G, G and G are regarded as c A operators on same normed space.

We

DehnitlOn 2.1: Assuming that the plant G is exactly modelled, i.e. G =G, let y (t) and y (t) A no o denote the outp.1ts of the open-loop system wlth and without load disturbance respectively, y (t) ne and y (t) denote those of the closed-loop system. c The 1M2 system in Fig.3(b) lS sald to reduce the sensitivity wlth respect to outp.1t disturbance if

11 Y ., ( t ) when

Y , ( t)

11 < 11

Y •• ( t ) - Y • ( t)

11

U.lb)

11 d (t) 11-> o.

DefinitlOn hl: SuAx>sing that the system is free of load disturbance. Let the plant G have possible perturbation G-G.+LlG

(2.17)

or (2.18)

G-G.(I+LlG)

Let

open

0

loop system wlth and without plant perturbation, y t:.c

and y

(2.20) Fran this result we can see that small values of 111~AGc 11 are deslrable for achieving good sensitivity reduction. Ccrrpared wlth (2.15), we can see ' that although the design is not exactly minimislng the operator norm 111~AGc 11 but to minimize the dlstance of the image of

1~

G with the specified

A c

inp.1t. This may indicate the system has minilrum sensltlvitles when the slgnals are of this specific type, namely the set-point varylng type. If we see the sensitivity reductlon property as an iIrportant function of a feedback system, then this discussion may imply the design of IMC system enhences th1S functl0n to same optimum. Whlle some similar analysls on the classlc configuration as in Flg. 1 may sh<::M that this property are not necessarily present even if the deslgn criterion lS of the type of (2.2).

ill

Th~

IMC: scheme

in

the

prese~-""

Jar~e

l.2'lg

have 9229. performance large !.fOdelliT19 errors gistl!l"pance. ~ 2.~

",,~D

and

Consldering system equation of the IMC system ( 2.21> where G,

G and G may be seen as operators or zA c transformations. If the closed-loop system is -1

robustly stable, and G =G c A

then

Y=R 1.e. perfect tracking irrespective any modelling error or load dlsturbance. However, even though in the deslgn, G lS meant to be the inverse of the c model, the exact inverse is not achievable due to realizab111ty, control energy constraints and stabihty reasons. But ln the frequency domain -1

G =G may be achieved in some ilrpJrtant frequenc A C1es. For ex~le, 1f F(l>=l. Wh1Ch 1S eas11y satlshed by choosing F(z) as 1n (2.10), then -1

Y (t) and y (t) denote the outp.1ts of t:.o

137

denote those of the closed-loop system. The

c system ln Flg.3(b) is sald to reduce the sensitlvity wlth respect to plant perturbatl0n if

(1) by the design. Th1S sh<::Ms that G (l>=G c A the inp.1t and the d1sturbances are set-po1nt changes, the steadv state error 1S zero 1n the case of any modelling error and set-point load d1sturbance of any magn1tude.

I~

11 when

Y ., (t) -

y, (t) 11 < 11

Y •• (t) -

y.C t) 11

3.

(2.19)

11 Ll G 11 -> O.

We have the followlng result concerning the sensitlvity reductl0n of the I~ system. ?~positlon

2.1 : assuming thatG,

In the 1M2 system 1n Fig.3(b), G and G are stable ln the 1 / 0 c A sense with respect to a normed space, and the 1nduced operator norm lS subord1nate, then the closed loop system reduces the sensitivltles w1th respect to both load disturbance and plant dlsturbance lf and only if

ADAPTIVE 1M2 SYSTE:1 AND ITS

ROBLS'ThfSS

The propertles dlSCUSSed 1n 2.2 1nd1cate that good response may be ach1eved 1n the I~ system when there exist- modelllng errors prov1ded- that the system is stable. In many adapt1ve control applicatl0ns, there are two cases 1n Wh1Ch modelling errors exist. The f1rst one is that at the early stage, the model converges gradually to the real plant thus modell1ng error eX1sts 1nevitablly. In the second case, the nodel structure mav be chosen to be sll%>ler than t.he p!"nt Rtmctllrp, the t'efo re the raxiel w,d 1 not p:)ssilJl y converge to the rea 1 plant; although forttmdtely 1n many appl1cat1ons the model will converge(3). A suff1c 1cnt

J.

138

Gu a n. L. Wang and D. H. Owens

robustness condition developed inl31 is useful to derive a switch onl off algorithm for lMC systems. This condition basically needs a robust stab~lity condition of the time invariant system. 3.1 1~

Robllirt:ly Systems

c4 Non-adaptive

S_~l.H!y ~J:ldit_~on

The system is stable if and only

~f

S(z)-[I +G,(G-GA)r'

has

all

of its poles

inside the

13.1 )

that is equivallent to S (z) has no zeros outside the unite circ le, which is satisfied if 13. 2 )

for all z belonging to the outside of the unit circle. Since G, G and G are assumed to be A c stable hence analytic outside the unit circle, according to the maximum roodulus principle, we have

hl:

An

IMC system will be

3.2

Robustness'!r:l<1

.l\daptIve - iM:Sv~tems

robustly

I G,(z)(G(z)- G .(z») 1 1%1_1 < 1

%-.

Q:>.nditi2!}!,;

of

A

at t~ k, produced by any estimation scheme, e.g. RLS. The structure i~ essentially arbitrarily chosen with restricted complexity and IS not therefore necessarily related in any way t o the underlying system structure. The internal roodel adaptive controller G (k,z) is then at time k given by C G,(k .z) - G~~(k .z)F(z)

where

G

13.8 )

(k,z) is the invertible part of

G

Aand

the

G (k,z) c

(k,z)

A

structure of F(z) is c hosen such that is proper or strictly proper for eac h k.

(3.3)

The control signal u(k) can be produced according to the internal roodel controller structure illustrated in Fig.3(b).

(3.4)

The stability conditions of the adaptive control systems are follOWing theorem.

and

limIG,(G-G.)I<1

~}t.sb g~ / og

stage b~g Cl ;

Internal roodel adapt~ve control system will be deSIgned on the baSIS of the certaInty equivalence pr~nciple[71. It includes the roodul e of parameter estimat~on and the roodule of internal roodel controller des~gn scheme which has been dISCUSSed prev ~ously. Let G (k,z) denote the estimated r-ode l

circle,

unit

-1

Proposition stable if

may surq::>ly be : at the Imtial adaptatIon when the roodelllng error IS big, choose a as the error IS reduced, choose a small Cl •

internal roodel given by the

It can be seen in (2.7) and (2.3) that 13.5)

(Note for the real-parameter plant, catplex zeros or poles a~r in conjugate pairs). And if lilnIG(z) 1=0, which is satisfied by the structure such as in ( 2.1> , (2.10), we have

by choosing a fi 1ter

as

in

Theorem 3.2: If the process plant G(z) and the ChOsen--filter F(z) are stable, then the internal roodel adaptive control system will be robustly stable in the sense that bounded reference input w~ 11 produce bounded system output, provided that

.

there exists k such that for a ll k>k and sufficiently small 6 the following conaitIons hold true, (i) G (k,z) is point wisely stable A

Theorem

hl:

An lMC

system is robustly stable if

(u)

~~rIG , (k.z)[G(z)-GA(k.Z)ll< 1

13.9)

(3.6) (iii) 11 9 (k) 11 H w' 11 9 (k) - 9 (k - 1 ) 11 :S 6 Note that this formula of the stability condition does not need the factorization of G (z), hence A

the testing of it may be sinl>lified.

where roodel

El

13 .10)

(k) is the identified parameters in G (k,z), 1I· 11 is a Vector norm on A

the the

Euclldean space. The effects of the filter on robustness is clearly expressed by (3.6). For example

F(z)-

satisfies

I

(I-a)z-I _I

1 - az

F(z)

1 1%1_1

.0
13.7)

S I. and as a-+ 1.F(z) -+ QV z~ 1,

which indicates that large Cl will inl>rove stability. On the other hand, large Cl will worsen the perfonnance generally. this represents a general phencrnenon that the perfonnance requirement and robustness are conflict objectives. The relation is s:iJtt:>ly represented by the parameter Cl. If Cl is nade available for tuning such as done in (4)' it may be a powerful method for achieving good performance in the adaptive system. A tuning guide-line

Proof: (neglected for brevity) The condition (iii) , the boundedness and slow variation of the identified parameters, can be ensured by using same roodified estimators, for example, Goodwin and his colleagues (1985) introduced a deadzone in a recurSIve least-square-like estllTlator to cope with unroodelled dynamics and unroodelled disturbances. This roodification ensures the boundedness and slow variation of the identifled parameters in the face of unroode11ed factors. CondIt~ons (i) .and (ii) are the point-wise robustness condItIons in the nonadaptive case. If the mathematical roodel of the plant is given, the point-wise robustness conditions can be

Internal Model Adaptive Control Systems checked. However, if the plant is unknown, an extra identifier can be used to estimate" the unknown system based on an assumed correctly structured m:Jdel G (k,z), then COnditlOrt (ii) in B

the theorem can process transfer rrodel G (K,Z).

be assessed

by replacing the function G(z) by the estimated

B

Generally speaking, the satisfaction of point-w1se robustness cond1tions is not sufficient to guarantee the bounded inp.It / bounded outplt stability of the internal m:Jdel adaptive control system. But if the point-wise robustness conditions are known to be satisfied at time k in the adaptive control

o

system, the adaptive mechanisms can be switched off and the adaptive controller can be replaced by a f1xed compensator bound by the adaptive mechamsms at k. The point-wise stability 1S hence

o

turned into BIBO stability of time 1nvariant system. If the plant is linear time varying, the adaptive mechanisms can be switched on when the po1nt-wise robustness conditions are violated. The switch off / switch on algor1thm is given as follows. ~~

Algorithm

Choose

E

; 0,

o

k>O and

o

repeat

the

following computation for every sampling period. Step 1: Identify the unknown system G based on an arbitrarily chosen m:Jdel structure G • A

Step 2: Design the internal m:Jdel adaptive controller as (3.1U

Step 3 : Identify the unknown system G based on an assumed accurate m:Jdel structure G • B

Step 4: If k>k , compute

-0

fJ (k)

- sup 1G, (k • z) [ G ,( k , z) - G A (k • z)]1

'"

Z-gJ".O~w~2n

(3.12)

and perform the following (i) If fJ(k)
I-£ogo to step one.

4.
5.REFERENCES III

c.

E. Garcia and M. Morar1(a) 'Internal m:Jdel control, I. A unifY1ng review and some new results', Ind. Eng. cher:>. Process Des. Dev., 1982, 21, pp.308-323. -

139

(b) 'Internal m:Jdel control. 11. Des1gn procedure for llUlt1vanable systems', Ind. Eng. Chem. Process Des. Dev., 1985, 24, pp.472-484. 121 M. Morari and B. R. Holt, 'DeSlgn of res1lient processing plants-VI. A general fr~rk for the assessment of the effect of rlght-half-plane zeros on dynamlc resilience'. Chern1cal Engineering Sc1ence , Vol. 40, ~. 1. pp.59-74,1985. 131 D. H. OWens and L. Wnag. 'SW1tch off I sw1tch on algor1thms for robust adapt1ve ocntrol', Research Report, Dept. of Mathemat ics, Cm v. of Strathclyde. Also appears 1n the 8th IFAC S\~­ S1um on Ident1f1cation and parameter estunation, Be1jing , 1988. 141 L. Wang and D. H. OWens. 'Robust pole-ass1qnment adapt1ve controllers with on-line turn1ng parameters', lEE Control Conference. Oxford, 1988. 15 I H. Kwakernaak and R. S1nva, 'Llnear optimal control systems'(New York, W111ey, 1972). 161 J. C. W1 11 ems , 'The analysis system' (M.I.T. Press, 1971).

of

feedback

171 Y. Bar-Sholom and E. Tse, 'Dual effect, certa1nty equ1valence and separat10n 1n stochastic control', IEEE Trans. Aut. Contr •• Vol. 19, No. 5, 1974 .