Discrete Direct Multivariable Adaptive Control*

opyrigh t © IFAC Adaptive Syste ms in Contro l nd Signa l Processing. San Francisco. USA 1983

DISCRETE DIRECT MULTIVARIABLE ADAPTIVE CONTROL I. Bar-Kana and H . Kaufman f'-1nl m "I. COlI/jJl/ln. I/Ilfl Sy"II'IIIS fllgilll'f'rillg [)r/)(nllw'lIl. R f'II.,.,d"n j 'oiYlnhllir I mlill/lf'. Tun. '\T 1218IJ. 1 'SA

Abs t ract. Dir ect multivariable model reference adaptive control (D~ll1RAC) procedures have been successf u lly used in continuous - time systems. Previous discrete ve r s i ons of the algori t hm required apriori knowledge of a stabilizing feedback gain matrix for the controlled plant and also an estima t e of the ideal model following control in order to guarantee a bounded output tracking error . In this paper a discrete version of the D~ll1RAC algo r ithm is shown to guarantee s t ability of the adaptive system as well as asymp t otically perfect output model following provided that the sys t em satisfies a positive realness co ndition . No a priori information is needed for implementation. Keywords . Adaptive control; reduced order model ; discrete-time systems; multivariable control sys t ems; linear systems; nonlinear equations; stability; invariance principle.

*

This materia l is based upon work supported by the Nat i onal Science Foundation under Grant No. EC580-l6l73. 1.

INTRODUCTION

equations:

A simple direct multivariable model reference adaptive control (D~AC) procedure developed by Sobel (1980) and by Sobel, Kaufman and tlabius(1980), and extended by Bar - Kana and i(aufman (1982a, 1982b) was shown to guarantee stability of continuous - time adaptive control systems. Discrete ve r sions of the adaptive algorithm required a priori knowledge of a stabilizing feedback gain matrix and also an estimate of the ideal model following controller in order to guarantee boundedness of the output tracking error.

x (k+l) p y (k) p

(1)

C x (k) + D u (k) p p p p

(2)

and the plant output y (k) is requic2d to p

follow the output Ym(k) of the asymptotically stable referenc e model : x (k+1)

m

A x (k) + B u (k) mm mm

(3)

(4)

In this paper it is shown that a new discrete version of the DMMRAC a l gorithm guarantees the stability of the adapt i ve systems as well as asymptotically perfect output model follow ing, provided that there exists a constant gain feedback matrix (unknown a nd not needed for implementation) such t hat t he equivalent closed-loop system is (simply rather than strictly) positive real. No a prio r i infor mation about the contro l led system is required for implementation of t he algori t hm .

where it is permissible to have dim(x ) »dim(x ) p m

(5)

We represent the input commands um(k) as outputs of a command generating system of the form v (k+l ) m u (k)

m

It is also shown that some systems which do not satisfy the posi t ive realness condition (like non - minimum phase systems) can be controlled by using a supplimentary d ir ec t input-ou t put feed forward gain matr i x. 2.

A x (k) + B u (k) p p p p

(6)

A v (k) v m C v

v m

(7)

(k)

where Av is not necessarily stable.

The

representation (6) - (7) is only needed for the subsequent analysis. The matrices Av and C v

are unknown and only measurements of the input um(k) a r e permit t ed.

FOR}ruLATION OF THE PROBLEM

When the refe r ence model (3) i s supplied

The controlled plant i s represented by the

357

35 8

I. Bar-Kana and H. Kaufman

with a n input o f the form (6)-(7) its soluti on can be writt e n as x (k) = E v (k) + Ak 0 m m m 0

A solution for Equation (20) exists, in general, if dim[v I < dim[u I + dim[x I. * m m m In that case x (k) defined by Equation (11)

(8)

p

c an eventually satisfy Equation (1) as

By s ub s titutin g a so lution of the form (8) i n t o Equ a ti on ( 3 ) it can be shown that the ma tri x E sa ti s fi es th e equation

+

A E - EA m v

=0

B C m v

and, also, y * (k) p

=

m

~,

~.

We thus require that the actual trajectory x (k) of the pla nt satisfy :

(9)

p

a nd t ha t

oo

y (k) as

~

x (0) - Ev (0 ) m m

(k)

x

(10)

~

P

Fo r th e s ub se qu e nt s t a bilit y a nal ysis we defin e some bound ed " idea l c urv es " x* (k) of the f orm p

x * (k) as k P

~

(21)

00

*

y (k)

(22)

y (k)

p

P

3.

THE DISCRETE ADAPTIVE ALGORITHM

( 11)

The s t a te error is defined as

s uc h th a t x * (k ) sa ti s fie s th e pla nt equation

e (k ) = x * (k ) - x (k)

p

(1 ) whe n th e pla nt i s fo r ce d by an "ide al

x m

(1 2)

u m

At th e same time it i s des ir ed th a t x

p

e (k ) = y (k) - y (k) y m p

* (k)

P

x * (k ) + D u * (k) = C x (k) +

P P

P P

mm

u (k) = K(k+l) r (k) p

m m

!l u (k) = y ( k )

(13)

m

[ (ApXll - Xl l Am + BpKx)E + ApX12 - X12Av

r T(k) I:. [ eT (k) y

xT(k) m

uT(k)1 m

(26)

K(k) = [Ke( k)

K (k) x

K (k)1 u

(27)

a nd K(k+l) = KI(k+l ) + Kp(k+l)

(28)

KI(k+l)

KI(k) + ey(k)rT(k)T

(29)

K (k+l)

e (k)r (k)T

p

C e (k) - D (K(k+l)-K)r(k)-D K e (k) p x p p e y

x (k+l ) =A x (k ) +B u * (k) - (A Xll-XllA +B -K )Ak 0

v * (k)=v

· p·m

(k) + (C

p

P

- -C Xll+D K

m P x

k

)A 0 m m 0

p x

m

(16) ( 1 7)

Equa tions ( 14) - ( 15 ) ca n b e simplified. we def in e

If

x

X E + X 12 ll

(18)

K

KE+

Ku Cv

( 19)

x

r\

~p

B~ I

DJ

Ixl r

XA

1

1_ J jw: J K

m

mcv

0

-

(20)

(cp X11 +D p Kx -Cm)

Ak 0 n 0

or e (k ) y

an d u se Equ a ti on ( 9 ) , then Equation s (14)( 15) be come

k

- D u (k)- (C Xll+D K - C ) A 0 P P P P x m m0

then t he id ea l c urves (11 ) sa ti s f y the equa ti ons

P P

(30)

-

(15 )

P P

-

e (k) = C x * (k ) + D u * ( k) - C x (k) y p p p p p p

[CC p X1l +Dp Kx - Cm)E+Cp X12 +Dp Ku Cv - DmCv Iv m(k)=O

P

T

Y

From Equation ( 24) we get

(14 )

*

(25)

where

In gene r a l c ur ves def ine d by Equat io n (11 ) canno t s imult a neo us l y s atis f y th e plant e qu a t io n ( 1) and the output e qua ti on (13) . Howeve r, by compa r i n g t he di ffe r e nce e qua ti on ob t a ine d by s ubs t i tuting Equa ti on (11) into Eq ua t ion (1) with t he e qua ti on obt a ine d by direc t d iffe r enc i ng of Equation ( 11) it can be s hown th a t, i f th e fo llowi n g r e lations ho l d

*

(24)

The a daptive a lgorithm gene rates the following plant c ontro l:

sa ti sfy th e output tr ac king e qua ti on y * ( k) = C

(23)

p

a nd th e output tra cking error is then

p

u * ( k) = -K x (k ) + K u ( k) P

P

x

inp u t" con t r ol u* (k) of the form

Cp e x (k)-D p (K(k+l)-K)r(k)+EAmk 0

0

(31)

where K

C p

i5

P

u

I

(32)

K

I:.

)-1 C (I + D p e p

(33)

I:.

(I + D

K

(34)

P e

) -1 D

P

Dis c r e t e Direct Multivariable Adaptive Control

~ (I+D K ) -1 (C Xll+D K - C ) Ak p e

p

p x

m

m

<5 0

(35)

he difference equation of the s tate error is hen

_rT(k)(K(k+l) _K)T [DT(STS) + (STS)D ](K(k+l )-K)· p

x

p

p

T

r(k) - eT(k)(STS)e (k)r (k)(2T+T)r(k)

x * (k+l) - x (k+l)

• (k+l)

359

y

p

y

- 2r T (k))K(k+l) - K)T (ST S) EAk 0

~ Px*(k)+B u*(k ) - (A Xll-XllA +B K ) Ak ~ p P P P m P x m ~o

m

0

(45)

If the following re l a t ions are satisfied ·A x

p p

x

(k)

-

(k+l)

B u (k)

p p

A e

p x

(46)

B (K(k+l)-K) r(k)

(k)

;\?rA

p

p

k -Bp Ke e y (k) - (A p Xll-XI1A m+B pKx )A m 50

.

k A e (k) - B (K(k+l ) - K) r (k) +FA 8 (37) p x p m0

, (k+l ) x

(48) we finally get 6.V(k) =_ [e T (k)L T_r T (k)(K(k+l ) _K)T WT] . x

"he r e

[Le (k)-W(K(k+l)-K)r(k)] x

zec - Ke (I + DpKe ) -1 ,

L,

.\ = A P

B p

(38)

BK C P ec P

P

6.

(47)

(36)

iubs tituti ng ev(k) from Eq ua ti on (29) finally si ves

P

p

(39)

B (I-K D ) p e p

(40) ( 49)

-6

-

-

-

F=(A Xll - XllA +B K ) - B K (C Xll+D K - C ) P m P x P ec P p x m ( 41) STABILITY ANALY SI S

4.

k -2e (k+l)PFA 0 x m 0

The following disc r e te quadratic Lya punov function is used t o prove stability of the adaptive sys t em r e pr esented by Equation (29) and Equation (37) :

Note that 6.V(k) is not necessarily negative de finite or semidefinite, due to the last two terms in Equation (49). However, 6.V(k) can still be use d to prove stabili t y of the sys tem. To this end, by using Equation (37) and Equation (42) it can be shown (Bar-Kana, 1983) tha t , i f lie (k) 11 or IIK(k+l)-KII x

become large enough, we ge t from Equation ( 49)

V(k) = eT(k) Pe (k) x

x

+tr[S(KI(k) - K)T -l (KI(k) - K)TST]

(42)

c,\,(k)

V(k+l) - V(k)

( 43)

~\,(k)

eT(k+l) Pe (k+l ) - eT(k) Pe (k) x x x x

V(k+l) < V(k)+aV(k) 11 A 11 k for some a. > 00 (50) -

or V( k+l) < (l+all A 11 k)V(k) -

V( k)

Equation ( 29) and Eq uati on (37) a nd manipulating the resulting algeb r a i c exp r ess i ons , we get:

- T= e (k )(A P A -P) e (k)

L ~(k)

x

P

p

x

- 2e I (k) AlpB ( K(k+l ) - K) r(k) x

p

p

~

W(k)

(52)

(44)

By s ub s tit uti ng ex(k+l ) and KI(k+ l) f rom

T

(51)

m

It i s clear that

+tr[S(KI(k+l) - K)T - l (KI(k+l) - K)TST] -tr [S(KI(k+l) - K)T - l(K I (k) - K)TST]

m

where W(k) is defined by the difference equation h'(k+l)=(l+all A 11k)W(k) ; W(O) = V(O) m

( 53)

[sing Equation (53) fo r k=0,1,2 ... , it can be seen that W(k+l)=(l+all A Il k) (1 +a I I A Ilk- I) . .. (l+a)W(O) m

m

(54)

and from (54)

+rI (k) (K(k+l)-K)I B Ip B (K(k+l ) - K)r(k) p

p

(l+a ll A 11 n- 2) ... (l+a.11 A 11 )(l+a)W(O) (55) m

m

360

I . Bar-Kana and H. Kaufman ~

We can apply the ratio test t o the series ob t a ined from Eq . (55) fo r k ~ 70 , t o show that the se ries converges and that W(k) is bounded. From Eq. (52) it can be seen th at V(k) is bounded for all k; th en , the quad ratic fo rm Eq . (42) of V(k) gua rante es th at the ga ins KI(k), t he s tate error ex(k), and the ou t put error ey(k) a r e bounded. In th a t case the la s t two term s in t:q . (49) van ish as k ~ "' . This fact permits a subsequen t applicat i on of a modified La Sa ll e ' s Invariance Principle for discrete nonlinear nonautonomous systems (LaSalle, 1977; Bar-Kana, 1983) to ge t the following theorem of s tabi lity for the discrete adaptive sys tem.

P

C

Co nditi ons E'ls. (46)-(48) are equ i valent to requirin g th a t th e closed loop input-output tran sfer function

z (z)

o + p

-

-

C (zI - A) p

p

- 1-

B

P

(56)

be (simply r ather than s trict ly) posi tive real. Special at t ention must be pa i d t o sys t ems with D =0. As see n from Eq. (48), in that case the Gisc r e t e pos itive realness condition canno t be sa ti sfied . Therefore, we may try to use s up pleme t a r y g ains D and D in o r der to satisfy t he posi tivit y cgnd i tioWs .

X

XA

I

I

,

I,

"

i...

( 61)

C E m

I

P~

v

0 C ~

m v -l

~ote

that the above relation has more equa tions than variables and it does not have a solution, in general . For the particular case of asymptotically constant inputs

=

lim um(k)

=

L

(62)

constant

k~..c

we ge t by subst it uting Av Eqs. (14) - (15) :

!

Then, all sta t es a nd ga i ns of the adapt i ve system are bounded and the output trackin g error vanishes asymptot i cally .

0

..,

l

pi

0

p

0

Theorem Assume th a t t he r e exist a positive de f inie matrix P and a ga in matrix Ke (not needed for implementation) such that relations Eqs . (46)(48) a r e sa ti sfied ; also, assume that Equations (18)-(20) have a solu ti on for t he gain matrices XII' x12 ' Kx ' and Ku ·

B

A

A X P

+

E

K - )(A

P

i

C

in

I

v

v

I c X- C E m p

I

o ,

L

o

(63)

0 K- D C

m v

P

L

...J

where U is a vect o r o f c o nstant coeff i c i ent. Equations (66) hav e a solutio n in genera l if dim [u (k)] > 1 or if L = O. \!hen m

dim [ u (k)]

an unkllol<'n v a lue lJ

~

ill

which

s a tisfies Eq . (6 3 exists dnd mi g ht b e f o und experiCle nta Ily . r J.

A minimum- phase and two non-m i nimum phase un stable systems are used fo r applica ti on of th e digital D111'IRAC t o regula t e and con tr ol dig it al control sys t ems . An implicit loop was imp l emen t ed, in o r der to satisfy Eqs . (24)-(30) . The r eference model is represented in all cases by

l1

o x

m

i

,I x (k) m I

(k+l

"I

+

u

.

OJ

. 25

L.

0

I

m

(k)

(64)

1 J

However, even in that case we want I.j x (k)

-.8

C x (k) p P

~

(65)

m

(57)

C x (k) as k m m

Example 1

and ge t i nstead The f i rst example is the minimum phase plant

+

C x (k) p p

as k

->-

0 u (k) p p

C x (k) m m

+

D u (k) m m

(58)

'"

x (k+l) p

p

...-. 882 4

In general , condit i on (SS) does not imply cond ition (57), because, i f we want C x (k) m m

C x (k) p p

( 59)

then Eq.

+ '

I

I

~l_

1 .8797J

(k)

iu

P (66)

\' (k) ' p

=

[

2. ] x (k) p

- 1.

+

2. u (k) p

(67)

The adapta ti on gain matrices defined in Eqs. (29)-(30) are

and D u (k) p p

(k)

x

I

=

o

1

,0 i

-+

D u (k) m m

(20) mus t be repla ced by

( 60)

T

=

T

=

.1 I

(68)

A sinusoidal input was used to control the system. The results of the d i g it al s i mula tion are represented in Fig. 1. Resul ts show good output model tracking.

Discrete Direct Multivariable Adaptive Control

Such a value for 0 was found experimentally m to be -.08.

Example I I The second example is an unstable plant which, without the direct feedforward, would have been non-minimum phase ~

I 0 u

x (k+l) p

p

(k)

l-.625 (69) Y (k) p

1.] x, (k) - 6. u (k) ,p

[-2.

=

(70)

The adaptation gain matrices are (71)

T=T=I.

The results of the servo following test with unit step input are shown in Fig. 2 and with a sinusoidal input are represented in Fie. 3. Observations show good output model following. Example III The third example is a non-minimum phase unstable system represented by

L625

x (k+l) P

:.J xp(k) {:}p(k)

( 72)

(73)

1.] x (k)

Y (k) = [ 2. p

p

The system (72)-(73) does not satisfy the positive realness conditions (46)-(48), therefore Eq. (73) is replaced by y

p

(~)

=

[2.

1.] x (k) + 7u (k) p p

(76)

such that the new system satisfies the positivity conditions. ror the subsequent presentation of the results we define the following values (k)

C x (k) P P

(75)

v" m(k)

C x (k) m m

(76)

ey (k)

y-m (k)

(77)

y

p

- YP (k)

The adaptation gain matrices are T = T

(78)

.1 1.

The results of the servo following test with a unit step input are shown in !"igs. 4-5 and with a sinusoidal input are shown in Figs. 6-7. Observations indicate that Yp(k) ~ Ym(k) while e (k) is finite and bounded. Since y

361

e

dim (u (k» = 1 we do not expect (k) to m Y vanish, as shown by Eq. (63) and the conclusiond following it. However since the system is stable, a gain 0 exists for step inputs such that Eq. (63) rrs satisfied.

The results of the servo following test with a step input for the adjusted model are represented in Figs. 8-9. Observations show that both conditions (59) and (60), are satisfied in this case. CONCLUSIONS In this paper the feasibility of direct multivariable adaptive model reference procedures was extended for discrete-time systems. The stability of the adaptive system is guaranteed provided that a positive realness condition is satisfied. No a priori information is required for implementation of the algorithm. A supplementary direct feed forward gain may be used to control systems like non-minimum phase which to not satisfy the positive realness conditions. This extension, together with the simplicity of implementation and the low order of the controller make the DHhRAC algorithm a useful adaptive control method suitable expecially for large scale systems.

Bar-Kana, I., H. Kaufman (1982). Hodel Reference Adaptive Control for Time-Variable Input Commands. Proc. 1982 Conf. on Inf. Sciences and Systems, Princeton, NJ. Bar-Kana, I., 11. Kaufman (1982). Hultivariable Direct Adaptive Control for a General Class of Time-Variable Commands, Proc. 21st IEEE CDC, Fl. Bar-Kana, I. (1983). Direct Multivariable Hodel Reference Adaptive Control with Applications to Large Structural Systems, Ph.D. Thesis, RPI, Troy, NY. LaSalle, J. P. (1977). The Stability of Dynamical Systems. SI~~. Sobel, K. (1980). tlodel Reference Adaptive Control for Multi-Input Multi-Output Systems. Ph.D. Thesis, RPI, Troy, NY. Sobel, K., H. Kaufman and L. Mabius (1980). Model Reference Output Adaptive Control Systems Without Parameters Identification. Proc. 18th IEEE CDC, Fl.

362

I. Bar-Kana and H. Kaufman !,

!'

;.\-'1

1",

/

'

/

~l

!l

!~

..

!! '

!'

~

Fi~.

1: Example I Sinusoidal input tracking. Outputs y (t), y (t) m p

..

Fig. 2: Example 11 Step input trackin g Outp uts y (t), Y (t) m p

Fi g . 3 : Example 11 Sinusoidal input tr ackin g Outputs y Ct) , y Ct) p

m

!~ ~I

!I

,, ~!!

~I

!J 'J

Fig. 4: Example III Step input tracking Output y (t), y (t) m p

Fig. 5: Example III Step input tracking Outouts ym (t), Y- p (t) •

Fig. 6:

Example III Sinusoidal input tr ac kin g Outputs y Ct) , y (t) m

!l

p

!i

!.

Fig. 7: Example III Sinusoidal input tracking Outputs y (t), y (t) m p

Fig. 8: Example III D = -.08 m Outputs ym(t), y (t) p

Fig. 9:

Example III D = -.08 m

Outputs

v- m (t) ,y p Ct)