Control Structure of Stabilizing Controller for the Minimum Phase Systems and Design Method of Adaptive Control Systems

Copyright «) IFAC Large Scale Systems: Theory and Applications, Bucharest, Romania, 2001

CONTROL STRUCTURE OF STABILIZING CONTROLLER FOR THE MINIMUM PHASE SYSTEMS AND DESIGN METHOD OF ADAPTIVE CONTROL SYSTEMS Kou Yamada'

• Gunrna Univ(,1'sily. / -.1)- / Tf'njinc!w, Kiryu. Japan

A bstract.: In tb(' presellt paper. Wf' study Oil stabilizing controllers for the minimum phase s.vstelll~. At hrst, W(' clarify t.11(' cOlltrol structure of stabilizing controllers for t.hE' minimum pha.'i(' syst.ems. If t.he existence of t.he infinite gain loop is allowable, we can design control system such t.hat the control system stabilizes any minimum phase plant evell if t.he codficiellts of the plant are all unknown . The control structure for t.ll!-' minimulll ph
1. INTRODUCTION

In t.11P presellt paper, we examin!' a design method of (:olltrol syst.em for t1u' TllillinllllH pha.<;p syst.ems. In past studies on control design nwthods , lllany argument have been mad!' ullder the a.'5sulllptioll that. the plant. is of millilrlUlll pha.<;e, for example dec:oupling control(P.L.Falb and W.A.Wolovich. 1!)67: E.G.Gilbert, 1!)(i!)), model mat.ching control. inverse system (T.Yoshika.wa alld T.Sugie, 1982: Yalllada and Watanabe, 1!J!J2) , model feedback control (Narikiyo ami Izumi , 19m) , simple adaptive control (K.Sobel allCl L.Mabius, 1!)82) . model rden'lIce' adapt.ive control (K.S. Narendm. 1!)H9: S. Ma.';udda. 1!m4). Ahllost. all above desigll methods adopt. tl)(' id!'a. such t.hat. all of zeroes of t.hf' plallt. aff~ cancellf'o by th(' po It's of t.11(' cont.roller. To achipvp int.f'mally st.ability condit.ion. all of z('ro('S of t.h(' plant. llIUSt 1)(' in tll(' open ldt half planf'. That. is the plallt. is II('c('ssary t.o 1)(' of minilllulll phase. '1'll('s(' llIethods US(' only CJlI(' charact.erist.ics of t.1J(' millilllUIll phase sllch that tilt' minilllulII phas(' syst.em has no zpro in tll(' dosed right. half planf'. Ot.her dmract.t'ristics has not. 1)(,pn used and darifi('d. Considpring the ("()nt.rol charad.('ristics f(JI' t.II(' mini mu III pha.<;(' syst.em , it is natural t.o st.art dis-

cuss ions from the parametrization for the class of all stabilizing controllers for the minimum phase syst.ems. The parametrization for the class of all stabilizing controllers for the minimum phase systems Wc1.'5 first solved by Glaria and Goodwin (Glaria and Goodwin, 1994). Glaria and Goodwin gave explicit parametrization of all proper stabilizing controllers for the minimum phase systems. For the strictly proper system, their parametrization includes improper controllers. Yamada expanded t.he parametrization by Glaria and Goodwin and gave explicit parametrization for the class of all of proper stabilizing controllers for the minimum phase systems (Yamada, 2000). The purpose of this paper is to clarify the cont.rol struct.lIf!' for the minimum phase systems and t.o give a design method using its control st.ructure . At first , we clarify the control struct.ure for the minimum phase systems using the paranwt.rizat.ion by Glaria and Goodwin (Glaria and Goodwin, 1994; Yamada, 2000). It is shown t.hat if thp existence of the infinite gain loop is allowable. then we can design control system such t.hat t.hf' control system stabilizes any minimum pha.<;e plant. That is, we can design control syst.em asy m ptotically stable even if the coefficients of the plant are all unknown . In some cases, the existence

of the infinite gain loop is not allowable. When the existence of the infinite gain loop is not allowable, the control structure for the minimum phase system is given. Using this structure, we propose a simple design method of adaptive control systems. The stability of the control system designed by proposed method is clarified.

u

Notations

R R(s)

y

u

G(s)

Fig. 1. Control structure for the minimum phase biproper system Fig. 2 to hold u in Fig. 1 to be equal to u in Fig. 2 . It is interesting that to construct the control

y

2. PRELIMINARY

G(s)

In the present section, we summarize the parametrization for the class of all stabilizing controllers for the minimum phase systems (Glaria and Goodwin, 1994; Yamada, 2000) and present the control structures of the control system for the minimum phase systems.

-'

y = G(s)u { u = -C(s)y .

(1)

Fig. 2. Similarity transformation of Fig.1

Here, G(s) E R(s) is the single-input/singleoutput linear minimum phase system. That is, G(s) has no zero in the closed right half plane. C (s) E R( s) is the controller, y E R is the output and u E R is the input. When G(s) is biproper, that is the degree of numerator of G(s) is equal to that of denominator of G(s), the necessary and sufficient condition that the control system in (1) is asymptotically stable is summarized as following theorem.

system in Fig. 2 ,the dynamics of the plant G(s) is not required. That is, the control structure in Fig. 2 can always stabilize any minimum phase biproper plant G(s). The control system in Fig. 2 has the infinite gain loop. From the practical point of view, the infinite gain loop is not desirable. That is, in many cases, the infinite gain loop is not allowable. Then to stabilize the minimum phase system G(s), it is necessary to do system identification or adaptation. When the infinite gain loop is not allowable, the control system in Fig. 1 is rewritten by Fig. 3 to hold u in Fig. 1 to be equal to u in Fig. 3 . Here, Ql (s) E, RH00'

Theorem 1. If G(s) is of mmlmum phase and bi proper, then the proper controller C (s) stabilizes the unity feedback control system (1) and all of the transfer function (1 + G(s)C(S))-l, C(s)(1 +G(s)C(S))-I, G(s)(1 + G(s)C(S))-1 and G(s)C(s)(l + G(s)C(S))-1 are proper if and only if C (s) is written by

u

1-.-~

"

-I

1

C(s) = Q(s) - G(s) w-oo

•

---OrBI4--4-

Let us consider the unity feedback control system in

lim (1 + G(jw)C(jw))

G(s) ~

Q(s)

field of real numbers space of all real-rational transfer functions field of asymptotically stable proper rational transfer function square integrable functions field of bounded functions

1

1

1

(2)

y

••

I

! Q,(s) !

-I- 0

-

where l/Q(s) is any minimum phase and biproper rational function. (Glaria and Goodwin, 1994; Yamada, 2000) According to Theorem 1, the control structure to stabilize the minimum phase biproper system G(s) is shown in Fig. 1 . Fig. 1 is rewritten by

Fig. 3. Control system for the biproper system

Q2(S)

582

E

RHoo satisfies

Q(s) = G(s)Qt(s)

+ Q2(S) .

(3)

u

Any asymptotically stable Q(s) can always be factorized as above equation. The parametrization of all QI(S) and Q2(S) is written by

0] [D(S)] lfQJ(s)]_[ Q2(S) - Q(s) + -N(s) L(s),

1

1

= - Q(s) {(I - Qds))y - Q2(S)U) + Q(s) r =

L(s)D(s) - 1 Q(s) - L(s)N(s) Q(s) Y+ Q(s) U (9)

(4)

Let W1 ( t) and W2 denote

where, L(s) E RHoo is any rational function, N(s) E RHoo and D(s) E RHoo is coprime factor of G(s) over RHoo to satisfy

G(s) = N(s) D(s) .

(10)

(5)

and

In the present section, it is shown that if the control system in Fig. 2 is allowable, then we can design control system such that the unity feedback control system in (1) is asymptotically stable. \Vhen the control system Fig. 2 is not allowable , we can use the control structure in Fig. 3 .

[

1:

1

u,

respectively. fh and fh are settled to satisfy (12)

In the next section, we give a design method of adaptive control system using control structure in Fig. 3 .

and

fhwz(t) = (Q(s) - L(s)N(s)) u, 3. MODEL REFERENCE ADAPTIVE CONTROL

1 1 1 U = Q(s/IWdS) + Q(s/ZWZ(s) + Q(s{(14)

G( s) is assumed to be biproper minimum phase system. Gm(s) denotes reference model written by

From the assumption that all of coefficients of the plant G(s) are unknown, BI and Bz are unknown parameters. Let estimated values of BI and Bz be denoted by rh and Bz , respectively. Then we have adaptive control law by

(6)

(7)

where r is reference input, Ym is the output of the reference model. Without loss of generality, r is bound and continuous signal, that is r is differentiable. In addition, without loss of generality, the coefficients of the maximum order of both the numerator of Gm(s) and denominator of Gm(s) are positive.

Next, we describe a estimation law of 8 1 and 8z· In order to obtain a estimation law, a estimation model for (15) is required. In control law (14), it replace I / Q(s)r with I/Q(s)y. We have

The problem considered in the present paper is to design control system to satisfy lim (y(t) - Ym(t)) = 0

y = Q(s)u - B1wI(s) - 8zwz(s).

(8)

Let

and all of signals in the control system are bounded under the assumption that all coefficients of both numerator and denominator of G (s) are unknown. From Fig. 3 written by

(13)

respectively. Here ,\ 1 (s) is polynomial of the denominator of L (s )D( s), ,\z (s) is polynomial of the denominator of Q(s) - L(s)N(s), nl is degree of the denominator of L(s), nqis degree of the denominator of Q(s). (9) is rewritten by

In the present section, we give a design method of adaptive control system using control structure described in the previous section.

t~oo

(11)

sn+nl+n"

~I ,

6 and

ZJ

(16)

be (17)

(18)

and (4) , the control input u is and

583

v = Q(s)u,

(19)

respectively. Then (16) is rewritten by

is satisfied, where (28)

(20)

The error e between the output y and the output of the reference model Yrn is written by

and 0= [0 1 O2

e = Y - Yrn = V - (h6(s) - (h6(s).

From (27), we have (21)

= v(t) - 81';1(t) - 82 6(t).

The error ee between e and ee(t)

(30)

+ .;T(

a

From the assumption that f in (27) is positive definite diagonal matrix, V defined by

(22)

V

e is written by

= e(t) - e(t) = (1';1 + (2';2 = (.;,

ee f ';

(=_

To identify (h and (h in (21), following method can be used . Let e be estimation value of e (21). We have e(t)

(29)

] •

= ~(Tf-l(

(31)

2

is positive definite. This yields

.

V = (23)

e2

eT < O. a+'; .;-

(32)

Therefore V(t) converges finite number. This implies that ( is bounded. From (23) and (30) , ( E Loo is satisfied. From (32) , the function

where (24)

_ ] V(t)dt = ] (25)

o

ee

(

0

~)

2 dt

(a+.;T.;)

= - V(oo) + V(O)

and

is bounded. This implies (i

= 8i

-

Oi (i

= 1,2).

ee

(33)

1

E L2.

Since

L2 ,

(34)

(a+.;T.;)2

(26)

(23) is linear function of 01 and O2 . We have adaptive law for (23) by

ee(t) = m(t) ( E

(27)

1

(a + .;T.;) 2 , m(t)

E

Loo, [) E Loo, ( E L2 n Loo, [) E L2 n Loo hold.

Since [} is bounded and the reference input r is continuous function, all signals in the control system is included in the class PIO,oo]'

where f is any positive definite diagonal matrices and a > O.

Next, it is shown that all signals in the control system are bounded. Contraposition is shown, that is, if the input u and the output Y diverge, then it cause several contradictions. Applying Lemma 2.6 in the book by Narendra and Annaswamy (K.S. Narendra, 1989) to (10) and (11), then we have

Thus proposed adaptive control law is summarizes as follows . • parameter estimation law is given by (27) • control law is given by (15) Stability of the control system using proposed adaptive control law is summarized as following theorem.

supllwl(r)1I = 0 [suP1y(r)l]

Theorem 2. G(s) is assumed to be of minimum phase and biproper. If the control law is given by (15) and parameter estimation law is given by (27), then all signals in the feedback loop are bounded and (8) holds.

T~t

(35)

T~t

and sup Ilw2(r)11 = 0 [sup IU(T)I] . T~t

(Proof) From (24),(26),

T~t

Since Wl includes direct pass term of y ,

584

(36)

sup ly(r)1 = 0 [SUp IIWl(r)11] T~t

(37)

(48)

sup Ilw2(r)11 = 0 [sup IIv(r)lI] T~t

T~t

T~t

is satisfied. From (35) and (48), we have

is satisfied. Therefore sup ly(r)1 '" sup Ilwl (r)1I

(38)

( 49)

sup Ilw(r)11 = 0 [sup IIz(r)ll] ,

T~t

T:S;t

T~t

holds. From (14), Q(s)u is rewritten by

T:S;t

where (50)

From Lemma2.6 in Narendra and Annaswamy (K.S. Narendra, 1989), {j E Loo, T E Loo and (38), we have

~~~ IQ(s)u(r)1 = 0 [~~~ Ilw(r)ll] ,

From mum From (K.S.

the assumption that G(s) is of mlDlphase, (s + a)G(s) is of minimum phase. Lemma 2.7 in Narendra and Annaswamy Narendra, 1989),

(40)

where is satisfied. Above equation and (48) yield (41) Since Q(s)u include the direct pass term of u, Therefore sup lu(r)1 T~t

=0

[sup IQ(s)u(r)l]

(42)

T~t

sup IIw(r)11 '" sup Ily(r)1I T~t

is satisfied. This implies sup IQ(s)u(r)1 '" SUp lu(r)l. T~t

sup Ilw2(r)1I (53) T9

holds. Using Lemma 2.6 in Narendra and Annaswamy (K.S. Narendra, 1989),

(43)

T:S;t

sup lu(r)l, sup Ilv(r)ll, sup

Here v denotes

T~t

1

v(t) = - - u . s+a

T~t

11~(r)11 = 0

[sup ly(r)l] T~t

(44)

is satisfied.

e is

On the other hand, from (22), (19) and (9), written by

y(t) = (s + a)G(s)v(t).

(45)

, L(s)D(s) - 1 e = L(s)N(s) y

Applying Lemma 2.6 in Narendra and Annaswamy (K.S. Narendra, 1989) to above equation, we have

sup lIiJ(r)l1

T~t

(54)

Y is rewritten using v by

T~t

rv

T~t

= 0 [sup IIw(r)ll] .

1

+ L(s)N(s) T

-

(1.6 (t)

-Q(S){j2~2(t).

(55)

Using Lemma 2.9 in Narendra and Annaswamy (K.S. Narendra, 1989), we have

(46)

T~t

Since (11) is rewritten by

le(t)1

= 0

[~~~ 118(r)lI] + [!~~ Ily(r)ll]

+0

(47)

0

[~~ IIT(r)ll] .

Form (53) and the assumption that and )d s) is polynomial of denominator of Q( s) L(s)N(s), ~~s) E RHoo. Therefore from Lemma2.6

le(t)1 =

0

[sup Ily(r)lI] T:S;t

in Narendra and Annaswamy (K.S. Narendra, 1989),

holds. From (21) and (23),

585

(56) T

is bounded, (57)

y(t) :::; Ym(t)

+ 0[SUp IIY(T)II] + m(t) (0" + e{) ~

design control system such that the control system stabilizes any minimum phase plant without using all of coefficients of plant. That is, we can design control system asymptotically stable even if the coefficients of the plant are all unknown. When the existence of the infinite gain loop is not allowable, the control structure for the minimum phase system was given. Using this structure, we proposed a design method of adaptive control system that guarantee the stability of the control system. We can expand the proposed method for the multivariable strictly proper systems. The detail of the design method of adaptive control system for the multivariable strictly proper system is omitted to limit of space. This research was supported by Takayanagi Foundation for electronics science and technology

T$t

(58)

is satisfied. From m(t) E L2 and Lemma 2.9 in Narendra and Annaswamy (K.S . Narendra, 1989)

is satisfied. This equation contradict (53) . Thus it is proved that all signals in the control system are bounded. From the fact that all signals in the control system and (57), we have lim e(t)

t~ oo

= 0,

(60)

5. REFERENCES

and ~ is bounded . From (23),

ee is written by

E.G .Gilbert (1969) . The decoupling of multivariable systems by state feedback . SIAM J. on Control 7, 50-64. Glaria, J .J . and G.C. Goodwin (1994) . A parametrization for the class of all stabilizing controllers for linear minimum phase systems. IEEE Trans. on AC 39, 433- 434. K.S. Narendra, A.M.Annaswamy (1989). Stable adaptive systems,Prentice Hall. Prentice Ha" New Jersey. K .Sobel, H.Kaufman and L.Mabius (1982). plicit adaptive control for a class of mimo i;,:. _. terns. IEEE Trans . Aerospace and Electronic Systes AES-18, 576-589. Narikiyo, T . and T . Izumi (1991) . On model feedback control for robot manipulators. J, of Dynamic Systems, Measurement, and l trol 113, 371- 378. P.L.Falb and W .A.Wolovich (1967) . Decoupling in the design and synthesis of multi variable control systems. IEEE Trans. on Automatic Control 12, 651-659. S. Masudda, A. Inoue, V.T. Kroumov K. Sugimoto (1994). A design method for model reference adaptive control systems including fixed compensator using coprime factorization approach. Transaction of the Society of Instrument and Control Engineers 30, 417426. T.Yoshikawa and T .Sugie (1982). Filtered inverses with application to servo systems. Transaction of the Society of Instrument and Control Engineers 18, 792- 799. Yamada (2000) . A parametrization for the class of all proper internally stabilizing controllers for linear minimum phase systems. Submitted for publication. Yamada, K. and K. Watanabe (1992) . State space design method of filtered inverse systems. Transaction of the Society of Instrument and Control Engineers 28, 923- 930.

(61)

Since ( is bounded, since

ee is also bounded. From (34), (62)

rh(t) is bounded . Using Lemma 2.12 in Narendra and Annaswamy (K.S. Narendra, 1989) lim m(t)

t~oo

=

0

(63)

is satisfied . From above discussions , we have lim ee (t)

t-> oo

=0

(64)

and lim (y(t) - Ym(t))

t-> oo

= O.

We have thus proved this Theorem 2.

(65)

I

Using similar procedure, we can design adaptive control system for the minimum phase strictly proper system. The detail of the design method of adaptive control system for the minimum phase system is omitted for limit of space.

4. CONCLUSION In the present paper, we clarified the characteristics of the control structure for the minimum phase systems and gave a design method using its characteristics. It was shown that if the existence of the infinite gain loop is allowable, then we can

586