IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993 27 1

gain K guarantees stability with uncertainty bound F. If this same gain is used in an observer-based control law, what is the magnitude of the uncertainty for which stability is still guaran- teed? Theorem 1 provides a sufficient condition for answering this question and Theorem 2 shows that if some additional conditions are imposed, stability can be guaranteed for uncer- tainty bounds arbitrarily close to T.. Through an example, we also show how these results can be used to design observer-based robust controllers.

The observers designed by our technique to fully recover the uncertainty bound are often high gain. The effect of this high- gain observer is to decrease the disturbance rejection properties of the control system. This trade-off is described in Theorem 4. Finally, if there is no measurement disturbance, then it is possible to fully recover the uncertainty bound with an observer-based control law while simultaneously maintaining the additive disturbance rejection bound. This result is given as Corollary 1.

ACKNOWLEDGMENT

The authors wish to thank Prof. M. Corless and anonymous reviewers for valuable suggestions.

REFERENCES B. R. Barmish and A. R Galimidi, “Robustness of Luenberger observers: linear systems stabilized via nonlinear control,” Auto- matica, vol. 22, pp. 413-423, 1986. V. W. Breinl and G. Leitmann, “State feedback for uncertain dynamical systems,” Appl. Math. Comput., vol. 22, pp. 65-87, 1987. I.R. Petersen, “A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 904-907, Sept. 1985. M. Tahk and J. L. Speyer, “Modeling of parameter variations and asymptotic LQG synthesis,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 793-801, Sept. 1987. I. R. Petersen and C. V. Hollot, “High gain observers applied to problems in stabilization of uncertain linear systems, disturbance attenuation and Hm optimization,” Int. J. Adaptive Confr. Signal Processing, vol. 2, pp. 347-369, 1988. F. Jabbari and W. E. Schmitendorf, “Robust linear controllers using observers,” IEEE Trans. Automat. Contr., vol. 36, pp.

I. R. Petersen, “Disturbance attenuation and H, optimization: A design method based on the algebraic Riccati equation,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 427-429, May 1987. P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust stabiliza- tion of uncertain linear systems: Quadratic stabilizability and H, control theory,” IEEE Trans. Automat. Contr., vol. 35, pp. 356-361, Mar. 1990. R. J. Veillette, J. V. Medanic, and W. R. Perkins, “Robust stabi- lization and disturbance rejection for systems with structured un- certainty,” presented at the 28th IEEE Conf. Decision Contr., Tampa, FL, Dec. 1989, pp. 936-941. L. Xie and C. E. deSouza, “Robust H, control for linear systems with norm-bounded time-varying uncertainty,” presented at the 29th IEEE Conf. Decision Contr., Honolulu, HI, Dec. 1990, pp.

I. R. Petersen and C. V. Hollot, “A Riccati equation approach to the stabilization of uncertain linear systems,” Automatica, vol. 22, pp. 397-411, July 1986. W. E. Schmitendorf, “Designing stabilizing controllers for uncer- tain systems using the Riccati equation approach,” IEEE Trans. Automat. Contr., vol. 33, pp. 376-379, Apr. 1988. I. R. Petersen, “A stabilizing algorithm for a class of uncertain linear systems,” Syst. Contr. Lett., vol. 8, pp. 351-357, 1987. B. R. Barmish, “Necessary and sufficient conditions for quadratic stabilizability of an uncertain system,” J . Optimiz. Theory Appl., vol.

1509-1514, 1991.

1034-1035.

46, pp. 399-408, Aug. 1985.

[15] V. Kucera, “A contribution to matrix quadratic equations,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 344-347, June 1972.

[16] B. A. Francis, “The optimal linear quadratic time invariant regula- tor with cheap control,” IEEE Trans. Automat. Contr., vol. AC-24, pp. 616-621, Aug. 1979.

A Robust Indirect Adaptive Regulation Algorithm with Self-Excitation Capability

Hua-Jye Peng and Bor-Sen Chen

Abstract-In this note, we present a robust version of the adaptive control algorithm established by Kreisselmeier [ l l . We know that if the conventional dead-zone adaptive law is used in the adaptive system, it is troubleseome that the size of the unmodeled dynamics, denoted by E, must be within the chosen dead-zone size for robustness. In this robust version, by introducing a positive design parameter E , the robust stabil- ity can be achieved without this constraint. It is also shown that the plant output and control input will converge asymptotically within a bound proportional to max{eE,E} := E,, where c is a finite positive constant. Moreover, in the ideal case (i.e., E = O), the plant output and control input will converge to zero under some condition.

I. INTRODUCTION

Recently, many papers concerned about robustness of the adaptive control system have been presented. In [2], the authors classify methods for ensuring robustness into two types. In the first type, the structures of the controller and adaptive law are kept the same as in the ideal case (i.e., the case of no plant uncertainty), but some signals in the adaptive loop are made dominantly rich such that the robust stability is guaranteed. In the second type, the structure of the controller is also the same as in the ideal case, but the adaptive law is modified. For implementation, most of those adaptive laws in the second type, e.g., dead zone, fixed-o, and E,-modification adaptive law, need the prediction of the size of the unmodeled dynamics. In the conventional dead-zone adaptive law, the dead-zone size must be larger than the size of the unmodeled dynamics. Once a dead-zone size is chosen, the degree of robustness of the adap- tive system is restricted within the dead-zone size. Other adap- tive laws such as fixed-o and E,-modification adaptive laws also have this drawback. In [5], it is shown that the switching o-mod- ification adaptive law can be implemented without this kind of prediction. In our proposed algorithm, the robust stability is also ensured without this constraint that the size of the unmodeled dynamics must be within a certain chosen dead-zone-type size.

A direct adaptive control algorithm with self-excitation capa- bility is firstly presented in [3]. Because this algorithm cannot ensure robustness, a robust version is established in [6] to overcome the unmodeled dynamics. Since the algorithm of [6] includes a conventional dead zone, the above-mentioned draw- back also exists. Kreisselmeier’s adaptive control algorithm in [ l] is an indirect version with self-excitation capability, but the robust stability is not ensured either. In this note, by introducing a positive design parameter Z, we establish a robust version of

Manuscript received September 29, 1989; revised August 10, 1990,

The authors are with the Department of Electrical Engineering, Tsing

IEEE Log Number 9203147.

June 14, 1991, and November 8, 1991.

Hua University, Hsin-Chu 30043, Taiwan, Republic of China.

0018-9286/93$03.00 0 1993 IEEE

272 IEEE TRANSACTlONS O N AUTOMATIC CONTROL. VOL. 38, NO. 2, FEBRUARY 1993

the adaptive control algorithm in [l], but the above-mentioned drawback is avoided. Moreover, it is shown in this robust vcrsion that the plant output and control input will converge asymptoti- cally within a bound that is proportional to max{ce, E ) := E,,,;

where c is a finite positive constant, and E is the size of the unmodeled dynamics. Moreover, in the ideal case (i.e., E = 01, the plant output and control input will converge to zero under some condition. Although the intention of using a self-excitation signal (internal impulse signal) and the ideas adopted in [7] are similar to ours, respectively, the approach and technical details are different.

The remaining contents are organized as follows. The pro- posed indirect adaptive control algorithm is established in Sec- tion 11. In Section 111, robustness and performance of the adaptive system are studied. Conclusion is given in Section IV.

11. THE INDIRECT ADAPTWE CONTROL ALGORITHM

A. Description of the Plant

Let d represent the unit delay operator. The plant to be controlled is defined by the following equations (also shown in [61):

A ( d ) l ( t ) = u ( t ) (2 . la)

(2.lb) Y ( f > = B ( d ) i ( t ) + 77,(1) where

A ( d ) = 1 + a,d + ..' +and" (2.lc)

B ( d ) = b,d + ... +b,d" (2.ld)

< ( t ) is the partial state; q , ( f ) is the unmodeled dynamics; and u( t ) and y ( t ) are the measurable plant input and output, respec- tively.

At this point, some notations are introduced for convenience and transparency: 1) g ( C ) denotes the smallest singular value of matrix C; 2) A(C) denotes the smallest eigenvalue of matrix C; and 3) let s ( t ) be a time signal. If s ( t ) is uniformly bounded

I l S L := supls(t)l . It 0

The truncated signal s,l(t) is defined as follows:

i.e., s,l(t) is obtained by truncating s( t> at TI. For s,l(t)

l/s,lll% := sup I S T , ( t ) l . O s t s 7 ,

0. = [ a , ;.., a,, b , ;.., b,]'

be the true parameter vector. We make the necessary assump- tions about the plant as follows.

Let

Assumption 2. I : a) A ( d ) and B ( d ) are coprime. b) llOp/l I k,] < x, where I I . I / denotes the Euclidean norm of a

c) For all t , (q,(t)( I EP(t), where E > 0 and P ( t ) is defined by vector.

P ( t + 1) = @ ( t ) + max{lu(t)l + lY( t ) l , 11 (2.2) P(0) = k , (0 < k , < x), where r E (0, l), and k , is chosen sufficiently large according to the initial states of the plant. 0

The plant (2.1) can be rewritten as

Y ( t > = 4 7 ( t ) 0 p + V ( f ) (2.3a)

where

77(t) = A(d)VI(t) (2.3b)

+ ( t > = [ - y ( t - I);.., - y ( t - n ) , u ( t - I);.., u ( t - TI)] ' .

(2 .3~)

B. SelfExcitation Signal / I /

Let

f ( t ) f ( t - 1) ... f ( t - 2n) - 1 f(t - 4 n ) ... I : f(t - 2n)

F ( t ) =

where f(t) is an arbitrary function of time. Moreover, f ( t ) is uniformly bounded and chosen such that g ( F ( R T ) ) 2 const > 0, where T 2 M + 1, M 2 4n, k = 0,1,2;.., . The self-excitation signal is

4 t ) = f ( t > e ( t > (2.4)

where e ( t ) is an error signal to be defined below.

C. Error Signal

Let 6,,(t) = [Z,( t ) , . . . , Z n ( t ) , 6,(t);.. , &,(t)]' be the estimate of B,, at time t , and A * ( d ) = 1 + aT;d + ... +a;,,d2" be a chosen Hurwitz polynomial. We construct signals j ( t ) and U ( t ) by the following equations:

A * ( d ) j ( t ) = b ( t , d ) v ( t ) (2.5a)

A"(d)U( t ) = i ( t , d ) v ( t ) (2.Sb)

where

g ( t , d ) = 6 , ( t ) d + i 2 ( t ) d 2 + ... +6,( t )d"

a ( t , d ) = 1 + 6 , ( t ) d + ... +Z, ( t )d" .

(2%)

(2.5d)

Also, let I

Z ( t ) = ( l y ( t - i ) - Y ( t - i)l + / u ( t - i ) - U ( t - i)l) I= 1

(2.6a)

P ( t ) = / I P,llx (2.6b)

Z p ( t > = Z ( t ) / P ( t > ( 2 . 6 ~ )

Z b ( 1 ) = C p ( t ) / ( l + E B ( t ) ) . (2.6d)

The error signal is defined as follows: e(0) is given and

e ( t + 1)

Z ; ( t + l ) P ( t + l), if Z i ( t + 1) > E

if Z b ( t + 1) 5 E

and t = kT

and t = kT = lo, e ( t > , otherwise

(2.7)

where E > 0 is a design parameter.

D. Identification of the Plant

Define

@ p ( t > := [ 4 ( t ) > . . . > 4(t - M ) I / P ( t ) (2.8a)

Y,(t) := [ y ( t ) ; . . , y ( t - M ) ] ' / p ( t ) . (2.8b)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993 273

The adaptive law for adjusting ip(t) is chosen as follows. Let

6;(t + 1) = Gp(t ) + [ cul + @fi(t)@pT(t)] - I

. a0(t)[ Y p ( t ) - @ l ( t ) i p ( t ) ] , a > 0. (2.9a)

Gp is adjusted by the following equation:

G p ( f + 1)

6i ( t + l ) ,

& ( t ) ? otherwise

if Z i ( t - T + 1) > E and t = ( k + l ) T

(2.9b) -i

where ip(0) can be arbitrarily given.

E. Adaptation of the Controller Coeficients

and B(d) , there exist two unique polynomials The similar work is shown in [l]. By the coprimeness of A ( d )

(2.10a)

(2 .lob)

D ( d ) = 1 + d,d + ... +d,d"

N ( d ) = n,d + ... +n,d"

such that

A ( d ) D ( d ) + B ( d ) N ( d ) = A * ( d ) .

b ( t , d ) = 1 + d ; ( t ) d + ... +i,(t)d" n i ( t , d ) = ri,(t)d + *.. +ri,(t)d"

(2.10c)

Let polynomials

(2.11a)

(2.1 lb)

be the estimates of D ( d ) and N ( d ) at time t , respectively. Define vector

i , ( t) = [io(t), i,(t);.., c i , ( t ) , ril(t);.., ri,(t)lT. (2.12a)

Letting S(Op( t ) ) = be the ?ylvester matrix constituted by the coefficients of polynomials A ( t , d ) and B(t , d), and defining 0, = [ l , dl;.., d,, nl;.., nnIT and a* = [ l , a;;.., a;,]T, we have

S T ( O,)O, = a* . (2.12b)

The adaptive law for adjusting O,(t) is chosen as follows: Let

G;(t + 1) = i c ( t ) + [ y l + s ^ ( t ) $ T ( t ) ] - l

. i ( t ) [ a * - s " T ( t ) 6 c ( t ) ] , y > 0. (2.13a)

0, is adjusted by the following equation:

i,(f + 1)

i ; ( t + l ) , and t = ( k + l ) T

if Il&'(t+l)II 5 k c , Z i ( t - T + l ) > E , (2.13b) = [ G,( t ) , otherwise

where k, is an upper bound of ~ ~ O , ~ ~ , and &0) can be arbitrarily given. Note that k , is bounded from a) and b) in Assumption 2.1. In this not:, it is assumed that k, is known a priori. Consequently, 11 O,(t)ll is uniformly bounded.

F. Control Law

The control law u( t ) is generated by the following equation:

b(t, d ) u ( t ) = - f i ( t , d ) y ( t ) + v ( t ) . (2.14)

G. Remarks

a) Note that if the initial states of the plant are unknown, they can be arbitrarily given for implementation without affecting the results below. In the following, we assume they are known a priori for convenience and brevity.

b) E used in (2.7) and (2.9b) acts as a dead zone, but it is emphasized that it is not necessary to care whether the size of unmodeled dynamics is within it for implementation. Thus, it is different from the dead zone used in the conventional adaptive control algorithm. In (2.13b), k, is needed to be known a priori. By using this supplementary knowledge and the self-excitation signal, the robust stability of the adaptive system can be proved in Section 111.

111. ROBUSTNESS AND PERFORMANCE ANALYSIS

The following lemmas are necessary for proving the main result in this note. Although the ideas are similar to ones in [71, the approach and technical details are different.

such that for all t , 7 2 0

Lemma 3.1: There exist 0 < c1,c2 < 00, 1 5 cg <

lu(t)l 5 C l P ( t ) > I Y ( t ) l 5 c , P ( t ) * P ( t + 7) 5 c m t > .

Proof This lemma is similar to [3, proposition 21 and, hence, the proof is omitted here.

Let A = {k 2 1: Zi(kT - T + 1) > E ] and 6(kT) =

A( afi( kT)@T(kT)). Then we have the following result. Lemma 3.2: Despite whether p ( t ) is uniformly bounded, there

exists c1 > 0 such that 6(kT) 2 1. > 0, for every E < c1 and k E A.

Proof The proof given here is adapted from [l]. Conse- quently, only different pivotal equations and explanations are derived and given, respectively.

Letting $ ( t ) = [ u ( t ) , u(t - I);.., u(t - n), y ( t - I);.., y ( t - n)]' and x ( t ) = [ l ( t ) , l ( t - I);.., l(t - 2n)lT, we have from (2.1) that

- + ( t ) = W t ) + % ( t )

where S = S(OJ and - v;(t) = [0,.-,0, v,(t - l), . . . , v1(f - 41'.

By Lemma 3.1, c) in Assumption 2.1, and noting that S is nonsingular

1177;(t)ll 5 J t ; d ( t ) , Ilx(t)ll 5 k l l m ) l l +kzll%t)ll I k 3 P ( t ) >

0 < k , , k,, k, < m. (3.1)

Next, note that there exists a full-rank matrix L such that

+ ( t ) = L $ ( t ) . It follows that

1 M

Now, it is concluded via (3.1) and (3.2) that if

1 M

p 2 ( k T ) i = o x(kT - i )xT(kT - i) > 0

274 IEEE TRANSACTIONS O N AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993

$,T(t)Sx(t) + f i ( t , d ) q l ( t ) = u ( t ) . (3.4) b) From-(3.5d)

Let q;( t ) = [vJt);.. , ql(t - 2n)I7, X ( t ) = [ X ( f ) , . . . , x(t - 2n)l, and f ( t ) = [f(t);.., f(t - 2n)]'. Since the controller (&t, d), $(t, d ) } is frozen and e ( t ) is constant during [kT - T + 1, k T ] , (3.4) gives

[ i ~ ( k T ) S X ( t ) l / p ( k T ) + [ f i ( k T , d)?7;' ' ( t ) l / p (kT)

= f T ( t ) e p ( k T ) , t E [ k T - T + I , k T ]

where e,(t) := e ( t ) / p ( t ) . Now, by nofing \ I $ ( f ) \ l I kiE@i) (0 < k: < CO), uniform boundedness of O , ( t ) , e p ( k T ) > zi. ( P ( k T - T + l) /p(kT)) 2 Z . C : - ~ (by Lemma 3.1), Lemma 3.1, and following the similar lines like those in [l], for every unit vector w , it can be concluded that

kT 1 ( x T ( t ) w ) 2 > k,E2 + ksE, 0 < k,, k, < m.

p 2 ( k T > r = k T - 4 n

Thus, tee rest$ is establish_ed fr2m the above equation.

results for if and iC.

Q.E.D. Let Op = Op - O,, and 0, = 0, - 0,. We have the following

Lemma 3.3: a) If E < E ] , then Il6,(t)ll is uniformly bounded. b) If E < and A is an infinite set, then

c) There exists 0 < e2 I E ] such that if A is an infinite set, then VE < e2

lim A(i , ) = lim k h E / [ z - (1 + j i / a ) - ' ] = k , ~ ( 1 + a@). 4- 2'1

By taking cq = k , (1 + a/ j i ) and the comparison theorem [81, the result is established.

c) From the result of b) above, lim sup, ,,l$t) - SI1 5 k,E (0 < k , < E). Thus, if E is sufficiently small, S ( t ) is nonsin- gular after some finite time. The result is established by follow- ing the similar lines, like those in [l]. Q.E.D.

Remark 3.1: From Lemma 3.3, without the knowledge of k,, Il&)ll is uniformly bounded under the condition E < E ] . But if k , is known a priori and (2.9b) is replaced by the following equation:

" ( t + 1)

GJt + l ) , if )I $( t + 1))) 5 k, , Z i ( t - T + 1) > E , and t = ( k = l ) T = I &( t ) , otherwise

(2.Yb)'

it can be ensured that Il6,,(t)ll is uniformly bounded for any E .

However, we will get the similar results in spite of which is used. n o

We establish the property of C h ( t ) as follows. Lemma 3.4: If E < E ~ , then

lim sup E h ( k T + 1) I em. (3.6) k + =

Proof: By contradiction, it is assum_ed that (3.:) is not true. Thus, A must be an infinite set. Let A*(t , d ) = D(t , d ) A ( d ) + $t , d ) B ( d ) - A*(& From (3.31, we have

lim sup (1 i,(t>)I I c s E , 0 < cs < x. r + =

Proof: a) Let A = (io, i1 , i2; . . , ) . In A , 1 I io < i, < i, < ... .

From (2.3), (2.Y), and the definition of &(t) , we have that Vq A * ( d ) i ( t ) = - A * ( t , d ) i ( t ) + u ( t ) - $ ( t , d ) q l ( t ) . (3.7)

Note that @&,T) is uniformly bounded by Lemma 3.1 and ~ ~ ? j p ( i q T ) ~ ~ I k i e (0 < ki < =)by c) in Assumption 2.1 and (2.3b).

definition of e ( t ) , Iu(t) l I If(t)l P( t ) , . Vt . Consequently, from Lemma 3.3, c) in Assumption 2.1, the definition of P(t ) , and the

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993 275

exponential stability of A*(d)-’ , we have that

lim sup I(u - ii)(t)l I k , ~ P ( t ) ,

limsup l ( y - y ) ( t ) l I k , ~ P ( t ) ,

0 < k , <

0 < k , < m

t + m

1 - m

if E < E,. Then

Iim sup ~ ( t ) I c e p ( t ) , c = T . ( k , + k , ) . f + m

Thus

limsupCi(t) = l imsupCB(t)/(l + C B ( t > ) I c e / ( l + c e ) . 1- m t + m

Since c/(l + C E ) I c

lim supCi(t) I C E .

This gives a contradiction, and, hence, the proof is completed. Q.E.D.

f’ m

Now, we can establish the main result in this note. Theorem 3.1: There exists E * > 0 such that if E, < E * , then

0 < c6 < 00. (3.9)

Proofi At first, by Lemma 3.2, it is noted that if E < E,,

then the parameter estimates of the controller and plant are uniformly bounded. Thus, any signal is finite over any finite interval if E < e2. The following derivations will be established on the condition E < e2.

From Lemma 3.4, there exists a sequence {e,(kT + 11, k 2 1: E , ( ~ T + 1) -+ 0 as k + m} such that V k 2 1

u(t ) and y ( t > are uniformly bounded and

limsup Iu(t)I,limsup I y ( t ) l I c ~ E , , t + m I’m

C g k T + 1) = CB(kT + 1)/(1 + CB(kT + 1))

I ( E , + E,(kT + 1)).

CB(kT + 1) I €,/(l - E,) + E4(kT + 1)

It follows that if E, < 1, then

where

E , ( ~ T + 1) = E,(kT + 1)/(1 - E,)

+ C@(kT + l)E,(kT + 1)/(1 - E,) .

Note that e4(kT + 1) -+ 0 as k + 00. Define E; := ~ , / ( l - E,).

By Lemma 3.1, it also follows that V t E [kT - T + 1, k T ]

I(u - Zi>(t)l, KY -”I I d ( k T + 1) I ( E L + E4(kT+ l ) ) P ( k T + 1)

I cl(€; + E,(kT + l))P(kT - T + 1).

This means

I(u - W t ) l > l ( Y -”I I cT<.; + % ( t ) ) P ( t ) , V t

(3.10)

where E&) + 0 as t -+ m. Next, by (2.5) and Lemma 3.4

I c ( t ) l , IY ( t ) l I k l O ( ~ , + ~ ~ ( t ) ) P ( t ) , 0 < klo < 00 (3.11)

where E&) -+ 0 as t -+ m. Combining (3.10) and (3.11), we have

lu(t)l, l Y ( t ) l I <.,‘E; + kl”% + E 7 ( f ) ) P ( t ) (3.12)

where ~ , ( t ) --$ 0 as t + m. Let A’ := { t : P ( t ) = p ( t ) } :=

{ to , t , , t , ; . . , 1, where to < t , < t , < ... . Define Z, := CTE; +

k l O ~ , . From (2.2), it is obtained that

P ( ‘ i + I ) I d ( t i ) + + E 7 ( t ) ) P ( t i ) 7 ‘1 < 4 t 1 ) + 2(Z, + E 7 ( t ) ) P ( 4 ) + 1

p ( t , + , ) = [ a + 2(?, + E 7 ( t ) ) l d t , ) + 1,

for all t , , t ,+ 1. Let the sequence { p(t,): i = 0,1,2;..,} satisfy

d t o ) = P < t J (3.13)

and E; = sup{^,,,: E, < 1 and cT~,/(l - E , ) + k , , , ~ , < (1 - r)/2}. Take E * = min{E;, C E ~ } . Thus, if E, < E * , then p( t , ) is uniformly bounded, since the system defined by (3.13) asymptoti- cally converges to an exponentially stable LTI system. Thus, the uniform boundedness of P ( t ) is implied by the comparison theorem [SI. From (3.12), (3.9) is established by taking c6 =

[c,T/(1 - E,) + k, , I . II PlL. Q.E.D. Proposition 3.1: If E = 0 and Z < E * , then A is a finite set.

Proot The proof is by contradiction. Assume A is infinite. By Lemma 3.3, IIO,(t)ll + 0 and IlO,(t)ll + 0 as t + m. From (3.8) and the uniform boundedness of u ( t ) and y ( t ) , y - J + 0 and U - U -+ 0 as t -+ W. Thus, C; ( t ) + 0 as t + 00. This means A is finite and gives a contradiction. Q.E.D.

Corollaly 3.1: Suppose E = 0 and ? < E * . By Prpposition 3.1, we let i, be the maximum element of A. If A(d)D(i*T + 1, d ) + B ( d ) N ( i * T + 1, d ) is a strict Hurwitz polynomial, then lu(t)l + 0 and ly(t)l + 0 as t + m.

Proofi Since the controller parameters are frozen after i,T + 1, the closed-loop system defined by (2.1) and (2.14) is LTI and asymptotically stable with v ( t ) = 0 after i,T + 1. Thus, the result holds. Q.E.D.

Remark 3.2: In the ideal case (i.e., E = O), besides the condi- tion, which is stated in Corollary 3.1, being satisfied, the pro- posed adaptive algorithm cannot ensure that lu(t)l and ly(t)l will converge to zero. n o

IV. CONCLUSION

In this note, a robust version of the adaptive control algorithm in [l] is established. Theorem 3.1 tell us the stability of the adaptive system is guaranteed if E, is sufficiently small. The feature of the proposed adaptive control algorithm is that E is chosen without the constraint that the size of the unmodeled dynamics must be within it. In [5], by using the switching a-mod- ification adaptive law, the adaptive control algorithm there also possesses this feature. The plant output and control input will converge asymptotically within a bound proportional to E,. But in the idea case (i.e., E = O), except some condition, which is stated in Corollary 3.1, being satisfied, the proposed algorithm cannot ensure that they will converge to zero.

REFERENCES G. Kreisselmeier, “An indirect adaptive controller with a self- exitation capability,” IEEE Trans. Automat. Contr., vol. 34, pp. 524-528, 1989. P. Ioannou and J. Sun, “Theory and design of robust direct and indirect adaptive control schemes,” Int. J . Contr., vol. 47, pp.

G. Kreisselmeier and M. C. Smith, “Stable adaptive regulation of arbitrary nth-order plants,” IEEE Trans. Automat. Contr., vol.

C . A. Desoer and M. Vidyasagar, Feedback System: Input-Output Properties. New York Academic, 1975. P. A. Ioannou and K. S. Tsakalis, “A robust direct adaptive con- troller,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 1033-1043, 1986.

775-813, 1988.

AC-31, pp. 299-305, 1986.

276 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 2, FEBRUARY 1993

[6] F. Giri, J. M. Dion, L. Dugard, and M. M'Saad, "Robust pole placement direct adaptive control," ZEEE Trans. Automat. Contr.,

F. Giri, M. M'Saad, J. M. Dion, and L. Dugard, "On the robustness of discrete-time adaptive linear controllers," in Proc. 1988 ZFAC Workshop Robust Adaptiue Contr., Newcastle, Australia, Aug. 1988. R. K. Miller and A. N. Michel, Ordinary Diflerential Equations. New York: Academic, 1982.

vol. 34, pp. 356-359, 1989. [7]

[8]

Solutions of the Equation AV + BW = W and Their Application to Eigenstructure

Assignment in Linear Systems

Gaung-Ren Duan

Abstract-Two new simple, complete, analytical, and restriction-free solutions with complete and explicit freedom of the matrix equation A V + B W = VF are proposed. Here [A B1 is known and is control- lable, and F is in the Jordan form with arbitrary given eigenvalues. Based on the proposed solutions of this matrix equation, a complete parametric approach for eigenstructure assignment in linear systems via state feedback is proposed, and two new algorithms are presented. The proposed solutions of the matrix equation and the eigenstructure assign- ment result are generalizations of some previous results and are simpler and more effective.

I . INTRODUCTION

The matrix equation

with [ A B ] being known and controllable, and F being in the Jordan form with arbitrary given eigenvalues, has close relations with many problems in linear control theory, such as the eigen- value assignment problem, the state observer design problem, and the eigenstructure assignment problem [1]-[5], and has been studied by many authors [l], [6]-[8]. Tsui [ l ] has summarized the existing solutions to this equation, and also has presented an attractive analytical and restriction-free solution with explicit freedom. To derive the solution of (1) proposed in [l], one needs to carry out an orthonormal similarity transformation and an inverse matrix, and solve series of liner equation groups.

In this note, another two new solutions of (1) are proposed, which are complete, analytical, and restriction free. They are linearly expressed by a group of parameter vectors, which repre- sent the freedom of the solution and are in clear neat forms. To obtain our solutions, one needs only to carry out a series of matrix elementary transformations. Thus, comparing with Tsui's solution, our solutions are simpler, need less computational work, and also possess good numerical property. Based on the proposed solutions of (11, the problem of eigenstructure assign- ment via state feedback for linear systems is considered, and two new algorithms are obtained, which are simpler than those proposed in [9]-[ll] and also eliminate the conditions required in [9]-[ll] on the closed-loop poles.

Manuscript received June 15, 1990; revised April 5 , 1991 and January

The author is with the Department of Control Engineering, Harbin

IEEE Log Number 9203148.

3, 1992.

Institute of Technology, Harbin, People's Republic of China.

11. PROBLEM OF EIGENSTRUCTURE ASSIGNMENT

Given the following time-invariant completely controllable linear system:

x = R x + B u (2) where x E R", U E R' are, respectively, the state vector and the input vector; A , B are known matrices of appropriate dimen- sions; and B is of full rank. If a linear state feedback control law

u = K r (3) is applied to system (2), a closed-loop system is obtained in the following form:

i = A , x , A , = A + BK. (4) Let I' = (si , s, E C, i = 1, 2;. . ,n ' , 1 I n' I n} be the set of

eigenvalues of matrix A,, which is symmetric about the real axis; and denote the algebraic and geometric multiplicity of si by m, and qi, respectively, then in the Jordan form of matrix A,, there are q, Jordan blocks, associated with si, of orders p i j , j = 1, 2;", qi, and the following relations hold:

pil + p , , + ... +piq, = m,, m, + m2 + ... +m,, = n. (5)

Again, denote the eigenvector and generalized eigenvectors of A , associated with si by U:, k = 1, 2;.., p i j , j = 1, 2;.., ql, then we have

( A + BK - S J ) ~ ; = .;=o (6) k = 1,2;..,pij, j = l , 2 ; . . , q i , i=1,2; . . ,n ' .

If we make the following conventions throughout this note: 1. Any group of vectors x$ k = 1,2;.-,pij, j = 1, 2;.., qi,

2. Any group of vectors {xk} are uniquely corresponding to i = 1, 2;.., n' may be denoted by {xfr,.}.

a matrix X = [xfr,] in the following manner: '1.

x = [XI x, ... X".] ( 7 )

xi = [ X i , x,, ... XI,,] (8)

X l j = [ X i j x; ... X C t l ] . (9)

Then the problem of eigenstructure assignment via state feed- back for system (2 ) can be stated as follows.

Given set r, and integers p i j , qi, mi, j = 1, 2;..,qi, i = 1, 2;.., n' satisfying (51, to characterize the following set:

E = ( ( V , K ) I V = [ U ; ] E C"'", K E R""

satisfying (6) and det ( V ) # 0).

Writing the equations in (6) in the following matrix form:

A V + B K V = VF (10)

W = W (11)

where F E C"'" is the Jordan form of A,. Denoting

and introducing the following two constraints concerning the linear independency of {U;} and the realness of matrix K :

C,: de t (V) # 0

C,: U; = ij[:, for s,=s, [9,1O]

then the solution to the above eigenstructure assignment prob- lem has the following representation:

E = { ( V , K ) l K = WV-I, ( V , W ) E U }

0018-9286/93$03.00 0 1993 IEEE