Lyapunov-based adaptive control of MIMO systems

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Automatica 39 (2003) 1251 – 1257 www.elsevier.com/locate/automatica

Brief Paper

Lyapunov-based adaptive control of MIMO systems
Ramon R. Costaa;∗ , Liu Hsua , Alvaro K. Imaia , Petar Kokotovi.cb a Department

b Center

of Electrical Engineering, COPPE/UFRJ, P.O. Box 68504, 21945 970 Rio de Janeiro, Brazil for Control Engineering and Computation, University of California, Santa Barbara, CA 93106, USA

Received 8 November 2000; received in revised form 1 August 2001; accepted 25 February 2003

Abstract The design of Model-Reference Adaptive Control for MIMO linear systems has not yet achieved, in spite of signi6cant e7orts, the completeness and simplicity of its SISO counterpart. One of the main obstacles has been the generalization of the SISO assumption that the sign of the high-frequency gain (HFG) is known. Here we overcome this obstacle and present a more complete MIMO analog to the well known Lyapunov-based SISO design which is signi6cantly less restrictive than the existing analogs. Our algorithm makes use of a new control parametrization derived from a factorization of the HFG matrix Kp = SDU , where S is symmetric positive de6nite, D is diagonal, and U is unity upper triangular. Only the signs of the entries of D or, equivalently, the signs of the leading principal minors of Kp , are assumed to be known. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Lyapunov methods; Adaptive control; Multivariable systems; High-frequency gain; Factorization

1. Introduction In this paper we develop a multivariable (MIMO) analog of one of the oldest single-variable (SISO) results in adaptive control—the Lyapunov-based model-reference (MRAC) design of minimum phase linear systems with relative degree one. Initiated by Butchart and Shackcloth (1965), and forcefully articulated by Parks (1966), the Lyapunov-based adaptive design is one of the early applications of the celebrated (Strict) Positive Real Lemma. Research in adaptive control of MIMO linear systems continues to be active. Early results, including Monopoli and Hsing (1975) and Elliott and Wolovich (1982) and many others, were surveyed by Dugard and Dion (1985) and summarized in several textbooks (Ioannou & Sun, 1996; Narendra & Annaswamy, 1989; Sastry & Bodson, 1989).
Research supported in part by the National Science Foundation under grant ECS-9812346, the Air Force OEce of Scienti6c Research under grant F49620-95-1-0409, CAPES, CNPq, FAPERJ and PRONEX. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Bernard Brogliato under the direction of Editor Robert R. Bitmead. ∗ Corresponding author. E-mail addresses: [email protected] (R. R. Costa), liu@coep. ufrj.br (L. Hsu), [email protected] (A. K. Imai), petar@seidel. ece.ucsb.edu (P. Kokotovi.c).

They have recently been extended and applied to Jight control, visual servoing and other challenging tasks (Bodson & Groszkiewicz, 1997; Hsu & Aquino, 1999; Hsu, Costa, & Aquino, 2000; Zergeroglu, Dawson, de Queiroz, & Behal, 1999). In spite of all this progress, a complete or at least satisfactory MIMO generalization of the Lyapunov-based direct adaptive control has not yet been achieved even for the relative degree one case. Indeed, the existing direct MIMO MRAC schemes require much more stringent assumptions on the plant than in the SISO case. The main stumbling block is the high-frequency gain (HFG) matrix Kp . For a MRAC design using the direct adaptation approach, restrictive assumptions on the prior knowledge of Kp have been made. For example, in Ioannou and Sun (1996) it was assumed that a matrix Sp is known such that Kp Sp = (Kp Sp )T ¿ 0. On the other hand, using the indirect adaptation approach, one must require that the estimate of Kp be nonsingular at all times. In de Mathelin and Bodson (1995) this singularity is avoided by using hysteresis and projection and only an upper bound on the norm of Kp is needed as prior knowledge. A more general solution was proposed in Weller and Goodwin (1994). They introduced a new plant parametrization in which the HFG matrix is factored as Kp = LU , where L is unity lower triangular and U is upper triangular. The singularity of the estimate of Kp is avoided by constraining

0005-1098/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00085-2

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R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257

the diagonal entries of the estimate of U to be away from zero, which requires the knowledge of their lower bounds. Recently, a similar LU factorization was shown to be a key procedure in circumventing the usual restrictive prior assumptions required in direct MRAC design of a 2 × 2 visual servoing system in Hsu and Aquino (1999) and Hsu et al. (2000). The same visual servoing problem was solved in Zergeroglu et al. (1999) using a factorization of the form Kp = SU , where S is symmetricpositive de6nite. An extension of the LU factorization to m × m systems, in which Kp is the only unknown quantity, was presented in Hsu and Costa (1999). The essential feature of the recent parametrizations employing factorizations of Kp is that they focus on the quantities whose signs (or lower bounds) need to be known. In Hsu and Aquino (1999), Hsu and Costa (1999) and Zergeroglu et al. (1999) the common assumption is the knowledge of the signs of the diagonal entries of U or, equivalently, the signs of the leading principal minors of Kp . This assumption is analogous to the knowledge of the sign of the HFG in the SISO case and will be extended in this paper to more general MIMO systems. In this paper, we employ the factorization Kp = SDU , introduced by Morse (1993), where S is symmetric positive de6nite, D is diagonal, and U is unity upper-triangular. This factorization of Kp is convenient because of the distinct role played by each of its factors S, D and U . The role of S is to assure that WM (s)S is SPR, where WM (s) is the desired reference model. We prove that such a matrix S exists. The role of D is to make possible a straightforward extension of the SISO assumption about the sign of the HFG: only the sign of the diagonal entries of D are assumed to be known. Finally, the role of U is to eliminate the possibility of static loops and thus assure that the adaptive control law is well de6ned. The new parametrization based on the factorization Kp =SDU resulted in a more complete MIMO analog of the SISO adaptive Lyapunov-based design which is signi6cantly less restrictive than the existing analogs.

2. Preliminaries Let {A; B; C} be a realization of a strictly proper and nonsingular m × m rational transfer function matrix G(s) = C(sI − A)−1 B. • The observability index of the pair {C; A} (A ∈ Rn×n ; C ∈ Rm×n ) is the smallest integer , (1 6  6 n), such that T
O = C T (AC)T · · · (A−1 C)T (1) has full rank. The observability index has a nice system interpretation: ( − 1) is the largest number of derivatives of y required to determine the initial condition (Kailath, 1980). In other words, it gives information about the order

of the state variable 6lters required in the structure of a MIMO MRAC design. • If det(CB) = 0 we say that G(s) has relative degree 1 and the nonsingular matrix Kp = CB is referred to as the HFG matrix. This is a natural generalization of the relative degree one condition for SISO systems (Bodson & Groszkiewicz, 1997).

3. Problem statement For an observable and controllable MIMO linear time-invariant plant given by an m × m transfer matrix G(s), y = G(s)u;

(2)

we make the following assumptions: (A1) The transmission zeros of G(s) have negative real parts. (A2) G(s) has relative degree 1. (A3) The observability index  of G(s) is known. (A4) The signs of the leading principal minors of the HFG matrix Kp are known. The minimum phase assumption (A1) is fundamental in MRAC framework. With assumption (A2) we focus on the simplest case amenable to Lyapunov based designs. Assumption (A3) can be weakened to require only the knowledge of an upper bound on , as in Ioannou and Sun (1996), which, however, would increase the order of the 6lters and the number of parameters. While Kp is nonsingular, some of its leading principal minors may be zero. Without the key assumption (A4), Weller and Goodwin (1994) solved this problem using a complicated scheme with switching logic to adaptively select an ordering of the outputs for which the leading principal minors of the Kp are nonzero. With assumption (A4) we require an increase of apriori information to achieve simplicity of the resulting MRAC scheme and a clear understanding of its properties. Signi6cant applications in which this type of apriori information is available, such as visual servoing, make assumption (A4) meaningful and motivate the development of a simple MRAC scheme for MIMO systems. The adaptive control objective is to achieve asymptotic tracking e(t) = y(t) − yM (t) → 0

as t → ∞;

(3)

where yM ∈ Rm is the output of the reference model yM = WM (s)r;

(4)

and r ∈ Rm is a piecewise continuous uniformly bounded signal. To select our reference model, we recall the fact that det(CB) = 0 implies that G(s) can be rendered diagonal by dynamic feedback (Rugh, 1993). Thus, without loss of

R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257

generality, we select a diagonal SPR reference model   1 WM (s) = diag ; s + ai

(5)

where ai ¿ 0, (i = 1; : : : ; m).

If G(s) is known, then a control law which achieves matching between the closed-loop transfer matrix and WM (s), ∗

y = G(s)u = WM (s)r = yM ;

(6)

u∗ = 1∗T !1 + 2∗T !2 + 3∗ y + 4∗ r =  ∗T !;

(7)

where  ∗T = [1∗T 2∗T 3∗ 4∗ ] and the regressor vector ! = [!1T !2T yT r T ]T are de6ned by

A(s) u; (s)

!1 =

3∗ ∈ Rm×m ; A(s) y; (s)

!2 =

A(s) = [I Is · · · Is−2 ]T ; (s) =

0

+

1s

4∗ = Kp−1 ;

!1 ; !2 ∈ Rm(−1) ;

I ∈ Rm×m ;

+ · · · + s−1 is Hurwitz:

The matched closed-loop system is obtained setting u = u∗ . When !1 , !2 , y, and r in (7) are expressed in terms of u∗ , u∗ = 1∗T

(8)

the matching equation becomes A(s) A(s) − 2∗T G(s) − 3∗ G(s) (s) (s)

= 4∗ WM−1 (s)G(s):

(9)

Right multiplying both sides of the above equation by u, one gets u=

∗T

!−

Kp−1 r

+

Kp−1 WM−1 (s)G(s)u:

(12)

where (t) is an estimate of  ∗ . With (12) the output error is a linear function of the parameter error ˜ =  −  ∗ , e = WM (s)Kp [˜ T !]:

(13)

(3) Assuming that sign(Kp ) is known, stability and convergence of e(t) are assured by the update law (SISO case) (14)

The MIMO generalizations of these steps, discussed in the textbooks (Ioannou & Sun, 1996; Narendra & Annaswamy, 1989; Sastry & Bodson, 1989), can be brieJy summarized as follows. For Step 1 a diagonal reference model WM (s) is selected as in (5). For Step 2 the matrix version of the control law (12) is employed, where  is a matrix, while ! is an enlarged vector. The MIMO error equation retains the same form of the SISO error equation (13), except that WM (s), Kp and ˜ are matrices. For Step 3 di7erent assumptions about Kp have been made. In Sastry and Bodson (1989), the diEculty is avoided by assuming that Kp is known. Unknown Kp was considered in Ioannou and Sun (1996) under the assumption that a matrix Sp is known such that Kp Sp = (Kp Sp )T ¿ 0. This assumption, however, is too restrictive. 5. Gain factorization

A(s) ∗ A(s) G(s)u∗ u + 2∗T (s) (s)

+ 3∗ G(s)u∗ + 4∗ WM−1 (s)G(s)u∗ ;

I − 1∗T

(1) WM (s) is a scalar transfer function chosen to be SPR. (2) The adaptive control law is

˙ = −! sign(Kp )!e:

is given by Sastry and Bodson (1989).

1∗ ; 2∗ ∈ Rm(−1)×m ;

form as the well known SISO error equation. Let us recall the main steps in the SISO design:

u =  T !;

4. Review of previous designs

1253

(10)

Finally, upon multiplying (10) by WM (s)Kp , rearranging and using e = y − yM , y = G(s)u, yM = WM (s)r, one obtains the output error equation 
e = WM (s)Kp u −  ∗T ! : (11) Except for the fact that WM (s) and Kp are matrices and e and u are vectors, this MIMO error equation has the same

To derive our new parametrization, we 6rst perform a factorization of the in6nite frequency gain Kp in which we introduce a positive diagonal matrix D+ as a free parameter. For this we need the following lemma adapted from Morse (1993). Lemma 1. Every m × m real matrix Kp with nonzero leading principal minors "1 ; "2 ; : : : ; "m can be factored as Kp = SDU;

(15)

where S is symmetric positive de8nite, D is diagonal, and U is unity upper triangular. Proof. Since the leading principal minors of Kp are nonzero, there exists a unique factorization (Strang, 1980) Kp = L1 Dp LT2 ; where L1 and L2 are unity lower triangular and   "2 "m Dp = diag "1 ; ; : : : ; : "1 "m−1

(16)

(17)

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R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257

Factoring Dp as Dp = D+ D;

(18)

where D+ is a diagonal matrix with positive entries, we T rewrite (16) as Kp =L1 D+ LT1 L−T 1 DL2 , so that (15) is satis6ed by S = L1 D+ LT1 ;

T U = D−1 L−T 1 DL2 :

(19)

Remark. In Morse (1993), the matrix D is made of diagonal entries +1 or −1. Our factorization Kp = SDU is not unique because the positive diagonal matrix D+ , introduced in (18), is a free parameter. Example 1. To illustrate the patterns for each of the factors S, D, and U , we consider   k11 k12 Kp = : k21 k22 The LDU factorization (16) gives     "1 0 1 0 ; ; Dp = L1 = "2 0 l1 1 "1

 L2 =

1

0

l2

1

0

;

(23)

Next, we introduce new parameter vectors '∗i via the identity T ∗T ∗T ['∗T 1 (1 '2 ( 2 · · · ' m (m ]

≡ K1 !1 + K2 !2 + K3 y + K4 r + (I − U )u:

(24)

In addition to the concatenated ith rows of the matrices K1 , K2 , K3 , K4 , each row vector '∗T includes the unknown i entries of the ith row of (I −U ). The corresponding regressor vectors are (1T = [!T u2 u3 · · · um ]; .. . (25)

The error equation (23) has thus been brought to the new form T ∗T ∗T e = (WM (s)S)D(u − ['∗T 1 (1 '2 (2 · · · 'm (m ] ): (26)

In this new parametrization the adaptive control law is u = ['T1 (1 'T2 (2 · · · 'Tm (m ]T ; (20)

6. Control parametrization We now employ the SDU factorization of Kp to derive a new form of the error equation. Substituting Kp = SDU in (11) and using (7) we obtain e = WM (s)SD (21)

A further re6nement of this expression will make sure that the control law is well-de6ned. With the decomposition Uu = u − (I − U )u;

−K3 y − K4 r − (I − U )u]:

(mT = [!T ]:

1

×[Uu − U1∗T !1 − U2∗T !2 − U3∗ y − U4∗ r]:

e = WM (s)SD[u − K1 !1 − K2 !2

(2T = [!T u3 · · · um ];



where l1 = k21 ="1 and l2 = k12 ="1 , and, for   + 0 d1 ; D+ = 0 d+ 2 −1 the SDU factorization (15) yields D = D+ Dp ,   + + d 1 l1 d1 ; S= + + 2 d 1 l 1 d2 + d + 1 l1   d+ 1 l1 " 2 1 l − 2  2  d+ U = 2 "1  :

entries of U are incorporated in the new parametrization by de6ning K1 = U1∗T , K2 = U2∗T , K3 = U3∗ , and K4 = U4∗ , and rewriting (21) as

(22)

where (I − U ) is strictly upper triangular, it is possible to de6ne the control signal u as a function of (I − U )u. No static loops can appear, because u1 depends on u2 ; : : : ; um , while u2 depends on u3 ; : : : ; um , and so on. The unknown

(27)

where 'i are the estimates of '∗i . Compared with the control law (12), this new control makes use of a larger number of parameters. The key feature of the new error equation (26) is that the diagonal matrix D appears in the place of the Kp , and an assumption can now be made about the signs of its entries d1 ; : : : ; dm . It may seem that this advantage comes at a price: the SPR condition is to be satis6ed by WM (s)S, rather than by WM (s) alone. Indeed, WM (s) being SPR jointly with S = S T being positive de6nite does not imply that WM (s)S is SPR. Fortunately, we can show that for any W (s) in (5), a positive de6nite S = S T exists such that WM (s)S is SPR. To prove this, we employ the property that S in Kp = SDU is not unique. For the ease of understanding, we 6rst consider the 2 × 2 case. Example 2. Consider WM (s) = diag{1=(s + a1 ); 1=(s + a2 )}, a1 ; a2 ¿ 0, and let {A; S; I } be a minimum realization of WM (s)S, where   −a1 0 A= ; S = S T ¿ 0: 0 −a2

R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257

By the SPR Lemma, WM (s)S is SPR if and only if P = P T ¿ 0 and Q = QT ¿ 0 exist such that AP + PA = −2Q;

(28)

PS = I:

(29)

In the special case, when WM (s) has two identical entries, a1 = a2 = a ¿ 0, then P = S −1 and Q = aS −1 . This means that WM (s)S = 1=(s + a)S is SPR for any S = S T ¿ 0. When a1 = a2 , then there exist symmetric positive definite S which do not satisfy the SPR condition (28), (29). We need to prove that there exists at least one S in the form (19) which satis6es (28), (29). Using the explicit expression for S given in (20), we get −1  + d+ d1 1 l1 −1 P=S = + 2 d+ d+ 1 l1 2 + d 1 l1   1 l21 l1 + −   d+ d+ d+  2 2  = 1 ;  1  l1 − + d2 d+ 2 + which is positive de6nite for all l1 , because d+ 1 ¿ 0, d2 ¿ 0 by construction. Then   a1 a1 l21 l1 (a1 + a2 ) −   d+ + d+ 2d+   2 2 Q = −(PA + AP)=2 =  1    l1 (a1 + a2 ) a2 − + + 2d2 d2

will be positive de6nite and, hence, WM (s)S will be SPR, if + 2 2 d+ 1 ¡ 4a1 a2 d2 =[l1 (a1 − a2 ) ]. This proves that, for any 2 × 2 unity lower triangular matrix L1 , there exists a matrix D+ such that WM (s)S = WM (s)L1 D+ LT1 is SPR. Let us generalize this result. Lemma 2. For any A = diag{−ai }, ai ¿ 0; (i = 1; : : : ; m), and any m × m unity lower triangular matrix L1 , there + exists D+ = diag{d+ i }, di ¿ 0, such that WM (s)S = (sI − A)−1 L1 D+ LT1

(30)

is SPR. Proof. See proof in Hsu, Costa, Imai and Kokotovic (2001). 7. Adaptive control Combining the state x ∈ Rn of the plant (2) with the 6lter states !1 , and !2 , we de6ne X = [xT !1T !2T ]T ∈ Rn+2m(−1) . With XM we denote the state of the corresponding nonminimal realization CM (sI − AM )−1 BM of WM (s)S where CM BM = S. Then, the state error z = X − XM and the output

1255

error e in (26) satisfy T ∗T z˙ = AM z + BM D(u − ['∗T 1 (1 · · · 'm (m ] );

e = CM z:

(31)

T Because WM (s)S is SPR, there exist matrices PM = PM ¿0 T and QM = QM ¿ 0 satisfying

ATM PM + PM AM = −2QM ;

(32)

T : PM BM = CM

We design an update law for the control parameters vectors 'i in the adaptive control (27) in a complete analogy with SISO adaptive case. We use the Lyapunov function

 m  1 −1 T T z PM z + V= (33) !i |di |'˜ i '˜ i ; 2 i=1

where '˜ i = 'i − '∗i are the parameter errors, di are the entries of D, and !i ¿ 0 are adaptation gains. The time derivative of (33) along the trajectories of the error system (31) yields V˙ = −z T QM z + z T PM BM D['˜ T1 (1 · · · '˜ Tm (m ]T +

m 

˜ T ˜˙ !−1 i |di |'i 'i

i=1

= −z T QM z +

m 

˜T ˜˙ !−1 i |di |'i [!i sign(di )ei (i + 'i ]:

i=1

An update law which renders V˙ nonpositive, V˙ = −z T QM z, is '˙ i = '˜˙ i = −!i sign(di )ei (i

(i = 1; : : : ; m):

(34)

Thus, the adaptive control (27) and the update law (34) guarantee '˜ i ; 'i ∈ L∞ and z ∈ L∞ ∩ L2 . Because z = X − XM and XM are bounded, X is also bounded and, consequently, y, !1 and !2 are bounded. Since r(t) is uniformly bounded by assumption, ! is bounded. To prove that (1 ; : : : ; (m are bounded and, hence, u is also bounded, we return to (25). The advantage of the structure of (25), resulting from the control parametrization, is that (m =! being bounded, implies that um ='Tm (m is bounded. T Therefore (m−1 = [!T um ] is bounded. Repeating this argument we show that um−1 ; : : : ; u2 ; u1 are all bounded. Therefore, all the signals in the closed-loop system are bounded. This also implies that z, ˙ e, ˙ '˙ i and consequently VU are all uniformly bounded. Finally, the usual argument invoking Barbalat’s Lemma proves that z(t); e(t) → 0 as t → ∞. We have thus obtained the following result. Theorem 1. Consider system (2) and the reference model (4). Suppose that assumptions (A1) – (A4) hold. If r(t) is piecewise continuous and uniformly bounded, then the

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R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257

adaptive control (27) with update law (34) assures that all the closed loop signals are uniformly bounded and the tracking error e(t) converges to zero. Design example. Consider the following plant transfer  macos(0) sin(0) trix G(s)=(1=(s+1))Kp ; Kp = being −sin(0) cos(0) a rotation matrix. A similar problem appears in planar visual servoing with uncalibrated camera where 0 corresponds to the unknown misalignment angle between the camera and workspace coordinate systems (Hsu & Aquino, 1999; Zergeroglu et al., 1999). We note that, even for the limited range −90◦ ¡ 0 ¡ 90◦ , there is no matrix S such that Kp S is symmetric and positive de6nite in any open interval within the given range of 0. In contrast, Kp satis6es "1 ¿ 0 and "2 ¿ 0 for the whole range. We assume that only Kp is unknown and de6ne WM (s)=diag{1=(s+1); 1=(s+1)}. Hence, we set K1 =K2 =K3 =0 and the adaptive control (27) reduces to u1 = 'T1 (1 ; (1T = [r1 r2 u2 ]; u2 = 'T2 (2 ; (2T = [r1 r2 ], and the update laws are '˙ 1 = −!1 (1 e1 ; '˙ 2 = −!2 (2 e2 . We emphasize that the signs of the entries of D are the only information about Kp and its factors, required to implement this adaptive control design. Simulation results can be found in Hsu et al. (2001). 8. Conclusion We presented a complete MIMO analog of the SISO MRAC design for minimum phase systems with relative degree one. The simplicity of this design is a consequence of a new control parametrization. The key ingredients of the new control parametrization are the SDU factorization of the multi-variable high frequency gain Kp together with the decomposition Uu = u − (I − U )u. The nonuniqueness of the SDU factorization was shown to be essential to assure the SPR condition of the transfer function matrix WM (s)S. The analog of the SISO assumption that the sign of the high frequency gain is known, is the assumption about the signs of the leading principal minors of Kp . This appears to be the least amount of a priori information needed for Lyapunov-based adaptive design of relative degree one MIMO systems. Acknowledgements The authors are grateful to Professor Graham C. Goodwin for helpful comments on a early version of this paper. References Bodson, M., & Groszkiewicz, J. E. (1997). Multivariable adaptive algorithms for recon6gurable Jight control. IEEE Transactions on Control Systems and Technology, 5(2), 217–229.

Butchart, R. L., & Shackcloth, B. (1965). Synthesis of model reference adaptive control systems by Lyapunov’s second method. In Proceedings of the 1965 IFAC symposium on adaptive control, Teddington, UK. de Mathelin, M., & Bodson, M. (1995). Multivariable model reference adaptive control without constraints on the high-frequency gain matrix. Automatica, 31(4), 597–604. Dugard, L., & Dion, J. M. (1985). Direct adaptive control for linear multivariable systems. International Journal of Control, 42(6), 1251–1281. Elliott, H., & Wolovich, W. A. (1982). A parameter adaptive control structure for linear multivariable systems. IEEE Transactions on Automatic Control, 27(2), 340–352. Hsu, L., & Aquino, P. L. (1999). Adaptive visual tracking with uncertain manipulator dynamics and uncalibrated camera. In Proceedings of the IEEE conference on decision and control, Phoenix, December 1999 (pp. 1248–1253). Hsu, L., & Costa, R. R. (1999). MIMO direct adaptive control with reduced prior knowledge of the high frequency gain. In Proceedings of the IEEE conference on decision and control, Phoenix, December 1999 (pp. 3303–3308). Hsu, L., Costa, R. R., & Aquino, P. L. S. (2000). Stable adaptive visual servoing for moving targets. In Proceedings of the American control conference, Chicago, June 2000 (pp. 2008–2012). Hsu, L., Costa, R. R., Imai, A. K., & Kokotovic P. (2001). Lyapunov based adaptive control of mimo systems. In Proceedings of the American control conference, Arlington, July 2001 (pp. 4808– 4813). Ioannou, P., & Sun, K. (1996). Robust adaptive control. Englewood Cli7s, NJ, USA: Prentice-Hall PTR. Kailath, T. (1980). Linear systems. Englewood Cli7s, NJ: Prentice-Hall. Monopoli, R. V., & Hsing, C. C. (1975). Parameter adaptive control of multivariable systems. International Journal of Control, 22(3), 313–327. Morse, A. S. (1993). A gain matrix decomposition and some ot its applications. Systems and Control Letters, 21, 1–10. Narendra, K., & Annaswamy, A. (1989). Stable adaptive systems. Englewood Cli7s, NJ: Prentice-Hall. Parks, P. C. (1966). Lyapunov redesign of mode reference adaptive control systems. IEEE Transactions on Automatic Control, 11, 362–367. Rugh, W. J. (1993). Linear systems theory. Englewood Cli7s, NJ: Prentice-Hall. Sastry, S. S., & Bodson, M. (1989). Adaptive control: Stability, convergence and robustness. Englewood Cli7, NJ: Prentice-Hall. Strang, G. (1980). Linear algebra and its applications (2nd ed.). New York: Academic Press, Inc. Weller, S. R., & Goodwin, G. C. (1994). Hysteresis switching adaptive control of linear multivariable systems. IEEE Transactions on Automatic Control, 39(7), 1360–1375. Zergeroglu, E., Dawson, D. M., de Queiroz, & M. S., Behal, A. (1999). Vision-based nonlinear tracking controller with uncertain robot-camera parameters. In International conference on advanced intelligent mechatronics, Atlanta, September 1999 (pp. 854 –859).

Ramon R. Costa received the D.Sc. degree in Electrical Engineering from COPPE/Federal University of Rio de Janeiro in 1990. In the period 1999 –2000 he was a visiting scholar in the Center for Control Engineering and Computation at the University of California, Santa Barbara (UCSB). His research interests include adaptive control, sliding mode control, and robotics.

R. R. Costa et al. / Automatica 39 (2003) 1251 – 1257 Liu Hsu received the B.Sc. and M.Sc. degrees in electrical engineering from the Instituto Tecnol.ogico de Aeron.autica (ITA), Brazil, in 1968 and 1970, respectively. His doctoral research was carried out at the LASS/CNRS, Toulouse, and he received the Docteur d’Etat degree from the Universit.e Paul Sabatier, Toulouse, in 1974. In 1975 he joined the Graduate School and Research in Engineering of the Federal University of Rio de Janeiro (COPPE/UFRJ). His current research interests include adaptive control systems, variable structure systems, stability and oscillations of nonlinear systems and their applications to industrial process control, industrial robotics and underwater robotics. Dr. Hsu is a member of the Sociedade Brasileira de Autom.atica and of the IEEE Control Systems Society. He is an associate editor of the scienti6c journal of the Brazilian Society for Automation “Controle & AutomaZca˜ o”. Liu Hsu is a full member of the Brazilian Academy of Sciences. Alvaro Koji Imai received the B.Sc. and M.Sc. degrees in mechanical engineering from the Instituto Militar de Engenharia (IME), Rio de Janeiro, Brazil, in 1992 and 1997, respectively. Since 1999, he is a Ph.D. student in electrical engineering at Federal University of Rio de Janeiro. His research interests include multivariable adaptive control and modeling of mechanical systems.

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Petar V. Kokotovic has been active for more than four decades as control engineer, researcher and educator, 6rst in his native Yugoslavia and then, from 1966 through 1990, at the University of Illinois, where he held the endowed Grainger Chair. In 1991 he joined the University of California, Santa Barbara, where he directs the Center for Control Engineering and Computation. Professor Kokotovic has supervised 30 Ph.D. students and 20 postdoctoral coworkers. With them he has co-authored eight books and numerous articles contributing to sensitivity analysis, singular perturbation methods, large scale systems and robust adaptive and nonlinear control. Professor Kokotovic is also active in industrial applications of control theory. As a consultant to Ford he was involved in the development of the 6rst series of on-board computer controls and at General Electric he participated in powere systems studies. Professor Kokotovic is a Fellow of IEEE, and a member of National Academy of Engineering, USA. He received the 1983 and 1993 Outstanding IEEE Transactions Paper Awards and presented the 1991 Bode Prize Lecture. He is the recipient of the 1990 IFAC Quazza Medal, the 1995 IEEE Control Systems Award, the 2002 IEEE James H. Mulligan, Jr. Education Medal, and the 2002 ACC Richard E. Bellman Control Heritage Award.