JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 103, No. 2, pp. 421-439, NOVEMBER 1999

Dynamic Robust Output Min-Max Control for Discrete Uncertain Systems1 N. SHARAV-SCHAPIRO,2 Z. J. PALMOR,3 AND A. STEINBERG4

Communicated by G. Leitmann Abstract. Min-max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min-max control is a static state feedback law. Recently, the applicability conditions of discrete static min-max control through the output have been derived. In this paper, the results for output static min-max control are further extended to a class of output dynamic min-max controllers, and a general parametrization of all such controllers is derived. The dynamic output min-max control is shown to exist in many circumstances under which the output static min-max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min-max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min-max control, is derived. The latter is shown to be a particular case of the dynamic minmax control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observercontroller has the same stability properties as those of the full state feedback min-max control. Key Words. Robust control, discrete-time systems, min-max control, output dynamic control, insensitive observers. 1. Introduction Consider the following uncertain discrete-time system with matched uncertainties: 1

A preliminary version of this paper was presented at the 2nd IFAC Symposium on Robust Control Design (ROCOND'97), Budapest, Hungary, 1997. 2 System Engineer, Guidance and Control Group, RAFAEL—Israel Armament Development Authority, Israel. 3 Professor, Faculty of Mechanic: I Engineering, Technion—Israel Institute of Technology, Haifa, Israel. 4 Senior Lecturer, Faculty of Aerospace Engineering, Technion—Israel Institute of Technology, Haifa, Israel.

421 0022-3239/99/1100-0421$16.00/0 © 1999 Plenum Publishing Corporation

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where xeR n , ueR m , and n(x(k), k) is unknown but assumed to be cone bounded. That is, there exist constants p1 and p0 such that In Eqs. (1)-(2), n(x(k), k) represents the lumped uncertainties (Refs. 1-2). The nominal system is defined by and is assumed to be stabilizable. Corless (Ref. 3) derived a state feedback min-max controller for the system (1) which is given by where and P > 0 is the solution of the following discrete Riccati equation: for a positive-definite matrix Q. The min-max controller (4)-(6) will be referred to as the state Riccati min-max controller (RMMC). SharavSchapiro et al. (Ref. 1) revealed the relations between the above RMMC and the Lyapunov based min-max controller derived by Manela (Ref. 4). In Ref. 5, the relation between the min-max controller and the discrete positive real property is shown. The RMMC (4) stabilizes the nominal system (3) and has the following properties. Define where P>0, Q>0 satisfy the Riccati equation (6) and A.(X), k(X) denote the smallest and largest eigenvalue of a square matrix X, respectively. If then: (i)

In the absence of external disturbances (p0 = 0), the RMMC guarantees the asymptotic stability of system (1). (ii) In the presence of external disturbances (p 0 =0), the RMMC assures ultimate boundedness. For more details, see Refs. 3-4. In practice, only a part of the state vector is available for control. Hence, the problem of the output stabilizing control for the system (1) was considered in Refs. 6-8. Given the following output:

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where yeW, the existence conditions of an output static RMMC were derived. It is clear that the RMMC (4)-(6) can be applied via the output (9) if and only if there exist matrices F, P>0, Q>0 that solve the Riccati equation (6) under the following equality constraint:

It was shown in Refs. 1, 9-10 that such matrices F, P>0, Q>0 exist if and only if there exists a matrix *F such that the control law

stabilizes the nominal system (3), where G(z) is the transfer function matrix of the nominal system, given by

The control law (11) is then the output static RMMC. For square systems, i.e.,p = m; the above-mentioned condition becomes extremely simple, since the control law (11) reduces to

and a search for the matrix *P is not required. In Ref. 1, it was assumed that the nominal system was controllable and observable. This assumption was relaxed in Ref. 10, and only stabilizability and detectability were assumed. Note that there is no need to compute the matrices P, F, Q for checking the existence or for the design of the output static RMMC. The latter matrices need to be computed when the allowable bound on the uncertainties (7) is required. This paper treats square systems (i.e., p=m); recent results derived for the static output Riccati min-max control are extended to a family of output dynamic controllers. It turns out that the output static RMMC is a particular case of the dynamic controller treated in this paper. The output dynamic controller has several advantages. First, it adds degrees of freedom in the control design. Second, as will be shown in this paper, it almost always exists, even when an output static RMMC does not exist. Third, even when the latter does indeed exist, the allowable bound on the uncertainties may be increased by the output dynamic controller. It is interesting to note that the results for discrete systems derived in this paper are in contrast to those of the continuous case. For continuous square systems, the nonexistence of a static output min-max controller implies the nonexistence of a dynamic output min-max controller (Refs. 1112). The rest of the paper is organized as follows. Section 2 treats square systems, the output dynamic RMMC is defined, and the conditions for its

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existence are derived. In addition, a parametrization of all output dynamic RMMC is given. In Section 3, an observer which is insensitive to the matched uncertainties is presented, and the applicability conditions of this observer are derived. When these conditions are met, the combination of the observer and the static output RMMC assures the same stability properties as those of the full state feedback. The latter observer-controller is shown to be a particular case of the output dynamic RMMC. A simple example demonstrating the main results of the paper is presented in Section 4. Section 5 summarizes the results of the paper. A preliminary version of this paper was presented at ROCOND'97 (Ref. 13). 2. Output Dynamic RMMC In this section, square systems (i.e., p=m) of the form (1) with a given output (9) are considered, and an output dynamic RMMC is defined. Theorem 2.1 gives the three conditions that such a controller satisfies; based on these conditions, a necessary and sufficient condition for the existence of an output dynamic RMMC is derived in Theorem 2.2. In addition, the theorem presents the parametrization of all output dynamic RMMC. The following assumptions are used in the sequel: (Al) {A, B, C} is stabilizable and detectable, (A2) B, C are full rank, (A3) A is nonsingular, and the following question is introduced: do there exist dynamic controllers which guarantee the existence of an output static RMMC for the resulting closed loop? Refer to the scheme in Fig. 1: the dynamic controller K(z) is

Fig. 1. Block diagram of the system closed with a dynamic controller K(z) and a static output RMMC applied to the compensated system 2.

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applied to the original system (1) and the resulting closed loop is denoted by E. The transfer function of the nominal part of E is denoted by G(z),

where G(z) is the transfer function matrix of the nominal system given in (12). The system 2 will be referred to as the compensated system. Before deriving the existence conditions of an output static RMMC for the system E, it is shown first that, if the original system (1) satisfies the matching conditions, so does the compensated system 2. Suppose that the controller K(z) is of dimension nc and has the following state space representation:

where x c eR nc ,y cf eR m . The input to the system (1) is the sum of the outputs from the controller K(z) and from the output static RMMC designed for £, and it is given by

where umm (k) is the output of the static RMMC designed for the system SL Substitution of (16) and (17) into the system equations (1) leads to

Defining the augmented state vector XT=[XT, xTc] yields the following state space representation of the compensated system E:

where

Therefore, the system t has the same form as the system (1), and its matched uncertainty is bounded by the following bound:

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In the following, conditions on the original uncertain system (1) guaranteeing the existence of a static output RMMC for the compensated system (19) are derived. When these conditions are satisfied, there is no need to design separately k(z) and the RMMC for the compensated system (19). The overall controller, denoted K0(z), can then be designed in one step; K0 (z) will be referred to as the output dynamic RMMC. Next, a precise definition for the output dynamic RMMC is given. Definition 2.1. K0(z) is an output dynamic RMMC if it can be represented as the sum of a dynamic controller K(z) and an output static RMMC for the compensated system £; see Fig. 1. The next theorem states the necessary and sufficient conditions for K0 (z) to be an output dynamic RMMC. Theorem 2.1. Consider the system (1), (9) with G(z) represented by (12), the transfer function matrix of the nominal part. A controller K0(z) is an output dynamic RMMC if and only if it satisfies the following three conditions: (a) K0(z) stabilizes the nominal part of the original system (3). (b) K0 (z) has no poles and no zeros at the origin. (c) K0(z) satisfies

Proof. Necessity. Suppose that there exists an output dynamic RMMC which is the sum of a dynamic controller K(z) and an output static RMMC for the compensated system. The output static RMMC stabilizes the nominal part of the compensated system; consequently, K0(z) stabilizes the nominal part of the original system, hence condition (a). In Ref. 1, it was shown that a necessary condition for the existence of an output static RMMC for the system £ is that its nominal part has no zeros at the origin. The zeros of the compensated system £ consists of the poles of the controller K(z) and the zeros of the original nominal system (3). Consequently, G(z) has no zeros at the origin [G(0) is nonsingular] and K(z) has no poles at the origin. K 0 (z) is the sum of K(z) and a static controller; therefore, K(z) and k 0 (z) have the same poles; hence, K0(z) has no poles at the origin. Substitution of z = 0 into (14) yields and since G(0) is nonsingular, the output static RMMC for the compensated system is given by [see (13)]

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By Definition 2.1, K0 (z) is the sum of the dynamic controller K(z) and the static controller [see (25)]. Thus,

Substitution of z = 0 into (26) yields (23), hence condition (c). Since by Assumption A3, G(z) has no poles at the origin, then (23) implies that K0(z) is nonsingular; therefore k 0 (z) has no zeros at the origin, hence condition (b). Sufficiency. Suppose that there exists K 0 (Z) that satisfies all three conditions of the theorem. It will be shown that K0 (z) is a sum of a dynamic controller K(z) and a static RMMC for the compensated system. Let us define K(z) as follows:

hence,

It remains to show that G-1 (0) is the output static RMMC for the compensated system. With K(z) in (27), the transfer function of the compensated system £ is (see (14))

As was mentioned in the introduction, there exists an output static RMMC for the system £ if and only if G-1 (0) asymptotically stabilizes the nominal part of 2. G-1(0) is then the output static RMMC for E. Substitute z = 0 into (29) to get

Since k 0 (z) satisfies (23), it follows that

Therefore, if there exists an output static RMMC for the compensated system, then it is equal to G-1 (0) as required. Next, it is shown that indeed there exists an output static RMMC for the compensated system. That is, G - 1 (0) stabilizes 2. The overall controller KT(z), which is applied to the original system, is the sum of the dynamic controller K(z) and the static

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RMMC, and is given by Note that (32) is identical to K0(z) [see (28)]; therefore, according to condition (a), it stabilizes the nominal part of the original system (3). Hence, G-1 (0) stabilizes £ and it is an output static RMMC for the compensated system. Next, the conditions for the existence of an output dynamic RMMC are derived, and the parametrization of all output dynamic RMMC is presented. This parametrization is based on the Youla parametrization of all stabilizing controllers for linear time invariant (LTI) systems (Ref. 14), which is briefly presented. Given a nominal system (3), its transfer function G(z) [see (12)] may always be decomposed by a right [left] coprime factorization rcf [1cf]. That is, where N(z), M(z), M(z), N(z) are proper and stable transfer matrices. Then, the set of all stabilizing controllers H(z) for G(z) is parameterized by either

or

where Qr(z), Q l (z) are any proper stable transfer matrices and U0(z), V 0 (z), U0(z), V0(Z) are proper stable transfer matrices which satisfy the Bezout identities

Using the Youla parametrization, the next theorem presents a necessary and a sufficient condition for the existence of an output dynamic RMMC for the original system (1). Theorem 2.2. Consider the system (1) with the Assumptions A1-A3. There exists an output dynamic RMMC K0(z) (see Definition 2.1) if and only if the nominal part of the system has no zeros at the origin [det N(0)=0].

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The set of all output dynamic RMMC K0 (z) is parameterized by either

or here, s is any positive integer; Ql(z), Q r ( z ) are any stable transfer matrices with no poles or zeros at the origin such that z - s Q r ( z ) , z - s Q l ( z ) are proper transfer matrices that satisfy (35) or (37) respectively; V0(z), U 0 (z), V0(z), U 0 (z) are proper stable transfer matrices which satisfy the Bezout identities. Proof. Necessity. This was shown in the proof of Theorem 2.1. Sufficiency. It is shown that, if the system (3) has no zeros at the origin, then there exists an output dynamic RMMC. To this end, it will be shown that (40) satisfies all three conditions (a)-(c) of Theorem 2.1. The proof for (41) is readily obtained using the same argumentations. Condition (a).

Define

Substitution of (42) to (34) yields (40). Hence, the parametrization in (40) is a subclass of all stabilizing controllers, and condition (a) follows. Conditions (b), (c).

Substitute z = 0 in (40) to obtain

Since N(0) and Qr(0) are both nonsingular [the original system and Qr(z) have no zeros at the origin], then Ko(0) exists and

hence condition (c) follows. Since the original system has no zeros and poles at the origin, so does k0(0), and condition (b) is satisfied. So far, it was proven that (40) is an output dynamic RMMC. In order to prove that (40) is the parametrization of all output dynamic RMMC, one needs to show that every controller that satisfies conditions (a)-(c) can be parameterized according to (40). To this end, suppose that there exists a controller Ko(0) that satisfies conditions (a)-(c). According to (a), k0(0) can be parametrized by (34), the set of all stabilizing controllers. In the following, it is shown that, if the controller given in (34) has no poles and no zeros at the origin [condition (b)], then in order to satisfy (23) [condition (c)], it is necessary to choose Qr(z) as in (42). To this end, substitute z = 0

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in (34) and equate it to G -1 (0), Condition (b) implies that V 0 (0) + N(0)Qr(0) is nonsingular. Now, assume that it is also finite. It will be shown by contradiction that the above term must be infinite, and since Vo(0) and N(0) are finite, it implies that Qr(z) is not analytic at the origin. To this end, postmultiply both sides of (45) by V0(0) + N(0)Qr(0) to obtain Substitution of z = 0 and (46) into the identity (39) yields Since M(0), N(0), M(0), N(0) are all nonsingular, then (33) implies that the left-hand side of (47) is identical to the zero matrix; hence, a contradiction arises. Therefore, Qr(z) is not analytic at the origin, and it must have at least one pole at the origin. Hence, it has the form of (42). Substitution of z = 0 into (40) implies that a necessary condition for K0(z) not to have poles and zeros at the origin is that Qr(z) does not have such poles and zeros. D So far, it has been shown that the output dynamic RMMC K0 (z) can be designed in one step, without an explicit expression of the compensated system 2. However, for the computation of the bound on the allowable uncertainties [see (7)-(8)], the state space representation of the compensated system £ is required. The state space realization {A, B, C] of 2 is given in (20) in terms of the state space realization {Ac, B c ,C c ,D c } of K(z). In the following, the relation between the state space realizations of K(z) and K0 (z) is derived. Let the state space representation of K0(z) be given by {A 0 , B0, C0, D0}. Then, according to (27), it is clear that the triple {Ac, Bc, Cc} is identical to the triple {A0, B0,C0}, and that Dc is given by It is useful to note that, since K 0 (Z) satisfies (23), D0 and Dc satisfy

3. Insensitive Observer

The previous section presented the parametrization of all output dynamic RMMC. There are many degrees of freedom in the design of the

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controller, and clearly each design may lead to a different bound on the allowable uncertainties. In this section, we present a special asymptotic observer, which is insensitive to the matched uncertainties. The existence conditions of such an observer are quite restrictive. However, when these conditions are satisfied, it is shown that, by applying the state RMMC to the estimated state vector, the stability properties of the overall system are similar to those achieved using a full state feedback. That is, the bound on the allowable uncertainty is known in advance and is identical to the bound achieved by the full state feedback. Furthermore, it will be shown that the overall controller, which is based on the combination of the above mentioned observer and the state RMMC, is a particular case of the family of all output dynamic RMMC, presented in the previous section (provided that the nominal system has no zeros at the origin). We treat systems of the form (1) that satisfy the following conditions: (dl) CB is full column rank. (d2) The nominal part of the system is of minimum phase; i.e., all zeros are inside the unit disc. For systems satisfying the above conditions, an asymptotic observer which is insensitive to the matched uncertainties is derived. The observer introduced here is similar to the observer derived by Kudva (Ref. 15) for continuous systems. It was shown there that, if a continuous system satisfies the above two conditions, then there exists a reduced-order Luenberger observer which asymptotically estimates the state vector without using any input information. Therefore, such an observer can achieve asymptotic estimation of the state of a system with matched uncertainties without any information of the matched uncertainty 77 (Ref. 17). It is easy to show that the conditions for the existence of such an asymptotic observer for discrete systems with matched uncertainties are identical to those of the continuous case, i.e., conditions (d1), (d2). A brief presentation of the structure of the insensitive observer follows. Given the system (1), the objective is to design an asymptotic observer which is insensitive to the matched uncertainty 77. We introduce the reducedorder Luenberger observer and show the conditions under which it will be an asymptotic observer even when r\ is unknown. Chose CeRn-pxn such that the matrix [ C T \ C T ] T is nonsingular. Define yeRn-p as follows:

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to obtain

or

If Y (k) denotes the estimation of y(k), then the estimated state vector is given by A Luenberger observer of (n—m)th order is constructed as follows:

where H is the observer gain and

Since r)(x(k), k) is unknown, the above observer will be an asymptotic observer if the observer gain H can be chosen such that

Using the same arguments as in the proof in Ref. 15, it can be shown that, if the system satisfies conditions (d1) and (d2), then a gain matrix H satisfying (59) can always be found. More specifically, it can be seen from (58) that, if CB is of full rank, then H satisfying T = 0 can always be found; in fact, for square systems, H is unique. In addition, if one selects such an H, then all the poles of the observer (eigenvalues of Aob) are the zeros of the original system. Therefore, if conditions (d1), (d2) are satisfied, then one can always choose H such that F = 0 and Aob is asymptotically stable. That is, we can construct an asymptotic observer, which is insensitive to the matched uncertainty 77. Suppose now that conditions (d1), (d2) are satisfied and that H satisfying (59) is selected. The stability of the system (1) with the state RMMC applied to the estimated state vector is investigated next. It will be proven that these stability properties are as those of the full state feedback. To this end, suppose that the state RMMC is applied to the estimated state vector

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as follows:

where Fmm is the gain of the output static RMMC given by Then, (8) is also the bound on the allowable uncertainty assuring asymptotic (practical) stability of the overall system with the observer and the controller (60). That is, the bound is identical to that of the full state feedback. To see this, define the estimation error by Substitution of y(k) from (1) into (60) and using the definition of e(k) in (62) and that of j ( k ) in (51) yield However, according to the definition of L2, L1 given in (52), it is clear that therefore, The overall system with the observer and the RMMC applied to the estimated state is given by

and since Aob is asymptotically stable, then the estimation error goes to zero independently of the uncertainty 77. From (66), it follows that the dynamics of the state vector is given by where Using the bound (2) on 77, it follows that n(x(k), e(k), k) is bounded by

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or where

For the stability analysis of the system, choose the quadratic Lyapunov function

with P>0 that solves the Riccati equation (6). Substitution of (61), (67) into the difference of the Lyapunov function yields

For any positive matrix and any vector tf>, it is known that

Therefore,

and with the substitution of (69),

Suppose now that the system (1) is robustly stabilizable (asymptotically or ultimately bounded) by the state RMMC; i.e., condition (8) is satisfied. It is next shown that the system (67) is also asymptotically (practically) stable. From (8), it follows that the coefficient of ||x(k)||2 on the right-hand side of (75) is negative. Therefore, the following conclusions can be drawn: (a)

If in the original system p0 = 0, then since e(k)->0, so does p 0 (k) [see (71)]. Therefore, there exists k1 such that, for any k > k 1 , A V(k) < 0; hence, the system is asymptotically stable. (b) If in the original system p 0 =0, then po(k)->p 0 . Therefore, there exist k2 and rmax such that, for any k>k2 and any x(k) that satisfy ||x(k)||>r max , AF(k)<0; hence, ultimate boundedness (practical stability) is guaranteed. Finally, we derive the state space realization of the overall controller Koc(z), i.e., the observer plus the state RMMC applied to the estimated state vector. It will be shown that this controller is a particular case of the dynamic controller presented in the previous section. That is, when the original system (3) has no zeros at the origin (the necessary condition for the existence of

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an output dynamic RMMC), then for any P that solves the unconstrained Riccati equation, the corresponding Koc(z) is an output dynamic RMMC. To this end, we write the observer equations (54) and (55) in a standard form. By defining (55) reduces to

Since the state RMMC is applied to the estimated state vector [see (60)], the realization of the overall controller Koz(z) (observer plus RMMC) is given by

where Fmm is given in (61) and P is the solution of the Riccati equation (6) for any Q > 0. The next lemma proves that, when the system satisfies the existence conditions for an asymptotic observer [i.e., conditions (d1) and (d2)], then the combination of the insensitive observer and the state RMMC applied to the estimated state vector [i.e., Koc(z)] is a particular case of the output dynamic RMMC. Lemma 3.1. Consider the system (1) with Assumptions A1-A3. If the nominal system satisfies conditions (dl)-(d2) and has no zeros at the origin, then the combination of the insensitive observer and the state RMMC applied to the estimated state vector [i.e., Koc(z) in (79)] is an output dynamic RMMC, according to Definition 2.1. Proof. The proof will be based on Theorem 2.1; i.e., it will be shown that, under the conditions of the lemma, Koc(z) [see (79)] satisfies all three conditions of Theorem 2.1. Condition (a). As was proven in this section, Koc(z) asymptotically stabilizes the nominal part of the system. Condition (b). Since the zeros of the original nominal system are the poles of the observer, and if the nominal system has no zeros at the origin, then Koc(z) has no poles at the origin. In the following, it will be shown that Koc(z) satisfies condition (c) and therefore Koc(0) is-nonsingular; hence, it also has no zeros at the origin.

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Condition (c). From (58), we have Substitution of (80) into (56) and (57) yields

where W is defined by Using (79), it follows that

In order to verify that (23) is satisfied, Koc (0) is found next,

Postmultiplication of both sides of (88) by G(0) = -CA -1B yields

Using (64), and noting that [I-L2(CWAL2) follows that

-1

CWA]L2 is equal to zero, it

But since WB=0 [see (83)], (90) reduces to Therefore, Koc(0) is nonsingular, i.e., Koc{z) has no zeros at the origin; hence, condition (b) is satisfied. In addition (91) is identical to (23), by replacing Koc (z) with Ko (z); hence, condition (c) is satisfied, and according to Theorem 2.1, Koc(z) is an output dynamic RMMC. D Remark 3.1. Note that, according to Lemma 3.1, the observer controller is a particular case of the output dynamic RMMC only if the original

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nominal system has no zeros at the origin. If this is not the case, then it does not fall under that category, but still is an output dynamic controller.

4. Example

For demonstrating the properties of the dynamic controllers developed in the paper, the following uncertain matched system, taken from Ref. 4, is used: where

and it is given that Note that, for this example, where C, is given by (7). Maximizing (7) subject to (6) leads to £ = 0.5, which determines the best bound for the uncertainties under the state RMMC. Now, assume that the output equation is From (13), it follows that an output static RMMC exists for (92), (93), (96) iff

Two cases are considered. Case 1. c0 = 1, C1 = 0. In this case, an asymptotic observer which is insensitive to uncertainties does not exist because CB is not of full rank. However, an output static RMMC does exist. To find the best bound on the allowable uncertainties, one maximizes (7) subject to (6) and (10), and then £ = 0.32 is obtained. Note that this value is smaller than the one obtained for the state RMMC. Next, consider the dynamic controller

which is an output dynamic RMMC since it satisfies the three conditions of

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Theorem 2.1. To compute the uncertainty bound f, in this case one maximizes (7) subject to (6) and (10) corresponding to £ (see Fig. 1). This maximization can be done easily by applying the LMI Toolbox, and then £ = 0.4 is obtained. The latter bound is larger than the one obtained for the output static RMMC but is smaller than that of the state RMMC. Case 2. c0 = 1, c1 = 6. It is clear from (97) that an output static RMMC does not exist. However, an insensitive asymptotic observer does exist and guarantees the same bounds on uncertainties as the state RMMC, that is, E = 0.5. Note that, besides the observer, other output dynamic RMMC's, may be used in this case at the cost of lower bounds on the uncertainties.

5. Summary

The paper introduced a class of output dynamic min-max controllers for discrete systems with matched uncertainties, and a general parametrization of all controllers in that class has been presented. It has been shown that, except for systems with (nominal) zeros at the origin, the output dynamic min-max controllers exist for all other systems. Hence, these controllers are applicable in many circumstances under which the static output min-max controllers do not exist. When both the static and dynamic controllers exist, the latter usually allows for broader bounds on the uncertainties. Another class of output dynamic min-max controllers has been derived in the paper from a different direction. This class of controllers, which consists of insensitive observers and a state feedback, has been shown to exist under quite restrictive conditions. However, when these conditions are satisfied, they permit bounds on uncertainties like those of the static state min-max controllers. It was further shown that the latter controllers are a particular case of the first class, provided that the nominal system has no zeros at the origin.

References 1. SHARAV-SCHAPIRO, N., PALMOR, Z. P., and STEINBERG, A., Output Stabilizing Robust Control for Discrete Uncertain Systems, Technical Report TME-440, Faculty of Mechanical Engineering, Technion—Israel Institute of Technology. Haifa, Israel, 1996. 2. GUTMAN, S., Uncertain Dynamical Systems: A Lyapunov Min-Max Approach, IEEE Transactions on Automatic Control, Vol. 24, pp. 437-443, 1979.

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