A Unied Approach on Stability Robustness for Uncertainty. Descriptions based on Fractional Model Representations

Transcription

1 A Unied Approach on Stability Robustness for Uncertainty Descriptions based on Fractional Model Representations R.A. de Callafon z P.M.J. Van den Hof P.M.M. Bongers x Mechanical Engineering Systems and Control Group, Delft University of Technology, Mekelweg, 68 CD Delft, The etherlands. Internal Report -480, October 994 Abstract The powerful standard representation for uncertainty descriptions in a basic perturbation model as introduced in Doyle (98) can be used to attain necessary and sucient conditions for stability robustness within various uncertainty descriptions. In this paper these results are applied to uncertainty descriptions that are based on fractional model representations. Special attention will be given to upper bounds on additive coprime factor perturbations which could have been obtained by identication techniques. ecessary and sucient conditions for stability robustness will be derived and compared with well known stability robustness results based on -gap and gap metric distance measures. Based on this comparisment a result on conservatism is presented here. Introduction In the research area of system identication and model based control design it is widely recognized that models found by system identication and used for control design are necessarily approximative. On the one hand exact modelling can be impossible or too costly and on the other hand control design methods can become unmanageable if they are applied to models of high complexity. The approximation aspects give rise to a possible mismatch between the actual plant and the model being used, which has to be dealt with. In a model based control design paradigm, the design is based on a (necessarily) approximative model, which will be denoted by ^P. An apparently succesfull control design leads z The work of Raymond de Callafon is sponsored by the Dutch Systems and Control Theory etwork x ow with the Unilever Research, Technical Application Unit, Vlaardingen, The etherlands

2 to a controller C ^P, having some desired closed loop properties for the feedback controlled model ^P, but due to the possible mismatch between the actual plant Po and the model ^P, a verication of these desired closed loop properties is preferred before implementing the controller C on the actual plant ^P P o. In this paper the discussion is directed towards the verication of one of the most important closed loop properties: stability. To evaluate stability when the controller C is being applied to the plant ^P P o, a characterization of the discrepancy between the plant P o and the model ^P can be used (Doyle, 98; Francis, 987; Doyle et al., 99). Since the real plant P o is unknown, the discrepancy in general is characterized by a so called uncertainty set, denoted with P. Typically an uncertainty set P is dened by the (nominal) model ^P, which is found by physical modelling or identication techniques, and some bounded `area' around it (Doyle et al., 99). The uncertainty set P itself reects all possible perturbations of the (nominal) model ^P that may occur. Some typical examples of commonly used unstructured uncertainty sets that are bounded by using an H -norm can for example be found in Doyle and Stein (98), Doyle et al. (99) or the tutorial paper in Kwakernaak (99) and are summarized below: Additive uncertainty set: P A ( ^P ; VA ; W A ; ) := fp j P = ^P + A ; with kv A A W A k g: () Multiplicative (output) uncertainty set: P M ( ^P ; VM ; W M ; ) := fp j P = [I + M ] ^P ; with kv M M W M k g: () Additive (right) coprime fractional uncertainty set, where ^P has been factorized rst in a right coprime factorization ^P := ^ ^D and the set is dened as P F ( ^; ^D; V F D ; V F ; W F ; ) := fp j P = [ ^ + ][ ^D + D ] ; with 4 V F D 0 0 V F WF g: () The arguments (; ;...) in the denition of the uncertainty sets is used to clarify the main elements that construct the norm bounded uncertainty set under consideration and will be used throughout this paper. Clearly, by dening the uncertainty set in such a way that at least the plant P o P, the evaluation of stability properties of the closed loop system when applying the controller C to the plant ^P P o can be overestimated by the evaluation of

3 the stability properties when applying C ^P to all plants P P. In this way the evaluation of stability of the closed loop system with P P is based on a complete set and is called stability robustness (Doyle et al., 99). In this perspective, special attention will be given in this paper to an uncertainty set P F which is characterized by perturbations on an additive coprime factor description of the nominal model ^P, like in (). The specic application of the uncertainty set description of () will be motivated by the favourable properties it has over a standard additive () or multiplicative () uncertainty set description. The outline of this paper will be as follows. In section some preliminary notations and denitions will be given. In section some results on stability robustness using the powerful perturbation model (Doyle, 98) will be summarized. This perturbation model can represent the information of any possible bounded uncertainty set P very accurately and gives rise to an unied approach to handle stability robustness. Section 4 contains the results of applying this unied approach to additive uncertainty descriptions on fractional model representations like in () and favourable properties are illuminated. The link with gap and -gap based stability robustness results is discussed in section and a result on conservatism will be presented. This paper ends with some concluding remarks. Preliminaries. Feedback conguration Throughout this paper, the feedback conguration of a plant P and a controller C is denoted by T (P ; C) and dened by the feedback connection structure depicted in gure. r - u - y P e 6 C? y c u c?e + r Fig. : feedback connection structure T (P ; C) of a plant P and a controller C To ensure wether the feedback connection T (P ; C) is well dened, the requirement of wellposedness (Boyd and Barrat, 99; pp. ) is needed and will be assumed throughout this paper. Denition. The feedback connection T (P ; C) depicted in gure is said to be wellposed if det[i + CP ] 6 0.

4 4 In gure the signals u and y reect respectively the inputs and outputs of the plant P. The signals u c and y c are respectively the inputs and outputs of the controller C ^P, and r and r are external reference signals. The closed loop dynamics of the closed loop system T (P ; C) can be described by the mapping of [r r ] T to [y u] T which is given by the transfer function matrix T (P ; C): T (P ; C) := 4 P I h [I + CP ] C I i : (4) The controller C based on a nominal model ^P will be denoted by C ^P, while the actual plant under consideration is denoted by P o. Since the controller C ^P will be applied on both the actual plant P o and the model ^P, according to the feedback structure given in gure, the closed loop dynamics of the two feedback connection structures is described respectively by the two transfer function matrices T (P o ; C ^P ) and T ( ^P ; C ^P ).. orms and sets In this section some basic denitions of signal and systems norms that will be used in this paper have been summarized. The attention is restricted to an l -norm on discrete time signals and an H -norm on discrete time linear time invariant systems. For more detailed denitions one is referred to Francis (987) or Desoer and Vidyasagar (97). The notation of a norm function in this paper will be as follows. Let X be a linear space over the eld of real or complex numbers and let x X, then kxk X is used to denote the norm of x dened on X. An explicit denition of a norm function can be found in Francis (987). ow let fx(t)g be a discrete time signal with x(t) IR n 8t f0; ;...; g and fx(t)g X. With the trace operator trfg on x(t)x T (t) trfx(t)x T (t)g := X diagfx(t)x T (t)g = consider the following norm function on X : kxk X := kxk = vu u t X t=0 nx k=0 x k (t) trfx(t)x T (t)g () All sequences fx(t)g X for which () is bounded are said to be an element of the normed space l, which is a complete normed space or Banach space (Francis, 987). The norm dened on l is called the l -norm or -norm and denoted with k k in the sequel. For a linear operator G mapping l to l, an induced -norm based on () can be dened as follows: kgxk kgk ;ind := sup : (6) x6=0 kxk

5 If the operator G : l! l is BIBO stable (see denition.4) the induced -norm in (6) will be bounded and is equal to the Hilbert or spectral norm kgk ;ind = p max fg Gg (7) where p max fg Gg is the maximum eigenvalue of G G and denotes the complex conjugate transposed. If G denotes a multivariable discrete time transfer function G(z) with z = e i!,! [0; ), taking the supremum of p max fg(e i! ) G(e i! )g = fg(e i! )g over!, where fg(e i! )g denotes the maximum singular value of G(e i! ), will yield the spectral norm of the system G and is dened as the H -norm: kgk := sup fg(e i! )g (8)![0;) All transfer function matrices G that have a bounded H -norm are said to be an element of the Hardy space IRH (Francis, 987).. Factorizations and coprimeness Using the theory of fractional representations, a plant P is expressed as a ratio of two stable proper mappings and D. Similarly as in Vidyasagar (98) the following denitions for coprimeness and coprime factorization will be used, where IRH denotes the set of all rational stable transfer functions. Denition. Let ; D IRH, then the pair (; D) is called right coprime over IRH if there exist X; Y IRH such that X + Y D = I: The pair (; D) is a right coprime factorization (rcf) of P if P = D for detfdg 6 0 and (; D) is right coprime over IRH. Basically, a coprime factorization of P over IRH means that and D are stable and do not have common unstable zeros. If and D do have common unstable zeros, they must cancel in the so called Bezout factors X and Y to satisfy X + Y D = I. However, this is not possible since X; Y IRH. Additional to the denition of coprime factorization the notion of normalized coprime factors will be used in the sequel. Denition. A right coprime factorization (; D) is called a normalized right coprime factorization (nrcf) if it satises + D D = I where denotes the complex conjugate transposed.

6 6 For (normalized) left coprime factorizations dual denitions exist. State space formulae for coprime factors can be found in ett et al. (984). State space solutions to the computation of normalized coprime factors can for example be found in Meyer and Franklin (987), Vidyasagar (988) or Bongers and Bosgra (99) for continues time and in Meyer (990) or Bongers and Heuberger (990) for discrete time systems..4 Stability Throughout this paper, stability of a map is dened by the BIBO (bounded input, bounded output) property of the map and the following denition will be used (Francis, 987). Denition.4 Let U and Y be linear spaces having respectively norm functions k k U dened on U and k k Y dened on Y. Let G be an operator mapping U onto Y, then G is called BIBO stable if kgk Y;U;ind <, where kgk Y;U;ind := kguk Y sup uu;u6=0 kuk U Following the work of Doyle et al. (99), U and Y are taken to be complete normed space l in the sequel and kgk Y;U;ind will be the spectral norm given in (7). For G IRH this is the H -norm dened in (8). For stability of an interconnection structure like the feedback system given in gure the following denition of internal stability (Desoer and Chan, 97) will be used. Denition. Consider a well posed feedback connection T (G; K) of the two systems G and K given in gure (a). The feedback system T (G; K) is called internally stable if the mapping from the signals [r r ] T to [u u ] T, introduced in gure (b), is BIBO stable. - r e G - u - G + + K (a) 6 K (b) u + e? + r Fig. : general feedback connection structure With denition. the following result for the internal stability of the feedback system T (P ; C) depicted in gure can be obtained.

7 7 Lemma.6 The feedback system T (P ; C) is internally stable if and only if the transfer function matrix is BIBO stable. H(P ; C) := 4 H H = 4 [I + P C]?[I + P C] P H H [I + CP ] C [I + CP ] Proof: The mapping from the signals [r r ] T to [u c u] T given in gure is given by the transfer function matrix H(P ; C). ) According to denition., T (P ; C) is internally stable if this map is stable, hence H(P; C) must be BIBO stable. ( If H(P ; C) is BIBO stable, all elements in H(P ; C) will be BIBO stable, hence the mapping from the signals [r r ] T to [u c u] T is BIBO stable. If both C ^P and P are unstable, all four transfer function matrices in lemma.6 must be checked. However if at least C ^P Theorem.7 Let P, C ^P is internally stable if and only if [I + P C ^P ] is stable. or both C ^P and P are stable, less work has to be done: of the feedback system T (P ; C ^P ) be BIBO stable, then T (P; C) Proof: ) If T (P ; C ^P ) internally stable, all elements in H(P ; C ) must be BIBO stable, ^P hence H will be BIBO stable. ( With H = [I + P C ^P ] = I? [I + P C ^P ] P C ^P the elements of H(P ; C ) can be ^P rewritten into I + H C ^P = H I? P H = H (9) I? P H C ^P = H With P, C and ^P H BIBO stable (9) shows that H, H and H will BIBO stable and hence T (P ; C ^P ) is internally stable. With lemma.6 for internal stability and denition. for left and right coprime factorizations, the following general theorem for internal stability of a closed loop system based on coprime factorizations can be derived. Theorem.8 Let P = D = ~ D ~ where (; D) is a rcf and ( ~ D; ~ ) a lcf of P. Let C = ^P cd c = Dc ~ c ~ where ( c ; D c ) is a rcf and ( Dc ~ ; c ~ ) a lcf of C ^P. The following statements are equivalent i. the feedback system T (P ; C ^P ) given in gure is internally stable

8 8 ii. T (P; C) dened in (4) is BIBO stable iii. is BIBO stable, with := h ~Dc h iv. ~ is BIBO stable, with ~ := ~D ~ c i 4 D ~ i 4 D c Proof: i, ii: From gure and lemma.6 it can be seen that 4 y u = 4?I 0 = 0 I 4?I 0 0 I + 4 r 4 u c u H(P; C) 4 r r 0 c + 4 r 0 = T (P; C) 4 r Hence T (P; C) is BIBO stable if and only if H(P; C) is BIBO stable and the feedback connection of P and C ^P is internally stable if and only if H(P; C) is BIBO stable. i, iii,iv: See Vidyasagar (98), Bongers (994) or Schrama (99).. Distance measures Fractional representations have a close relation with approximation in the graph topology. The graph topology is the weakest topology in which a variation of the elements of a stable feedback conguration around their nominal values, preserves stability of that closed loop system (Vidyasagar et al., 98). In this perspective the elements of the stable conguration system are characterized by stable fractional representations of the plant P and the controller C, while the graph topology can be induced by a distance measure. Distance measures or metrics (Kelley, 96; pp. 8), like the graph metric introduced in Vidyasagar (984) or the gap metric introduced in Zames and El-Sakkary (980) induce this graph topology and are dened as follows (Georgiou, 988). Denition.9 Let the distance between two plants P and P in the graph and gap metric respectively be denoted by d(p ; P ) and (P ; P ), and let P i = i D i a nrcf for i [; ], then d(p ; P ) is dened by d(p ; P ) := maxf ~ d(p ; P ); ~ d(p ; P )g; with ~d(p i ; P j ) := inf QIRH ; kqk 4 D i i? 4 D j j Q r where ( i ; D i ) is Given two topologies O and O, O is said to be weaker than O if O is a subcollection of O, see also Vidyasagar et al. (98)

9 9 and (P ; P ) is dened by (P ; P ) := maxf ~ (P ; P ); ~ (P ; P )g; with ~ (P i ; P j ) := inf QIRH 4 D i i? 4 D j j Q The distance (P ; P ) [0; ] is called `the gap' between the plants P, P. ote that the computation of a nrcf and the minimization in denition.9 are needed to compute the distance between the plants P and P. In the distance measures given in denition.9 the closed loop operation of the plants P and P, induced by the controller C, is not taken into account. Based on this observation the so-called -gap ~ (P ; P ) is employed in Bongers (994). Denition.0 Let P i = i D i, where ( i ; D i ) is a nrcf for i [; ]. Let C = D ~ c c ~ be any controller, where ( Dc ~ ; c ~ ) is a nlcf, that creates a stable closed loop system T (P ; C), then ~ (P ; P ) is dened as with = [ ~ Dc D + ~ c ]. ~ (P ; P ) := inf 4 D QIRH? 4 D Q The dierence between ~ (P ; P ) and ~ (P ; P ) is the additional shaping of the nrcf of P with into a rcf ( ; D). In this way := ~ D c D + ~ c = I, with =, D = D, which is used to consider the closed loop operation of P induced by the controller C being employed. This makes the distance between P and P dependent on the nrcf of the controller C. However, the distance measure ~ (P ; P ) is not a metric since ~ (P ; P ) 6= ~ (P ; P ) due to the inuence of the controller C (Bongers, 994). Robustness considerations. Basic perturbation model In the analysis of robustness properties, the information about uncertainty can be represented in a so called basic perturbation model (Doyle, 98). This perturbation model can represent the information of any possible bounded uncertainty set very accurately and gives rise to an unied approach to handle stability robustness. The information about the uncertainty set is represented in the following way. For stability robustness, the stable closed loop system T ( ^P ; C ) created by the nominal model ^P ^P and the model based controller C ^P is considered to have a set of inputs d and outputs z, which is used to represent

10 0 an H -norm bounded uncertainty set P on the model ^P (and/or the controller C ^P ) of the closed loop system T ( ^P ; C ^P ). By the introduction of stable weightings in the map from d onto z of the closed loop system T ( ^P ; C ^P ), a weighted stable closed loop system is created. This weighted closed loop system is denoted by M( ^P ; C ^P ) and the uncertainty is represented by pulling out any uncertainties in the weighted closed loop system M( ^P ; C ^P ) into a system. If M IRH is used to denote the corresponding stable transfer function from d to z, this yields the so called basic perturbation model T (M; ) as indicated in gure. - M d z Fig. : Basic perturbation model T (M; ) Due to the weightings being introduced, one is always able to normalize the H -norm bound on the uncertainty that has been pulled out (see the examples in section ). Due to this generalization the H -norm bound on the uncertainty can be represented by kk and helps in formulating standard results for testing robustness properties (Doyle et al., 99). In order to keep track of the size of the uncertainty in the results on stability robustness, the weighted uncertainty is assumed to be atted and bounded by kk : (0) where is any real and strict positive number.. Stability Robustness For stability robustness the internal stability of the closed loop system when applying a controller C ^P to all plants P P must be evaluated. Since the uncertainty set P has been represented in the basic perturbation model T (M; ), the internal stability of the closed loop system given in gure must be checked. For this purpose, additional signals can be introduced in the basic perturbation model T (M; ), similar as in denition., and is depicted in gure 4. Unfortunately, only an upper bound like (0) is known for, making

11 - e r u - M + e? r u + Fig. 4: Additional signals for internal stability of T (M; ) the direct application of lemma.6 inappropriate to prove internal stability for T (M; ). Instead, the small gain theorem (Desoer and Vidyasagar, 97) can be used. Theorem. Let U be a linear space with a norm function k k U dened on U. Let G be an operator, mapping U to U and let G be BIBO stable, then a sucient condition for the unity feedback loop of gure (K I) to be internally stable is kgk U;ind = kguk U sup < uu;u6=0 kuk U Proof: Can be found by application of the xed point theorem see Luenberger (969) or eustadt (976) and has been worked out in Desoer and Vidyasagar (97)... Sucient condition By application of the small gain theorem the following sucient condition for internal stability of the basic perturbation model T (M; ) can be found. Theorem. Let M, IRH, then a sucient condition for the basic perturbation model T (M; ) to be internally stable is and with the upper bound (0) this implies kmk kk < kmk < Proof: Using gure 4 the following equations can be obtained a similar proof can also be found in Doyle et al. (99) u = r + r + Mu u = r + Mr + Mu ()

12 Similar to the work of Doyle et al. (99) we let r, r, u and u to be signals in l. Since both M and IRH, M and M in () are IRH and BIBO stable. Applying theorem. with u ; u U = l on both equations in () yields the following sucient condition for the mapping from [r r ] T to [u u ] T to be stable: kmk ;ind < and kmk ;ind < With M; M IRH this can be replaced by (see (8)) fm(e i! )(e i! )g < 8! and f(e i! )M(e i! )g < 8! () Using the well known property of singular values fm(e i! )(e i! )g fm(e i! )gf(e i! )g 8! f(e i! )M(e i! )g fm(e i! )gf(e i! )g 8! the inequality fm(e i! )gf(e i! )g < 8! () implies both inequalities in () and reects a sucient condition for a frequency dependent upper bound on the perturbations that do not destabilize the basic perturbation model. Furthermore, if the uncertainty has been atted to (0) by the introduction of scalings in M, then fm(e i! )gf(e i! )g fm(e i! )g < 8! and the sucient condition for internal stability of T (M; ) becomes sup! fm(e i! )g = kmk <... ecessary condition The inequality in theorem. is a sucient condition for internal stability of the basic perturbation model T (M; ). It can be shown (Maciejowski, 989) that it is possible to nd perturbations that violate theorem., but do not destabilize T (M; ). However, if all perturbations can occur for which the bound (0) holds, theorem. becomes sucient as well as a necessary condition for internal stability of T (M; ): Theorem. Let M, IRH with kk. The basic perturbation model T (M; ) is internally stable 8 kk if and only if kmk < Proof: ) Proven in theorem. ( Has been worked out in Maciejowski (989) or Doyle et al. (99) and is based on proving the existence of a destabilizing perturbation if kmk = + ".

13 In cases that has some special structure, e.g. a block diagonal structure, theorem. is no longer a necessary but only a sucient condition. This is due to the fact that all perturbations satisfying (0) are no longer admissible due to the special structure of. In that case the application of the structured singular value fg will lead to necessary and sucient conditions. The usage of structured singular value is beyond the scope of this paper and for details one is referred to Doyle (98) or Doyle et al. (98). 4 Stability robustness for uncertainty descriptions based on fractional model representations 4. Motivation The results of section on stability robustness can be applied to various H -norm bounded uncertainty by rewriting the uncertainty description into the basic perturbation model T (M; ). In this section the application of the results of section are applied to coprime factor uncertainties. In here the model ^P will be represented in a right coprime factorization dened in denition. only, since dual results can be found for left coprime factorizations. A crucial assumption of the theorems in section on stability robustness to hold, is IRH for all kk. In case of the representation of an additive () or multiplicative () uncertainty set in the basic perturbation model T (M; ), this assumption implies the condition that the location of all unstable poles of P are assumed to be xed. Furthermore, P = ^P + A cannot create or delete unstable poles in P, but P = [I + M ] ^P may be able to delete unstable poles in P. Since the location of the unstable poles remains xed in P M, this gives rise to the dicult problem of unstable pole/zero cancelation in P M. Therefore the stable zeros in P M are not allowed to move out of the stability region. Additive perturbations on coprime factorizations () are more exible and allow changes in both the number and the locations of poles and zeros anywhere in C (Chen and Desoer, 98). Moreover, fractional representations have a close relation with approximation in the graph topology. Distance measures (or metrics) like the graph and gap metric given in denition.9 induce this same graph topology and can also be used to evaluate stability robustness properties of a closed loop system (Vidyasagar, 984; El-Sakkary, 98; Vidyasagar and Kimura, 98). 4. Basic perturbation model Let a model ^P = ^ ^D, where ( ^; ^D) is a rcf, and a controller C ^P = ~ D c ~ c, where ( Dc ~ ; c ~ ) is a lcf, create an internally stable closed loop system T ( ^P ; C ^P ) according to

14 4 lemma.6. The main issue being considered here is stability robustness, which is the ability of the controller C ^P to create an internally stable closed loop system T (P; C ) for ^P all plants P P F. In this case the set P F is being characterized by additive perturbations on the right coprime factorization ( ^; ^D) similar as in () and formulated as follows P F ( ^; ^D; D ; ) := fp j 9 ; D IRH such that P = D and F := := 4 D IRH such that? 4 ^D ^ with 8 < : f D(e i! )g j D (e i! )j f (e i! )g j (e i! )j g (4) where D and reect knowledge of a frequency dependent upper bound on D and separately. This knowledge can for example be found by identication experiments (de Callafon et al., 994). The following subsections contain necessary and sucient conditions on the size of the uncertainty set P F in order to have a controller C ^P that robustly stabilizes all plants P P F. These conditions are found in two ways. Firstly, by rewriting the problem into the powerful standard representation for uncertainty descriptions using the basic perturbation model of gure. Secondly, by using results based on the distance measures given in denition.9 and denition.0. The conditions for stability robustness based on the basic perturbation model T (M; ) given in theorem. can be used by rewriting the additive uncertainties on the coprime factors ( ^; ^D) into T (M; ), where the uncertainty F in (4) will be atted to (0). ow let V F D (e i! ), V F (e i! ) and W F (e i! ) be stable and stably invertible frequency dependent weighting functions, the uncertainty F := 4 V F D 0 0 V F in (4) can be atted to (0) by dening WF ; such that kk : () and thereby reformulating the denition of the set P F ( ^; ^D; D ; ) of (4), using the normalized uncertainty description (), into P F ( ^; ^D; VF D ; V F ; W F ; ) := fp j 9 ; D IRH such that P = D and := 4 V F D 0 0 V F := 4 D? 4 ^D with ^ WF ; such that kk g (6)

15 - d -? d-? ? V F D? d V F D D ^D W F 6 W F d? - ^ V F V F +?? z d C ^P??d + Fig. : Additive perturbations on a right coprime factorization The result of the introduction of the weighting functions is depicted in gure and produces z = M 4 d ; with d h M =?W [ ^D F + C ^] ^P I =?W C ^P F h ~Dc ~ c i 4 V i 4 V F D 0 F D 0 0 V F 0 V F ; = [ ~ Dc ^D + ~ c ^]: (7) With the results in theorem.8 and V F D (e i! ), V F (e i! ), W F (e i! ) being stable and stably invertible, it follows that both M and IRH and theorem. can be used for testing stability robustness for an unstructured (weighted) uncertainty F. Theorem 4. Let ^P = ^ ^D, where ( ^; ^D) is a rcf, and a controller C ^P be such that T ( ^P ; C ^P ) is BIBO stable. Let P F ( ^; ^D; V F D ; V F ; W F ; ) be given by (6), then the closed loop system T (P ; C ^P ) is BIBO stable 8P P F ( ^; ^D; VF D ; V F ; W F ; ) if and only if h < W [ ^D F + C ^] ^P I C ^P i 4 V F D 0 0 V F Proof: Can be found directly by substituting () and (7) in theorem. and the result that kmk < is equivalent to < kmk.

16 6 It should be noted that for an uncertainty F having a special structure, a similar result holds by applying the results based on the structured singular value fg (Doyle, 98), where fg is computed with respect to the structure of the uncertainty. A related example is the case of a multivariable model ^P, where for each transfer function [ ^P ] i;j = [ ^ ^D ] i;j information on F i;j, or Di;j and i;j separately, is available. In this case the uncertainty F will have a diagonal structure in which each diagonal element satises a bound similar as in () and the computation of the structured singular value fg becomes unavoidable to obtain necessary and sucient conditions for stability robustness. Stability robustness based on distance measures. Introduction In results based on distance measures like the gap or -gap given in denition.9, an other approach to stability robustness tests is being followed. Firstly, one is interested in nding the largest distance max from a model ^P to a plant P for which stability of T (P ; C ^P ) still holds. This largest distance max from the model ^P, provided by the distance measure being used, will induce a set P stab of plants P with stable T (P ; C ^P ). Secondly, stability of T (P ; C ) for all plants ^P P in a prespecied set P F can now be analyzed by checking P F P stab. If ^P P F this in turn can be done by checking wether the distance of ^P to the boundary of P F, provided by the distance measure being used, is smaller then max.. Stability robustness in the gap metric The gap metric, as dened in denition.9, can be used to specify a so called gap-based uncertainty set P. With this distance measure the following result can be obtained for stability robustness in the gap metric. Theorem. Let ^P and a controller C ^P be such that T ( ^P ; C ^P ) is BIBO stable. Let P ( ^P ; max ) be given by P ( ^P ; max ) := fp j ( ^P ; P ) max g (8) then the closed loop system T (P ; C ^P ) is BIBO stable 8P P ( ^P ; max ) if and only if max < kt ( ^P ; C ^P )k Proof: Can be found in Georgiou and Smith (990a).

17 7 An alternative proof of theorem. can also be found by using theorem 4., since it can be shown that an uncertainty set bounded by a directed gap metric, corresponds to a special choice of the weighting functions and the coprime factorization of the model ^P in set characterization of (6). This links the set P ~ ( ^P ; max ) bounded by a directed gap metric with the set P F ( ^; ^D; V F D ; V F ; W F ; max ) of (6) bounded by weighted additive coprime factor perturbations. This seems to be trivial, but in order to show this rst the following lemmas and corollary will be needed in order to give an alternative proof of theorem.. Lemma. Let ; D IRH (not necessarily coprime) then a plant P can be characterized by the factorization (; D) with P = D if and only if 9 Q IRH such that where ( n ; D n ) is a nrcf of P. 4 D = n Q Proof: Let ( n ; D n ) be a nrcf of P. ( There exists a Q IRH and hence = n Q IRH, D = D n Q IRH and D = n QQ D n = n D n = P. ) Since ( n ; D n ) is a (n)rcf of P, 9 X, Y IRH such that X n + Y D n = I. Using = n Q and D = D n Q and X; Y IRH makes X + Y D = [X n + Y D n ]Q = Q. Since, D IRH, Q IRH and hence 9 Q IRH, namely Q = X + Y D. Lemma. gives a parametrization of all stable (not necessarily coprime) factors, D of a plant P = D in terms of the nrcf ( n ; D n ) of the plant P and a free parameter Q IRH. This gives rise to an alternative representation of the uncertainty set P F ( ^; ^D; V F D ; V F ; W F ; ) given in (6) and is summarized in the following corollary. Corollary. The set P F ( ^; ^D; VF D ; V F ; W F ; ) given in (6) is equivalent to the set P F ( ^; ^D; V F D ; V F ; W F ; ) := fp j 9 Q IRH such that P = [ n Q][D n Q] := with ( n ; D n ) a nrcf of P and 4 V F D 0 0 V F := n Q? 4 ^D ^ with WF ; such that kk g (9)

18 8 Proof: ) Consider a plant P P F ( ^; ^D; V F D ; V F ; W F ; ) in (6), then there exist stable (not necessarily coprime) factors, D of the plant P such that P = D and = 4 D? 4 ^D ^ with := 4 V F D 0 0 V F WF ; such that kk Using the result of lemma., the existence of such stable factors, D is equivalent to the existence of a Q IRH where the and D are represented in terms of the nrcf ( n ; D n ) of P and a Q IRH, yielding 4 D = while P = D = n QQ D n = n D n and = n n Q Q? 4 ^D ^ with := 4 V F D 0 0 V F WF ; such that kk making P P F ( ^; ^D; V F D ; V F ; W F ; ) in (9). ( Consider a P P F ( ^; ^D; VF D ; V F ; W F ; ) in (9), then 9 Q IRH such that kk : With lemma., := n Q IRH, D := D n Q IRH such that P = D and = 4 D? 4 ^D ^ with := 4 V F D 0 0 V F WF satises kk. Hence, there exists ; D IRH such that P = D and kk, making P P F ( ^; ^D; VF D ; V F ; W F ; ) in (6).

19 9 It should be noted that Q IRH in the characterization of the set P F in (9) does not aect the elements P of the set P F and thereby the set P F itself, since it cancels out in the operation P = [ n Q][D n Q] and hence it does not appear in the argument list (; ;...) of the set P F. ow it can be shown that the set P F ( ^; ^D; V F D ; V F ; W F ; ) in (6), along with a special choice of the weighting functions V F D, V F and W F and the coprime factorization ( ^; ^D) of the model ^P, is equivalent to a set bounded by the directed gap between ^P and P, and is given in the following lemma. Lemma.4 Let the set P F ( ^; ^D; VF D ; V F ; W F ; ) be given by (6) with the conditions ( ^; ^D) is a nrcf of the model ^P V F D = I, V F = I and W F = I then P F ( ^; ^D; V F D ; V F ; W F ; ) is equivalent P ~ ( ^P ; ) := fp j ~ ( ^P ; P ) g. Proof: With corollary. the set P F ( ^; ^D; V F D ; V F ; W F ; ) given in (6) is equivalent to the set P F ( ^; ^D; V F D ; V F ; W F ; ) in (9) and hence again the equivalency between the two sets P F ( ^; ^D; VF D ; V F ; W F ; ) in (9) and P ~ ( ^P ; ) needs to be proven. ) Consider a P P F ( ^; ^D; V F D ; V F ; W F ; ) in (9), then 9 Q IRH such that kk : In order to nd a Q IRH such that kk, Q can be taken to be Q = arg min kk : QIRH Using V F D = I, V F = I and W F = I, in (9) becomes = = n Q? 4 ^D ^ and min QIRH kk = min QIRH k? k = min QIRH 4 ^D ^? n Q : Using the condition that ( ^; ^D) is a nrcf of the model ^P, min QIRH 4 ^D ^? n Q = ~ ( ^P ; P )

20 0 by denition.9, making P P ~ ( ^P ; ) ( Consider a P P ~ ( ^P ; ), then ~ ( ^P ; P ) and ~ ( ^P ; P ) = min 4 ^D QIRH ^? n Q by denition.9, where ( ^; ^D) must be a nrcf of the model ^P and ( n ; D n ) a nrcf of the plant P. For V F D = I, V F = I and W F = I, min 4 ^D QIRH ^? n Q = min 4 QIRH D = min QIRH kk and hence 9 Q IRH such that kk, making P P F ( ^; ^D; VF D ; V F ; W F ; ) in (9) under the conditions that ( ^; ^D) is a nrcf of ^P and V F D = I, V F = I and W F = I. Lemma.4 relates the set dened by a gap metric bound with the set P F a special choice of the weighting functions V F D, V F, W F in (6) by and the coprime factorization ( ^; ^D) of the model ^P. This gives rise to an unied approach to handle sets of plants that are bounded by a gap metric, since it corresponds to a special choice of the weighting functions and the coprime factorization of the model being used. With this unied approach and the representation of P ~ theorem. as follows. in lemma.4, it is possible to give an alternative proof of Proof: (alternative proof of theorem.) Dening to be equal to max, the set fp j ~ ( ^P ; P ) max g is equivalent to the set P F ( ^; ^D; VF D ; V F ; W F ; max ) given in (6) with kk max and the conditions of lemma.4. For P P F ( ^; ^D; VF D ; V F ; W F ; max ) in (6) the necessary and sucient condition for stability robustness can be found in theorem 4., which modies for kk max and the conditions given in lemma.4 into max < [ ^D + C ^P ^] h I C ^P i Again using the condition that ( ^; ^D) is a nrcf, kt ( ^P ; C ^P )k = 4 ^ ^D [ ^D + C ^P ^] h I C ^P i = [ ^D + C ^P ^] h I C ^P i making max < kt ( ^P ; C ^P )k

21 a necessary and sucient condition. It remains to show that ( ^P ; P ) = ~ ( ^P ; P ). Application of Bode's sensitivity integral (Maciejowski, 989) yields kt ( ^P ; C ^P )k k[i +C ^P ^P ] k > and hence for stability ~ ( ^P ; P) max < kt ( ^P ; C ^P )k < If ~ ( ^P ; P ) < then ( ^P ; P) = ~ ( ^P ; P ) (Georgiou, 988) and max < kt ( ^P; C ^P )k is the necessary and sucient condition to stabilize all plants P P where P is given by P ( ^P ; max ) := fp j ( ^P ; P) max g. Finally it should be noted that the gap and graph metric are induced by the same topology and are uniformly equivalent (Georgiou, 988; Packard and Helwig, 989). Therefore stability robustness in the graph metric yields a similar result as mentioned in theorem... Stability robustness in the -gap In a similar way as for the gap-based result in theorem., for the -gap the following result on stability robustness can be obtained. Theorem. Let ^P and a controller C ^P be such that T ( ^P ; C ^P ) is BIBO stable. Let P ~ ( ^P ; ;max ) be given by P ~ ( ^P ; ;max ) := fp j ~ ( ^P ; P) ;max g (0) then the closed loop system T (P ; C ^P ) is BIBO stable 8P P ~ ( ^P ; ;max ) if and only if ;max < Proof: Sucient conditions can be found in Bongers (99) or Bongers (994). ecessary and sucient conditions can be found by again using the unied approach of representing the set P ~ by the set P F in (6) by a special choice of the weighting functions V F D, V F, W F and the coprime factorization ( ^; ^D) of the model ^P. Following the alternative proof of theorem., rstly the relation between the set dened by a -gap bound and the set P F in (6) is summarized in the following lemma. Lemma.6 Let the set P F ( ^; ^D; V F D ; V F ; W F ; ) be given by (6) under the conditions ( ^; ^D) is a nrcf of the model ^P

22 V F D = I, V F = I, and W F = with := [ ~ D c ^D + ~ c ^] wherein ( ~ Dc ; ~ c ) is a nlcf of any controller C ^P = ~ Dc ~ c that creates a stable closed loop system T ( ^P ; C ^P ) then P F ( ^; ^D; VF D ; V F ; W F ; ) is equivalent to P ~ ( ^P ; ) := fp j ~ ( ^P ; P ) g. Proof: With corollary. the set P F ( ^; ^D; V F D ; V F ; W F ; ) given in (6) is equivalent to the set P F ( ^; ^D; VF D ; V F ; W F ; ) in (9) and hence the equivalency between the two sets P F ( ^; ^D; VF D ; V F ; W F ; ) in (9) and P ~ ( ^P ; ) needs to be proven. ) Consider a P P F ( ^; ^D; V F D ; V F ; W F ; ) in (9), then 9 Q IRH such that kk : In order to nd a Q IRH such that kk, Q can be taken to be Q = arg min kk : QIRH Using V F D = I, V F = I and W F =, in (9) becomes = with Q := Q IRH and hence min QIRH kk = = min QIRH k? k = n Q? 4 ^D ^ min 4 ^D QIRH ^ Using the condition that ( ^; ^D) is a nrcf of the model ^P, min 4 ^D QIRH ^? n by denition.0, making P P ~ ( ^P ; ) ( Consider a P P ~ ( ^P ; ), then ~ ( ^P ; P) and ~ ( ^P ; P ) = min 4 ^D QIRH ^? n Q = ~ ( ^P ; P )? Q n Q : by denition.0, where ( ^; ^D) must be a nrcf of the model ^P and ( n ; D n ) a nrcf of the plant P. For V F D = I, V F = I and W F =, min 4 ^D QIRH ^? n Q = min QIRH 4 4 ^D ^ = min QIRH kk? n Q

23 with Q := Q IRH and hence 9 Q IRH such that kk, making P P F ( ^; ^D; V F D ; V F ; W F ; ) in (9) under the conditions that ( ^; ^D) is a nrcf of ^P and the weightings V F D = I, V F = I and W F =. With lemma.6 the following proof can be given for the necessary and sucient conditions for stability robustness mentioned in theorem.. Dening to be equal to ;max the set P ~ ( ^P ; ;max ) in theorem. is equivalent to the set P F ( ^; ^D; V F D ; V F ; W F ; ) given in (6) under the conditions given in lemma.6 with kk ;max. For P P F ( ^; ^D; V F D ; V F ; W F ; ) in (6) the necessary and sucient condition for stability robustness can be found in theorem 4., which modies for kk ;max and the conditions given in lemma.6 into ;max < [ ^D + C ^P ^] h I Using the condition that ( D ~ c ; ~ c ) is a nlcf of C, ^P h i [ ^D + C ^] ^P I = making C ^P ;max < C ^P i h i ~Dc c ~ = a necessary and sucient condition to stabilize all plants P P ~ ( ^P ; ;max ). In case of the -gap, the nrcf of ^P is shaped or weighted by (see denition.9) to account for the closed loop operation of the model ^P. However, in lemma.6 the stable and stably invertible weighting lter W F = is used to weight the uncertainty in (6). It can be shown that choosing any stable and stably invertible lter W to shape or weight a rcf ( ^; ^D) of the model ^P, is equivalent to the choice of the weighting WF = W on the uncertainty in P F of (6) and is stated in the following corollary. Corollary.7 Consider any W; W IRH, V, V IRH, V D, V IRH D and any rcf ( ; D) of the model ^P. Let PF; be given by P F ( ^; ^D; VF D ; V F ; W F ; ) in (6) under the conditions ( ^; ^D) is the rcf of ^P with ^ := and ^D := D. V F D = V D, V F = V, and W F = I and let P F; be given by P F ( ^; ^D; V F D ; V F ; W F ; ) in (6) under the conditions ( ^; ^D) is the rcf of ^P with ^ := W and ^D := DW. V F D = V D, V F = V, and W F = I

24 4 then P F; is equivalent to P F;. Proof: ) Let P o P F; then 9 o ; D o IRH such that P o = o D o and kk = 4 V D 0 0 V W = 4 V D 0 0 V 4 D o o W? W Taking ^D := DW, ^ := W where W; W IRH yields ( ^; ^D) to be a rcf of ^P. Dening D := D o W IRH, := o W IRH yields D = o W W D o = o D o = P and hence 9 ; D IRH such that P = D and 4 V D 0 0 V 4 D o o W? W = = 4 V D 0 0 V 4 V D 0 0 V 4 D making P P F;. ( Let P P F; then 9 o ; D o IRH such that P = o D o and kk = 4 V D 0 0 V I = 4 V D 0 0 V 4 D o o 4 ^D ^ I = kk? D? 4 W Dening D := D o W IRH and := o W IRH yields D = o D o = P and hence 9 ; D IRH such that P = D and 4 V D 0 0 V 4 D o o D? 4 W = = 4 V D 0 0 V 4 V D 0 0 V 4 D W? W W = kk making P P F;. In fact V D, V IRH is not a requirement to prove corollary.7, but the lters in (6) are dened to be stable and stably invertible. Finally, with corollary.7 the result given in lemma.6 can also be given in the following way. Lemma.8 Let the set P F ( ^; ^D; VF D ; V F ; W F ; ) be given by (6) under the conditions ( ^; ^D) is the rcf of the model ^P with ^ :=, ^D := D wherein ( ; D) is a nrcf of ^P and := [ ~ D c ^D + ~ c ^] with ( ~ Dc ; ~ c ) a nlcf of any controller C ^P = ~ D c ~ c that creates an internally stable closed loop system T ( ^P ; C ^P )

25 V F D = I, V F = I, and W F = I then P F ( ^; ^D; VF D ; V F ; W F ; ) is equivalent to P ~ ( ^P ; ) := fp j ~ ( ^P ; P ) g. Proof: Can be found directly by using lemma.6 and the result given corollary.7 for W =, V = I and V D = I. Finally, it should be noted that the weighting functions V F D and V F D are used to incorporate information of an upper bound on D and seperately, whereas this information is not being used in the gap results. The introduction of weightings in the gap metric has also been studied in Geddes and Postlethwaite (99), Georgiou and Smith (990b) or Qui and Davidson (99). In Geddes and Postlethwaite (99) a multiplicative uncertainty description on the nrcf ( ^; ^D) of the model ^P is being used, leading to an uncertainty structure F having a diagonal form. Due to the diagonal form only necessary and sucient conditions based on the structured singular value fg can be obtained. The weightings in the weighted gap of Georgiou and Smith (990b) have to be dened a posteriori which makes the choice of the weighting functions, to access robustness issues on the basis of a weighted gap, not a trivial task. Information on the size of the coprime factor perturbations can be used in the weighted pointwise gap metric dened in Qui and Davidson (99), but still an ecient computational method for pointwise gap metric is not available yet..4 Conservatism issues For the uncertainty set P F ( ^; ^D; VF D ; V F ; W F ; ) in (6), the necessary and sucient conditions for stability robustness are stated in theorem 4.. The necessary and sucient conditions for the uncertainty set P ( ^P ; max ) in (8) and P ~ ( ^P ; ;max ) in (0) bounded respectively by the gap and the -gap are summarized in theorem. and theorem.. Since all the results are necessary and sucient, no conservatism is introduced in the test for checking stability robustness itself. Therefore, conservatism can be introduced only in the specic description of the uncertainty being used, making the test for stability robustness related to it apparently conservative (Hsieh and Safonov, 99). In this perspective the concept of conservatism is an intrinsic property of the uncertainty set being used and the following denition of conservatism of an uncertainty set description can be given. Denition.9 Let C ^P be a controller that robustly stabilizes a set P of plants P, hence T (P ; C ^P ) is internally stable 8P P, then the set P is said to be conservative if 9 P such that P P and T (P ; C ^P ) is internally stable 8P P. With denition.9 the conservatism aspects of the dierent uncertainty sets discussed in this paper can be evaluated. For a (nominal) model ^P and a controller C ^P, the sets

26 6 of plants that are robustly stabilized by the controller and their relation is stated in the following theorem. Theorem.0 Let a model ^P = ^ ^D, where ( ^; ^D) is a rcf, and a controller C ^P be such that T ( ^P ; C ^P ) is BIBO stable, then the following sets of plants are robustly stabilized by the controller C ^P : P F := P F ( ^; ^D; V F D ; V F ; W F ; ) from (6) with h < W [ ^D F + C ^] ^P I P := P ( ^P ; max ) from (8) with C ^P i 4 V F D 0 0 V F P ~ := P ~ ( ^P ; ;max ) from (0) with max < kt ( ^P; C ^P )k ;max < and are interrelated by P P ~ P F () Proof: The sets can be found straightforwardly by application of respectively theorem 4., theorem. and theorem.. To show (), rstly P P ~ is proven by the implication P P ) P P ~. Let P P then 9 Q IRH such that 4 ^D ^? n Q < kt ( ^P ; C ^P )k () where ( ^; ^D) is taken to be a nrcf of ^P and ( n ; D n ) is a nrcf of P. With := [ ~ Dc ^D+ ~ c ^] where ( ~ D c ; ~ c ) is a nlcf of C ^P kt ( ^P; C ^P )k = k k and with () this implies 4 ^D ^? n Q 4 ^D ^? n Q k k < kt ( ^P ; C ^P )k k k = () a proof on this part can also be found in Bongers (99)

27 7 with Q := Q. The left hand side of () equals the denition of the -gap given in denition.0 and hence ~ ( ^P ; P ) < making P P ~. To show P ~ P F, the implication P P ~ ) P F needs to be proven. ow let P P ~, then 9 Q IRH such that 4 ^D ^? n Q < (4) by denition.0, where ( ^; ^D) must be a nrcf of the model ^P and ( n ; D n ) a nrcf of the plant P. For the special choice of the weighting functions V F D = I, V F = I, W F = and the fact h [ ^D + C ^] ^P I C ^P i = h ~Dc ~ c i where ( ~ D c ; ~ c ) is a nlcf of C ^P, it follows that there should 9 Q IRH in (6) and a that satises < W F h ~Dc h [ ^D + C ^] ^P I i c ~ 4 I 0 0 I C ^P i 4 V F D 0 = 0 V F in order to have P P F. With V F D = I, V F = I and W F =, (4) modies into 4 4 ^D ^? n Q = kk < with Q = Q IRH and hence 9 Q IRH such that kk, with < making P P F under the conditions that ( ^; ^D) is a nrcf of ^P and V F D = I, V F = I, W F =. With the denition of conservatism in denition.9 it can be seen that the set P F along with the stability robustness test given in theorem 4. is the one that is the least conservative. Intuitively this is also clear, since the gap metric does not take into account the closed loop operation of the plants P in the set, induced by the controller C being ^P used. This drawback has been rectied by the use of the -gap and this renement makes P P ~, see also Bongers (99). Unfortunately both the gap and -gap based result do not take into account the specic shape of the additive perturbations D and = of the right coprime factorization of the model ^P as can be done in (6). This renement again yields P ~ P F and hence in this framework of additive coprime factor perturbations the set P F and the corresponding stability robustness test given in theorem 4. can considered to be the least conservative.

28 8 6 Conclusion The powerful standard representation for uncertainty descriptions in a basic perturbation model as introduced in Doyle (98) can be used to attain necessary and sucient conditions for stability robustness within various uncertainty descriptions. In this paper these results are applied to uncertainty descriptions based on fractional model representations and special attention is given to upper bounds on additive coprime factor perturbations obtained by identication techniques. The usage of the basis perturbation model to represent additive coprime factor perturbations leads to necessary and sucient conditions for stability robustness. In this way an unied approach to handle additive coprime factor perturbations can be derived which yields a manageable and comprehensive way to relate gap and -gap based uncertainty sets to (weighted) additive coprime factor perturbations. Based on this framework necessary and sucient conditions for gap and -gap based uncertainty sets and their interrelation can be found relatively easily. Based on the interrelation, conservatism issues are discussed, leading to the observation that the usage of the basis perturbation model to represent weighted additive coprime factor perturbation along with the corresponding stability robustness test, is the least conservative. References Bongers, P.M.M. (99). On a new robust stablity margin. Recent Advances in Mathematical Theory of Systems, Control, etworks and Signal Processing, Proc. of the Int. Symposium MTS{9. pp. 77{8. Bongers, P.M.M. (994). Modeling and Identication of Flexible Wind Turbines and a Factorizational Approach to Robust Control. PhD thesis. Delft University of Technology, Mech. Eng. Systems and Control group. Bongers, P.M.M. and O.H. Bosgra (99). ormalized coprime factorizations for systems in generalized state space form. IEEE Trans. on Automatic Control, Vol. 8. pp. 48{0. Bongers, P.M.M. and P.S.C. Heuberger (990). Discrete normalized coprime factorization and fractional balanced reduction. Lecture otes in Control and Information Sciences, Vol. 44. pp. 07{. Boyd, S.P. and C.H. Barrat (99). Linear Controller design { Limits of Performance. Englewood Clis, Prentice Hall. Chen, C.T. and C.A. Desoer (98). Algebraic theory for robust stability of interconnected systems: necessary and sucient conditions. In Proc. IEEE Conference on Decision and Control. pp. 49{494.

29 9 de Callafon, R.A., P.M.J. Van den Hof and D.K. de Vries (994). Identication and control of a compact disc mechanism using fractional representations. In Proc. 0th IFAC Symp. on System Identication, Vol.. pp. {6. Desoer, C.A. and M. Vidyasagar (97). Feedback Systems: Input{Output Properties. Academic Press, ew York. Desoer, C.A. and W.S. Chan (97). The feedback interconnection of linear time invariant systems. Journal of the Franklin Institute. pp. {. Doyle, J.C. (98). Analysis of feedback systems with structured uncertainties. IEE Proc. on Control Theory and Applications, part D, Vol. 9. pp. 4{0. Doyle, J.C. and G. Stein (98). Multivariable feedback design: concepts for a classical/modern synthesis. IEEE Trans. on Automatic Control, Vol. 6. pp. 4{6. Doyle, J.C., B.A. Francis and A.R. Tannenbaum (99). Feedback Control Theory. MacMillan Publishing Company, Y, USA. Doyle, J.C., J.E. Wall and G. Stein (98). Performance and robustness analysis for structured uncertainty. In Proc IEEE Conference on Decision and Control. pp. 69{66. El-Sakkary, A.K. (98). The gap metric: Robustness of stabilization of feedback systems. IEEE Trans. on Automatic Control, Vol. 0. pp. 40{47. Francis, B.A. (987). A course in H control theory. Lecture notes in control and information sciences, Vol. 88, Springer Verlag, Berlin. Geddes, E.J.M. and I. Postlethwaite (99). The weigthed gap metric and structured uncertainty. In Proc. American Control Conference. pp. 8{4. Georgiou, T.T. (988). On the computation of the gap metric. Systems & Control Letters, Vol.. pp. {7. Georgiou, T.T. and M.C. Smith (990a). Optimal robustness in the gap metric. IEEE Trans. on Automatic Control, Vol.. pp. 67{686. Georgiou, T.T. and M.C. Smith (990b). Robust control of feedback systems with combined plant and controller uncertainty. In Proc. American Control Conference. pp. 009{ 0. Hsieh, G.C. and M.G. Safonov (99). Conservatism of the gap metric. IEEE Trans. on Automatic Control, Vol. 8. pp. 94{98. Kelley, J.L. (96). General Topology. Springer Verlag, ew York. Kwakernaak, H. (99). Robust control and H optimization{tutorial paper. Automatica, Vol. 9. pp. {7. Luenberger, D.G. (969). Optimization by Vector Space Methods. John Wiley, ew York. Maciejowski, J.M. (989). Multivariable Feedback Design. Addison{Wesley Publishing Company, Wokingham, UK.