1Structural Instability and Minimal RealizationsL.H. Keel and S.P. BhattacharyyaAbstract—In this paper we consider the structural stability of anunstable plant feedback stabilized by a controller. It is shown thatinfinitesimal parameter perturbations in the plant transfer matrix candestabilize the stable feedback control system obtained from a minimalrealization of the “nominal plant”. The remedy for such structuralinstability consists in using a minimal realization in which the dimensionof the “antistable part” is maximized over the uncertain p arametervalues. This requires knowledge of the “correct class” of parametrizationsto which the nominal plant transfer matrix belongs.I. INTRO DUCTIONThe dynamic models that arise in control engineering depend onvarious system parameters which are imprecisely known at best andare subject at least to small perturbations. Nevertheless, controllerdesign typically proceeds by setting these parameters to nominalvalues, designing a controller for the nominal system, and validatingits performance over the expected uncertainty set. In the context oflinear multivariable control theory, this often amounts to startingwith a parametrized plant transfer matrix P(s, p), setting p = p0the nominal value, constructing a minimal state space realization ofP0(s) = P(s,p0) and designing a feedback controller that stabilizesthis minimal realization (see the classic works [1], [2], [3], [4], [5],[6]).The purpose of this note is to point out a potential pitfall in thisprocedure. Specifically, we show that there are situations in whichstabilization of any minimal state space realization of P(s,p0) willlead to a closed-loop system which becomes unstable f or arbitrarilysmall perturbations of p0. This occurs because the McMillan degreeof P(s, p) is, in general, discontinuous with respect to p at p0. Wealso show that such a catastrophic failure can be avoided if t he correctclass of parametrizations to which P0(s) belongs is known.Let ν+[P(s, p)] and ν−[P(s, p)] be the McMillan degrees of theantistable (all RHP poles) and stable (all LHP poles) components ofP(s, p), respectively. Accordingly, let ν+maxbe the maximal McMil-lan degree of the antistable component of P(s,p) under the givenparametrization. It can be easily found by arbitrarily perturbing p0by a small amount within its perturbation class, since the McMillandegree drops only on an algebraic variety. By stabilizing a perturbednominal plant whose minimal realization has ν+maxunstable poles,the structural instability discussed above can be overcome as thecontroller also stabilizes a “ball” of plants centered at the newnominal. Such a stability ball around the perturbed nominal planthowever cannot include the original system if ν+[P(s, p0)] < ν+max.II. MOTIVATIONBy way of motivation, consider the transfer functionP0(s) =2(s + 1)(s − 1)...1s − 1·· ··· ···· ·· ··· ····1s − 1...1s − 1which represents an unstable plant to be stabilized by feedback. Theorder of a mi nimal realizati on of P0(s) is 2, and it can be stabilizedL.H. Keel is with Department of Electrical & Computer Engineering,Tennessee State University, Nashville, TN 37209.S.P. Bhattacharyya is with Department of Electrical & Computer Engineer-ing, Texas A&M University, College Station, TX 77843by a compensator C0, say of order q. Now suppose that P0(s) perturbstoP1(s) = P (s,δ) =2(s + 1)(s − 1)...1 + δs − 1·· ··· ···· ·· ··· ····1s − 1...1s − 1where δ is a real parameter perturbation. It is easily seen that theclosed-loop system with compensator C0and plant P1(s) is unstablewith a closed-loop pole near s = 1. Moreover, this occurs for everynominally stabilizing controller C0of P0(s), and for infinitesimallysmall perturbations δ.To avoid the undesirable situation discussed above, it is necessaryto know the class of uncertain systems to which P0(s) belongs, and tostabilize a perturbed version of P0(s) whi ch has the generic maximalorder of unstable poles in the perturbation class. In the exampleabove, the nominal plant should have been chosen after perturbation(P1(s) with δ 6= 0) and realized minimally to be of order 3, and astabilizing controller C1designed for it. The controller C1remainsstabilizing under small perturbations. It is important to note howeverthat such a stability “ball” around the perturbation cannot include thenominal system P0(s)!III. NOMINAL AND PARAMETRIZED MODELSConsider a plant parametrized by a family of rational propertransfer function matrices, P(s,p) where p is an ℓ dimensional realparameter vector which ranges over an uncertainty set Ω ⊂ Rℓ. Weassume that the coefficients of the transfer functions in P(s, p) arecontinuous functions of p and that P(s, p) has a state space real-ization [A(p), B(p),C(p),D(p)] with matrix entries being continuousfunctions of p. Let p = p0be the nominal parameter and denoteP (s,p0) = P0(s).Now let ν[P(s, p)] denote the McMillan degree [7] of P(s, p). Fol-lowing [8], decompose P(s,p) into its stable and antistable (all polesunstable) componentsP(s, p) = P−(s,p)+ P+(s,p)and letνP+(s,p) =: ν+[P(s, p)]νP−(s,p) =: ν−[P(s, p)].When the context is clear, we write ν(p), ν+(p) etc. instead ofν[P(s, p], ν+[P(s, p)].In general the functions ν(p), ν+(p), ν−(p) are discontinuousfunctions of p. Moreover t he generic, maximal McMillan degr eedepends on the specific structure of the parametrization as we showbelow.Example 1: Continuing with our previous plantP0(s) =2(s + 1)(s − 1)...1s − 1·· ··· ···· ·· ··· ····1s − 1...1s − 1(1)consider, for example, the four parametrized families to which P0(s)2might belong:P1(s,a) =2 + a1(s + 1)(s −1 + a2)1 + a1s − 1 + a21 + a1s − 1 + a21 + a1s − 1 + a2, (2)where a = [a1a2], a0= [0 0]P2(s,b) =2 + b1(s + 1)(s −1 + b5)1 + b2s − 1 + b51 + b3s − 1 + b61 + b4s − 1 + b6, (3)where b = [b1b2·· · b6], b0= [0 0 ··· 0]P3(s,c) =2 + c1(s + 1)(s −1 + c5)1 + c2s − 1 + c51 + c3s − 1 + c51 + c4s − 1 + c6, (4)where c = [c1c2·· · c6], c0= [0 0 ··· 0]P4(s,d) =2 + d1(s + 1)(s −1 + d5)1 + d2s − 1 + d61 + d3s − 1 + d71 + d4s − 1 + d8, (5)where d = [d1d2·· · d8], d0= [0 0 · ··