POLYNOMIAL AND RATIONAL MATRIX INTERPOLATION: SYSTEMS AND CONTROL APPLICATIONS
Zhiqjang Gao · 2009
In this paper,a theory ofpolynomial andrational matrixinterpolation ishintrodued andapplied toproblems in systemsandcontrol. Thepolynomial matrixinterpolation theory is firstoutlined andthenapplied tosolving matrix equations; itisalsousedinpoleassignment andothercontrol problems. Rational matrix interpolation is alsodiscussed and itisusedtosolve rational matrix equations including themodel matching problem. I. INTRODUCTION A theoryofpolynomial andrational matrix interpolation isbriefly outlined in this paperanditsapplication tocertain systemsandcontrol problems isdiscussed; full details can be foundin(21). Manysystemandcontrol problems can beformulated in termsofmatrixequations wherepolynomial or rational solutions withspecific properties areofinterestItis known that equationsinvolving justpolynomials can besolved by either equating coefficients ofequal power oftheindeterminate s orequivalently byusingthevalues obtained whenappoprate values fors aresubstituted in thegivenpolynomials; inthe latter case one uses results from thetheory ofpolynomial interpolation. Similarly one may solvepolynomial matrix equations usingthetheoryofpolynomial matrix interpolation presented here(alsoin(1-7), (21)); thisapproach has significant advantages andthese arediscussed below.Rational, mostlyscalar interpolation hasbeenofinterest tosystemsand controlresearchers recenly.Note thattherational interpolation results presented hereare distinct fromother literature results as theyrefer tomatrix caseandconcentrate on fundanental representation questions. Otherresults inthe literature attempt tocharacterize rational functions thatsatisfy crtin interpolation constraints andareoptimal insome sense andso theyrather complement ourresults thancompetewith them. In this paperpolynomial matrix interpolation ofthetype Q(sj)aj= bj,whereQ(s)isa matrix andaj,bjvectors, is introduced as a generalization of thescalarpolynomial interpolation oftheformq(sj) = bj.Thisgeneralization appearstobewellsuited tostudy andsolvea variety of multivariable systemandcontrol problems. The original motivation forthedevelopment ofthe matrixinterpolation theory was tobeabletosolvepolynomial matrixequations, whichappearinthetheory ofSystems andControl andin particular theDiophantine equation; theresults presented here andin (21) however go wellbeyond solving thatequation. Theuse ofinterpolation typeconstraints insystemand control theoryisfirst discussed anda numberofexamples are presented.