IBC:A WorkingToolforRobustParametric Identification
Roberto Tempo · 1992
Ini thisl)aler. weexjlore theutility ofInformationB!asr(I (Cnn#pIftity (113C') forcontrol-oriented robust parametric idewtification inthepresence ofnoisy data. Inadditioll. weshlow that anmiber ofIBCalgorithms andtheir associatled worst-case errors arecomputable bymeansof vertexoptimilization). optimization tool. InaHilbert space setting, Melkman andMicchelli [41, studied global optimality properties of smoothing algorithms; seealso theworkofTempo[.5] for simplifications andapplications ofthese ideas inasystem identification framework. Formoredetails onthis subject, thereader canrefer tothesurvey papers onrobust parametric identification ofNorton [6], Walter andPietLahanier [7] andMilanese andVicino [8]. Finaly, wemention thatanewline ofresearch onrobust nonparametric identification problems wasrecently initiated byHelmicki, Jacobson andNett[9]. I.PRELIMINARIES Thlaimii ofthis paper istoshowthatInformation!Iaird (Cnmphfxity (113C) isa.usefuil working toolforrol)ust paraniietric idenitification problems. Recently developed intheareaofcomputer science, IBCisageneral tlheor studying thecomplexity ofproblems approximately solved forthepresence ofpartial and/or contaminiatedI information; seethebookofTraub, Wasilkowski anici Woiniakowski [1] andreferences therein. Typical ap pl irations iiiclhide dist iibutied computation, clock- synchroniization. solution ofnonlinear equations andlarge linear systens. l)epending onthespecific problem, different settings are StLdied: worst-case, average-case, probabilistic adh(l asVmpt)otiC. Ineachsetting. thetaskistodevelop op1tinlal algor itht hmsandtocompute theassociated errt'ts andcomputational complexity. Inthis paper, wewill focus only ontheworst-casw setting andstudy anumber ofBCWalgorithms having different optimality properties and(I different comptutational complexity: spline, projection, smoothing. interpolatory andcentral Theutility ofIBCforcontrol-oriented system identification problems ismotivated withtheaidofanillustrative example: Theparameter identification ofFinite Impulse Response (FIR) plants. Tothis end.weintroduce thekey concept ofpIaram(ftr information set; that is, thesetofall paraTmeters oftheFIRplant compatible withtheapriori iniformationi anid withtheavailable data.WVith this concept inminid, weshowthat theIBCalgorithms listed above ranbecomputed bysolv ingavertex optimization problem.-Roughly speaking, this meansthattheparameter estimation oftheFIRplant requires onlv asearch overa finite set ofvertices andanoptimization overacontinuum isniot reclilired. Theoptimally estimated parameters of theFIRplant andtheassociated estimation errors canbe usedasasecond step todesign arobust Linear Quadratic Regulator (LQR). Werecall that someoftheideas expressed inthis paperar'e notentirely new.Forexample, inthespecial case wlhen theinlput sequience isfixed, Milanese andTempo[2] proved thatcentral algorithms forrobust parametric estimnationi arecomputable bvlinear programming. Within thlis frameNork. TempoandWNJasilkowski [3] haveshown that pIrojectionl algoritlhms areconstructable withthesame