2 STEM R∈ ALIZATION USE SD OF M TO FWD A,,C SQvARE SVD (-1)=U2y ∑- AXA DIAGONAL MATRIX WHERE A= RANK(M(i-s)) →RAEj2 U2T 、 CAN VSE T=工 ME T2V AN GET -c FRoM FIRST Ny RoWs oF U2T 8 FRoM FIRsT n y COLS oF T2 FIND A BY SOLVING =M;升AE;=U2了A12∥护 A=TEUM2Ga-V2T
2-n: ISSUES WITH THIS ALGoRITHM NEE0 To RECORD THE MARKoV PARAMETERS 工 THEORY M1 j-1)WoULD BE DF RANKAN s WOULD THEN KNow DIMENS(ON aF THE SYSTEM(A MATRIx) PROBLEM:N1j-) Is USUALLY FULL RANK Due TO SENSoR NoIsE AND NONLINERITIES KHEN Y6U CALCULATE THE SINGULAR VALVES You FIND THAT 0能G(E)=(《 +。, BUT SMALL MORE REALISTIC TRUNCATE MODEL sZE升T“ 了0EAL So THAT 06(x2)=(2,4) DYNAMics AssocITED WITH THeSE Sv's ARE A BLENo oF NoISE, NoNLINEARI nES, ETC. Do yoU WANT THESE IN THE MODEL
#9 State Space Models Thanks to Bart demoor, P. Van Overschee, Bo Wahlberg, and M. Jansson LL 208-211 section 10.6
Fall9 9 E21192 Introduction o Assumed truth model form 1= ACk+ Buk+Wk Uk Crk t Duk +Uk isn×1,yism×1 and u is r×1 w(process noise)and v(sensor noise) are assumed to be sta tionary, zero-mean, white Gaussian noises R=8 i. e. in this case we explicitly include the noises Objectives: Use the measured data yk, uk, k=l,., N to 1. Estimate the system order n 2. Estimate a model that is similar to the true description 3. Estimate the noise covariances so that we can design a kalman
Fall9 9 E21193 Basic point: given the state response of the system (ak), it is a simple linear regression to find the plant model matrices A, B, C, D Reason: If k known Vk, then we can rewrite Axk+ bur+ Crk+ Duk +Uk Yk=6重k+Ek where A B Ck yk C D k Could then estimate the covariance matrix using the square of the model residuals(as we did before R 1 N ∑Ek and then use this to solve for the Kalman filter gain K Primary motivation for Subspace approach: If we can develop a reasonable estimate for the state rk from the measured data, then it is relatively easy to develop a model of the plant model matrices A, B, C, D