Analysis of Emerson s Multiple Model Interpolation Estimation Algorithms: The MIMO Case

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1 Technica Report PC Anaysis of Emerson s Mutipe Mode Interpoation Estimation Agorithms: The MIMO Case João P. Hespanha Dae E. Seborg University of Caifornia, Santa Barbara February 0, 004

2 Anaysis of Emerson s Mutipe Mode Interpoation Estimation Agorithms: The MIMO Case Abstract We extend to Mutipe-Input/Mutipe-Output MIMO processes the previous anaysis of Emerson s Moving Mutipe Mode Interpoation MMMI agorithm for parameter estimation and compare it with standard east-squares estimators. We consider a MIMO ARX process mode amenabe to Mode Predictive Contro MPC sunthesis. Our anaysis proves that for MIMO ARX process modes with affine unknown parameters, the ony equiibrium point of Emerson s MMMI estimation agorithm is precisey the east-squares estimate of the parameters. MIMO mutipe-modes interpoation We consider a genera MIMO ARX mode for the process of the form yk = τ y A i θ p yk i + τ u B i θ p uk i + nk, k {,, 3,... }, where yk R ny denotes the output, uk R nu the input, nk R p output measurement noise, and A i θ p, B i θ p matrices of coefficients that depends on an unknown parameter vector θ p that beongs to a parameter set P R n. The process mode, can be written in the regression form yk = Cθ p ϕk + nk, k {,, 3,... }, where ϕk denotes the regression matrix defined by yk yk. ϕk := yk τ y R nyτy+nuτu, 3 uk. uk τ u and Cθ p := [ A θ p A τy θ p B θ p B τu θ p ] R ny nyτy+nuτu. 4 We assume that we have avaiabe a set of data { } yk, ϕk : k =,,..., L 5

3 coected over a time-window of ength L. By stacking the outputs and regression vectors as foows we can write the process mode as where Y := [ y y yl ] R ny L, 6 Φ := [ ϕ ϕ ϕl ] R nyτy+nuτu L, 7 Y = Cθ p Φ + N, 8 N := [ n n nl ] R ny L, 9 is a matrix with measurement noise. The vaue of N is not avaiabe to estimate p. The data-set is processed mutipe times by examining its fits with respect to a finite bank of modes that varies from iteration to iteration. We denote by Mi := {θm i, θ m i,..., θm m i} P 0 the vaues for the parameters for the bank of modes used in the ith iteration. The estimate Y mi of Y based on the th mode during the ith iteration is defined by and the corresponding prediction error is given by Ym i = Cθ m i Φ, {,,..., M}. E mi := Y mi Y, {,,..., M}. The sum-of-squares error SSE i for the th mode during the ith iteration is given by SSE i := L ymik yk = trace [ Ymi Y Ymi Y ], 3 k= and we define the corresponding performance index J i by J i := SSE i. 4 Based on these definitions, we construct a mutipe mode interpoation MMI estimator by ˆθ p i := J iθmi J. 5 i Exampe. For a one-step deay system with unknown gain θ p [, 0], we have ϕk := uk, Cθ p := θ p, P := [, 0] 6 eading to a mode simiar to the one considered in the August 8, 003 report with τ = : yk = θ p uk + nk, p [, 0]. 7 3

4 Exampe. For a system with unknown gain θ [, 0] and unknown deay θ {,, 3}, we woud have ϕk := [ uk uk uk 3 ], [ θ 0 0 ] θ = Cθ, θ := [ 0 θ 0 ] θ =, [ 0 0 θ ] p = 3 P := [, 0] {,, 3} 8 eading to θ uk θ = yk = nk + θ uk θ =, θ [, 0]. 9 θ uk 3 θ = 3 Moving mutipe-modes interpoation The moving mutipe-modes interpoation MMMI estimation agorithm is defined as foows:. Set i = 0 iteration index. Set = parameter index 3. Compute the MMI estimate ˆθ p i based on the famiy of mode defined by the candidate parameters θ mi 4. Compute a new famiy of modes by computing a new set of candidate parameters θ mi + centered at the th parameter in ˆθ p i: θmi + = ˆθ M + p i + e, 0 where e denotes the th eement of the canonica basis of R n 5. Increment i and moduo n and go to 3 unti there is no significant change in ˆθ p i. We are assuming here that the number of modes M is odd and there is a constant spacing among the mode vaues for the th parameter. According to the MMMI agorithm we obtain ˆθ p i + = = θ mi + J i + J i + ˆθp i + J i + e J i + = ˆθ J i + p i + J e 3 i + where the performance indexes J i + are given by 4. 4

5 3 Anaysis We start with some preiminary resuts needed to anayze Emerson s MMMI agorithm. 3. Preiminaries Lemma. Given a positive semi-definite quadratic function f : R n R, if there exists a vector x 0 R n, a basis {v : =,,..., n} for R n, and positive scaars α i, β i, i {,,..., m} such that m α i fx0 + β i v fx 0 β i v = 0, {,,..., n}, 4 then x 0 is a minimum of f. Proof of Lemma. A generic positive semi-definite quadratic function is of the form fx = x x Qx x + c, x R n, 5 where Q is a n n positive semi-definite matrix, c a scaar, and x one of the minima of Q. Because of 4, we concude that n α i x0 + β i v x Qx 0 + β i v x x 0 β i v x Qx 0 β i v x = 0, 6 n α i β i x 0 x Qv = γx 0 x Qv = 0,. 7 where γ = n α iβ i > 0 and the v form a basis of R n, we concude that But then which means that x 0 is aso a minimum of f. x 0 x Q = 0. 8 fx 0 = x 0 x Qx 0 x + c = c = fx, 9 When the parametrization C is affine, the sum-of-square errors for an arbitrary vaue θ P of the parameter: SSEθ := trace [ CθΦ Y CθΦ Y ], 30 is a quadratic function of θ. In this case, the foowing resut is a consequence of Lemma : Coroary. When C is affine, if there exists a vector θ 0 R n, a basis {v : =,,..., n} for R n, and positive scaars α i, β i, i {,,..., m} such that m α i SSEθ0 + β i v SSEθ 0 β i v = 0, {,,,..., n}, 3 then θ 0 is a east-squares estimate of θ. This impicity assumes that P is the whoe R n 5

6 3. Equiibrium Assume that the MMMI agorithm converges to some vaue ˆθ p. Since ˆθ p must be a fixed-point of 3 for every, we concude that at equiibrium J J e = 0 M M + J = 0, {,,..., n}, 3 where J denote the asymptotic vaue of J i for the parameter index as i, i.e., J = im i SSE i = im i SSE θ mi = SSE ˆθp e Separating the summation into terms with smaer, equa, and arger than, 3 can be re-written as M + J + M + M + J M + M + J M + + M = + M + J = 0, 34 J = 0 35 J J = 0,. 36 Using 33, we further concude that M + which can be written compacty as SSE ˆθp + e SSE ˆθp = 0,, e 37 α SSE ˆθp + β e SSE ˆθp β e = 0,. 38 where α := SSE ˆθp + e SSE ˆθp e > 0, β := M + > Since {e : =,,..., n} is a basis of R n, the foowing resut can be obtained from Coroary : Theorem Equiibrium. When C is affine, any equiibrium point of the MMMI agorithm is a east-squares estimate of the parameter θ. 6

7 4 Concusions The resuts in the report [] were generaized for the MIMO case and with an arbitrary number of modes. In particuar, we showed that for MIMO ARX process modes with affine unknown parameters, the ony equiibrium point of Emerson s MMMI estimation agorithm is precisey the east-squares estimates of the parameters. The affine parameterization was used in the proofs but it does not seem necessary. In fact, it suffices that the sum-of-squares error function satisfy the condition that appears in Lemma for the function f. Future work incude i the study of the convergence properties of the agorithm, ii the characterization of non-affine parameterizations for which the agorithm sti converges to a east-squares estimate, and iii the characterization of non-affine parameterizations for which the agorithm converges to a vaue cose to the east-squares estimate. References [] J. P. Hespanha and D. E. Seborg. Anaysis of Emerson s MMI estimation agorithm. Technica Report PC , University of Caifornia, Santa Barbara, Aug

Analysis of Emerson s MMI Estimation Algorithm

Technical Report PC-03-0808 Analysis of Emerson s MMI Estimation Algorithm João P. Hespanha Dale E. Seborg University of California, Santa Barbara August 8, 003 Abstract In this report we analyze Emerson