On the Equivalence of OKID and Time Series Identification for Markov-Parameter Estimation

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1 On the Equivalence of OKID and Time Series Identification for Markov-Parameter Estimation P V Albuquerque, M Holzel, and D S Bernstein April 5, 2009 Abstract We show the equivalence of Observer/Kalman Identification OKID and Time Series Identification TSI for Markov parameters PhD Student, University of Michigan, Ann Arbor, MI PhD Student, University of Michigan, Ann Arbor, MI Professor, Aerospace Engineering Department, University of Michigan, Ann Arbor, MI, 48109

2 1 Introduction Introduce paper here 2 Markov Parameter Estimation using OKID Observer/Kalman System Identification OKID involves adding an observer into an observable discrete-time system in order to help determine the Markov parameters The method for doing so, as seen in [2, is clearly derived in this section 21 Discrete-Time Linear System Consider the linear discrete-time system xk + 1 = Axk + Buk, 1 yk = Cxk + Duk, 2 k 0, xk R n, yk R p, and uk R m, and A, B, C, and D are real matrices of corresponding sizes We assume that A, C is observable For nonnegative integers k and r we have xk + r = A k xr + yk + r = CA k xr + A i 1 Buk + r i, 3 CA i 1 Buk + r i + Duk + r, 4 0 = 0 Now let s r Then, lining up 4 for k = 0,, s r yields Y [r:s = G [0:s r X r,s + H [0:s r U r,s, 5 Y [r:s [ yr yr + 1 yr + 2 ys R p s r+1, G [0:s r [ C CA CA 2 CA s r R p ns r+1, H [0:s r [ H 0 H 1 H 2 H s r [ D CB CAB CA s r 1 B R p ms r+1, 2

3 xr xr X r,s 0 Rns r+1 s r+1, 0 0 xr ur ur + 1 us 0 ur us 1 U r,s Rms r+1 s r ur 22 Adding an Observer to the System Adding and subtracting F yk, F R n p, to the right hand side of 1 yields which, with 2, can be written as xk + 1 = Axk + Buk + F yk F yk 6 = A + F Cxk + B + F Duk F yk, 7 xk + 1 = Axk + Bvk, 8 yk = Cxk + Dvk, 9 A A + F C R n n, B [B + F D F R n m+p, D [D 0 R p m+p, [ uk vk R m+p yk For nonnegative integers k and r we have xk + r = A k xr + yk + r = CA k xr + A i 1 Bvk + r i, 10 CA i 1 Bvk + r i + Dvk + r 11 Now, let s r Then, in analogy with 5, we have Y [r:s = G [0:s r X r,s + H [0:s r V r,s, 12 3

4 G [0,s r [ C CA CA 2 CA s r R p ns r+1, H [0:s r [ H 0 H 1 H 2 H s r [ D CB CAB CA s r 1 B R p m+ps r+1, vr vr + 1 vs 0 vr vs 1 V r,s Rm+ps r+1 s r vr Let r 0 and r + n < s so that n < s r Then, Y [r:s = [ Y [r:r+n 1 Y [r+n:s R p s r+1, G [0:s r = [ G [0:n 1 G [n:s r R p ns r+1, H [0:s r = [ H [0:n+1 H [n:s r R p m+ps r+1, V r,s = vr vr + 1 vr + n 1 vr + n vr + n + 1 vs 1 vs 0 vr vr + n 2 vr + n 1 vr + n vs 2 vs vr vr + 1 vr + 2 vs n vs n vr vr + 1 vs n 1 vs n vr vs n vs n vr vs n vr V r,r+n 1 V r+n,s 0 m+p n 0 m+ps n r n 0 m+ps n r 1 V r,s n 1 R m+ps r+1 s r+1, 4

5 V r+n,s R m+pn+1 s r n+1 is given by vr + n vr + 2n V r+n,s v2r + 2n s vr + n, s < r + 2n, vr vs n vr + n vr + 2n V r+n,s, s = r + 2n, vr vr + n vr + n vr + 2n vs V r+n,s, s > r + 2n, vr vr + n vs n Note that for s = n + 1, and thus r = 0, V 0,0 denotes v0 From 12, we have Y [r+n:s = G [n:s r X r+n,s + H [0:n+1 V r+n,s + H [n:s r [0 m+ps n r 1 V r,s n 1, 13 Note that, since H 0 = D = [D 0, D can be replaced by D, and thus columns p + 1,, 2p of H [0:n+1 and rows m + 1,, m + p of V r+n,s can be deleted In addition, by defining H [0:n+1 [ D H 1 H 2 H n+1 R p m+pn+m, un + r un r us V vn 1 + r vn + r vs 1 r+n,s Rm+pn+m s r n+1 vr v1 + r vs n Since A, C is observable, we choose F such that A is nilpotent Hence for all k n, it follows that A k = 0 so that H k = 0 for all k n + 2 Therefore, for k n, 10 and 11 can be written as xk + r = yk + r = minn,k minn,k A i 1 Bvk + r i, 14 CA i 1 Bvk + r i + Dvk + r 15 5

6 Based on the nilpotent assumption, G [n,s r = 0 p ns r n+1, H [n:s r = 0 p m+ps r n 1 Thus 13 can be rewritten as Y [r+n:s = H [0:n+1V r+n,s 16 If V r+n,s has full column rank, then the unique solution of 16 is given by H [0:n+1 = Y [r+n:s V + r+n,s, 17 + is the generalized inverse When u and y are corrupted by noise, 17 provides a least squares estimate 23 Deriving H i from H [0:n+1 Let k 1 Then H k = CA k 1 B = [β k α k, β k CA + F C k 1 B + F D, 18 α k CA + F C k 1 F 19 Next, for k 3 we have H k = CA k 1 B, 20 = C A + F C F C A k 2 B = C A + F CA F CA A k 3 B = C A + F CA + F C F C F CA A k 3 B = C A + F C 2 A + F CF C F CA A k 3 B 2 = C A + F C 2 A + F C i 1 F CA 2 i A k 3 B 21 6

7 More generally we have H k = C A + F C j j A + F C i 1 F CA j i A k j+1 B = C A + F C j A j A + F C i 1 F CA j+1 i A k j+2 B = C A + F C j A + F C F C j A + F C i 1 F CA j+1 i A k j+2 B = C A + F C j+1 A + F C j F C j A + F C i 1 F CA j+1 i A k j+2 B = C A + F C j+1 j+1 A + F C i 1 F CA j i A k j+2 B 22 Using 22, 21 becomes k 1 H k = C A + F C k 1 A + F C i 1 F CA k i 1 B k 1 = CA + F C k 1 B C A + F C i 1 F CA k i 1 B = α k D + CA + F C k 1 B k 1 C A + F C i 1 F CA k i 1 B + α k D = β k k 1 CA + F C i 1 F CA k i 1 B + α k D k 1 = β k α i H k i + α k H 0 Thus, β k + k α i H k i, k = 1,, n; H k = n α i H k i, k n

8 3 Alternative Derivation of OKID Using a Pseudo- FIR Model The discrete-time state space model 1-2 can be represented by the infinite impulse response IIR model y k = α 1 y k α n y k n + β 0 u k + β 1 u k β n u k n, 24 for k n Rearranging 24, we obtain y k = H 0 v k + + H n v k n, 25 H k [ β k α k, 26 and from 24, α 0 = 0 Furthermore, note that 25 can be written as which has the least-squares solution and ˆα 0 is constrained to be zero Y [r:s = H [0:s r V r:s, 27 Ĥ [0:s r = V + r:sy [r:s, 28 Ĥ k [ ˆβk ˆα k, 29 Next, note that if y 0,, y µ is the impulse response of a system, then y k H k Impulsing 24, we obtain H 0 = β 0, H 1 = α 1 y 0 + β 1 u 0 = α 1 H 0 + β 1, H 2 = α 1 y 1 + α 2 y 0 + β 2 u 0 = α 1 H 1 + α 2 H 0 + β 2 H 3 = α 1 y 2 + α 2 y 1 + α 3 y 0 + β 3 u 0 = α 1 H 2 + α 2 H 1 + α 3 H 0 + β 3 β k + k α k H k i, 0 k n, H k = α k H k i, k > n 8

9 Thus using the estimates 29, the Markov parameters can be estimated recursively by ˆβ k + k ˆα k Ĥ k i, 0 k n, Ĥ k = 30 ˆα k Ĥ k i, k > n 4 Conclusion Note that 17 and 28 are reshuffled representations of the same equation While OKID and TSI both set up the V matrix differently, each contains the same information and thus the least square approximation of H must be identical This conclusion is verified in [1 Also, OKID and TSI share a very similar method of obtaining the estimates of the Markov parameters in 23 and 30 References [1 S Barnett Inversion of partitioned matrices with patterned blocks International Journal of Systems Science, 142: , 1983 [2 ; Wei Chen John Valasek Observer/kalman filter identification for online system identification of aircraft Journal of Guidance, Control, and Dynamics, : ,

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