The sample complexity of worst-case identification of FIR linear systems

We consider the problem of identification of linear systems in the presence of measurement noise which is unknown but bounded in magnitude by some /spl delta/>0. We focus on the case of linear systems with a finite impulse response (FIR). It is known that the optimal identification error is related (within a factor of 2) to the diameter of a so-called uncertainty set and that the latter diameter is upper-bounded by 2/spl delta/, if a sufficiently long identification experiment is performed. If the identification error is measured with respect to the l/sub 1/ norm, we establish that, for any K/spl ges/1, the minimal length of an identification experiment that is guaranteed to lead to a diameter bounded by 2K/spl delta/ behaves like 2/sup Nf(1/K)/, when N is large, where N is the length of the impulse response and f is a positive function known in closed form. We contrast this with identification in H/sub /spl infin//, where an experiment of length O(N/sup 3/) suffices. While the framework is entirely deterministic, our results are proved using probabilistic tools.<<ETX>>

[1]  K. Poolla,et al.  On the time complexity of worst-case system identification , 1994, IEEE Trans. Autom. Control..

[2]  M. Milanese,et al.  Optimal finite-sample experiment design in worst case l/sub 1/ system identification , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[3]  C. Jacobson,et al.  Worst case system identification in l 1 ne-equation> 1 : optimal algorithms and error bounds , 1992 .

[4]  P. Khargonekar,et al.  Linear and nonlinear algorithms for identification in H/sub infinity / with error bounds , 1992 .

[5]  John N. Tsitsiklis,et al.  Worst-Case Identification of Nonlinear Fading Memory Systems , 1992, 1992 American Control Conference.

[6]  P. Khargonekar,et al.  Robust convergence of two-stage nonlinear algorithms for identification in H ∞ , 1992 .

[7]  Guoxiang Gu,et al.  A class of algorithms for identification in H∞ , 1992, Autom..

[8]  J. Tsitsiklis,et al.  Optimal asymptotic identification under bounded disturbances , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[9]  P. Mäkilä.  Robust identification and Galois sequences , 1991 .

[10]  Carl N. Nett,et al.  Control oriented system identification: a worst-case/deterministic approach in H/sub infinity / , 1991 .

[11]  P. Khargonekar,et al.  Linear and nonlinear algorithms for identification in H∞ with error bounds , 1991, 1991 American Control Conference.

[12]  M.A. Dahleh,et al.  Robust and Optimal Identification in the l1 Norm , 1991, 1991 American Control Conference.

[13]  J. Partington,et al.  Robust approximation and identification in H∞ , 1991, 1991 American Control Conference.

[14]  C. Jacobson,et al.  Worst Case System Identification in l1: Optimal Algorithms and Error Bounds , 1991, 1991 American Control Conference.

[15]  Arthur J. Helmicki,et al.  Identification in H ∞ : linear algorithms , 1990 .

[16]  A. Helmicki,et al.  Identification in H∞: a robustly convergent, nonlinear algorithm , 1990, 1990 American Control Conference.

[17]  A. Helmicki,et al.  Identification in H∞: linear algorithms , 1990, 1990 American Control Conference.

[18]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[19]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[20]  Rogelio Lozano,et al.  Reformulation of the parameter identification problem for systems with bounded disturbances , 1987, Autom..

[21]  R. R. Bahadur.  Some Limit Theorems in Statistics , 1987 .

[22]  R. Tempo,et al.  Optimal algorithms theory for robust estimation and prediction , 1985 .

[23]  Toby Berger,et al.  Review of Information Theory: Coding Theorems for Discrete Memoryless Systems (Csiszár, I., and Körner, J.; 1981) , 1984, IEEE Trans. Inf. Theory.

[24]  M. Milanese,et al.  Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators , 1982 .

[25]  Henryk Wozniakowski,et al.  A general theory of optimal algorithms , 1980, ACM monograph series.

[26]  G. Zames.  On the metric complexity of casual linear systems: ε -Entropy and ε -Dimension for continuous time , 1979 .