On the Approximation of Toeplitz Operators for Nonparametric $\mathcal{H}_{\infty}$-norm Estimation

Given a stable SISO LTI system <tex>$G$</tex>, we investigate the problem of estimating the <tex>$\mathcal{H}_{\infty}$</tex>-norm of <tex>$G$</tex>, denoted <tex>$\Vert G\Vert_{\infty}$</tex>, when <tex>$G$</tex> is only accessible via noisy observations. Wahlberg et al. [1] recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left <tex>$n\times n$</tex> corner <tex>$T_{n}$</tex> of the Toeplitz (convolution) operator associated to <tex>$G$</tex>. In this paper, we prove sharp non-asymptotic bounds on the necessary length <tex>$n$</tex> needed so that <tex>$\Vert T_{n}\Vert$</tex> is an <tex>$\varepsilon$</tex>-additive approximation of <tex>$\Vert G\Vert_{\infty}$</tex>. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same <tex>$\varepsilon$</tex>-additive approximation.

[1]  H. Widom.  Asymptotic behavior of block Toeplitz matrices and determinants. II , 1974 .

[2]  A. Böttcher,et al.  Notes on the asymptotic behavior of block TOEPLITZ matrices and determinants , 1980 .

[3]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[4]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity /-bounded uncertainty , 1989 .

[5]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions , 1990 .

[6]  M. Steinbuch,et al.  A fast algorithm to computer the H ∞ -norm of a transfer function matrix , 1990 .

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

[8]  L. Ljung,et al.  Asymptotic properties of the least-squares method for estimating transfer functions and disturbance spectra , 1992, Advances in Applied Probability.

[9]  A. Böttcher,et al.  Norms of Inverses, Spectra, and Pseudospectra of Large Truncated Wiener-Hopf Operators and Toeplitz Matrices , 1997 .

[10]  A. Böttcher,et al.  On the condition numbers of large semidefinite Toeplitz matrices , 1998 .

[11]  Alexander Goldenshluger,et al.  Nonparametric Estimation of Transfer Functions: Rates of Convergence and Adaptation , 1998, IEEE Trans. Inf. Theory.

[12]  A. Böttcher,et al.  Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis , 2000 .

[13]  Jie Chen,et al.  Control-oriented system identification : an H[infinity] approach , 2000 .

[14]  A. Zeevi,et al.  NONASYMPTOTIC BOUNDS FOR AUTOREGRESSIVE TIME SERIES MODELING , 2001 .

[15]  B. Pasik-Duncan.  Control-oriented system identification: An H∞ approach , 2002 .

[16]  Erik Weyer,et al.  Finite sample properties of system identification methods , 2002, IEEE Trans. Autom. Control..

[17]  Mathukumalli Vidyasagar,et al.  A learning theory approach to system identification and stochastic adaptive control , 2008 .

[18]  Bo Wahlberg,et al.  Non-parametric methods for L2-gain estimation using iterative experiments , 2010, Autom..

[19]  Bo Wahlberg,et al.  Analyzing iterations in identification with application to nonparametric H∞-norm estimation , 2012, Autom..

[20]  P. Olver.  Nonlinear Systems , 2013 .

[21]  Moritz Hardt,et al.  The Noisy Power Method: A Meta Algorithm with Applications , 2013, NIPS.

[22]  Benjamin Recht,et al.  Non-Asymptotic Analysis of Robust Control from Coarse-Grained Identification , 2017, ArXiv.

[23]  Tengyu Ma,et al.  Gradient Descent Learns Linear Dynamical Systems , 2016, J. Mach. Learn. Res..