A probabilistic point of view on peak effects in linear difference equations

Abstract It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that “solutions of stable difference equations probably experience peak”. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.

[1]  S. A. Kuruklis,et al.  The Asymptotic Stability of xn+1 − axn + bxn−k = 0 , 1994 .

[2]  Natalia Dudarenko,et al.  Large Deviations in Discrete-time Systems with Control Signal Delay , 2019, ICINCO.

[3]  Eduardo F. Camacho,et al.  Robust Design Through Probabilistic Maximization , 2018 .

[4]  G. Smirnov,et al.  Stabilization Problems with Constraints: Analysis and Computational Aspects , 1998 .

[5]  Solutions of Discrete Time Linear Systems: Upper Bounds on Deviations , 2018, 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC).

[6]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[7]  P. S. Shcherbakov,et al.  Upper Bounds on Peaks in Discrete-Time Linear Systems , 2018, Autom. Remote. Control..

[8]  M. M. Kogan,et al.  Synthesis of multipurpose linear control laws of discrete objects under integral and phase constraints , 2011 .

[9]  P. Apkarian,et al.  Optimizing the Kreiss Constant , 2019, SIAM J. Control. Optim..

[10]  Dmitry V. Balandin,et al.  Finite-horizon multi-objective generalized H2 control with transients , 2019, Autom..

[11]  James F. Whidborne,et al.  On the Minimization of Maximum Transient Energy Growth , 2007, IEEE Transactions on Automatic Control.

[12]  Boris T. Polyak,et al.  Large deviations for non-zero initial conditions in linear systems , 2016, Autom..

[13]  Georgi V. Smirnov,et al.  Transient Response in Matrix Discrete-Time Linear Systems , 2019, Autom. Remote. Control..

[14]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[15]  Juan J. Flores,et al.  On the hyperbox: hyperplane intersection problem , 2009 .

[16]  Pavel Shcherbakov,et al.  On peak effects in discrete time linear systems , 2017, 2017 25th Mediterranean Conference on Control and Automation (MED).

[17]  P. S. Shcherbakov,et al.  Peak effects in stable linear difference equations , 2018, ArXiv.

[18]  B. T. Polyak,et al.  Optimal design for discrete-time linear systems via new performance index , 2001 .

[19]  S. Elaydi.  An introduction to difference equations , 1995 .

[20]  Hideaki Matsunaga,et al.  Exact stability criteria for delay differential and difference equations , 2007, Appl. Math. Lett..

[21]  Pavel Shcherbakov.  The Trinomial Equation xk+l-axk + bXk-n = 0: Analysis of the Nonasymptotic Behavior of Solutions , 2019, 2019 27th Mediterranean Conference on Control and Automation (MED).

[22]  Mikhail V. Khlebnikov,et al.  Large deviations in linear control systems with nonzero initial conditions , 2015, Autom. Remote. Control..

[23]  K. Glover,et al.  Bounded peaking in the optimal linear regulator with cheap control , 1978 .

[24]  Boris Polyak,et al.  Superstable Linear Control Systems. I. Analysis , 2002 .

[25]  Giuseppe Carlo Calafiore,et al.  RACT: Randomized Algorithms Control Toolbox for MATLAB , 2008 .

[26]  Boris Polyak,et al.  Non-monotone Behavior of the Heavy Ball Method , 2018, Difference Equations and Discrete Dynamical Systems with Applications.

[27]  Graziano Chesi,et al.  Convergent upper bounds of peak response of LTI and polytopic LTV systems through LMIs , 2020, Autom..

[28]  Boris Polyak.  Extended Superstability in Control Theory , 2004 .

[29]  N. Saha.  Clinical Pharmacokinetics and Drug Interactions , 2018 .

[30]  Fabrizio Dabbene,et al.  On the Generation of Random Stable Polynomials , 2011, Eur. J. Control.

[31]  Solutions of Stable Difference Equations Probably Experience Peak , 2020 .

[32]  B. Ross Barmish,et al.  The uniform distribution: A rigorous justification for its use in robustness analysis , 1996, Math. Control. Signals Syst..

[33]  Christophe Andrieu,et al.  An improved method for uniform simulation of stable minimum phase real ARMA (p,q) processes , 1999, IEEE Signal Processing Letters.

[34]  Benjamin C. Kuo,et al.  AUTOMATIC CONTROL SYSTEMS , 1962, Universum:Technical sciences.

[35]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[36]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[37]  Mario Sznaier,et al.  Peak Estimation Recovery and Safety Analysis , 2021, IEEE Control Systems Letters.

[38]  F. Dannan,et al.  The Asymptotic Stability of , 2004 .

[39]  Minyue Fu,et al.  Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis , 1998, Math. Control. Signals Syst..

[40]  A. Fam.  The volume of the coefficient space stability domain of monic polynomials , 1989, IEEE International Symposium on Circuits and Systems,.

[41]  R. Tempo,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems , 2004 .