Stability analysis and guaranteed cost control for stochastic nonlinear quadratic systems

In this paper we extend the guaranteed cost control approach for nonlinear quadratic systems (NLQSs), developed by the same authors in some recent papers, to the stochastic framework. In particular, we consider a stochastic NLQS in the Itô's form and provide a sufficient condition for the existence of a state feedback controller guaranteeing, with a certain risk factor α ∈ [0,1), that the closed loop system satisfies, for any initial condition belonging to a given polytopic set, an assigned bound for a given quadratic cost; the condition requires to solve a feasibility problem constrained by linear matrix inequalities. The proposed theory is then illustrated by an example concerning the design of optimal strategies for the removal of malicious software in computer networks.

[1]  F. Amato,et al.  Domain of attraction and guaranteed cost control for non-linear quadratic systems. Part 2: controller design , 2013 .

[2]  Weihai Zhang,et al.  Finite‐Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control‐Dependent Noise , 2013 .

[3]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

[4]  M. Tomizuka,et al.  Linear quadratic design of decoupled preview controllers for robotic arms , 1985 .

[5]  T. K. C. Peng,et al.  Adaptive Guaranteed Cost of Control of Systems with Uncertain Parameters , 1970 .

[6]  D. Gangsaas,et al.  Application of Modern Synthesis to Aircraft Control : Three Case Studies , 2001 .

[7]  V. A. Belyakov,et al.  Linear quadratic Gaussian controller design for plasma current, position and shape control system in ITER , 1999 .

[8]  O. Rössler.  Chaotic Behavior in Simple Reaction Systems , 1976 .

[9]  O. Patrick Kreidl,et al.  Feedback control applied to survivability: a host-based autonomic defense system , 2004, IEEE Transactions on Reliability.

[10]  T. Başar,et al.  Optimal and robust epidemic response for multiple networks , 2009 .

[11]  F. Kozin,et al.  Stability of the linear stochastic system , 1972 .

[12]  D. McFarlane,et al.  Optimal guaranteed cost control and filtering for uncertain linear systems , 1994, IEEE Trans. Autom. Control..

[13]  Alan J. Laub,et al.  The LMI control toolbox , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[14]  Kai Liu.  Stochastic Stability of Differential Equations in Abstract Spaces , 2022 .

[15]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[16]  Francesco Amato,et al.  Guaranteed cost control for uncertain nonlinear quadratic systems , 2014, 2014 European Control Conference (ECC).

[17]  Bruno Siciliano,et al.  Modelling and Control of Robot Manipulators , 1997, Advanced Textbooks in Control and Signal Processing.

[18]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[19]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[20]  S. Aachen.  Stochastic Differential Equations An Introduction With Applications , 2016 .

[21]  Francesco Amato,et al.  Domain of attraction and guaranteed cost control for non-linear quadratic systems. Part 1. Analysis , 2012 .

[22]  Vijay Vittal,et al.  Application of the normal form of vector fields to predict interarea separation in power systems , 1997 .

[23]  F. Amato,et al.  On the Region of Asymptotic Stability of Nonlinear Quadratic Systems , 2006, 2006 14th Mediterranean Conference on Control and Automation.

[24]  D. Gangsaas,et al.  Application of modem synthesis to aircraft control: Three case studies , 1986 .