Stabilizing And Optimal Control For Linear Systems In The Presence Of Colored Multiplicative Noise With Applications To Networked Control Systems

Recently,stochastic noises are not at all new phenomena to the scientific and engineering communities.It is known that the white multiplicative noise is a class of important stochastic noises,which can be used to model,for example,the round-off error in floating point arithmetic numerical calculations.As a result,research on linear systems with white multiplicative noise is an important project for control theory,and a great number of research results on stability,optimal control,etc.,of the system has been reported.Powered by the computer and network technologies,a huge amount of research interests has been attracted on the networked control systems in which the control and feedback signals are transmitted by communication channels.Many studies of networked control have found that the white multiplicative noise is a powerful tool to model uncertainties induced by the communication channels,e.g.,erasure and possibly fading channels.However,there is no classical noise can be used to characterize the channel uncertainty induced by the random transmission delays.Therefore,a class of multiplicative uncertainties,which can be used to analytically model the channel with random transmission delays and packet dropouts,is introduced in the thesis.Then the stabilization and optimal control of linear feedback systems with such uncertainty is addressed.The main contents of the thesis are as follows.1)A class of stochastic systems with finite impulse response is proposed,followed by analyzing their properties in the perspective of impulse response.The proposed stochastic system is decomposed into two parts,namely a deterministic mean system and an induced uncertainty.With the uncertainty being characterized by the energy spectral density of its impulse response,an important concept named coefficient of frequency variation is introduced.The input-output property of the uncertainty is also derived in frequency domain.Moreover,when applying the proposed stochastic system to modeling communication channels,this work may be the first to obtain the frequency property of the random transmission delay channel.2)Consider the proposed stochastic system as a correlated(colored)multiplicative noise of a single-input control system,a small gain theorem which ensures the mean-square internal stability of the closed-loop system is proposed by means of spectrum analysis,which illustrates the significance of the coefficient of frequency variation.Based on the mean-square small gain theorem,the fundamental conditions of mean-square stabilizability are developed.These conditions,both necessary and sufficient,provides a fundamental limit imposed by the system’s unstable poles,nonminimum phase zeros and input delay(i.e.,relative degree),and the coefficient of frequency variation of the colored multiplicative noise.It also reveals how the interplay between the coefficient of frequency variation and the system’s unstable poles affects the mean-square stabilizability of the closed-loop system.Furthermore,the set of all linear controllers which stabilize the closed-loop system in the mean-square sense is provided based on the mean-square small gain theorem.3)An optimal control problem of a linear system with colored multiplicative noise is raised and,by virtue of the mean-square small gain theorem,converted into an optimal control problem of an auxiliary system with i.i.d.multiplicative noise,where the latter can be solved using a mean-square stabilizing solution of a modified algebraic Riccati equation(MARE).A necessary and sufficient condition for the existence of the stabilizing solution of the MARE is also presented.The result is applied to deal with the average control power minimization problem for networked feedback systems subject to erasure channel.Classical results on the mean-square stability conditions for such systems are recovered,and the explicit form of the optimal average control power for the networked control system is obtained.4)On the basis of the stability criterion for single-input systems,the mean-square small gain theorem for linear multi-input systems with structured colored multiplicative noise is provided.By virtue of controller parametrization,the mean-square stabilizability condition is provided,in which the interplay between the system’s characteristic and the coefficient of frequency variation of each noise is illustrated explicitly.