Study On Control And Quantization For Linear Systems With Multiplicative Noise

The control problem for stochastic uncertain systems with multiplicative noises havebeen fnding many applications in a broad range such as aerospace, mechanical, chemicalreactor and economic systems etc. For linear systems, the diferences between the mul-tiplicative and additive noises are: due to the existence of the former, stability for thesystems can be undermined, moreover, nonlinearity property is presented. As a conse-quence, the control problem with multiplicative noises is more complicated than additivenoise(for example linear quadratic Gaussian (LQG) problem), some issues in stabiliza-tion and optimal control problems for such systems are still opened. Therefore, duringlast four decades, control problem for stochastic uncertain systems with multiplicativenoises have attracted many research interests. In particular, the recent development innetworked control area shows that the multiplicative noise model may be an efcien-t way in modeling channel uncertainties, such as, quantization errors, packet loss, theconstraints on signal-to-noise ratios and bandwidth limits etc. Thus, corresponding net-worked control systems(NCSs) are equivalently converted into stochastic control systemswith multiplicative noises, furthermore, methods of analysis and design for stochasticcontrol theory are available to such systems. Up to now, Optimal design problem viastate feedback For linear stochastic control systems with state and control multiplicativenoises stay unsettled as a great challenge. Moreover, for quantized control systems underthe given fnite bit rate, by using dynamic quantizer, most of the existing works concen-trated on the quantized feedback stabilization for deterministic and stochastic systemswith additive noises. However, the works related to stochastic control systems with mul-tiplicative noises are in its beginning stage, and static quantizer mainly is confned toscalar linear systems. Hence, the existing control and quantization problems which havebeen presented above for stochastic systems, remain to be perfected further.The dissertation studies control and quantization problems, in which stochasticcontrol and quantization theory are adopted. The main contents of this dissertationare summarized as follows: the frst part presents an penetrating overview for lineardiscrete-time stochastic control systems with multiplicative noises. The second part s-tudies mean-square stabilization and optimal control problems via state feedback for lin-ear discrete-time systems with state and control multiplicative noises. In the third part,based on quantization and encoding theory, our work focuses on the smallest feedbackdata rate above which a given dynamical stochastic systems with multiplicative noises can be stabilized. In the fourth part, a static periodic quantization scheme which solvesfeedback stabilization problem under the smallest data rate is proposed.Firstly, a survey for linear stochastic systems with multiplicative noises is presentedby following the latest development in networked control feld. In the existing works,there were few papers which are concerned with overview for such systems, due to theimportance for this feld, it is necessary for us to present a survey. we review several recentresults on stability analysis, mean-square stabilization, optimal control and estimationfor linear stochastic uncertain systems with multiplicative noises in a systematic manner.For some problems, pose some new viewpoints and clarify the intrinsic relation betweenthem. Besides, state feedback design algorithms are investigated.Secondly, the optimal design problems via state feedback for linear discrete-timemultivariable systems with multiplicative noises is solved thoroughly, a sufcient andnecessary condition for the optimal design via state feedback is presented. Concretelyspeaking, we frst show that in general the optimal mean-square stabilization problemvia state feedback amounts to solving a generalized eigenvalue problem, i.e., a globaloptimal solution exists and can be solved by a set of linear matrix inequalities(LMI) andline search technique. Next, a sufcient and necessary condition for the optimal controlstabilizing systems is given. Finally, we formulated H2optimal design via state feedbackfor multivariable discrete-time system with control multiplicative noises, It is shown thatthe optimal mean-square stabilization problem is equivalent to a class of optimal mean-square stabilization problem, therefore, the H2optimal control problem can also be solvedby a generalized eigenvalue approach.And then, by adopting upper-bound sequence encoding scheme, feedback stabiliza-tion problems under fnite data rate is studied for a class of second order stochasticsystems with structural multiplicative noises, a sufcient and necessary condition whichstabilizes systems is obtained, from which we have: not only unstable eigenvalue forsystems matrix, channel uncertainties and noises variance but also included angle ofeigenvector are related to the smallest feedback data rate above which a given dynamicalstochastic systems can be stabilized. Furthermore, by applying adaptive quantizationscheme, a sufcient condition which stabilize scalar linear stochastic systems with multi-plicative noises is derived.Finally, a static periodic quantization scheme diferent from before is presented,smallest feedback data rate above which a given second order systems can be stabilizedin practical stability sense is obtained, from which we can see: smallest feedback data rateis piecewise function of unstable eigenvalue for systems matrix. By using algebraic and geometric methods, respectively, the optimization problems for controller which satisfesspecifc conditions are solved.