Stabilization Of Networked Systems Based On Stochastic Control For Time-Delay Systems

This thesis mainly investigates the stabilization problem for networked control systems (NCSs) based on the stochastic control theory of time-delay system. NC-Ss are spatially systems that integrate sensors, controllers, actuators with network channels. Recently, with the rapid evolution of the internet technology, NCSs have attracted considerable attention. However, the limited communication resource in-troduces some inevitable problems, including transmission delay and data packet dropout. These transmission delay and packet dropout issues can degrade the per-formance of NCSs and even destabilize the whole system. In fact, the stabilization problems of the NCSs with simultaneous transmission delay and packet dropout are challenging, difficulty and unsolved. This can be ascribed to the fundamental diffi-culty of stochastic control, i.e., the separation principle does not hold. In this the-sis, by introducing delay-dependent algebraic Riccati equation (DARE) and delay-dependent Lyapunov equation (DLE), we give the necessary and sufficient stabiliz-ing conditions for the first time, which solves this stabilization problem completely. In the meantime, this thesis, based on the stochastic control theory for time-delay systems, has important theoretical value and practical significance for the study of network systems.The main academic contributions are listed as follows in the order of chapters:1. We study the stabilization problem for discrete-time stochastic system with input delay and multiplicative noise. On the one hand, using the Lyapunov stabil-ity theory and linear matrix inequalities (LMIs), we give the sufficient stabilizing condition. On the other hand, by constructing the coupled Lyapunov equations, we give the necessary stabilizing condition, which is the basis to establish the Lyapunov stabilizing criterion for the NCSs below. At last, with the augmented method, we prove that the original system is equivalent to some augmented delay-free stochastic system, and then give the necessary and sufficient stabilizing condition.2. We are concerned with the stabilization problem of NCSs with stochastic control theory for time-delay system. We investigate the stabilization problem for discrete-time NCSs with simulta-neous transmission delay and packet dropout, and present two different stabi-lizing conditions. One is the Riccati equation based stabilizing condition. It is shown that the NCS is stabilizable if and only if the DARE has a unique stabilizing solution. Moreover, we propose the analytical form of the optimal and stabilizing control, which is a function of the conditional expectation of the state. The other is the Lyapunov stabilizing criterion, which shows that the NCS is stabilizable if and only if the DLE has a positive-definite solution. This result is actually in accordance with the classical result for a delay-free system. We study the maximum packet dropout rate problem. Firs, by introducing a proper Lyapunov function, we obtain the existence theorem of the maximum packet dropout rate. Then, for the general case, on account of the Lyapunov stabilizing criterion, we derive a quasi-convex optimization algorithm to com-pute the value of the maximum packet dropout rate. Specifically, for the scalar case, the NCS is shown to be stabilizable if and only if the transmission delay and the packet dropout rate satisfy a simple algebraic inequality. If the trans-mission delay is known a priori, the maximum packet dropout rate is given explicitly in terms of the transmission delay and unstable eigenvalue of the system matrix. If the packet dropout rate is known a priori, the maximum al-lowable delay bound is also given explicitly, which is uniquely determined by the packet dropout rate and unstable eigenvalue of the system matrix. Consider the continuous-time NCSs with simultaneous transmission delay and signal attenuation, we derive the necessary and sufficient stabilizing condition-s. For the scalar case, we give the explicit value of the minimum mean-square capacity and maximum allowable delay bound.3. We investigate the stabilizing solution to the DARE with operator spectral theory. First, by introducing the delay-dependent Lyapunov operator and the notion-s of operator spectrum and spectral radius, we study the relationship between the stabilizing solution and maximum solution, which shows that the stabilizing solu-tion, if exists, is unique and coincides with the maximum solution. Then, using the semi-definite programming (SDP) theory, one LMI algorithm for computing the sta-bilizing solution is introduced. Second, we show the existence condition in terms of the delay-dependent Lyapunov operator and unobservable mean-square eigenvalue, under which the general DARE has a unique stabilizing solution. At last, we pro-pose the asymptotic behavior of the stabilizing solution and a convergence algebraic algorithm for computing its explicit value.The main academic innovations are as:For the NCSs with simultaneous trans-mission delay and packet dropout, by introducing the DARE, we propose the nec-essary and sufficient stabilizing condition for the first time, and give the analytical form of the optimal and stabilizing controller; We establish the Lyapunov stabiliz-ing criterion, which shows that the NCS is stabilizable if and only if the DLE has a positive-definite solution. Utilizing this result, we introduce a quasi-convex opti-mization algorithm to compute the value of maximum packet dropout rate; For scalar case, it is shown that the NCS is stabilizable if and only if the transmission delay and the packet dropout rate satisfy a simply algebraic inequality. Then, we give the ex-plicit value of the maximum packet dropout rate and the maximum allowable delay bound respectively; We investigate the stabilizing solution to the DARE and derive the existence condition in terms of the delay-dependent Lyapunov operator and un-observable mean-square eigenvalue, under which the general DARE has a unique stabilizing solution. Besides, we give the equivalent relation between the stabilizing solution and maximum solution.