Analysis And Design For Linear Discrete-time Control System With Network-induced Multiplicative Noises

Compared to typical linear certain systems, the linear systems with multiplicative noises, as a special kind of stochastic systems, probably represent a broader category of practical processes, and their control problem has been found many applications in a broad range such as aerospace, chemical reactor, economic systems and mechanical etc. For linear systems, due to the existence of multiplicative noises, stability can be tremendously undermined. Moreover, nonlinearity property is also introduced to those systems. As a consequence, control problem for those systems with multiplicative noises is more complicated than that for systems with additive noises. So far, some issues on stabilization and optimal control problem for them are still opened. Therefore, during last decades, control problem for stochastic uncertain systems with multiplicative noises have attracted many research interests. In particular, the recent development in networked control area shows that the multiplicative noise model may be an efficient way in modeling communication channel uncertainties, such as packet loss, quantization errors, channel fading, the constraints on signal-to-noise ratios and bandwidth limits etc. Thus, by this model we can equivalently convert networked control systems into stochastic uncertain systems with network-induced multiplicative noises. Furthermore, existing analysis and design methods for stochastic control theory are available to such systems. In this thesis, we study the control problem for linear discrete-time systems with network-induced multiplicative noises, including mean-square stability, mean-square stabilization, the H2 optimal control in mean-square sense as well as LQR optimal quadratic regulator design, mean-square detectability of multi-output systems and discrete-time periodic square wave signal optimal tracking, etc. This study focuses mainly on the following five aspects:Firstly, stability for networked control systems with quantized control signal is studied. A typical configuration of networked feedback control systems is considered, in which the control signal related to the output of the plant is sent to the plant via network. The network under consideration is modeled as the sum of a logarithmic quantizer and a noise free communication channel and the quantization error is formulated as a white noise process. Thus the quantized control system under consideration is actually a linear stochastic system with network-induced multiplicative noises.Different from the existing works, the output-feedback controller in this study is formulated by Youla parameterization via coprime factorization of given system to be controlled in time domain. Following the stochastic small gain theorem, a sufficient and necessary condition for second-order stochastic input-output stability is proposed. This result gives an upper bound for the variance of quantization error and its value is only determined by the unstable poles of given plant.Secondly, mean-square stabilization via output feedback for a networked MIMO feedback system over a packet dropping communication channel which consists of several parallel sub-channels is studied. In our work, we focus on two special cases:One is that the plant is a non-minimum phase plant with relative degree one and each non-minimum phase zero is associated with one control input channel of the plant. The other is that the plant is a minimum phase plant with relative degree one. For the former case, the output-feedback controller is formulated by Youla parameterization via a new and so-called upper triangular coprime factorization of given system to be controlled, a sufficient condition in mean-square stabilization problem is presented in terms of the interaction between channel’s SNR and characteristics of the plant. That is, a critical value (lower bound) for each subchannel’s capacity is requested in mean-square stabilization via output feedback. For the latter case, the minimum total channel capacity for mean-square stabilization is given and its allocation to sub-channels is expressed. The result shows that, for this case, the total capacity for mean-square stabilization via output feedback must be greater than a minimum value which is only determined by the product of the plant’s unstable poles.Thirdly, optimal H2 control design and LQR optimal regulation problem for networked linear discrete-time feedback control system with state and input control dependent multiplicative noises are studied. The quantization errors and/or uncertainties of communication channels are modeled as multiplicative noises. By this model, we can convert networked feedback control system with quantization effects (or input multiplicative noises) into stochastic uncertain system with network-induced multiplicative noises. For optimal H2 control design, following the stochastic small gain theorem, the necessary and sufficient condition for the existence of positive semi-definite solution to a modified algebraic Riccati equation (MARE), which is determined the state feedback gain of optimal controller, is presented. Simultaneously, a design method for optimal state-feedback-gain-coefficient is formulated. For LQR optimal regulation problem, taking a quadratic cost function as measured performance of system, and similarly following the stochastic small gain theorem, the necessary and sufficient condition for the existence of positive semi-definite solution to a modified algebraic Riccati equation (MARE), which is determined the state feedback gain of optimal regulation controller, is presented. Simultaneously, a design method for optimal state-feedback-gain-coefficient is formulated. Finally, numerical simulations of the first and the second corollary are carried out and their results demonstrate the validity of design method for optimal state-feedback-gain-coefficient.And then, mean-square detectability of discrete-time multi-output systems over stochastic multiplicative channels is studied, where a multiplicative white noise model the unreliability of output channels. Based on the bisection technique, we obtain the critical value (lower bound) of mean-square capacity for guaranteeing the mean-square detectability for the single packet case, and give a sufficient and necessary condition on overall mean-square capacity for mean-square detectability in terms of the Mahler measure or topological entropy of the plant for the m-parallel multiple packets transmission strategy, under the assumption that the given network resource can be allocated among all the output channels. Finally, Applications in erasure-type channel and channel with stochastic sector-bounded uncertainty are provided to demonstrate the results which are consistent with the existing literature. It is turned out that, like classic control systems, the relationship between mean-square stabilization and detectability for networked control systems remain dual.Finally, the tracking problem in linear time invariant SISO discrete-time networked control systems with periodic signal reference input is studied. The most important difference from existing works on the tracking problem is that the reference input signal considered in this study is not the common non-periodic signal (e.g., a step signal) considered in most of the existing works, but a discrete-time periodic signal, whose waveform shapes is same and repeat in each period, accordingly the power of input signal in each period is fixed. So we study the system response to the power of input signal based on power spectrum. The tracking performance is measured by the power of the tracking error between the plant output and the reference, and the optimal tracking performance is thus measured by the mean power of the tracking error. In the networked control systems under consideration, packet dropouts may be only exist in the forward communication channel. And the packet dropouts may be look as a mixture of deterministic signal and stochastic signal, thus the error caused by packet dropouts is assumed to be a product of the original signal and a white noise. By applying Parseval identity and Wiener-Khinchin theorem as well as norm matrix theory, the lower bound of the performance in tracking is derived in terms of the characteristics of the plant and the uncertain type of signal. Numerical simulation results demonstrate the effectiveness in tracking periodic signal under the control of optimal tracking controller presented in this study. Consequently, it demonstrates the correctness of our result. Finally, the impact of the poles of plant Gc, the the poles of controller Kf and fundamental period N on the tracking performace (tracking error) is studied.