Robust stability and stabilization of linear systems over networks

Abstract

A fundamental issue in networked control systems (NCSs) is stabilization with information constraints in the communication channels. This thesis proposes several novel methodologies for stability analysis and stabilization of linear systems under different channel models.

Firstly, a design method of output feedback controllers is proposed to stabilize a linear time-invariant (LTI) system over fading channels. It is shown that a sufficient and necessary condition for the existence of such controllers can be obtained by solving a convex optimization problem in the form of a semidefinite program. This condition is obtained by reformulating mean square stability as asymptotic stability of a suitable matrix comprising plant, controller and channel, and by introducing modified Hurwitz and Schur stability criteria.

Secondly, uncertainty in the plant is further investigated. Specifically, it is assumed that the plant is affected by polytopic uncertainty and is connected to the controller in a closed-loop over fading channels. First, it is shown that the robust stability in the mean square sense of the uncertain closed-loop NCS is equivalent to the existence of a Lyapunov function in a certain class. Second, it is shown that the existence of a Lyapunov function in such a class is equivalent to the feasibility of a set of linear matrix inequalities (LMIs). Third, it is shown that the proposed condition can be exploited for the synthesis of robust controllers ensuring robust stability of the closed-loop NCS.

Thirdly, an uncertain discrete-time NCS subject to input and output quantization and packet loss is studied. First, a necessary and sufficient condition in terms of LMIs is proposed for the quadratic stability of the closed-loop system with double quantization and norm bounded uncertainty in the plant. Moreover, it is shown that the proposed condition can be exploited to derive the coarsest logarithmic quantization density under which the uncertain plant can be quadratically stabilized via quantized state feedback. Second, a new class of Lyapunov function which depends on the quantization errors in a multilinear way is developed to obtain less conservative results. Lastly, the case with both quantization and packet loss is further considered.

In the end, this thesis investigates the minimum average transmit power required for mean-square stabilization of a discrete-time LTI system over a time-varying block-fading channel with additive white Gaussian noise (AWGN). It is assumed that the transmit power is allowed to vary with channel state. Both the case of independent and identically distributed (i.i.d.) fading and fading subject to a Markov chain are considered. Necessary and sufficient conditions for mean-square stabilization are derived based on information-theoretic and control-theoretic methods, and it is shown that the minimum average transmit power to ensure stabilizability can be obtained by solving a geometric program.

In each part, illustrative examples are provided to demonstrate the effectiveness of the proposed results.