Towards a packet-based control theory - Part I: Stabilization over a packet-based network

In this paper, we study the classical problem of stabilizing a Linear Time Invariant (LTI) system in a packet-based network setting. We assume that the LTI system is unstable but both controllable and observable. The state information is transmitted to the controller over a packet-based network. We also assume that there is a perfect link from the controller to the plant. We give a set of sufficient conditions under which the system can be stabilized for a given data rate C. In particular, these conditions can yield an upper bound on the minimum C for which the system can be stabilized. A recursive encoding-decoding scheme and an associated control law are proposed to achieve stability for rate exceeding this bound. An optimal bit allocation problem is investigated in which we ask about how to allocate the bits in a single packet for a subsystem of a general LTI system such that a minimum upper bound on the data rate is achieved. We then formulate the optimal bit allocation problem as a Linear Matrix Inequality (LMI) optimization problem which can be solved efficiently using standard Semi-definite Programming (SDP) solvers. Examples and simulations are given to demonstrate the results.