Quantized stabilization of linear systems: complexity versus performance

Quantized feedback control has been receiving much attention in the control community in the past few years. Quantization is indeed a natural way to take into consideration in the control design the complexity constraints of the controller as well as the communication constraints in the information exchange be- tween the controller and the plant. In this paper, we analyze the stabilization problem for discrete time linear systems with multi- dimensional state and one-dimensional input using quantized feedbacks with a memory structure, focusing on the tradeoff between complexity and performance. A quantized controller with memory is a dynamical system with a state space, a state updating map and an output map. The quantized controller complexity is modeled by means of three indexes. The first index L coincides with the number of the controller states. The second index M is the number of the possible values that the state updating map of the con- troller can take at each time. The third index is the number  N of the possible values that the output map of the controller can take at each time. The index  N corresponds also to the number of the possible control values that the controller can choose at each time. In this paper, the performance index is chosen to be the time T needed to shrink the state of the plant from a starting set to a target set. Finally, the contraction rate C, namely the ratio between the volumes of the starting and target sets, is introduced. We evaluate the relations between these parameters for various quantized stabilizers, with and without memory, and we make some comparisons. Then, we prove a number of results showing the intrinsic limitations of the quantized control. In particular, we show that, in order to obtain a control strategy which yields arbitrarily small values of T/ln C (requirement which can be interpreted as a weak form of the pole assignability property), we need to have that LN/ln C       is big enough.