On time-varying bit-allocation maintaining input-output stability: A convex parameterization

In [9], we constructed a parameterization of general time-varying quantizers. The construction is general in that it can have infinite memory and be time-varying in that the strategy it follows in allocating a total of R bits to its inputs, is a function of time. We derived sufficient conditions for input-output stability as functions of the quantizer's time-dependent bit-allocation strategy for bounded reference inputs. Our generalized construction of the quantizer also led to the result that the set of allocation strategies that maintains stability for bounded inputs is convex, allowing the search for the most efficient strategy to ensure stability to be formulated as a convex optimization problem. In this paper, we extend our stability analysis and derive sufficient conditions for decaying reference signals. We further show that the set of allocation strategies that maintains stability remains convex, as in the case for bounded inputs. We then compute optimal bit-allocation strategies for a class of finite-memory quantizers for various plant and controller pairs, and observe that the most efficient strategies are non-trivial and time-varying. Throughout, we consider a system in which the plant and feedback controller are separated by a noiseless finite-rate communication channel.