On Infimum Quantization Density for Multiple-input Systems

This paper deals with quadratic stabilization of discrete-time linear time-invariant systems, when the control is based on a static (or memoryless) quantized measurement of the state. A measure of quantization density is utilized in accordance with previous definitions in the literature. Based on this quantization density measure, the paper finds, for multiple-input systems that can be stabilized using a one-dimensional subspace of the input space, the infimum quantization density over all state quantizers that are quadratically stabilizing with respect to a given control Lyapunov function. This result is shown to differ from a previously published result. This discrepancy is explored by means of a numerical example that shows that, whereas the previously published result yields an inconsistent value of the density, our result provides a suitable one. The paper thus corrects the previously published result on infimum quantization density for a given control Lyapunov function, for the class of multiple-input systems considered.