On the Stabilization of Linear Systems Under Assigned I/O Quantization

Abstract

This paper is concerned with the stabilization of discrete-time linear systems with quantization of the input and output spaces, i.e., when available values of inputs and outputs are discrete. Unlike most of the existing literature, we assume that how the input and output spaces are quantized is a datum of the problem, rather than a degree of freedom in design. Our focus is hence on the existence and synthesis of symbolic feedback controllers, mapping output words into the input alphabet, to steer a quantized I/O system to within small invariant neighborhoods of the equilibrium starting from large attraction basins. We provide a detailed analysis of the practical stabilizability of systems in terms of the size of hypercubes bounding the initial conditions, the state transient and the steady state evolution. We also provide an explicit construction of a practically stabilizing controller for the quantized I/O case

Left invertibility of discrete systems with finite inputs and quantized output

“…Quantized control systems have been attracting increasing attention of the control community in recent years (see [7], [27], [21] and references therein). The mathematical operation of quantization and the possibility of considering only finite inputs occurs in many communication and control systems.…”

Global state feedback stabilisation for a class of high-order nonlinear systems with quantised input and state

“…Recently, some corresponding works on input and output quantised feedback systems have been presented (see e.g. Coutinho et al, 2010;Ishii & Bassar, , 2005;Picasso & Bicchi, 2007;Richter & Misawa, 2003;Xia, Yan, Shi, & Fu, 2013;Yue, Peng, & Tang, 2006). In Richter and Misawa (2003), using a uniform quantiser, a global asymptotic stability condition was obtained for linear discrete time systems with quantised input and state measurements.…”

“…The problem of designing quantisers for the stabilisation of a remote linear control system was studied in Ishii and Bassar (2005), in which the sensor and the actuator are connected to the controller over a shared network. Nevertheless, all the systems considered in Richter and Misawa (2003), Coutinho et al (2010), Yue et al (2006), Picasso and Bicchi (2007) and Ishii and Bassar (2005) are linear and completely known or suffer from uncertainties composed of external disturbances only. In practice, it is often required to consider the case when the plant to be controlled is nonlinear and uncertain.…”

Left invertibility of output-quantized systems: An application to cryptography

“…In quantized systems the hybrid nature is given by the existence of both continuous variables (state-space variables), and discrete valued variables (input and output variables). Quantized control systems have been attracting increasing attention in recent years ( [1,3,7,21,25]). The mathematical operation of quantization and the possibility of considering only finite inputs have practical and technological motivations in control with discrete sensors and/or actuators and control under communication constraints (in large-scale systems): see for example [2,6,15,26,27] and reference therein.…”