Stability and equilibria of linear control systems under input and measurement quantization

Scope and method of study. The problems of characterization of equilibria and stability analysis of a class of systems with quantization are treated in this work. System configurations considered include the main cases of open-loop stable linear plants under full state feedback, and under output feedback with dynamic compensation. The state feedback case is divided into sub-cases according to the type of quantization present in the system. The theoretical tools most predominantly used are those of Absolute Stability and Discrete Positive Real Theory. Standard results from these theories are expanded and modified to suit the needs of the particular problems. Standing assumptions include the open-loop stability of the plant and controller, in addition with properness conditions in specific cases.; Findings and conclusions. The sub-case of quantized input with precise state measurements, termed QI case, is amenable to explicit solution of the equilibrium equations. This knowledge is used in obtaining a necessary and sufficient condition for the origin to be the only equilibrium point. The stability problem in QI systems is analyzed directly using available tools of Absolute Stability and Discrete Positive Real theory. The main contribution to the stability analysis of QI systems is a parameterization of stabilizing feedback gains. For unstable continuous-time systems, a modified quantized feedback law is considered that can stabilize the system at the expense of chattering control. The equilibrium equations for the sub-case of quantization at the input and the state measurements, denoted QIQM, do not have a closed-form solution. A graphical construction is proposed that can be used in finding all equilibrium solutions of a QIQM system of arbitrary order. The stability problem cannot be directly analyzed using the standard tools of DPR theory or Absolute Stability. A system transformation is introduced that puts the system in a form similar to the Luré problem, where the sector nonlinearity is multiplicatively perturbed by a bounded function of the state. A result stating conditions for the stability of such systems is developed, and its use is not limited to systems with quantization. The stability analysis of QIQM systems culminates in a simple stability test in the frequency domain. The design problem in QIQM systems remains difficult, and only a method of gain scaling is presented. It is also shown that the parametric behavior of the system with respect to changes in gain scaling displays bifurcations. The sub-case of quantized input with precise output measurement and dynamic compensation, called QI0, reduces to it state-space counterpart, QI. The same is true for systems with no input quantization and quantized output feedback, termed IQO. The case of quantization at plant input and output, called QIQO, is more difficult to analyze. The equilibrium equations do no, have a closed-form solution, thus only an upper bound on the number of solutions given, along with a sufficient condition for the origin to be only equilibrium point is given. The stability analysis has been carried out by means of the Small Gain Theorem.