A New Framework for Stability Analysis of Quantized Feedback Systems

A quantized feedback system is a control system in which finite-level quantization of signal values is involved in the feedback loop. Quantized feedback is found in many engineering systems including mechanical systems and networked systems because multilevel-valued devices such as A/D (Analog-toDigital) converters, on/off switching actuators, and digital communication networks are widely used in these systems. The purpose of this thesis is to establish a unified approach for the stability analysis of quantized feedback systems. In this thesis, we first provide the motivation for developing a new framework for the stability analysis of quantized feedback systems. The stability analysis of quantized feedback systems is difficult in many cases because the traditional small gain theorem cannot be successfully applied owing to the nonlinearities caused by the finite-level quantization. This thesis shows the difficulties involved by using two examples: an uncertain networked control system with a rate-limited communication channel and a feedback system involving a uniform quantizer. Through the stability analysis of these examples, we discuss the need for introducing a new notion of stability and a new framework for analysis. A new framework for the stability analysis of quantized feedback systems is then developed. In particular, we introduce a new notion of small l signal l stability in this thesis. This is a practical and reasonable notion for the stability of quantized feedback systems; in contrast, classical stability notions such as finite gain l stability and asymptotic stability are occasionally too strong and not achievable in the presence of finite-level quantization of signals. The small level theorem is derived to give a sufficient condition for a feedback system to be small l signal l stable. A new class of uncertainty, level bounded uncertainty, is also introduced in this thesis. This is useful in approximating some classes of nonlinearities that include quantization errors. Using all these new notions and theorems, we provide a mathematical framework for the stability analysis of nonlinear systems that is applicable to a wide class of quantized feedback systems. i Finally, the use of the proposed framework is demonstrated by addressing two important issues related to quantized feedback systems: robust stabilization of an uncertain networked control systems over a rate-limited communication channel and stability analysis of a networked control system that is affected by finite-level quantization and packet dropouts. In the first example, quantitative analysis is provided for the combined effect of quantization and the model uncertainty in the system dynamics on the stability of the entire networked control system. In the second example, we elucidate the effect of quantization and packet dropouts on the stability of the overall networked control system. While the quantitative analyses of the stabilities of these two networked control systems have remained unresolved problems, the proposed framework provides us with a systematic approach for solving both of these important issues.