On design of finite-level quantization feedback control

This paper studies quantized feedback control of discrete-time linear systems using a finite-level quantizer. Motivated by the fact that most feedback communication channels allow a moderate bit rate, we are not particularly concerned with the problem of finding the minimum bit rate of feedback for a given control objective. Instead, we assume that a moderate bit rate is available. We introduce a dynamic scaling method and combine it with a known logarithmic quantization method. Using this approach, satisfactory control of linear systems can be achieved using a quantizer with a moderate number of quantization levels. Two main advantages of this approach are 1) it is very easy to implement, and 2) the closed-loop system behaves as if there was no limitation on the number of quantization levels when the state of the system is within a “normal” operating range. These features are important for practical applications of quantized feedback control. I. I NTRODUCTION Motivated by the fact that more and more control designs are implemented using digital communication links, a lot of research has been devoted to quantized feedback control in recent years; see, e.g., [1]-[14]. In quantized feedback control, the feedback signal is quantized and then coded for transmission. From the control design point of view, a fundamental problem is how to design a feedback controller and a quantizer jointly in order to achieve a certain control objective. Two kinds of quantizers can be deployed in quantized feedback design. A static quantizer is a memoryless nonlinear function, whereas a dynamic quantizer uses memory and thus can be much more complex and more powerful. Existing work using static quantizers includes, e.g., [1], [2], [3], [4], [5]. For quadratic stabilization of a linear syste m using state feedback, it is shown in [1] that the optimal static quantizer is alogarithmic quantizer. This result is generalized in [3] to a number of output feedback problems using a sector bound approach, where logarithmic quantizers are also shown to be optimal. When a dynamic quantizer is allowed, it is shown in [6] (also see [7]) that stabilization of a SISO LTI system (in some stochastic sense) can be achieved using only a finite number of quantization levels with the minimum number of quantization levels (also known as the minimum feedback information rate) explicitly related to the unstable poles of the system. Another type of dynamic quantizers Minyue Fu is with School of EE&CS, University of Newcastle, Callaghan, N.S. W. 2308 Australia (email: minyue.fu@newcast le.edu.au). Lihua Xie is with School of EEE, Nanyang Technological Unive rsity, Singapore (email: elhxie@ntu.edu.sg). The work of L. Xie was supported by Agency for Science, Techno logy and Research, Republic of Singapore under SERC Grant 052 101 0037. uses dynamic scaling in conjunction with a static quantizer . That is, the input signal is pre-scaled so that its range is more suitable for quantization. Noticeable work along this line includes [11]-[14]. In [11], it is pointed out that if a system is not excessively unstable, by employing a quantizer with various sensitivity a feedback strategy can be designed to bring the closed-loop state arbitrarily close to zero for an arbitrarily long time. The idea of quantizer with sensitivity is extended in [12] where it is shown that there exists a dynamic adjustment of the quantizer sensitivity and a quantized state feedback that asymptotically stabilizes the system. In the case of output feedback, a local (or semi-global) stabilization result is obtained. This paper is primarily inspired by the work of [6] but also motivated by its limitations. Although it is shown in [6] that stabilization of a linear system can be achieved by feedback with only a finite number of bits per sample and this number is typically very small, the encoder-decoder scheme used for proving this result is impractical and nonrobust. It is impractical because the encoder-decoder pair is a very nonlinear operator which would typically result in a large overshoot, and it is non-robust because even a very small amount of noise in the system can drive the closed-loop unstable. In this paper, we propose a simple dynamic scaling method for a logarithmic quantizer. A dynamic scaling factor is simply adjusted up or down depending whether the input signal to the quantizer is too “small” or too “large”. Using this dynamic scaling method, we show that a linear system can be asymptotically stabilized using a logarithmi c quantizer with only a finite number of quantization levels. This number turns out to be very moderate (typically a few bits to a few bytes) and is usually very compatible to the minimum information rate given in [6]. The main advantage of the proposed scheme is that the system behaves as if there were an infinite number of logarithmic quantization levels when the initial state is “moderate” in size, i.e., th e state would converge exponentially. Only when the initial state is very large, a transient period of overshoot can be present. The region of exponential convergence can be easily increased by using more quantization levels, and the number of feedback information bits grows only at a log(log(·)) rate when the size of this region increases. Since most digital communication channels can easily handle a few tens of bytes per sample, the proposed scheme should be very practical. Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24-28, 2006 FrP12.2