On The Absolute Stability Approach To H_∞ Control Of Quantized Feedback Control Systems

There is an increasingly keen research interest in quantized feedback control systems, which is mainly the result of the wide application of digital computers in control systems and the development of network based control theory. In such systems, the measurement and control signals cannot be directly transmitted before being quantized which indicates that the data are only available with finite precision. Thus, analysis and design of quantized feedback control systems are of great theoretical and practical importance.Many approaches have been developed to handle problems in quantized feedback control systems. For stabilization of discrete-time single-input single-output linear systems, it has been shown that the coarsest quantizer that quadratically stabilizes such a linear system is logarithmic. The results, however, are difficult to extend to the multiple-input case. By noting that the quantization error can be treated as uncertainty or nonlinearity and can be bounded by a sector bound, a sector bound approach to quantized feedback control problem is proposed. The classical sector bound approach also proved to be nonconservative for the analysis and design problems and it is able to convert many quantized feedback design problems to the well-known robust control problems with sector bound uncertainties, which can be solved by tools for the standard control problem. Another feature of the sector bound approach is that there is no difficulty in extending the results to the multiple-input case. By recognizing that only quadratic Lyapunov function is used in the classical sector bound approach, a quantization-dependent Lyapunov function approach which can lead to less conservative results is proposed in the literature. The main difficulty coming from this approach is that it leads to a large number of linear matrix inequalities (LMIs). The quantized control problem has also been considered in these two frameworks. H_∞H_∞The performance analysis of quantized feedback control systems is revisited in this paper. By applying some intrinsic properties of the logarithmic quantizer and the sector bounded conditions, we present a new approach to H_∞H_∞performance analysis of control systems with input quantization. The novelty is that we have observed a very interesting geometric property of the logarithmic quantizer based on which a more general Popov-type Lyapunov function is constructed for the quantized feedback systems. The results are expressed in linear matrix inequalities and are valid for both single-input and multiple-input discrete-time linear systems with logarithmic quantizers. Our method is different from the quantization-dependent Lyapunov function approach in that the number of the LMIs in our new results increases linearly with the number of input channels.The proposed method can render generally less conservative results than those in the quadratic Lyapunov function framework and quantization-dependent Lyapunov function framework, which is shown both through theoretical analysis and numerical examples.