Optimal Relation Between Quantization Precision and Sampling Rates

In this chapter, we refine the model of resource-constrained systems to take into account quantization aspects and mainly focus on performance considerations in presence of information limitations. The communication constrains are modeled at the bit level, in bits per second. By using this modeling, we have to determine the control inputs that have to be updated as well as their quantization precision. In order to limit the inherent complexity of the proposed protocol, we suppose that the quantization precision choices of the input control signals belong to a reduced finite set. At each sampling period, quantization possibilities may be chosen from this set. In the case of allocation of communication resources based on the "per symbol" paradigm, the information exchange is modeled at the symbol level. The quantization of measurements and control commands is thus implicitly taken into account. The general model is given in Fig. 7.1.In this model, the communication channel can transmit at most R bits per time unit. Because of these resource limitations, measurements and control commands must be encoded (as a flow of symbols) before their transmission and decoded at their reception. Various coding techniques may be employed. A fundamental question is to determine the necessary and/or sufficient data-rate allowing the existence of a coder, a decoder and a controller that achieve the stabilization of the system.In general, increasing the sampling frequency improves the disturbance rejection abilities whereas increasing the quantization precision improves the steady state precision. However, when the bandwidth is limited, increasing the sampling frequency necessitates the reduction of the quantization precision. In the opposite, augmenting the quantization precision requires the lowering of the sampling frequency.Motivated by these observations, an approach for the dynamical on-line assignment of sampling frequencies and control inputs quantization is proposed. This approach, which is based on the model predictive control (MPC) philosophy, enables to choose the sampling frequency and the quantization levels of control signals from a predefined set, in order to optimize the control performance. Naturally, handling dynamically the quantization precision requires some communication resources and some extra computational resources. Consequently, we have to jointly handle the computational complexity, the protocol bandwidth consumption and performance[GRAPHICS]improvements. In order to limit the inherent complexity of the proposed protocol, we suppose that the quantization choices of the input control signals belong to a reduced finite set, which may be chosen by the designer in order to ensure the stability and to comply with the computational requirements. At each sampling period, quantization possibilities may be chosen from this set. This contrasts with the approach of [99], where the quantization precision of control signals is fixed. The proposed approach aims to capture the intuitive notion that high sampling rates improve the disturbance rejection and the transient behavior whereas the fine quantization improves the static precision near the origin [80]. The proposed method allows to dynamically choosing the pertinent control information to send, knowing the plant state and subject to the communication constraints.