Hybrid feedback stabilization of systems with quantized signals

Abstract

This paper is concerned with global asymptotic stabilization of continuous-time systems subject to quantization. A hybrid control strategy originating in earlier work (Brockett and Liberzon, IEEE Trans. Automat. Control 45 (2000) 1279) relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system, the measured output, or the control input. The analysis involves merging tools from Lyapunov stability, hybrid systems, and input-to-state stability.

Introduction

In the classical feedback control setting, the output of the process is assumed to be passed directly to the controller, which generates the control input and in turn passes it directly back to the process. In practice, however, this paradigm often needs to be re-examined because the interface between the controller and the process features some additional information-processing devices. These considerations arise, for example, in networked control systems; see the articles in Bushnell (2001) and the references therein.

One important aspect to take into account in such situations is signal quantization. We think of a quantizer as a device that converts a real-valued signal into a piecewise constant one taking on a finite set of values. Quantization may affect the process output (this happens, for example, when the output measurements to be used for feedback are obtained by using a digital camera, stored in the memory of a digital computer, or transmitted over a digital communication channel). It may also affect the control input (examples include the standard PWM amplifier and the manual transmission on a car).

We assume that the given system evolves in continuous time. In the presence of quantization, the state space (or the input space) of the system is divided into a finite number of quantization regions, each corresponding to a fixed value of the quantizer. At the time of passage from one quantization region to another, the dynamics of the closed-loop system change abruptly. Therefore, systems with quantization can be naturally viewed as hybrid systems, i.e., systems described by a coupling between continuous and discrete dynamics.

There are two well-studied phenomena which account for changes in the system's behavior caused by quantization. The first one is saturation: if the signal is outside the range of the quantizer, then the quantization error is large, and the control law designed for the ideal case of no quantization leads to instability. The second one is deterioration of performance near the equilibrium: as the difference between the current and the desired values of the state becomes small, higher precision is required, and so in the presence of quantization errors asymptotic convergence is impossible. These phenomena manifest themselves in the existence of two nested invariant regions such that all trajectories of the quantized system starting in the bigger region approach the smaller one, while no further convergence guarantees can be given.

A standard assumption made in the literature is that parameters of the quantizer are fixed in advance and cannot be changed by the control designer; see, among many sources, (Chou, Chen, & Horng, 1996; Delchamps, 1990; Raisch, 1995; Feng & Loparo, 1997; Sur & Paden, 1998; Lunze, Nixdorf, & Schröder, 1999). There has been some research concerned with the question of how the choice of quantization parameters affects the behavior of the system (Wong & Brockett, 1999; Åström & Bernhardsson, 1999; Liberzon & Brockett, 2000; Elia & Mitter, 2001; Ishii & Francis, 2002). In this paper, building on the earlier work reported in (Brockett & Liberzon, 2000; Liberzon, 2000), we adopt the approach that it is possible to vary some parameters of the quantizer in real time, on the basis of collected data. For example, if a quantizer is used to represent a camera, this corresponds to zooming in and out, i.e., varying the focal length, while the number of pixels of course remains fixed. This approach fits into the framework of control with limited information: the state of the system is not completely known, but it is only known which one of a fixed number of quantization regions contains the current state at each instant of time. The quantizer can be thought of as a coder that generates an encoded signal taking values in a given finite set. By changing the size and relative position of the quantization regions—i.e., by modifying the coding mechanism—we can learn more about the behavior of the system, without violating the restriction on the type of information that can be communicated to the controller. This will help us overcome the two difficulties described above.

The quantization parameters will be updated at discrete instants of time (these switching events will be determined by the values of a suitable Lyapunov function). This results in a hybrid quantized feedback control policy. There are several reasons for adopting a hybrid control approach rather than varying the quantization parameters continuously. First, in specific situations there may be some constraints on how many values these parameters are allowed to take and how frequently they can be adjusted. Thus a discrete adjustment policy is more natural and easier to implement than a continuous one. Secondly, the analysis of hybrid systems obtained in this way appears to be more tractable than that of systems resulting from continuous parameter tuning. In fact, we will see that invariant regions defined by level sets of a Lyapunov function provide a simple and effective tool for studying the behavior of the closed-loop system. This also implies that precise computation of the switching times is not essential, which makes our hybrid control policies robust with respect to certain types of time delays (such as those associated with periodic sampling).

The recent paper (Brockett & Liberzon, 2000) thoroughly investigates the hybrid control methodology outlined above in the context of the feedback stabilization problem for linear control systems with output (or state) quantization. It is shown there that if a linear system can be stabilized by a linear feedback law, then it can also be globally asymptotically stabilized by a hybrid quantized feedback control policy. The control strategy is usually composed of two stages. The first, “zooming-out” stage consists in increasing the range of the quantizer until the state of the system can be adequately measured; at this stage, the system is open-loop. The second, “zooming-in” stage involves applying feedback and at the same time decreasing the quantization error in such a way as to drive the state to the origin. The developments of Brockett and Liberzon (2000) were restricted to quantizers that give rise to rectilinear quantization regions.

The present work generalizes the contributions of Brockett and Liberzon (2000) in three different directions. First, we consider more general types of quantizers, with quantization regions having arbitrary shapes as in Lunze et al. (1999). This extension is useful in several situations. For example, in the context of vision-based feedback control mentioned earlier, the image plane of the camera is divided into rectilinear regions, but the shapes of the quantization regions in the state space which result from computing inverse images of these rectangles can be rather complicated. The so-called Voronoi tessellations suggest that, at least in two dimensions, it may be beneficial to use hexagonal quantization regions rather than more familiar rectangular ones (Du, Faber, & Gunzburger, 1999). We will demonstrate that the principal findings of Brockett and Liberzon (2000) are still valid in this more general setting.

Another goal of this paper is to address the quantized feedback stabilization problem for nonlinear systems. It can be shown via a linearization argument that by using the approach of Brockett and Liberzon (2000) one can obtain local asymptotic stability for a nonlinear system, provided that the corresponding linearized system is stabilizable (Hu, Feng, & Michel, 1999). Here we are concerned with achieving global stability¹ results. We will show that the techniques developed in Brockett and Liberzon (2000) can be extended in a natural way to those nonlinear systems that are input-to-state stabilizable with respect to measurement disturbances. We thus reveal an interesting interplay between the problem of quantized feedback stabilization, the theory of hybrid systems, and topics of current interest in nonlinear control design. A preliminary investigation of these questions has been reported in Liberzon (2000), but only for state quantizers with rectilinear quantization regions.

Finally, in this paper we develop analogous results for systems with input quantization, both linear and nonlinear. In view of the examples given earlier, this expands the potential applicability of the hybrid quantized feedback control techniques. We discover that the analysis of systems with input quantization can be carried out quite similarly to the state quantization case. This analysis also yields a basis for comparing the effects of input quantization and state quantization on the performance of the system. As we will see, for nonlinear systems the case of state quantization presents a greater challenge from the viewpoint of control design.

As in Brockett and Liberzon (2000), all control laws are constructed explicitly. All vector fields and feedback laws are assumed to be sufficiently regular (e.g., smooth). Solutions to all differential equations are well defined, with the understanding that they are to be interpreted in the sense of Filippov (1988) if necessary (control strategies described in this paper do not rely on chattering and the analysis of the resulting closed-loop systems does not explicitly require a concept of generalized solution). With some abuse of terminology, we call a closed-loop hybrid system globally asymptotically stable if the origin is a globally asymptotically stable equilibrium of the continuous dynamics. We denote by |·| the standard Euclidean norm in $\text{R}{}^{\mathrm{n}}$ and by ||·|| the corresponding induced matrix norm in $\text{R}{}^{\mathrm{n\times n}}$. A function $\text{α}\text{:}\text{[0,∞)→[0,∞)}$ is said to be of class$\text{K}{}_{\mathrm{\infty }}$ if it is continuous, strictly increasing, and such that α(0)=0 and α(r)→∞ as r→∞.