Stabilization of linear systems with limited information

Abstract

In this paper, we show that the coarsest, or least dense, quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulator (LQR) problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized state estimators. This leads to the design of hybrid output feedback controllers. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We generalize the definition of density of quantization to the density of sampling and quantization in a natural way, and search for the coarsest sampling and quantization scheme that ensures stability. We show that the resulting optimal sampling time is only function of the sum of the unstable eigenvalues of the continuous time system, and that the associated optimal quantizer is logarithmic with the logarithmic base being a universal constant independent of the system. The coarsest sampling and quantization scheme so obtained is related to the concept of minimal attention control recently introduced by Brockett. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system. This final result provides a way to practically implement the theory developed in this paper.

Capacity of Communication Channels

“…Due to the enormous growth in communication technology, there has been a significant interest in the problem of control and state estimation via limited capacity communication channels in recent years (see, e.g., Delchamps [1990], Brockett and Liberzon [2000], Elia and Mitter [2001], Petersen and Savkin [2001], Ishii and Francis [2002], Liberzon [2003], Savkin and Petersen [2003], De Persis and Isidori [2004], Nair and Evans [2004], Matveev and Savkin [2005] Savkin [2007, 2008]). Minimum capacity of the communication channels required for state estimation and control has been investigated in, e.g., Nair and Evans [2004], Savkin [2006].…”

Networked stabilization for multi-input systems over quantized fading channels

“…We assume that A has no eigenvalues on the unit circle throughout the paper. If A has eigenvalues on the unit circle, a standard trick (see for example [4,7,13]) can be used to modify the proof. The inequality (22) can be written as…”

“…Different from [4,7], the sector bounded uncertainty will be assumed to be nonlinear time-varying and dynamic (NTVD) with the unique equilibrium point at the origin having norm bounds (under zero initial condition)…”

“…Both have been investigated extensively. Elia and Mitter are the first to study logarithmic quantization [4], who obtain the quadratic stabilizability for SI systems under state feedback. The same problem is studied in [7,9] for MI systems, and a nice analytic solution is obtained in [13].…”

Joint Optimization of Wireless Communication and Networked Control Systems

“…For example, [WB99] and [NE98] established various closed-loop stability conditions involving the feedback data rate and eigenvalues of the openloop system, while [BM97,TSM98] studied control with communication constraints within the classical linear quadratic Gaussian framework. Closely related is also the research on control with quantized feedback information, see [Cur70,Del90,KH94,BL00,EM01].…”