Limitations for nonlinear stabilization over uncertain channels

We study the problem of mean-square exponential incremental stabilization of nonlinear systems over uncertain communication channels. We show the ability to stabilize a system over such channels is fundamentally limited and the channel uncertainty must provide a minimal Quality of Service (QoS) to support stabilization. The smallest QoS necessary for stabilization is shown as a function of the positive Lyapunov exponents of uncontrolled nonlinear systems. The positive Lyapunov exponent is a measure of dynamical complexity and captures the rate of exponential divergence of nearby system trajectories. One of the main highlights of our results is the role played by nonequilibrium dynamics to determine the limitation for incremental stabilization over networks with uncertainty

Data Rate Limitations for Stabilization of Uncertain Systems over Lossy Channels

This paper considers data rate limitations for mean square stabilization of uncertain discrete-time linear systems via finite data rate and lossy channels. For a plant having parametric uncertainties, a necessary condition and a sufficient condition are derived, represented by the data rate, the packet loss probability, uncertainty bounds on plant parameters, and the unstable eigenvalues of the plant. The results extend those existing in the area of networked control, and in particular, the condition is exact for the scalar plant case

Random-Time, State-Dependent Stochastic Drift for Markov Chains and Application to Stochastic Stabilization Over Erasure Channels

It is known that state-dependent, multi-step Lyapunov bounds lead to greatly simplified verification theorems for stability for large classes of Markov chain models. This is one component of the "fluid model" approach to stability of stochastic networks. In this paper we extend the general theory to randomized multi-step Lyapunov theory to obtain criteria for stability and steady-state performance bounds, such as finite moments. These results are applied to a remote stabilization problem, in which a controller receives measurements from an erasure channel with limited capacity. Based on the general results in the paper it is shown that stability of the closed loop system is assured provided that the channel capacity is greater than the logarithm of the unstable eigenvalue, plus an additional correction term. The existence of a finite second moment in steady-state is established under additional conditions.Comment: To appear in IEEE Transactions on Automatic Contro

Optimal Causal Rate-Constrained Sampling of the Wiener Process

We consider the following communication scenario. An encoder causally observes the Wiener process and decides when and what to transmit about it. A decoder makes real-time estimation of the process using causally received codewords. We determine the causal encoding and decoding policies that jointly minimize the mean-square estimation error, under the long-term communication rate constraint of R bits per second. We show that an optimal encoding policy can be implemented as a causal sampling policy followed by a causal compressing policy. We prove that the optimal encoding policy samples the Wiener process once the innovation passes either √(1/R) or −√(1/R), and compresses the sign of the innovation (SOI) using a 1-bit codeword. The SOI coding scheme achieves the operational distortion-rate function, which is equal to D^(op)(R)=1/(6R). Surprisingly, this is significantly better than the distortion-rate tradeoff achieved in the limit of infinite delay by the best non-causal code. This is because the SOI coding scheme leverages the free timing information supplied by the zero-delay channel between the encoder and the decoder. The key to unlock that gain is the event-triggered nature of the SOI sampling policy. In contrast, the distortion-rate tradeoffs achieved with deterministic sampling policies are much worse: we prove that the causal informational distortion-rate function in that scenario is as high as D_(DET)(R)=5/(6R). It is achieved by the uniform sampling policy with the sampling interval 1/R. In either case, the optimal strategy is to sample the process as fast as possible and to transmit 1-bit codewords to the decoder without delay

Optimal Causal Rate-Constrained Sampling of the Wiener Process

We consider the following communication scenario. An encoder causally observes the Wiener process and decides when and what to transmit about it. A decoder makes real-time estimation of the process using causally received codewords. We determine the causal encoding and decoding policies that jointly minimize the mean-square estimation error, under the long-term communication rate constraint of $R$ bits per second. We show that an optimal encoding policy can be implemented as a causal sampling policy followed by a causal compressing policy. We prove that the optimal encoding policy samples the Wiener process once the innovation passes either $\sqrt{\frac{1}{R}}$ or $-\sqrt{\frac{1}{R}}$, and compresses the sign of the innovation (SOI) using a 1-bit codeword. The SOI coding scheme achieves the operational distortion-rate function, which is equal to $D^{\mathrm{op}}(R)=\frac{1}{6R}$. Surprisingly, this is significantly better than the distortion-rate tradeoff achieved in the limit of infinite delay by the best non-causal code. This is because the SOI coding scheme leverages the free timing information supplied by the zero-delay channel between the encoder and the decoder. The key to unlock that gain is the event-triggered nature of the SOI sampling policy. In contrast, the distortion-rate tradeoffs achieved with deterministic sampling policies are much worse: we prove that the causal informational distortion-rate function in that scenario is as high as $D_{\mathrm{DET}}(R) = \frac{5}{6R}$. It is achieved by the uniform sampling policy with the sampling interval $\frac{1}{R}$. In either case, the optimal strategy is to sample the process as fast as possible and to transmit 1-bit codewords to the decoder without delay

Minimum data rate for stabilization of linear systems with parametric uncertainties

We study a stabilization problem of linear uncertain systems with parametric uncertainties via feedback control over data-rate-constrained channels. The objective is to find the limitation on the amount of information that must be conveyed through the channels for achieving stabilization and in particular how the plant uncertainties affect it. We derive a necessary condition and a sufficient condition for stabilizing the closed-loop system. These conditions provide limitations in the form of bounds on data rate and magnitude of uncertainty on plant parameters. The bounds are characterized by the product of the poles of the nominal plant and are less conservative than those known in the literature. In the course of deriving these results, a new class of nonuniform quantizers is found to be effective in reducing the required data rate. For scalar plants, these quantizers are shown to minimize the required data rate, and the obtained conditions become tight

Tracking an Auto-Regressive Process with Limited Communication per Unit Time

Samples from a high-dimensional AR[1] process are observed by a sender which can communicate only finitely many bits per unit time to a receiver. The receiver seeks to form an estimate of the process value at every time instant in real-time. We consider a time-slotted communication model in a slow-sampling regime where multiple communication slots occur between two sampling instants. We propose a successive update scheme which uses communication between sampling instants to refine estimates of the latest sample and study the following question: Is it better to collect communication of multiple slots to send better refined estimates, making the receiver wait more for every refinement, or to be fast but loose and send new information in every communication opportunity? We show that the fast but loose successive update scheme with ideal spherical codes is universally optimal asymptotically for a large dimension. However, most practical quantization codes for fixed dimensions do not meet the ideal performance required for this optimality, and they typically will have a bias in the form of a fixed additive error. Interestingly, our analysis shows that the fast but loose scheme is not an optimal choice in the presence of such errors, and a judiciously chosen frequency of updates outperforms it