Song Fang

Control over Additive White Gaussian Noise Channels: Bode-Type Integrals, Channel Blurredness, Negentropy Rate, and Beyond

Abstract This paper aims at developing Bode-type integrals for control systems over additive white Gaussian noise channels, by way of deriving information theoretic equalities and inequalities. The integrals characterize the fundamental performance trade-offs of such networked feedback systems with linear time-invariant plants and causal stabilizing controllers. We propose two new notions to facilitate our development: channel blurredness and negentropy rate. The channel blurredness provides an alternative measure for the quality of communication channels to the conventional notion of channel capacity. The negentropy rate, on the other hand, relates the entropy rate of a stochastic process to its power spectrum. Both notions are shown to be closely relevant to networked feedback systems. Indeed, the Bode-type integrals developed herein are seen to depend on the channel blurredness of the communication channel, as well as the negentropy of the exogenous disturbances.

Control Performance Measures

This chapter introduces the control performance measures including the \(\mathcal {H}_{\infty }\) norm and power gain. The \(\mathcal {H}_{\infty }\) norm is a well-established performance measure in robust control. The power gain, on the other hand, serves as a worst-case measure in power reduction of a system. It is shown that power gain is closely related to the entropy rate increase/decrease for signals passing through the system. This measure is examined for several classes of systems, and is seen to reduce to the \(\mathcal {H}_{\infty }\) norm for LTI systems. We also generalize power gain to MIMO systems.

Bounds on Estimation Error

In this chapter, we show that our preceding analysis can be applied to estimation problems as well. The results can be viewed as implications of the performance bounds on power gain and in variance minimization presented in the previous two chapters. In particular, we derive fundamental estimation bounds for estimation systems that are not necessarily LTI with noises that are not necessarily white Gaussian. The bounds are seen to be tight in the particular case of a scalar LTI system with white Gaussian noises, as verified by the benchmark given by the renowned Kalman filter.

Continuous-Time Systems

In this chapter, we develop parallel results for continuous-time systems. We first introduce the continuous-time versions of negentropy rate, Gaussianity-whiteness, and power gain, based on which the continuous-time versions of Bode-type integrals and power gain bounds are obtained. We shall focus on SISO systems only. While the results in Chaps. 5 and 6 can be extended almost entirely, the present chapter concerns systems with no communication channels.

Bode-Type Integrals

In this chapter, we develop Bode-type integrals for networked feedback systems, i.e., feedback control systems containing communication channels. The integrals explicitly quantify the disturbance attenuation properties of the networked feedback systems over different frequencies. Fundamentally, the integrals depend on not only the plant’s properties, but also the communication channels’ characteristics, no matter how well the controller may be designed. We consider SISO systems in this chapter; MIMO systems will be addressed in a subsequent chapter.

Bounds on Power Gain

In this chapter, we derive bounds on power gain for networked feedback systems. The lower bounds, which are based directly upon the corresponding Bode-type integrals in the previous chapter, characterize the worst-case disturbance attenuation properties of such systems. We consider only SISO systems in this chapter; MIMO systems will be discussed in the next chapter.

Information Measures and Spectral Analysis

In this chapter we introduce the notations and background knowledge that will be used in this book. In particular, the key notions from information theory and stochastic processes, e.g., entropy, mutual information, channel capacity, “water-filling” power allocation, stationarity and asymptotic stationarity, are exposed concisely together with discussions on their properties. In addition, we propose a measure of how Gaussian and white an asymptotically stationary stochastic process is, which is therefore termed Gaussianity-whiteness. Towards this end, we first introduce the notion of negentropy rate, which provides a measure of non-Gaussianity for asymptotically stationary stochastic processes. Then, by combining negentropy rate with spectral flatness in a non-trivial way, the Gaussianity-whiteness is defined. Properties of the notions are also analyzed.

Tradeoffs in Networked Feedback Systems: From Information-Theoretic Measures to Bode-Type Integrals

This paper investigates performance limitation issues of multivariable networked feedback systems with multiport communication channels. We develop a general information-theoretic paradigm for analyzing the limitations and tradeoffs imposed by communication channels, which is enabled by the development of new information measures and Bode-type integral inequalities. The integral inequalities quantify the tradeoffs in disturbance attenuation for broad classes of systems consisting of linear time-invariant plants and causal, possibly nonlinear, time-varying stabilizing controllers communicating over general noisy channels with causal encoders and decoders. The channel blurredness, a newly developed information measure for the quality of multiport communication channels, is used to characterize the effect of communication channel constraints on the integrals and henceforth the tradeoff in disturbance attenuation. For several canonical channels with power constraints, such as additive white Gaussian noise channels, additive colored Gaussian noise channels, and fading channels, we derive explicit forms for the channel blurredness, which, together with the Bode-type integral inequalities, show that to mitigate the noise effect on performance tradeoff, the power of the communication channels must be distributed in ways fundamentally different from the Shannon’s classical “water-filling” power allocation policy.

Design constraints and limits of networked feedback in disturbance attenuation: An information-theoretic analysis

Abstract In this paper we investigate the intrinsic design constraints and performance limits of networked control systems. We propose new information measures and correspondingly, develop an information-theoretic paradigm for analyzing the performance trade-offs and limits in disturbance attenuation over information-constrained networked feedback, which is enabled by a cohesive development of new information measures, Bode-type integral inequalities, and performance bounds. The integrals and bounds incorporate the information measures and serve to quantify the trade-offs and limits in disturbance attenuation for broad classes of networked feedback systems consisting of linear time-invariant plants and causal, possibly nonlinear, time-varying stabilizing controllers communicating over general noisy channels with causal encoders and decoders. The notion of negentropy rate is introduced to address general, non-Gaussian disturbances. The channel blurredness, a newly proposed information measure for the quality of communication channels, is used to characterize the effect of communication channel noises on the integrals and henceforth the trade-offs in disturbance attenuation. Bounds on the power gain, a novel disturbance attenuation measure tailored for performance analysis of networked control systems, provide the fundamental limits of disturbance attenuation achievable by networked feedback.