Information Flow in Linear Systems

Modern control systems have their own unique features that distinguish them from classical control systems. In many cases, there inherently exist multiple distributed controllers and so communication networks can be introduced to connect them. Due to these features, it is challenging to design efficient controller and transceivers for modern control systems. Practically, we must answer questions like ``How reliable do we need the communication channels to be to achieve the desired control performance?", ``What information should be exchanged between controllers?" and ``What are the optimal controller and transceiver structures?" All these practical equations are related to one theoretical question ``Can we understand the information flows between controllers?" In other words, the controllers communicate with each other explicitly through the communication networks and also implicitly through the plants, and we have to understand this information flow for control. In this thesis, we consider three seemingly simple but fundamental problems to understand explicit and implicit information flows for control, as initial building blocks for a theory that we hope will eventually lead to novel and efficient designs for modern control systems./nIn the first technical chapter, we consider Kalman filtering problems when the observations are intermittently erased or lost. Practically, this problem is the simplest model for the packet losses that can happen in communication networks connecting distributed controllers. Theoretically, by relating the erasure probability of the channel with the stability of the control system, we can measure the minimum quality requirements for uncoded information that has to flow to stabilize the system. It was known that the Kalman filtering estimates are mean-square unstable when the erasure probability is larger than some critical value, and stable otherwise. But what that critical value actually is has been open for years. Unlike prior work that tried to connect with Lyapunov stability, we connect with observability to completely characterize the critical erasure probability. We introduce a new concept of \textit{eigenvalue cycles} which captures the periodicity of systems, and characterize the critical erasure probability based on this. It is also proved that eigenvalue cycles can be easily broken if the original physical system is considered to be continuous-time --- randomly-dithered nonuniform sampling of the observations makes the critical erasure probability almost surely $\frac{1}{|\lambda_{max}|^2}$, the best that could be hoped for even with arbitrarily complex coding. This implies that the rank of the observability gramian can be thought of as the amount of information that the estimator learns about linear systems, and nonuniform sampling helps maximize that rank. Furthermore, different subspaces of the states can be thought of as the source of information flows and separated as long as they belong to different eigenvalue cycles./nIn the second technical chapter, to understand implicit information flows we consider distributed linear systems without explicit communication networks. To do this, we build a unified view of both network coding and decentralized control. Precisely speaking, we consider both as linear time-invariant systems by appropriately restricting channels and coding schemes of network coding to be linear time-invariant, and the plant and controllers of decentralized control to be linear time-invariant as well. First, we apply linear system theory to network coding. We introduce a novel technique that we call \textit{Network Linearization}. This technique gives a way of converting an arbitrary relay network to an equivalent acyclic single-hop relay network. Based on network linearization, we prove that the fundamental design limit, mincut, is achievable by a linear time-invariant network-coding scheme regardless of the network topology. Unlike previous approaches relying on graph theory, we use linear system theory and linear algebra, and exploit the fact that there can be multiple network representations for a given algebraic transfer function. For broadcast and unicast problems, unintended messages at receivers turn out to be modeled as secrecy constraints after network linearization./nHaving built a linear-systems view of network coding, we turn it around to view decentralized linear control systems. We argue that linear time-invariant controllers in a decentralized linear system ``communicate" via linear network coding to stabilize the plant. To justify this claim, we revisit classical stabilizability results concerning fixed modes. We give an algorithm to ``externalize" the implicit communication between the controllers that we believe must be occurring to stabilize the plant. Based on this, we show that the stabilizability condition for decentralized linear systems comes from an underlying communication limit, which can be described by the algebraic mincut-maxflow theorem. With this re-interpretation in hand, we also consider \textit{stabilizability over LTI networks} to emphasize the connection with network coding. These results confirm the intuition that there are implicit information flows in distributed control systems which we can visualize. Moreover, the rank of subspaces are the proper measure of information for linear systems when we consider stabilizability./nIn the third and fourth technical chapters, we go beyond stabilizability and study how the size of implicit information flows constrain the optimal control performance. To do this, we must allow arbitrary controllers without imposing linearity constraints. In particular, we focus on scalar decentralized average-cost infinite-horizon LQG problems with two controllers. For fast-dynamics systems --- when the eigenvalue of the system is large ---, it is shown that the best linear controllers' performance can be an arbitrary factor worse than the optimal nonlinear controller performance./nTo understand the required nonlinearity in such control systems, we caricature bit-levels of the states as different subspaces, and the rank of those subspaces as the amount of information. In other words, we take a \textit{linear view of nonlinearity}. Based on this insight, we propose a simple set of finite-dimensional nonlinear controllers, and prove that the proposed set contains easy-to-find approximately optimal strategies that achieve within a constant ratio of the optimal quadratic cost. The insight for the nonlinear strategies comes from revealing the relationship between implicit information flow in control and wireless information flow. More precisely, we discuss a close relationship between the high-SNR limit in wireless communication and the fast-dynamics case in decentralized control, and justify how the proposed nonlinear control strategy can be understood as exploiting a kind of generalized degree-of-freedom gain in wireless communication theory. For a rigorous justification of this argument, we develop new mathematical tools and ideas. We extend Witsenhausen's counterexample to MIMO (multiple-input multiple-output) Witsenhausen's counterexamples, just as wireless communication extends the scalar AWGN (additive white Gaussian noise) channel to MIMO channels and from there, eventually tackles multi-terminal problems. To reveal the relationship between infinite-horizon problems and generalized MIMO Witsenhausen's counterexamples, we introduce the idea of \textit{geometric slicing} that plays a role like that of cut-set bounds in communication theory. To analyze nonlinear strategy performance, we introduce an approximate-comb-lattice model for the relevant random variables. For the slow-dynamics cases --- when the eigenvalue of the system is small ---, we prove that single-controller optimal strategies ---linear strategies--- are constant-ratio optimal among all distributed control strategies./nUnderstanding the nature of information flow for control should eventually lead to a unified theory for control and communication. We believe the parallelism between control information flow and wireless information flow is not just a coincidence but strong evidence for such a unified theory. However, still lots of concepts and ideas in control and communication remain separate and have not been related --- for example, secrecy, interference alignment, and scaling laws in communication. Therefore, further research is required to continue uncovering the fundamental relationship between control and communication. Moreover, we also have to think how to leverage this understanding in practical system designs, and how to build efficient distributed controllers and transceivers for modern control systems.

Advisor: Sahai, Anant