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Consider a control problem with a communication channel connecting the observer of a linear stochastic system to the controller. The goal of the controller is to minimize a quadratic cost function in the state variables and control signal, known as the linear quadratic regulator (LQR). We study the fundamental tradeoff between the communication rate $r$ b/s and the expected cost $b$. We obtain a lower bound on a certain rate-cost function, which quantifies the minimum directed mutual information between the channel input and output that is compatible with a target LQR cost. The rate-cost function has operational significance in multiple scenarios of interest: among others, it allows us to lower-bound the minimum communication rate for fixed and variable length quantization, and for control over noisy channels. We derive an explicit lower bound to the rate-cost function, which applies to the vector, non-Gaussian, and partially observed systems, thereby extending and generalizing an earlier explicit expression for the scalar Gaussian system, due to Tatikonda et al. [S. Tatikonda, A. Sahai, and S. Mitter, “Stochastic linear control over a communication channel,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1549–1561, Sep. 2004.]. The bound applies as long as the differential entropy of the system noise is not $-\infty$. It can be closely approached by a simple lattice quantization scheme that only quantizes the innovation, that is, the difference between the controller's belief about the current state and the true state. Via a separation principle between control and communication, similar results hold for causal lossy compression of additive noise Markov sources. Apart from standard dynamic programming arguments, our technical approach leverages the Shannon lower bound, develops new estimates for data compression with coding memory, and uses some recent results on high resolution variable-length vector quantization to prove that the new converse bounds are tight. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long-standing problems of linear quadratic regulation (LQR) control and stabilization for a class of discrete-time stochastic systems involving multiplicative noises and input delay. These fundamental problems have attracted resurgent interests due to development of networked control systems. An explicit analytical expression is given for the optimal LQR controller. More specifically, the optimal LQR controller is shown to be a linear function of the conditional expectation of the state, with the feedback gain based on a Riccati-ZXL difference equation. It is also shown that the system is stabilizable in the mean-square sense if and only if an algebraic Riccati-ZXL equation has a particular solution. These results are based on a new technical tool, which establishes a non-homogeneous relationship between the state and the costate of this class of systems, and the introduction of a new Lyapunov function for the finite-horizon optimal control design. [ABSTRACT FROM PUBLISHER]

We consider finite-horizon LQR control with limited controller-system communication. Within a time-horizon T, the controller can only communicate with the system d