LQG Control With Minimum Directed Information: Semidefinite Programming Approach

Abstract

We consider a discrete-time Linear-Quadratic-Gaussian (LQG) control problem in which Massey's directed information from the observed output of the plant to the control input is minimized while required control performance is attainable. This problem arises in several different contexts, including joint encoder and controller design for data-rate minimization in networked control systems. We show that the optimal control law is a Linear-Gaussian randomized policy. We also identify the state space realization of the optimal policy, which can be synthesized by an efficient algorithm based on semidefinite programming. Our structural result indicates that the filter-controller separation principle from the LQG control theory, and the sensor-filter separation principle from the zero-delay rate-distortion theory for Gauss-Markov sources hold simultaneously in the considered problem. A connection to the data-rate theorem for mean-square stability by Nair & Evans is also established.

Theoretical foundations for layered architectures and speed-accuracy tradeoffs in sensorimotor control

“…We adapt the results of control under information constraints (39)(40)(41)(42) to explicitly characterize the impact of information constraints, delay, and actuator saturation. Specifically, let P : R + → R + and G : R + → R + be functions given by P (λ) = {1 − λ + a 2 λ + 4λ + (a 2 λ − λ + 1) 2 }/2, G(λ) = a 2 P 3 (a 2 Λ + a 2(T +1) − Λ)/(P 2 + 2λP + λ 2 + λa 2 P 2 ), where Λ = a 2(T +1) /(2 2R − a 2 ).…”

“…To this end, we apply the tools from robust control theory, which characterizes the performance of a feedback system using its input-output relation (44,45). The emerged expressions are reminiscent of the bounds in a stochastic system derived using information theoretic arguments (40,41,46) but they can be derived using the time-domain arguments that rely on linear algebra only. In section 1, we derive the fundamental limits in system performance as a function of the delay and data rate in the deterministic worst-case setting.…”

Directed information and privacy loss in cloud-based control

“…A related optimization problem to (11) is already considered in [42], where the only difference is that the optimization domain considered there is D {P Ut|X t ,U t−1 } T t=1 . Notice that D ⊂ D since every element in D can be, by compositions of stochastic kernels, mapped to an element of D .…”

“…Notice that D ⊂ D since every element in D can be, by compositions of stochastic kernels, mapped to an element of D . In [42], it is shown that the optimal solution P Ut|X t ,U t−1 in D can be realized by the interconnection of a linear sensor, Kalman filter, and a controller as shown in Fig. 4.…”

Sequential Empirical Coordination Under an Output Entropy Constraint

“…5] to reconstruction problems with causality constraints (additional relevant works, motivated by video and perceptual coding, include [23]- [25]). Moreover, when the reconstruction is fed into a controller that can act on the information source, and if one can establish some form of the separation principle between estimation and control, the methods of sequential rate-distortion theory can be brought to bear on the problem of optimal quantizer design for this problem of control under communication constraints [4], [26]- [34].…”