Rate-Cost Tradeoffs in Control

Abstract

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 bits/sec 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 el al. [2]. The bound applies as long as the differential entropy of the system noise is not −∞. 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 variablelength vector quantization to prove that the new converse bounds are tight.

Predictive Coding and Control

“…Papers [17], [18] consider the cost-rate tradeoff in linear quadratic control. This problem seeks the lowest average bitrate, R(b), channel required to achieve a specified LQ performance b. Stavrou et al [19] treat a related Kalman filtering problem and seek the minimal data rate required to achieve a specific distortion or mean squared error between the plant state and the receiver-side Kalman filter.…”

“…Each of [17], [18], [19] and ourselves has a single bandlimited forward channel and a high-fidelity return channel, used in [19] for communication of the receiver state estimate and in [17], [18] and here to communicate the control signal. Kostina and Hassibi [17] and Tanaka et al [18] require stabilizability of the plant system's [A, B] pair and adjust the coding to accommodate the plant feedback stabilization as part of their calculation. This is evident in their inherent satisfaction of Tatikonda's and Mitter's [20] and Nair's and Evans' [4] lower bound on the bit rate based on the unstable eigenvalues of A.…”

“…Tanaka et al [18] and Stavrou et al [19] incorporate the communication of the receiver's state estimate of the plant state to compute their state innovations, which is then quantized in the coder. We note too that the encoding strategy in [17] also is based on quantizing then entropy coding the state innovations between the true plant state and the receiver's estimate and can accommodate partial state measurements with Gaussian noise. Initial values, channel noise and quantization error force the state estimates at the transmitter and receiver to differ.…”

“…• Our problem focus is to optimize the plant performance given the communications structure based on predictive coding. This is in contrast to [17], [18], [19] where the control performance is specified and communications required to achieve this is then designed. A number of papers are dedicated to reconstruction of the conditional density of either the plant state or the predictor state using methods allied with Kalman filtering [5], Bayesian filtering [6] and particle filtering [7].…”

Experimental and educational platforms for studying architecture and tradeoffs in human sensorimotor control

“…A heavily studied and central topic in neuroscience is speed-accuracy tradeoffs (SATs) [1][2][3][4][5]. At the neuroncomponent level, the resource limitations (space and metabolic costs) of the brain impose severe speed/accuracy constraints in neural signaling [6], as well as in the muscle actuation [7][8][9].…”