Optimal Unpredictable Control for Linear Systems

TL;DR

This paper introduces a method for making linear system outputs unpredictable by adding stochastic control inputs, optimizing the distribution to degrade adversarial inference while maintaining control performance.

Contribution

It formulates novel stochastic optimization problems for optimal unpredictable control distribution and provides solutions with theoretical and numerical methods, including an unpredictable LQR.

Findings

Optimal control input distribution outperforms Gaussian and Laplace distributions.

The proposed methods effectively increase unpredictability against adversaries.

Simulations validate the effectiveness of the unpredictable control algorithm.

Abstract

In this paper, we investigate how to achieve the unpredictability against malicious inferences for linear systems. The key idea is to add stochastic control inputs, named as unpredictable control, to make the outputs irregular. The future outputs thus become unpredictable and the performance of inferences is degraded. The major challenges lie in: i) how to formulate optimization problems to obtain an optimal distribution of stochastic input, under unknown prediction accuracy of the adversary; and ii) how to achieve the trade-off between the unpredictability and control performance. We first utilize both variance and confidence probability of prediction error to quantify unpredictability, then formulate two two-stage stochastic optimization problems, respectively. Under variance metric, the analytic optimal distribution of control input is provided. With probability metric, it is a non-convex optimization problem, thus we present a novel numerical method and convert the problem into a solvable linear optimization problem. Last, we quantify the control performance under unpredictable control, and accordingly design the unpredictable LQR and cooperative control. Simulations demonstrate the unpredictability of our control algorithm. The obtained optimal distribution outperforms Gaussian and Laplace distributions commonly used in differential privacy under proposed metrics.