LQG for Constrained Linear Systems: Indirect Feedback Stochastic MPC with Kalman Filtering

TL;DR

This paper develops a stochastic model predictive control method for linear systems with Gaussian noise, combining Kalman filtering and indirect feedback SMPC to ensure probabilistic constraints are satisfied.

Contribution

It introduces a novel SMPC approach that guarantees recursive feasibility and chance constraint satisfaction, recovering the LQG solution under certain conditions.

Findings

The proposed method ensures non-conservative constraint satisfaction.

Recursive feasibility is established through specific initialization.

The approach recovers the unconstrained LQG solution when applicable.

Abstract

We present an output feedback stochastic model predictive control (SMPC) approach for linear systems subject to Gaussian disturbances and measurement noise and probabilistic constraints on system states and inputs. The presented approach combines a linear Kalman filter for state estimation with an indirect feedback SMPC, which is initialized with a predicted nominal state, while feedback of the current state estimate enters through the objective of the SMPC problem. For this combination, we establish recursive feasibility of the SMPC problem due to the chosen initialization, and closed-loop chance constraint satisfaction thanks to an appropriate tightening of the constraints in the SMPC problem also considering the state estimation uncertainty. Additionally, we show that for specific design choices in the SMPC problem, the unconstrained linear-quadratic-Gaussian (LQG) solution is recovered if it is feasible for a given initial condition and the considered constraints. We demonstrate this fact for a numerical example, and show that the resulting output feedback controller can provide non-conservative constraint satisfaction.