Linear quadratic control for discrete-time systems with stochastic and bounded noises

This paper focuses on the linear quadratic control (LQC) design of systems corrupted by both stochastic noise and bounded noise simultaneously. When only of these noises are considered, the LQC strategy leads to stochastic or robust controllers, respectively. However, there is no LQC strategy that can simultaneously handle stochastic and bounded noises efficiently. This limits the scope where existing LQC strategies can be applied. In this work, we look into the LQC problem for discrete-time systems that have both stochastic and bounded noises in its dynamics. We develop a state estimation for such systems by efficiently combining a Kalman filter and an ellipsoid set-membership filter. The developed state estimation can recover the estimation optimality when the system is subject to both kinds of noise, the stochastic and the bounded. Upon the estimated state, we derive a robust state-feedback optimal control law for the LQC problem. The control law derivation takes into account both stochastic and bounded-state estimation errors, so as to avoid over-conservativeness while sustaining stability in the control. In this way, the developed LQC strategy extends the range of scenarios where LQC can be applied, especially those of real-world control systems with diverse sensing which are subject to different kinds of noise. We present numerical simulations, and the results demonstrate the enhanced control performance with the proposed strategy.

💡 Research Summary

The paper addresses the linear‑quadratic control (LQC) problem for discrete‑time linear systems whose process and measurement noises consist of both stochastic (Gaussian) and bounded (ellipsoidal) components. Existing approaches treat these two noise types separately: stochastic LQG controllers assume purely Gaussian disturbances, while robust min‑max controllers assume only bounded uncertainties. When both types coexist—as is common in real‑world applications such as autonomous vehicles—pure LQG yields over‑optimistic state estimates and the resulting controller can become unstable, whereas pure robust designs become overly conservative and sacrifice performance.

Mixed‑noise model. The authors decompose the process noise (w_k) and measurement noise (v_k) as
(w_k = w_k^s + w_k^b,; v_k = v_k^s + v_k^b),
where (w_k^s, v_k^s) are zero‑mean Gaussian with covariances (P_{w_k}, P_{v_k}), and (w_k^b, v_k^b) belong to known ellipsoids (E(0,M_{w_k})) and (E(0,M_{v_k})). The initial state is modeled analogously. This “mixed‑noise” representation captures both statistical uncertainty and hard bounds within a single framework.

Hybrid state estimator. To estimate the state under mixed noise, the paper proposes a novel estimator that fuses the Kalman filter (optimal for Gaussian noise) and the set‑membership (ellipsoidal) filter (optimal for bounded noise). The estimator maintains three quantities at each time step: a mixed centre (\hat{x}{k|k}), a covariance matrix (P{k|k}) for the stochastic part, and a shape matrix (M_{k|k}) for the bounded part. The prediction step propagates the centre with the system dynamics, updates the covariance by the usual Kalman prediction, and expands the ellipsoidal shape using a Minkowski sum of the transformed previous ellipsoid and the process‑noise ellipsoid.

During the measurement update, a gain matrix (\Gamma_k) is chosen to minimise a combined cost
(V_k = \operatorname{tr}(P_{k|k}) + \operatorname{tr}(M_{k|k})),
i.e., the sum of the stochastic error variance and the volume (trace) of the bounded error ellipsoid. The optimal (\Gamma_k) is derived analytically, involving scalar weighting parameters (p_k) and (q_k) that balance the relative importance of the two error types (expressed in equations (23)–(24)). Consequently, the estimator yields a less optimistic estimate than a pure Kalman filter while avoiding the excessive conservatism of a pure set‑membership filter.

Optimal LQC law. With the mixed estimate in hand, the control problem is reformulated as a min‑max‑expectation optimisation:
\