[Paper Review] Anytime Proximity Moving Horizon Estimation: Stability and Regret

Meriem Gharbi, Bahman Gharesifard+1 more|arXiv (Cornell University)|Jun 25, 2020

Advanced Control Systems Optimization4 citations

TL;DR

This paper proposes an anytime proximity moving horizon estimation (pMHE) scheme for constrained discrete-time linear systems that delivers stable state estimates after any number of optimization iterations. By combining a proximal point algorithm with a Luenberger observer-based a priori estimate for warm-starting, the method ensures global exponential stability of estimation errors and achieves sublinear regret that decreases with more iterations, enabling real-time, computationally efficient estimation with theoretical performance guarantees.

ABSTRACT

In this paper, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme which employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret which can be rendered smaller by increasing the number of optimization iterations can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.

Motivation & Objective

• To address the computational burden of online moving horizon estimation (MHE) in constrained linear systems.
• To develop a real-time, anytime MHE scheme that guarantees stability after any number of optimization iterations.
• To provide performance guarantees via regret analysis, showing sublinear regret that decreases with more iterations.
• To ensure stability without requiring full convergence of the optimization algorithm at each time step.
• To demonstrate that the proposed scheme outperforms standard pMHE by reducing regret through improved warm-starting.

Proposed method

• The method employs a proximity-based optimization formulation using a Bregman distance centered on the current iterate, enabling fast, incremental updates.
• A proximal point algorithm is used as the underlying optimization scheme, generalized to allow for mirror descent-like steps with flexible Bregman distances.
• The algorithm is warm-started using a stabilizing a priori estimate derived from a Luenberger observer, which ensures stability even with few iterations.
• The estimation error dynamics are analyzed via Lyapunov methods to establish global exponential stability under minimal assumptions.
• Regret is defined as the difference in accumulated cost between the iterative estimate and a comparator sequence, and upper bounds are derived for this performance metric.
• The scheme is designed to be anytime: stability and performance are guaranteed after any number of iterations, including just one.

Experimental results

Research questions

• RQ1 Can a computationally efficient MHE scheme be designed that guarantees stability after any number of optimization iterations?
• RQ2 How does the performance of the iterative pMHE scheme, measured by regret, scale with the number of optimization iterations?
• RQ3 Can the stabilizing effect of the Luenberger observer be effectively leveraged in a warm-start strategy without compromising stability or performance?
• RQ4 What theoretical guarantees can be provided for estimation error stability and regret in a constrained, discrete-time linear system under limited computation?
• RQ5 Does the proposed scheme outperform standard pMHE in terms of regret when the same number of iterations is used?

Key findings

• Global exponential stability of the estimation error is guaranteed after any number of optimization iterations, even with just one iteration per time step.
• The regret of the scheme grows sublinearly with time, specifically bounded by O(√T), implying that the average regret R(T)/T tends to zero as T → ∞.
• Increasing the number of optimization iterations per time step reduces the regret, and the scheme with 20 iterations achieves lower regret than the full-optimization pMHE algorithm in [21].
• The warm-start strategy based on the Luenberger observer enables faster convergence and lower regret compared to schemes that fix the Bregman center to the a priori estimate at each iteration.
• Numerical simulations confirm that the average regret decreases over time and that the regret bound is tighter when more iterations are performed.
• The proposed anytime pMHE scheme outperforms the standard pMHE algorithm in [21] in terms of regret, even though it does not compute the full solution at each step.