TY - JOUR

T1 - A Lyapunov function for robust stability of moving horizon estimation

AU - Schiller, Julian D.

AU - Muntwiler, Simon

AU - Köhler, Johannes

AU - Zeilinger, Melanie N.

AU - Müller, Matthias A.

PY - 2023/12

Y1 - 2023/12

N2 - We provide a novel robust stability analysis for moving horizon estimation (MHE) using a Lyapunov function. Additionally, we introduce linear matrix inequalities (LMIs) to verify the necessary incremental input/output-to-state stability (δ-IOSS) detectability condition. We consider an MHE formulation with time-discounted quadratic objective for nonlinear systems admitting an exponential δ-IOSS Lyapunov function. We show that with a suitable parameterization of the MHE objective, the δ-IOSS Lyapunov function serves as an M-step Lyapunov function for MHE. Provided that the estimation horizon is chosen large enough, this directly implies exponential stability of MHE. The stability analysis is also applicable to full information estimation, where the restriction to exponential δ-IOSS can be relaxed. Moreover, we provide simple LMI conditions to systematically derive δ-IOSS Lyapunov functions, which allows us to easily verify δ-IOSS for a large class of nonlinear detectable systems. This is useful in the context of MHE in general, since most of the existing nonlinear (robust) stability results for MHE depend on the system being δ-IOSS (detectable). In combination, we thus provide a framework for designing MHE schemes with guaranteed robust exponential stability. The applicability of the proposed methods is demonstrated with a nonlinear chemical reactor process and a 12-state quadrotor model.

AB - We provide a novel robust stability analysis for moving horizon estimation (MHE) using a Lyapunov function. Additionally, we introduce linear matrix inequalities (LMIs) to verify the necessary incremental input/output-to-state stability (δ-IOSS) detectability condition. We consider an MHE formulation with time-discounted quadratic objective for nonlinear systems admitting an exponential δ-IOSS Lyapunov function. We show that with a suitable parameterization of the MHE objective, the δ-IOSS Lyapunov function serves as an M-step Lyapunov function for MHE. Provided that the estimation horizon is chosen large enough, this directly implies exponential stability of MHE. The stability analysis is also applicable to full information estimation, where the restriction to exponential δ-IOSS can be relaxed. Moreover, we provide simple LMI conditions to systematically derive δ-IOSS Lyapunov functions, which allows us to easily verify δ-IOSS for a large class of nonlinear detectable systems. This is useful in the context of MHE in general, since most of the existing nonlinear (robust) stability results for MHE depend on the system being δ-IOSS (detectable). In combination, we thus provide a framework for designing MHE schemes with guaranteed robust exponential stability. The applicability of the proposed methods is demonstrated with a nonlinear chemical reactor process and a 12-state quadrotor model.

KW - eess.SY

KW - cs.SY

KW - Robust stability

KW - Stability criteria

KW - Estimation

KW - Observers

KW - state estimation

KW - Incremental system properties

KW - moving horizon estimation

KW - Noise measurement

KW - Standards

KW - Lyapunov methods

KW - moving horizon estimation (MHE)

UR - http://www.scopus.com/inward/record.url?scp=85161059119&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2202.12744

DO - 10.48550/arXiv.2202.12744

M3 - Article

VL - 68

SP - 7466

EP - 7481

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 12

ER -