Anytime algorithms for estimation-based model predictive control

• Applicant: Professor Dr.-Ing. Christian Ebenbauer
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 399211811

• Final Report Year: 2024

Final Report Abstract

The project dealt with the state estimation problem of linear and nonlinear constrained discretetime systems within the novel framework of proximity moving horizon estimation, or pMHE. A key innovation which has led to the formulation of the presented pMHE approaches as well as to many of the obtained theoretical results was linking MHE to proximal methods. We derived for the proposed proximity-based MHE schemes and algorithms desirable theoretical properties including stability, robustness, and performance guarantees, which hold for any horizon length and irrespectively of the convex stage cost being used in the underlying optimization problem. The first part of the project contributed to the existing research on constrained MHE by proposing a proximity-based formulation and analysis of the estimation problem. A major theme here was to tailor the design of the pMHE problem to the considered class of dynamical systems and devise suitable sufficient conditions for the exponential stability of the resulting estimation error. More specifically, we provided specific design approaches for the Bregman distance and the a priori estimate that are based on the Luenberger observer, the Kalman filter, and the extended Kalman filter depending on the considered system class. The obtained stability properties were established due to the novel proof technique of employing the Bregman distance used in the pMHE problem as a Lyapunov function. In addition, we proved for linear systems that the pMHE scheme is input-to-state stable with respect to additive process and measurement disturbances. Moreover, we showed under suitable assumptions that pMHE can be interpreted as a Bayesian estimator with stability guarantees. In the second part of the project, we proposed a novel pMHE iteration scheme in which a suboptimal state estimate is computed at each time instant after an arbitrary and limited number of optimization algorithm iterations. The optimization algorithm is based on the mirror descent method which generalizes the gradient descent methods. We emphasized that the a priori estimate is the key factor that provides guaranteed stability in our work, despite using it only to warm-start the algorithm, which renders its bias fading away through further iterations. By taking the dynamics of the employed optimization algorithm into account in the stability analysis, we obtained anytime MHE algorithms where stability of the resulting estimation error is ensured after any number of iterations. In addition, the performance of the pMHE iteration scheme was characterized by the resulting real-time regret, based on which we showed that performance can be improved by increasing the number of optimization algorithm iterations. Finally, we applied in the third part of the project the proposed anytime pMHE algorithm in the context of MHE-based output feedback MPC.

Publications

• A proximity moving horizon estimator based on Bregman distances and relaxed barrier functions. 2019 18th European Control Conference (ECC). IEEE.
  Gharbi, Meriem & Ebenbauer, Christian
  
• Proximity moving horizon estimation for linear time-varying systems and a Bayesian filtering view. 2019 IEEE 58th Conference on Decision and Control (CDC), 3208-3213. IEEE.
  Gharbi, Meriem & Ebenbauer, Christian
  
• A proximity moving horizon estimator for a class of nonlinear systems. International Journal of Adaptive Control and Signal Processing, 34(6), 721-742.
  Gharbi, Meriem & Ebenbauer, Christian
  
• An iteration scheme with stability guarantees for proximity moving horizon estimation. 2020 European Control Conference (ECC), 973-978. IEEE.
  Gharbi, Meriem & Ebenbauer, Christian
  
• Anytime MHE-based output feedback MPC. IFAC-PapersOnLine, 54(6), 264-271.
  Gharbi, Meriem & Ebenbauer, Christian
  
• Anytime Proximity Moving Horizon Estimation: Stability and Regret for Nonlinear Systems. 2021 60th IEEE Conference on Decision and Control (CDC), 728-735. IEEE.
  Gharbi, Meriem; Gharesifard, Bahman & Ebenbauer, Christian
  
• Proximity Moving Horizon Estimation for Discrete-Time Nonlinear Systems. IEEE Control Systems Letters, 5(6), 2090-2095.
  Gharbi, Meriem; Bayer, Fabia & Ebenbauer, Christian
  
• Anytime Proximity Moving Horizon Estimation: Stability and Regret. IEEE Transactions on Automatic Control, 68(6), 3393-3408.
  Gharbi, Meriem; Gharesifard, Bahman & Ebenbauer, Christian
  
• Synthesis of constrained robust feedback policies and model predictive control. 2024 European Control Conference (ECC), 3476-3483. IEEE.
  Gramlich, Dennis; Scherer, Carsten W.; Häring, Hannah & Ebenbauer, Christian