Offline and Online Use of Interval and Set-Based Approaches for Control and State Estimation: A Selection of Methodological Approaches and Their Application

TL;DR

This paper surveys offline and online interval and set-based methods for robust control and state estimation under bounded and epistemic uncertainty, emphasizing parameter identification, guaranteed model predictive control, interval observers, and mixed set-stochastic approaches. It presents structured frameworks such as predictor--corrector parameter identification, interval-based variable-structure control with barrier functions, and interval-valued MPC, and demonstrates applications to battery state estimation and iterative-learning observers. The contributions include methodological overviews, practical implementation guidelines, and connections between set-based guarantees and online control/estimation tasks, highlighting the potential for verified performance in complex, uncertain systems. The work underlines the practical significance of combining interval calculus, ellipsoidal methods, and stochastic representations to achieve robust, real-time decision-making in domains like energy systems and distributed control.

Abstract

Control and state estimation procedures need to be robust against imprecisely known parameters, uncertainty in initial conditions, and external disturbances. Interval methods and other set-based techniques form the basis for the implementation of powerful approaches that can be used to identify parameters of dynamic system models in the presence of the aforementioned types of uncertainty. Moreover, they are applicable to a verified feasibility and stability analysis of controllers and state estimators. In addition to these approaches which are typically used offline for analysis of system models designed with classical floating point procedures, interval and set-based methods have also been developed in recent years, which allow to directly solve the associated design tasks and to implement reliable techniques that are applicable online, i.e., during system operation. The latter approaches include set-based model predictive control, online parameter adaptation techniques for nonlinear variable-structure and backstepping controllers, interval observers, and fault diagnosis techniques. This paper provides an overview of the methodological background and reviews numerous practical applications for which interval and other set-valued approaches have been employed successfully.

Figures (6)

• Figure 1: Predictor--corrector scheme for combined state estimation and parameter identification RAUH2020112484.
• Figure 2: Simulation-based distinction between possibly feasible, guaranteed feasible, and infeasible system parameterizations rauhdoetschelsofc4RAUH2020112484.
• Figure 3: Application scenario: Swing-up of an inverted pendulum according to Fnadi2023.
• Figure 4: Overall structure of the interval-based nonlinear model predictive control approach, extended by an underlying pre-stabilization of the plant dynamics.
• Figure 5: Estimation of the open-circuit voltage and the state of charge.