Feedback Stabilization Using Koopman Operator

Abstract

In this paper, we provide a systematic approach for the design of stabilizing feedback controllers for nonlinear control systems using the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this paper is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Simulation results are presented to verify the main results on the design of stabilizing feedback controllers and the data-driven aspect of the proposed approach.

A Convex Data-Driven Approach for Nonlinear Control Synthesis

“…For the data-driven approximation of Koopman operators and subsequent P-F operators, we adopt the algorithmic techniques in [12,26,27]. Specifically, we leverage the numerical algorithm in [27] to directly approximate Koopman generators.…”

“…Time derivatives of the states ẋ can be accurately estimated using numerical algorithms, as shown in [28,29]. Additionally, the pair {x, ẋ} in (26) do not have to be from a single trajectory; it can be a concatenation of multiple experiment/simulation trajectories.…”

Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems

“…This may lead to spurious dynamics, which may subsequently lead to unstable controllers. To address this issue, Huang et al 24 proposed a stabilizing feedback controller which relies on control Lyapunov function (CLF) and thus achieves stabilization in the truncated Koopman eigenfunction space. The authors comment on optimality of the controller using the principle of inverse optimality; however, it does not account for explicit state and input constraints.…”

Efficient Identification of Linear Evolutions in Nonlinear Vector Fields: Koopman Invariant Subspaces

“…Literature Review: The eigenfunctions of the Koopman operator [1], [2] evolve linearly in time, and hence its eigendecomposition can be used to analyze and predict the behavior of dynamical systems [3]- [5]. This can simplify identification [6] and control [7]- [11] of nonlinear systems. Traditional roadblocks to the widespread use of the Koopman operator have been its infinite-dimensional nature and the lack of practical methods to find representations for it.…”