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Control-Coherent Koopman Modeling: A Physical Modeling Approach

H. Harry Asada, Jose A. Solano-Castellanos

TL;DR

This work extends Koopman operator theory to controlled, non-autonomous systems by introducing a Control-Coherent Koopman Model (CCK) that preserves an exact input structure without relying on data-driven fitting of the input matrix. By partitioning the state into an actuator-driven subsystem and the remainder, and enforcing linear actuation with observables that include actuator states, the authors derive a lifted linear model $z_{t+1}=Az_t+Bu_t$ with a physically coherent $B$. The method is applied to robot-arm dynamics and validated through numerical simulations on a two-link arm with MPC, where CCK outperforms standard DMDc and bilinear models, and demonstrates the critical importance of using a coherent input matrix for control design. The approach promises broad applicability to control synthesis for nonlinear systems by bridging Koopman theory with actuator-informed modeling, enabling accurate and reliable controller design.

Abstract

The modeling of nonlinear dynamics based on Koopman operator theory, which is originally applicable only to autonomous systems with no control, is extended to non-autonomous control system without approximation to input matrix B. Prevailing methods using a least square estimate of the B matrix may result in an erroneous input matrix, misinforming the controller about the structure of the input matrix in a lifted space. Here, a new method for constructing a Koopman model that comprises the exact input matrix B is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact control matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. The proposed method is applied to multi degree-of-freedom robotic arms and multi-cable manipulation systems. Model Predictive Control is applied to the former. It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate control matrix B does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct B matrix.

Control-Coherent Koopman Modeling: A Physical Modeling Approach

TL;DR

This work extends Koopman operator theory to controlled, non-autonomous systems by introducing a Control-Coherent Koopman Model (CCK) that preserves an exact input structure without relying on data-driven fitting of the input matrix. By partitioning the state into an actuator-driven subsystem and the remainder, and enforcing linear actuation with observables that include actuator states, the authors derive a lifted linear model with a physically coherent . The method is applied to robot-arm dynamics and validated through numerical simulations on a two-link arm with MPC, where CCK outperforms standard DMDc and bilinear models, and demonstrates the critical importance of using a coherent input matrix for control design. The approach promises broad applicability to control synthesis for nonlinear systems by bridging Koopman theory with actuator-informed modeling, enabling accurate and reliable controller design.

Abstract

The modeling of nonlinear dynamics based on Koopman operator theory, which is originally applicable only to autonomous systems with no control, is extended to non-autonomous control system without approximation to input matrix B. Prevailing methods using a least square estimate of the B matrix may result in an erroneous input matrix, misinforming the controller about the structure of the input matrix in a lifted space. Here, a new method for constructing a Koopman model that comprises the exact input matrix B is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact control matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. The proposed method is applied to multi degree-of-freedom robotic arms and multi-cable manipulation systems. Model Predictive Control is applied to the former. It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate control matrix B does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct B matrix.
Paper Structure (5 sections, 32 equations, 5 figures, 1 table)

This paper contains 5 sections, 32 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND AND PROBLEM FORMULATION
  3. CONTROL-COHERENT KOOPMAN MODELING
  4. APPLICATION TO ROBOT ARM DYNAMICS
  5. NUMERICAL SIMULATION: A TWO-LINK ROBOT ARM

Figures (5)

  • Figure 1: Dynamic modeling of the ith joint actuator with torsional compliance at the power train.
  • Figure 2: Configuration range for a two-degree-of-freedom robotic arm following three circular trajectories.
  • Figure 3: MPC control comparison of (a) Control-Coherent Koopman, (b) DMDc, and (c) Bilinear Bruder2021AdvantagesDynamicsb model for circular trajectories with different radii. Tracking performance is summarized in Table \ref{['tab:Comparison']}.
  • Figure 4: Histogram comparison shows almost identical prediction accuracy between (a) Control-Coherent Koopman, (b) DMDc, and (c) Bilinear Koopman Bruder2021AdvantagesDynamicsb.
  • Figure 5: MPC control comparison of DMDc and a hybrid of $A_{DMDc}$ and $B_{CCK}$.

Theorems & Definitions (1)

  • proof