Enhanced Koopman Operator Approximation for Nonlinear Systems Using Broading Learning System

TL;DR

This work tackles the difficulty of controlling nonlinear systems by leveraging Koopman operator theory to linearize dynamics and addressing EDMD's sensitivity to basis-function choice. It introduces BLS-EDMD, which uses Broad Learning System to learn data-driven basis functions, yielding a high-dimensional predictor that enhances both accuracy and robustness. The predictor feeds into BE-MPC, a model predictive controller that optimizes tracking and control effort over a horizon with linear-like predictions in the lifted space. Through simulations on a forced van der Pol oscillator and a Deep Sea Rescue Vehicle, the approach demonstrates superior prediction performance and precise depth control, suggesting strong practical potential for complex nonlinear, high-dimensional systems.

Abstract

Traditional control methods often show limitations in dealing with complex nonlinear systems, especially when it is difficult to accurately obtain the exact system model, and the control accuracy and stability are difficult to guarantee. To solve this problem, the Koopman operator theory provides an effective method to linearise nonlinear systems, which simplifies the analysis and control of the system by mapping the nonlinear dynamics into a high-dimensional space. However, the existing extended dynamical mode decomposition (EDMD) methods suffer from randomness in the selection of basis functions, which leads to bias in the finite-dimensional approximation to the Koopman operator, thus affecting the accuracy of model prediction. To solve this problem, this paper proposes a BLS-EDMD method based on the Broad learning system (BLS) network. The method achieves a high-precision approximation to the Koopman operator by learning more accurate basis functions, which significantly improves the prediction ability of the model. Building on this, we further develop a model predictive controller (MPC) called BE-MPC. This controller directly utilises the high-dimensional and high-precision predictors generated by BLS-EDMD to predict the system state more accurately, thus achieving precise control of the underwater unmanned vehicle (UUV), and its effectiveness is verified by simulation.

Figures (5)

• Figure 1: The hidden layer $\varphi$ of the Broad Learning System network consists of the feature layer $Z^n$ and the enhancement layer $H^n$. The input $x$ is enriched by $\varphi$ to better represent the feature.
• Figure 2: The BLS-EDMD method plays a role in lift the dimensionality of both $\Phi_B$ and $\tilde{\Phi}_B$ during the training process. In the training part, the ridge regression operation is performed by boosting states $x$ and $y$ to obtain $\mathcal{K_B}$,In the prediction part, $x_k$ lift states $x_k\tilde{\Phi}(x_k)$ is mapped by $\mathcal{K_B}$ to $\hat{y}_k\tilde{\Phi}(y_k)$ and decoded by $\tilde{\Phi}_B$ to get $\hat{y}_k$.
• Figure 3: Prediction comparison based on forced van der Pol oscillator: The initial condition $x_0 =[-0.6,0.2] ^T$ set in the interval [-1,1] and a simulation time of $3$s, we performed a prediction comparison. In this process, the control input u(t) is set to be a square wave signal with a period of 0.3 s and unit amplitude.
• Figure 4: Robustness test based on forced van der Pol oscillator: The initial condition ${x_0} = [-0.4, 0.2]^T$ is set in the interval $[-0.5,0.5]$, and the simulation time is $1$s. We perform a robustness comparison. In this process, the control input u(t) is set to be a square wave signal with a period of 0.3 s and unit amplitude.
• Figure 5: BE-MPC based UUV dive depth $50$ m position task: rudder angle $[-30^\circ, 30^\circ]$.

Theorems & Definitions (2)

• Remark 1
• Remark 2