Adaptive Koopman Embedding for Robust Control of Complex Nonlinear Dynamical Systems

TL;DR

This work tackles robustness of Koopman-based linear embeddings for nonlinear dynamics by introducing an adaptive architecture that combines offline nominal lifting with online corrections. An autoencoder-based lifting learns a nominal $z = \varphi(x)$ and identifies matrices $A,B,C$ offline, while a feed-forward online network learns corrections $\Delta A,\Delta B$ to handle disturbances and model drift within a control framework. The approach is validated in MPC-based tracking across coupled pendulums, a 3R serial manipulator, and a planar quadrotor, showing substantial improvements in tracking performance under parametric changes, measurement noise, wind, and non-parametric disturbances. Theoretical results (Theorems 1–2) justify compensating uncertain dynamics within linear/bilinear Koopman representations under invariant subspace assumptions, and the method reduces reliance on large offline datasets by exploiting online adaptation. Overall, the adaptive Koopman framework enables robust, real-time control of diverse nonlinear systems using computationally efficient linear models, expanding the practical reach of Koopman-based control in robotics.

Abstract

The discovery of linear embedding is the key to the synthesis of linear control techniques for nonlinear systems. In recent years, while Koopman operator theory has become a prominent approach for learning these linear embeddings through data-driven methods, these algorithms often exhibit limitations in generalizability beyond the distribution captured by training data and are not robust to changes in the nominal system dynamics induced by intrinsic or environmental factors. To overcome these limitations, this study presents an adaptive Koopman architecture capable of responding to the changes in system dynamics online. The proposed framework initially employs an autoencoder-based neural network that utilizes input-output information from the nominal system to learn the corresponding Koopman embedding offline. Subsequently, we augment this nominal Koopman architecture with a feed-forward neural network that learns to modify the nominal dynamics in response to any deviation between the predicted and observed lifted states, leading to improved generalization and robustness to a wide range of uncertainties and disturbances compared to contemporary methods. Extensive tracking control simulations, which are undertaken by integrating the proposed scheme within a Model Predictive Control framework, are used to highlight its robustness against measurement noise, disturbances, and parametric variations in system dynamics.

Figures (8)

• Figure 1: Illustration of the proposed adaptive embedding along with their neural network architectures. The proposed architecture combines online and offline learning. An offline model (shown in grey-colored blocks) is initially leveraged to train a nominal model by minimizing $L_{nom}$. The dataset $\mathcal{D}_{nom}$, generated from nominal system dynamics, consists of snapshot matrices of states $\boldsymbol{X}_i$ and $\boldsymbol{Y}_i$ together with control inputs $\boldsymbol{U}_i$ for each trajectory $i$, used to train the nominal model. The online network (shown in green colored blocks) consists of the pre-trained lifting block and an adaptive neural network block. The adaptive network is employed to update the nominal Koopman model by adapting to any change in dynamics that happens post-training or unmodeled dynamics unaccounted for in the training dataset. The online adaptation module minimizes the adaptation losses over the window of $w$ latest datasets collected online during deployment. For time-step $k$, the online dataset $\mathcal{D}_{k, online}$ includes snapshot matrices $\boldsymbol{Z}_{k,obs},\boldsymbol{\Delta Z}_{k,obs},$ and $\boldsymbol{U}_{k,obs}$ which contain past $w$ values of the actual lifted state (obtained after lifting the measured state $\boldsymbol{x}_{obs}$) of the system ($\boldsymbol{z}_{k,obs}$), prediction error ($\boldsymbol{\Delta{z}}_k = \boldsymbol{{z}}_{k,obs} - \boldsymbol{\hat{z}}_k$), and control inputs ($\boldsymbol{u}_k$).
• Figure 2: Illustration of the adaptive Koopman architecture embedded in a closed loop control framework. During the offline training phase, input-output data from the nominal system is used to train the nominal Koopman Neural networks, which, upon training, gives the model matrices $\boldsymbol{A}, \boldsymbol{B}$, $\boldsymbol{C}$ and lifting function $\boldsymbol{\varphi}(.)$. During the deployment phase, the online adaptive architecture uses the difference between predicted lifted values, $\boldsymbol{\hat{z}}_{k}$ and actual lifted values, $\boldsymbol{z}_{k, obs}$ to calculate the required correction to the Koopman matrices in the form of $\boldsymbol{\Delta A}$ and $\boldsymbol{\Delta B}$. The controller (MPC in the present study) uses the updated Koopman model matrices to compute the required control output $\boldsymbol{u}_k$ required to drive $\boldsymbol{x}_{k,obs}$ to its reference value $\boldsymbol{x}_{ref}$. This computed control input $\boldsymbol{u}_k$ is applied to the real system (with uncertainties), and the resulting state is measured.
• Figure 3: Simulation results of the proposed linear(left)/bilinear(right) adaptive Koopman framework for the controlled synchronous revolution of the coupled pendulum system with $\boldsymbol{N=5}$. The figure presents tracking performance with the nominal system dynamics modified by $\delta = 40 \%$ change in system parameters, and the measured state, $\boldsymbol{x}_{obs}$, is corrupted with the Gaussian noise with SNR = $30$ dB. The adaptive Koopman algorithm is integrated with MPC to drive the coupled pendulum system to track a reference angular velocity $\boldsymbol{\dot{\theta}} = 40$ rad/s. a) Evolution of $\boldsymbol{\theta}$ and $\boldsymbol{\dot{\theta}}$ for the nominal linear Koopman and the linear adaptive Koopman algorithm. b) Evolution of $\boldsymbol{\theta}$ and $\boldsymbol{\dot{\theta}}$ for the nominal bilinear Koopman and the bilinear adaptive Koopman algorithm.
• Figure 4: Robustness analysis of the proposed adaptive Koopman framework against parametric uncertainty of range $\boldsymbol{\delta \in \{0,5,10,15,20,25,30,35,40}\}$ and measured states $\boldsymbol{x}_{obs}$ corrupted by Gaussian noise with SNR in range $\{\text{no noise}\boldsymbol{, 40, 35, 30, 25, 20}\}$ dB for solving controlled synchronization problem with $\boldsymbol{\dot{\theta}}_{ref} = \boldsymbol{40}$ rad/s. a) Percent decrease in the average RMS tracking error in $\boldsymbol{\theta}$ for (left: linear, right: bilinear) adaptive Koopman framework compared to nominal Koopman framework. b) Percent decrease in the average RMS tracking error in $\boldsymbol{\dot{\theta}}$ for (left: linear, right: bilinear) adaptive Koopman framework compared to nominal Koopman framework. c) Average RMS tracking error $(e_{\boldsymbol{\theta}})$ in $\boldsymbol{\theta}$ for (left: linear, right: bilinear) adaptive Koopman framework. d) Average RMS tracking error $e_{\boldsymbol{\dot{\theta}}}$ in $\boldsymbol{\dot{\theta}}$ for (left: linear, right: bilinear) adaptive Koopman framework.
• Figure 5: Comparison of average computation time per iteration between linear and bilinear nominal Koopman framework with their corresponding adaptive Koopman formulation for solving controlled synchronization problem. The box plot is generated from simulation data using $42$ different trials with different values of $\delta$ and noise levels.