Latent-Space Non-Linear Model Predictive Control for Partially-Observable Systems

TL;DR

This work presents a scalable control framework based on nonlinear Model Predictive Control for high-dimensional dynamical systems by integrating data-driven reduced order modelling, control in a latent space, and state estimation within a unified formulation.

Abstract

This work presents a scalable control framework based on nonlinear Model Predictive Control for high-dimensional dynamical systems. The proposed approach addresses the key challenges of model scalability and partial observability by integrating data-driven reduced order modelling, control in a latent space, and state estimation within a unified formulation. A predictive model is constructed via Operator Inference on a Proper Orthogonal Decomposition basis, yielding a compact latent representation that captures the dominant system dynamics. State estimation is achieved through an Unscented Kalman Filter, which reconstructs the latent space from sparse and noisy measurements, enabling closed-loop control. The input signals are computed directly in the reduced-order latent space, improving computational efficiency with negligible impact on predictive capability. The methodology is validated on the one- and two-dimensional Kuramoto--Sivashinsky equations, serving as benchmarks for chaotic and spatially-extended systems. Numerical experiments demonstrate that the proposed framework achieves accurate stabilisation. Overall, the framework provides a practical approach for nonlinear control of complex, high-dimensional systems where full-state measurements are often inaccessible or infeasible.

Figures (8)

• Figure 1: Schematic of the control algorithm. Top: Data-driven model identification. An offline dataset of trajectories with $M$ snapshots is collected with open-loop control, and a low-dimensional latent space is constructed from dominant Proper Orthogonal Decomposition modes. The system dynamics are learned directly in this latent space via Operator Inference in a non-intrusive manner and including nonlinear dependencies. Bottom: Control procedure. An Unscented Kalman Filter estimates the latent coordinates using partial and noisy state measurements to correct the model predictions. The posterior Kalman estimate is used to initialise the predictive model at each control step. Within Model Predictive Control, the surrogate model generates open-loop latent-state predictions (solid blue line; dots indicate discrete time samples) over a prediction horizon $w_p$ (red shaded area). These predictions are used to optimise the control sequence over the control horizon $w_c$ (purple shaded area). The dashed red line indicates the target latent trajectory.
• Figure 2: Left panels show the training dataset used for system identification via operator inference. The top row displays the state map, i.e. the spatio-temporal evolution of the state, while the bottom row shows the open-loop actuation. The training dataset is shown for the first $300~t.u.$. The right panels show the model evaluation on an ensemble of $250$ trajectories. The top row presents the prediction error ($\epsilon_k^e$) over time, with the average as a solid line and the $\pm\sigma$ confidence region shaded. The middle and bottom rows show the predicted state and the prediction error for a trajectory with median ensemble error. Each prediction starts from the ground-truth state vector.
• Figure 3: A posteriori Unscented Kalman Filter prediction of the state vector for controlled trajectories. Panels on the left show the ground truth of the state map and the a posteriori prediction for a case with $4$ sensors (indicated by red triangles in the top figure), measurement noise intensity $\sigma_{\nu^y} = 0.1$, and measurement sampling period $T_s = 0.1$. Panels on the right show the average prediction error $\bar{\epsilon^e}$ over the time interval $[50,\,100]$, for different configurations of noise levels, number of sensors, and update frequencies. For each case, the initial condition of the prediction is estimated using a Gaussian Process Regression based on the sensor measurements available at the initial time step.
• Figure 4: Results of the application of Model Predictive Control with an Unscented Kalman Filter state estimator. The control is performed around the three unstable equilibrium points of the $1$D Kuramoto-Sivashinsky system, denoted as $E_1$ (left column), $E_2$ (middle column), and $E_3$ (right column). The first three rows correspond to a configuration with $4$ sensors, measurement noise $\sigma_{\nu^y} = 0.1$, and sampling interval $T_s = 0.1$. The first row reports the control error of the true state with respect to the target (red solid line), while the blue line refers to the full-state feedback case for comparison. The second and third rows display the state evolution at two time instants, $t_A$ and $t_B$, under partial observability (red) and full observability (blue). The last two rows present the ensemble-averaged error ($\bar{\epsilon^c}$) computed over the final $5~t.u.$ of control.
• Figure 5: Representation of a subset of the training dataset used for system identification in the $2$D Kuramoto–Sivashinsky case. The central panel depicts the time evolution of the state energy over the interval $t_k \in [450,\,550]$. The green-shaded region corresponds to uncontrolled dynamics, while the unshaded region includes random, smooth actuation. The top and bottom panels display the state field (blue–green colourmap) at selected time instants within the uncontrolled ($t_A$, $t_B$, $t_C$, $t_D$) and controlled ($t_E$, $t_F$) regions. For times $t_D$ and $t_F$, the corresponding actuation maps (blue–red colourmap) are also shown. Yellow crosses indicate the actuator centres, and the contour of each actuator region corresponding to one standard deviation of the Gaussian is shown.