Informative Input Design for Dynamic Mode Decomposition

TL;DR

The paper tackles efficient system identification for high-dimensional dynamics by integrating informative input design into the Dynamic Mode Decomposition with control (DMDc) framework. It formulates an approximate convex optimization problem that minimizes the trace of the estimation error covariance, enabling scalable planning of future inputs under state and control constraints. The approach extends to reduced-order models via DMDc and offers two optimization formulations: a semidefinite program using $\mathrm{tr}(\hat{W}^{-1})$ and a faster linear program using $-\mathrm{tr}(\hat{W})$, with a convex-concave procedure to handle nonconvexity. Validation across fluid- and aircraft-systems (WaterLily, Aerobench, and X-Plane) demonstrates improved identification accuracy with less data, and the authors provide open-source implementations to facilitate adoption in industry and research. Real-time online planning capabilities further highlight the method's practicality for adaptive, data-driven control.

Abstract

Efficiently estimating system dynamics from data is essential for minimizing data collection costs and improving model performance. This work addresses the challenge of designing future control inputs to maximize information gain, thereby improving the efficiency of the system identification process. We propose an approach that integrates informative input design into the Dynamic Mode Decomposition with control (DMDc) framework, which is well-suited for high-dimensional systems. By formulating an approximate convex optimization problem that minimizes the trace of the estimation error covariance matrix, we are able to efficiently reduce uncertainty in the model parameters while respecting constraints on the system states and control inputs. This method outperforms traditional techniques like Pseudo-Random Binary Sequences (PRBS) and orthogonal multisines, which do not adapt to the current system model and often gather redundant information. We validate our approach using aircraft and fluid dynamics simulations to demonstrate the practical applicability and effectiveness of our method. Our results show that strategically planning control inputs based on the current model enhances the accuracy of system identification while requiring less data. Furthermore, we provide our implementation and simulation interfaces as an open-source software package, facilitating further research development and use by industry practitioners.

Figures (5)

• Figure 1: Schematic of the informative input design for DMDc pipeline, demonstrated by controlling the rotation of a cylinder in a fluid. Data are collected from the system for the $n$ state variables and $m$ control inputs. Next, DMDc is performed to construct a reduced order model of the system. The covariance matrix $\Gamma$ is constructed and is then used to plan future informative control inputs.
• Figure 2: Predicted and actual vorticity field for fluid flow past a cylinder at $\mathrm{Re}=100$ after collecting data from the informative inputs of the SDP method. The four plots show the control inputs planned by each of the four methods. Blue indicates the initial data set and orange indicates the informative inputs that were designed.
• Figure 3: Example of the control inputs generated by each of the four methods for the F-16 Aerobench simulations. The right plots show the objective and solution time comparison of the four methods across 100 simulation runs.
• Figure 4: Top: image of the Cessna 172 aircraft used for the experiment in the X-Plane simulator with illustrations showing possible future trajectories of the system with uncertainty bounds. Bottom: plots of the state variables throughout the experiment (roll, pitch, sideslip, roll rate, pitch rate, yaw rate, velocity in the x,y,z directions, angle of attack, elevator, aileron, rudder, and throttle positions). Blue: initial data. Orange: data collected during execution of informative inputs. Markers: indicate when each system identification replanning epoch occurred.
• Figure 5: Optimization progression of the control inputs using both the SDP and LP formulation with the corresponding objective values shown for the F-16 example using the Aerobench simulator.