On excitation of control-affine systems and its use for data-driven Koopman approximants

TL;DR

This work addresses data-driven identification of control-affine (bilinear) dynamics within the Koopman/EDMD framework by formulating an affine data-fitting regression and deriving bounds that depend on the minimal singular value $\sigma_{\min}(V)$. It develops excitation strategies to maximize $\sigma_{\min}(V)$, including a necessary condition for optimality and a subspace-angle framework for sequential data collection, to improve sample efficiency and conditioning. The results are instantiated in bilinear EDMDc, with extensions to generator EDMD and kernel EDMD, and accompanied by uniform error bounds for the kernel-EDMD control setting. The methodology is demonstrated on a nonholonomic robot, showing that carefully designed input strategies yield more accurate Koopman surrogates and more reliable data-driven control performance.

Abstract

The Koopman operator and extended dynamic mode decomposition (EDMD) as a data-driven technique for its approximation have attracted considerable attention as a key tool for modeling, analysis, and control of complex dynamical systems. However, extensions towards control-affine systems resulting in bilinear surrogate models are prone to demanding data requirements rendering their applicability intricate. In this paper, we propose a framework for data-fitting of control-affine mappings to increase the robustness margin in the associated system identification problem and, thus, to provide more reliable bilinear EDMD schemes. In particular, guidelines for input selection based on subspace angles are deduced such that a desired threshold with respect to the minimal singular value is ensured. Moreover, we derive necessary and sufficient conditions of optimality for maximizing the minimal singular value. Further, we demonstrate the usefulness of the proposed approach using bilinear EDMD with control for non-holonomic robots.

Figures (7)

• Figure 1: Graphical abstract of the motivational background considering error bounds in bilinear (g)EDMDc and kEDMD and the proposed framework for data-fitting of control-affine mappings.
• Figure 2: Box plots of $\frac{1}{\sqrt{d}} \sigma_\mathrm{min}(V)$ for $m = 4$ and $d \in \{5, 6, 7, 8, 9, 10, 15, 20, 25\}$ with $u_i$ drawn i.i.d. and uniformly from the set $[-0.5, 0.5]^4$ without (left) and with normalization ($\|u_i\| = 1$; right).
• Figure 3: Left: Illustration of the choice of orthogonal input vectors proposed in Corollary \ref{['cor:scaling']}, where $u^{(j)}$ denotes the respective direction in the input space. Right: Illustration of the simplicial choice of input vectors considered in Proposition \ref{['prop:simplex']}, where the angle $\theta$ between the vectors equals $120^\circ$. Both choices yield an optimal excitation for the depicted case of $m=2$.
• Figure 4: Left: Surface plot of the function $\Theta$ in \ref{['eq:theta']} for the case $m=2$ on the box $[-5,5]^2\subset\mathbb R^2$ with peak at $-\mathds{1}_2 = [-1,-1]^\top$. The affine subspace on which $\Theta$ vanishes is indicated as line red. Right: Choosing $u_0 = -(u_1+u_2)$ for two randomly given $u_1$ and $u_2$ to maximize the subspace angles for $m=2$, see Theorem \ref{['t:main']}.
• Figure 5: Minimum singular values over the observation points which are sorted w.r.t. $\sigma_{\min}$ for ${d}_i+1=m+1=3$ neighbors (left) and empirical cumulative distribution function of the normalized minimum singular values for $U_{\textnormal{r}}$, $U_\triangle$, and $U_\perp$ for a differing number $({d}_i+1)\in\{3,\, 4,\, 5,\, 6,\, 10,\, 20,\, 30\}$, where the line opacity is decreased for increasing ${d}_i$ (right).

Theorems & Definitions (26)

• Proposition 2.1: Error bound
• proof
• Lemma 3.1
• proof
• Proposition 3.2: Excitation by scaling
• proof
• Corollary 3.3: Orthogonal inputs
• proof
• Remark 3.4: Balanced normalized tight frames: BNTF
• Proposition 3.5: Simplex vertices as inputs