Optimal Mode Decomposition for Control

TL;DR

This work extends optimal mode decomposition (OMD) to controlled systems (OMDc), unifying identification and reduction in a single optimization on the Grassmann manifold to produce compact, faithful reduced-order models for high-dimensional dynamics. Building on DMD with control (DMDc), it analytically eliminates system matrices to derive closed-form expressions for the reduced dynamics, then optimizes the projection subspace $L$ on $\mathcal{G}(n,r)$ via a Grassmannian conjugate gradient method. An implementation strategy using a thin QR factorization reduces computational cost to $O(m)$ per iteration, making the approach viable for large-scale PDE-based problems. The method is demonstrated on coupled diffusion equations for wood-chip drying, where a 10-mode reduced model accurately captures the dynamics and provides a practical tool for industrial drying simulations. The results highlight that OMDc yields reduced dynamics differing from DMDc, reflecting a more optimal joint treatment of projection and dynamics, with clear paths to parametric extensions and Grassmann-manifold interpolation for design and control tasks.

Abstract

We present an extension of optimal mode decomposition (OMD) for autonomous systems to systems with controls. The extension is developed along the same lines as the extension of dynamic mode decomposition (DMD) to DMD with control (DMDc). DMD identifies a linear dynamic system from high-dimensional snapshot data. DMD is often combined with a subsequent reduction by a projection to a truncated basis for the space spanned by the snapshots. In OMD, the identification and reduction are essentially integrated into a single optimization step, thus avoiding the somewhat adhoc decoupled, a posteriori reduction that is necessary if DMD is to be used for model reduction. DMD was devised for autonomous systems and later extended to DMD for systems with control inputs (DMDc). We present the analogous extension of OMD to OMDc, i.e. OMD for systems with control inputs. We illustrate the proposed method with an application to coupled diffusion-equations that model the drying of a wood chip. Reduced models of this type are required for the efficient simulation of industrial drying processes.

Figures (5)

• Figure 1: Illustration of the Grassmann manifold $\mathcal{G}(n,r)$. The red dots mark elements $\mathcal{L}$ and $\mathcal{L}^\prime$. Their linear interpolation is the dotted blue line which leaves $\mathcal{G}(n,r)$. The geodesic with direction $\mathcal{H}$ is given as the solid red line, see \ref{['eq:geodesic']}.
• Figure 2: The wood chip is surrounded by hot dry air with temperature $T_\infty(t)$ and water vapor density $\varrho_\infty(t)$. A temperature distribution $T(x,t)$ and a moisture distribution $X(x,t)$ describe the interior the wood chip. During drying, the wood chip absorbs heat from the surrounding air and emits vapor. The pore structure, that actually exists in a wood particle, which is partially filled with water, is homogenized with spatially continuous quantities for the numerical treatment.
• Figure 3: Mean temperature of the wood chip over time.
• Figure 4: Mean moisture of the wood chip over time.
• Figure 5: Eigenvalues of the identified linear system for DMDc and OMDc in the complex plane. The unit circle is drawn in black for reference.