Direct data-driven interpolation and approximation of linear parameter-varying system trajectories

TL;DR

Addressing missing data in $LPV$ trajectories, this paper develops a direct data-driven interpolation method for $LPV$-$SA$ systems. It leverages a kernel-based LPV representation and a data-driven Hankel construct $\mathcal{H}_L(\cdot)$ together with the generalized persistence of excitation (GPE) condition to characterize and recover admissible interpolants from a data-dictionary $\mathcal{D}_{N_d}$. A practical algorithm is provided to compute the interpolant from data, with existence and uniqueness guaranteed by three criteria, and the approach is demonstrated on a mass-spring-damper example and extended to LPV-control and nonlinear embeddings via scheduling maps. The work shows that, given sufficient rich data, exact interpolation is possible, while insufficient data yields non-uniqueness that can be exploited for data-driven trajectory planning and approximation, potentially extendable to noisy data.

Abstract

We consider the problem of estimating missing values in trajectories of linear parameter-varying (LPV) systems. We solve this interpolation problem for the class of shifted-affine LPV systems. Conditions for the existence and uniqueness of solutions are given and a direct data-driven algorithm for its computation is presented, i.e., the data-generating system is not given by a parametric model but is implicitly specified by data. We illustrate the applicability of the proposed solution on illustrative examples of a mass-spring-damper system with exogenous and endogenous parameter variation.

Figures (5)

• Figure 1: The true trajectory $w_{[1,L]}$ of the MSD system ( ), the given points $w_{\mathbb{I}_\mathrm{g}}$ (indicated by $\ast$), and the missing points $w_{\mathbb{I}_\mathrm{m}}$ (indicated by $\circ$).
• Figure 2: The true trajectory $w_{[1,L]}$ of the MSD system ( ), the given points~$w_{\mathbb{I}_\mathrm{g}}$ (indicated by $\ast$), and the interpolated trajectory $\tilde{w}_{[1,L]}$ ( ) that is obtained from Algorithm \ref{['alg:interp']}.
• Figure 3: The true trajectory $w_{[1,L]}$ of the MSD system ( ) displayed with the given points~$w_{\mathbb{I}_\mathrm{g}}$ (indicated by $\ast$), where $K = 10$. Additionally, some valid interpolants of $w_{\mathbb{I}_\mathrm{g}}$ are shown in blue ( ). For this case, Condition \ref{['con:int:unique']} is not satisfied and there are multiple (in fact, an infinite number of) valid solutions for $\tilde{w}_{[1,L]}$, characterized as in Remark \ref{['rem:minsamp']}.
• Figure 4: Solving the interpolation problem in a data-driven control context, where we solved \ref{['eq:intcon']} for (exponentially) increasing $Q$, shown in a gradient from blue ($Q=10^{-4}$) to red ($Q=1$). The reference points for $u$ and $y$ are indicated with '$\ast$'.
• Figure 5: Solving the interpolation problem for an LPV embedding of a nonlinear system in the data-driven control context of Example 2. The solution obtained with fmincon is shown in black ( ). The solution obtained using the SQP method is shown in blue ( ). The (semi) transparent blue lines correspond to intermediate solutions of the iterative SQP method, where a higher opacity corresponds to a higher iteration. The reference points for~$u$ and $y$ are indicated with '$\ast$'.