Robust targeted exploration for systems with non-stochastic disturbances

TL;DR

The paper addresses robust parameter identification for uncertain LTI systems with energy-bounded disturbances by designing targeted exploration inputs. It develops a data-dependent uncertainty bound and derives spectral-content–based sufficient conditions, culminating in an SDP that minimizes exploration energy while guaranteeing a prescribed parameter accuracy. The approach accommodates non-stochastic disturbances and bounded nonlinearities, using multi-sine inputs across chosen frequencies and convex relaxations to enforce LMIs. Numerical results on a nonlinear chain demonstrate a priori guarantees and practical efficiency, with clear advantages over naive exploration. This work provides a principled, robust framework for optimal experiment design under worst-case disturbances and opens avenues for dual-control integration.

Abstract

In this paper, we introduce a novel targeted exploration strategy designed specifically for uncertain linear time-invariant systems with energy-bounded disturbances, i.e., without making any assumptions on the distribution of the disturbances. We use classical results characterizing the set of non-falsified parameters consistent with energy-bounded disturbances. We derive a semidefinite program which computes an exploration strategy that guarantees a desired accuracy of the parameter estimate. This design is based on sufficient conditions on the spectral content of the exploration data that robustly account for initial parametric uncertainty. Finally, we highlight the applicability of the exploration strategy through a numerical example involving a nonlinear system.

Figures (5)

• Figure 1: Illustration of the exploration input energy $\gamma_\mathrm{e}^2$, in comparison with the disturbance energy bound $\gamma_\mathrm{w}$ for the initial uncertainty level $\|D_0\|=10^2$.
• Figure 2: Illustration of the a posteriori guaranteed bound on the squared error of the parameters $\|G \cdot P\|$, in comparison with the desired bound on the squared error $\|D_\mathrm{des}^{-1}\|$ for different initial uncertainty bounds $D_0^{-1}$.
• Figure 3: Illustration of the a posteriori guaranteed bound on the squared error of the parameters $\|G \cdot P\|$ for both targeted and naive exploration with same input energy, for different initial uncertainty bounds $D_0^{-1}$.
• Figure 4: Illustration of the distribution of the input energy $\gamma_\mathrm{e}^2$ for different initial uncertainty bounds $D_0^{-1}$.
• Figure 5: Illustration of input energy $\gamma_\mathrm{e}^2$ for different desired uncertainty levels $\|D_\mathrm{des}\|$.

Theorems & Definitions (11)

• Remark 3
• Remark 5
• Lemma 6
• Theorem 7
• Lemma 9
• Lemma 10
• Proposition 11
• Lemma 12
• Proposition 13
• Theorem 14