Robust targeted exploration for systems with non-stochastic disturbances

We propose a novel targeted exploration strategy designed specifically for uncertain linear time-invariant systems with energy-bounded disturbances, i.e., without any assumptions on the distribution of the disturbances. We use classical results characterising 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.

💡 Research Summary

The paper addresses the problem of designing an informative experiment (targeted exploration) for linear time‑invariant (LTI) systems when the disturbances are not stochastic but are only known to be energy‑bounded. Unlike classical optimal experiment design, which typically assumes i.i.d. Gaussian noise and builds confidence ellipsoids based on statistical properties, this work adopts a set‑membership perspective: given a finite data record and a known bound γ_w on the total disturbance energy, the set of parameters that are consistent with the data (the “non‑falsified” set) can be described exactly as an ellipsoid centered at the least‑squares estimate θ̂_T with shape matrix P = (ΦΦᵀ)⁻¹⊗I_{n_x} and radius G = γ_w + ‖θ̂_T‖_{P⁻¹}² – ‖X‖². This ellipsoid depends on the data and on the unknown disturbance, which makes the design of an experiment more challenging.

The goal is to choose an input sequence u_k over a horizon T such that the estimated parameters satisfy a user‑specified accuracy requirement (θ_tr – θ̂_T)ᵀ(D_des⊗I_{n_x})(θ_tr – θ̂_T) ≤ 1, where D_des ≻ 0 encodes the desired precision. The authors restrict the input to a sum of sinusoids at L distinct frequencies ω_i, i = 1,…,L, with amplitudes ū(ω_i). The amplitudes are collected in a diagonal matrix U_e = diag(ū(ω_1),…,ū(ω_L)). The total input energy is bounded by a scalar γ_e, which can be expressed as a linear matrix inequality (LMI) via the Schur complement, yielding the matrix S_energy(γ_e, U_e) ≽ 0.

To relate the input amplitudes to the state trajectories, the paper uses spectral‑line theory. For each frequency ω_i, the steady‑state response of the state is expressed as a linear combination of the input amplitude and the disturbance amplitude: \bar{x}(ω_i) = V_{x,i} ū(ω_i) + Y_{x,i} \bar{w}(ω_i) + \bar{x}{err}(ω_i). Here V{x,i} = (e^{j2π ω_i}I – A_tr)⁻¹B_tr and Y_{x,i} = (e^{j2π ω_i}I – A_tr)⁻¹. The transient error term \bar{x}{err} decays as 1/√T and is assumed negligible for sufficiently large T. Stacking all frequencies gives \bar{X} = V{x,tr} U_e + Y_{x,tr} W, where W contains the (unknown) disturbance spectral amplitudes.

The central technical contribution is Theorem 7, which provides a sufficient LMI condition (inequality (15) in the paper) that guarantees the accuracy requirement. This condition involves the data matrices Φ and X quadratically, which in turn depend on the unknown amplitudes U_e and the disturbance. By substituting the spectral‑line expressions for Φ and X, the authors derive explicit LMI constraints that are linear (or convex quadratic) in U_e and γ_e, while still accounting for the worst‑case disturbance within the energy bound.

Because the original sufficient condition is non‑convex, the authors propose a convex relaxation that yields a semidefinite program (SDP). The SDP simultaneously minimizes the input energy γ_e and enforces the LMI derived from Theorem 7, together with the energy‑bound LMI S_energy(γ_e, U_e) ≽ 0. The resulting optimization problem is tractable with standard SDP solvers.

A numerical example demonstrates the method on a nonlinear system that is linearized for the purpose of experiment design. Starting from an initial uncertainty set Θ_0, the SDP computes sinusoidal inputs that drive the system so that the final parameter estimate lies inside the desired precision ellipsoid, while using significantly less energy than a naïve uniform excitation. The example also shows robustness: even if the disturbance is adversarial (as long as its total energy respects γ_w), the guaranteed bound on the estimation error holds.

In summary, the paper introduces a robust, non‑stochastic targeted exploration framework for LTI systems with energy‑bounded disturbances. By leveraging set‑membership estimation, spectral‑line analysis, and LMI‑based SDP design, it provides a systematic way to compute low‑energy excitation signals that ensure a prescribed parameter‑estimation accuracy, without relying on probabilistic disturbance models. This approach broadens optimal experiment design to settings where disturbances are deterministic, possibly adversarial, and only known through energy constraints, making it highly relevant for safety‑critical applications such as robotics, power systems, and aerospace where worst‑case guarantees are essential.