Determining optimal input-output properties: A data-driven approach

Abstract

Due to their relevance in systems analysis and (robust) controller design, we consider the problem of determining controltheoretic system properties of an a priori unknown system from data only. More specifically, we introduce a necessary and sufficient condition for a discrete-time linear time-invariant system to satisfy a given integral quadratic constraint (IQC) over a finite time horizon using only one input-output trajectory of finite length. Furthermore, for certain classes of IQCs, we provide convex optimization problems in form of semidefinite programs (SDPs) to retrieve the optimal, i.e. the tightest, system property description that is satisfied by the unknown system. Finally, we provide bounds on the difference between finite and infinite horizon IQCs and illustrate the effectiveness of the proposed scheme in a variety of simulation studies including noisy measurements and a high dimensional system.

Data-driven inference on optimal input-output properties of polynomial systems with focus on nonlinearity measures

“…To apply both congruence transformation in the sequel, we calculate for [16] (Section 4.2). Applying the congruence transformation with Y to X 0 yields (37) and applying the congruence transformations with diag(Y, I nz , I nx ) to (40) with Y is invertible as V is invertible yields (41) with…”

“…Before Theorem 4 is employed for a numerical example, some comments are appropriate. First, Lemma 4 is equivalent to Theorem 4 while the matrix inequalities (37) and (39) are linear in the optimization variables X, Y −1 , τ Σ1 , . .…”

“…The focus of this section is the extension of Theorem 4 to determine for polynomial systems (7) more general optimal input-output properties specified by certain classes of time domain hard IQCs similar to [37]. Subsequent, we highlight the flexibility of the presented framework, e.g., by the consideration of non-polynomial nonlinear system dynamics which are polynomial sector-bounded.…”

“…Proof: The claim follows analogously to Theorem 4. Applying a Schur complement on (46) as in [16] (Lemma 4.2), then use the congruence transformation from Theorem 4 including Y, and thereafter exploit the generalized S-procedure from Lemma 4 yields that (37) and (46) imply…”

Data-Driven Controller Design via Finite-Horizon Dissipativity

“…It was successfully applied in several fields, for example in datadriven simulation (Markovsky and Rapisarda (2008)), system analysis (e.g. ; Romer et al (2019); Koch et al (2020)), controller design (e.g. Markovsky and Rapisarda (2007); ; De Persis and Tesi (2020); Berberich et al (2020b)) and model predictive control (Coulson et al (2019); Berberich et al (2020a)).…”