Construction of Worst-Case Input Signals for Discrete-Time Linear Time-Varying Systems

Code-Level Formal Verification of Ellipsoidal Invariant Sets for Linear Parameter-Varying Systems

“…, n δ and all integers k ≥ 0. The analysis results in [17] are used to determine the state-invariant and output-bounding ellipsoids of system G. To apply these results, system G is first expressed as a linear fractional transformation (LFT) on uncertainties, where the uncertainties in this case are the static linear time-varying perturbations δ i for i = 1, . .…”

“…Namely, an IQC multiplier is used to define a quadratic constraint that the input and output signals of the uncertainty operator must satisfy. In the work [17], this quadratic constraint must be satisfied at every time instant and is hence referred to as a pointwise IQC. A pointwise IQC is more restrictive than the standard IQC [23], which involves an infinite summation of quadratic terms.…”

“…However, the uncertainty set in our problem admits a pointwise IQC characterization. The approach in [17] allows representing the exogenous input d as a pointwise bounded signal, where its value lies in some closed, convex polytope Γ for all time instants or an ellipsoid E. While the analysis conditions provided in [17] are generally nonconvex, they can be relaxed into convex conditions by applying the multiconvexity relaxation technique [1], along with gridding. Thus, the positive definite matrices defining the state-invariant and output-bounding ellipsoids can be obtained by solving semidefinite programs.…”

“…Specifically, we formally verify that, if the state of the system lies in an ellipsoidal invariant set at the initial time, then it resides in this set at all time instants, and, further, the output of the system resides in another ellipsoid at all time instants as well for all permissible pointwise-bounded inputs and parameter trajectories. These sets are obtained by applying new results developed in [17] for computing state-and outputbounding sets for discrete-time uncertain linear fractional transformation (LFT) systems using pointwise integral quadratic constraints (IQCs) to characterize the uncertainties and the S-procedure. Uncertainties that admit pointwise IQC characterizations include static linear time-invariant and time-varying perturbations, sector-bounded nonlinearities, and uncertain time-varying time-delays.…”

“…Additionally, we utilize WP, a Frama-C plugin based on the weakest precondition calculus and deductive methods, to transform annotations and code into proof objectives. Thus, the software verification in our case focuses on translating the guarantees obtained at the algorithmic level, using the analysis results from [17], and expressing them at the code level. Then, we revalidate the invariant properties at the code level using Alt-Ergo-Poly [28], an extension of the SMT solver Alt-Ergo [8] with a sound Sum-of-Squares solver [27,22], to discharge positive polynomial constraints.…”