[Paper Review] Parabolic Set Simulation for Reachability Analysis of Linear Time Invariant Systems with Integral Quadratic Constraint

Paul Rousse, Pierre-Loïc Garoche+1 more|arXiv (Cornell University)|Feb 28, 2019

Fault Detection and Control Systems30 references4 citations

TL;DR

This paper proposes a parabolic set simulation method for reachability analysis of linear time-invariant (LTI) systems subject to integral quadratic constraints (IQC), using time-varying paraboloidal overapproximations that touch the reachable set boundary via solutions to an initial value problem involving a differential Riccati equation. The approach enables stable, scalable, and bounded overapproximation of reachable sets for both stable and unstable systems, outperforming traditional ellipsoidal methods in handling complex constraints.

ABSTRACT

This work extends reachability analyses based on ellipsoidal techniques to Linear Time Invariant (LTI) systems subject to an integral quadratic constraint (IQC) between the past state and disturbance signals , interpreted as an input-output energetic constraint. To compute the reachable set, the LTI system is augmented with a state corresponding to the amount of energy still available before the constraint is violated. For a given parabolic set of initial states, the reachable set of the augmented system is overapproximated with a time-varying parabolic set. Parameters of this paraboloid are expressed as the solution of an Initial Value Problem (IVP) and the overapproximation relationship with the reachable set is proved. This paraboloid is actually supported by the reachable set on so-called touching trajectories. Finally , we describe a method to generate all the supporting paraboloids and prove that their intersection is an exact characterization of the reachable set. This work provides new practical means to compute overapproximation of reachable sets for a wide variety of systems such as delayed systems, rate limiters or energy-bounded linear systems.

Motivation & Objective

• To address the lack of effective reachable set characterization for LTI systems with integral quadratic constraints (IQC), which model delays, rate limiters, and energy bounds.
• To extend set-based simulation techniques beyond ellipsoidal methods to handle non-convex and unstable systems under IQC.
• To develop a scalable, numerically stable method that guarantees bounded overapproximations of reachable sets using paraboloidal envelopes.
• To enable practical reachability analysis for large-scale systems where HJB and moment-based methods fail due to scalability issues.

Proposed method

• The method augments the LTI system with a state representing the integral term in the IQC, transforming the constraint into a state-space form.
• It constructs time-varying paraboloidal overapproximations of the reachable set that touch the boundary along 'touching trajectories' satisfying the IQC.
• Parameters of each paraboloid are derived from solutions to an initial value problem (IVP) involving a differential Riccati equation (DRE), ensuring tightness and contact with the reachable set.
• The reachable set is represented as the intersection of uncountably many such supporting paraboloids, with a finite subset used for practical overapproximation.
• A numerical integration scheme based on the Chandrasekhar method is adapted to solve the DRE even when the solution is not sign-definite.
• An algorithm computes a minimal-volume paraboloid enclosing the intersection of computed paraboloids, improving computational efficiency.

Experimental results

Research questions

• RQ1 Can paraboloidal overapproximations provide a tighter and more scalable alternative to ellipsoidal methods for reachability analysis under IQC constraints?
• RQ2 How can time-varying paraboloids be parameterized to ensure they touch the boundary of the reachable set while respecting integral quadratic constraints?
• RQ3 What conditions ensure boundedness and existence of the overapproximation for unstable LTI systems under IQC?
• RQ4 How can the method be adapted to handle large-scale systems where traditional HJB or moment-based methods fail due to computational intractability?
• RQ5 Can the method be extended to systems with multiple IQCs or mixed 2-norm and ∞-norm constraints?

Key findings

• The method successfully overapproximates the reachable set of LTI systems with IQC constraints using time-varying paraboloids, even for unstable systems.
• The reachable set is exactly represented as the intersection of uncountably many supporting paraboloids, each corresponding to a touching trajectory.
• The approach guarantees bounded overapproximations through careful selection of scaling functions and initial parameters.
• Numerical results show the method scales favorably for systems with less than a hundred states, outperforming HJB and moment-based methods in computational efficiency.
• The implementation is open-source and leverages a modified Chandrasekhar method for solving the DRE, even when the solution is not sign-definite.
• The method enables the computation of minimal-volume enclosing paraboloids, reducing computational cost while preserving tightness.