Parabolic Set Simulation for Reachability Analysis of Linear Time Invariant Systems with Integral Quadratic Constraint

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.

[1]  Alex M. Andrew,et al.  Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics , 2002 .

[2]  Pravin Varaiya,et al.  On Ellipsoidal Techniques for Reachability Analysis. Part I: External Approximations , 2002, Optim. Methods Softw..

[3]  Ulf T. Jönsson.  Robustness of trajectories with finite time extent , 2002, Autom..

[4]  I. Petersen,et al.  Recursive state estimation for uncertain systems with an integral quadratic constraint , 1995, IEEE Trans. Autom. Control..

[5]  Alexandre M. Bayen,et al.  Aircraft Autolander Safety Analysis Through Optimal Control-Based Reach Set Computation , 2007 .

[6]  F. Gouaisbaut,et al.  Stability analysis of time‐delay systems via Bessel inequality: A quadratic separation approach , 2018 .

[7]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis of Discrete-Time Linear Systems , 2007, IEEE Transactions on Automatic Control.

[8]  Khalik G. Guseinov.  Approximation of the attainable sets of the nonlinear control systems with integral constraint on controls , 2009 .

[9]  Dimitri Peaucelle,et al.  Integral Quadratic Separators for performance analysis , 2009, 2009 European Control Conference (ECC).

[10]  Peter Seiler,et al.  Finite Horizon Robustness Analysis of LTV Systems Using Integral Quadratic Constraints , 2017, Autom..

[11]  Demetrios Lainiotis.  Generalized Chandrasekhar algorithms: Time-varying models , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[12]  Didier Henrion,et al.  Convex Computation of the Region of Attraction of Polynomial Control Systems , 2012, IEEE Transactions on Automatic Control.

[13]  C. Kenney,et al.  Numerical integration of the differential matrix Riccati equation , 1985 .

[14]  Milan Korda,et al.  Moment-sum-of-squares hierarchies for set approximation and optimal control , 2016 .

[15]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[16]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[17]  A. Helmersson.  An IQC-Based Stability Criterion for Systems with Slowly Varying Parameters , 1999 .

[18]  Vladimír Kucera,et al.  A review of the matrix Riccati equation , 1973, Kybernetika.

[19]  A. Rantzer,et al.  System analysis via integral quadratic constraints , 1997, IEEE Trans. Autom. Control..

[20]  Mikhail I. Gusev,et al.  On Extremal Properties of Boundary Points of Reachable Sets for a System with Integrally Constrained Control , 2017 .

[21]  F. L. Chernousko.  What is Ellipsoidal Modelling and How to Use It for Control and State Estimation , 1999 .

[22]  Pravin Varaiya,et al.  Reach Set Computation Using Optimal Control , 2000 .

[23]  Decision Systems.,et al.  Integral quadratic constraints for systems with rate limiters , 1997 .

[24]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[25]  H. Hermes,et al.  Foundations of optimal control theory , 1968 .

[26]  Ian R. Petersen,et al.  Model validation for robust control of uncertain systems with an integral quadratic constraint , 1996, Autom..

[27]  A. Megretski.  KYP Lemma for Non-Strict Inequalities and the associated Minimax Theorem , 2010, 1008.2552.

[28]  Friedemann Schuricht.  Ordinary differential equations with measurable right-hand side and parameters in metric spaces , 2006 .

[29]  Pierpaolo Soravia.  Viscosity solutions and optimal control problems with integral constraints , 2000 .

[30]  B. Krogh,et al.  Hyperplane method for reachable state estimation for linear time-invariant systems , 1991 .