Automatic Synthesis of Piecewise Linear Quadratic Invariants for Programs
Among the various critical systems that are worth to be formally analyzed, a wide set consists of controllers for dynamical systems. Those programs typically execute an infinite loop in which simple computations update internal states and produce commands to update the system state. Those systems are yet hardly analyzable by available static analysis method, since, even if performing mainly linear computations, the computation of a safe set of reachable states often requires quadratic invariants.
In this paper we consider the general setting of a piecewise affine program; that is a program performing different affine updates on the system depending on some conditions. This typically encompasses linear controllers with saturations or controllers with different behaviors and performances activated on some safety conditions.
Our analysis is inspired by works performed a decade ago by Johansson et al, and Morari et al, in the control community. We adapted their method focused on the analysis of stability in continuous-time or discretetime settings to fit the static analysis paradigm and the computation of invariants, that is over-approximation of reachable sets using piecewise quadratic Lyapunov functions.
Keywordsformal verification static analysis piecewise affine systems piecewise quadratic lyapunov functions
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- [All09]Allamigeon, X.: Static analysis of memory manipulations by abstract interpretation — Algorithmics of tropical polyhedra, and application to abstract interpretation. PhD thesis, École Polytechnique, Palaiseau, France (November 2009)Google Scholar
- [BGLM05]Biswas, P., Grieder, P., Löfberg, J., Morari, M.: A Survey on Stability Analysis of Discrete-Time Piecewise Affine Systems. In: IFAC World Congress, Prague, Czech Republic (July 2005)Google Scholar
- [CC77]Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Conference Record of the Fourth Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Los Angeles, California, pp. 238–252. ACM Press, New York (1977)CrossRefGoogle Scholar
- [CH78]Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Aho, A., Zilles, S., Szymanski, T. (eds.) POPL, pp. 84–96. ACM Press (1978)Google Scholar
- [FR94]Filé, G., Ranzato, F.: Improving abstract interpretations by systematic lifting to the powerset. In: Logic Programming, Proc. of the 1994 International Symposium, Ithaca, New York, USA, November 13-17, pp. 655–669 (1994)Google Scholar
- [Joh03]Johansson, M.: On modeling, analysis and design of piecewise linear control systems. In: Proc. of the 2003 International Symposium on Circuits and Systems, ISCAS 2003, vol. 3, pp. III–646–III–649 (May 2003)Google Scholar
- [MFTM00]Mignone, D., Ferrari-Trecate, G., Morari, M.: Stability and stabilization of piecewise affine and hybrid systems: An lmi approach. In: Proc. of the 39th IEEE Conference on Decision and Control, vol. 1, pp. 504–509 (2000)Google Scholar
- [Min01]Miné, A.: A new numerical abstract domain based on difference-bound matrices. In: Danvy, O., Filinski, A. (eds.) PADO-II 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001)Google Scholar
- [RJGF12]Roux, P., Jobredeaux, R., Garoche, P.-L., Feron, E.: A generic ellipsoid abstract domain for linear time invariant systems. In: Dang, T., Mitchell, I. (eds.) HSCC, pp. 105–114. ACM (2012)Google Scholar