Pierre-Loïc Garoche

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. n nIn 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. n nOur 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.

A Property-Driven Approach to Formal Verification of Process Models

More and more, models, through Domain Specific Languages (DSL), tend to be the solution to define complex systems. Expressing properties specific to these metamodels, and checking them, appear as an urgent need. Until now, the only complete industrial solutions that are available consider structural properties such as the ones that could be expressed in OCL. There are although some attempts on behavioural properties for DSL.

Property-based Polynomial Invariant Generation Using Sums-of-Squares Optimization

While abstract interpretation is not theoretically restricted to specific kinds of properties, it is, in practice, mainly developed to compute linear over-approximations of reachable sets, aka. the collecting semantics of the program. The verification of user-provided properties is not easily compatible with the usual forward fixpoint computation using numerical abstract domains.

Static safety for an actor dedicated process calculus by abstract interpretation

The actor model eases the definition of concurrent programs with non uniform behaviors. Static analysis of such a model was previously done in a data-flow oriented way, with type systems. This approach was based on constraint set resolution and was not able to deal with precise properties for communications of behaviors. We present here a new approach, control-flow oriented, based on the abstract interpretation framework, able to deal with communication of behaviors. Within our new analyses, we are able to verify most of the previous properties we observed as well as new ones, principally based on occurrence counting.

Formal Analysis of Robustness at Model and Code Level

Robustness analyses play a major role in the synthesis and analysis of controllers. For control systems, robustness is a measure of the maximum tolerable model inaccuracies or perturbations that do not destabilize the system. Analyzing the robustness of a closed-loop system can be performed with multiple approaches: gain and phase margin computation for single-input single-output (SISO) linear systems, mu analysis, IQC computations, etc. However, none of these techniques consider the actual code in their analyses. The approach presented here relies on an invariant computation on the discrete system dynamics. Using semi-definite programming (SDP) solvers, a Lyapunov-based function is synthesized that captures the vector margins of the closed-loop linear system considered. This numerical invariant expressed over the state variables of the system is compatible with code analysis and enables its validation on the code artifact. This automatic analysis extends verification techniques focused on controller implementation, addressing validation of robustness at model and code level. It has been implemented in a tool analyzing discrete SISO systems and generating over-approximations of phase and gain margins. The analysis will be integrated in our toolchain for Simulink and Lustre models autocoding and formal analysis.

Automatic synthesis of k-inductive piecewise quadratic invariants for switched affine control 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 discrete-time settings to fit the static analysis paradigm and the computation of invariants, that is over-approximation of reachable sets using piecewise quadratic Lyapunov functions. This approach has been further extended to consider k-inductive properties of reachable traces (trajectories) of systems. The analysis has been implemented in Matlab and shown very good experimental results on a very large set of synthesized problems.