Reachability Analysis of Multi-affine Systems

Theoretical Computer Science, 2016

Reachability for piecewise affine systems is known to be undecidable, starting from dimension 2. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region-to-region bounded time versions leads to NP-complete or co-NP-complete problems, starting from dimension 2. We also prove that a bounded precision version leads to P SP ACE-complete problems.

Theoretical Computer Science, 2007

In this work we are concerned with the formal verification of two-dimensional non-deterministic hybrid systems, namely polygonal differential inclusion systems (SPDIs). SPDIs are a class of nondeterministic systems that correspond to piecewise constant differential inclusions on the plane, for which we study the reachability problem. Our contribution is the development of an algorithm for solving exactly the reachability problem of SPDIs. We extend the geometric approach due to Maler and Pnueli [MP93] to non-deterministic systems, based on the combination of three techniques: the representation of the two-dimensional continuous-time dynamics as a one-dimensional discrete-time system (using Poincaré maps), the characterization of the set of qualitative behaviors of the latter as a finite set of types of signatures, and acceleration used to explore reachability according to each of these types.

2004

We describe techniques to generate useful reachability information for nonlinear dynamical systems. These techniques can be automated for polynomial systems using algorithms from computational algebraic geometry. The generated information can be incorporated into other approaches for doing reachability computation. It can also be used when abstracting hybrid systems that contain modes with nonlinear dynamics. These techniques are most naturally embedded in the hybrid qualitative abstraction approach proposed by the authors previously. They also show that the formal qualitative abstraction approach is well suited for dealing with nonlinear systems.

Computer Aided Verification, 2020

Reachability analysis is a critical tool for the formal verification of dynamical systems and the synthesis of controllers for them. Due to their computational complexity, many reachability analysis methods are restricted to systems with relatively small dimensions. One significant reason for such limitation is that those approaches, and their implementations, are not designed to leverage parallelism. They use algorithms that are designed to run serially within one compute unit and they can not utilize widely-available high-performance computing (HPC) platforms such as many-core CPUs, GPUs and Cloud-computing services. This paper presents PIRK, a tool to efficiently compute reachable sets for general nonlinear systems of extremely high dimensions. PIRK can utilize HPC platforms for computing reachable sets for general highdimensional non-linear systems. PIRK has been tested on several systems, with state dimensions up to 4 billion. The scalability of PIRK's parallel implementations is found to be highly favorable. Keywords: Reachability analysis • ODE integration • Runge-Kutta method • Mixed monotonicity • Monte Carlo simulation • Parallel algorithms A. Devonport and M. Khaled-Contributed equally.

1997

Hybrid systems possess continuous dynamics defined within regions of state spaces and discrete transitions among the regions. Many practical control verification and synthesis tasks can be reduced to reach ability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are defined by piecewise constant vector fields and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector fields. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector fields geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satisfiability of quantified boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli [2] that proves computational undecidability for three-dimensional constant-derivative systems.

2008 American Control Conference, 2008

In this paper, we consider discrete-time continuous-space Piecewise Affine (PWA) systems with uncertain parameters, and study temporal logic properties of their trajectories. Specifically, given a PWA system with polyhedral parameter uncertainties and a Linear Temporal Logic (LTL) formula over linear predicates in its state variables, we attempt to find the largest region of initial states from which all trajectories of the system satisfy the formula. Our method is based on the iterative computation and model checking of finite transition systems simulating the original PWA system. We illustrate our method by computing the basins of attraction for the two equilibria of a PWA model of a two-gene network.

2004

In this paper we present an abstraction method for nonlinear continuous systems. The main idea of our method is to project out some continuous variables, say z, and treat them in the dynamics of the remaining variables x as uncertain input. Therefore, the dynamics of x is then described by a differential inclusion. In addition, in order to avoid excessively conservative abstractions, the domains of the projected variables are divided into smaller regions corresponding to different differential inclusions. The final result of our abstraction procedure is a hybrid system of lower dimension with some important properties that guarantee convergence results. The applicability of this abstraction approach depends on the ability to deal with differential inclusions. We then focus on uncertain bilinear systems, a simple yet useful class of nonlinear differential inclusions, and develop a reachability technique using optimal control. The combination of the abstraction method and the reachability analysis technique for bilinear systems allows to treat multi-affine systems, which is illustrated with a biological system.

Lecture Notes in Computer Science, 2020

Reachability analysis techniques aim to compute which states a dynamical system can enter. The analysis of systems described by nonlinear differential equations is known to be particularly challenging. Hybridization methods tackle this problem by abstracting nonlinear dynamics with piecewise linear dynamics around the reachable states, with additional inputs to ensure overapproximation. This reduces the analysis of a system with nonlinear dynamics to the one with piecewise affine dynamics, which have powerful analysis methods. In this paper, we present improvements to the hybridization approach based on a dynamics scaling model transformation. The transformation aims to reduce the sizes of the linearization domains, and therefore reduces overapproximation error. We showcase the efficiency of our approach on a number of nonlinear benchmark instances, and compare our approach with Flow*.

Proceedings of the 41st IEEE Conference on Decision and Control, 2002.

Given a multi-affine system on an AE-dimensional rectangle, the problem of reaching a particular facet, using multiaffine state feedback is studied. Necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the rectangle, and a method for constructing a multi-affine state feedback solution is presented. The technique is applied to the control of hybrid models of bioregulatory networks.

Ural Mathematical Journal, 2020

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval...