In this paper a novel computational technique for finite discrete approximation of continuous dynamical systems suitable for a significant class of biochemical dynamical systems is introduced. The method is parameterized in order to affect the imposed level of approximation provided that with increasing parameter value the approximation converges to the original continuous system. By employing this approximation technique, we present algorithms solving the reachability problem for biochemical dynamical systems. The presented method and algorithms are evaluated on several exemplary biological models and on a real case study. This is a full version of the paper published in the proceedings of CompMod 2011.
Discrete approximation methods are commonly used in continuous and hybrid systems analysis (see @cite_19 for an overview regarding reachability) to handle the uncountability of the state space. Direct methods work on the original system and rely on a successor operation iteratively computing the reachable set whereas indirect methods abstract from the continuous model by a finite structure for which the analysis is simpler. Our method belongs to the latter class, since it uses numerical simulations and creates the abstraction automaton. Considering a fixed set of initial conditions, there is a certain overhead with generating states of the automaton in comparison with simple numerical simulations. However, the advantage of constructing the automaton is obtaining a global view of the dynamics. Moreover, in addition to rectangular abstraction, the automaton is augmented with weighted transitions which represent quantitative information describing volumes of subsets of initial conditions belonging to attraction basins of different parts of the phase space.
@cite_19
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We investigate techniques for automatically generating symbolic approximations to the time solution of a system of differential equations. This is an important primitive operation for the safety analysis of continuous and hybrid systems. In this paper we design a time elapse operator that computes a symbolic over-approximation of time solutions to a continuous system starting from a given initial region. Our approach is iterative over the cone of functions (drawn from a suitable universe) that are non negative over the initial region. At each stage, we iteratively remove functions from the cone whose Lie derivatives do not lie inside the current iterate. If the iteration converges, the set of states defined by the final iterate is shown to contain all the time successors of the initial region. The convergence of the iteration can be forced using abstract interpretation operations such as widening and narrowing. We instantiate our technique to linear hybrid systems with piecewise-affine dynamics to compute polyhedral approximations to the time successors. Using our prototype implementation TIMEPASS, we demonstrate the performance of our technique on benchmark examples. A new approach is presented for computing approximations of the reached sets of linear hybrid automata. First, we present some new theoretical results on termination of a class of reachability algorithms, which includes Botchkarev's, based on ellipsoidal calculus. The main contribution of the paper is a revised reachability computation that avoids the approximations caused by the union operation in the discretized flow tube estimation. Therefore, the new algorithm may classify as unreachable states that are reachable according to the previous algorithm because of the looser over-approximations introduced by the union operation. We implemented the new reachability algorithm and tested it successfully on a real-life case modeling a hybrid model of a controlled car engine. Many applications in science and engineering lead to models which require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden’s family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. Abstract We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.