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.
Direct methods are mostly based on hybridization realized by partitioning the system state space into domains where the local continuous behaviour is linearized @cite_14 . This method, in an improved form, has been applied to non-linear biochemical dynamical systems @cite_7 . In general, direct methods give good results for low-dimensional systems and small initial sets. In comparison with indirect approaches, they are computationally harder. From this viewpoint, our approach lies between both extremes.
@cite_14 @cite_7
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This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. In this article, we describe some recent results on the hybridization methods for the analysis of nonlinear systems. The main idea of our hybridization approach is to apply the hybrid systems methodology as a systematic approximation method. More concretely, we partition the state space of a complex system into regions that only intersect on their boundaries, and then approximate its dynamics in each region by a simpler one. Then, the resulting hybrid system, which we call a hybridization, is used to yield approximate analysis results for the original system. We also prove important properties of the hybridization, and propose two effective hybridization construction methods, which allow approximating the original nonlinear system with a good convergence rate. We propose an abstraction method for medium-scale biomolecular networks, based on hybrid dynamical systems with continuous multi-affine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easily automated. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An approximation for an ordinary differential equation model of the lac operon is constructed, and it is shown that the abstraction passes the same experimental tests as were used to validate the original model. The well studied biological system exhibits bistability and switching behaviour, arising from positive feedback in the expression mechanism of the lac operon. The switching property of the lac system is an example of the major qualitative features that are the building blocks of higher level, more coarse-grained descriptions. The present approach is useful in helping to correctly identify such properties and in connecting them to the underlying molecular dynamical details. Reachability analysis together with the knowledge of the steady-state structure are used to identify ranges of parameter values for which the system maintains the bistable switching property. Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform.