Computing the viability kernel is key in providing guarantees of safety and proving existence of safety-preserving controllers for constrained dynamical systems. Current numerical techniques that approximate this construct suffer from a complexity that is exponential in the dimension of the state. We study conditions under which a linear time-invariant (LTI) system can be suitably decomposed into lower dimensional subsystems so as to admit a conservative computation of the viability kernel in a decentralized fashion in subspaces. We then present an isomorphism that imposes these desired conditions, most suitably on two-time-scale systems. Decentralized computations are performed in the transformed coordinates, yielding a conservative approximation of the viability kernel in the original state space. Significant reduction of complexity can be achieved, allowing the previously inapplicable tools to be employed for treatment of higher dimensional systems. We show the results on two examples including a 6-D system.
In @cite_22 , an approximate dynamic programming technique is presented that, although still grid-based, enables a more efficient computation of the viability kernel. The viability kernel (similarly to @cite_4 ) is expressed as the zero sublevel set of the value function of the corresponding optimal control problem. It is assumed that the value function, which is a viscosity solution of a HJB PDE, is differentiable everywhere on the constraint set. The PDE is then discretized and the resulting value function is numerically computed on a grid using a function approximator such as the @math -nearest neighbor algorithm. The error-bounded approximation is not conservative (it is an over-approximation) but converges to the true viability kernel in the limit as the number of grid points goes to infinity.
@cite_4 @cite_22
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In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces. The solution of a particular Hamilton-Jacobi (HJ) partial differential equation (PDE) provides an implicit representation of reach sets and tubes for continuous systems with nonlinear dynamics and can treat inputs in either worst-case or best-case fashion; however, it can rarely be determined analytically and its numerical approximation typically requires computational resources that grow exponentially with the state space dimension. In this paper we describe a new formulation - also based on HJ PDEs - for reach sets and tubes of systems where some states are terminal integrators: states whose evolution can be written as an integration over time of the other states. The key contribution of this new mixed implicit explicit (MIE) scheme is that its computational cost is linear in the number of terminal integrators, although still exponential in the dimension of the rest of the state space. Application of the new scheme to four examples of varying dimension provides empirical evidence of its considerable improvement in computational speed. This paper presents a bounded model checking tool called @math for hybrid systems. It translates a reachability problem of a nonlinear hybrid system into a predicate logic formula involving arithmetic constraints and checks the satisfiability of the formula based on a satisfiability modulo theories method. We tightly integrate (i) an incremental SAT solver to enumerate the possible sets of constraints and (ii) an interval-based solver for hybrid constraint systems (HCSs) to solve the constraints described in the formulas. The HCS solver verifies the occurrence of a discrete change by using a set of boxes to enclose continuous states that may cause the discrete change. We utilize the existence property of a unique solution in the boxes computed by the HCS solver as (i) a proof of the reachability of a model and (ii) a guide in the over-approximation refinement procedure. Our @math implementation successfully handled several examples including those with nonlinear constraints. We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier-Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods-since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.