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.
Recently, we presented a connection between the viability kernel and efficiently-computable classes of reachability constructs known as maximal reachable sets. Owing to this connection, scalable numerical algorithms (collectively referred to as @cite_48 ) such as @cite_6 @cite_27 @cite_1 @cite_13 @cite_2 @cite_36 @cite_17 , originally developed for maximal reachability, can now be used to approximate the viability kernel. We presented two algorithms for LTI systems with convex constraints based on piecewise ellipsoidal representations @cite_20 and support vectors @cite_0 that have polynomial complexity. In contrast to these results, the technique presented here reduces the complexity indirectly by decentralizing computations. The benefit of this approach is that it allows useful features of Eulerian methods such gradient-based control synthesis and handling of arbitrarily shaped nonconvex constraints be taken advantage of.
@cite_36 @cite_48 @cite_1 @cite_6 @cite_0 @cite_27 @cite_2 @cite_13 @cite_20 @cite_17
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Abstract While a number of Lagrangian algorithms to approximate reachability in dozens or even hundreds of dimensions for systems with linear dynamics have recently appeared in the literature, no similarly scalable algorithms for approximating viable sets have been developed. In this paper we describe a connection between reachability and viability that enables us to compute the viability kernel using reach sets. This connection applies to any type of system, such as those with nonlinear dynamics and or non-convex state constraints; however, here we take advantage of it to construct three viability kernel approximation algorithms for linear systems with convex input and state constraint sets. We compare the performance of the three algorithms and demonstrate that the two based on highly scalable Lagrangian reachability–those using ellipsoidal and support vector set representations–are able to compute the viability kernel for linear systems of larger state dimension than was previously feasible using traditional Eulerian methods. Our results are illustrated on a 6-dimensional pharmacokinetic model and a 20-dimensional model of heat conduction on a lattice. We present a connection between the viability kernel and maximal reachable sets. Current numerical schemes that compute the viability kernel suffer from a complexity that is exponential in the dimension of the state space. In contrast, extremely efficient and scalable techniques are available that compute maximal reachable sets. We show that under certain conditions these techniques can be used to conservatively approximate the viability kernel for possibly high-dimensional systems. We demonstrate the results on two practical examples, one of which is a seven-dimensional problem of safety in anesthesia. The thesis describes a new method for obtaining minimizers for optimal control problems whose minima serve as control policies for guiding nonlinear dynamical systems to achieve prescribed goals under imposed trajectory and actuator constraints. One of the major contributions of the present work resides in the approximation of such minimizers by piecewise polynomial functions expressed in terms of a linear combination of non-uniform rational B-spline (NURBS) basis functions and the judicious exploitation of the properties of the resulting NURBS curves to improve the computational effort often associated with solving optimal control problems for constrained dynamical systems. In particular, by exploiting the two structures combined in a NURBS curve, NURBS basis functions and an associated union of overlapping polytopes constructed from the coefficients of the linear combination, we are able to separate an optimal control problem into two subproblems | guidance and obstacle avoidance, making the original problem tractable. This is accomplished by laying out the union of overlapping polytopes in such a way that they delineate a section of space that avoids all obstacles and then manipulating the NURBS basis functions to obtain trajectories that are guaranteed to remain bounded by this section of space without explicitly including the conjunction of disjunctions naturally induced from obstacles into the guidance problem. In addition, we show how one can construct systematically a feasible region that corresponds to a NURBS parameterization starting from an ordered union of pairwise adjacently overlapping nonempty compact convex sets. Specifically, we show how to setup a nonlinear programming problem to solve for the feasible region in terms of an ordered union of pairwise adjacently overlapping polytopes with nonempty interiors by maximizing the sum of their volumes and starting from a feasible region described by an ordered union of pairwise adjacently overlapping nonempty convex compact simi-algebraic sets. Finally, we show how this strategy can be implemented practically for an autonomous system traversing an urban environment. Finally, this work culminated in the filing of patent US20070179685 on behalf of Northrop Grumman for the Space Technology sector and in the development of the NURBS-based OTG software package. This C++ package contains the theoretical results of this thesis in the form of an object-oriented implementation optimized for real-time trajectory generation. We propose an algorithm which performs a progressive approximation of a viability kernel, iteratively using a classification method. We establish the mathematical conditions that the classification method should fulfil to guarantee the convergence to the actual viability kernel. We study more particularly the use of support vector machines (SVMs) as classification techniques. We show that they make possible to use gradient optimisation techniques to find a viable control at each time step, and over several time steps. This allows us to avoid the exponential growth of the computing time with the dimension of the control space. It also provides simple and efficient control procedures. We illustrate the method with some examples inspired from ecology