Computation of an over-approximation of the backward reachable set using subsystem level set functions (original) (raw)
Algorithms for Computing Reachable Sets and Control Sets
IFAC Proceedings Volumes, 2001
Recently, considerable progress has been made in the numerical computation of reachable sets and control sets. It is the purpose of this paper to survey some of these developments. In general, computation of reachable sets and control sets is a very difficult problem, since the objects which are to be computed have full dimension in the state space. In turns out that certain reformulated problems which in many cases give the desired objects are numerically easier to handle.
arXiv (Cornell University), 2022
This work presents a method of efficiently computing inner and outer approximations of forward reachable sets for nonlinear control systems with changed dynamics and diminished control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking or inflating a precomputed reachable set based on prior knowledge of the system's trajectory deviation growth dynamics, depending on whether an inner approximation or outer approximation is desired. These dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated on the nominal system using nominal control inputs, and by the impaired system based on the diminished set of control inputs, respectively. The dynamics depend on the given Hausdorff distance bound between the nominal set of admissible controls and the possibly unknown impaired space of admissible controls, as well as a bound on the rate change between the nominal and off-nominal dynamics. Because of its computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to on-board fault-tolerant path planning and failure recovery. In addition, the proposed algorithm does not require convexity of the reachable sets unlike our previous work, thereby making it suitable for general use. We raise a number of implementational considerations for our algorithm, and we present three illustrative examples, namely an application to the heading dynamics of a ship, a lower triangular dynamical system, and a system of coupled linear subsystems.
Reachability Analysis of Nonlinear Systems Using Conservative Approximation
2003
In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C 2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.
Determination of Inner and Outer Bounds of Reachable Sets Through Subpavings
1 Abstract The computation of the reachable set of states of a given dynamic system is an important step to verify its safety during operation. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function, and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by another guaranteed integration method. The methods developed were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step worse results are obtained.
Data-Driven Reachability Analysis with Christoffel Functions
2021 60th IEEE Conference on Decision and Control (CDC), 2021
We present an algorithm for data-driven reachability analysis that estimates finite-horizon forward reachable sets for general nonlinear systems using level sets of a certain class of polynomials known as Christoffel functions. The level sets of Christoffel functions are known empirically to provide good approximations to the support of probability distributions: the algorithm uses this property for reachability analysis by solving a probabilistic relaxation of the reachable set computation problem. We also provide a guarantee that the output of the algorithm is an accurate reachable set approximation in a probabilistic sense, provided that a certain sample size is attained. We also investigate three numerical examples to demonstrate the algorithm's capabilities, such as providing nonconvex reachable set approximations and detecting holes in the reachable set.
Recent progress in continuous and hybrid reachability analysis
2006
Set-based reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachability analysis has been limited so far to simple systems, both in the continuous and the hybrid domain. In this paper we present recent advances that, in combination, significantly improve this applicability, and allow us to find better balance between computational cost and accuracy. The presentation covers, in a unified manner, a variety of methods handling increasingly complex types of continuous dynamics (constant derivative, linear, nonlinear). The improvements include new geometrical objects for representing sets, new approximation schemes, and more flexible combinations of graph-search algorithm and partition refinement. We report briefly some preliminary experiments that have enabled the analysis of systems previously beyond reach.
Overapproximating the reachable sets of LTI systems through a similarity transformation
2010
We present a decomposition method for complexity reduction in reachability analysis and controller synthesis based on a series of transformations. The decomposition is guaranteed to yield weakly-coupled (lower dimensional) subsystems with disjoint control input across them. Reachable sets, computed independently for each subsystem, are backprojected and intersected to yield an overapproximation of the actual reachable set. Using an example we show that significant reduction in the computational costs can be achieved. This technique has considerable potential utility for use in conjunction with computationally intensive reachability tools.
Hybrid Control Synthesis for Eventuality Specifications Using Level Set Methods
International Journal of Control, 2004
This paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications. Given a hybrid system modelled by a hybrid automaton and a target set of states, the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time. Due to the existence of set-valued disturbance inputs, the problem is studied in a game-theoretic framework. Having shown that a least restrictive solution does not exist, we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property. To implement the algorithm, reachable sets of pursuit-evasion differential games need to be computed. For that reason level set methods are employed, where the boundary of the reachable set is characterized as the zero level set of a Hamilton-Jacobi equation. The procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology. Finally, to demonstrate the practical character of our results, a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology.
PIRK: Scalable Interval Reachability Analysis for High-Dimensional 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.
Verification by approximate forward and backward reachability
1998
Approximate reachability techniques trade o accuracy for the capacity to deal with bigger designs. In this paper, we extend the idea of approximations using overlapping projections to symbolic backward reachability. This is combined with a previous method of computing overapproximate forward reachable state sets using overlapping projections. The algorithm computes a superset of the set of states that lie on a path from the initial state to a state that violates a speci ed invariant property. If this set is empty, there is no possibility of violating the invariant. If this set is non-empty, it may be possible to prove the existence of such a path by searching for a counter-example. A simple heuristic is given, which seems to work well in practice, for generating a counter-example path from this approximation. We evaluate these new algorithms by applying them to several control modules from the I/O unit in the Stanford FLASH Multiprocessor.