Reachability of Nonlinear Systems With Unknown Dynamics (original) (raw)
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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.
On Reachable Sets for a Class of Nonlinear Systems with Constraints
Journal of Mathematical Analysis and Applications, 1999
The determination of the reachable set for a class of nonlinear systems with control and state trajectory constraints is investigated. The main result links this problem with the determination of the set of admissible controls, for which procedures already exist. The paper also gives a procedure to generate an admissible control which steers the system to a reachable state.
On Estimates of Reachable Sets of Multidimensional Control Systems
Proceedings of the 18th IFAC World Congress, 2011
The paper is devoted to the problem of construction of external estimates for the reachable set of a multidimensional control system by means of vector estimators. A considered system permits a decomposition into several independent subsystems (like for linear subsystems), which are connected to each other by means of nonlinear interconnections. For each of the subsystems, an external estimate of the reachable set is assumed to be known. An estimate for the reachable set of the combined system is given on the base of estimates for subsystems and use of analogs of vector Lyapunov functions.
On reachability analysis for nonlinear control systems with state constraints
Dynamical Systems and Differential Equations, AIMS Proceedings 2015 Proceedings of the 10th AIMS International Conference (Madrid, Spain)
The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set for a nonlinear inequality. A procedure to remove state constraints is proposed; this procedure consists in replacing a primary system by an auxiliary system without state constraints. The equations of the auxiliary system depend on a small parameter. It is shown that a reachable set of the primary system may be approximated in the Hausdorff metric by reachable sets of the auxiliary system when the small parameter tends to zero. The estimates of the rate of convergence are given.
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.
Nonlinear Systems: Approximating Reach Sets
2004
We describe techniques to generate useful reachability information for nonlinear dynamical systems. These techniques can be automated for polynomial systems using algorithms from computational algebraic geometry. The generated information can be incorporated into other approaches for doing reachability computation. It can also be used when abstracting hybrid systems that contain modes with nonlinear dynamics. These techniques are most naturally embedded in the hybrid qualitative abstraction approach proposed by the authors previously. They also show that the formal qualitative abstraction approach is well suited for dealing with nonlinear systems.
Inner and outer reachability for the verification of control systems
Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control
We investigate the information and guarantees provided by different inner and outer approximated reachability analyses, for proving properties of dynamical systems. We explore the connection of these approximated sets with the maximal and minimal reachable sets of Mitchell [31], with an additional notion of robustness to disturbance. We demonstrate the practical use of a specific computation of these approximated reachable sets. We revisit in particular the reachavoid properties.
Set-Membership Computation of Admissible Controls for Trajectory Tracking
HAL (Le Centre pour la Communication Scientifique Directe), 2017
In a context of predictive control strategy, this paper addresses the computation of admissible control set for a trajectory tracking over a prediction horizon. The proposed method combines numerical methods based on set-membership computation and control methods based on a flatness concept. It makes possible i) to provide a guaranteed computation of admissible controls, ii) to deal with uncertain reference trajectory, iii) to reduce the time complexity of the algorithm compared to the existing approach. Simulations illustrate the efficiency of the developed methods in two different cases. For Single Input-Single Output (SISO) systems, generalized affine forms are computed otherwise a Branch & Prune algorithm with an inner inclusion test is used for Multi Inputs-Multi Outputs (MIMO) systems. The computational time is reduced significantly compared to the one required by the existing approach.
The method of uniform monotonous approximation of the reachable set border for a controllable system
Journal of Global Optimization, 2015
A numerical method of a two-dimensional non-linear controllable system reachable set boundary approximation is considered. In order to approximate the boundary right piecewise linear closed contours are used: a set of broken lines on a plane. As an application of the proposed technique a method of finding linear functional global extremum is described, including its use for systems with arbitrary dimensionality. Keywords Reachable set • Optimal control • Non-linear dynamic system 1 Introduction Reachable sets (RS) are among classical objects of investigation in the optimal control theory. The possibility to successfully operate with RS of controllable systems can greatly simplify the solutions of a number of traditional extreme problems, such as local and global search for a functional extremum, parametric identification, optimal control synthesis, system phase state estimation, control normalization, trajectories pencil control, and others. The problem of finding the RS is closely related to another classical problem of constructing integral funnels of differential inclusions [4
Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers
2021
Neural Networks (NNs) can provide major empirical performance improvements for robotic systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of closed-loop systems with NN controllers. Recent work provides bounds on these reachable sets, yet the computationally efficient approaches provide overly conservative bounds (thus cannot be used to verify useful properties), whereas tighter methods are too intensive for online computation. This work bridges the gap by formulating a convex optimization problem for reachability analysis for closed-loop systems with NN controllers. While the solutions are less tight than prior semidefinite programbased methods, they are substantially faster to compute, and some of the available computation time can be used to refine the bounds through input set partitioning, which more than overcomes the tightness gap. The proposed framework further considers systems with measurement and process noise, thus being applicable to realistic systems with uncertainty. Finally, numerical comparisons show 10× reduction in conservatism in 1 2 of the computation time compared to the state-of-theart, and the ability to handle various sources of uncertainty is highlighted on a quadrotor model.