Sets Of Nonlinear Control Systems Research Articles

Online Guaranteed Reachable Set Approximation for Systems With Changed Dynamics and Control Authority

This work presents a method of efficiently computing inner and outer approximations of forward reachable sets for nonlinear control systems with changed dynamics and 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 by control inputs from the nominal and 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. 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.

Towards optimal space-time discretization for reachable sets of nonlinear control systems

Towards optimal space-time discretization for reachable sets of nonlinear control systems

Online Inner Approximation of Reachable Sets of Nonlinear Systems with Diminished Control Authority.

This work presents a method of efficiently computing inner approximations of forward reachable sets for nonlinear control systems with diminished control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking a precomputed convex reachable set based on a priori knowledge of the system's trajectory deviation growth dynamics. The trajectory deviation growth dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated by control inputs from the nominal and diminished set of control inputs, respectively. These growth dynamics are a function of a given Hausdorff distance bound between the nominal convex space of admissible controls and the possibly unknown impaired space of admissible controls. Because of its relative computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to onboard fault-tolerant path planning and failure recovery. We consider the implementation of the approximation procedure by way of numerical integration and a root finding scheme, and we present two illustrative examples, namely an application to a control system with quadratic nonlinearities and aircraft wing rock dynamics.

Enclosing Reachable Sets for Nonlinear Control Systems using Complementarity-Based Intervals

Abstract Enclosing the reachable sets of nonlinear control systems is useful in state estimation and safety verification for chemical process models. We present a new approach for computing time-varying interval bounds for ordinary differential equation models based on differential inequalities. Instead of using interval arithmetic like established approaches, we instead generate bounding information for the right-hand side (RHS) function by optimizing its convex relaxations. Complementarity formulations are explored, and are found to be particularly beneficial if the RHS function is quadratic and we employ the αBB relaxations.

Optimization-based subdivision algorithm for reachable sets

<p style='text-indent:20px;'>Reachable sets for nonlinear control systems can be computed via the use of solvers for optimal control problems. The paper presents a new improved variant which applies adaptive concepts similar to the framework of known subdivision techniques by Dellnitz/Hohmann. Using set properties of the nearest point projection, the convergence and rigorousness of the algorithm can be proved without the assumption of diffeomorphism on a nonlinear mapping. The adaptive method is demonstrated by two nonlinear academic examples and for a more complex robot model with box constraints for four states, two controls and five boundary conditions. In these examples adaptive and non-adaptive techniques as well as various discretization methods and optimization solvers are compared. <p style='text-indent:20px;'>The method also offers interesting features, like zooming into details of the reachable set, self-determination of the needed bounding box, easy parallelization and the use of different grid geometries. With the calculation of a 3d funnel in one of the examples, it is shown that the algorithm can also be used to approximate higher dimensional reachable sets and the resulting box collection may serve as a starting point for more sophisticated visualizations or algorithms.

On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

Approximation of reachable sets using optimal control and support vector machines

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.

Параллельный алгоритм приближенного построения множеств достижимости нелинейных управляемых систем

Параллельный алгоритм приближенного построения множеств достижимости нелинейных управляемых систем

Bounds on reachable sets using ordinary differential equations with linear programs embedded

Abstract This work considers the computation of time-varying enclosures of the reachable sets of nonlinear control systems via the solution of an initial value problem in ordinary differential equations (ODEs) with linear programs (LPs) embedded. To ensure the numerical tractability of such a formulation, the properties of the ODEs with LPs embedded are discussed including existence and uniqueness of the solutions of the initial value problem in ODEs with LPs embedded. This formulation is then applied to the computation of rigorous componentwise time-varying bounds on the states of a nonlinear control system. The bounding theory used in this work exploits physical information to yield tight bounds on the states; this work develops a new implementation of this theory. Finally, the tightness of the bounds are demonstrated for a model of a reacting chemical system with uncertain rate parameters.

Optimization‐based subdivision algorithm for reachable sets

AbstractThis paper shows how an optimization‐based approach to calculate reachable sets can be improved by using a subdivision‐algorithm. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Bounds on Reachable Sets Using Ordinary Differential Equations with Linear Programs Embedded

Abstract This work considers the computation of time-varying enclosures of the reachable sets of nonlinear control systems via the solution of an initial value problem in ordinary differential equations (ODEs) with linear programs (LPs) embedded. To ensure the numerical tractability of such a formulation, the properties of the ODEs with LPs embedded are discussed including existence and uniqueness of the solutions of the initial value problem in ODEs with LPs embedded. This formulation is then applied to the computation of rigorous componentwise time-varying bounds on the states of a nonlinear control system. The bounding theory used in this work exploits physical information to yield tight bounds on the states; this work develops a new implementation of this theory. Finally, the tightness of the bounds are demonstrated for a model of a reacting chemical system with uncertain rate parameters.

Bounds on the reachable sets of nonlinear control systems

The computation of rigorous enclosures of the reachable sets of nonlinear control systems is considered, with a focus on applications for which speed is crucial. Low computational costs make interval methods based on differential inequalities an attractive option. Unfortunately, such methods are prone to large overestimation and often produce useless results in practice. From physical considerations, however, it is common that some crude set is known to contain the reachable set. We establish a general bounding result, based on differential inequalities, which enables the effective use of such sets during the bounding procedure. In the case where this set is a convex polyhedron, an efficient implementation using interval computations is developed. Using readily available physical information from practical examples, this method is shown to provide significant advantages over alternative methods in terms of both efficiency and accuracy.

SET-VALUED DYNAMICS IN PROBLEMS OF MATHEMATICAL THEORY OF CONTROL PROCESSES

The dynamics and properties of set-valued states of differential control systems with uncertainties in initial data are studied. It is assumed that the dynamical system has a special structure, in which nonlinear terms in the right-hand sides of related differential equations are quadratic in state coordinates. We construct external and internal ellipsoidal estimates of reachable sets of nonlinear control system and find differential equations of proposed ellipsoidal estimates of reachable sets of nonlinear control system. The results obtained for quadratic system nonlinearities are extended to other types of control systems under uncertainty.

On external estimates for reachable sets of nonlinear control systems

The paper is devoted to the problem of constructing external estimates for reachable sets of a nonlinear control system. The estimates are constructed in the form of level sets of smooth functions in the space of states satisfying differential inequalities. In the system under consideration, the linear part is found, for which the corresponding functions are assumed to be known. The method proposed for estimating trajectories of a nonlinear system is based on modifying estimates for the linear part and on applying the comparison principle.

An algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls

In this paper, an algorithm for approximate calculation of the attainable sets of nonlinear control systems with integral constraint on controls is presented. A numerical example of the computer application of this algorithm is also given.

10.1007/s11488-008-1001-3

The method of sections for constructing reachable sets of nonlinear control systems with constraints on the control and system state is considered. The proposed algorithms are based on the application of complexes of programs whose functional content contains methods for calculating an optimal control for various constraints on control and system coordinates. The problems of many-processor computing technology are discussed and the results of numerical experiments are presented.

An algorithm of the method of sections and program tools for constructing reachable sets of nonlinear control systems

The method of sections for constructing reachable sets of nonlinear control systems with constraints on the control and system state is considered. The proposed algorithms are based on the application of complexes of programs whose functional content contains methods for calculating an optimal control for various constraints on control and system coordinates. The problems of many-processor computing technology are discussed and the results of numerical experiments are presented.

On the Continuity Properties of the Attainable Sets of Nonlinear Control Systems with Integral Constraint on Controls

The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radiusμ0inLp([t0,θ];ℝm) (p∈(1,+∞)). Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.

Differential-geometric and group-theoretical methods in optimal control theory

Problems arising in the geometry of attainable sets of nonlinear control systems are considered. Old and new results are discussed.

Estimating Reachable Sets

Two methods are presented for investigating reachable sets for nonlinear control systems. One method, based on a reachability maximum principle, lacks appropriate boundary conditions if the reachable set is not closed. The main result of the paper is an approximate method employing a Lyapunov-type function and an associated optimization problem, both involving a parameter vector. For each value of the parameter vector the resulting estimate for the reachable set (and the intersection of all such estimates) is guaranteed to contain the actual reachable set. The method is applicable to systems of any dimension and does not require integration of the equations of motion.