Sets For Linear Control Systems Research Articles

Dispersive problem and control sets of linear control systems on Lie groups

This paper is dedicated to the study of dispersiveness and controllability of linear control systems on Lie groups. Dispersiveness means absence of recursiveness, that is contrary to the existence of control set. A linear control system on a Lie group associates with a derivation operator. For a linear system with stable derivation, it is shown that the system is dispersive if and only if the trajectories through the neutral element have no limit at infinity. As a consequence, a nonempty limit set at the neutral element is a necessary condition for the linear system to admit a control set or to be controllable. In the nondispersive case, the control sets are described by the Lie subgroup of all recurrent points of the automorphism flow of the system. If the derivation operator is asymptotically stable, the central limit set at the neutral element is the unique control set of the system.

Control Sets of Linear Control Systems on ℝ2. The Complex Case

This paper explicitly computes the unique control set D with the non-empty interior of a linear control system on ℝ2, when the associated matrix has complex eigenvalues. It turns out that the closure of D coincides with the region delimited by a computable periodic orbit 𝒪 of the system.

Approximate calculation of reachable sets for linear control systems with different control constraints

The paper considers the problem of approximate construction of reachability sets for a linear control system, when the control action is constrained simultaneously by geometric and several integral constraints. A variant of the transition from a continuous to a discrete system is proposed by uniformly dividing the time interval and replacing the controls at the step of dividing them with their mean values. The convergence of the reachability set of the approximating system to the reachability set of the original system in the Hausdorff metric is proved as the discretization step tends to zero, and an estimate is obtained for the rate of convergence. An algorithm for constructing the boundary of reachable sets based on solving a family of conic programming problems is proposed. Numerical simulation has been carried out.

Inverse variational principles for control systems

Inspired by Katok’s idea in defining measure-theoretic entropy, we introduce several measure-theoretic invariance entropies in control systems via spanning sets. Properties of them are investigated. Especially, we show that the measure-theoretic invariance entropies of a given set can be characterized by its relatively invariant subsets. More importantly, we obtain two inverse variational principles connecting measure-theoretic invariance entropy and topological invariance entropy, which help us to establish two formulae for the measure-theoretic invariance entropies of control sets for linear control systems, including linear continuous-time and discrete-time systems.

Control Sets of Linear Control Systems on Matrix Groups and Applications

We review recent results on control sets of linear control systems on matrix groups. We also mention some applications of systems with algebraic structures.

A numerical method for the approximation of reachable sets of linear control systems

Abstract In this article, a numerical method for the approximation of reachable sets of linear control systems is discussed. First a continuous system is transformed into a discrete one with Runge–Kutta methods. Then based on Benson’s outer approximation algorithm for solving multiobjective optimization problems, we propose a variant of Benson’s algorithm to sandwich the reachable set of the discrete system with an inner approximation and an outer approximation. By specifying an approximation error, the quality of the approximations measured in Hausdorff distance can be directly controlled. Furthermore, we use an illustrative example to demonstrate the working of the algorithm. Finally, computational experiments illustrate the superior performance of our proposed algorithm compared to a recent algorithm in the literature.

Explicit formulas for ellipsoids approximating attainable sets: II

Ellipsoidal estimates of attainable sets for linear control systems that are globally optimal in the sense of the ellipsoid volume are studied. The following two basic results are reported: a rapidly converging iteration method for explicit determination of optimal ellipsoids is described, and asymptotics of parameters of optimal ellipsoids for an autonomous control system for large time of motion is determined.

Approximation of reachable sets by direct solution methods for optimal control problems

A numerical method for the approximation of reachable sets of linear control systems is discussed. The method is based on the formulation of suitable optimal control problems with varying objective function, whose discretization by Runge–Kutta methods leads to finite-dimensional convex optimization problems. It turns out that the order of approximation for the reachable set depends on the particular choice of the Runge–Kutta method in combination with the selection strategy used for control approximation. For an inappropriate combination, the expected order of convergence cannot be achieved in general. The method is illustrated by two test examples using different Runge–Kutta methods and selection strategies, in which the run times are analysed, the order of convergence is estimated numerically and compared with theoretical results in similar areas.

LQR optimization of linear system switching

Considers offline optimization of a switching sequence for a given finite set of linear control systems, together with joint optimization of control laws. A linear quadratic full information criterion is optimized and dynamic programming is used to find an optimal switching sequence and control law. The main result is a method for efficient pruning of the search tree to avoid combinatoric explosion. A method to prove optimality of a found candidate switch sequence and corresponding control laws is presented.

Under-estimates of reachable sets for linear control systems

A new algorithm is presented for providing under-estimates of the reachable set from the origin for a class ofn-dimensional linear systems with bounded controls. This algorithm is based on the novel approach of choosing a feedback control which makes all the eigenvalues of the closed loop system unstable. Results from feedback control and Liapunov stability theory are then used to formulate the problem as the minimization of a nonlinear function subject to constraints on certain matrices. The solution of this optimization problem provides an under-estimate of the reachable set in the form of ann-dimensional ellipsoid. Examples of both continuous and discrete-time systems are presented to illustrate the method. Comparison with existing exact results for some 2-dimensional systems shows that the method provides good approximations in these cases.