Largest Invariant Set Research Articles

Invariant regions in piecewise linear area-preserving map

This paper studies regular islands (invariant regions) associated with elliptic periodic points of a piecewise linear area-preserving map. We reveal why regular island chains are interspersed in a chaotic sea, which is typical of the behavior exhibited by the area-preserving map. We find that regular islands may be ellipses or polygons and this depends on whether the rotation number of the rotation matrix that defines the linear map is irrational or rational. We also present a method for determining the boundary of these regular islands and chaotic seas (the largest invariant set). We show that critical lines play an important role in determining the boundaries of regular islands and chaotic seas.

Dual-mode global stabilization of high-order saturated integrator chains: LMI-based MPC combined with a nested saturated feedback

This paper considers the problem of high-performance global stabilization of an integrator chain via a bounded control at the presence of input disturbance. While nested saturated feedback (NSF) is known as the most inspiring existing solution in the literature, we shall highlight the inherent shortcomings of this approach which cause a poor performance in terms of convergence rate. Then, a novel dual-mode control scheme combining an improved NSF law with a linear matrix inequality (LMI)-based model predictive controller (MPC) is developed to overcome the weaknesses of pure NSF. By offline calculations, a set of nested robust invariant attraction regions and their attributed feedback gains are computed respecting the known bound for disturbance. In online implementation, an improved NSF law is adopted (as the first mode) to bring states within the largest invariant set. The proposed LMI-based state feedback law is applied thereafter (as the second mode) to bring state vector toward the origin with a fast convergence rate. In both modes, the well-known extended state observer (ESO) is employed to effectively suppress the external disturbance. The superiority of the proposed globally stabilizing scheme in terms of faster convergence and improved disturbance rejection ability is demonstrated comparing performances of foremost existing solutions in case of a triple integrator chain.

An Interpolation Technique for Input Constrained Robust Stabilization

This paper presents an interpolation technique of two given robust stabilizing gains for input constrained uncertain systems. The proposed interpolated feedback gain is computed at every sampling instance by solving an optimization problem with a single decision variable. Use of the interpolated feedback gain can bring about not only large invariant set but also good control performance. Moreover, it is shown that the feasible and invariant set yielded by the proposed control is the convex hull of the two invariant sets by the known controls. The simulation results show that the proposed interpolation based feedback gain indeed results in both large invariant set and good control performance.

Population Control of Equilibrium States of Quantum Systems via Lyapunov Method

摘要: This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results. 关键词: Quantum system / population control / Bloch vector / Lyapunov function / invariant set

Population Control of Equilibrium States of Quantum Systems via Lyapunov Method

Abstract This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results.

Bounding the errors for convex dynamics on one or more polytopes

We discuss the greedy algorithm for approximating a sequence of inputs in a family of polytopes lying in affine spaces by an output sequence made of vertices of the respective polytopes. More precisely, we consider here the case when the greed of the algorithm is dictated by the Euclidean norms of the successive cumulative errors. This algorithm can be interpreted as a time-dependent dynamical system in the vector space, where the errors live, or as a time-dependent dynamical system in an affine space containing copies of all the original polytopes. This affine space contains the inputs, as well as the inputs modified by adding the respective former errors; it is the evolution of these modified inputs that the dynamical system in affine space describes. Scheduling problems with many polytopes arise naturally, for instance, when the inputs are from a single polytope P, but one imposes the constraint that whenever the input belongs to a codimension n face, the output has to be in the same codimension n face (as when scheduling drivers among participants of a carpool). It has been previously shown that the error is bounded in the case of a single polytope by proving the existence of an arbitrary large convex invariant region for the dynamics in affine space: A region that is simultaneously invariant for several polytopes, each considered separately, was also constructed. It was then shown that there cannot be an invariant region in affine space in the general case of a family of polytopes. Here we prove the existence of an arbitrary large convex invariant set for the dynamics in the vector space in the case when the sizes of the polytopes in the family are bounded and the set of all the outgoing normals to all the faces of all the polytopes is finite. It was also previously known that starting from zero as the initial error set, the error set could not be saturated in finitely many steps in some cases with several polytopes: Contradicting a former conjecture, we show that the same happens for some single quadrilaterals and for a single pentagon with an axial symmetry. The disproof of that conjecture is the new piece of information that leads us to expect, and then to verify, as we recount here, that the proof that the errors are bounded in the general case could be a small step beyond the proof of the same statement for the single polytope case.

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincare map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers' favour.

Nonlinear model predictive control with enlarged terminal sets using support vector machine

In this paper, model predictive control (MPC) of nonlinear systems subject to input and state constraints is considered, for which nominal closed-loop stability is guaranteed. We propose the use of a large terminal invariant set and an estimate of the terminal cost to reduce the online computational burden of MPC. These terminal sets and costs are learned off-line via support vector machine method. Its main advantage with respect to other well-known techniques is the reduction of online computational effort by relaxing the terminal constraints. An example illustrates the efficiency of the approach.

Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard

AbstractThe billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set M0 of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding ofM0 [Mor91] and estimates using convex fronts related to the derivative of the billiard ball map [Sto03] to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.

Adaptive tracking control for cooperative multiple robot systems

This study presents a novel adaptive controller for cooperative multiple robot systems to obtain both zero motion and force tracking errors. First, by introducing the concepts of force decomposition and virtual control input, we reformulate the control problem of cooperative manipulators as a problem of controlling a single constrained manipulator. Next, a stable subspace is defined by both motion and internal force errors. This subspace includes both zero motion and force errors as its largest invariant set. The controller is then designed to drive the error dynamics into the stable subspace. A non‐adaptive and an adaptive approach both systematically show the asymptotic motion/internal force tracking. Note here that the zero force tracking error is achieved without needing the persistent excitation (PE) condition in the adaptive control scheme. Furthermore, based on a decentralized method, the proposed controllers are implemented in multiprocessor systems to balance the computation load. Finally, two cooperative planar robots are used to demonstrate the developed methodology where the simulation results illustrate the expected performance.

On enlarging the basin of attraction for linear systems under saturated linear feedback

We consider the problem of enlarging the basin of attraction for a linear system under saturated linear feedback. An LMI-based approach to this problem is developed. For discrete-time system, this approach is enhanced by the lifting technique, which leads to further enlargement of the basin of attraction. The low convergence rate inherent with the large invariant set (hence, the large basin of attraction) is prevented by the construction of a sequence of invariant ellipsoids nested within the large one obtained.