A proximal characterization of the reachable set

We show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a Hamilton-Jacobi equation involving proximal normals.     

Time-scale decomposition of the reachable set of constrained linear systems

We consider a linear control system with a multiparameter singular perturbation representing multiple time scales and with constraints for the control and the slow state. The Hausdorff limit of the reachable set when the small parameters tend to zero is found. The result provides a basis for a time-scale approximation of the reachable set.on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances

In this paper, improved conditions are proposed for finding an ellipsoidal bound of reachable sets which bound the state trajectories of uncertain dynamic systems with time-varying delays and bounded peak disturbances. A more generalized time-derivative condition is considered in comparison with previous work. By the use of convex-hull properties and application of the Lyapunov method, sufficient conditions for finding an ellipsoidal bound of reachable sets of the concerned systems are established in terms of linear matrix inequalities (LMIs). Two numerical examples are included to show the superiority and effectiveness of our results.     

On the reachable set for third-order linear discrete-time systems with positive control

In this paper we study the geometrical properties of the set of reachable states of a single-input third-order discrete-time linear system with positive controls. This set is a cone and we give a complete geometrical characterization of this set when the system has all real eigenvalues. More in detail, we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations. Moreover, we provide necessary and sufficient conditions for finite time reachability of every reachable state and characterize the minimum number of steps needed to reach every state in terms of eigenvalue locations.     

On the reachable set of inflated singularly perturbed differential equations

Singularly perturbed differential equations with slow and fast subsystems are under consideration. It is well-known that the invariant probability measures of the unperturbed fast subsystems produce a limiting system for the slow subsystem. Unfortunately, the limiting system is non-smooth in general and might be too big since it also generates trajectories that are not related to the singularly perturbed system. However, if the flows produced by the unperturbed fast subsystems are chain transitive an inflation of the singularly perturbed system can be used in order to approximate all trajectories of the limiting system. Moreover, it turns out that the reachable sets of the limiting system are contained in the reachable sets of the slightly inflated singularly perturbed system.     

Boundary regularity of reachable sets of control systems

The reachable set of a differential inclusion has nonsmooth boundary in general. Now we give sufficient conditions on the compact initial set and the differential inclusion (a.e.) so that this topological boundary has “positive erosion”, i.e. it satisfies a uniform interior sphere condition. The proof is based on the close relation between the semiconcave value function of a suitable Mayer problem and its epigraph with positive erosion.     

Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays

The problem of reachable set estimation of linear uncertain polytopic time-varying delay systems subject to bounded peak inputs is studied in this paper. The delays considered in this paper are assumed to be non-differentiable and vary within an interval where the lower and upper bounds are known. Based on the Lyapunov–Krasovskii method and delay decomposition technique, a sufficient condition for the existence of a ball that bounds the reachable set of the system is proposed in terms of matrix inequalities containing only one scalar which can be solved by using an one-dimensional search method and Matlab’s LMI Toolbox and allow us to find the smallest radius. A numerical example is given to illustrate the effectiveness of the proposed result.     

Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances

We consider the problem of finding an ellipsoidal reachable set that bounds the states of time-delayed linear systems with bounded peak disturbances. The considered time-delay is time-varying but it has an upper bound in magnitude and rate. Based on the modified Lyapunov-Krasovskii(L-K) type functional, we derive a delay-dependent result expressed in the form of matrix inequalities containing only one non-convex scalar. The result is extended to uncertain time-delayed linear systems. Finally, we show the usefulness of our result by a comparative numerical example.     

On the reachable set for third-order linear discrete-time systems with positive control: The case of complex eigenvalues

In this paper, we study the geometrical properties of the set of reachable states of a single input third-order discrete-time linear system with positive controls. This set is a cone and we give a complete geometrical characterization of this set when the system has a complex conjugate pair of eigenvalues. More in detail, we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations.     

Reachable set bounding for delayed systems with polytopic uncertainties: The maximal Lyapunov-Krasovskii functional approach

This paper deals with the problem of reachable set estimation for time delay systems subject to both polytopic parameter uncertainties and bounded peak inputs. The maximal Lyapunov-Krasovskii functional is constructed as the pointwise maximum of a family of Lyapunov-Krasovskii functionals. Each functional corresponds to a vertex of uncertain polytope. Some criteria bounding the reachable set are derived. This approach shows great advantages over the traditional methods based on the common Lyapunov functionals. By introducing a modified integral inequality, the limitation imposed on the derivative of time delay being less than one is relaxed. Furthermore, the obtained results involve less computational burden when the number of vertices of the polytope is small. Two examples are given to illustrate the theoretical results.     

Topological properties of reachable sets and the control of quantum bits

We consider the problem of controlling the state of a two-level quantum system (quantum bit) via an externally applied electro-magnetic field. The describing model is a bilinear right-invariant system whose state varies on the Lie group of 2×2 special unitary matrices. We study the topological structure of the reachable sets. If two or more independent controls are used, then every state can be achieved in arbitrary time. However, this is no longer true if only one control is available and, in this case, we give an exact characterization of states reachable in arbitrary time. We prove small time local controllability for any state and the existence of a critical time which is the smallest time after which every transfer of state is possible. We provide upper and lower bounds for such a time. The mathematical development is motivated by the problem of manipulating the state of a quantum bit. Every transfer of state may be interpreted as a quantum logic operation and not every logic operation can be obtained in arbitrary time. The analysis we present provides information about the feasibility of a given operation as well as estimates for the speed of a quantum computer.     

Reachability of linear switched systems: Differential geometric approach

The paper investigates the structure of the reachable set of linear switched systems. The structure of the reachable set is determined using techniques from classical nonlinear systems theory, namely, the theory of orbits developed by H. Sussman and the realization theory for nonlinear systems developed by B. Jakubczyk.     

A unified algorithm to determine the reachable and dexterous workspace of parallel manipulators

This paper presents a novel idea for determining the reachable and dexterous workspace of parallel manipulators. Both the reachable workspace and dexterous workspace are utmost important for optimal design and performance comparison of manipulators, because each configuration or point in this region has specified kinematic dexterity by the designer. We propose a uniform algorithm, called stratified workspace boundary determining algorithm (SWBDA), which considers various physical and contrived constraints. The problems of determining the reachable and dexterous workspace boundaries are defined and the unified method is applied to solve all the problems of this kind. The validity and efficiency of the proposed method are verified with two kinds of representative parallel manipulator, since their relational results were studied in literatures. Finally, the advantages of the proposed method are summarized by comparing with other methods.     

The relaxed optimal control problem for Mean-Field SDEs systems and application

The present study deals with a new approach of optimal control problems where the state equation is a Mean-Field stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient depends on the term control. Our consideration is based on only one adjoint process, and the necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle are obtained, with application to Linear quadratic stochastic control problem with mean-field type.     

Abstractions of Hamiltonian control systems

Given a control system and a desired property, an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail. In previous work, abstractions of linear and nonlinear control systems were considered while preserving reachability properties. In this paper, we consider the abstraction problem for Hamiltonian control systems, where, in addition to the property of interest we also preserve the Hamiltonian structure of the control system. We show how the Hamiltonian structure of control systems can be exploited to simplify the abstraction process. We then focus on local accessibility preserving abstractions, and provide conditions under which local accessibility properties of the abstracted Hamiltonian system are equivalent to the local accessibility properties of the original Hamiltonian control system.     

Steps in the creation of energy-conservation control systems on a set of operating states

Problems related to the creation of energy-conservation systems used in control of dynamic objects are considered, in particular, the formalization of the optimal control problem, the set of the object’s operating states, analysis and synthesis of the optimal control, techniques of increasing the efficiency of energy-conservation control systems, and the introduction and maintenance of energy-conservation control systems in the course of operation.     

基于N步不变集的时滞切换系统的饱和控制

构造离散时滞切换系统的不变集,提出基于N步不变集的切换控制器设计方法,估计执行器饱和非线性的吸引域范围。首先,考虑时滞的影响,选取依赖于时滞的Lyapunov函数,构造时滞切换系统的不变集,并将其表达为若干个椭球集的凸组合,椭球集的个数与时滞常数相关。其次,在系统的前N个采样时刻,分别施加不同的饱和约束,求解得到一组椭球集,椭球集的个数与常数N相关,而每一步计算得到的椭球集均为时滞切换系统的不变集。再将N个不变集用一组凸包系数拟合,即可获取较大的吸引域估计。最后,在满足平均驻留时间约束的条件下设计切换律,并设计状态反馈控制器,保证闭环系统渐近稳定。控制器的求解转化为线性矩阵不等式的可行性问题。仿真结果验证了所提方法的可行性和有效性。     

Reach set computation and control synthesis for discrete-time dynamical systems with disturbances

The paper is devoted to the formulation of the reachability problem for discrete-time dynamical systems with disturbances. The concept of maxmin and minmax forward and backward reach sets is addressed. Invariance of the backward reach set is discussed. The emphasis of the paper is on discrete-time linear systems, for which the ellipsoidal computational method is described. The synthesis of maxmin and minmax closed-loop control for steering the system to a given target set using ellipsoidal backward reach set approximations is explained. The ellipsoidal method covered in the paper is implemented in the Ellipsoidal Toolbox for MATLAB, a popular collection of ellipsoidal calculus routines freely available online.     

A boundary equation of the principal period-2 component in the degree-n bifurcation set

The degree-n bifurcation set is a generalized Mandelbrot set for the complex polynomial P _c (z)=z ⁿ +c. The boundary of the principal period-2 component in the degree-n bifurcation set is first defined and then formulated by a parametrization of its image, which is the unit circle under the multiplier map. We investigate the boundary equation using the geometric symmetry of the degree-n bifurcation set with respect to rays of symmetry in the complex plane. In addition, an algorithm drawing the boundary curve with Mathematica codes is proposed.     

On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems

A fairly general class of nonlinear plants can be modeled as fuzzy systems, i.e., as a time-varying convex combination of “vertex” linear systems. As many linear LMI control results naturally generalize to such fuzzy systems, LMI formulations for fuzzy control became the tool of choice in the 1990s. Important results have since been obtained in the fuzzy arena, although significant sources of conservativeness remain. This paper reviews some of the sources of conservativeness of fuzzy control designs based on the linear vertex models instead of the original nonlinear equations. Then, ideas that may overcome some of the conservativeness issues (but increasing computational requirements) are discussed. Recently, the sum of squares paradigm extended some linear results to polynomial systems; this idea can be used for the so-called fuzzy polynomial systems that are also discussed in this work.