Nonsmooth dynamics of a 3D rigid body on a vibrating plate

In this work, we study a 3D rigid body, which is in a shape similar to a dumbbell, bouncing on a harmonically vibrating plate. The system involves two contact sets whose states vary with the relative motion between the plate and the dumbbell. These states are closely related to the physical properties of contact surfaces, and can be identified using the relationships of the relative kinematics. Under certain states, system-singular modes will likely occur due to the absence of the tangential compliance in Coulomb’s friction. Resolutions for these system singularities are given, and an integrated model, taking a hierarchical structure adaptable to the state variations, has been developed. Within an impact-free state, the contact forces to drive the motion of the dumbbell are obtained using an LCP (linear complementary problem) formulation. For the system in an impact state, the post-impact outputs to serve as the initial conditions for the subsequent motion can be calculated based on the new theory developed in impact dynamics. Specifying the dumbbell with initial states to bounce in a vertical plane, this model was justified by the comparison of our results and the experimental findings in other work. As certain periodic behaviors appear in the planar dynamics, the system in a 3D scenario also reveals intriguing patterns in the trajectories of the dumbbell’s mass center as projected onto the horizontal plane. They include a closed circular orbit, a dog-legged path, and a straight line.     

Reachable sets of second-order switched systems

This paper studies the reachability of second-order switched nonlinear systems in which the switching law is the only control action. For some initial conditions we find states that are reachable from the initial condition. We also give the switching law that drives the system from the initial state to the final state. Moreover, for switched systems with limit cycles, it is proved that the reachable set is totally determined for each switched convergent state. Finally, several examples and a real application are presented in order to illustrate the results.     

On the design and synthesis of limit cycles using switching linear systems

This article deals with the design and synthesis of limit cycles for a class of switched linear systems. This work is motivated by an application in the context of electrical networks, but the methodologies can be applied in other engineering fields as well. Several methods for the design and synthesis of limit cycles are presented. Based on a monodromy matrix associated with a periodically switched system, a design method and two feedback strategies are developed. The first strategy is based on linear feedback design using pole placement. The second strategy is based on the observation that certain state-dependent switching strategies can be implemented by means of a simple nonlinear output feedback controller. Advantages of this latter strategy are not only the ease with which the switching strategy can be implemented, but also the fact that classical techniques may also be used to ascertain the stability of the resulting limit cycle. Our next contribution is the development of a novel frequency domain-based approach to limit cycle design. This approach is based on the observation that the existence of certain limit cycles can be deduced from an infinity of circle criteria generated by a family of periodic systems. By making use of recent results, this observation can be used to develop a one-parameter spectral search to deduce the approximate frequencies of feasible limit cycles. Having selected a frequency of oscillation, one may then make use of the aforementioned nonlinear elements to realise a switching strategy that generates a stable limit cycle with given frequency and amplitude. Examples are given to illustrate the effectiveness of the approaches presented.     

Semi-globally tracking control for uncertain non-affine self-restructuring systems with state constraints

It is still an open problem to achieve tracking control with state constraints if the initial states are out of the constrained region. This underlying problem becomes more challenging if the system suffers from modeling uncertainties and external disturbances. In this paper, we investigate such problem for a class of uncertain nonaffine self-restructuring systems subject to state constraints in the absence of any limitation on the initial values of states. In order to steer the original system states into the desired constraint region in finite time, certain auxiliary system is firstly constructed via introducing auxiliary variables. By resorting to some auxiliary nonlinear functions, the original system with state constraints is transformed into an unconstrained one in the sense that the constraint problem of the original system is equivalent to ensuring the stability of the transformed one. By following the backstepping design technique, an adaptive control scheme is developed that enables the states to enter into the constrained region for any initial point within any fixed time and then never violate it. All the closed loop signals are ensured to be semi-globally uniformly ultimately bounded. Finally, one numerical simulation example is presented to verify the theoretical analysis.     

Stabilizable regions of receding horizon predictive control with input constraints

Stabilizable regions of receding horizon predictive control (RHPC) with input constraints are examined. A feasible region of states, which is spanned by eigenvectors of the closed-loop system with a stabilizing feedback gain, is derived in conjunction with input constraints. For states in this region, the feasibility of state feedback is guaranteed with the corresponding feedback gain. It is shown that an RHPC scheme with adequate finite terminal weights can guarantee stability for any initial state which can be steered into this region using finite number of control moves in the presence of input saturation. This methodology results in feasible regions which are infinite (in certain directions) even in the case of open-loop unstable systems. It is shown that the proposed feasible regions are larger than the ellipsoidal regions which were suggested in earlier works. We formulated the optimization problem in LMI so that it can be solved by semidefinite programming.     

Disappearance of entanglement: a topological point of view

We give a topological classification of the evolution of entanglement, particularly the different ways the entanglement can disappear as a function of time. Four categories exhaust all possibilities given the initial quantum state is entangled and the final one is not. Exponential decay of entanglement, entanglement sudden death and sudden birth can all be understood and visualized in the associated geometrical picture - the polarization vector representation. The entanglement evolution categories of any model are determined by the topology of the state space and the dynamical subspace, the limiting state and the memory effect of the environment. Transitions between these types of behaviors as a function of physical parameters are also possible. These transitions are thus of topological nature. The symmetry of the system is also important, since it determines the dimension of the dynamical subspace. We illustrate the general concepts with a visualizable model for two qubits, and give results for extensions to N-qubit GHZ states and W states.     

Synthesizing object life cycles from business process models

Unified modeling language (UML) activity diagrams can model the flow of stateful business objects among activities, implicitly specifying the life cycles of those objects. The actual object life cycles are typically expressed in UML state machines. The implicit life cycles in UML activity diagrams need to be discovered in order to derive the actual object life cycles or to check the consistency with an existing life cycle. This paper presents an automated approach for synthesizing a UML state machine modeling the life cycle of an object that occurs in different states in a UML activity diagram. The generated state machines can contain parallelism, loops, and cross-synchronization. The approach makes life cycles that have been modeled implicitly in activity diagrams explicit. The synthesis approach has been implemented using a graph transformation tool and has been applied in several case studies.     

利用有向环的性质求解可达关系

不确定规划研究的最终目标是求出规划解,但是由于缺少引导信息,直接求规划解会导致大量的无用状态和动作被搜索。获得状态间的可达关系可以避免冗余计算。目前求可达关系的方法效率较低,因此设计了一种求可达关系的新方法。将不确定状态转移系统抽象成一个图,在这个图中,查找状态之间的可达信息是否形成一个有向环。若存在一个有向环,说明环内每两个状态之间都有可达关系。将其中一个状态作为父节点,并且将这个环内所有状态的可达关系记录在父节点中,通过访问父节点的可达信息更新环内状态的可达信息,减少了许多无用的状态和动作被搜索。实验结果表明,所设计的算法不仅能得到更全面的可达关系,而且效率也高于已有的算法。     

A simplified method of solution for the short cycle creep-plasticity problem

This paper describes a method for estimating the long-term effects on structures under cyclic changes of loading. For loads less than a certain critical amplitude (shakedown limit), the stress in the structure will a symptote to a cyclic stationary state consisting of an elastic part in response to the cyclic loading, plus a system of self-equilibrating residuals constant in time. It is shown that corresponding to this cyclic stationary state, the creep energy dissipation per cycle of loading is a maximum. Instead of following the exact time history to reach this state, in this paper it is found by a procedure of successive approximations. It corrects the admissible residual stress distribution at the beginning of a cycle by the creep and plastic strains accumulated over an entire cycle, which are in general not compatible, and requires additional self-equilibrating stresses to give an elastic strain distribution such that the total strain satisfies compatibility. The steady state is reached when no further correction is necessary. Convergence may be accelerated by a suitable choice of initial starting value, and by an artificial choice of the cycle time for the best computational convenience, upon which the steady-state solution can be proved to be independent. The procedure is a powerful device to obtain the cyclic steady-state solution, which will give an upper bound to the creep deformation per cycle and may also be used to find the shakedown limit. The formulation of the procedure in conjunction with the finite element method is given in detail and results of a few examples of the analysis are shown.     

Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions     

More efficient on-the-fly LTL verification with Tarjan's algorithm

State-of-the-art algorithms for on-the-fly automata-theoretic LTL model checking make use of nested depth-first search to look for accepting cycles in the product of the system and the Büchi automaton. Here, we present two new single depth-first search algorithms that accomplish the same task. The first is based on Tarjan's algorithm for detecting strongly connected components, while the second is a combination of the first and Couvreur's algorithm for finding acceptance cycles in the product of a system and a generalized Büchi automaton. Both new algorithms report an accepting cycle immediately after all transitions in the cycle have been investigated. We show their correctness, describe efficient implementations and discuss how they interact with some other model checking techniques, such as bitstate hashing. The algorithms are compared to the nested search algorithms in experiments on both random and actual state spaces, using random and real formulas. Our measurements indicate that our algorithms investigate at most as many states as the old ones. In the case of a violation of the correctness property, the algorithms often explore significantly fewer states.     

Qubit–qubit entanglement dynamics control via external classical pumping and Kerr nonlinearity mediated by a single detuned cavity field powered by two-photon processes

The nonlinear time-dependent two-photon Hamiltonian of a couple of classically pumped independent qubits is analytically solved, and the corresponding time evolution unitary operator, in an exact form, is derived. Using the concurrence, entanglement dynamics between the qubits under the influence of a wide range of effective parameters are examined and, in detail, analyzed. Observations analysis is documented with aid of the field phase-space distribution Wigner function. A couple of initial qubit states is considered, namely similar excited states and a Bell-like pure state. It is demonstrated that an initial Bell-like pure state is as well typical initial qubits setting for robust, regular and a high degree of entanglement. Moreover, it is established that high-constant Kerr media represent an effective tool for generating periodical entanglement at fixed time cycles of maxima reach unity forever when qubits are initially in a Bell-like pure state. Further, it is showed that the medium strength of the classical pumping stimulates efficiently qubits entanglement, specially, when the interaction occurs off resonantly. However, the high-intensity pumping thermalizes the coherent distribution of photons, thus, the least photons number is used and, hence, the least minimum degree of qubits entanglement could be created. Furthermore, when the cavity field and external pumping are detuned, the external pumping acts like an auxiliary effective frequency for the cavity, as a result, the field Gaussian distribution acquires linear chirps, and consequently, more entanglement revivals appear in the same cycle during timescale.     

The influence of limit cycle topology on the phase resetting curve

Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study. We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.     

Asymptotic distributions of quantum walks on the line with two entangled coins

We advance the previous studies of quantum walks on the line with two coins. Such four-state quantum walks driven by a three-direction shift operator may have nonzero limiting probabilities (localization), thereby distinguishing them from the quantum walks on the line in the basic scenario (i.e., driven by a single coin). In this work, asymptotic position distributions of the quantum walks are examined. We derive a weak limit for the quantum walks and explicit formulas for the limiting probability distribution, whose dependencies on the coin parameter and the initial state of quantum walks are presented. In particular, it is shown that the weak limit for the present quantum walks can be of the form in the basic scenario of quantum walks on the line, for certain initial states of the walk and certain values of the coin parameter. In the case where localization occurs, we show that the limiting probability decays exponentially in the absolute value of a walker??s position, independent of the parity of time.     

The Lagrange problem on an optimal column: old and new results

The paper is devoted to the optimization and post-buckling behavior of columns elastically supported at both ends. The unimodal solutions are analyzed, and it is shown that for nonzero support stiffnesses they are not optimal. The bimodal formulation of the problem is set up. By using analytical expressions for bimodal columns obtained earlier, the bimodal optimal solutions are integrated for different values of the support stiffnesses. With the assumption of geometrical nonlinearity, the post-buckling behavior of the bimodal optimal columns is studied. It is shown that the initial post-buckling behavior is governed by four supercritical solutions emanating from the trivial equilibrium state at the critical load. The stability of the new equilibrium states is investigated by using the second variation of the total potential energy. It is shown that only two post-buckling equilibrium states are stable while the other two are unstable, this conclusion being valid for all considered values of the support stiffnesses. An important limit case of a clamped–simply supported column that has caused debate in many publications is analyzed.     

用滑动模态实现一类非完整动力学系统的指数镇定

对于基变量具有对称性的一类确定非完整动力学系统,直接基于该系统模型,利用 滑动模态的方法设计了指数镇定控制器,得到了状态空间中的一个区域.当系统初始状态位 于该区域内时,系统状态是指数收敛的,而当初始状态不在该区域内时,可用非零的开环常值 控制使系统状态在有限时间内进入这一区域.因此,它是一种简洁的全局镇定律.最后,用仿 真结果验证了这一方法的有效性.     

Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems

This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.     

Power-Law Decay in the Ionization of H Rydberg Atoms

We consider a classical study of the ionization of hydrogen Rydberg atoms by circularly polarized microwaves (CPM) in the frequency regime, where typically diffusive manner of the ionization is observed [1,2]. The CPM field pulse shape and the atomic initial state influence on the survival probability, S, are investigated in the two-dimensional (2D) Hamiltonian model [3]. This is motivated by the fact that it is possible experimentally to prepare circular states, e.g. by the crossed fields method [4]. For such states and for states with not too large eccentricity the simplified 2D model should be a good approximation of the three-dimensional life for high enough states. Ionization of highly excited hydrogen atoms by linearly polarized microwaves (LPM) has been studied in the last twenty years (e.g. see [5,6]). The very first experimental results [7] were explained theoretically [8] using Monte-Carlo classical simulations. Classically, the ionization occurs due to the break up of the Kolmogorov-Arnold-Moser (KAM) tori when the microwave amplitude is large enough, hence the ionization threshold can be associated with the onset of classical chaos. Therefore, in the presence of an external periodic force we can observe the escape process of an electron to the continuum from a certain phase-space region defined by the initial conditions. For Hamiltonian systems, the escape may be slowed down significantly due to the presence of the remnants of KAM tori (Cantori) and one can observe the power-law (algebraic) decay S ~ t-z from the region containing KAM stability islands rather than exponential decay S ~ exp(-t), in the limit of large time t.     

Design of orbitally stable zero dynamics for a class of nonlinear systems

A method for the generation of globally attractive limit cycles for nonlinear systems is presented. It consists in designing an output that, when regulated through a suitable feedback, forces a limit cycle in the zero dynamics. Conditions are then given to ensure that a globally attractive limit cycle in the zero dynamics results in a globally attractive limit cycle in the whole system. The method is illustrated on the torque control of an induction motor.