Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

Sliding Mode Control with Minimization of the Regulation Time in the Presence of Control Signal and Velocity Constraints

In this paper, the design of the terminal continuous-time sliding mode controller is presented. The influence of the external disturbances is considered. The robustness for the whole regulation process is obtained by adapting the time-varying sliding line. The representative point converges to the demand state in finite time due to the selected shape of the nonlinear switching curve. Absolute values of control signal, system velocity and both of these quantities are bounded from above and considered as system constraints. In order to evaluate the dynamical performance of the system, the settling time is selected as a quality index and it is minimized. The approach presented in this paper is particularly suited for systems in which one state (or a set of states) is the derivative of the other state (or a set of states). This makes it applicable to a wide range of electromechanical systems, in which the states are the position and velocity of the mechanical parts.

A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ( $N$ ) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

Switching Characteristics Optimization of Two-Phase Interleaved Bidirectional DC/DC for Electric Vehicles

In electric vehicles (EVs), bidirectional DC/DC(Bi-DC/DC) is installed between the battery pack and the DC bus to step up the voltage. In the process of mode switching under step signal, the Bi-DC/DC will be affected by a large current inrush which threatens the safety of the circuit. In this paper, a Bi-DC/DC mode switching method based on the optimized Bézier curve is proposed. The Boost and Buck modes can be switched based on the proposed method with fast and non-overshoot switching performance. The experimental results show that the mode switching can be finished in 4 ms without overshoot based on the optimal switching curve.

Optimal routeing in two-queue polling systems

We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

TWO-CLASS ROUTING WITH ADMISSION CONTROL AND STRICT PRIORITIES

We consider the problem of routing and admission control in a loss system featuring two classes of arriving jobs (high-priority and low-priority jobs) and two types of servers, in which decision-making for high-priority jobs is forced, and rewards influence the desirability of each of the four possible routing decisions. We seek a policy that maximizes expected long-run reward, under both the discounted reward and long-run average reward criteria, and formulate the problem as a Markov decision process. When the reward structure favors high-priority jobs, we demonstrate that there exists an optimal monotone switching curve policy with slope of at least −1. When the reward structure favors low-priority jobs, we demonstrate that the value function, in general, lacks structure, which complicates the search for structure in optimal policies. However, we identify conditions under which optimal policies can be characterized in greater detail. We also examine the performance of heuristic policies in a brief numerical study.