Piecewise Polynomial Systems Research Articles

Fault reconstruction‐based robust tracking control design for periodic piecewise polynomial systems with disturbances

AbstractIn this article, we address the topics of fault reconstruction and robust fault‐tolerant tracking control for periodic piecewise polynomial uncertain systems with time‐varying delays, faults and external disturbances. In detail, the periodic piecewise polynomial learning observer system is configured with the intent to concurrently reconstruct the faults and immeasurable states of the system that is being considered. Precisely, the key intent of this study centers around in devising a robust fault‐tolerant tracking control to achieve the tracking performance of the system as intended with mixed and passivity performance. To this end, a robust fault‐tolerant tracking control is designed by incorporating the reconstructed faults and error feedback, which aids in vouching for the outcomes that were desired. Besides, by building the periodic piecewise polynomial Lyapunov–Krasovskii functional and making use of the matrix polynomial lemma and Wirtinger's inequality, certain adequate conditions are procured in the frame of linear matrix inequalities. Subsequently, the required gain values for the devised controller and the configured observer are found by using the established linear matrix inequalities. Ultimately, in order to attest the controller's intrinsic potential and relevance, numerical example along with the simulation results is supplied.

Disturbance suppression based quantized tracking control for periodic piecewise polynomial systems

The investigation presented here deals with the issue of disturbance suppression and the design of robust tracking control with quantization mechanism for periodic piecewise polynomial systems prone to parameter uncertainties, time delays and external disturbances by means of a proportional integral observer (PIO)-based parallel equivalent input disturbance (PEID) approach. Primarily, in the PEID technique, in order to reduce estimate errors, the PEID notion has been put forward, wherein these errors are perceived as artificial disturbances and a chain of EID compensators has been employed to make amends for them. Meanwhile, PIO’s integrating part aids in blending a relaxing variable into the system’s layout, which allows for greater flexibility in the system’s framework while rendering the system more robust. Subsequently, with the information of estimates from PEID and PIO, disturbance suppression-based quantized tracking control is designed, which simultaneously makes the system states follow the reference states and mitigates the disturbances from the system. Further, the input signals are quantized before being sent as a result of the limited capacity of the channel via which the data is transmitted. Subsequently, through configuring a periodic piecewise polynomial matrix and blending Lyapunov stability theory with the matrix polynomial lemma, adequate requirements are derived in the framework of linear matrix inequalities which ensure the desired outcomes. After which, the requisite controller and observer gain matrices are generated by solving the stated linear matrix inequality-based relations. Ultimately, the simulation portion offers a numerical example that verifies the potential of the acquired findings.

Heteroclinic loop bifurcations by perturbing a class of Z2-equivariant quadratic switching Hamiltonian systems with nilpotent singular points

This paper investigates the problem of heteroclinic loop bifurcation by perturbing a class of Z2-equivariant quadratic switching systems with nilpotent singular points. Firstly, it provides sufficient and necessary conditions for the occurrence of a heteroclinic loop. Next, the system is perturbed by piecewise polynomial systems of degree n≥1. The paper then considers the lower bound for the maximum number of limit cycles that bifurcate from the generalized heteroclinic loop, finding at least n+[n2] limit cycles.

Reference management-based resilient control for delayed periodic piecewise polynomial systems under proportional integral observer

This article examines the proportional integral observer (PIO)-based resilient tracking control problem for periodic piecewise polynomial systems (PPPSs) in the presence of time delay and disturbances. To be precise, the PPPSs are formulated by dividing the fundamental period of periodic systems into numerous subintervals, each of which is defined using matrix polynomial functions. Further, the PIO strategy is employed to estimate the states of the undertaken system with high precision wherein the integral term in PIO enhances the system's design flexibility and also increases its robustness. Taking advantage of these characteristics, a PIO-based tracking control is configured with gain perturbations to track the desired reference states. Moreover, to attenuate the implications made by disturbances in considered system design, H ∞ performance is employed. Furthermore, in order to establish adequate conditions in the form of linear matrix inequalities, the time-varying polynomial Lyapunov–Krasovskii functional is implemented. Then, the time-varying gain matrices of the control scheme and PIO configuration are computed by solving the obtained criteria through the MATLAB platform. Concludingly, to evidence for the discoveries' potential and usefulness, a numerical example is offered.

Input–output finite-time stabilisation for periodic piecewise polynomial systems with nonlinear actuator faults: an observer-based approach

This paper is concerned with the input–output finite-time (IO-FT) stabilisation problem for a class of continuous-time periodic piecewise polynomial systems (PPPSs) with immeasurable states and external disturbances via state estimation-based robust reliable controller. Firstly, to reflect the actuality, a parameter uncertainty that exhibits the random nature and the actuator faults with nonlinear character are considered in the addressed PPPSs and the control scheme, respectively. In detail, the randomness of uncertain parameters is portrayed by the stochastic variable and it is presumed to pursue the Bernoulli distributions. Secondly, to estimate the immeasurable states of PPPSs, a periodic piecewise polynomial observer is designed depending on the output of PPPSs. The main intent of this article is to devise a state estimation-based robust reliable controller to ascertain the IO-FT stabilisation of the PPPSs. Moreover, by bridging the Lyapunov stability theory, linear matrix inequality technique and IO-FT stability theory, the required IO-FT stabilisation conditions in the frame of linear matrix inequality are procured for the system under consideration. Eventually, simulation results of the addressed PPPSs are shown in line with the proposed analytical findings, revealing the competence, inherent capability and utility of the devised control protocol.

The number of limit cycles for regularized piecewise polynomial systems is unbounded

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as ϵ→0. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called VI3). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.

Tracking control design for periodic piecewise polynomial systems with multiple disturbances

AbstractThe present study examines the state tracking problem of periodic piecewise polynomial systems subject to multiple disturbances and actuator faults. Specifically, the fundamental period is partitioned into several subintervals and the piecewise matrix polynomial functions can be put together to approximate the dynamics over each period, which are characterized by Bernstein polynomials. Moreover, the system dynamics contains both matched and mismatched disturbances, wherein the matched disturbance is unknown and resulted from some exogenous system, and mismatched case is norm‐bounded. The control law is configured by integrating the output of the disturbance observer with state‐feedback reliable control law. Particularly, a disturbance observer is designed to estimate the disturbance caused by an exogenous system. Besides, by considering time‐varying polynomial Lyapunov function and making use of Bernstein polynomial approach, a set of sufficient conditions is derived which are expressed in terms of linear matrix inequalities. Further, by virtue of MATLAB software the desired controller and observer gain matrices can be computed. In conclusion, the potential and significance of the theoretical findings are confirmed by presenting a numerical example with simulation results.

Fast Computation of Tight Funnels for Piecewise Polynomial Systems

Fast Computation of Tight Funnels for Piecewise Polynomial Systems

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

Comparison of two polynomial approaches in performance analysis for periodic piecewise polynomial systems

In this paper, the theory and effectiveness of two polynomial approaches are compared in the analysis of L2-L∞ and H∞ performance for a type of periodic piecewise polynomial systems, where the time-varying subsystems can be characterized in Bernstein polynomials. Using the Bernstein polynomial-based lemma and the existing lemma concerning the negativity/positivity of matrix polynomial functions, sufficient conditions are established in tractable forms aimed at the global asymptotic stability and performance analysis. Four cases of optimization constraints are considered based on the proposed conditions. The performance indices obtained via the four cases are compared through a numerical example, and the lower conservatism achieved by the proposed Bernstein polynomial approach is demonstrated.

Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary

Continuing the investigation for the piecewise polynomial perturbations of the linear center ẋ=−y,ẏ=x from Buzzi et al. (2018) for the case where the switching boundary is a straight line, in this paper we allow that the switching boundary is non-regular, i.e. we consider a switching boundary which separates the plane into two angular sectors with angles α∈(0,π] and 2π−α. Moreover, unlike the aforementioned work, we allow that the polynomial differential systems in the two sectors have different degrees. Depending on α and for arbitrary given degrees we provide an upper bound for the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center using the averaging method up to any order. This upper bound is reached for the first two orders. On the other hand, we pay attention to the perturbation of the linear center inside this class of piecewise polynomial Liénard systems and give some better upper bounds in comparison with the one obtained in the general piecewise polynomial perturbations. Again our results imply that the non-regular switching boundary (i.e. when α≠π) of the piecewise polynomial perturbations usually leads to more limit cycles than the regular case (i.e. when α=π) where the switching boundary is a straight line.

Local cyclicity in low degree planar piecewise polynomial vector fields

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, Mpc(n), with degrees 2, 3, 4, and 5. More concretely, Mpc(2)≥13,Mpc(3)≥26,Mpc(4)≥40, and Mpc(5)≥58. The computations use parallelization algorithms.

A Bernstein Polynomial Approach to Estimating Reachable Set of Periodic Piecewise Polynomial Systems

In this article, a Bernstein polynomial approach is first applied to the estimation of reachable set for a class of periodic piecewise polynomial systems, whose subsystems are time-varying and can be expanded to Bernstein polynomial forms. A lemma on the negativity/positivity for a class of Bernstein polynomial matrix functions is presented, which can provide a feasible set larger than that by the existing method. Based on the integration of the presented lemma and the theory of matrix polynomials, two tractable sufficient conditions are developed. For comparison of conservatism, the reachable set estimation is achieved through optimizing the ellipsoidal bounding region. Four sets of constraints with different conservatism are derived and compared. The effectiveness and superiority of the Bernstein polynomial approach in reachable set estimation are demonstrated via an illustrative example. The results show that the proposed approach enables lower conservatism in reachable set estimation, providing an intuitive route to tackle time-varying parameter products with high powers.

Stability analysis problems of periodic piecewise polynomial systems

This paper studies the stability analysis problems of periodic piecewise systems, in which subsystems are given in the time-varying polynomial forms. A Lyapunov function with continuous time-varying Lyapunov matrix is adopted, which relaxes constraints on the variation of Lyapunov function in each subsystem. Using the time interval information, the exponential stability and stabilizing controller synthesis are studied. The results provide a possible alternative method to study the general periodic time-varying systems, which may further support the analysis and synthesis of general periodic systems. The effectiveness of the proposed method is validated and illustrated via numerical examples.

Limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincaré cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

Piecewise Polynomial Model Reduction Method for Nonlinear Systems in Time Domain

AbstractModel order reduction (MOR) of nonlinear systems draws great attention in the past several decades. This paper presents a new MOR method in time domain for nonlinear dynamical systems. The new algorithm is based on the combination of the Taylor series expansion for the state variable x(t) and the trajectory piecewise polynomial technique. Firstly, the nonlinear system is approximated by a piecewise polynomial representation. Then, based on the Taylor series coefficients of x(t), we formulate the projection matrix V for the piecewise polynomial system and the compact model of the piecewise polynomial system is obtained in the following. Besides, error estimation and stability analysis are also presented in this paper. Finally, two nonlinear systems are tested to verify the effectiveness of the algorithm.

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

Bifurcation of limit cycles at infinity in piecewise polynomial systems

In this paper, we study bifurcation of limit cycles from the equator of piecewise polynomial systems with no singular points at infinity. We develop a method for computing the Lyapunov constants at infinity of piecewise polynomial systems. In particular, we consider cubic piecewise polynomial systems and study limit cycle bifurcations in the neighborhood of the origin and infinity. Moreover, an example is presented to show 11 limit cycles bifurcating from infinity.

Modelling of Driver’s Stopping Maneuver Based on Expression as Hybrid Dynamical System under Three-Dimensional Virtual Reality Equipment

This paper presents a new framework to understand the modeling strategy of the human driving behavior based on the expression as Hybrid Dynamical System(HDS) model focusing on the driver’s stopping maneuver. The driving data are collected by using the three-dimensional driving simulator based on CAVE Automatic Virtual Environment(CAVE), which provides stereoscope immersive virtual environment. In our modeling, the control scenario of the human driver, that is, the mapping from the driver’s sensory information to the operation of the driver such as acceleration, braking and steering, is expressed by Piecewise Polynomial System (PWPS) model. Since the PWPS model includes both continuous behavior given by polynomials and discrete logical conditions, it can be regarded as a class of Hybrid Dynamical System(HDS). The key idea to solve the identification problem for the PWPS model is formulated as the Mixed Integer Linear Programming(MILP) by transforming the switching conditions into binary variables. From the obtained results, it is found that the driver appropriately switches the ‘control law’ according to the following scenario : At the beginning of the stopping maneuver (just after finding the stopping point), the driver decelerate the vehicle based on the acceleration information, and then switch th toe control law based on the distance to the stop line. These results enable us to capture not only the physical meaning of the driving skill(polynomials), but also the decision-making aspect(switching conditions) in the driver’s stopping maneuver.

The maximum number and its distribution of singular points for parametric piecewise algebraic curves

The piecewise algebraic curve, as the zero set of a bivariate spline function, is a generalization of the classical algebraic curve. Based on the previous method presented by Lai et al. (2009), we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems. In this article, we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. This method also produces necessary and sufficient conditions of its parameters must be satisfied. An illustrated example shows that our method is flexible.