Switching Manifold Research Articles

A Fuzzy Design for a Sliding Mode Observer-Based Control Scheme of Takagi-Sugeno Markov Jump Systems under Imperfect Premise Matching with Bio-Economic and Industrial Applications

Fuzzy theory is widely studied and applied. This article introduces an adaptive control scheme for a class of non-linear systems with Markov jump switching. The introduced scheme supposes that the system is submitted to external disturbances under imperfect premise matching. By using discrete-time Takagi–Sugeno fuzzy models, a sliding mode observer-based control scheme is utilized to estimate unmeasured states of the system. We build two fuzzy switching manifolds for the disturbance and sliding mode observer systems. Then, a linear matrix inequality-based criterion is developed using slack matrices. This criterion proves that the sliding mode dynamics are robustly admissible under an H-infinity performance often used in control theory. Hence, new adaptive sliding mode controllers are synthesized for the disturbance and sliding mode observer systems. This allows the reachability of pre-designed sliding surfaces to be guaranteed. Finally, experimental numerical illustrations on a bio-economic system and a tunnel diode circuit are presented to show potential applications, as well as validating the effectiveness of the scheme proposed in the present investigation.

Bifurcation of periodic orbits and its application for high-dimensional piecewise smooth near integrable systems with two switching manifolds

In this paper, we study the bifurcation of periodic orbits for high-dimensional piecewise smooth near integrable systems defined in three regions separated by two switching manifolds. We assume that the unperturbed system has a family of periodic orbits which cross two switching manifolds transversely. The expression of Melnikov function is derived based on the first integral. And the conditions of periodic orbits bifurcated from a family of periodic orbits for the high-dimensional piecewise smooth near integrable system are obtained. The theoretical results are applied to the bifurcation analysis of periodic orbits of two-degree-of-freedom piecewise smooth system of nonlinear energy sink. The periodic orbits configurations are presented with numerical method and the number of periodic orbits is three.

Global dynamics for a class of tristable system with negative stiffness

Tristable system with negative stiffness has attracted extensive attention in the low frequency vibration isolation and vibration energy harvester. As a low frequency vibration isolator, it can achieve high static stiffness and low dynamic stiffness. As a vibration energy harvester, it had a wider bandwidth for resonance than the bistable one. The introduction of negative stiffness may induce subharmonic resonance and chaos in the tristable system. Chaos usually brings disorder to mechanical vibration system. Subharmonic resonance plays the negative effect on low frequency vibration isolation because they will transfer the high frequency energy of the system to the low frequency, but it is beneficial to broaden the working frequency band of vibration energy harvester. In this paper, the subharmonic bifurcation and chaos of a class of tristable system with negative stiffness are studied. The piecewise linearized systems are established to approximate the system with tristable potential. In order to conduct Melnikov analysis, the homoclinic-heteroclinic orbits and periodic orbits for the unperturbed piecewise linearized system are obtained respectively. The subharmonic Melnikov method for nonsmooth systems with four switched manifolds is developed. The thresholds for homoclinic-heteroclinic chaos and subharmonic resonance are derived by using non-smooth Melnikov method. It provides a theoretical support not only for design of the vibration energy harvester to obtain wider working frequency band, but also for design of the vibration isolation system to avoid high frequency energy transfer to low frequency. Moreover, the phenomena for infinite subharmonic bifurcations to chaos from odd order subharmonic orbit and the coexistence for chaotic and subharmonic attractors are revealed. The subharmonic Melnikov method with four switching manifolds developed in this paper lays a foundation for the subharmonic resonance analysis of other nonsmooth tristable systems.

Chaos emerges from coexisting homoclinic cycles for a class of 3D piecewise systems

Accurately detecting a homoclinic cycle and chaos in a concrete system takes a huge challenge. This paper gives a sufficient condition of coexisting homoclinic cycles (degenerate) with respect to a saddle-focus in a class of three-dimensional piecewise systems with two switching manifolds. Furthermore, by constructing the Poincaré map, it is rigorously shown that chaotic behavior is induced by the coexisting cycles without necessary symmetry in the considered system, which is evidently different from the usual dynamics in smooth system theory that a pair of symmetrical homoclinic cycles can generate chaos but the asymmetric ones cannot. It finally provides an example to illustrate the effectiveness of the results established.

Melnikov-type method for a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation: Homoclinic orbits.

The Melnikov method is extended to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation when an unperturbed system is a piecewise Hamiltonian system with a homoclinic orbit. The homoclinic orbit continuously crosses the first switching manifold and transversally jumps across the second switching manifold by the impulsive effect. The trajectory of the corresponding perturbed system crosses the first switching manifold by applying the reset map describing the impact rule instantaneously. Then, the random Melnikov process of such systems is derived and the criteria for the onset of chaos with or without noise excitation are established. In addition, the complicated dynamics of concrete piecewise-smooth systems with or without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations are investigated by this extended method and numerical simulations.

Single Phase Second Order Sliding Mode Controller for Mismatched Uncertain Systems with Extended Disturbances and Unknown Time-Varying Delays

In this paper, a novel single phase second order sliding mode controller (SPSOSMC) is proposed for the mismatched uncertain systems with extended disturbances and unknown time-varying delays. The main achievements of this study consist of three tasks: 1) a reaching phase in conventional sliding mode control (CSMC) technique is removed to ensure the global stability of the system; 2) an influence of the undesired high-frequency oscillation phenomenon in control input is vanished; 3) an exogenous perturbation is generally extended to the k-order disturbance of state variable. Firstly, a single phase switching manifold function is defined to eliminate the reaching phase in CSMC. Secondly, an unmeasurable state variable is estimated by using the proposed reduced-order sliding mode observer (ROSMO) tool. Next, a SPSOSMC is built based on the help of ROSMO tool and output information only. Then, a sufficient condition is established by employing the linear matrix inequality (LMI) technique and Lyapunov function theory such that the resulting sliding mode dynamics is asymptotically stable. Finally, a numerical example is simulated via the well-known MATLAB software to validate the effectiveness of the proposed technique.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths for a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin–Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an n-dimensional system with a switching manifold in the drift that forms an (n−1)−dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use Γ−convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Attenuated-chattering adaptive second order variable structure controller for mismatched uncertain systems

In this paper, an attenuated-chattering adaptive second order variable structure controller (ACASOVSC) is proposed for mismatched uncertain systems using a Moore-Penrose inverse method. The key achievements of this study include three tasks: 1) influence of the chattering in control input is diminished, 2) finite-time convergence of system states is guaranteed, and 3) external disturbance is generally assumed to be unknown in advance. Firstly, a switching manifold which comprises only output information is defined. Secondly, a reduced-order variable structure estimator (ROVSE) with lower dimension is designed to reduce the computation burden and enhance the robustness. Thirdly, an adaptive approach is used to guess the upper bound of the unknown exogenous disturbance. Next, an ACASOVSC is investigated for attenuating the chattering phenomenon and stabilizing the system. Then, a novel linear matrix inequality (LMI) constraint by the Lyapunov technique is given such that the plant is entirely invariant to matched uncertainties and asymptotically stable. Finally, a mathematical illustration is simulated, which exhibits the usefulness and the feasible application of the proposed method.

Dimension reduction for slow-fast, piecewise-linear ODEs and obstacles to a general theory

This paper concerns systems of ODEs that are continuous, piecewise-linear (with two pieces) and slow-fast. The limiting slow dynamics occurs on a piecewise-linear critical manifold that corresponds to equilibria of the fast dynamics (layer equations). When the critical manifold is attracting for each piece of the layer equations, one would like to know when the full dynamics of the system near this manifold is a regular perturbation of the limiting slow dynamics (reduced system). We identify three phenomena unique to the piecewise-linear setting that obstruct such an extension of Fenichel theory: at the boundary (switching manifold) the equilibrium of the layer equations can be unstable, coincide with another attractor, or attract orbits in a chaotic fashion. The first two phenomena (at least) can inhibit a regular perturbation and lead to the creation of canards. If a regular perturbation does occur for the piecewise-linear approximation to a boundary equilibrium bifurcation, dimension reduction can be achieved and we illustrate this with a three-dimensional model of ocean circulation. We also introduce a slow-fast version of the observer canonical form for piecewise-linear ODEs that simplifies the analysis.

Adaptive Neuro Sliding Mode Control of Superconducting Magnetic Energy Storage System

ABSTRACT In this paper, an adaptive RBF neural network-based sliding mode controller is developed for superconducting magnetic energy storage (SMES) installed in a wind–diesel power system. Due to sudden load and wind power variations in a wind–diesel power system, power imbalances may force the system frequency to deviate from its nominal value and drive the system to operate in an unstable mode. Therefore, in a wind–diesel power system using a converter interface, a fast-acting and high power density SMES device is interconnected to carry the required power exchange whenever power imbalances occur. Based on switching manifold design, a sliding mode controller is developed to control the charging and discharging operations of SMES coil as per the power requirements, and the same is achieved by controlling the converter. A neural network using a radial basis function (RBF) is developed to estimate the unknown function of the system. Lyapunov stability analysis is conducted to guarantee the asymptotic stability of the system. MATLAB simulations are carried out and are presented to show the improved performance with the system exposed to disturbances in load and wind power.

Correction to: The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a Switching Manifold

Correction to: The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a Switching Manifold

Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on . For such a map f, this is achieved by constructing three objects. First, is trapping region for f. Second, is a finite set of words that encodes the forward orbits of all points in . Finally, is an invariant expanding cone for derivatives of compositions of f formed by the words in . The existence of , , and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.

Limit cycles in Filippov systems having a circle as switching manifold.

It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here, we study the limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle when both linear differential systems are centers. Our main results show that such discontinuous piecewise differential systems can have zero, one, two, or three limit cycles, but no more limit cycles than three.

Dynamics and stability of non-smooth dynamical systems with two switches

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. In this paper, we leverage a methodology that, through blow-ups and singular perturbation, allows to extend Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, nonlinear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

Bursting oscillations with adding-sliding structures in a Filippov-type Chua’s circuit

In this paper, a modified Chua’s circuit, possessing an external excitation current and a piecewise nonlinear resistor, is considered to investigate bursting oscillations and the dynamical mechanism in the Filippov system, focusing on the effects of sliding bifurcations on the bursting dynamics. Five typical representative bursting oscillations are observed when there is a gap between the external excitation frequency and the natural one. Codimension-1 bifurcations of the fast subsystem are discussed by regarding the whole excitation term as a bifurcation parameter. The necessary conditions of conventional bifurcations of the equilibria and the bifurcations of boundary equilibria are obtained via theoretical analysis. Both the local adding-sliding bifurcation, crossing-sliding bifurcation, non-smooth fold bifurcation, the fold bifurcation of non-smooth limit cycle in the fast subsystem and the adding-sliding bifurcation in the slow–fast coupling system are observed via numerical method. Based on slow–fast analysis method and the bifurcation analysis, the dynamical mechanism is discovered. Research found that the non-smooth fold bifurcation of the boundary equilibrium can lead to jumping behaviors of the trajectory from the switching manifold to the attractors of the subsystem. The crossing-sliding bifurcation may not cause the transition between spiking state and quiescent state. The local adding-sliding bifurcation in the fast subsystem can lead to adding-sliding bifurcation of non-smooth limit cycle in the coupling system. There may exist adding-sliding structure in bursting oscillations before or after the adding-sliding bifurcation.

Dynamic performance improvement of wind-diesel power system through robust sliding mode control of hybrid energy storage system

Off-grid users can be provided with electricity via a hybrid integration of wind power generators and a diesel system functioning as a backup supply. However, due to wind power fluctuations and rapid load changes, system voltage and frequency variances may exceed permitted limits, resulting in aberrant system behavior. Therefore, to improve the dynamic performance of the wind-diesel power system, a hybrid energy storage system (HESS) made of battery and superconducting magnetic energy storage is installed with the system via a converter interface. Based on the switching manifold design, a sliding mode controller with the super-twisting feature is developed over the hybrid energy storage system (HESS) to carry out the required amount of power exchanges with the system accomplished through the control of converter operation. Lyapunov stability analysis is conducted to guarantee the asymptotic stability of the system. MATLAB simulations are performed to validate the improved performance of the system with the proposed scheme.

Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function

<p style='text-indent:20px;'>In this paper, we study the chaotic behavior of a class of hybrid piecewise-smooth system incorporated into an impulsive effect (HPSS-IE) under a periodic perturbation. More precisely, we assume that the unperturbed system with a homoclinic orbit, it transversally jumps across the first switching manifold by an impulsive stimulation and continuously crosses the second switching manifold. Then the corresponding Melnikov-type function is derived. Based on the new Melnikov-type function, the bifurcation and chaotic threshold of the perturbed HPSS-IE are analyzed. Furthermore, numerical simulations are precisely demonstrated through a concrete example. The results indicate that it is an extension work of previous references.</p>

SMC for Semi-Markov Jump Cyber-Physical Systems Subject to Randomly Occurring Deception Attacks

This brief investigates the sliding mode control (SMC) for semi-Markov jump cyber-physical systems (S-MJCPSs) under randomly occurring deception attacks (RODA). It is assumed that the adversaries inject false data into the control signal in probability. First, a mode-dependent sliding surface is designed under the framework of integral form. Second, the criteria to ensure stochastic stability (SS) of the corresponding system are proposed, which establishes the theoretical framework of semi-Markov Lyapunov function. Third, an appropriate SMC law under attacks and uncertainties is constructed to drive the states of S-MJCPSs onto the switching manifold in finite time. Finally, the simulation study on a single-link robot arm model (SLRAM) is provided to demonstrate the obtained results.

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

<p style='text-indent:20px;'>In this paper we consider an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold <inline-formula><tex-math id="M2">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> which is an intersection of two hyperplanes <inline-formula><tex-math id="M3">\begin{document}$ \Sigma_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \Sigma_2 $\end{document}</tex-math></inline-formula>. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of <inline-formula><tex-math id="M5">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, converging to <inline-formula><tex-math id="M6">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> as both of the two regularization parameters go to <inline-formula><tex-math id="M7">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> by applying implicit function theorem and geometric singular perturbation theory.</p>

A novel framework to finite-time tracking of nonlinear MIMO systems based on fast terminal sliding-mode method

In this paper, a novel approach is proposed to design a robust finite-time tracking controller for uncertain affine multi-input multi-output (MIMO) systems. In the proposed approach, a fast terminal sliding mode (FTSM) controller is designed, based on the input–output relation of the MIMO nonlinear systems. The main advantage of the proposed method is its capability to handle the internal dynamics of the system. Moreover, in the proposed controller, the state variables of the internal dynamics of the system are not necessary to be measured. To realize the finite-time convergence of the output variables, a set of switching manifolds with a recursive procedure is utilized. Finally, the robust stability and tracking efficiency of the proposed control law are shown through computer simulations.