Common Equilibrium Point Research Articles

Practical stabilisation of discrete-time switched nonlinear systems without a common equilibrium and using hybrid switching functions

This paper presents a new practical stabilization method for discrete-time switched nonlinear systems without a common equilibrium point among all modes. The proposed method has two main features. First, the number of nonlinear terms in the stability conditions is independent of the number of switched system modes. The result is that the computational cost is reduced, and it can be applied to switched systems with a large number of subsystems. Second, using switched Lyapunov functions, hybrid switching laws are designed to preserve the practical stability property in both the closed-loop state-dependent switching function and the open-loop time-dependent one. As a result, when the state data is lost during sensor or communication failures, the controller can change the control scheme from a closed-loop state-dependent switching function to an open-loop time-dependent one without loss of stability property. Numerical examples are presented to illustrate the validity and efficiency of the proposed method.

Dynamic output feedback based fault tolerant control for continuous-time switched affine systems via reduced-order observer

In this study, the issue on dynamic output feedback fault-tolerant controller design is addressed for a class of continuous-time switched affine systems in the presence of actuator faults. The existence of affine terms gives rise to the lack of a common equilibrium point and further makes it more difficult to propose the desired output-feedback-based fault-tolerant control scheme. Firstly, the actuator fault estimation is achieved through a reduced-order observer design approach. Then, a kind of dynamic output feedback controller is constructed to perform the fault tolerance goal by utilizing fault estimation information. Moreover, the mode-dependent average dwell time switching constraint is borrowed to avoid the chattering owing to the high-frequency switching. The sufficient conditions with less conservative are derived to guarantee the practical exponential stability of closed-loop system with a prescribed L∞ gain. Finally, both the effectiveness and validity of the designed fault-tolerant control strategy are illustrated via a case study of a buck-boost DC-DC converter.

Bounded output‐feedback consensus‐based formation control of nonholonomic vehicles

AbstractIn multi‐agent systems, most commonly, consensus consists in all agents converging to a common equilibrium point. For nonholonomic vehicles, which move on a Cartesian plane and occupy a physical space, it is more appropriate to speak of consensus‐based formation, which implies that the vehicles are required to converge to a relative position in a given formation while, distributively, agreeing on the center of such formation and on their orientation. In this paper we solve such a problem via a smooth, time‐varying, output‐feedback control; more precisely, without velocity measurements. In addition, and this is our main contribution, the controller is designed to satisfy pre‐imposed bounds to avoid input saturation. To the best of the authors' knowledge this is the first controller that solves the consensus‐based formation problem in networks of nonholonomic vehicles via a bounded, output‐feedback controller. The paper also provides a simulation comparison between our proposal and its unbounded counterpart.

Dynamic output feedback [formula omitted] control of switched affine systems: An event-triggered mechanism

This work focuses on the dynamic output feedback L∞ control for switched affine systems with dwell time constraints. The presence of affine terms leads to the absence of a common equilibrium point and further makes it more difficult to design dynamic output feedback controllers and avoid the triggering Zeno behavior. To save resources and eliminate the effects of affine terms, first, an improved event-triggered condition with a constant and output information is provided. It is proved that the proposed triggering condition can prevent Zeno behavior from happening by obtaining a minimal inter-event interval. Then construct a novel dynamic output feedback switched affine controller, which skillfully adopts the mode-dependent average dwell time approach with Lyapunov stability theory to obtain a more rigorous and complete control strategy. It not only accommodates the unavailable state information and network transmission restrictions, but also achieves the practical stability of the closed-loop system while ensuring the L∞ performance from external disturbance to the output. Finally, the efficiency of the proposed algorithm is illustrated by simulating the buck-boost DC–DC converter.

Strict Lyapunov Functions for Dynamic Consensus in Linear Systems Interconnected Over Directed Graphs

We study dynamic consensus for general networked (homogeneous) linear autonomous systems, that is, it is only assumed that they are stabilizable. Dynamic consensus pertains to a general form of consensus in which, as a result of the systems’ interactions, they exhibit a rich collective dynamic behavior. This generalizes the classical consensus paradigm in which case all systems stabilize to a common equilibrium point. Our main statements apply to systems interconnected over generic directed connected graphs and, most significantly, the proofs are constructive. Indeed, even though our controllers are reminiscent of others previously used in the literature, to the best of our knowledge, we provide for the first time in the literature strict Lyapunov functions for fully distributed consensus over generic directed graphs.

Almost Output Regulation of Switched Affine Systems and Its Application to a Circuit Model

This brief is concerned with the almost output regulation issue of switched affine systems. For switched affine systems, affine terms make the absence of the common equilibrium point, which also brings the distinct and difficulty on the analysis of output regulation performance. Moreover, in contract to the classical output regulation problem, a more extensive situation where the controlled plant and exosystem suffer from the unknown external interferences is concerned. Accordingly, the concept of the almost output regulation problem of switched affine systems is presented. Then, the co-design method of the switching law and regulators is established to guarantee the almost output regulation property. The given co-design method gets ride of the requirement for the almost output regulation performance of each subsystem. At last, the employed electronic circuit model substantiates the applicability and validity of the developed results.

Regulation cooperative control for heterogeneous uncertain chaotic systems with time delay: A synchronization errors estimation framework

This paper addresses the regulation cooperative control problem of heterogeneous uncertain chaotic systems with nonlinear dynamics and time delay. By proposing a robust estimation framework for synchronization error level, this paper investigates the case that complete synchronization for heterogeneous uncertain chaotic systems cannot be achieved since multiple heterogeneous uncertain chaotic systems do not have a common equilibrium point. By introducing a virtual leader node, it is proven that the regulation cooperative control for heterogeneous uncertain chaotic systems can be achieved with an error level by constructing an appropriate synchronization tracking condition with free parameters. To demonstrate the proposed approach in engineering applications, it is applied to the secure communication with multiple heterogeneous uncertain chaotic systems, in which each chaotic synchronization state is treated as an information signal that is injected into the transmitter and simultaneously transmitted to the receiver. Simulation results on heterogeneous uncertain chaotic systems and secure communication consisting of six nonidentical Chua’s circuits are presented to demonstrate the effectiveness of the proposed approaches.

Region stability analysis and tracking control of memristive recurrent neural network

Memristor is firstly postulated by Leon Chua and realized by Hewlett–Packard (HP) laboratory. Research results show that memristor can be used to simulate the synapses of neurons. This paper presents a class of recurrent neural network with HP memristors. Firstly, it shows that memristive recurrent neural network has more compound dynamics than the traditional recurrent neural network by simulations. Then it derives that n dimensional memristive recurrent neural network is composed of 22n2 sub neural networks which do not have a common equilibrium point. By designing the tracking controller, it can make memristive neural network being convergent to the desired sub neural network. At last, two numerical examples are given to verify the validity of our result.

Generating Chaos from Two Three-Dimensional Rigorous Linear Systems via a Novel Switching Control Approach

By constructing two three-dimensional (3D) rigorous linear systems, a novel switching control approach for generating chaos from two linear systems is presented. Two 3D linear systems without any constant term have only one common equilibrium point that is the origin. By employing an absolute-value switching law, chaos can be generated by switching between two linear systems. Basic dynamical behaviors of the systems are investigated in detail. Numerical examples illustrate the effectiveness of the presented approach.

Proximal point algorithm for infinite pseudo-monotone bifunctions

In this paper, first we study the weak convergence of the proximal point algorithm for an infinite family of equilibrium problems of pseudo-monotone type in Hilbert spaces. Then with additional conditions on the bifunctions, we prove the strong convergence for the family to a common equilibrium point. We also study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point of the family of infinite pseudo-monotone bifunctions without any additional assumptions on the bifunctions. A concrete example of a family of pseudo-monotone bifunctions is also presented.

On the emergence of chaos in dynamical networks

We investigate how changes of specific topological features result on transitions among different bounded behaviours in dynamical networks. In particular, we focus on networks with identical dynamical systems, synchronised to a common equilibrium point, then a transition into chaotic behaviour is observed as the number of nodes and the strength of their coupling changes. We analyse the network's transverse Lyapunov exponents (tLes) to derive conditions for the emergence of bounded complex behaviour on different basic network models. We find that, for networks with a given number of nodes, chaotic behaviour emerges when the coupling strength is within a specific bounded interval; this interval is reduced as the number of nodes increases. Furthermore, the endpoints the emergence interval depend on the coupling structure of network. We also find that networks with homogeneous connectivity, such as regular lattices and small-world networks are more conducive to the emergence of chaos than networks with heterogeneous connectivity like scale-free and star-connected graphs. Our results are illustrated with numerical simulations of the chaotic benchmark Lorenz systems, and to underline their potential applicability to real-world systems, our results are used to establish conditions for the chaotic activation of a network of electrically coupled pancreatic β-cell models.