Singular System Theory Research Articles

Hamilton-Jacobi Treatment of Singular Systems Using Fractional Calculus

In this paper, the theory of fractional singular systems is investigated with second-order derivatives. The fractional Hamilton–Jacobi treatment of these systems is examined. The fractional Hamilton–Jacobi partial differential equations (FHJPDEs) are constructed. The (FHJPDEs) can be solved to obtain the fractional Hamilton–Jacobi function. By building the fractional Hamilton–Jacobi function, the equations of motion can be obtained.

Fuzzy tracking control of singular multi-agent systems under switching topology

<p>The consensus tracking problem of leader-follower multi-agent systems (MASs) with singular structures on jointly connected topology is studied in this paper. To achieve the objective of consensus tracking, a distributed adaptive control protocol is formulated to adjust the coupling weights among the agents using the adaptive rate, where the adaptive protocol can be implemented by each agent in a fully distributed manner without using any global information. A fuzzy logic system method is used to deal with the nonlinear terms in response to the limitations of nonlinear system analysis. The consensus tracking problem is transformed into an error system stability analysis, and two sufficient conditions are provided to guarantee the control objective based on Lyapunov stability theory and singular system theory. Finally, the effectiveness of this method is verified through a simulation example.</p>

Lie symmetries and conserved quantities of fractional nonconservative singular systems

AbstractIn this paper, according to the fractional factor derivative method, we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space. First, fractional calculus is calculated by using the fractional factor, and the fractional equations of motion are derived by using the differential variational principle. Second, the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained. Furthermore, the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge‐generating function that fulfills the structural equation, which conforms to the Noether criterion equation. Finally, we present an example of a calculation. The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems, but because of increased invariance restriction, the nonconservative forces do not change the form of conserved quantity; meanwhile, the fractional factor method has high natural consistency with the integral calculus, so the theory of integer‐order singular systems can be easily extended to fractional singular Lagrange systems.

Approach to Energy Objects’ Dynamics Modelling Based on Singular Systems’ Elements

Over the last decades there has been substantial progress on the development of theory and numerical methods for singular systems (known also as descriptor systems, semistate systems, differential alge-braic systems, generalized state-space systems, etc.). The need for such methodsarisen primarily from the increased practical interest for a more general system description which takes the intrinsic physical sys-tem model structure into account. Besides that, many physical process-es are most naturally and easily modelled as mixed systems of differen-tial and algebraic equations (DAE). As the title implies the paper de-scribes the singular systems theory application in power systems dy-namics simulation, particularly considered an alternative method for energy systems’ mathematical models formulation based on the singu-lar systems theory elements with some indicative examples illustrating feasibility and efficiency of this approach

Consensus tracking for singular multi‐agent systems with Lipschitz nonlinear dynamics and external disturbances

AbstractThe consensus tracking control problem for singular multi‐agent systems (SMASs) with Lipschitz nonlinearities and unknown disturbances is investigated. An extended state observer using only the relative output information is introduced and then a distributed consensus protocol is proposed, whose performance is analyzed using matrix analysis, singular system theory and Lyapunov stability theory. Sufficient conditions for ensuring consensus tracking with disturbance rejection of linear and Lipschitz nonlinear SMASs are, respectively, obtained under directed networks. Specifically, an equivalent formulation of achieving consensus tracking for SMASs is established, and the design procedures of gain matrices in the proposed consensus protocols are presented by the linear matrix inequality solver. Finally, simulation examples are provided to illustrate the theoretical results.

Guaranteed‐performance consensus for nonlinear singular multiagent systems with directed topologies

AbstractThe guaranteed‐performance consensus problems for nonlinear singular multiagent systems with directed topologies in both leader‐following case and leaderless case are investigated, where the directed interaction topologies contain a directed spanning tree. For situations with and without the leader, two different quadratic performance functions are proposed based on state errors. The state errors in the leader‐following case are the state differences between the followers and the leader while in the leaderless case are the state differences between agents with adjacent numbers. First, by exploiting state transformation, the consensus problems are transformed into the stability problems of the corresponding error systems. Second, on the basis of singular systems theory, Laplacian matrix properties, and linear matrix inequality techniques, sufficient conditions for guaranteed‐performance consensus are derived for both leader‐following and leaderless cases. Third, the explicit expressions of the guaranteed‐performance cost are obtained. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

Admissible Bipartite Consensus in Networks of Singular Agents Over Signed Graphs

This brief focuses on the network of singular agents with unknown disturbances. An admissible bipartite consensus problem is considered under cooperative-competitive interactions modeled by signed graphs. A distributed control algorithm derived by the extended state observer is designed, based on which singular multi-agent systems achieve not only disturbance rejection but also admissible bipartite consensus. It is shown that, by using singular system theory and matrix theory, the designed controller is solvable for admissible bipartite consensus under the assumption of connection and structural balance. Finally, the simulation example illustrates the feasibility of the proposed method.

Стійкість двоконтурних систем керування напругою DC-DC перетворювача

The paper is devoted to research and development of the cascaded DC-link voltage control systems for DC-DC boost converters whose mathematical model is highly nonlinear and non-minimum phase. The relevance of the analysis method for classic control systems for DC-DC boost converters is substantiated, which are similar to vector-controlled electric drives systems if the speed controller is replaced by DC-link voltage controller. It is shown that the reduced-order solution of initial nonlinear system dynamics can be obtained giving the time-scale separation between the input current and the DC-link voltage control processes based on the singular perturbation systems theory. In its turn, it is concluded that the reduced-order system dynamics is locally (asymptotically) stable if cascaded control algorithm is applied. The similarity of the time-scale separation conditions for the cascaded control systems with proportional and proportional-integral DC-link voltage controller is shown. The time-scale separation of the control processes is achieved if inner current regulation loop is much faster than the outer voltage regulation loop. The simulation study demonstrates that if the time-scale separation conditions between the input current and the DC-link voltage regulation are met then the full-order system dynamics of the DC-link voltage regulation of DC-DC boost converters is close to the reduced-order system dynamics. The control algorithm has a typical structure of the modern controlled converters applied in hybrid energy storage systems for electric vehicles. From the results of the simulation study, it follows that the influence of the inner resistance of the input inductance on the system stability and quality indicators of the DC-link voltage control process is insignificant for the investigated DC-DC boost converter.

PD-Type Iterative Learning Control for Uncertain Spatially Interconnected Systems

This paper puts forward a PD-type iterative learning control algorithm for a class of discrete spatially interconnected systems with unstructured uncertainty. By lifting and changing the variable of discrete space model, the uncertain spatially interconnected systems is converted into equivalent singular system, and the general state space model is derived in view of singular system theory. Then, the state error and output error information are used to design the iterative learning control law, transforming the controlled system into an equivalent repetitive process model. Based on the stability theory of repetitive process, sufficient condition for the stability of the system along the trial is given in the form of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed algorithm is verified by the simulation of ladder circuits.

Diseño e implementación de un controlador no-lineal para un brazo robótico con articulaciones flexibles y enlaces rígidos

Los sistemas robóticos se están volviendo cada vez más complejos y las teorías de la ley de control tradicional pierden robustez, lo que aumenta la dificultad con la que se puede controlar el robot para interactuar con el entorno que lo rodea. El objetivo de este trabajo de investigación es el estudio de sistemas no lineales complejos con la particularidad de tener articulaciones flexibles y enlaces rígidos. Dicha flexibilidad provoca un comportamiento interesante en los sistemas robóticos porque duplica el número de variables involucradas en la tarea de control. Se han llevado a cabo varios estudios en la investigación de robots flexibles, sin embargo, la mayoría de ellos utilizan el marco clásico de Euler-Lagrange para describir los sistemas mecánicos. Este trabajo se ha centrado en la implementación de controladores no lineales dentro del marco portuario-hamiltoniano, y en la teoría de sistemas de perturbación singular a escala múltiple. En este sentido, la descripción matemática de dos leyes de control diferentes propuestas por [1] y [2] se presentan y adaptan a la planta física del brazo robótico Quanser de dos grados de libertad. Además, las ecuaciones de los controladores port-hamiltonianos propuestos se han implementado en una simulación para probar la validez de las leyes de control para la configuración rígida y flexible del robot. Finalmente, los controladores se han implementado en la planta física del brazo robótico para validar experimentalmente la teoría matemática de control propuesta. La implementación experimental de los controladores puerto-hamiltonianos propuestos mostró una mejora en el control del error de posición para la configuración rígida y flexible en comparación con un controlador de referencia propuesto por el fabricante del brazo robótico, con una tasa de error para el valor RMS de la señal inferior al 1,2% del valor RMS de la trayectoria deseada. Otros estudios y pruebas experimentales deben estar dirigidos a la implementación de controladores porto-hamiltonianos para lograr una tasa de error aún más baja.

Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity

For the nonlinear Schrodinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For \begin{document}$ m>1 $\end{document} and \begin{document}$ 0 , corresponding to the level curves given by \begin{document}$ H(\psi, y) = 0 $\end{document} , the exact explicit bounded traveling wave solutions can be given. For \begin{document}$ m = 1 $\end{document} , corresponding all bounded phase orbits and depending on the changes of system's parameters, all exact traveling wave solutions of the equation can be obtain.

Guaranteed‐performance consensus tracking of singular multiagent systems with Lipschitz nonlinear dynamics and switching topologies

SummaryIn this paper, the problem of guaranteed‐performance consensus tracking of continuous‐time singular multiagent systems with Lipschitz nonlinearities and switching topologies is investigated. Consideration is that the interaction of the concerned agent network is described by a set of directed graphs with the union graph having a directed spanning tree rooted at the leader. To establish the guaranteed‐performance criterion, a quadratic performance function is constructed by utilizing the consensus errors among all agents. Then, a consensus protocol that collects the local information from neighboring agents is proposed to achieve consensus tracking and to guarantee the consensus regulation performance of the multiagent systems. On the basis of nonsingular transformation approach, singular systems theory, and Lyapunov stability analysis, the concerned guaranteed‐performance consensus tracking problem is cast into the admissibility analysis for an equivalent kind of switched singular consensus error system. Furthermore, sufficient conditions on the guaranteed‐performance consensus tracking protocol design are formulated in terms of linear matrix inequalities. Finally, numerical examples are employed to demonstrate the effectiveness of the theoretical results.

Observer-based Intermittent Consensus Control of Nonlinear Singular Multi-agent Systems

This paper studies the leader-following consensus for a class of nonlinear singular multi-agent systems. Unlike the related works, the case that states of agents are unavailable and communication channel among agents may be subjected to frequently attacks is considered, where agents can only communicate with their neighbors on some disconnected time intervals. Based on graph theory, singular system theory and multi-Lyapunov function approach, the observer-based intermittent feedback control protocol and sufficient conditions are given to ensure the existence and uniqueness of solutions and consensus of the considered systems. Besides, a description of state jumps of the systems at the instants of communication changes is presented. Finally, a numerical example is given to illustrate the feasibility of the result.

Leader‐following consensus of nonlinear singular multiagent systems with intermittent communication

This paper investigates the leader‐following consensus of nonlinear singular multiagent systems with intermittent communication on a fixed undirected topology. Unlike the existing works on singular multiagent systems that the communication among agents is continuous, in this work, agents can only communicate with their neighbors on some disconnected time intervals. Based on algebraic graph theory, singular system theory and multi‐Lyapunov function approach, the cooperative intermittent control protocol and sufficient conditions are proposed to guarantee the existence and uniqueness of solutions and leader‐following consensus of the systems. Furthermore, a description of state jumps at the instants of communication interrupt and restoration is presented. Finally, numerical examples are provided to illustrate the effectiveness of the theoretical analysis.

Two side observer design for singular distributed parameter systems

We consider the problem of state observation for singular distributed parameter system (SDPS) with singular derivative term. For the nonhomogeneous boundary, we define the SDPS observer and investigate the exponential stability of the error SDPS with SDPS theory. Considering the uncertainty, the robustness of proposed SDPS observer is investigated by singular system theory. As an application, the state observation of distributed building automatic temperature system is considered.

Event‐triggered Guaranteed Cost Consensus of Networked Singular Multi‐agent Systems

AbstractThe problem of event‐triggered guaranteed cost consensus of discrete‐time singular multi‐agent systems with switching topologies is investigated in this paper. To save the limited network communication bandwidth of multi‐agent systems, a novel event‐triggered networked consensus mechanism is proposed. Based on the graph theory and singular system theory, sufficient conditions of guaranteed‐cost consensus of discrete‐time singular multi‐agent systems are derived and given in the form of the linear matrix inequalities, respectively. A co‐design approach of the multi‐agent consensus gain matrix and the event‐triggered parameters is presented. Furthermore, based on the approach of second class equivalent transformation for singular systems, the cost function is determined, and an explicit expression of consensus functions is presented. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.

Noether symmetries and conserved quantities of fractional nonconservative singular systems

The Noether theory of fractional nonconservative singular systems is studied based on fractional factor derivative method in form space. The Lagrange equations with fractional factor are established through the variational principle. The criterion equation and the conserved quantities are further studied according to the fractional order Hamilton action quantity maintain invariance under the infinitesimal transformation. Finally, an example is given to illustrate the application. The results show that comparing with the conservative systems, the nonconservative forces have impact on the Noether identity, but because of enhancing the invariance condition, it does not change the form of Noether type conserved quantities, at the same time, we use fractional factor method to study the nonconservative singular systems, some conclusions are highly natural consistent with the classical integer order singular systems, so the fractional factor can establish the connection between the fractional order systems and the integer order systems.

P-type iterative learning control algorithm for a class of linear singular impulsive systems

This paper is focused on the iterative learning control problem for linear singular impulsive systems. For the purpose of tracking the desired output trajectory, a P-type iterative learning control algorithm is investigated for such system. Based on the fundamental property of singular impulsive systems and the restricted equivalent transformation theory of singular systems, the convergence conditions of the tracking errors for the system are obtained in the sense of λ norm. Finally, the validation of the algorithm is confirmed by a numerical example.

Analysis and Control of the Singular System Model of Aphid Ecosystems

Considering the change of the parameter related to the natural enemy population and the impact on the aphid populations in the fold catastrophe manifold, the singular system model of aphid ecosystems is proposed. Combining singular system theory with catastrophe theory, the corresponding dynamics behaviors and the existence conditions of the impasse points are given by using the qualitative analysis. The biological significance of the analytical results is also discussed. The controllers are designed to make the aphid populations stabilize the refuge level by releasing natural enemy. Some numerical simulations are carried out to prove the results.

A new highly anti-interference regularization method for ill-posed problems

The solution of inverse problems has many applications in mathematical physics. Regularization methods can be applied to obtain the solution of ill-conditioned inverse problems by solving a family of neighboring well-posed problems. Thus, it is significant to investigate the regularization methods to increase the accuracy and efficiency of the solution of inverse problems. In this work, a new regularization filter and the related regularization method based on the singular system theory of compact operator are proposed to solve ill-posed problems. The Cauchy problem of Laplace equation of the first kind is a kind of well-known ill-posed problem. Numerical tests show that the proposed regularization method can solve the Cauchy problems more efficiently under a proper selection of regularization parameters. Numerical results also show that the proposed method is especially effective in solving ill-posed problems with big perturbations.