Real Jordan Form Research Articles

Limit cycles in 4-star-symmetric planar piecewise linear systems

Our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is xy=0. We consider the symmetric case. That is, one vector field defined in the odd quadrants and the other in the even ones. We deal with equilibrium points of center-focus type, with matrices in real Jordan form, in each vector field when the infinity is monodromic. In this case, we provide the center classification at infinity, we prove that the maximum order of a weak focus is five. Moreover, we show the existence of systems exhibiting five limit cycles bifurcating from infinity.

Canonical heights and monomial maps: On effective lower bounds for points with dense orbit

We prove, for the canonical height defined by Silverman [10] on monomial maps, the existence of effective lower bounds for heights of points with Zariski dense orbit, for cases with endomorphisms induced by matrices with real Jordan form.

A Truncated Prediction Approach to Consensus Control of Lipschitz Nonlinear Multiagent Systems With Input Delay

This paper deals with the consensus control problem for Lipschitz nonlinear multiagent systems with input delay. A prediction of the agent state over the delay period is approximated by the zero input solution of the agent dynamics. The structure of a linear state feedback control algorithm is assumed for each agent based on such approximated state prediction. By transforming the Laplacian matrix into the real Jordan form, sufficient conditions are established under which the multiagent systems under the proposed control algorithms achieve global consensus. The feedback gain is then designed by solving these conditions with an iterative linear matrix inequality procedure. A simulation study is given to validate the proposed control design.

Stochastic consensus control with finite frequency specification for Markov jump networks

SummaryA stochastic finite frequency consensus protocol for directed networks with Markov jump topologies and external disturbances is proposed in this paper. By introducing the frequency band information into consensus control design, the disagreement dynamics of interconnected networks asymptotically converge to zero with an improved level of disturbance attenuation in the specific frequency band. In addition, the new model transformation approach is presented by exploring certain features of Laplacian matrix in real Jordan form, which leads to more generality of the designed protocol. A numerical example validates the potential of the developed results. Copyright © 2015 John Wiley & Sons, Ltd.

Stochastic finite‐time consensualisation for Markov jump networks with disturbance

This study is devoted to the finite-time consensus control for directed networks with stochastic Markov jump topologies and external disturbances. The purpose of the study is to design a control protocol to ensure that the disagreement dynamics of interconnected networks stay in a given bound over a finite-time interval rather than asymptotically converge to zero in infinite settling time. Through utilisation of certain features of Laplacian matrix in real Jordan form, sufficient conditions for the existence of finite-time consensus protocol is derived by allowing Lyapunov function to increase in a fixed-time interval. Finite-time convergence result for stochastic consensus problem is validated via a simulation study.

Robust H ∞ control for time‐delay networked control systems with probability constraints

In this study, the probability-guaranteed H ∞ control of networked control systems (NCSs) with time-varying networked-induced delays is addressed. The time-varying delays are modelled as polytopic uncertainty by a new approximation method based on real Jordan form. The system uncertainty lies in a parameter hyper-rectangle box which can shrink, i.e. the probability can be less than one with a better disturbance attenuation level. By the uniformity principle and the one-to-one correspondence between the time delays and the vertices of polytopic system, a sufficient condition for the NCS to achieve prescribed performance with a given probability is derived. The controller synthesis problem is also proposed to meet the probabilistic requirement by solving a set of bilinear matrix inequalities iteratively. A uniform distribution is assumed for all the delays and thus the maximal probability guaranteeing the given performance can be calculated.

Consensus control of a class of Lipschitz nonlinear systems

This paper deals with consensus control of a class of nonlinear systems with Lipschitz nonlinearities. Certain features of the Laplacian matrix are explored in the real Jordan form to identify conditions for global consensus control. Under the identified conditions, consensus control and stability of the proposed control are analysed in time domain through Lyapunov functions in real domain. The proposed control uses relative state information of the system. A simulation study is included to demonstrate the proposed control designs with some simulation results shown.

ON THE CONSTRUCTION OF TRANSFORMATIONS OF LINEAR HAMILTONIAN SYSTEMS TO REAL NORMAL FORMS

A technique for constructing the transformations to real Hamiltonian normal forms of linear Hamiltonian systems via permutation matrices is presented. In particular, a method is shown for obtaining the symplectomorphism between the symplectic basis of the real Jordan form to the standard symplectic basis in which the real Hamiltonian normal form resides. All possible degeneracies are accounted for since the algebraic and geometric multiplicities of nonsemisimple eigenvalues are not restricted, including the "difficult" cases of zero and imaginary eigenvalues. Since the normal forms are not unique, several possible arrangements of the suggested transformations are given which result in the various normal forms derived previously as well as in a few new ones for degenerate cases which have not appeared before.

A WAVE-BASED CONTROLLER DESIGN FOR GENERAL FLEXIBLE STRUCTURES

This paper treats a travelling-wave approach to suppressing vibration of general flexible structures. This approach aims to minimize all of the reflective waves at actuator positions located at the structural boundaries. A variation of the transfer matrix method shows the property that the elastic motion is obtained by superposing the waves travelling in a flexible structure; this transfer matrix method is based on the finite element method for structural analysis. Moreover, the method gives the propagation and scattering relations of the waves in the structure. Since these relations are described by a complex-valued function with respect to Laplace variable, they are transformed into a real-valued form to design a controller by a lot of state–space methods. This transformation is given by diagonalizing the unity transfer matrix into a real-Jordan form. The problem is then formulated as an Hxoptimization problem to find a compensator minimizing the reflective waves at the actuators. The designed compensator is based only on the scattering relations at the controller positions and on the sensor-input/controller-output relations. A multispan flexible beam is used to verify the validity of the present approach. It is numerically shown that the approach is able to achieve good damping improvement in the closed-loop system.