Discrete Periodic Systems Research Articles

Structural properties of linear discrete periodic systems based on fully actuated system approaches

AbstractIn this paper, the structural properties of linear discrete‐periodic systems are investigated using fully actuated system approaches and the related criteria are presented, with emphasis on the observability and controllability criteria based on fully actuated system approaches. Both the high‐order fully measured and high‐order fully actuated models of linear discrete periodic time‐varying systems are constructed, and it is shown that a sufficient and necessary condition for the linear discrete periodic system to be observable (controllable) is that it can be equivalently transformed into a step‐forward high‐order fully measured (fully actuated) model and a step‐backward high‐order fully measured (fully actuated) model. Finally, a numerical example is offered to demonstrate the validity and feasibility of the approach.

Controllability and control for discrete periodic systems based on fully-actuated system approach

The problem of controllability and control of linear discrete-periodic systems is investigated in this paper. The high-order fully-actuated models for linear discrete periodic time-varying systems are constructed, and a controllability criterion based on fully-actuated system models is proposed. On this basis, stabilization as the fundamental issue is studied and periodic state-feedback control laws are designed via fully-actuated systems approach and parametric design, which converted the original problem into the pole assignment problem for linear constant systems. Finally, a numerical example is given to verify the validity and feasibility of the proposed method.

Pulsating traveling fronts of a discrete periodic system with a quiescent stage

In this paper, we study the pulsating traveling fronts for a spatially discrete periodic reaction–diffusion system with a quiescent stage. It is known that there exists a critical number [Formula: see text] (called minimal wave speed) such that a pulsating traveling front exists if and only if its speed is above [Formula: see text]. In this paper, we derive a uniqueness theorem for supercritical pulsating traveling fronts. Further, we show that all supercritical pulsating traveling fronts are exponentially stable.

A parametric poles assignment algorithm for high-order linear discrete periodic systems

The periodic state feedback pole assignment problem of high-order periodic discrete systems is investigated, and the pole assignment problem for such systems is transformed into a class of problems for resolving periodic Sylvester matrix equations with constraints. Using the technique of cyclic lifting, such equations can be transformed into high-order time-invariant Sylvester matrix equations and solved using a parametric method. Lastly, a numerical example is provided to demonstrate the validity of the procedure.

Stability regions of discrete linear periodic systems with delayed feedback controls

We propose a geometric method to determine the stability region of the zero solution of a linear periodic difference equation via the delayed feedback control (briefly, DFC) with the commuting feedback gain. For the equation, our method is more effective than the Jury criterion. First, we give a relationship, named the C-map theorem, between the characteristic multipliers of an original equation and those of the equation via DFC. Next, we show the existence and m-starlike property, defined in this paper, of an m-closed curve induced from the C-map. Using this result, we prove that the region enclosed by the m-closed curve is the stability region of the zero solution of the equation via DFC.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of $\mathcal{A}$ and $\mathcal{B}$ (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of $\mathcal{A}$ (see Theorem 4.1) by the method of variation and approximation theory (iterative method), and an explicit expression of $\mathcal{B}$ by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system by our algorithm even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

Propagation of Non-Adiabatic Heterogeneous Solid Combustion Waves with Effect to Internal Microstructure and Heat Release

ABSTRACT Propagation of combustion front with effect to the external heat loss is studied by the two-dimensional discrete periodic and disorder systems. Discrete periodic systems are modeled by the regular arrangement of burnt and unburnt point heat sources characterized by the identical rate of heat release and different heat loss parameter. Discrete disorder systems, namely, S1, S2 and S3 are considered with the concatenation of regular burnt and unburnt point heat sources distributed by different shaping parameter of beta distribution function. The unburnt point heat sources in a discrete disorder system are characterized by the different heat loss parameter and incorporation of two categories of distributions for the rate of heat release namely (i) gamma (a = 20) and (ii) gamma (a = 2). The oscillations in ignition time delay, with the manifestation of bifurcation pattern, have been detected for discrete periodic systems. In a discrete disorder system, the external heat loss plays a crucial role in developing the fingering instabilities and determining the early onset of unstable combustion, quenching and combustion limit. Degree of randomness in the distribution of heat release for the disorder system affects the duration of combustion, burn rate without affecting the combustion limit. Computed values of burn rate and combustion limit from the discrete combustion model have been compared with the available data from experiment.

A recursive identification algorithm for linear discrete periodic systems

The parameter identification problem of linear discrete period time-varying (LDPV) systems is studied in this paper and a two-step identification algorithm based on the recursive least squares principle is proposed. The recursive principle-based identification algorithm is mostly used to deal with the parameter identification of time-invariant systems. Combine with the cyclic reconstruction method, the LDPV system is converted to a linear time-invariant system. Then, based on this and combined with recursive least squares identification, a discrimination algorithm is proposed for the parameter estimation of the LDPV system. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithm.

Output feedback based poles configuration for LDP systems with varying state and input dimensions

The problem of pole configuration via periodic output feedback for linear discrete periodic systems with time-varying dimensions of states and input spaces is addressed. By some mathematical derivation, the considered problem can be reduced into solving a special Sylvester matrix equation from which the periodic output feedback gains can be separated and computed explicitly. Based on this, a detail parametric algorithm is presented, by which countless periodic output feedback gains can be generated by choosing different free parameter matrices. Furthermore, the proposed approach is used to achieve robustness performance and robust poles configuration algorithm is provided. Numerical examples are employed to illustrate the effectiveness of the proposed algorithms.

Surface bound states in the continuum in Dyakonov structures

Surface bound states in the continuum (SBICs) have been found to occur in diverse settings, but so far always at the interface of nonhomogeneous media, such as discrete lattices or periodic systems. Here, we show that they can also exist at the interface of homogeneous media, resulting in unique SBICs. Specifically, we found that, contrary to general belief, leaky Dyakonov states exist at the interface between materials that exhibit opposite signs of anisotropy. In addition, properly breaking the anisotropy symmetry leads to the formation of both guided states and also SBICs embedded within the continuum. A direct implication of our finding is the possibility to create SBICs and Dyakonov states in a whole new class of materials and metamaterials.

Nondispersive One-Way Signal Amplification in Sonic Metamaterials

Parametric amplification---injecting energy into waves via periodic modulation of system parameters---is typically restricted to specific multiples of the modulation frequency. However, broadband parametric amplification can be achieved in active metamaterials which allow local parameters to be modulated both in space and in time. Inspired by the concept of luminal metamaterials in optics, we describe a mechanism for one-way amplification of sound waves across an entire frequency band using spacetime-periodic modulation of local stiffnesses in the form of a traveling wave. When the speed of the modulation wave approaches that of the speed of sound in the metamaterial---a regime called the sonic limit---nearly all modes in the forward-propagating acoustic band are amplified, whereas no amplification occurs in the reverse-propagating band. To eliminate divergences that are inherent to the sonic limit in continuum materials, we use an exact Floquet-Bloch approach to compute the dynamic excitation bands of discrete periodic systems. We find wide ranges of parameters for which the amplification is nearly uniform across the lowest-frequency band, enabling amplification of wave packets while preserving their speed, shape, and spectral content. Our mechanism provides a route to designing acoustic metamaterials which can propagate wave pulses without losses or distortion across a wide range of frequencies.

Existence of a Ground-State and Infinitely Many Homoclinic Solutions for a Periodic Discrete System with Sign-Changing Mixed Nonlinearities

Existence of a Ground-State and Infinitely Many Homoclinic Solutions for a Periodic Discrete System with Sign-Changing Mixed Nonlinearities

Exact asymptotic behavior of pulsating traveling waves for a periodic monostable lattice dynamical system

This paper is concerned with the pulsating traveling waves of a periodic lattice dynamical system with monostable nonlinearity. We first establish the exponential upper and lower bounds of the pulsating wave profiles at minus infinity. Then we prove the uniqueness result and derive the asymptotic behavior of all non-critical monostable pulsating traveling waves. This might be the first time to obtain the exact asymptotic behavior of the pulsating traveling waves for periodic discrete systems towards the unstable steady state.

Transitivity and sensitivity for the $ p $-periodic discrete system via Furstenberg families

<abstract><p>The consistency and implication relation of chaotic properties of $ p $-periodic discrete system and its induced autonomous discrete system are obtained. The chaotic properties discussed involve several types of transitivity and some stronger forms of sensitivity in the sense of Furstenberg families.</p></abstract>

Search for Cycles of Nonlinear Periodic Discrete Systems Using the Method of Average Predictive Control

Search for Cycles of Nonlinear Periodic Discrete Systems Using the Method of Average Predictive Control

Continuum Limit and Homogenization of Stochastic and Periodic Discrete Systems – Fracture in Composite Materials

AbstractThe limiting behaviour of a one‐dimensional discrete system is studied by means of Γ‐convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard‐Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms tends to infinity. During that limit, a homogenization process takes place. The limiting functional is discussed, especially with regard to fracture. Secondly, we consider a rescaled version of the problem, which yields a limiting energy of Griffith's type consisting of a quadratic integral term and a jump contribution. The periodic case can be found in [8], the stochastic case in [6,7].

A robust reduced-order observers design approach for linear discrete periodic systems

This paper investigates the problem of designing reduced-order observers for linear discrete-time periodic (LDP) systems. In case that the linear discrete-time periodic system is observable, an algebraic equivalent system is obtained by non-singular linear transformation, and the partial states to be observed are separated simultaneously. Then the considered problem is transformed into the problem of solving a class of periodic Sylvester matrix equation and an iterative algorithm for periodic reduced-order state observers design is derived. In addition, robust consideration based on periodic reduced-order state observers for LDP systems is also conducted. At last, one numerical example is worked out to illustrate the effectiveness of the proposed approaches.

Effect of disorderness in two-dimensional discrete adiabatic combustion waves in a heterogeneous solid medium under stable regime.

The effect of disorderness for the propagation of stable combustion waves in a two-dimensional discrete adiabatic system are studied by considering two systems, namely, (i) periodic and (ii) disorder. The discrete periodic system is modeled by regular arrangement of burnt and unburnt point heat sources. In contrast, the discrete disordered system is modeled by the concatenation of regular burnt and randomly distributed unburnt heat sources. Four different kinds of discrete disordered systems (S1, S2, S3, and S4) are modeled by varying the joint frequency count of unburnt heat sources without changing the domain size of the system. Results show that the disordered structure of the discrete system plays a crucial role in the combustion dynamics, shape of the combustion wave, and it affects the propagation of stable combustion waves with the manifestation of sparse and chaotic finger patterns. Quantitative analysis performed for the width and roughness of combustion front show that the disordered structure and ignition temperature (ε) of the discrete system plays a crucial role. The burn rates of discrete system are sensitive to change in structure of the system and ε. It is observed that the burn rates obtained for the discrete disordered system is always less than that of burn rates of the discrete periodic system. The burn rates calculated for the present model are compared with the available experimental data on combustion of thermite systems.

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

In analogy to the Kucera–Youla parametrization, we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption, the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case, the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogs for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita’s theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.