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A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.

The active disturbance rejection control (AD RC), first proposed by Jingqing Han in late 1980s, is a powerful control technology being able to deal with external disturbances and internal uncertainties in large scale for control systems in engineering applications. This survey paper will articulate, from a theoretical perspective, the origin, ideology and progress of ADRC for not only uncertain finite-dimensional systems but also uncertain infinite-dimensional ones. Some recent theoretical developments, general framework and unsolved problems of ADRC for finite-dimensional systems with mismatched disturbances and uncertainties by output feedback, uncertain finite-dimensional stochastic systems, uncertain infinite-dimensional systems described by both the wave equation and the fractional-order partial differential equation are successively addressed, from which we see the challenges and opportunities for this remarkable emerging control technology to various types of control systems.

In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

We address the output feedback stabilization for a Euler–Bernoulli beam equation with boundary moment control and disturbance. The stabilization of this system has been studied in Guo et al. (J Dyn Control Syst. 2014;20:539–58), where the controller is based on full state feedback. In order to derive the output feedback controller, we design a new disturbance estimator to estimate the total disturbance in the sense that the estimation error signal belongs $L^{2}(0,\infty )$ , and it decays exponentially if the initial state is smooth. Using the estimated total disturbance, we propose a control law to stabilize the system. Using admissibility theory, we show that the closed-loop system is exponentially stable and the signals in the disturbance estimator in the closed-loop are proved to be bounded.

This study aims to address the impulsive controllability of linear fractional impulse controlled systems. The authors establish the Kalman rank criterion to guarantee the impulsive controllability of the considered system. They also present the minimum number of impulsive controls for a realisation of such controllability. An example is finally presented to demonstrate the effectiveness of the authors' results.

This paper presents a parameter-dependent design of feedback compensator with space-varying gains for Mittag-Leffler stabilization of linear time fractional parabolic MIMO partial differential equations subject to space-varying diffusion and reaction coefficients. In the proposed design method, under a boundedness assumption, the reaction coefficient is written in a parametric form. By using the parametric form for the reaction coefficient and multiple non-collocated observation outputs, an observer-based state feedback compensator with space-varying gains is then constructed such that the resulting closed-loop coupled equations are Mittag-Leffler stable. By applying the Lyapunov technique with Caputo fractional derivative and variants of Poincaré–Wirtinger’s inequality, a sufficient condition for the existence of such feedback compensator is presented in terms of standard linear matrix inequalities. Finally, simulation results are presented to support the proposed design method.

In this paper, we investigate the event-triggered boundary feedback control problem for an unstable time fractional wave equation with unknown perturbation at the boundary. To cope with the instability of system when there is no disturbance, the backstepping method is adopted to convert the original unstable system into a stable system. An UDE-based estimator based on low-pass filter is proposed to estimate unknown time-varying input disturbance. With the estimation of disturbance, the event-triggered boundary feedback controller is proposed. It is shown that the event-triggered strategy could asymptotically stabilize system and a positive lower bounded of minimum inter-event time is ensured to exclude the Zeno phenomenon.

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

This paper considers performance output tracking and disturbance rejection for a boundary controlled one-dimensional Euler–Bernoulli beam equation suffered unknown external bounded disturbance. By first proposing a disturbance estimator and then based on this disturbance estimator, we construct a servomechanism to track the reference signal and reject the external disturbance. Four control objectives are achieved: a) the output is exponentially tracking the reference signal; b) the disturbance is rejected; c) all the states of internal-loops are bounded; d) when the disturbance and reference are disconnected, the closed-loop is exponentially stable. Finally, the state of the system is shown to be exponentially tracking the reference state.

In this paper, we pay attention to the boundary feedback stabilization of a one-dimensional flexible string system with the exogenous disturbance that acts on the control end. The active disturbance rejection control (ADRC) and disturbance observer-based control (DOBC) are introduced to reduce the vibration caused by external disturbance. By utilizing Lyapunov method and constructing a suitable Lyapunov functional, we conclude that system is uniformly bounded stable. The numerical simulations are also presented to demonstrate the validity of both methods.

This paper is concerned with the adaptive error feedback regulator problem whereby the 1D Euler--Bernoulli beam equation with general unknown amplitude harmonic disturbance is rejected and general ...

In this study, the asymptotical stability for several classes of fractional order differential systems with time delay is investigated. The authors first present an integral inequality by which the Halanay inequality is extended to fractional order case. Based on the generalised Halanay inequality, the authors establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable. It is worth to note that these stability conditions are easy to check without resorting to the solution expression of the systems.

The stability analysis of fractional-order systems with unbounded delay remains an open problem. In this paper, we firstly explore two new integral inequalities. Using these two integral inequalities obtained, the Halanay inequality with unbounded delay is extended to Caputo fractional-order case and Riemann–Liouville fractional-order case. Finally, several examples are presented to illustrate the effectiveness and applicability of the fractional Halanay inequalities in obtaining the asymptotic stability conditions of fractional-order systems with unbounded delay.

This paper concerns a n-dimensional spherically symmetric model for the combustion of a viscous, compressible, radiative-reactive gas with a chemical kinetics equation. Under suitable assum...

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the ...

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observer-based output feedback are designed to stabilize the systems.

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

In this paper, we consider boundary output feedback stabilization for a multidimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance by using the hidden regularity of the wave equation. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in L2(0, ∞) but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded.

We solve the output regulator problem for a system described by a Neumann boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measured and its observation operator is assumed to be admissible for a certain Co-semigroup that is related to the original system, while the measurements used for control are located at the boundary First we show that the open-loop system is well-posed. Then we find a state feedback control law to solve the output regulator problem by the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on this observer, we propose an output feedback controller to achieve output regulation.

We consider performance output tracking for a boundary controlled multi-dimensional heat equation with possibly unknown internal nonlinear uncertainty and external disturbance on one part and control on the rest of the boundary. The active disturbance rejection control (ADRC) approach is adopted in investigation. With partial boundary measurement only, we propose an extended state observer which is used to estimate both system state and total disturbance, design a servomechanism and an output feedback controller. Three control objectives have been achieved: a) the performance output is exponentially tracking the arbitrary given reference signal; b) all internal signals are uniformly bounded; c) the closed-loop system is internally asymptotically stable whenever both the reference signal and the disturbance vanish or belong to the spaces H1(0, ∞; L1(Γ 1 )) and L2(0, ∞; L2(Γ 0 )), respectively. In addition, when boundary measurement is suffered from the noise, it is shown that the proposed controller is robust to the measurement noise.

In this paper, we propose a new output feedback control law to exponentially stabilize an one-dimensional wave equation with unknown anti-damping coefficient subject to general external disturbance entering the control end. With the proposed control law, although the anti-damping parameter is unknown and the disturbance is presented, it is shown that the resulting closed-loop system still decays exponentially. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

This work aims to design the disturbance rejection controllers for three classes of fractional heat equations. Based on Filippov’s theory, the existence conclusion for the partial differential inclusion solution (PDIS) is established for fractional heat equations with discontinuous boundary conditions. Boundary control strategies are designed directly without the use of any robust control method to respectively achieve the power-law type stabilization and the asymptotical stabilization for fractional heat equations without and with time delay, respectively. A numerical example is included to illustrate the obtained results.

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

In this paper we solve the tracking and disturbance rejection problem for an unstable wave equation with reference and disturbance signals that are generated by a linear exosystem whose eigenvalues are located on the imaginary axis. We consider both unmatched disturbance at the boundary and also in the domain, and also matched disturbance at the control boundary. To make the control design more easy, the in-domain disturbance is first shifted to the control boundary by introducing a new variable transformation, and the anti-stable boundary term is then transformed into a dissipative boundary term by constructing a transport equation. We find an ordinary differential equation (ODE) with delay that describes the relationship between the output tracking error and the coupled signals consisting of the disturbance and reference signal, by using Riemann variables. Based on this ODE with delay, we apply the internal model principle to find the dynamic controller that achieves the output regulation while rejecting the disturbances, and uses only the tracking error as input.

This paper investigates the existence of solutions for infinitely many points boundary value problems at resonance with \(\mathrm{dimker}\,L=2\) regarding fractional differential equations on an unbounded domain. By the well-known coincidence degree theory of Mawhin, it is shown that the considered system under certainly conditions admits at least one solution. An example to illustrate the applicability of the given conditions is also given.

In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for the fractional three-point boundary value problems in $\mathbb{R}^n$, where the dimension of the kernel of fractional differential operator with the boundary conditions can take any value in $\{1,2,\ldots,n\}$. This is our novelty. Several examples are presented to illustrate the result.

In this research we are concerned with a switched system that switches very fast between two infinite-dimensional LTI subsystems, such that the overall system spends on average an equal amount of time in both possible subsystems. We ask if it is possible to approximate the switched system with an average system that is the limit of our switched system when the switching period tends to zero. We are mainly concerned with the situation when both subsystems are described by a contraction semigroup on a Hilbert space. We examine in particular the case when both subsystems are obtained from a basic subsystem via different output feedback operators.

We solve the output tracking and disturbance rejection problem for a plant described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞ (0, ∞) and L∞ (0, ∞), respectively. Generally, these signals cannot be generated by an exogenous system. We explore an approach based on proportional control. We show that the proportional low gain controller can achieve exponential output tracking while rejecting the disturbance. We solve the problem in three steps. In the first step, we convert the original system without disturbance into the two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. In the second step, based on our recent results on disturbance estimators, we apply the estimation/cancellation strategy to cancel to the external disturbance and to track the reference asymptotically. In the third step, we design a controller for the state observer system. Since no explicit disturbance appears in this system, the disturbance is exactly compensated, and the output signal to be controlled is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals. Numerical experiments demonstrate the effectiveness of the proposed control law.

Consider a nonlinear distributed parameter system (DPS) described by ẋ (t) = Ax(t)+Bu(t)+B N N(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators 𝕋 on a Hilbert space X, B and B N , defined on Hilbert spaces U and U N , respectively, are admissible control operators for 𝕋 and N : X × [0,∞) ↦ U N is continuous in t and Lipschitz in x, with Lipschitz constant L N independent of t. Thus B and B N can be unbounded as operators from U and U N to X, in which case the nonlinear term B N N(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ∈ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ∈ L2([0, τ];U). We suppose that there exist linear operators F and F b such that (A, [B B N ], F) and (−A; [B B N ], F b ) are regular triples and A + BF Λ and −A + BF b,Λ are generators of strongly continuous semigroups 𝕋f and 𝕋b on X such that ∥𝕋 t f∥ · ∥𝕋 t b∥ decays to zero exponentially. We prove that if L N is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

We consider boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation. By the ADRC approach, a state feedback disturbance estimator with time-varying gain is designed to estimate the disturbance. It is shown that the closed-loop system is asymptotically stable by canceling the disturbance in the feedback loop with its online estimation. In the second part, the SMC is applied to reject the disturbance. The well-posedness of the closed-loop system via SMC is proven, and the monotonicity of the "reaching condition" is presented without differentiating the sliding mode function which may not always exist for the weak solution. The numerical experiments are presented to illustrate the convergence and the peaking value reduction caused by the constant high gain. In addition, the control energies are compared numerically for two approaches.

We consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An disturbance estimator is designed to estimate, in real time, the disturbance, and the disturbance is canceled in the feedback loop with its approximation. All subsystems in the closed-loop are shown to be asymptotically stable. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

In this paper we are concerned with the output regulation problem for a system (plant) described by a boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measurable and its output operator is assumed to be admissible for a certain C 0 -semigroup that is related to the original system whereas the measurement is located at the boundary. First we show that the open-loop system is well-posed. We construct a state feedback control law to solve the output regulation problem, using the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on these observers, we design an output feedback controller that achieves output regulation.

This paper investigates the controllability of fractional damped dynamical systems with distributed delays. A necessary and sufficient condition for the controllability of linear fractional damped dynamical system with distributed delay in control is established, and a sufficient condition for the nonlinear system is also established. Two examples are given to illustrate the theorems.

The main aim of this paper is to study the global existence of solutions of initial value problems for nonlinear fractional differential equations(FDEs) on the half-axis, which is fundamental in the basic theory of FDEs and important in stability analysis of this kind of equations. In this paper, we are concerned with the nonlinear FDE D 0 + α x ( t ) = f ( t , x ) , t ∈ ( 0 , + ∞ ) , 0 α ≤ 1 , where D 0 + α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition lim t → 0 + t 1 − α x ( t ) = u 0 . By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained to guarantee the global existence of solutions on the interval [ 0 , + ∞ ) . Moreover, in the case α = 1 , existence results of solutions of initial value problems for ordinary differential equations on the half-axis are also obtained. An interesting example is also included.

We are concerned with the nonlinear (n+1)-term fractional integrodifferential inclusions L(D)u∈F(t,u,Tu,Su), a.e. t∈[0,1], where L(D)=Dα-bnDβn-bn-1Dβn-1-⋯-b1Dβ1, 0<β1<β2<⋯<βn-1<βn<1<α<2, b1,b2,…,bn are constant coefficients, and (Tu)(t)=∫01k(t,s)u(s)ds, (Su)(t)=∫0tl(t,s)u(s)ds, subject to the nonlocal conditions u(0)=0, u(1)=∑i=1mγiu(ηi). The existence results are obtained by using two fixed-point theorems due to Bohnenblust-Karlin and Covitz-Nadler, respectively. Our results partly generalize and improve the known ones.

In this paper, we solve the output tracking and disturbance rejection problem for a system described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞[0, ∞) and L∞[0, ∞), respectively. Generally, these signals cannot be generated from an exosystem. We explore an approach based on proportional control. It is shown that a proportional gain controller can achieve exponentially the output tracking while rejecting disturbance. Our method consists of three steps: first, we convert the original system without disturbance into two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. Second, based on our recent result on disturbance estimator, we apply the estimation/cancellion strategy to cancel to the external disturbance and to track the reference asymptotically. Third, we design a controller using a state observer. Since disturbance does not appear in the observer explicitly (the disturbance is exactly compensated), the controlled output signal is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals.

In this paper, we investigate the existence of solutions for multi-point boundary value problems at resonance concerning fractional differential equation on a half-line. Our analysis relies on the coincidence degree of Mawhin. As an application, an example is presented to illustrate the main results. 2010 Mathematics Subject Classification. 34A08; 34B10; 34B40.

To investigate the effect of hydrogen inhalation on acute lung injury after hemorrhagic shock in rats.Twenty-four adult male Sprague-Dawley (SD) rats were equally randomized into three groups: sham operation group, model group and hydrogen-treatment group. Pressure-controlled hemorrhagic shock and resuscitation model was reproduced by blood-letting for 1 hour followed by fluid replacement for 2 hours. The rats in model group received a mixture of 50% oxygen-50% nitrogen during the process. The rats in hydrogen-treatment group received inhalation of a mixture of 2% hydrogen-48% nitrogen-50% oxygen 10 minutes before fluid replacement till the end of resuscitation. The arterial blood samples were collected for the measurement of arterial partial pressure of oxygen (PaO₂) before exsanguination, 1 hour after shock, 1 hour and 2 hours after fluid replacement. Blood and lung tissues were collected at the end of experiment, and tumor necrosis factor-α (TNF-α) and interleukin-6 (IL-6) levels in plasma, lung wet/dry weight ratio (W/D), malondialdehyde (MDA) content, superoxide dismutase (SOD) and myeloperoxidase (MPO) activity in the lung tissue were determined. The lung tissue was subjected to pathological examination.At the end of fluid replacement, compared with model group, hydrogen could significantly reduce pulmonary edema (lung W/D ratio: 4.72 ± 0.12 vs. 4.94 ± 0.14, P0.05), inhibit oxidative stress (MDA: 0.55 ± 0.09 nmol/mg vs. 0.72 ± 0.08 nmol/mg, P0.05), enhance antioxidant activity (SOD activity: 79.53 ± 14.33 U/mg vs. 59.55 ± 9.07 U/mg, P0.05), reduce the release of pro-inflammatory cytokines (TNF-α: 55.58 ± 10.06 ng/L vs. 66.58 ± 5.17 ng/L; IL-6: 23.00 ± 2.77 ng/L vs. 27.09 ± 2.46 ng/L, both P0.05) and inhibit neutrophil infiltration (MPO: 1.05 ± 0.18 U/g vs. 1.40 ± 0.14 U/g, P0.05). It alleviated the damage to lung tissue, and then improved the lung function (PaO₂: 146.3 ± 22.1 mm Hg vs. 123.6 ± 16.0 mm Hg, P0.05).Hydrogen treatment can alleviate acute lung injury as a result of hemorrhagic shock and resuscitation.

In this paper we study the existence and continuation of solution to the general fractional differential equation (FDE) with Riemann–Liouville derivative. If no confusion appears, we call FDE for brevity. We firstly establish a new local existence theorem. Then, we derive the continuation theorems for the general FDE, which can be regarded as a generalization of the continuation theorems of the ordinary differential equation (ODE). Such continuation theorems for FDE which are first obtained are different from those for the classical ODE. With the help of continuation theorems derived in this paper, several global existence results for FDE are constructed. Some illustrative examples are also given to verify the theoretical results.