Order Delay System Research Articles

Global fractional Halanay inequalities approach to finite-time stability of nonlinear fractional order delay systems

In this paper, we investigate the robust finite time stability of fractional order systems with time varying delay and nonlinear perturbation. We improve the sufficient condition for general fractional delay systems by utilizing the special structure of a singular Gronwall inequality. For stable fractional delay systems, our approach bases on a global Halanay type inequality in differential and integral forms. A sharper delay dependent sufficient condition for robust finite time stability of such systems is formulated in terms of the Mittag-Leffler functions and the delayed size. The connection between the new sufficient condition and the previous results is compared and discussed thoroughly.

Mikhailov stability criterion for fractional commensurate order systems with delays

In this paper, we present the extension of the Mikhailov stability criterion to linear fractional commensurate order systems with delays of the retarded type. The extension is obtained by generalizing the Mikhailov stability criterion of fractional commensurate order and integer order delay systems. The validity of the results is illustrated by means of several examples.

Optimal control of nonlinear fractional order delay systems governed by Fredholm integral equations based on a new fractional derivative operator

We aim to develop a direct transcription approach for solving a notable category of optimal control problems governed by nonlinear fractional Fredholm integral equations having delays in both input and output signals. The foundation of the new methodology is based on a multi domains decomposition scheme by utilizing the fractional-order Legendre functions. A new fractional derivative operator associated with the fractional basis is introduced by using the Caputo fractional derivative operator. With the use of derivative and delay operators, one can transform the dynamical system related to the fractional control problem into a new system containing algebraic equations. A wide variety of challenging test problems are studied to provide a detailed explanation of the designed approach.

An approximation method and PID controller tuning for systems having integer order and non-integer order delay

In this study, an algebraic approximation method is proposed to investigate the stability analysis and time domain response of non-integer order delay systems. Besides, stabilizing Proportional Integral Derivative (PID) controllers are computed in the (kp,ki) plane for the fixed value of kd and a simple PID controller tuning method for the systems with integer order delay and non-integer order delay is presented. For this purpose, to obtain stable PID controller parameters in the (kp,ki) plane, the necessary equations are obtained using the stability region concept. A new design technique is proposed that facilitates the selection of the controller parameters from this region. This design technique is compared with the weighted geometrical center method (WGC). Both methods are used for the first time to the systems with non-integer order delay. Comparisons are also made with some other tuning methods in the literature. Considering the reasonable results obtained, the proposed method can be seen as a very advantageous design technique. Moreover, it provides an easy design procedure without any performance criteria restrictions such as gain margin, phase margin, maximum overshoot, etc. The only criterion here is to obtain the stability region. The simulation results include a detailed comparison of the WGC and the proposed method. Although the WGC seems to be superior in terms of performance, the computational load is considerably higher than the proposed method, which makes the proposed method quite sufficient for a simple and acceptable start.

Stability Regions of Fractional First Order Controllers Applied to Fractional Order Delay Systems

This paper focuses on the problem of stabilizing fractional order time delay systems by fractional first order controllers. A solution is proposed to find the set of all stability regions in the controller’s parameter space. The D-decomposition method is employed to find the real root boundary and complex root boundaries which are used to identify the stability regions. Illustrative examples are given to show the effectiveness of the proposed approach, and it is remarked that the stability region obtained for the fractional order controller is larger than the non-fractional controller.

Stabilization Domains for Second Order Delay Systems

In this paper, an analytic condition is given for determining delayed positive feedback controller for stabilizing an oscillatory system. The $\tau $ -decomposition and $D$ -decomposition methods are employed in deriving this condition. The obtained results are then used to stabilize second order delay systems by a proportional controller. Under-damped and over-damped systems are treated separately, where the Smith-predictor structure is used in the over-damped case to obtain the stability conditions. Illustrative examples are given to show the effectiveness of the proposed approach.

Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation

Time delay occurs naturally due to the limited bandwidth of any real-world system. However, this problem can deteriorate the system performance and can even result in system instability. Input saturation is also an essential issue due to the energy constraint in real actuators that makes the control design procedure more difficult. This article concerns with the stability of uncertain fractional order (FO) delay systems of neutral type including structured uncertainties, distributed delays and actuator saturation. A Lyapunov–Krasovskii functional allows the formulation of the conditions to insure the asymptotic robust stability of such systems via the linear matrix inequalities (LMI) and to compute the gain of a state feedback controller. In addition, by using the cone complementarity linearization method, we obtain the controller gains that extend the domain of attraction. Several simulations validate the theoretical analysis.

Chaos Analysis of Fractional-order System with Delay Based on Adomian Decomposition

In this paper, fractional-order Lü chaotic system with nonlinear delay term is proposed, and the fractional-order Lü chaotic system with time-delay is numerically decomposed by Adomian decomposition. At the same time, the influence of fractional-order number on the system is verified. Then the bifurcation diagram, complexity and attractor phase diagram of the system are analyzed by Matlab software. The numerical simulation results show that the 0.9order system has rich dynamic characteristics, which provides a theoretical basis for parameter selection when fractional order delay system is applied to multimedia encryption.

Some New Results for the Sobolev-Type Fractional Order Delay Systems with Noncompact Semigroup

The topological structure of solution sets for the Sobolev-type fractional order delay systems with noncompact semigroup is studied. Based on a fixed point principle for multivalued maps, the existence result is obtained under certain mild conditions. With the help of multivalued analysis tools, the compactness of the solution set is also obtained. Finally, we apply the obtained abstract results to the partial differential inclusions.

Model identification of critically damped second order plus time delay systems

ABSTRACTIn the present work, a simple method is proposed to calculate the parameters of a critically damped second-order transfer function model from the fractional step response of higher order systems. Five simulation examples are given to show the simplicity of the proposed method. The method makes use of a unique property of a critically damped second order plus time delay system (Harriott, Process Control, McGraw Hill, New Delhi), that, a time of 2.58τ is required to reach a fractional step response of 0.73. The time required to reach 0.35 (or any value) of fractional step response is also required. The derivation of the equations for the model parameters is simpler than that proposed by Rangaiah and Krishnaswamy (Ind. Engg. Chemistry Research, V 33, 1994, 1867–1871). The ISE and IAE values for the present method are compared with that of Rangaiah and Krishnaswamy. The method is also modified to get the model parameters of a FOPTD system: .

Proportional integral-fractional filter controller design for first order lag plus time delay system

ABSTRACTIn this work, a design approach of proportional integral-fractional filter (PI-FF) controller for first order plus time delay (FOPTD) system is proposed in order to enhance the feedback control system performances characteristics. The controller design method is drawn up such that the transfer function of the overall closed-loop system is equivalent to the transfer function of the general fractional Bagley–Torvik reference model whose behaviour ranges from relaxation to oscillation for different values of the fractional order derivative and the damping ratio-like parameter. The tuning parameters of the PI-FF controller are derived analytically from the FOPTD process model and the general fractional Bagley–Torvik reference model parameters. Illustrative examples were presented to test the effectiveness and the usefulness of the proposed PI-FF controller on the feedback control system performance characteristics enhancement.

Proportional integral-fractional filter controller design for first order lag plus time delay system

ABSTRACTIn this work, a design approach of proportional integral-fractional filter (PI-FF) controller for first order plus time delay system (FOPTD) is proposed in order to enhance the feedback control system performance characteristics. The controller design method is drawn up such that the transfer function of the overall closed-loop system is equivalent to the transfer function of the general fractional Bagley–Torvik reference model whose behaviour ranges from relaxation to oscillation for different values of the fractional order derivative and the damping ratio-like parameter. The tuning parameters of the PI-FF controller are derived analytically from the FOPTD process model and the general fractional Bagley–Torvik reference model parameters. Illustrative examples were presented to test the effectiveness and the usefulness of the proposed PI-FF controller on the feedback control system performance characteristics enhancement.

Identification of multi-delay systems using orthogonal hybrid functions in state space environment

In this paper, a set of orthogonal hybrid functions (HF) is utilised for the identification of non-homogeneous as well as homogeneous multi-delay systems. The HF set works with function samples and reconstructs a function in a piecewise linear manner. This reduces the mean integral square error (MISE) of approximation and also the computational burden compared to other competitive methods like Walsh analysis, block pulse analysis, etc. in general. The presented technique uses a simple algorithm for time-delay system identification. After introducing the HF domain approximation and integration, the HF domain theory of identification of multi-delay systems is presented. To support the theory, many numerical examples are treated to identify both first and second order delay systems. The presented tables and graphs prove that the results obtained are highly reliable to establish the validity of the proposal.

Identification of multi-delay systems using orthogonal hybrid functions in state space environment

In this paper, a set of orthogonal hybrid functions (HF) is utilised for the identification of non-homogeneous as well as homogeneous multi-delay systems. The HF set works with function samples and reconstructs a function in a piecewise linear manner. This reduces the mean integral square error (MISE) of approximation and also the computational burden compared to other competitive methods like Walsh analysis, block pulse analysis, etc. in general. The presented technique uses a simple algorithm for time-delay system identification. After introducing the HF domain approximation and integration, the HF domain theory of identification of multi-delay systems is presented. To support the theory, many numerical examples are treated to identify both first and second order delay systems. The presented tables and graphs prove that the results obtained are highly reliable to establish the validity of the proposal.

Chaotic analysis of fractional Willis delayed aneurysm system

The dynamic system of Willis aneurysm (WAS) has played an important role in theoretical and clinical research of cerebral aneurysms. Fractional differential is an effective mathematical tool that can describe the cerebral aneurysm system accurately and profoundly. However, the existing fractional Willis aneurysm system (FWAS) cannot describe the delayed aneurysm rupture of unknown cause in reality. Therefore, by introducing the time-delay factors into the existing fractional Willis aneurysm system as a rational extension, a new fractional Willis aneurysm system with time-delay (FWASTD) is proposed in this paper.First, FWASTD is introduced in the context, and the comparison of time sequences map between FWAS and FWASTD proves that FWASTD is feasible in the depiction of time-delay situations. The bifurcation diagram and the largest Lyapunov exponent diagram as well as the phase diagram of fractional order also confirm the chaotic characteristics of the FWASTD.Then, the classical analysis methods in chaotic dynamics, such as time series diagram, phase diagram and Poincar section are used to analyze FWASTD in detail. When studying the diagrams of time-delay factors for the important physiological parameters of the system, we find that blood flow resistance coefficient can exert a remarkable effect on the system stability under time-delay. Besides, the experimental results show that the FWASTD becomes stable with the increase of blood flow resistance under a certain condition. Usually, promoting thrombosis is a kind of adjunctive therapy in clinic for cerebral aneurysm. The results of this part can accord with the treatment in clinic and has great significance in clinical diagnosis.Finally, as the chaotic state of the time-delay system indicates that cerebral aneurysm is in a dangerous situation, the primary task of the control for this new system is to achieve stability rather than synchronization. The stability theory of fractional time-delayed system is adopted in a strict proof of the uniqueness of solution for the FWASTD. To make FWASTD stable, a corresponding linear controller is designed based on the stability theory of fractional order delay system. The numerical simulation indicates that the linear controller can control the blood flow velocity and speed up the periodic fluctuation within a small range, which illustrates that it is not easy to rupture the cerebral aneurysm. We also make self-synchronization control between FWASTD and FWAS just in case that the coefficients of the system are not clear.The research results in this paper, to some extent, can serve as theoretical guidance for the clinical diagnosis and the treatment of aneurysm.

FPGA-Based Model-Free Nonlinear Control Approach with Application to First Order Delays System

FPGA-Based Model-Free Nonlinear Control Approach with Application to First Order Delays System

Design SPARTAN FPGA-Based PD Controller for FOD Systems

It goes without saying that, there is quite a diverse mixture of the linear controller. Nevertheless, I assume the most famous would probably be Proportional Integral (PI) controller. The things with PI controller are that most of PI controllers are reduction the error. As well as PI controllers, another kind of linear controller worth mentioning could be Proportional Derivative (PD) controller and the unique characteristic of PD controller is that PD controllers are high speed and independent of system’s dynamic modeling. In addition, there are the usual things like PID controllers and PI2D controllers. The main objective of this paper designs a minimum delay proportional-derivative (PD) controller to the control of first order delay (FOD) system. First order delay system has delay time about 4 seconds in certain and about 19 seconds in limited uncertain condition. To improve the flexibility, design high-speed and low-cost controller, the micro-electronic device (FPGA-Based) controller is used in this research. The proposed design is 30-bits FPGA-based controller for inputs and 35-bits for output. In this research, the maximum frequency is 63.629 MHz and the minimum period is 15.716 ns, the minimum input arrival time before the clock is 4.362 ns and the maximum output required time after clock is19.727 ns. In this algorithm, the delay time for the derivative algorithm is 15.526 ns which 87.8% is a logic delay and 12.2% is route delay.

Design Single Chip Micro-Based Controller for First Order Delays System

Design a delay free FPGA-based Proportional-Integral-Derivative (PID) controller to control of first order delay system is the main objective in this research. In order to provide high performance micro-based controller, FPGA-based PID controller based on design Derivative and Integral algorithm is selected. Conventional PID controller is a stable linear type model-free controller that reduces the delay time in first order delay system. This controller has acceptable performance in presence of uncertainty (e.g., overshoot=0%, rise time=0.8 second, steady state error = 1e-9 and RMS error=1.8e-12). In this research, linear controller need real time mobility operation, and one of the most important devices which can be used to solve this challenge is Field Programmable Gate Array (FPGA). FPGA can be used to design a controller in a single chip Integrated Circuit (IC). In HDL based derivative algorithm the minimum input arrival time before clock is 16.466 ns and the maximum frequency is 60.73 MHz, but in the best design action, the maximum frequency to design this single chip algorithm should be 63.629 MHz. In HDL integral algorithm the minimum input arrival time before clock is 15.599 ns and the maximum frequency is 64.1 MHz, but in the best design action, the maximum frequency to design this single chip algorithm should be 178.190 MHz.

Design Sensor-less PID Filter Controller for First Order Delays System

The dynamics of a first order delay system is highly nonlinear, time variant, uncertain and coupling effects. The main objectives to control of first order delay system are time response and acceleration measurements. The problem of acceleration measurements can be reduced, based on design sensor-less Proportional-Integral-Derivative (PID) filter controller in this research. Assuming unstructured uncertainties and structure uncertainties can be defined into one term and considered as an uncertainty and external disturbance, the problem of computation burden and large number of parameters can be solved to some extent. To solve the uncertainties acceleration measurements play an important role. In order to design sensor-less PID filter controller, an accurate PD surface and the derivative of PD surface plays important role. To design an accurate PD surface, stable and tuning surface slope is needed to form the structure of main PID controller. In this algorithm, the derivative of PD surface computes the second derivation of error. Regarding to this method, the challenge of system uncertainties and time response have been solved based on sensor-less acceleration linear filter controller. As this point if s = K1e + e + K2Σe is chosen as desired surface, if the dynamic of first order delay is derived to surface then the linearization can be realized. Because, when the system dynamic is on the surface is used the derivative of surface S = K1e + e + K2e is equal to the zero that is a decoupled and linearized closed-loop systems dynamics. Linearization and decoupling by the above method can be obtained in spite of the quality of the first order delay dynamic model.

Design a Methodology to Model-Reference Control of First Order Delays System

Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most important challenging works. This research focuses on the design, and analysis of a model-reference sliding mode controller for first order delay system, in presence of uncertainties. In order to provide high performance nonlinear methodology, model-reference sliding mode controller is selected. Pure sliding mode controller can be used to control of partly known nonlinear dynamic parameters. Conversely, pure sliding mode controller is used in many applications; it has an important drawback namely; chattering phenomenon. To attenuation the chattering, new filter based high speed control technique is introduced. In this technique, two type derivative techniques are used to improve the rate of delay as well stability, robustness and chattering attenuation. This technique cased to improve the rate of delay compare with conventional PID controller and conventional sliding mode controller.