Unit Step Response Research Articles

Transient solution for the directional response at the focus of a paraboloidal reflector

A transient solution is presented for the directional response at the focus of a paraboloidal reflector due to an incident plane wave. The validity of the solution is associated with the Kirchhoff approximation used to determine the boundary condition on the surface of the reflector. This approximation requires the wavelengths to be short compared with the minimum radius of curvature, and the radius of the aperture, of the reflector. The solution in the far field due to a point source at the focus is related by reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of a function obtained for the unit step response with the time derivative of the waveform incident on the reflector. For a point source far removed from a shallow paraboloidal reflector, it is shown that the pressure in the far field reduces to the expression obtained by Morse (Vibration and Sound) for transient radiation from a baffled circular piston. Results are presented illustrating the angular dependence of the reflected pressure waveforms at the focus for incident plane waves that include an N wave and a tone burst.A transient solution is presented for the directional response at the focus of a paraboloidal reflector due to an incident plane wave. The validity of the solution is associated with the Kirchhoff approximation used to determine the boundary condition on the surface of the reflector. This approximation requires the wavelengths to be short compared with the minimum radius of curvature, and the radius of the aperture, of the reflector. The solution in the far field due to a point source at the focus is related by reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of a function obtained for the unit step response with the time derivative of the waveform incident on the reflector. For a point source far removed from a shallow paraboloidal reflector, it is shown that the pressure in the far field reduces to the expression obtained by Morse (Vibration and Sound) for transient radiation from a baffled circular piston. Results are presented illu...

Performance assessment of primary frequency control responses for thermal power generation units using system identification techniques

Primary frequency control (PFC) responses for thermal power generation units are concerned about whether changes of turbine rotating speeds lead to proper responses of generator active powers. A new method is proposed to estimate performance metrics for PFC responses via system identification techniques. Dynamic models are identified from data samples of the turbine rotating speed and the main steam pressure as two inputs, and the generator active power as the output. Hypothesis tests are formulated to evaluate the qualities of identified models. If model qualities are satisfactory and the contribution from the turbine rotating speed to the generator active power is significant, then performance metrics are estimated from a noise-free unit step response of the identified model. The proposed method removes three major limitations of the most widely-used method in contemporary industrial practices, namely, (i) performance metrics are obtained from all data samples of PFC responses, instead of some specific data samples being sensitive to noises; (ii) the proposed method is applicable to arbitrary-type PFC responses, not limited to step changes of the turbine rotating speed; (iii) noticeable effects of the main steam pressure are separated to avoid erroneous performance assessment results. Examples from thermal power generation units are provided to illustrate the effectiveness of the proposed method.

A New Variable Fractional-Order PI Algorithm

In this paper the authors present a novel control algorithm based on control error sign-dependent variable-fractional-order PI controller. The algorithm is being optimized via ITSE criterion for control error. It is tested both for unconstrained control signal and a more real-case scenario, i.e. ±2.5 saturation on control signal. The algorithm is tested for A-, B-, D-and E-type variable-order PI controllers and compared to basic PI and fractional-PI (FPI) controllers. Important parameters, including rise and settling time, overshoot and peak time of unit-step response, as well as graphical representation of unit-step response are presented. Conducted numerical simulations show some interesting behaviour of the A-type definition both in non-limited and limited control signal cases, i.e. switching between derivation and integration action. Moreover, collected unit-step response parameters indicate the A-type definition to be the best behaving in all considered criteria. However, some unwanted minor oscillations in the unit-step response are to be observed, whose origin will be investigated in further research.

PI-PD Controllers Design Using Bode’s Ideal Transfer Function

In this paper, a method is presented to design PIPD controllers for control system with time delay. The presented method takes Bode’s ideal transfer function as reference model and thus PI-PD controller parameters can be obtained by optimization. Bode's ideal transfer function gives the desired unit step response by setting two parameters. PI-PD parameters corresponding to the minimum value of the error which is the difference between the reference model and system to be controlled are obtained by using integral performance criteria which are frequently used in evaluating the performance of control systems. Matsuda's 4th order integer approximation is used in the Bode’s ideal transfer function. The reason of using Bode’s ideal transfer function instead of a standard second order system is that a fractional order plant represents a real system better than an integer order plant. Finally, the success of the method is demonstrated in the given example.

拉曼激光光学锁相环参数设计及仿真

The optical phase-locked loop(OPLL) technology is an effective-method to realize the phase coherent of laser beam. In consideration of the loop filter influence on OPLL, a new method was presented to obtain the optimized parameters of a second order passive loop filter. Firstly, a formula was deduced to design the parameters according to the definition of phase angle margin and mathematical model of open-loop transfer function. Based on MATLAB software, a parameter optimization algorithm was formulated. Secondly, in order to calculate the parameters of Raman laser optical phase-locked loop precisely, a experiment of Doppler-free absorption spectrum was designed to get the accurate gain parameter from the piezoelectric ceramic to feedback the laser. The overshoot and regulation time of unit step response were about 6.53% and 0.584s, respectively, in the systematic performance simulation of closed-loop optical phase-locked system. Finally, by using Simulink tools to establish models and simulating the optical phase-locked loop system module, the lasers' phase was locked at a high speed with 2s. Therefore, the method of OPLL parameters optimization has turned out reasonable. In addition, certain guidelines can be given for the circuit devise of the optical phase-locked loop in engineering applications.

Interactive educational tool for the design of compensators using frequency response analysis

This article provides a didactic platform that is developed to improve the teaching as well as learning of “design of compensators” in control systems engineering. The educational tool comprises a graphical user interface in MATLAB. The graphical user interface permits an individual to investigate various parameters involved in the design of the required compensator, which includes percentage overshoot, peak time, and velocity error constant and observe the results dynamically as the parameters change. The approach used in this article for the design of compensators is based on the frequency response analysis. One can obtain various plots including unit step response plot, Bode plot, and root locus plot for uncompensated and compensated systems to have a better understanding of the designed compensator.

On the stability of linear fractional-space neutron point kinetics (F-SNPK) models for nuclear reactor dynamics

The aim of this work is to investigate the stability of linear fractional-space neutron point kinetics (F-SNPK) models for nuclear reactor dynamics, using three methods: root locus, Bode plot and unit step response. The F-SNPK is based on an approximation non-Fickian that is modeled considering that the differential operator of neutron density current is of fractional order, known as anomalous diffusion exponent. Two rector geometries for different values of anomalous diffusion exponent where analyzed. The results obtained with F-SNPK were compared with the classical neutron point kinetic (CNPK) equations. It has been found that the F-SNPK models are open-loop as well as closed-loop stable for both the geometries. The models are also found to exhibit faster dynamics with increase in subdiffusivity.

Practical Tuning Algorithm of PDµ Controller for Processes with Time Delay

In this paper, a practical tuning algorithm of fractional order PD controller for processes with time delay using the weighted geometrical center (WGC) method is presented. This method is based on calculating of the stabilizing PDµ controller parameters region which is plotted using the stability boundary locus in the (kd,kp) plane and computing the weighted geometrical center of stability region.The important advantages of the proposed method are both calculating of controller parameters without using complex graphical methods and ensuring the stability of closed loop system. From the examples, it can be easily seen that this simple tuning method can perform quite reliable results in that unit step response.

A New Integer Order Approximation Table for Fractional Order Derivative Operators

There is considerable interest in the study of fractional order calculus in recent years because real world can be modelled better by fractional order differential equation. However, computing analytical time responses such as unit impulse and step responses is still a difficult problem in fractional order systems. Therefore, advanced integer order approximation table which gives satisfying results is very important for simulation and realization. In this paper, an integer order approximation table is presented for fractional order derivative operators (sa) where α ϵ R and 0 < α < 1 using exact time response functions computed with inverse Laplace transform solution technique. Thus, the time responses computed using the given table are almost the same with exact solution. The results are also compared with some well-known integer order approximation methods such as Oustaloup and Matsuda. It has been shown in numerical examples that the proposed method is very successful in comparison to Oustaloup’s and Matsuda’s methods.

Transient Analysis of Nonuniform Heat Input Propagation Through a Heat Sink Base

For thermal management architectures wherein the heat sink is embedded close to a dynamic heat source, nonuniformities may propagate through the heat sink base to the coolant. Available transient models predict the effective heat spreading resistance to calculate chip temperature rise, or simplify to a representative axisymmetric geometry. The coolant-side temperature response is seldom considered, despite the potential influence on flow distribution and stability in two-phase microchannel heat sinks. This study solves three-dimensional transient heat conduction in a Cartesian chip-on-substrate geometry to predict spatial and temporal variations of temperature on the coolant side. The solution for the unit step response of the three-dimensional system is extended to any arbitrary temporal heat input using Duhamel's method. For time-periodic heat inputs, the steady-periodic solution is calculated using the method of complex temperature. As an example case, the solution of the coolant-side temperature response in the presence of different transient heat inputs from multiple heat sources is demonstrated. To represent a case where the thermal spreading from a heat source is localized, the problem is simplified to a single heat source at the center of the domain. Metrics are developed to quantify the degree of spatial and temporal nonuniformity in the coolant-side temperature profiles. These nonuniformities are mapped as a function of nondimensional geometric parameters and boundary conditions. Several case studies are presented to demonstrate the utility of such maps.

A novel interval type-2 fractional order fuzzy PID controller: Design, performance evaluation, and its optimal time domain tuning

In this paper, a novel concept of an interval type-2 fractional order fuzzy PID (IT2FO-FPID) controller, which requires fractional order integrator and fractional order differentiator, is proposed. The incorporation of Takagi-Sugeno-Kang (TSK) type interval type-2 fuzzy logic controller (IT2FLC) with fractional controller of PID-type is investigated for time response measure due to both unit step response and unit load disturbance. The resulting IT2FO-FPID controller is examined on different delayed linear and nonlinear benchmark plants followed by robustness analysis. In order to design this controller, fractional order integrator-differentiator operators are considered as design variables including input-output scaling factors. A new hybridized algorithm named as artificial bee colony-genetic algorithm (ABC-GA) is used to optimize the parameters of the controller while minimizing weighted sum of integral of time absolute error (ITAE) and integral of square of control output (ISCO). To assess the comparative performance of the IT2FO-FPID, authors compared it against existing controllers, i.e., interval type-2 fuzzy PID (IT2-FPID), type-1 fractional order fuzzy PID (T1FO-FPID), type-1 fuzzy PID (T1-FPID), and conventional PID controllers. Furthermore, to show the effectiveness of the proposed controller, the perturbed processes along with the larger dead time are tested. Moreover, the proposed controllers are also implemented on multi input multi output (MIMO), coupled, and highly complex nonlinear two-link robot manipulator system in presence of un-modeled dynamics. Finally, the simulation results explicitly indicate that the performance of the proposed IT2FO-FPID controller is superior to its conventional counterparts in most of the cases.

Efficient Parameter Design Guideline for Closed-Loop Synchronization Architectures

In this letter, an efficient parameter design guideline is proposed for closed-loop synchronization architectures. First, the feasible ranges of loop parameters are directly provided in the z -domain according to the closed-loop stability. Second, depending upon unit step response characteristics of synchronization loops, accurate parameters are ascertained by minimizing the variance of the thermal noise induced phase error conditionally. Compared with available methods, this method not only has the features of better intuition but also does not need repeated trial and adjustment operations. Simulation results are presented to corroborate the effectiveness and feasibility of the proposed design guideline.

A novel robust fuzzy control of an uncertain system

This paper presents a robust control method combining the conventional proportional–integral–derivative (PID) scheme and the sliding mode fuzzy control scheme for a second-order non-linear system having uncertainties in the system dynamics. The goal of the proposed scheme is to force the response of the uncertain plant to follow that of the nominal model. The first phase of the design approach is to obtain a nominal PID controller for the nominal plant model. The poor performance of the sole PID scheme on the uncertain non-linear system motivates the proposal of the technique discussed here. To compensate for the deficiencies in the unit step response of the uncertain system, a fuzzy compensation scheme based on sliding mode control (SMC) is proposed and the PID loop is augmented by the proposed approach. It is shown that the performance with the proposed scheme is better than the sole PID-based control system. With the proposed technique, the response of the uncertain system converges to of the nominal system with admissible controller outputs. Furthermore, simulation results show that the proposed method produces consistent results even with noisy measurements.

The impact of product returns and remanufacturing uncertainties on the dynamic performance of a multi-echelon closed-loop supply chain

We investigate a three-echelon manufacturing and remanufacturing closed-loop supply chain (CLSC) constituting of a retailer, a manufacturer and a supplier. Each echelon, apart from its usual operations in the forward SC (FSC), has its own reverse logistics (RL) operations. We assume that RL information is transparent to the FSC, and the same replenishment policies are used throughout the supply chain. We focus on the impact on dynamic performance of uncertainties in the return yield, RL lead time and the product consumption lead time. Two outcomes are studied: order rate and serviceable inventory. The results suggest that higher return yield improves dynamic performance in terms of overshoot and risk of stock-out with a unit step response as input. However, when the return yield reaches a certain level, the classic bullwhip propagation normally associated with the FSC does not always hold. The longer remanufacturing and product consumption lead times result in a higher overshoot and a longer time to recover inventory, as well as more oscillation in the step response at the upstream echelons. We also study bullwhip and inventory variance when demand is a random variable. Our analysis suggests that higher return yield contributes to reduced bullwhip and inventory variance at the echelon level but for the CLSC as a whole the level of bullwhip may decrease as well as increase as it propagates along the supply chain. The reason for such behaviour is due to the interaction of the various model parameters and should be the subject of further analytical research. Furthermore, by studying the three-echelon CLSC, we produce a general equation for eliminating inventory offsets in an n-echelon CLSC. This is helpful to managers who wish to maintain inventory service levels in multi-echelon CLSCs.

Influences of input impedance of oscilloscope on the accuracy of switching overvoltage measuring system

The accuracy of switching overvoltage measuring system is significantly affected by the input impedance of oscilloscope. First, the principle of switching overvoltage is presented, followed by a brief introduction to the structure of measuring system. Using the laplace transform, the transfer function of the measuring system is then derived. Its frequency response, ranging from DC to 1 GHz, is investigated in detail. The influences of the input impedance on the amplitude and phase angle responses are clearly revealed. Two typical cases, i.e. unit step response and sinusoidal step response, are also considered in the study. To validate the above analysis, a series of experiments has been performed at a substation. Both the theoretical analysis and the experimental results indicate that the input impedance affects the accuracy of switching overvoltage measuring system significantly. In the case that the impedance is rather small, the voltage waveform displayed on oscilloscope would be considerably distorted from the one stressed on the line. The impedance should be greater than 107 Ω to satisfy the accuracy requirement. On the other hand, it can be also used as a direct indicator to assess the performance of oscilloscope.

Analytical and statistical approach for evaluating the effects of a river barrage on river–aquifer interactions

AbstractThe construction of a river barrage can increase groundwater levels upstream of the barrage during the rainy season. Analytical and statistical approaches were applied to evaluate the relationship between groundwater and river water at the Changnyeong–Haman river barrage in Korea using time series data of water level and electrical conductivity from June 2011 to September 2014. An artificial neural network based time series model was designed to filter out the effect of rainfall from the groundwater level data in the study area. Aquifer diffusivity and river resistance were estimated from the analytical solution of a one‐dimensional unit step response function by using the filtered groundwater level data. River resistance increased in response to groundwater level fluctuations. Cross‐correlation analyses between the groundwater and the river water showed that the lag time increased during the observation period for both the water level and the electrical conductivity while the cross‐correlation function declined for the same period. The results indicated that a constant river stage maintained at the river barrage can weaken the hydrologic stress and reduce the exchange of material between the river and the adjacent aquifer because of the deposition of fine sediment on the river bottom and walls. Copyright © 2016 John Wiley &amp; Sons, Ltd.

A compound compensation method for car-following model

A compound compensation mechanism was introduced into the car-following system. Two basic compensation methods were combined to generate a compound control strategy to improve the performance of the traffic flow system. The compensation effect was analyzed with unit step response in time domain and bode diagram in frequency domain, respectively. Two lemmas and one theorem were proved with the use of Routh criteria and small gain theorem. Numerical simulations were conducted in two situations under three types of condition. The simulation results verify the truth that with the increasing compensation parameters the stability of the car-following system is strengthened. It is shown that numerical results are in accordance with analytical results. In general, the performance of car-following model can be improved with an exterior control method.

Analysis of car-following model with cascade compensation strategy

Cascade compensation mechanism was designed to improve the dynamical performance of traffic flow system. Two compensation methods were used to study unit step response in time domain and frequency characteristics with different parameters. The overshoot and phase margins are proportional to the compensation parameter in an underdamped condition. Through the comparison we choose the phase-lead compensation method as the main strategy in suppressing the traffic jam. The simulations were conducted under two boundary conditions to verify the validity of the compensator. The conclusion can be drawn that the stability of the system is strengthened with increased phase-lead compensation parameter. Moreover, the numerical simulation results are in good agreement with analytical results.

A novel method for optimal model simplification of large scale linear discrete-time systems

An accurate and stability preserving model order reduction method for large scale linear discrete-time systems is presented in this paper. The denominator coefficients of the reduced order model are obtained by least squares matching of time-moment proportional's and Markov parameters of the original system and reduced order model about (z − 1), while the numerator coefficients are determined by minimising the integral square error (ISE) between the unit step responses of the original system and its reduced order approximate. Instead of evaluating ISE from time responses of the system and its approximate, a matrix formula is developed in this paper. Three examples are provided to show the efficacy of the proposed method.

Hydrocarbon reservoirs characterization by co-interpretation of pressure and flow rate data of the multi-rate well testing

Abstract Pressure transient behavior is among the most important information for characterizing a reservoir, forecasting its future performance, and designing an appropriate recovery scheme. Although a continuous real-time monitoring of reservoir bottom-hole pressure has become a routine task in intelligent wells, complete extraction of the potential information from these valuable sources of data may not be achieved by using traditional interpretation methods. Deconvolution transforms the pressure transient data related to the wells with variable production rates into an equivalent constant rate pressure data with duration equal to the whole duration of the multi-rate test i.e., unit step response. This technique can reveal high valuable information over a distance from the wellbore which may be several orders of magnitude greater than the radius of investigation of individual flow periods. In the present study, a robust and practical deconvolution methodology is developed for extracting the unit step response (USR) from synthetic, noisy and incomplete pressure transient histories pertaining to multi-rate data. Our proposed scheme calculates the USR from those multi-rate well testing data which may contain high levels of noises in both the flow rate and pressure data. A coupled wavelet transform/superposition theorem is the basis of the proposed method. The algorithm has shown an excellent performance for revealing reservoir/boundary models and their associated parameters.