Unit Step Response Research Articles

Nonlinear step responses and how to find them

AbstractThe unit step response is a standard tool for experimental identification; its shape is equivalent to a solution of an appropriate governing equation. In this study, we investigate two variants of nonlinear first-order governing equation, $${y'=k(1-y^\beta )/\beta }$$ y ′ = k ( 1 - y β ) / β , and mainly its rescaled version $${y'=k\left( 1-y^\beta \right) }$$ y ′ = k 1 - y β . We explore a continuum of response shapes, ranging from constant zero to constant one, including standard shapes such as $$1-\exp (-t)$$ 1 - exp ( - t ) and $${\tanh (t)}$$ tanh ( t ) , but also uncommon shapes such as a part of the logarithmic integral. Two methods for estimating the parameter $$\beta $$ β are proposed: one based on analyzing the curvature of a single response and the other based on analyzing initial slopes for multiple responses, while varying the amplitude of the step. Our investigation is conducted in the context of gas permeation through a homogeneous porous medium, where nonlinear response shapes have been observed. This context allows for a meaningful interpretation of the results, consistent with expectations for specific thermodynamic processes. However, neither method is constrained by this context, and either might be considered for any first-order system that fails to have amplitude-scale invariance or for which the shapes seem to exhibit an effect of saturation.

Research on the characteristics of electro-hydraulic position servo system of RBF neural network under fuzzy rules

A radial basis function neural network PID controller under fuzzy rules (FUZZY-RBF-PID) was designed for the electro-hydraulic position servo system under the influence of uncertain factors such as load mutation, and load stiffness change. Firstly, the mathematical model of the system is established, and the frequency domain and time domain analysis of the system are carried out. Secondly, based on the analysis results, a radial basis function (RBF) neural network PID controller is designed, and fuzzy rules are innovatively used to adjust the learning rate of PID parameters in the RBF neural network learning algorithm in real time. Thirdly, the simulation results show that under the action of the FUZZY-RBF-PID controller, the unit step response of the system has high steady-state accuracy, fast response speed, and under the condition of large load stiffness, the system can recover to the steady-state value faster after being disturbed. At the same time, when the input signal is the sinusoidal signal of 10 HZ, the system under the action of the FUZZY-RBF-PID controller has no obvious phase lag phenomenon, and the tracking error is minimal. The proposed method can effectively improve the comprehensive performance of the electro-hydraulic position servo system under the influence of uncertain factors.

A comparative study on PI-PD controller design using stability region centroid methods for unstable, integrating and resonant systems with time delay

In this paper, controller tuning methods based on stability region centroid methods reported in the literature are used to design PI-PD controllers for unstable, integrating and resonant systems with time delay. By analyzing the stability boundary locus (SBL) for the PD controller, which is utilized in the inner loop of this structure, the controller parameters are obtained using three methods which are the Weighted Geometric Center (WGC), Centroid of Convex Stability Region (CCSR), and Stability Triangle Approach (STA). These techniques were applied analytically, step by step. For the closed loop transfer function obtained in the inner loop of the controlled system, these three methods were utilized to design the PI controller in the outer loop, individually. Unit step responses of the controlled system, using PI-PD gains determined by each method, have been obtained. Furthermore, a disturbance of a certain time and amplitude was added to the systems to test the disturbance rejection behavior and the robustness performance of the methods. Perturbed responses were obtained through changing the model parameters at a certain rate. Time domain performance metrics were analyzed to compare the responses. The simulations were evaluated using settling time, rise time, and percentage overshoot as the assessment criteria. As a result of this study, the effectiveness of three methods, namely WGC, CCSR, and STA, in controller design for unstable, integrating, and resonant time-delay systems has been demonstrated. In addition, a comparison of time domain performance metrics is presented for the nominal and perturbed systems. Based on these comparisons, it is concluded that the methods outperform each other only in some time response performance measures. The presented results showed the advantages of these methods over each other in terms of some performance criteria. The contribution of this study to the literature is the comparative analysis of these three analytical methods.

Low voltage ride through enhancement of a permanent magnet synchronous generator based wind energy conversion system in an islanded microgrid: A dynamic matrix controlled virtual DC machine

In an isolated microgrid, the wind energy conversion system based on direct-drive permanent magnet synchronous generator may experience fluctuations in the DC bus voltage due to wind variations and grid voltage drop, thereby compromising system reliability. To address this issue, the scheme of the supercapacitor energy storage system is proposed. The existing double closed-loop PID control, which cannot provide inertia support, has shortcomings such as difficult parameter setting and slow dynamic response. To solve this problem, the dynamic-matrix-control based predictive control for a virtual DC machine method is proposed. Firstly, the concept of virtual DC machine control is firstly introduced in supercapacitor energy storage system. Secondly, according to the unit step response of virtual DC machine, the prediction model of virtual DC machine is established. Finally, the desired torque increment reference of virtual DC machine is calculated by the dynamic-matrix-control controller to change the input torque reference. Hardware-in-the-loop experiment verifies the effectiveness of the proposed method: Compared with traditional double closed-loop PID control, the maximum voltage fluctuation is reduced by 67.9 % under random wind fluctuation and the inhibition effect on DC voltage surge and plunge when fault occurs in the power grid is increased by 61.4 % and 57.9 %.© 2017 Elsevier Inc. All rights reserved.

Diffusive wave model in a finite length channel with a concentrated lateral inflow subject to different types of boundary conditions

The diffusive wave model is one of the simplified forms of Saint-Venant equations, and it is often used instead of the full model. In this paper, we present an analytical solution for the linearized diffusive wave model represented by a simultaneous system of two first-order partial differential equations focused on spatial variation of a lateral inflow in a finite channel. A concentrated lateral inflow from a small-width tributary is considered through the Dirac delta function. We use the Laplace transform method to solve these equations analytically. Two types of upstream boundaries are considered here in the form of a flow-discharge hydrograph and a flow-depth hydrograph, while keeping a flow-depth hydrograph as the downstream boundary. Using unit-step responses of the lateral inflow, the effect of different boundaries on the flow-depth responses and the flow-discharge responses is studied for different values of the Peclet number (Pe). The flow depth is observed to be more sensitive to the downstream boundary and other parameters used in this work. Consideration of the flow depth as the upstream boundary reflects the effect of all the parameters on the unit-step responses presented. These responses are compared with the available semi-infinite channel responses, which are found to be an inappropriate substitute for the finite channel responses for Pe<5 which implies that the downstream boundary cannot be ignored for these cases. However, for the case Pe>5, although the semi-infinite channel responses are found to satisfactorily estimate the discharge along the entire channel, they can approximate the flow depth at the locations closer to the upstream boundary only.

Inverse Estimation of Dynamic Heat Flux Using Proportional Integral Derivative Control

To improve long-term dynamic heat flux prediction using an embedded thermocouple, a novel inverse estimation approach using a proportional integral derivative (PID) control scheme is proposed. A feedforward model of the discrete integer-order transfer function was identified from a one-dimensional dynamic heat-flux sensor using the calibrated heat flux with varying amplitudes of the pseudo-random binary sequence and its single-temperature response data. Using the model and unit-step temperature response as the controlled plant and reference input signal, the PID controller with a low-pass filter was designed. Using the on-site temperature measurement data as the reference signal instead of the unit-step temperature, the time-history control output values, that is, the inverse estimated dynamic heat flux of the designed PID controller, were obtained. The comparison of the test results and the discussion indicate that 1) the proposed inverse method avoids the impact of the thermal inertia of the thermocouple and significantly alleviates the effects of measurement noise, and 2) in addition to high identification efficiency and measurement fidelity, further improvement in the effective measuring time of the dynamic heat flux was observed.

Interval Type 2 Adaptive Neuro-Fuzzy Inference System-Based Artificial Pacemaker Design and Stability Analysis.

This paper presents the design and simulation of an Interval type 2 fuzzy system (IT2FS) based, adaptive neuro-fuzzy inference system (ANFIS) pacemaker controller in MATLAB. After designing the type 1 fuzzy logic model, the stability of the designed system has been verified in the time-domain (unit step response). In previous works, the type 1 (IT1FS) model step response was analyzed. They are compared with the other proportional integral derivative (PID) and fuzzy models that only least-square-estimation and the backpropagation algorithms are used for tuning membership functions (MF) and generation of type 1 fis (fuzzy inference system) file. At current work, fuzzy C means (FCM) method shows better results than other methods have been used. The pacemaker controller determines the pacing rate and adjusts the heart rate of the patient for the reference input signal. The rise-time, overshoot and settling-time have been improved significantly.

Influence of models approximating the fractional-order differential equations on the calculation accuracy

In recent years, the usage of fractional-order (FO) differential equations to describe objects and physical phenomena has gained immense popularity. Due to the high computational complexity in the exact calculation of these equations, approximation models make the calculations executable. Regardless of their complexity, these models always introduce some inaccuracy which depends on the model type and its order. This article shows how the selected model affects the obtained solution. This study revisits seven fractional approximation models, known as the Continued Fraction Expansion method, the Matsuda method, the Carlson method, a modified version of the Stability Boundary Locus fitting method, and Basic, Refined, and Xue Oustaloup filters. First, this work calculates the steady-state values for each of the models symbolically. In the next step, it calculates the unit step responses. Then, it shows how the model selection affects Nyquist plots and compares the results with the plot determined directly. In the last stage, it estimates the trajectories for an example of an FO control system by each model. We conclude that when employing approximation models for fractional integro-differentiation, choosing the appropriate type of model, its parameters, and order is very important. Selecting the wrong model type or wrong order may lead to incorrect conclusions when describing a real phenomenon or object.

The modeling and cascade sliding mode control of a moving mass-actuated coaxial dual-rotor UAV

In this study, an attitude control scheme based on a three-track moving mass control mechanism is proposed to address the problems of the overcomplicated rotor components, low service life, and low reliability of coaxial dual-rotor unmanned air vehicles (UAV). The motion and aerodynamic models of a moving mass-actuated ducted coaxial dual-rotor UAV are derived. The rotational dynamic characteristics of a moving mass-actuated UAV (MAUAV) with different slider positions and mass ratios are analyzed. An attitude controller based on backstepping sliding mode control is designed to address the nonlinearity and uncertainty of the MAUAV rotation. Based on this, we developed a position controller using cascade sliding mode control. The simulation results demonstrate that the designed attitude controller can achieve a settling time of 1.438 s in the unit-step response and a steady-state error of less than 5% in the sinusoidal attitude-tracking experiment. Additionally, the designed position controller exhibited a better trajectory-tracking effect under different levels of gust disturbance than that of a linear quadratic regulator control-based position controller.

A novel floodwave response model for time-varying streambed conductivity using space-time collocation Trefftz method

Streambed conductivity can vary substantially over short time intervals during flood events, altering groundwater – surface water interactions. Traditional floodwave response models (FRM) developed by linking a unit step response function with the convolution integral are limited by the conditions of a stationary aquifer-stream system and a hydrostatic equilibrium initial condition. However, these two conditions can hardly be satisfied when streambed conductivity changes with time. As a result, in this study a new nonstationary FRM (NFRM) that can consider time-varying streambed conductivity was developed using a space–time collocation Trefftz method. In this new model, the aquifer was assumed homogeneous, and the groundwater flow was simplified to be one-dimensional and perpendicular to a partially penetrating river. The NFRM can be applied to both the forward calculation of groundwater responses to fluctuations of river stage and the inverse calculation of time-varying streambed conductivity. A wide range of synthetic cases generated by the numerical simulations of aquifer-stream interactions was used for validation. Good agreement between the results computed from the NFRM and the synthetic data can be found in all cases (Nash-Sutcliffe model efficiency coefficient > 0.90; Mean Absolute Relative Error < 0.03), showing the accuracy and robustness of the NFRM. The NFRM was then successfully applied to a study site on the Tallahatchie River (Mississippi, USA). The estimated streambed conductivity was found to vary between 10-3 m/d to 10-2 m/d over the studied period. It generally increased during high-stage events and decreased when the river stayed in a low-stage condition, but also generally depended on seepage direction with increases occurring while groundwater discharged to the river and decreases while river water recharged the aquifer.

An Approximation Method for Fractional-Order Models Using Quadratic Systems and Equilibrium Optimizer

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

Temperature response spatiotemporal correlation model of heat transfer system

For a linear heat transfer system, the unit step response of the temperature field to any system inputs, such as internal and external heat sources, is specific. This means that there is a definite temperature response correlation between any spatial points in the system spatiotemporally, which is independent of the system inputs. It is significant to achieve a quantitative description of the temperature response spatiotemporal correlation between the spatial points for solving the temperature field reconstruction problem of the heat transfer system with unknown inputs. In this paper, a temperature response spatiotemporal correlation model (RSTCM) of the heat transfer system is established originally to reveal the inherent quantitative relationship between the responses of any system spatial points theoretically. Firstly, an external description of the spatiotemporal mapping relationship of the heat transfer process is established according to the heat transfer equations. Furthermore, a basic RSTCM form is proposed and a response spatiotemporal correlation matrix is determined with the help of the external description. Finally, considering the ill-posed characteristic of the response spatiotemporal correlation problem, the regularization technology is employed to transform the original problem into a well-pose problem and to provide a response spatiotemporal correlation matrix with regularization. Effectiveness of the RSTCM is verified by the existing heat transfer experimental data. Besides, adaptability of the proposed model is demonstrated by the achieved numerical experiment results on a two-dimensional heat transfer system.

An analytical solution of fractional order PI controller design for stable/unstable/integrating processes with time delay

This paper aims to put forward an analytical solution for tuning parameters of a fractional order PI (FOPI) controller for stable, unstable, and integrating processes with time delay. Following this purpose, the analytical weighted geometrical center (AWGC) method has been extended to the design of fractional order PI controllers. To apply AWGC, the stability equations of the closed-loop system are written in terms of process and fractional order PI controller parameters. With the proposed method, the centroid can be calculated analytically, and the controller parameters can be easily calculated without the need of repetitive drawings of the stability boundary regions. Additionally, analytical equations are derived to calculate fractional integral order, ?, using the integral of squared error (ISE) objective function. The proposed analytical equations are simple and time-saving which might attract controller engineers for applying them on the industrial level. To show the efficiency of the suggested method compared to the given methods in the literature, several simulation examples are considered. Comparisons between the reported methods are figured out in terms of unit step responses for nominal and perturbed cases. Also, rise time, settling time, maximum sensitivity (Ms), integral of squared error (ISE), and total variation (TV) values are considered to compare the performance and robustness issues. Also, a real-time application of an inverted pendulum setup is deemed to prove the feasibility of the suggested method.

A Double-Loop Inertia Phase-Locked Loop With Antidisturbance Ability

Power grid disturbance usually degrades the performance of phase-locked loop (PLL), which may cause the stability issue of grid-connected inverter. Considering the power grid disturbance, including voltage amplitude, grid frequency, and grid stiffness, a double-loop inertia phase-locked loop (DLI-PLL) with antidisturbance ability is proposed. In addition, the mechanism and characteristics of the proposed DLI-PLL are analyzed based on the unit step response of transfer function. The inertia of the proposed DLI-PLL can be controlled by adjusting the control parameters. Compared with the single-loop inertia phase-locked loop, the proposed DLI-PLL have better grid synchronization performance in case of grid frequency fluctuation. Meanwhile, due to the inertial control of angular frequency, the designed DLI-PLL has good antidisturbance ability. Finally, experimental results validate the effectiveness of the proposed DLI-PLL.

System identification with piecewise-constant finite impulse response model and its statistical property

Finite impulse response (FIR) models have attracted much attention in system identification in recent years. However, finite impulse response models have many parameters, and it is sometimes difficult to identify them under conditions with limited computational resources. This paper considers a finite impulse response model with a reduced number of parameters. In particular, supposing the purpose of the model is to estimate the unit step response at a given time, it is shown that one of the desirable models for this purpose is a model in which the impulse response values are piecewise constants. The statistical properties of such a piecewise-constant FIR model obtained by system identification are clarified. The usefulness of the proposed method is confirmed through numerical examples.

Integral-proportional derivative controller tuning rules obtained from optimum responses for set-point tracking of unstable processes with time delay

This paper introduces the design of integral–proportional derivative (I-PD) controllers for unstable first-order plus time-delay and integrating-unstable first-order plus time-delay processes. Minimization of the error based on integral performance criteria is performed to derive analytical rules using curve fitting techniques. Obtained analytical formulas may be used to determine the necessary tuning parameters of the I-PD controller from the known plant transfer function parameters. Advantages of the introduced tuning of I-PD controllers are shown by comparisons with other existing ones in terms of unit step responses, measurement noise, control signals, perturbations in plant transfer function parameters, and integral performance indices. Results have shown that some significant advantages have been achieved with the introduced tuning of I-PD controllers when compared to others in the literature.

Exact Solutions of Fractional Order Oscillation Equation with Two Fractional Derivative Terms

The fractional oscillation equation with two fractional derivative terms in the sense of Caputo, where the orders alpha and beta satisfy 1<alpha le 2 and 0<beta le 1, is investigated, and the unit step response and two initial value responses are obtained in two forms by using different methods of inverse Laplace transform. The first method yields series solutions with the nonnegative powers of t, which converge fast for small t. The second method is our emphasis, where the complex path integral formula of the inverse Laplace transform is used. In order to determine singularities of integrand we first seek for the roots of the characteristic equation, which is a transcendental equation with four parameters, two coefficients and two noninteger power exponents. The existence conditions and properties of the roots on the principal Riemann surface are given. Based on the results on the characteristic equation, we derive these responses as a sum of a classical exponentially damped oscillation, which vanishes in an indicated case, and an infinite integral of the Laplace type, which converges fast for large t, with a steady component in the unit step response. Asymptotic behaviors of solutions for large t are derived as algebraic decays in negative power laws characterized by the orders alpha and beta. The fractional system exhibits a magical transition from oscillation to monotonic decay in negative power law.

Proportional-integral-derivative controller design for time-delay systems via stability region centroid

Design of proportional-integral-derivative (PID) controller with proportional, integral, and derivative gains given by , and respectively, for time-delay systems is presented in this study. The centroid of the convex stability region (CCSR) method in the - plane for fixed is used. PID controller design for time-delay systems in the - plane for a fixed and - plane for a fixed have been extensively researched. Despite the amenability of CCSR method to design of PID controller in the - plane for fixed , its application in this regard has not been given serious attention. The stability region in - plane for fixed was determined and the required controller gains in the region were determined using the CCSR method. Using the determined controller gains, the system closed loop unit step response for all the considered regions was plotted on same axes. Based on the obtained results, different combinations of controller gains can be implemented depending on the system time domain performance measures (TDPMs) requirements. However, selection of an appropriate controller gains combinations, requires compromise among any of the conflicting TDPMs.

Time-domain Analytical Method for Aeroelastic Analysis of High Aspect Ratio Wing Using Unsteady Indicial Aerodynamics

This paper deals with the development of a novel time-domain analytical method to perform an aeroelastic stability analysis of a high aspect ratio wing. The aerodynamic model in this paper is built upon the frequency domain of Theodorsen aerodynamics. An inverse Fourier transform technique is applied to convert the Theodorsen frequency domain transfer function to the Wagner time-domain response function. Later, lift and moment expressions, containing aerodynamic lag states are obtained using the indicial function approach which performs the convolution of the unit step response Wagner lift function with arbitrary input and its time derivative using the principle of superposition. Additional equations containing the derivative of aerodynamic lag states are coupled with previously obtained lift and moment expressions to fully get the complete unsteady aerodynamic model in the time-domain. This time-domain unsteady aerodynamic model is then coupled with a combined bending torsion beam finite element model, using the state space approach. A complex eigenvalue analysis of this state space system is performed, using MATLAB, to determine both the static and dynamic aeroelastic stability boundary. Finally, in this paper, aeroelastic stability analysis of the high aspect ratio wing is performed using the current method and results are validated with MSC Nastran aeroelastic analysis and those available in the literature.

Utilisation of Initialised Observation Scheme for Multi-Joint Robotic Arm in Lyapunov-Based Adaptive Control Strategy

In this paper, we present a modelling, dynamic analysis, and controller tuning comparison for a five-degree-of-freedom (DoF) multi-joint robotic arm based on the Lyapunov-based Adaptive Controller (LAC). In most pick-and-place applications of robotic arms, it is essential to control the end-effector trajectory to reach a precise target position. The kinematic solution of the 5-DoF robotic arm has been determined by the Lagrangian technique, and the mathematical model of each joint has been obtained in the range of motion condition. The Proportional-Integral-Derivative (PID) control parameters of the LAC have been determined by the Lyapunov stability approach and are initialised by four observation methods based on the obtained transfer function. The effectiveness of the initialised controller’s parameters is compared by a unit step response as the desired input of the controller system. As a result, the average error (AE) for Ziegler–Nichols is 6.6%, 83%, and 53% lower than for Pettit &amp; Carr, Chau, and Bucz. The performance of LAC for the robotic arm model is validated in a virtual 3D model under a robot operating system environment. The results of root mean square error by LAC are 0.021 (rad) and 0.025 (rad) for joint 1 and joint 2, respectively, which indicate the efficiency of the proposed LAC strategy in reaching the predetermined trajectory and the potential of minimizing the controller tuning complexity.