Time Delay Approximation Research Articles

An explicit tuning of the fractional order controller using a novel time delay approximation

An explicit tuning of the fractional order controller using a novel time delay approximation

Coupled LQG with robust control for milling chatter suppression

This paper applies a coupled control method of LQG and robust to suppress milling chatter. With two controllers working together, the coupled control method has strong robustness to reject disturbances. According to this superiority, the time delay item caused by the regenerative effect is regarded as a disturbance for simplifying the system model. Thus the designed controller has a relatively smaller order than other time delay approximation or discretization methods. Based on the designed active control system, simulated results show the coupled control method has better performance than the original LQG and robust control methods. In experiments, the coupled control method improves the average critical cutting depth from 1.5 mm to 6 mm with almost the same control voltage as LQG and robust control methods, while they only improve the corresponding depth to 4.5 mm and 3.5 mm respectively. Furthermore, the stability improvement is also larger than a recently published control method used in milling chatter suppression. The largest stable milling region improvement among these methods shows potential of the coupled control method in milling chatter suppression.

Stochastic delay differential equations: Analysis and simulation studies

Stochastic delay differential equations play an important role in modelling scientific and engineering systems. In this paper the partial differential equation satisfying the time evolution of the probability density function of the state variable governed by the stochastic delay differential equation (SDDE) with additive white noise and coloured noise perturbation is obtained under small time delay and small correlation time approximation using path integral formalism. This partial differential equation reduces to a Fokker–Planck equation for particular linear and non-linear SDDE. Fokker–Planck equation is then solved with reflecting boundary conditions to get the analytical solutions for Stationary Probability Density Function (SPDF). The analytical solutions for SPDF is compared with the SPDF obtained using simulation. Further the Mean First Passage Time (MFPT) of the bistable system is calculated for various values of time delay and noise strength and compared with that of the simulation. The MFPT for the time delayed system for the case of cubic potential driven by Gaussian and Levy noise as well as asymmetric bistable potential driven by a noise and periodic driving force has been investigated.

Cascade controllers design based on model matching in frequency domain for stable and integrating processes with time delay

PurposeThis paper aims to present an efficient and simplified proportional-integral/proportional-integral and derivative controller design method for the higher-order stable and integrating processes with time delay in the cascade control structure (CCS).Design/methodology/approachTwo approaches based on model matching in the frequency domain have been proposed for tuning the controllers of the CCS. The first approach is based on achieving the desired load disturbance rejection performance, whereas the second approach is proposed to achieve the desired setpoint performance. In both the approaches, matching between the desired model and the closed-loop system with the controller is done at a low-frequency point. Model matching at low-frequency points yields a linear algebraic equation and the solution to these equations yields the controller parameters.FindingsSimulations have been conducted on several examples covering high order stable, integrating, double integrating processes with time delay and nonlinear continuous stirred tank reactor. The performance of the proposed scheme has been compared with recently reported work having modified cascade control configurations, sliding mode control, model predictive control and fractional order control. The performance of both the proposed schemes is either better or comparable with the recently reported methods. However, the proposed method based on desired load disturbance rejection performance outperforms among all these schemes.Originality/valueThe main advantages of the proposed approaches are that they are directly applicable to any order processes, as they are free from time delay approximation and plant order reduction. In addition to this, the proposed schemes are capable of handling a wide range of different dynamical processes in a unified way.

A Novel PID Controller Cascaded With Higher Order Filter for FOPDT With Real Time Implementation

This work describes the design of Proportional-Integral-Derivative(PID) controller for stable and unstable First Order Processes with Dead Time(FOPDT). PID controller is augmented with a higher order filter. The PID and the filter parameters are analytically derived using polynomial method. Higher order time delay approximation has been considered for improved accuracy. Maximum sensitivity(MS) based analytical tuning procedure is presented and concrete tuning guidelines are derived. The suggested technique is validated against delay dominant and nonlinear processes. In terms of the various performance indices, the suggested technique is compared to existing methodologies. The proposed method is tested on a real time process to verify the practical application.

An alternative design approach for Fractional Order Internal Model Controllers for time delay systems

IntroductionFractional Order Internal Model Control (FO-IMC) extends the capabilities of the classical IMC approach into the generalized domain of fractional calculus. When dealing with processes that exhibit time delays, implementation of such controllers in a classical feedback loop requires the approximation of the fractional order terms, as well as of the corresponding time delays. ObjectivesThe present study proposes an alternative design procedure of FO-IMC controllers based on a novel approximation method of the process time delay, proving the efficiency of the proposed method and its suitability for time delay systems. MethodsThe generalized IMC control laws are obtained analytically, based on a novel approximation of time delay, the Non-Rational Transfer Function approach. ResultsSeveral numerical examples are chosen to illustrate the efficiency of the proposed approach. In addition, a vertical take-off and landing unit exhibiting second order plus time delay dynamics is chosen to experimentally validate the proposed control strategy. The obtained results are used to compare the proposed tuning strategy with a popular FO-IMC tuning approach, based on the Taylor series approximation of the time delay. ConclusionAll the chosen examples, both numerical and experimental ones, validate the proposed method. The overall closed loop results obtained with the proposed approach demonstrate an improved performance compared to the existing method. Ultimately, the purpose of the paper to provide an alternative design strategy that extends the existing FO-IMC control field is reached.

Optimised control using proportional-integral-derivative controller tuned using internal model control

Time delays are generally unavoidable in the designing frameworks for mechanical and electrical systems and so on.. In both continuous and discrete schemes, the existence of delay creates undesirable impacts on the under-thought which forces exacting constraints on attainable execution.The presence of delay confounds the design structure procedure also. It makes continuous systems boundless dimensional and also extends the readings in discrete systems fundamentally. As the Proportional-Integral-Derivative (PID) controller based on internal model control is essential and strong to address the vulnerabilities and aggravations of the model. But for an real industry process, they are less susceptible to noise than the PID controller.It results in just one tuning parameter which is the time constant of the closed-loop system λ, the internal model control filter factor.It additionally gives a decent answer for the procedure with huge time delays. The design of the PID controller based on the internal model control, with approximation of time delay using Pade’ and Taylor’s series is depicted in this paper. The first order filter used in the design provides good set-point tracking along with disturbance rejection.

Time delay and non-Gaussian noise-enhanced stability of foraging colony system

In view of the role of external noise, the statistical characteristics of foraging colony system with non-Gaussian noise and time delay are investigated. By means of the small time delay approximation and the Markov approximation, the expressions of stationary probability distribution and mean first passage time (MFPT) of foraging colony system are derived. Using the stochastic second-order Runge–Kutta algorithm, the stationary probability distribution of system, MFPT from the metastable state to stable state, and the probability density function (PDF) of the first passage time from the metastable state to stable state are stimulated. The results show that: (i) the external noise intensity α and the parameter q of departure from Gaussian noise can induce transition phenomenon; (ii) the MFPT as a function of α shows a minimum, which is enhanced by internal noise; (iii) the external noise can enhance stability of system; (iv) time delay-enhanced stability of system occurs.

Stochastic resonance in asymmetric time-delayed bistable system under multiplicative and additive noise and its applications in bearing fault detection

The asymmetric bistable system with time delays in the feedback force and random force under multiplicative and additive Gaussian noise is studied. Using the small time delay approximation approach and time-delayed Fokker–Planck equations (FPE), the signal-to-noise ratio (SNR) of the proposed stochastic system is obtained. The stochastic resonance (SR) phenomena influenced by parameters — including system parameters [Formula: see text], [Formula: see text], asymmetry parameter [Formula: see text], time delay [Formula: see text], strength [Formula: see text] of the time-delayed feedback, noise intensities [Formula: see text] and [Formula: see text] of multiplicative and additive noise, and correlation strength [Formula: see text] between two noises, are also analyzed by numerical simulations. Results demonstrate that the SR performance of the asymmetric bistable system is superior to one symmetric bistable system. Besides, both time delay and strength of time-delayed feedback could enhance the SR to some extent. Then, the asymmetric time-delayed bistable SR (ATDBSR) method is used to the bearing fault diagnosis. The engineering applications of the ATDBSR method are realized and the value of the method is verified by effective experimental results.

Control of master slave robotics system using optimal control schemes

This paper presents application of Proportional Integral Derivative (PID) and Non-linear PID (NPID) controllers to optimally operate the master slave robotic system. Teleoperation is widely used in different applications, such as surgical robots, underwater vehicles, power lines and even in space. However there are problems in teleoperation systems that may lead to degradation in system performance or even to system instability. This paper presents new and optimal control schemes that can satisfy the required performance, insure system stability and achieve zero tracking error in presence of constant time delay and model approximation. Optimal gains are obtained using the Genetic Algorithm in a systematic way that could be applied to other control schemes. The results proved the effectiveness of both control schemes than the previously applied scheme. The NPID control scheme has better performance, provided position tracking and achieved zero tracking error in less settling time than PID one. The results obtained by the presented control schemes are evaluated based on comparing the system performance using three different types of controllers which are P-like, Genetic PID and Genetic NPID. The study was carried out using MATLAB/SIMULINK 2017a.

Time delay and cross-correlated Gaussian noises-induced stochastic stability and regime shift between steady states for an insect outbreak system

In this paper, we focus on investigating the stochastic stability and the regime transition between the endangered state and the boom state for a time-delayed insect growth system driven by correlated external and internal noises. By use of the Fokker–Planck equation, the method of small time delay approximation and the fast descent method, we explore in detail the joint action of noise terms and time delay on the mean reproduction and depression time for the insect population. Our investigations indicate that the pseudo-resonance phenomenon of the mean first-passage time (MFPT) occurs because of the impact of different noises and time delay. Through the numerical calculation, it is discovered that multiplicative noise can speed up the shift of the insect population from the boom state to the endangered one, while the noise correlation and time delay can propel the insect system to evolve from the endangered state to the boom state and improve the biological stability. In addition, the impact of the additive noise on the stability of the biological system depends on the positive and negative situation of the noise correlation. On the other hand, during the process of suppressing the insect explosion, it is beneficial to the pest control to amplify the association noise strength and weaken the intensities of the multiplicative, additive noises and time delay. However, during the process of eliminating the pests, it can produce nice effect on the disinsection to increase time delay, the intensities of multiplicative and additive noises and weaken the strength of noise correlation.

Impact of time delay on population model with Allee effect

We investigate a general delayed stochastic single-species population system with Allee effect simulated by coupling between non-Gaussian colored noise and Gaussian white noise. Firstly, the Fokker–Planck function is derived through unified colored noise approximation (UCNA) and small time delay approximation. Then, the expression of stationary probability distribution is obtained. After that, the expression of mean first-passage time is calculated in order to quantify the transition between stable states. Besides, The influence of strong Allee effect and weak Allee effect on the stochastic population system is different. It is not conducive to the survival of the population when the intensity of multiplicative non-Gaussian colored noise is enhanced whether strong Allee effect or weak Allee effect exists. It is beneficial to the stability and survival for the population under strong Allee effect. Further, the phenomenon of resonant activation is firstly discovered in biological system with Allee effect. The increase of time delay will restrain the transition between stable states under strong Allee effect. When weak Allee effect exists, the increase of time delay will reduce stability of the population, even the population may be vulnerable to extinction. The results in our work provide effective guidance for future researches of ecosystem.

Stochastic resonance in two kinds of asymmetric nonlinear systems with time-delayed feedback and subject to additive colored noise

This paper attempts to investigate the stochastic resonance (SR) behaviors in two kinds of asymmetric nonlinear systems with time-delayed feedback driven by additive colored noise by virtue of two-state theory, small time delay approximation, path integral approach, and unified colored-noise approximation, where asymmetric nonlinear systems include asymmetric well depth and asymmetric well width alone. The characteristics of SR in two kinds of asymmetric systems are different for different asymmetric ratios and correlated times of additive colored noise. For asymmetric well width, optimal noise intensity is independent of asymmetric ratio and correlated time, whereas for asymmetric well depth it is closely related with asymmetric ratio and correlated time. However, optimal noise intensity is closely related with feedback intensity, and time-delay for two kinds of asymmetries. Even there exists the optimal feedback intensity, time delay and correlated time to make output SNR maximum. Above clues are helpful to achieve weak signal detection under strong background noise.

Analytical Design of Enhanced Fractional Filter PID Controller for Improved Disturbance Rejection of Second Order Plus Time Delay Processes

Abstract In this article, a modified fractional internal model control (IMC) filter structure is proposed to design a fractional filter Proportional-Integral-Derivative (FFPID) controller for improved disturbance rejection of second order plus time delay (SOPTD) processes. The proposed method aims at improving the disturbance rejection of slow chemical processes as the tuning rules for such processes are limited. The present design also considers the higher order approximation for time delay as it gives improved response for higher order processes. There is an additional tuning parameter in the proposed IMC filter apart from the conventional IMC filter time constant, which is tuned according to the derived formula. The additional adjustable parameter achieves the disturbance rejection and the closed loop stability. The simulation results have been performed for the same degree of robustness (maximum sensitivity, Ms) for a fair comparison. The results show an improved disturbance rejection for lag dominant and delay significant SOPTD processes with the proposed controllers designed using higher order Pade’s approximation of time delay than the proposed method using first order approximation and the conventional method. The closed loop robust performance is observed for perturbations in the process parameters and the performance is also observed for noise in the measurement. The robust stability analysis is carried out using sensitivity functions. In addition, the Ms range is also identified over which the system gives robust performance for the controllers designed using higher order pade’s approximation of time delay compared to conventional method.

Uncertainty analysis of transfer function of proton exchange membrane fuel cell and design of PI/PID controller for supply manifold pressure control

ABSTRACTProton Exchange Membrane (PEM) fuel cell is one of the possible green solutions for energy generation. The response of the fuel cell system depends on the air and hydrogen feed, flow and heat/ water management. The main control task is to maintain the supply manifold pressure at the cathode side of PEMFC. In this article, an approximation of First Order plus Time Delay (FOPTD) model is obtained from the PEMFC dynamic model by process reaction curve method. The analysis of uncertainty for the FOPTD model was carried out. From the FOPTD model, a simple PID controller was designed and compared with other techniques like: ZN-PI, Skogestad – Internal Model Control (SIMC)-PI, Improved SIMC-PID and Smith Predictor. Also, MPC and decoupler design were implemented. The simulation results are obtained using MATLAB® which shows that the response of supply manifold pressure is well controlled. The gain and phase margins were found out from the Bode plots of open loop transfer function and closed loop transfer functions for stability analysis.

Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models

This paper derives new formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and non-dominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. The accepted data-points from the stabilizing region in the design parameter space can then be mapped on to the PID controller parameter space, relating these two sets of parameters. The widest stability region is then used to find out the most robust solution which are investigated using an unsupervised data clustering algorithm yielding the optimal centroid location of the arbitrary shaped stability regions. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes. We also report, PID controller tuning rules for the robust stable solutions using the test-bench processes while also providing computational complexity analysis of the algorithm and carry out hypothesis testing for the distribution of sampled data-points for different classes of process dynamics and non-dominant pole types.

Stochastic resonance in a delayed triple-well potential driven by correlated noises.

In this paper, we investigate stochastic resonance (SR) in a delayed triple-well potential subject to correlated noises and a harmonic signal. The stationary probability density, together with the response amplitude of the system, is obtained by using the small time delay approximation. It is found that the time delay, noise intensities, and the cross-correlation between noises can induce the occurrence of the transition. Moreover, the appropriate choice of noise intensities and time delay can improve the output of the system, enhance the SR effect, and lead to the phenomenon of noise enhanced stability. Especially, the stochastic multi-resonance phenomenon is observed when the multiplicative and additive noises are correlated. Finally, the theoretical results are well verified through numerical simulations.

Double time-delays induced stochastic dynamical characteristics for a metapopulation system subjected to the associated noises and a multiplicative periodic signal

In this paper, we aim to investigating in detail the stability of the system, the mean extinction time and stochastic resonance (SR) phenomena caused by a multiplicative periodic signal for a dual time-delayed metapopulation system subjected to cross-correlated noises. By use of the fast descent method, the small time delay approximation method and the SR theory, we obtain the expressions of the steady state probability distribution function, the mean first-passage time and signal-to-noise ratio (SNR). Numerical results indicate that the multiplicative, additive and association noises together with time delay τ can all accelerate the transition from the stable state of big density to the extinction one and play significant roles in weakening the stability and shortening the mean extinction time of the metapopulation. In particular, the additive noise and time delay τ can result in the crash of the system, while another time delay θ can strengthen the biological system stability and extend the declining time for the population. On the other hand, with respect to the SNR, the figures show that time delay τ plays entirely antipodal roles in motivating stochastic resonance (SR) in a variety of different situations. Conversely, the multiplicative noise intensity Q and time delay θ all along produce negative effect on exciting the SR. Meanwhile, the increase of the weak additive noise intensity M can stimulate the SR phenomenon, but the bigger values of M will suppress the SNR and SR phenomenon. The strength of the noise correlation λ plays an important role in restraining the SR in most cases except that it does in the plot of SNR-Q.

Time-Delay Effects on Constitutive Gene Expression**Supported by the National Natural Science Foundation of China under Grant Nos. 11405223 and 81370970 and by the Youth Innovation Promotion Association of Chinese Academy of Sciences

The dynamics of constitutive gene expression with delayed mRNA degradation is investigated, where the intrinsic noise caused by the small number of reactant molecules is introduced. It is found that the oscillatory behavior claimed in previous investigations does not appear in the approximation of small time delay, and the steady state distribution still follows the Poisson law. Furthermore, we introduce the extrinsic noise induced by surrounding environment to explore the effects of this noise and time delay on the Fano factor. Based on a delay Langevin equation and the corresponding Fokker–Planck equation, the distribution of mRNA copy-number is achieved analytically. The time delay and extrinsic noise play similar roles in the gene expression system, that is, they are able to result in the deviation of the Fano factor from 1 evidently. The measured Fano factor for constitutive gene expression is slightly larger than 1, which is perhaps attributed to the time-delay effect.

Stationary properties of a saturation laser stochastic system with time delay

Stationary properties of a saturation laser stochastic system with the time delay are investigated. Making use the method of the small time delay approximation, we obtain the analytic expressions of the stationary probability distribution Pst(I) of the laser intensity, the nth power of the laser intensity 〈In〉, the normalized variance λ2(0), and the normalized skewness λ3(0) of the system. By numerical computation, we discuss effects of the delay time and the feedback strength on the stationary probability distribution Pst(I), the mean value of the laser intensity 〈I〉, the normalized variance λ2(0) and the normalized skewness λ3(0). Above threshold the laser system possesses the efficient laser intensity output, and the delay time and the feedback strength enchances the laser intensity output of the saturation laser system. The delay time increases the stability of the laser intensity output. In contrast, the feedback strength weakens the stability.