Third quadrant Nyquist point for the relay feedback autotuning of PI controllers

The original relay feedback autotuning method of Astrom and Hagglund [1] is based on the Nyquist point at the phase angle of −π (the critical frequency). Recently, Friman and Waller [8] showed that the critical frequency is too high to tune PI controllers and proposed an autotuning method that finds a Nyquist point at the third quadrant through the two-channel relay. Here, the method to find Nyquist points in the third quadrant is revisited and adaptive relay feedback method which can be applied to noisy processes is proposed. It is shown that the bandwidths of PI control systems and the first-order plus time delay model identifications support the Nyquist point at the third quadrant. Nyquist points at the third quadrant can be obtained by introducing a filter and hysteresis to the relay feedback loop. However, the filter time constant and the size of hysteresis should be adjusted iteratively because their phase shifts are dependent on the resulting relay oscillation frequency. Simulations show that this adaptive relay feedback method finds a given Nyquist point at the third quadrant accurately under noisy environments and provides excellent PI control systems.     

Relay feedback identification for processes under drift and noisy environments

Relay feedback identification methods are widely used to find the process ultimate information and tune proportional‐integral‐derivative controllers. The conventional relay feedback method has several disadvantages, which include poor estimates of the process ultimate information for low‐order processes, chattering of relay for noisy environments, and asymmetric relay responses for constant biases or slow drifts in the process outputs. Methods to mitigate each of the above disadvantages are available. However, a systematic method to treat all of them has not been studied yet. Here, simple relay feedback methods that resolve these problems by introducing band‐pass filters in the feedback loop are proposed. The high‐pass filter part in band‐pass filter removes a constant bias or low frequency drift, and the low‐pass filter part removes high frequency noise and high‐order harmonic terms in the relay feedback oscillation, resulting better estimates of the process ultimate information. Because filters used for the proposed methods are able to reject constant biases, the process steady state gains can be estimated without disturbing the relay feedback oscillations and first order plus time delay (FOPTD) models can be obtained by combining the process steady state gains with the relay oscillation information. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Assessment of controller performance: a relay feedback approach

Assessment of control loop performance is one of the important tasks to be carried out in a plant, regardless of the control strategy. The present work utilizes the shape information from single relay feedback test to assess the performance of a feedback system. The process considered is a typical first-order plus dead time process and the controller used is PI type of controller. An ideal relay is introduced in the feedback loop before the controller. The shape of the relay output characterizes the performance of the controller. The mismatch in the integral time can also be observed from the shape. Based on the shape information, analytical expressions for relay feedback responses are derived and straightforward procedures are evolved to assess the performance of the controller. The proposed scheme assesses the controller performance and computes the new tuning parameters, in case retuning of the controller is necessary. The robustness of the method is also tested for second-order plus dead time as well as for higher order systems. More importantly the present approach employs only one relay test for (1) assessment and (2) retuning of the controller. It provides a reliable way that is compatible even to a non-expert operator to assess the controller performance.     

Improved system identification method for Hammerstein-Wiener processes

We propose a new system identification method for Hammerstein-Wiener processes, in which an input static nonlinear block, a linear dynamic block, and an output static nonlinear block are connected in a series. The proposed method can estimate the model parameters in a very simple way without solving the full-dimensional nonlinear optimization problem by activating the process with a specially designed test signal, composed of a relay feedback signal, a binary signal and a multi-step signal. The proposed method analytically identifies the output nonlinear static function and the input nonlinear static function from the relay signal and the multi-step signal, respectively. The linear dynamic subsystem is identified from the relay feedback signal and the binary signal with existing well-established linear system identification methods. We demonstrate with a simple example that the proposed method can be successfully applied to identify the Hammerstein-Wiener-type nonlinear process.     

Identification of integrating and unstable processes from relay feedback

In this paper, the exact relay response expressions are derived for integrating and unstable 1st or 2nd order processes. Identification algorithms are subsequently developed in terms of the measurable parameters under relay feedback and fitting conditions of the limit cycle. It is demonstrated that the limit cycle can be undoubtedly formed for an integrating process under a biased/unbiased relay feedback test. A tighter limiting condition is given for identification of unstable processes with the standard relay feedback structure. Denoising methods are presented to cope with measurement noise. Illustrative examples are given to demonstrate the effectiveness and merits of the proposed identification algorithms.     

Estimation of a SOPTD transfer function model using a single asymmetrical relay feedback test

Using a single asymmetric relay feedback test, a method is proposed to identify four parameters of a transfer function model. The proposed method is used to identify all the parameters in a second order plus time delay model (SOPTD). The parameters estimated are of adequate accuracy for designing suitable controllers. The estimated SOPTD models have a step response behavior matching that of the actual process. The method can also be used for identifying open loop unstable transfer function models. For unstable systems, the closed loop step responses are compared. Simulation results are given for four case studies.     

改进的内模控制方法对一阶非自衡对象的控制研究

针对传统的内模控制方法在非自衡对象中应用的不足,提出一种改进的内模控制方法。该方法在传统的内模控制结构中添加了一个比例控制器和一个比例微分控制器,分别用于镇定不稳定对象和改善控制系统的抗干扰特性及鲁棒特性,并采用全极点近似法对被控对象的纯滞后项进行近似处理。改进的内模控制方法具有很好的目标跟踪特性和干扰抑制特性,各个控制器的设计简单,且整个控制系统只有一个调节参数。最后通过对一个一阶非自衡对象进行仿真,其结果说明了本文所提出的内模控制新方法的有效性。     

Tuning PI and fractional order PI controllers with an additional fractional order Pole

The PI controller with an additional pole (PI?+?P) already has been proposed to decrease the noise effect on the control signal. In this paper, a fractional order pole is employed to increase the PI?+?P controller performance. The fractional order is obtained by adjusting the Nyquist plot slope in gain crossover frequency. This condition as well as gain crossover frequency and phase margin specifications are utilized to design the PI controller augmented with an additional fractional order pole (PI?+?FO[P]). To design these two controllers, a first-order plus delay time (FOPDT) model is utilized. For plants that could not be described by this model, its fractional order version (FFOPDT) could be utilized. In this case, a FOPI?+?FO[P] controller is obtained that could improve the transient response of the closed-loop system. The numerical simulations accomplished on various plant models (including chemical plants) demonstrate the effectiveness of the proposed controller comparing with the PI?+?P, PID, FOPI, and fractional order proportional-integral-derivative controllers.     

Automatic tuning and gain scheduling for pH control

A framework is proposed for automatic tuning and gain scheduling in pH control. The nominal “tuning constants” and the “gain schedule” are found from one relay feedback experiment. The Gulaian—Lane titration curve model is modified so that the process titration curve is a linear combination of two base titration curves. The concept of selection of secondary measurements is extended for the selection of base titration curves. A criterion is also proposed whereby the base titration curve can be determined in an quantitative manner if prior knowledge about variations in the composition of the influent is available. With this model structure, the tuning constants and the gain schedule (from the estimated titration curve) can be determined from a relay feedback experiment. Simulation results show that a robust estimation of the process titration curve can be obtained even under severe process conditions. Furthermore, the proposed approach can overcome the realistic problem facing pH control in most wastewater treatment plants, that of rapid and large changes in influent composition, in a reliable manner.     

Identification and Control for Unstable Processes of Three Dynamic Types

This article classifies the process dynamics involving one or two unstable poles into three types, which are of paramount importance in controller design. Necessary and sufficient conditions are addressed for gain and relay stabilizability of such dynamics. Two nonlinear elements, one being a relay, are employed to excite rich input-output data for identification. A procedure is then developed to identify an unstable delayed model of appropriate order (second or third order), followed by the optimal PID controller design established based on the weighted ISE index. The resultant controller can cope with both load disturbances and set-point changes simultaneously and provide good robustness by satisfying the maximum gain margin criterion. The method is tested with good results on a nonlinear CSTR process.     

Identification and Control for Unstable Processes of Three Dynamic Types

This article classifies the process dynamics involving one or two unstable poles into three types, which are of paramount importance in controller design. Necessary and sufficient conditions are addressed for gain and relay stabilizability of such dynamics. Two nonlinear elements, one being a relay, are employed to excite rich input-output data for identification. A procedure is then developed to identify an unstable delayed model of appropriate order (second or third order), followed by the optimal PID controller design established based on the weighted ISE index. The resultant controller can cope with both load disturbances and set-point changes simultaneously and provide good robustness by satisfying the maximum gain margin criterion. The method is tested with good results on a nonlinear CSTR process.     

Process identification method using relay feedback and backward integrals

We propose a new process identification method that combines the two methods of the relay feedback to activate the process and the backward integrals to estimate the model parameters. Novel deviation variables are introduced to incorporate the case that the initial part of the process is unsteady-state without sacrificing the dynamic information included in the initial part, while the previous approaches assign zero-weighting to the initial parts, resulting in loss of the dynamic information included in the initial part. The final cyclic-steady-state part of the process input and output data is chosen as the reference of the deviation variables. The proposed method can estimate the model parameters analytically by using the backward integrals and the least squares method.     

Impedance spectroscopy analysis of glucose electro-oxidation on Ni-modified glassy carbon electrode

The electro-oxidation of glucose on nickel-modified glassy carbon electrode (GC/Ni) in a 1 M NaOH solution at different concentration of glucose was studied by the method of ac-impedance spectroscopy. In low concentration of glucose (<7 mM) two semicircles in the first quadrant of a Nyquist diagram were observed corresponding to charge transfer resistance and adsorption of intermediates. In higher concentration of glucose (>10 mM) negative resistance is observed in impedance plots as signified by semicircles terminating in the third quadrant. The impedance data in high concentration of glucose show different behavior at different applied anodic potential. The influence of the electrode potential on impedance pattern in high concentration of glucose is studied and a mathematical model was put forward to quantitative account for the impedance behavior of glucose oxidation. At potentials higher than 0.525 V/Ag-AgCl, a pseudoinductive behavior is observed but at higher than 0.4 V/Ag-AgCl, impedance patterns terminate in the second and third quadrants. The conditions required for this behavior are delineated with the use of the impedance model.     

一种分数阶PI~αD~β控制器的设计与化简

针对分数阶PIαDβ控制器数字实现相对复杂、结构调整相对困难这一情况,依据分数阶微积分算子最佳有理逼近原理,提出了一种逼近频带可选择的分数阶PIαDβ控制器设计与化简方法。这种分数阶控制器设计与化简方法可以方便地用程序来实现,为分数阶控制器的工程化应用提供了一个途径。     

IDENTIFICATION OF TRANSFER FUNCTION MODELS FROM THE RELAY FEEDBACK TEST

A MIMO process is considered to be composed of several single-input-multi-output (SIMO) processes. In this article, a method for identifying such SIMO process is devised. To identify each SIMO process, a single run of relay feedback experiment on a particular input is required. For each of such experiments, models of FOPDT (First-Order-Plus-Dead-Time) or of SOPDT (Second-Order-Plus-Dead-Time) are identified. Estimation of the parameters is carried out one-by-one and does not involve least square minimization. Direct application of these SIMO identification to MIMO processes is thus illustrated. The accuracy of each estimated gain can be controlled within one single run of experiment. A sufficient condition for the experiments to ensure robustness of the resulting model is presented. Thus, this proposed method can be applied to some MIMO processes even they are ill-conditioned.     

采用Butterworth传递函数设计二自由度纯滞后控制系统

采用Butterworth传递函数设计了纯滞后对象的状态反馈控制系统,针对纯滞后对象的状态反馈控制系统是单自由度的,提出一种采用Butterworth传递函数设计二自由度纯滞后控制系统的方法。用m阶模型逼近纯滞后因子,引入状态观测器,构成状态反馈子系统,再串入一个积分器,构成纯滞后过程的串级状态反馈控制系统,以n+1阶Butterworth传递函数为整个系统的目标函数,按状态反馈增益阵和状态观测器的协调优化设计状态反馈子系统。仿真结果表明了该方法设计的系统同时获得良好的给定值跟踪特性和干扰抑制特性。     

Relay feedback‐based time domain modelling of off‐diagonal elements of linear 2‐by‐2 MIMO systems

A novel approach to model the off‐diagonal closed‐loop transfer function of 2‐by‐2 multi input multi output (MIMO) system in time domain using ideal as well as biased relay feedback tests are presented. Analytical expressions are derived, for transfer function models of 2‐by‐2 MIMO system having individual transfer function elements of first‐order plus dead time (FOPDT) nature, using ideal and biased relay feedback tests in time domain and have been validated on 2‐by‐2 multivariable distillation column examples from the literature as well as on coupled tanks liquid level experimental set‐up. The results show that the derived expression matches with the original undesired relay response even without approximation of time delay in the denominator of closed‐loop transfer function of 2‐by‐2 MIMO system. These analytical expressions are useful to identify unknown system parameters. © 2012 Canadian Society for Chemical Engineering     

Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state

Application of relay feedback controllers to maintain process states near unstable steady states is extended to a second order system, the CSTR with an irreversible, exothermic reaction. The control scheme is successful provided the bandwith (the reactor temperature deviation which causes relay switching) is not chosen too large relative to the drive levels (the deviation in coolant temperature caused by relay switching). In this context Tsypkin's analytical methods for approximate solution of this and higher order problems are elaborated and tested. Tsypkin's technique proves superior to Describing Function analysis in accuracy and in bounds on the maximum permissible bandwidth.     

General rules for optimal tuning the PIλDµ controllers with application to first‐order plus time delay processes

For two main reasons optimal tuning the PI^λD^µ controllers is a challenging task: First, the search space is very large in dealing with such controllers, and second, there is not any generally applicable method for stability testing of the linear feedback systems containing both time delay and fractional‐order controllers. Hence, easy‐to‐use and effective rules for optimal tuning such controllers are highly demanded. In this paper, explicit formulas for optimal tuning the parameters of the PI^λD^µ controller, when it is applied in series with a first‐order plus time‐delay process in a standard output‐feedback system, are proposed. © 2012 Canadian Society for Chemical Engineering     

Enhanced frequency response estimator to guarantee pre-specified phase angle and static disturbance rejection with all harmonics removed

We propose a new frequency response estimation method in order to guarantee a pre-specified phase angle of the estimated model under static disturbance circumstances with rejecting harmonics completely. The proposed method uses one cycle of the conventional relay feedback signal followed by a sinusoidal signal. The sinusoidal signal in a cyclic steady state has no harmonics, resulting in exact frequency response estimates. Also, it guarantees the pre-specified phase angle and removes the effects of static disturbances by adjusting the reference value of the sinusoidal signal.