PID tuning rules for SOPDT systems: review and some new results

PID controllers are widely used in industries and so many tuning rules have been proposed over the past 50 years that users are often lost in the jungle of tuning formulas. Moreover, unlike PI control, different control laws and structures of implementation further complicate the use of the PID controller. In this work, five different tuning rules are taken for study to control second-order plus dead time systems with wide ranges of damping coefficients and dead time to time constant ratios (D/tau). Four of them are based on IMC design with different types of approximations on dead time and the other on desired closed-loop specifications (i.e., specified forward transfer function). The method of handling dead time in the IMC type of design is important especially for systems with large D/tau ratios. A systematic approach was followed to evaluate the performance of controllers. The regions of applicability of suitable tuning rules are highlighted and recommendations are also given. It turns out that IMC designed with the Maclaurin series expansion type PID is a better choice for both set point and load changes for systems with D/tau greater than 1. For systems with D/tau less than 1, the desired closed-loop specification approach is favored.     

An analytical method for PID controller tuning with specified gain and phase margins for integral plus time delay processes

In this paper, an analytical method is proposed for proportional-integral/proportional-derivative/proportional-integral-derivative (PI/PD/PID) controller tuning with specified gain and phase margins (GPMs) for integral plus time delay (IPTD) processes. Explicit formulas are also obtained for estimating the GPMs resulting from given PI/PD/PID controllers. The proposed method indicates a general form of the PID parameters and unifies a large number of existing rules as PI/PD/PID controller tuning with various GPM specifications. The GPMs realized by existing PID tuning rules are computed and documented as a reference for control engineers to tune the PID controllers.     

Design of PID controllers in double feedback loops for SISO systems with set-point filters

A PID controller is widely used to control industrial processes that are mostly open loop stable or unstable. Selection of proper feedback structure and controller tuning helps to improve the performance of the loop. In this paper a double-feedback loop/method is used to achieve stability and better performance of the process. The internal feedback is used for stabilizing the process and the outer loop is used for good setpoint tracking. An internal model controller (IMC) based PID method is used for tuning the outer loop controller. Autotuning based on relay feedback or the Ziegler-Nichols method can be used for tuning an inner loop controller. A tuning parameter (λ) that is used to tune IMC-PID is used as a time constant of a setpoint filter that is used for reducing the peak overshoot. The method has been tested successfully on many low order processes.     

Intuitive robust stability metric for PID control of self-regulating processes

Published methods establish how plant-model mismatch in the process gain and dead time impacts closed-loop stability. However, these methods assume no plant-model mismatch in the process time constant. The work presented here proposes the robust stability factor metric, RSF, to examine the effect of plant-model mismatch in the process gain, dead time, and time constant. The RSF is presented in two forms: an equation form and a visual form displayed on robustness plots derived from the Bode and Nyquist stability criteria. This understanding of robust stability is reinforced through visual examples of how closed-loop performance changes with various levels of plant-model mismatch. One example shows how plant-model mismatch in the time constant can impact closed-loop stability as much as plant-model mismatch in the gain and/or dead time. Theoretical discussion shows that the impact is greater for small dead time to time constant ratios. As the closed-loop time constant used in Internal Model Control (IMC) tuning decreases, the impact becomes significant for a larger range of dead time to time constant ratios. To complete the presentation, the RSF is used to compare the robust stability of IMC-PI tuning to other PI, PID, and PID with Filter tuning correlations.     

Improved automatic tuning of PID controller for stable processes

This paper presents an improved automatic tuning method for stable processes using a modified relay in the presence of static load disturbances and measurement noise. The modified relay consists of a standard relay in series with a PI controller of unity proportional gain. The integral time constant of the PI controller of the modified relay is chosen so as to ensure a minimum loop phase margin of 30. A limit cycle is then obtained using the modified relay. Hereafter, the PID controller is designed using the limit cycle output data. The derivative time constant is obtained by maintaining the above mentioned loop phase margin. Minimizing the distance of Nyquist curve of the loop transfer function from the imaginary axis of the complex plane gives the proportional gain. The integral time constant of the PID controller is set equal to the integral time constant of the PI controller of the modified relay. The effectiveness of the proposed technique is verified by simulation results.     

PI/PID controller design based on IMC and percentage overshoot specification to controller setpoint change

In this work, the normalized Internal Model Control (IMC) filter time constant is designed to achieve a specified value of the maximum sensitivity for stable first and second order plus time delay process models, respectively. Since a particular value of the maximum sensitivity results in an almost constant percentage overshoot to controller setpoint change, an empirical relationship between the normalized IMC filter time constant and percentage overshoot is presented. The main advantage of the proposed method is that only a user-defined overshoot is required to design a PI/PID controller. Simulation examples are given to demonstrate the value of the proposed method.     

Improved model reduction and tuning of fractional-order PI(λ)D(μ) controllers for analytical rule extraction with genetic programming

Genetic algorithm (GA) has been used in this study for a new approach of suboptimal model reduction in the Nyquist plane and optimal time domain tuning of proportional-integral-derivative (PID) and fractional-order (FO) PI(λ)D(μ) controllers. Simulation studies show that the new Nyquist-based model reduction technique outperforms the conventional H(2)-norm-based reduced parameter modeling technique. With the tuned controller parameters and reduced-order model parameter dataset, optimum tuning rules have been developed with a test-bench of higher-order processes via genetic programming (GP). The GP performs a symbolic regression on the reduced process parameters to evolve a tuning rule which provides the best analytical expression to map the data. The tuning rules are developed for a minimum time domain integral performance index described by a weighted sum of error index and controller effort. From the reported Pareto optimal front of the GP-based optimal rule extraction technique, a trade-off can be made between the complexity of the tuning formulae and the control performance. The efficacy of the single-gene and multi-gene GP-based tuning rules has been compared with the original GA-based control performance for the PID and PI(λ)D(μ) controllers, handling four different classes of representative higher-order processes. These rules are very useful for process control engineers, as they inherit the power of the GA-based tuning methodology, but can be easily calculated without the requirement for running the computationally intensive GA every time. Three-dimensional plots of the required variation in PID/fractional-order PID (FOPID) controller parameters with reduced process parameters have been shown as a guideline for the operator. Parametric robustness of the reported GP-based tuning rules has also been shown with credible simulation examples.     

Disturbance-rejection-based tuning of proportional–integral–derivative controllers by exploiting closed-loop plant data

A systematic data-based design method for tuning proportional–integral–derivative (PID) controllers for disturbance attenuation is proposed. In this method, a set of closed-loop plant data are directly exploited without using a process model. PID controller parameters for a control system that behaves as closely as possible to the reference model for disturbance rejection are derived. Two algorithms are developed to calculate the PID parameters. One algorithm determines the optimal time delay in the reference model by solving an optimization problem, whereas the other algorithm avoids the nonlinear optimization by using a simple approximation for the time delay term, enabling derivation of analytical PID tuning formulas. Because plant data integrals are used in the regression equations for calculating PID parameters, the two proposed algorithms are robust against measurement noises. Moreover, the controller tuning involves an adjustable design parameter that enables the user to achieve a trade-off between performance and robustness. Because of its closed-loop tuning capability, the proposed method can be applied online to improve (retune) existing underperforming controllers for stable, integrating, and unstable plants. Simulation examples covering a wide variety of process dynamics, including two examples related to reactor systems, are presented to demonstrate the effectiveness of the proposed tuning method.     

A simple nonlinear PD controller for integrating processes

Many industrial processes are found to be integrating in nature, for which widely used Ziegler–Nichols tuned PID controllers usually fail to provide satisfactory performance due to excessive overshoot with large settling time. Although, IMC (Internal Model Control) based PID controllers are capable to reduce the overshoot, but little improvement is found in the load disturbance response. Here, we propose an auto-tuning proportional-derivative controller (APD) where a nonlinear gain updating factor α continuously adjusts the proportional and derivative gains to achieve an overall improved performance during set point change as well as load disturbance. The value of α is obtained by a simple relation based on the instantaneous values of normalized error (e_N) and change of error (Δe_N) of the controlled variable. Performance of the proposed nonlinear PD controller (APD) is tested and compared with other PD and PID tuning rules for pure integrating plus delay (IPD) and first-order integrating plus delay (FOIPD) processes. Effectiveness of the proposed scheme is verified on a laboratory scale servo position control system.     

A new autotuning algorithm for PID controllers using dead-beat format

A novel algorithm for PID controllers based on dead-beat control and fuzzy inference mechanism is presented in this paper. The proposition is an extension of the work by the authors where the PI form of the algorithm was presented. The inclusion of the derivative term makes the method suitable for application in all types of processes including the ones having high rate disturbances. The proposed algorithm seems to be a complete and generalized PID autotuner as can be seen by the simulated and experimental results. In all the cases the method shows substantial improvement over the controller tuned with Ziegler Nichol's formula and the PI controller proposed in R. Bandyopadhyay, D. Patranabis, A fuzzy logic based PI autotuner, ISA Transactions 37 (1998) 227-235.     

Nonlinear control of an activated sludge aeration process: use of fuzzy techniques for tuning PID controllers

In this paper, several tuning algorithms, specifically ITAE, IMC and Cohen and Coon, were applied in order to tune an activated sludge aeration PID controller. Performance results of these controllers were compared by simulation with those obtained by using a nonlinear fuzzy PID controller. In order to design this controller, a trial and error procedure was used to determine, as a function of error at current time and at a previous time, sets of parameters (including controller gain, integral time and derivative time) which achieve satisfactory response of a PID controller actuating over the aeration process. Once these sets of data were obtained, neural networks were used to obtain fuzzy membership functions and fuzzy rules of the fuzzy PID controller.     

烧结混合料加水的模糊自适应分数阶PI~λD~u控制

烧结混合料加水系统具有大滞后、模型复杂的特性,且客观环境中存在干扰因素,传统的控制方法很难取得理想的控制效果。分数阶PIλDu控制器比常规PID控制器多了两个可调参数,具有更好的控制效果。在分析分数阶微积分的基础上,给出了分数阶微积分的数字实现,用分数阶PIλDu控制代替常规PID控制,结合模糊控制,首次提出了一种针对烧结混合料加水系统的模糊自适应分数阶PIλDu控制方法,利用模糊逻辑实现分数阶PIλDu控制参数的在线调整。并用MATLAB/simulink进行建模仿真。仿真结果验证该控制算法的有效性,能取得较好的控制效果。     

Model Based Tuning of Humidifying Processes with Transportation Lag

Abstract

A humidifying system with varying transportation lag was studied experimentally. Various models were tried and the system fitted a first order plus dead time model with an error of ±5%. The humidity was measured using an on‐line Yokogawa hygrometer. From the model parameters, various controllers, such as PI, Smith predictor, IMC, and IMC PID were analyzed using Matlab. The closed loop performance was studied for both regulator and servo problems. Based on rise time, settling time, and overshoot, the present study concludes that the IMC controller is best suited for this process.     

Enhanced disturbance rejection for open-loop unstable process with time delay

The present study suggests a disturbance estimator design method for application to a recently published, two-degree-of-freedom, control scheme for open-loop, unstable processes with time delay. A simple PID controller cascaded with a lead-lag filter replaces the high-order disturbance estimator for enhanced performance. A new analytical method on the basis of the IMC design principle, featuring only one user-defined tuning parameter, is developed for the design of the disturbance estimator. Several illustrative examples taken from previous works are included to demonstrate the superiority of the proposed disturbance estimator. The results confirm the superior performance of the proposed disturbance estimator in both nominal and robust cases. The proposed method also offers several important advantages for industrial process engineers: it covers several classes of unstable process with time delay in a unified manner, and is simple and easy to design and tune.     

A new design method for PI–PD control of unstable processes with dead time

PID controllers are still widely practiced in the industrial systems. In the literature, many publications can be found considering PID controller design for unstable processes. However, owing to the structural limitations of PID controllers, generally, good closed loop performance cannot be achieved with a PID for controlling unstable processes and usually a step response with a high overshoot and oscillation is obtained. On the other hand, PI–PD controllers are proved to give very satisfactory closed loop performances for unstable processes. The paper presents a simple design method to tune parameters of a PI–PD controller for the control of the unstable processes with time delay. The proposed method is based on plotting the stability boundary locus, which is a locus dependent on the parameters of the controller and frequency, in the parameter plane. The method uses a new concept named centroid of the convex stability region. Simulation examples and an experimental application are given to illustrate the superiority of the proposed method over some existing ones.     

Explicit analytical tuning rules for digital PID controllers via the magnitude optimum criterion

Analytical tuning rules for digital PID type–I controllers are presented regardless of the process complexity. This explicit solution allows control engineers 1) to make an accurate examination of the effect of the controller's sampling time to the control loop's performance both in the time and frequency domain 2) to decide when the control has to be I, PI and when the derivative, D, term has to be added or omitted 3) apply this control action to a series of stable benchmark processes regardless of their complexity. The former advantages are considered critical in industry applications, since 1) most of the times the choice of the digital controller's sampling time is based on heuristics and past criteria, 2) there is little a–priori knowledge of the controlled process making the choice of the type of the controller a trial and error exercise 3) model parameters change often depending on the control loop's operating point making in this way, the problem of retuning the controller's parameter a much challenging issue. Basis of the proposed control law is the principle of the PID tuning via the Magnitude Optimum criterion. The final control law involves the controller's sampling time T_s within the explicit solution of the controller's parameters. Finally, the potential of the proposed method is justified by comparing its performance with the conventional PID tuning when controlling the same process. Further investigation regarding the choice of the controller's sampling time T_s is also presented and useful conclusions for control engineers are derived.     

Tuning of an optimal fuzzy PID controller with stochastic algorithms for networked control systems with random time delay

An optimal PID and an optimal fuzzy PID have been tuned by minimizing the Integral of Time multiplied Absolute Error (ITAE) and squared controller output for a networked control system (NCS). The tuning is attempted for a higher order and a time delay system using two stochastic algorithms viz. the Genetic Algorithm (GA) and two variants of Particle Swarm Optimization (PSO) and the closed loop performances are compared. The paper shows that random variation in network delay can be handled efficiently with fuzzy logic based PID controllers over conventional PID controllers.     

A 2-Dof LQR based PID controller for integrating processes considering robustness/performance tradeoff

This paper focuses on the analytical design of a Proportional Integral and Derivative (PID) controller together with a unique set point filter that makes the overall Two-Degree of-Freedom (2-Dof) control system for integrating processes with time delay. The PID controller tuning is based on the Linear Quadratic Regulator (LQR) using dominant pole placement approach to obtain good regulatory response. The set point filter is designed with the calculated PID parameters and using a single filter time constant (λ) to precisely control the servo response. The effectiveness of the proposed methodology is demonstrated through a series of illustrative examples using real industrial integrated process models. The whole range of PID parameters is obtained for each case in a tradeoff between the robustness of the closed loop system measured in terms of Maximum Sensitivity (M_s) and the load disturbance measured in terms of Integral of Absolute Errors (IAE). Results show improved closed loop response in terms of regulatory and servo responses with less control efforts when compared with the latest PID tuning methods of integrating systems.     

Performance-based parameter tuning method of model-driven PID control systems

In this paper, performance-based parameter tuning method of model-driven Two-Degree-of-Freedom PID (MD TDOF PID) control system has been proposed to enhance the control performances of a process. Known for its ability of stabilizing the unstable processes, fast tracking to the change of set points and rejecting disturbance, the MD TDOF PID has gained research interest recently. The tuning methods for the reported MD TDOF PID are based on internal model control (IMC) method instead of optimizing the performance indices. In this paper, an Integral of Time Absolute Error (ITAE) zero-position-error optimal tuning and noise effect minimizing method is proposed for tuning two parameters in MD TDOF PID control system to achieve the desired regulating and disturbance rejection performance. The comparison with Two-Degree-of-Freedom control scheme by modified smith predictor (TDOF CS MSP) and the designed MD TDOF PID tuned by the IMC tuning method demonstrates the effectiveness of the proposed tuning method.     

Coordinating IMC-PID and adaptive SMC controllers for a PEMFC

For a Proton Exchange Membrane Fuel Cell (PEMFC) power plant with a methanol reformer, the process parameters and power output are considered simultaneously to avoid violation of the constraints and to keep the fuel cell power plant safe and effective. In this paper, a novel coordinating scheme is proposed by combining an Internal Model Control (IMC) based PID Control and adaptive Sliding Mode Control (SMC). The IMC-PID controller is designed for the reformer of the fuel flow rate according to the expected first-order dynamic properties. The adaptive SMC controller of the fuel cell current has been designed using the constant plus proportional rate reaching law. The parameters of the SMC controller are adaptively tuned according to the response of the fuel flow rate control system. When the power output controller feeds back the current references to these two controllers, the coordinating controllers system works in a system-wide way. The simulation results of the PEMFC power plant demonstrate the effectiveness of the proposed method.