New results on the robust stability of PID controllers with gain and phase margins for UFOPTD processes

This paper considers the problem of determining all the robust PID (proportional–integral–derivative) controllers in terms of the gain and phase margins (GPM) for open-loop unstable first order plus time delay (UFOPTD) processes. It is the first time that the feasible ranges of the GPM specifications provided by a PID controller are given for UFOPTD processes. A gain and phase margin tester is used to modify the original model, and the ranges of the margin specifications are derived such that the modified model can be stabilized by a stabilizing PID controller based on Hermite–Biehlers Theorem. Furthermore, we obtain all the controllers satisfying a given margin specification. Simulation studies show how to use the results to design a robust PID controller.     

A simple method of tuning parallel cascade controllers for unstable FOPTD systems

A simple method is proposed to design parallel cascade controllers for open loop unstable processes. A proportional (P)controller is considered for the secondary loop and a proportional integral (PI) controller is considered for the primary loop (P/PI control configuration). Coefficients of the corresponding powers of s (Laplace variable), in the numerator is matched with the coefficients of the corresponding powers of s in the denominator of a closed loop transfer function for a servo problem. Three simulation case studies are considered in this paper. The first case involves a stable secondary loop process and an unstable primary process, the second case involves both unstable primary and secondary processes and the third one, a simulation application to a nonlinear bioreactor model equations. For comparison purposes, P/PI controller design is also carried out by improved simultaneous relay autotuning method, synthesis method and minimizing ISE criterion method. It is found that the proposed method gives a better performance. Robust stability analysis using the complimentary sensitivity function is carried out. The present method is found to be more robust.     

Determination of all feasible robust PID controllers for open-loop unstable plus time delay processes with gain margin and phase margin specifications

This paper proposes a novel alternative method to graphically compute all feasible gain and phase margin specifications-oriented robust PID controllers for open-loop unstable plus time delay (OLUPTD) processes. This method is applicable to general OLUPTD processes without constraint on system order. To retain robustness for OLUPTD processes subject to positive or negative gain variations, the downward gain margin (GM_down), upward gain margin (GM_up), and phase margin (PM) are considered. A virtual gain-phase margin tester compensator is incorporated to guarantee the concerned system satisfies certain robust safety margins. In addition, the stability equation method and the parameter plane method are exploited to portray the stability boundary and the constant gain margin (GM) boundary as well as the constant PM boundary. The overlapping region of these boundaries is graphically determined and denotes the GM and PM specifications-oriented region (GPMSOR). Alternatively, the GPMSOR characterizes all feasible robust PID controllers which achieve the pre-specified safety margins. In particular, to achieve optimal gain tuning, the controller gains are searched within the GPMSOR to minimize the integral of the absolute error (IAE) or the integral of the squared error (ISE) performance criterion. Thus, an optimal PID controller gain set is successfully found within the GPMSOR and guarantees the OLUPTD processes with a pre-specified GM and PM as well as a minimum IAE or ISE. Consequently, both robustness and performance can be simultaneously assured. Further, the design procedures are summarized as an algorithm to help rapidly locate the GPMSOR and search an optimal PID gain set. Finally, three highly cited examples are provided to illustrate the design process and to demonstrate the effectiveness of the proposed method.     

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.     

An improved auto-tuning scheme for PI controllers

Ziegler-Nichols tuned PI and PID controllers are usually found to provide poor performances for high-order and nonlinear systems. In this study, an improved auto-tuning scheme is presented for Ziegler-Nichols tuned PI controllers (ZNPICs). With a view to improving the transient response, the proportional and integral gains of the proposed controller are continuously modified based on the current process trend. The proposed controller is tested for a number of high-order linear and nonlinear dead-time processes under both set-point change and load disturbance. It exhibits significantly improved performance compared to ZNPIC, and Refined Ziegler-Nichols tuned PI controller (RZNPIC). Robustness of the proposed scheme is established by varying the controller parameters as well as the dead-time of the process under control.     

PID controllers' fragility

In this paper, an index for measuring fragility of proportional integral derivative (PID) controllers is proposed. This index relates the losses of robustness of the control loop when controller parameters change, to the nominal robustness of the control loop. Furthermore, it defines when a PID controller is fragile, nonfragile or resilient.     

Set point weighted modified Smith predictor for integrating and double integrating processes with time delay

A simple method of designing the controllers for a modified form of Smith predictor is proposed for integrating and double integrating processes with time delay. The modified Smith predictor has two controllers, namely, a set point tracking controller and a load disturbance rejection controller for obtaining good set point tracking and load disturbance rejection, respectively. The set point tracking controller is designed using the classical direct synthesis method based on the process model without considering the time delay. The disturbance rejection controller is considered as a proportional-derivative (PD) controller and is designed using optimal gain and phase margin approaches. Set point weighting is considered for reducing undesirable overshoots and settling times in the modified Smith predictor. Guidelines are provided for selection of the desired closed loop tuning parameter in the direct synthesis method and the set point weighting parameter. The method gives significant load disturbance rejection performances. Illustrative examples are considered to show the performances of the proposed method. A significant improvement in control performance is obtained when compared to recently reported methods.     

一种模糊变系数PID控制器及其在水轮机调速器中的应用

根据水轮机调速器工况不断变化的特点，提出一种模糊变系数PID控制器，它以常规PID控制器为核心，利用模糊控制原理在线调整PID控制器的比例、积分及微分增益。通过水轮机调速器半物理仿真实验系统来检验该控制器的控制性能，并与普通的PID控制器进行比较，结果表明，该控制器有良好的控制性能，并具有抵抗系统参数变化的能力。该控制器算法简单，实时性强，适于微机调速器应用。     

Robust on-line relay automatic tuning of PID control systems

In this paper, a robust on-line relay automatic tuning method for PID control systems is developed which expand on the application domain of Astrom's renowned relay autotuning method. In the proposed configuration, a relay is applied to an inner loop of a controller-stabilised process in the usual manner. Using the induced limit cycle oscillations from the closed-loop system, the controller settings may be re-tuned non-iteratively to achieve enhanced performance without disrupting closed-loop control. Two control tuning methodologies are developed -- a direct and an indirect method based on an explicit process model. Simulation examples and a real-time experiment are provided to illustrate the practical appeal and potential advantages of the proposed method over the basic one.     

Design of a Drilling Torque Controller for a Machining Center

In the drilling process, cutting torque control through feedrate manipulation has significant benefits in prevention of tool breakage and reduction of machine tool vibration and tool wear. In this paper, a PID torque controller was designed for real time drilling torque control in a machining centre. The drilling torque was estimated from the spindle motor current and regulated by feedrate control. The plant including the feed drive system, cutting process, and spindle system was modelled for controller design. For a certain cutting condition, the Ziegler-Nichols method was used to determine the proportional gain and derivative and integral control action times of the controller. Also, the root locus plot was used to tune the proportional gain of the controller. A simple method was also suggested to obtain the tuned controller gain instead of the Ziegler-Nichols method and the root locus plot for an arbitrary cutting condition. Experimental works have been performed to verify the performance of the proposed controller. It is shown that the gain tuning is essential for enhancement of the controller performance and the designed controller can regulate the drilling torque well at a given reference level.Nomenclature G_fc(s) transfer function of combined system of feed drive and cutting process - G_s(s) transfer function of spindle system - G_p(s) transfer function of plant - G_CL(s) transfer function of closed loop system - G_con(s) transfer function of controller - _fc1, _fc2 time constants of combined system of feed drive and cutting process - _s time constant of spindle system - K controller gain - K_c cutting process gain - K_s spindle motor current gain - K_plant plant gain - K_cr critical gain - K_p proportional gain of controller - K_{p_tune} tuned proportional gain of controller - I_s spindle motor current - I_u,v,w u, v, and w phase current of spindle motor - I_idle idling current of spindle motor - damping ratio - T oscillation period of system response - T_c cutting torque - T_m motor torque - T_idle idling torque of spindle motor - T_i integral control action time - T_d derivative control action time     

Decentralized PI/PID controllers based on gain and phase margin specifications for TITO processes

In this paper, a decentralized PI/PID controller design method based on gain and phase margin specifications for two-input-two-output (TITO) interactive processes is proposed. The decouplers are designed for systems to minimize the interaction between the loops, and the first order plus dead time (FOPDT) model is achieved for each decoupled subsystem based on the frequency response fitting. An independent PI/PID controller is designed for each reduced order decoupled subsystem to obtain the desired gain and phase margins, and the performance is verified on the original interactive system to show the effectiveness of the proposed design method for the general class of TITO systems. Simulation examples are incorporated to validate the usefulness of the presented algorithm. An experimentation is performed on the Level-Temperature reactor process to show the practical applicability of the proposed method for the interactive system.     

Frequency response adaptation of PI controllers based on recursive least-squares process identification

The frequency domain approach to adaptive control or autotuning of proportional plus integral (PI) controllers is considered in this paper. The frequency response of the process is computed from an overparametrized recursive least-squares (RLS) model and used at sample intervals to tune the PI. By simulations it is shown that the frequency response estimates converge more quickly and are more constant in the presence of disturbances than the RLS coefficients from which they are calculated. The frequency response estimates are therefore more reliable for controller tuning. The method can handle concurrent gain variation of 50% and time constant or delay variations of 100%. The method is implemented and experimentally verified on a pilot distillation column.     

Evolutionary optimization of sliding mode controller for level control system

This paper accords the level control of single-input-single-output (SISO) level control system based on the fusion of sliding mode control (SMC) and evolutionary techniques or bio-inspired techniques. The non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective particle swarm optimization (MOPSO) are considered as two evolutionary techniques. Here, a comparative analysis of performances of an optimal proportional–integral (PI) controller, proportional–integral–derivative (PID) controller, conventional SMC, NSGA-II based tuned SMC and SMC parameter tuning using MOPSO algorithm has been carried out through MATLAB/SIMULINK. The objective functions, integral absolute error (IAE), integral squared error (ISE) and an integration of weighted objective function aggregated approach of the error performance indices, IAE and ISE are considered. Realistic conditions are used in a plant for testing the robustness of controller. The stability of the controller is successfully obtained which satisfies the Lyapunov stability criteria. Reduction in long settling time with tiny magnitude variations about an equilibrium point is achieved using bio-inspired techniques. The simulation as well as experimental results reveal that SMC parameter tuning based on NSGA-II algorithm gives a better performance as compared to the other design strategies.     

Extension of IMC tuning correlations for non-self regulating (integrating) processes

The filter term of a PID with Filter controller reduces the impact of measurement noise on the derivative action of the controller. This impact is quantified by the controller output travel defined as the total movement of the controller output per unit time. Decreasing controller output travel is important to reduce wear in the final control element. Internal Model Control (IMC) tuning correlations are widely published for PI, PID, and PID with Filter controllers for self regulating processes. For non-self regulating (or integrating) processes, IMC tuning correlations are published for PI and PID controllers but not for PID with Filter controllers. The important contribution of this work is that it completes the set of IMC tuning correlations with an extension to the PID with Filter controller for non-self regulating processes. Other published correlations (not based upon the IMC framework) for PID with Filter controllers fix the filter time constant at one-tenth the derivative time regardless of the model of the process. In contrast, the novel IMC correlations presented in this paper calculate a filter time constant based upon the model of the process and the user's choice for the closed-loop time constant. The set point tracking and disturbance rejection performance of the proposed IMC tunings is demonstrated using simulation studies and a bench-scale experimental system. The proposed IMC tunings are shown to perform as well as various PID correlations (with and without a filter term) while requiring considerably less controller action.     

An optimal PID controller via LQR for standard second order plus time delay systems

An improved tuning methodology of PID controller for standard second order plus time delay systems (SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement technique to obtain the desired performance measures. The pole placement method together with LQR is ingeniously used for SOPTD systems where the time delay part is handled in the controller output equation instead of characteristic equation. The effectiveness of the proposed methodology has been demonstrated via simulation of stable open loop oscillatory, over damped, critical damped and unstable open loop systems. Results show improved closed loop time response over the existing LQR based PI/PID tuning methods with less control effort. The effect of non-dominant pole on the stability and robustness of the controller has also been discussed.     

基于遗传禁忌算法的带式输送机直接转矩控制系统

带式输送机电气传动系统好坏直接关系到生产过程的安全可靠性和经济性。针对传统带式输送机电气系统启动加速度不平稳,容易引起皮带跑偏甚至撕裂皮带的安全事故,本文提出了一种改进遗传禁忌算法的PID控制器来实现直接转矩控制的转速闭环方案。该方案建立了误差绝对值时间积分性能指标的最小目标函数和禁忌变异算子的数学模型。结果表明,改进PID控制器具有超调量和调节时间小、抗干扰性强、鲁棒性好的优点,有助于解决系统皮带跑偏甚至撕裂的问题。     

Precision positioning of a DC-motor-driven aerostatic slide system

This paper deals with precision positioning in the presence of friction. The object studied is an aerostatic slide system driven by a DC motor with brushes that introduce friction to the system. For such systems, models that do not account for friction can only be used to describe the macrodynamic behavior. The microdynamic behavior is significantly different. Instead of designing two controllers for the different dynamics, a single-step precision positioning using a high-gain controller designed according to the macrodynamics alone is proposed. A proportional-integral-derivative (PID) controller with finite-gain derivative and a modified anti-windup integral reset is designed. Controller parameters are obtained by pole placement rather than manual tuning or other rules of thumb. Identification of the microdynamics is avoided. Single-step precision positioning and uniform response to different sizes of step input are achieved. This method is particularly effective when the friction has uncertainty. Experimental and simulated results indicate that the PID controller can provide a sufficiently high loop gain. In point-to-point positioning for step inputs from millimeter size down to submicrometer size, the positioning error is within ±2 nm and the response dynamics is satisfactory.     

基于FPGA的Fuzzy-PID温度控制器的设计与应用

针对某能源分析仪器中高温炉控制的要求,设计了一个基于模糊参数自整定的PID控制器,利用模糊控制所得的PID参数结果来调节PID参数变化,从而实现了PID参数的自动调节。结果显示此种模糊参数自整定的PID控制器对于高温炉系统中存在的非线性、大滞后、物理模型不精确等问题都有很好的解决效果。     

Small-signal modelling and control of photovoltaic based water pumping system

This paper studies small-signal modelling and control design for a photovoltaic (PV) based water pumping system without energy storage. First, the small-signal model is obtained and then, using this model, two proportional–integral (PI) controllers, where one controller is used to control the dc-link voltage and the other one to control the speed of induction motor, are designed to meet control goals such as settling time and peak overshoot of the closed loop responses. The loop robustness of the design is also studied. For a given set of system parameters, simulations are carried out to validate the modelling and the control design.