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.     

Nominal and robust stability regions of optimization-based PID controllers

In recent decades, several optimization-based methods have been developed for the proportional-integral-derivative (PID) controller design, and the common feature of these methods is that the controller has only one adjustable parameter. To keep the closed-loop systems stable is an essential requirement for the optimization-based PID controllers. In almost all these methods, however, no exact stability region for the single adjustable parameter was sketched. In this paper, using the proposed analytical procedure based on the dual-locus diagram technique, explicit stability regions of the optimization-based PID controllers are derived for stable, integrating, and unstable processes with time delay in the nominal and perturbed cases, respectively. It is revealed that the proposed analytical procedure is effective for the determination of the nominal and robust stability regions and it offers simplicity and ease of mathematical calculations over other available stability analysis methods. The results in this paper provide some insight into the tuning of the optimization-based PID controllers.     

H(infinity) PID controller design for runaway processes with time delay

This paper presents an efficient method for designing proportional-integra-derivative (PID) controllers for runaway processes with time delay. The method is developed based on the H(infinity) control theory in frequency domain. The constraints imposed by the internal stability and asymptotic properties of the closed-loop system are first investigated, a new procedure is then developed for analytically designing the controller, and simple design formulas are obtained. It is shown that the new controller can be designed to meet specified time domain performances. Typical design examples are provided to illustrate the proposed method.     

卡尔曼滤波器在PID控制器中的应用研究

本文建立球杆系统模型并分析系统中噪声,采用卡尔曼滤波器来克服球杆系统中的测量噪声和控制噪声对控制性能的影响,并在MATLAB环境下进行对比仿真试验,验证了滤波器对噪声的有效抑制作用。     

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.     

Dominant pole placement with fractional order PID controllers: D-decomposition approach

Dominant pole placement is a useful technique designed to deal with the problem of controlling a high order or time-delay systems with low order controller such as the PID controller. This paper tries to solve this problem by using D-decomposition method. Straightforward analytic procedure makes this method extremely powerful and easy to apply. This technique is applicable to a wide range of transfer functions: with or without time-delay, rational and non-rational ones, and those describing distributed parameter systems. In order to control as many different processes as possible, a fractional order PID controller is introduced, as a generalization of classical PID controller. As a consequence, it provides additional parameters for better adjusting system performances. The design method presented in this paper tunes the parameters of PID and fractional PID controller in order to obtain good load disturbance response with a constraint on the maximum sensitivity and sensitivity to noise measurement. Good set point response is also one of the design goals of this technique. Numerous examples taken from the process industry are given, and D-decomposition approach is compared with other PID optimization methods to show its effectiveness.     

Tuning PID controllers for higher-order oscillatory systems with improved performance

In this paper, model based design of PID controllers is proposed for higher-order oscillatory systems. The proposed method has no limitations regarding systems order, time delays and oscillatory behavior. The reduced model is achieved based on third-order modeling and selection of coefficients through the use of frequency responses. The tuning of the PID parameters are obtained from a reduced third-order model; the procedure seems to be simple and effective, and improved performance of the overall system can be achieved. Three simulation examples and one real-time experiment are included to demonstrate the effectiveness and applicability of the proposed method to systems with oscillatory behavior.     

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.     

一类时滞系统的PID稳定控制器设计

针对工业控制过程中常见的时滞系统,给出了快速计算PI和PID控制器所有稳定参数集的方法。推导了系统开环不稳定情形下稳定控制器存在的必要性条件．通过在该稳定参数区域上进行性能指标寻优,可以得到满足控制器设计性能指标要求的PID控制器参数区域。仿真实例表明了该方法的优越性。     

An improved auto-tuning scheme for PID controllers

An improved auto-tuning scheme is proposed for Ziegler–Nichols (ZN) tuned PID controllers (ZNPIDs), which usually provide excessively large overshoots, not tolerable in most of the situations, for high-order and nonlinear processes. To overcome this limitation ZNPIDs are upgraded by some easily interpretable heuristic rules through an online gain modifying factor defined on the instantaneous process states. This study is an extension of our earlier work [Mudi RK., Dey C. Lee TT. An improved auto-tuning scheme for PI controllers. ISA Trans 2008; 47: 45–52] to ZNPIDs, thereby making the scheme suitable for a wide range of processes and more generalized too. The proposed augmented ZNPID (AZNPID) is tested on various high-order linear and nonlinear dead-time processes with improved performance over ZNPID, refined ZNPID (RZNPID), and other schemes reported in the literature. Stability issues are addressed for linear processes. Robust performance of AZNPID is observed while changing its tunable parameters as well as the process dead-time. The proposed scheme is also implemented on a real time servo-based position control system.     

Thermal response simulation for tuning PID controllers in a 1016 mm guarded hot plate apparatus

A mathematical model has been developed and used to simulate the controlled thermal performance of a large guarded hot-plate apparatus. This highly specialized apparatus comprises three interdependent components whose temperatures are closely controlled in order to measure the thermal conductivity of insulation materials. The simulation model was used to investigate control strategies and derive controller gain parameters that are directly transferable to the actual instrument. The simulations take orders-of-magnitude less time to carry out when compared to traditional tuning methods based on operating the actual apparatus.The control system consists primarily of a PC-based PID control algorithm that regulates the output voltage of programmable power amplifiers. Feedback parameters in the form of controller gains are required for the three heating circuits. An objective is to determine an improved set of gains that meet temperature control criteria for testing insulation materials of interest.The analytical model is based on aggregated thermal capacity representations of the primary components and includes the same control algorithm as used in the actual hot-plate apparatus. The model, accounting for both thermal characteristics and temperature control, was validated by comparisons with test data.The tuning methodology used with the simulation model is described and results are presented. The resulting control algorithm and gain parameters have been used in the actual apparatus without modification during several years of testing materials over wide ranges of thermal conductivity, thickness, and insulation resistance values.     

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.     

基于LQR指标的直线伺服系统最优PID控制器设计

由音圈电动机驱动的直线伺服系统是实现非圆截面类零件数控车削加工的关键。基于二次型性能指标（LQR）设计了直线伺服系统的最优PID控制器。首先给出了直线伺服系统的双闭环PID控制结构，其次研究了基于二次型性能指标的数字PID参数最优整定方法。最后通过仿真实验验证了该控制方法的有效性。     

基于模糊自整定PID的注水流量控制系统

针对注水流量控制系统的控制对象数学模型难建立,同时由于井下环境复杂,控制系统存在严重的非线性的问题,采用模糊自整定PID控制算法,设计了基于TMS320F2812模糊自整定PID控制器。该验结果表明模糊自整定PID控制能够满足系统的控制精度要求,具有动态和稳态性能好等优点。     

A novel auto-tuning PID control mechanism for nonlinear systems

In this paper, a novel Runge–Kutta (RK) discretization-based model-predictive auto-tuning proportional-integral-derivative controller (RK-PID) is introduced for the control of continuous-time nonlinear systems. The parameters of the PID controller are tuned using RK model of the system through prediction error-square minimization where the predicted information of tracking error provides an enhanced tuning of the parameters. Based on the model-predictive control (MPC) approach, the proposed mechanism provides necessary PID parameter adaptations while generating additive correction terms to assist the initially inadequate PID controller. Efficiency of the proposed mechanism has been tested on two experimental real-time systems: an unstable single-input single-output (SISO) nonlinear magnetic-levitation system and a nonlinear multi-input multi-output (MIMO) liquid-level system. RK-PID has been compared to standard PID, standard nonlinear MPC (NMPC), RK-MPC and conventional sliding-mode control (SMC) methods in terms of control performance, robustness, computational complexity and design issue. The proposed mechanism exhibits acceptable tuning and control performance with very small steady-state tracking errors, and provides very short settling time for parameter convergence.     

On the fragility of fractional-order PID controllers for FOPDT processes

This paper analyzes the fragility issue of fractional-order proportional-integral-derivative controllers applied to integer first-order plus-dead-time processes. In particular, the effects of the variations of the controller parameters on the achieved control system robustness and performance are investigated. Results show that this kind of controllers is more fragile with respect to the standard proportional-integral-derivative controllers and therefore a significant attention should be paid by the user in their tuning.     

Design and implementation of a 2-DOF PID compensation for magnetic levitation systems

This paper employs a 2-DOF (degree of freedom) PID controller for compensating a physical magnetic levitation system. It is shown that because of having a feedforward gain in the proposed 2-DOF PID control, the transient performance of the compensated system can be changed in a desired manner unlike the conventional 1-DOF PID control. It is also shown that for a choice of PID parameters, although the theoretical loop robustness is the same for both the compensated systems, in real-time, 2-DOF PID control may provide superior robustness if a suitable choice of the feedforward parameter is made. The results are verified through simulations and experiments.     

Robust stabilizing first-order controllers for a class of time delay systems

In this paper, stabilizing regions of a first-order controller for an all poles system with time delay are computed via parametric methods. First, the admissible ranges of one of the controller’s parameters are obtained. Then, for a fixed value of this parameter, stabilizing regions in the remaining two parameters are determined using the D-decomposition method. Phase and gain margin specifications are then included in the design. Finally, robust stabilizing first-order controllers are determined for uncertain plants with an interval type uncertainty in the coefficients. Examples are given to illustrate the effectiveness of the proposed method.     

The generalized forced oscillation method for tuning PID controllers

In this paper a generalized tuning methodology for proportional–integral–derivative (PID) controllers is proposed. The methodology is akin to the Ziegler–Nichols forced oscillation method, inheriting fully its practical appeal, but can be applied to much more general classes of plants. This generalization is achieved by employing a relay with adjustable phase (RAP) in a relay feedback experiment, and the tuning consists of formulas based on measurements obtained from this experiment. Experimental results in a liquid processing plant with of-the-shelf industrial equipments illustrate the performance of the methodology and the practical procedures involved in its application, which can be fully automated.