PID controller frequency-domain tuning for stable, integrating and unstable processes, including dead-time

In the present paper a new tuning procedure is proposed for the ideal PID controller in series with the first-order noise filter. It is based on the recently proposed extension of the Ziegler-Nichols frequency-domain dynamics characterization of a process G_p(s). Measured process characteristics are the ultimate frequency and ultimate gain, the angle of the tangent to the Nyquist curve of the process at the ultimate frequency, and G_p(0). For a large class of processes the same tuning formulae can be effectively applied to obtain closed-loop responses with predictable properties. Load disturbance step responses without the undershoot and reference step responses with negligible overshoot are obtained by analyzing a test batch consisting of stable, integrating and unstable processes, including dead-time and oscillatory dynamics. The proposed tuning makes possible to specify the desired sensitivity to the high frequency measurement noise and the desired maximum sensitivity. Comparison with the optimal ideal PID controller in series with the first-order noise filter is presented and discussed. The extension of the proposed method to the PI controller tuning is direct. Comparison with the optimal PI controller is presented and discussed.     

PID controller design of TITO system based on ideal decoupler

The proposed method for designing multivariable controller is based on ideal decoupler D(s) and PID controller optimization under constraints on the robustness and sensitivity to measurement noise. The high closed-loop system performance and robustness are obtained using the same controller in all loops. The method is effective despite the values and positions of the right half plane zeros and dead-times in the process transfer function matrix G_p(s). The validity of the proposed multivariable control system design and tuning method is confirmed using a test batch consisting of Two-Input Two-Output (TITO) stable, integrating and unstable processes, and one Three-Input Three-Output (TITO) stable process.     

Robust PID controller tuning based on the constrained particle swarm optimization

This paper proposes a novel tuning strategy for robust proportional-integral-derivative (PID) controllers based on the augmented Lagrangian particle swarm optimization (ALPSO). First, the problem of PID controller tuning satisfying multiple H_∞ performance criteria is considered, which is known to suffer from computational intractability and conservatism when any existing method is adopted. In order to give some remedy to such a design problem without using any complicated manipulations, the ALPSO based robust gain tuning scheme for PID controllers is introduced. It does not need any conservative assumption unlike the conventional methods, and often enables us to find the desired PID gains just by solving the constrained optimization problem in a straightforward way. However, it is difficult to guarantee its effectiveness in a theoretical way, because PSO is essentially a stochastic approach. Therefore, it is evaluated by several simulation examples, which demonstrate that the proposed approach works well to obtain PID controller parameters satisfying the multiple H_∞ performance criteria.     

Domains of PID controller coefficients which guarantee stability and performance for LTI time-delay systems

In this work, the computation of the regions in the space of coefficients of a PID controller, inside which the controller stabilizes an LTI time-delay plant and guarantees a performance specification for the closed-loop system, is addressed. To ascertain the sensor noise attenuation, as a sample performance requirement, the H_∞-norm of the weighted complementary sensitivity function is kept below a desired bound. The problem of minimization of the H_∞-norm is transformed into the simultaneous stabilization of a family of quasipolynomials. Then, the stable domains of the PID coefficients are computed for these quasipolynomials using the parameter space approach. This work extends the previous results on the computation of the stability domains of a PID controller to ensure the desired performance specification for the closed-loop system.     

Series PID controller tuning based on the SIMC rule and signal filtering

The SIMC (Simple control) rule, proposed by Skogestad, is ineffective for a class of processes with oscillatory dynamics and processes defined by transfer functions obtained as a result of ideal decoupling of multiple-input multiple-output systems. For this class of stable processes it is proposed to apply a higher-order filtering to the open-loop process step response and to approximate the filtered step response with stable SOPDT models. These models are used to obtain a high performance/robustness tradeoff by the ideal series PID controllers, tuned by the SIMC rule, with the higher-order filter in the feedback loop. Parallel PID controllers, with higher-order noise filters, tuned by applying exact process frequency response and optimization under constraints on the robustness and sensitivity to measurement noise, are used to demonstrate merits of the proposed simple design and tuning of the series PID controller. Experimental results on a mechanical laboratory plant are presented in Appendix.     

Classification of dynamic processes and PID controller tuning in a parameter plane

A quadruplet, defined by the ultimate frequency ω_u, the ultimate gain k_u, the angle φ of the tangent to the Nyquist curve at the ultimate frequency and the gain G_p(0), is sufficient for classification of a large class of stable processes, processes with oscillatory dynamics, integrating and unstable processes G_p(s). From the model defined by the above quadruplet, a two parameter model G_n(s_n) is obtained by the time and amplitude normalizations. Two parameters of G_n(s_n), the normalized gain ρ and the angle φ, are coordinates of the classification ρ-φ parameter plane. Model G_n(s_n) is used to obtain the desired closed-loop system performance/robustness tradeoff in the desired region of the classification plane. Tuning procedures and tuning formulae are derived guaranteeing almost the same performance/robustness tradeoff as obtained by the optimal PID controller, applied to G_p(s) classified to the same region of the classification plane. Validity of the proposed method is demonstrated on a test batch consisting of stable processes, processes with oscillatory dynamics, integrating and unstable processes, including dead-time.     

H∞ filtering of networked discrete-time systems with random packet losses

This paper studies the H_∞ filtering problem for networked discrete-time systems with random packet losses. The general multiple-input-multiple-output (MIMO) filtering system is considered. The multiple measurements are transmitted to the remote filter via distinct communication channels, and each measurement loss process is described by a two-state Markov chain. Both the mode-independent and the mode-dependent filters are considered, and the resulting filtering error system is modelled as a discrete-time Markovian system with multiple modes. A necessary and sufficient condition is derived for the filtering error system to be mean-square exponentially stable and achieve a prescribed H_∞ noise attenuation performance. The obtained condition implicitly establishes a relation between the packet loss probability and two parameters, namely, the exponential decay rate of the filtering error system and the H_∞ noise attenuation level. A convex optimization problem is formulated to design the desired filters with minimized H_∞ noise attenuation level bound. Finally, an illustrative example is given to show the effectiveness of the proposed results.     

Multivariable PID control by decoupling

This paper presents a new methodology to design multivariable proportional-integral-derivative (PID) controllers based on decoupling control. The method is presented for general n × n processes. In the design procedure, an ideal decoupling control with integral action is designed to minimise interactions. It depends on the desired open-loop processes that are specified according to realisability conditions and desired closed-loop performance specifications. These realisability conditions are stated and three common cases to define the open-loop processes are studied and proposed. Then, controller elements are approximated to PID structure. From a practical point of view, the wind-up problem is also considered and a new anti-wind-up scheme for multivariable PID controller is proposed. Comparisons with other works demonstrate the effectiveness of the methodology through the use of several simulation examples and an experimental lab process.     

On the design of multivariable PID controllers via LMI approach

In this paper, we study the design problem of multivariable PID controllers which guarantee the stability of the closed loop systems, H₂ or H_∞ performance specifications, or maximum output control requirement, respectively. Algorithms based on iterative linear matrix inequality technique are developed to find the feedback gains of PID controllers corresponding to the above mentioned four cases. A numerical example on the design of PID controllers for aircraft is provided to illustrate the effectiveness of the proposed method.     

Measurement noise filtering for PID controllers

Measurement noise can generate undesired control activity resulting in wear of actuators and reduced performance. The effects of measurement noise can be alleviated by filtering the measurement signal. The design of the filter is then a trade-off; heavy filtering reduces the undesired control activity but performance is degraded. In this paper we discuss the trade-offs for PID control. Based on the insight gained we introduce two quantities that characterize the effect of measurement noise the SDU, which is a measure of noise activity analog to the IAE commonly used to characterize load disturbance response, and the noise gain k_n, which tells how fluctuations in the filtered measurement signal are reflected in variations of the control signal. Simple rules for choosing the filter time constant for PI and PID controllers are also given. The results are illustrated by simulations and lab experiments.     

Revisiting the Ziegler–Nichols process dynamics characterization

The Ziegler–Nichols process dynamics characterization is based on the estimation of the ultimate gain k_u and ultimate frequency ω_u. The angle φ of the tangent to the Nyquist curve at the frequency ω_u is introduced, as an additional parameter in the frequency domain. The essential dynamic characteristics of the process can be captured by using the tangent rule, proposed here as an extension of the Ziegler–Nichols approach. The validity of the tangent rule is confirmed by using the PID controller optimization on a test batch consisting of stable, integrating and unstable processes, including dead-time. Parameters k_u, ω_u and φ can be determined from the sustained oscillations, using the phase-locked loop identifier module.     

基于Maclaurin 展开的PID 设计与无模型自整定

针对自衡对象,提出一种基于期望模型的PID自整定方法,该方法无需被控对象的数学模型.利用Maclaurin展开技术,给出了PID控制器的整定公式;并通过开环阶跃响应,实现了PID控制器的无模型自整定.仿真结果表明,利用该自整定方法所得的PID能有效地提高高阶被控对象的系统性能;即使在噪声环境下,该方法仍具有很好的鲁棒性.     

Multiplicative Stochastic Systems with Multiple External Disturbances

With the methods of H₂/H_∞-control theory, in the presence of noise we solve the optimization problem for a multiplicative stochastic system with several external disturbances (the multiperturbation problem) and vector Wiener processes with arbitrary intensity matrices. We obtain matrix differential equations of Riccati type, reducing the original optimization problem to solving these equations.     

Distributed H_∞ PID Feedback for Improving Consensus Performance of Arbitrary-delayed Multi-agent System

The H∞ proportional-integral-differential(PID) feedback for arbitrary-order delayed multi-agent system is investigated to improve the system performance. The closed-loop multi-input multi-output(MIMO) control framework with the distributed PID controller is firstly described for the multi-agent system in a unified way. Then, by using the matrix theory, the prescribed H∞performance criterion of the multi-agent system is shown to be equivalent to several independent H∞ performance constraints of the single-input single-output(SISO) subsystem with respect to the eigenvalues of the Laplacian matrix. Subsequently, for each subsystem,the set of the PID controllers satisfying the required H∞ performance constraint is analytically characterized based on the extended Hermite-Biehler theorem. Finally, the three-dimensional set of the decentralized H∞ PID control parameters is derived by finding the intersection of the H∞ PID regions for all the decomposed subsystems. The simulation results reveal the effectiveness of the proposed method.     

Robust Unknown Input Observer Design for Linear Uncertain Time Delay Systems with Application to Fault Detection

In this paper, a novel approach is proposed to design a robust fault detection observer for uncertain linear time delay systems. The system is composed of both norm‐bounded uncertainties and exogenous signals (noise, disturbance, and fault) which are considered to be unknown. The main contribution of this paper is to present unknown input observer (UIO)‐based fault detection system which shows the maximum sensitivity to fault signals and the minimum sensitivity to other signals. Since the system contains uncertainty terms, an H_∞ model‐matching approach is used in design procedure. The reference residual signal generator system is designed so that the fault signal has maximum sensitivity while the exogenous signals have minimum sensitivity on the residual signal. Then, the fault detection system is designed by minimizing the estimation error between the reference residual signal and the UIO residual signal in the sense of H_∞ norm. A sufficient condition for the existence of such a filter is exploited in terms of certain linear matrix inequalities (LMIs). Application of the proposed method in a numerical example and an engineering process are simulated to demonstrate the effectiveness of the proposed algorithm. Simulation results show the validity of the proposed approach to detect the occurrence of faults in the presence of modeling errors, disturbances, and noise.     

Robust ℋ︁∞ PID control for multivariable networked control systems with disturbance/noise attenuation

In this paper, we study the design problem of PID controllers for networked control systems (NCSs) with polyhedral uncertainties. The load disturbance and measurement noise are both taken into account in the modeling to better reflect the practical scenario. By using a novel technique, the design problem of PID controllers is converted into a design problem of output feedback controllers. Our goal of this paper is two‐fold: (1) To design the robust PID tracking controllers for practical models; (2) To develop the robust ??_∞ PID control such that load and reference disturbances can be attenuated with a prescribed level. Sufficient conditions are derived by employing advanced techniques for achieving delay dependence. The proposed controller can be readily designed based on iterative suboptimal algorithms. Finally, four examples are presented to show the effectiveness of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Tuning PI/PID controllers for integrating processes with deadtime and inverse response by simple calculations

A simple calculation method of a practical PI/PID controller tuning for integrating processes with deadtime and inverse response based on a model is presented in this study. First, analytical expressions for PI/PID controller settings based on the model using a direct synthesis method for disturbance rejection (DS-d) are employed. Next, optimum tuning parameter (λ) for DS-d based on the model and minimum IAE criterion are obtained via the golden-section searching technique. These optimum λ data are then empirically correlated into two equations. Thus, PI/PID controller settings for the model can easily be obtained from the parameter λ using DS-d formulas. The advantage of the proposed method is that DS-d PI/PID settings could be expediently sought by simple calculations using these equations without any tedious design. Simulation results have demonstrated that the proposed tuning technique can perform better for load/disturbance changes than other available methods in the literature.     

Enhanced design of cascade control systems for unstable processes with time delay

In this paper, optimal H₂ internal model controller (IMC) is designed for control of unstable cascade processes with time delays. The proposed control structure consists of two controllers in which inner loop controller (secondary controller) is designed using IMC principles. The primary controller (master controller) is designed as a proportional-integral-derivative (PID) in series with a lead-lag filter based on IMC scheme using optimal H₂ minimisation. Selection of tuning parameter is important in any IMC based design and in the present work, maximum sensitivity is used for systematic selection of the primary loop tuning parameter. Simulation studies have been carried out on various unstable cascade processes. The present method provides significant improvement when compared to the recently reported methods in the literature particularly for disturbance rejection. The present method also provides robust closed loop performances for large uncertainties in the process parameters. Quantitative comparison has been carried out by considering integral of absolute error (IAE) and total variation (TV) as performance indices.     

Analytical IMC-PID design in terms of performance/robustness tradeoff for integrating processes: From 2-Dof to 1-Dof

This communication addresses the analytical PID tuning rules for integrating processes. First, this paper provides an analytical tuning method of two-degree-of-freedom (2-Dof) PID controller using an enhanced internal model control (IMC) principle. On the basis of the robustness analyses, the presented method can easily achieve the performance/robustness tradeoff by specifying a desired robustness degree. Second, an analytical tuning method of one-degree-of-freedom (1-Dof) PID also is proposed in terms of performance/robustness and servo/regulator tradeoffs, which are not commonly considered for 1-Dof controller design. The servo/regulator tradeoff is formulated as a constrained optimization problem to provide output responses as similar as possible to those produced by the 2-Dof PID controller. The presented PID settings are applicable for a wide range of integrating processes. Simulation studies show the effectiveness and merits of the proposed method.     

Stability issues of finite-precision controller structures for sampled-data systems

The paper investigates the sensitivity of closed-loop stability with respect to (w.r.t.) finite word length (FWL) effects in the implementation of the digital controller coefficients. Both the shift and delta operators are considered for controller parameterization. Two tractable lower-bound measures of closed-loop stability are studied, and the optimal realization of general FWL controller structures is formulated as a constrained non-linear optimization problem. The emphasis of the paper, however, is on the derivation of a new algorithmic approach for the optimal realization of FWL PID controller structures. It is shown that, for PID structures, the optimization can be decoupled into two unconstrained problems with a maximum of four independent variables. An optimization strategy is developed to provide an efficient computational method for searching the optimal FWL PID controller realization with maximum stability bound and minimum bit-length requirement. Simulation results involving an IFAC benchmark PID controller system are presented to illustrate the effectiveness of the proposed strategy.