Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID control with variable gains

The popular linear PID controller is mostly effective for linear or nearly linear control problems. Nonlinear PID controllers, however, are needed in order to satisfactorily control (highly) nonlinear plants, time-varying plants, or plants with significant time delay. This paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear PI, PD, and PID controllers with variable gains that can outperform their linear counterparts. In the present paper, we study the analytical structure of an important class of two- and three-dimensional fuzzy controllers. We link the entire class, as opposed to one controller at a time, to nonlinear PI, PD, and PID controllers with variable gains by establishing the conditions for the former to structurally become the latter. Unlike the results in the literature, which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables, this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types. Our structural results are thus more general and contain the existing ones as special cases. Two concrete examples are provided to illustrate the usefulness of the new results.     

Tuning fractional order proportional integral controllers for fractional order systems

In this paper, two fractional order proportional integral controllers are proposed and designed for a class of fractional order systems. For fair comparison, the proposed fractional order proportional integral (FOPI), fractional order [proportional integral] (FO[PI]) and the traditional integer order PID (IOPID) controllers are all designed following the same set of the imposed tuning constraints, which can guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. This proposed design scheme offers a practical and systematic way of the controllers design for the considered class of fractional order plants. From the simulation and experimental results presented, both of the two designed fractional order controllers work efficiently, with improved performance comparing with the designed stabilizing integer order PID controller by the observation. Moreover, it is interesting to observe that the designed FO[PI] controller outperforms the designed FOPI controller following the proposed design schemes for the class of fractional order systems considered.     

Decentralized robust PI controller design for an industrial boiler

This paper presents a new scheme to design decentralized robust PI controllers for uncertain LTI multivariable systems. Sufficient conditions for closed-loop stability and closed-loop diagonal dominance (almost decoupling) of a multivariable system are obtained. Satisfying these conditions and robust performance of the overall system are modeled as local robust performance problems. Then, by appropriately selecting the time constants of the closed-loop isolated subsystems in the IMC (Internal Model Control) strategy, the defined local robust performance problems are solved. To design a decentralized robust PI controller for a real industrial utility boiler, a control oriented nonlinear model for the boiler is identified. The nonlinearity of the system is modeled as uncertainty for a nominal LTI multivariable system. Using the new proposed method, a decentralized PI controller for the uncertain LTI model is designed. The designed controller is applied to the real system. The simulation results show the effectiveness of the proposed methodology.     

Design of linear time-invariant controllers for multirate systems

In this paper we discuss a frequency domain approach to model multirate single-input single-output (SISO) systems which facilitates design of linear time-invariant (LTI) controllers operating at the fast rate. To illustrate the approach we consider a dual-rate system with slow output measurements and fast control actions. We obtain necessary and sufficient conditions for the existence of stabilizing linear time-invariant (LTI) controllers for which model matching is also achieved at the fast rate with a desired single-rate system. Moreover, a solution to the problem of parameterizing the set of such LTI controllers is also given.     

Controller design for delay systems via eigenvalue assignment – on a new result in the distribution of quasi-polynomial roots

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment. First, a new result on the root distribution for a class of quasi-polynomials is developed based on the extension of the Hermite–Biehler theorem. Then, such result is applied to proportional–integral (PI) controller parameter design for a first-order plant with time delay through pole placement. The complete region of PI gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed-loop system with delay plant are assigned to desired positions in the complex plane. Furthermore, on the basis of the previous result, this paper also extended the PI control to the proportional–integral–derivative (PID) control. It is worth pointing out that this work aims to improve the performance of the closed-loop system on the premise of guaranteeing the stability.     

A novel hybrid DEPS optimized fuzzy PI/PID controller for load frequency control of multi-area interconnected power systems

In this paper, a novel hybrid Differential Evolution (DE) and Pattern Search (PS) optimized fuzzy PI/PID controller is proposed for Load Frequency Control (LFC) of multi-area power system. Initially a two-area non-reheat thermal system is considered and the optimum gains of the fuzzy PI/PID controller are optimized employing a hybrid DE and PS (hDEPS) optimization technique. The superiority of the proposed controller is demonstrated by comparing the results with some recently published modern heuristic optimization techniques such as DE, Bacteria Foraging Optimization Algorithm (BFOA), Genetic Algorithm (GA) and conventional Ziegler Nichols (ZN) based PI controllers for the same interconnected power system. Furthermore, robustness analysis is performed by varying the system parameters and operating load conditions from their nominal values. It is observed that the optimum gains of the proposed controller need not be reset even if the system is subjected to wide variation in loading condition and system parameters. Additionally, the proposed approach is further extended to multi-area multi-source power system with/without HVDC link and the gains of fuzzy PID controllers are optimized using hDEPS algorithm. The superiority of the proposed approach is shown by comparing the results with recently published DE optimized PID controller and conventional optimal output feedback controller for the same power systems. Finally, Reheat turbine, Generation Rate Constraint (GRC) and time delay are included in the system model to demonstrate the ability of the proposed approach to handle nonlinearity and physical constraints in the system model.     

A new approach to digital PID controller design

In this note, we present a new approach to the problem of designing a digital proportional-integral-derivative (PID) controller for a given but arbitrary linear time invariant plant. By using the Tchebyshev representation of a discrete-time transfer function and some new results on root counting with respect to the unit circle, we show how the digital PID stabilizing gains can be determined by solving sets of linear inequalities in two unknowns for a fixed value of the third parameter. By sweeping or gridding over this parameter, the entire set of stabilizing gains can be recovered. The precise admissible range of this parameter can be predetermined. This solution is attractive because it answers the question of whether there exists a stabilizing solution or not and in case stabilization is possible the entire set of gains is determined constructively. Using this characterization of the stabilizing set we present solutions to two design problems: 1) maximally deadbeat design, where we determine for the given plant, the smallest circle within the unit circle wherein the closed loop system characteristic roots may be placed by PID control and 2) maximal delay tolerance, where we determine, for the given plant the maximal-loop delay that can be tolerated under PID control. In each case, the set of controllers attaining the specifications is calculated. Illustrative examples are included.     

Control of LTI plants over erasure channels

This paper studies linear time-invariant (LTI) control architectures for LTI plants when communication takes place over a scalar erasure channel. Assuming i.i.d. data dropouts, we first show that such a channel is equivalent, in a second order moment sense, to an additive white noise channel subject to an instantaneous signal-to-noise ratio constraint. This key result is then exploited in two ways: First, we use it to characterize the set of all LTI controllers that achieve mean square stability in control architectures closed over scalar erasure channels. Second, we use it to show that the optimal design of LTI controllers over scalar erasure channels can be carried out by using tools from standard quadratic optimal control theory. We finally apply our results to the dynamic output feedback control of LTI single-input single-output (SISO) plants subject to data dropouts. In this case, we are able to establish closed form necessary and sufficient conditions on the minimal successful transmission probability that allows one to design mean square stabilizing controllers.     

Design of PID controllers for interval plants with time delay

First of all, the box theorem is extended to the interval plants with the fixed delay. An approach is presented to design the PID controller for interval plants with the fixed delay, which can obtain all of the stabilizing PID controllers. Then, using Hermite–Biehler theorem, extreme point results are provided by the virtual quasi-polynomials. When two virtual and two vertex quasi-polynomials corresponding to a Kharitonov-like segment plant are stable under a particular PID controller, it is sufficient that the same PID controller can stabilize this Kharitonov-like segment plant. The virtual quasi-polynomials are obtained in a simple way, and they are expressed in terms of the controller and the Kharitonov polynomials of the interval plants. A PID controller stabilizes interval plants with the fixed delay if it simultaneously stabilizes thirty-two quasi-polynomials. The example is given to illustrate the proposed method.     

A new class of nonlinear PID controllers with robotic applications

This paper introduces a new class of simple nonlinear PID controllers and provides a formal treatment of their stability analysis. These controllers are comprised of a sector-bounded nonlinear gain in cascade with a linear fixed-gain P, PD, PI, or PID controller. Three simple nonlinear gains are proposed: the sigmoidal function, the hyperbolic function, and the piecewise–linear function. The systems to be controlled are assumed to be modeled or approximated by second-order transfer functions, which can represent many robotic applications. The stability of the closed-loop systems incorporating nonlinear P, PD, PI, and PID controllers are investigated using the Popov stability criterion. It is shown that for P and PD controllers, the nonlinear gain is unbounded for closed-loop stability. For PI and PID controllers, simple expressions are derived that relate the controller gains and system parameters to the maximum allowable nonlinear gain for stability. A numerical example is given for illustration. The stability of partially-nonlinear PID controllers is also discussed. Finally, the nonlinear PI controller is implemented as a force controller on a robotic arm and experimental results are presented. These results demonstrate the superior performance of the nonlinear PI controller relative to a fixed-gain PI controller. © 1998 John Wiley & Sons, Inc. 15: 161–181, 1998     

Periodic Compensation of Continuous-Time Plants

This note presents a periodic compensator which achieves robust stability for single-input-single-output (SISO), linear time invariant (LTI) plants having both right-half plane (RHP) poles and zeros, a job LTI controllers fail to do. In addition, for strictly proper plants this controller achieves model matching ensuring at the same time that the periodic oscillations present in the plant output are insignificant in magnitude. The design steps are straightforward and linear algebraic in nature     

Synthesis of PID controllers for integral processes with time delay

This article deals with the problem of determination of the stabilizing parameter sets of Proportional‐Integral‐Derivative (PID) controllers for first‐order and second‐order integral processes with time‐delay. First, the admissible stabilizing range of proportional‐gain is determined analytically in terms of a version of the Hermite–Biehler Theorem applicable to quasi‐polynomials. Then, based on a graphical stability condition developed in parameter space, the complete stabilizing regions in an integral‐derivative plane are drawn and identified graphically, not calculated mathematically, by sweeping over the admissible range of proportional‐gain. An actual algorithm for finding the stabilizing parameter sets of PID controllers is also proposed. Simulations show that the stabilizing regions in integral‐derivative space are either triangles or quadrilaterals. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Operator-Based Robust Nonlinear Control for SISO and MIMO Nonlinear Systems With PI Hysteresis

In this paper, operator based robust nonlinear control for single-input single-output (SISO) and multi-input multi-output (MIMO) nonlinear uncertain systems preceded by generalized Prandtl-Ishlinskii (PI) hysteresis is considered respectively. In detail, by using operator based robust right coprime factorization approach, the control system design structures including feedforward and feedback controllers for both SISO and MIMO nonlinear uncertain systems are given, respectively. In which, the controller design includes the information of PI hysteresis and its inverse, and some sufficient conditions for the controllers in both SISO and MIMO systems should be satisfied are also derived respectively. Based on the proposed conditions, influence from hysteresis is rejected, the systems are robustly stable and output tracking performance can be realized. Finally, the effectiveness of the proposed method is confirmed by numerical simulations.      

Simultaneous stabilization of full state information Linear Time‐Invariant systems

Single necessary and sufficient stability conditions, are presented, for a common Linear Time‐Invariant (LTI) controller that simultaneously stabilizes each of a given family of LTI plants. It is assumed that the strictly proper, lumped and LTI plants have stabilizable realizations and are strongly stabilizable, and that the state‐space dimensions of the plants do not change, are even and are double the input‐space dimensions of the plants. The stabilizing controller is designed for a nominal plant and its free parameter is fixed assuring stability for all the plants, and assuring a stable controller. The stability conditions for simultaneous stabilization are based on the parity interlacing property, on a matrix that must be unimodular and on an analytical expression of the stabilizing controllers. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Feedback QoS control scheme for wireless network applications

Wireless LAN networking is an indispensable technology in an All-IP network architecture to satisfy the “anytime and anywhere” communication requirement of end users. This investigation proposes feedback controllers designing based on dynamic quality-of-service requirement for wireless LAN multimedia services. During the controllers design process, the time-domain is replaced by the s-domain, simplifying the calculation. This work presents three controllers namely proportional integral (PI), proportional derivative (PD) and proportional integral derivative (PID). Experimental results show that systems that employ the proposed controllers can quickly achieve the required system performance. Additionally, the PID controller has the best performance, and can improve delay performance by a rate 11.44% that without the feedback controller. The PI controller is superior to the PD controller. The delay when using the PD is 6.2% less than that achieved without the feedback controller.     

Independent design of multi-loop PI/PID controllers for interacting multivariable processes

The interactions between input/output variables are a common phenomenon and the main obstacle encountered in the design of multi-loop controllers for interacting multivariable processes. In this study, a novel method for the independent design of multi-loop PI/PID controllers is proposed. The idea of an effective open-loop transfer function (EOTF) is first introduced to decompose a multi-loop control system into a set of equivalent independent single loops. Using a model reduction technique, the EOTF is further approximated to the reduced-order form. Based on the corresponding EOTF model, the individual controller of each single loop is then independently designed by applying the internal model control (IMC)-based PID tuning approach for single-input/single-output (SISO) systems, while the main effects of the dynamic interactions are properly taken into account. Several illustrative examples are employed to demonstrate the effectiveness of the proposed method.     

时滞对象PID控制器的参数稳定域求解方法

研究时滞对象的PID控制器的参数稳定域的求解方法,通过引入Rekasius变换简化时滞特征多项式,并得到稳定域边界上PID控制器的3个参数所满足的方程.通过引入正弦变换,将对控制器参数在无穷范围内的遍历转为有限范围内的遍历,而特征多项式中的已知时滞被用以检验所遍历的点是否在稳定域边界上,在得到所有可能的稳定域边界后,使用Nyquist稳定性判据确定实际的稳定域.该方法适用于任意形式的单时滞对象,为实际应用中PID控制器的参数整定提供理论依据.     

Universal integral controllers for minimum-phase nonlinear systems

We consider a single-input-single-output (SISO) nonlinear system that has a well-defined normal form with asymptotically stable zero dynamics. Using only knowledge of the relative degree and the sign of the high-frequency gain, we design an output feedback integral controller that asymptotically regulates the output to a bounded time-varying reference signal with a constant limit. We give regional as well as semi-global results. We also show that, for relative degree one and two systems, the proposed integral controller reduces to the classical PI and PID controllers, respectively     

Tracking Performance Bound with Finite Control Energy and Erasure Channel Energy Constraint

This paper studies the tracking performance of the single‐input single‐output (SISO), finite dimensional, linear and time‐invariant (LTI) system under the channel input energy constraint over the Erasure channel. A new performance index is proposed which is minimized over all two‐degree‐of‐freedom stabilizing controllers. The explicit expressions of the lower bound of the performance index and the minimum of the signal‐to‐noise ratio are obtained. The results show that the performance bound is correlated to unstable poles, non‐minimum phase zeros and the packet loss probability. Finally, examples are given to validate the conclusions derived.     

Smith predictor based robust fractional order control: Application to water distribution in a main irrigation canal pool

This paper proposes a new methodology to design fractional integral controllers combined with Smith predictors, which are robust to high frequency model changes. In particular, special attention is paid to time delay changes. These controllers show also less sensitivity to high frequency measurement noise and disturbances than PI or PID controllers. This methodology is applied to design controllers for water distribution in a main irrigation canal pool. Simulated results of standard PI and PID controllers plus a Smith predictor, and the controller developed in this paper are compared when applied to the dynamical model of a real main irrigation canal pool showing that our controller exhibits better and more robust features than these. Moreover our controller is compared with other more complex control techniques as predictive control and robust H_∞ controllers, exhibiting better or similar performances than these.