Tomislav B. Šekara

Optimization of PID Controller Based on Maximization of the Proportional Gain Under Constraints on Robustness and Sensitivity to Measurement Noise

This technical note presents a new, simple and effective, four-parameters proportional-integral-derivative (PID) optimization method. The set of adjustable parameters is defined by the proportional gain k, integral gain ki, damping ratio of the controller zeros (DRCZ), and desired value of the sensitivity to measurement noise Mn. Given Mn and desired value of the maximum sensitivity Ms, for both maximization of k and maximization of ki, only three nonlinear algebraic equations need to be solved for a few values of DRCZ. Contrary to the method based on maximization of ki, in the method based on maximization of k the improvement of performance is obtained by decreasing DRCZ from 1 to the value corresponding to the minimum of the integrated absolute error (IAE). Moreover, this is achieved without deteriorating robustness to the model uncertainties, for a large class of stable processes. Compared to the recently proposed PID optimization methods, for the same Ms and Mn, lower values of IAE and M p are obtained by using the method presented here.

Optimal Reactive Compensators in Power Systems Under Asymmetrical and Nonsinusoidal Conditions

Based on the condition for the minimum rms value of line currents, a method for improving the power factor in power systems under asymmetrical and nonsinusoidal conditions is presented. This paper offers straightforward and convenient formulations of active currents for the three-phase four-wire, three-phase three-wire, and single-phase systems. Owing to the transformation matrix introduced in this paper, the formulation of active currents is valid regardless of whether the voltages and currents are balanced or unbalanced and it permits any point of the system to be chosen for the voltage reference. Observing the constraints related to the reactive compensators, the line current minimization procedure is used for the determination of optimal compensating capacitances for the three-phase four-wire, three-phase three-wire, and single-phase systems. The optimal reactive compensators based on up to six capacitors are proposed. The definitions of power factors before and after compensation are introduced. The influence analysis of nonactive power compensation on harmonic distortion of line currents has been also performed. Experimental results are obtained to confirm the validity and applicability of the proposed compensation procedure.

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.

PID Controller Tuning Based on the Classification of Stable, Integrating and Unstable Processes in a Parameter Plane

Classification of processes and tuning of the PID controllers is initiated by Ziegler and Nichols (1942). This methodology, proposed seventy years ago, is still actual and inspirational. Process dynamics characterization is defined in both the time and frequency domains by the two parameters. In the time domain, these parameters are the velocity gain Kv and dead-time L of an Integrator Plus Dead-Time (IPDT) model GZN(s)=Kvexp(-Ls)/s, defined by the reaction curve obtained from the open-loop step response of a process. In the frequency domain these parameters are the ultimate gain ku and ultimate frequency ωu, obtained from oscillations of the process in the loop with the proportional controller k=ku. The relationship between parameters in the time and frequency domains is determined by Ziegler and Nichols as

Optimization of distributed order fractional PID controller under constraints on robustness and sensitivity to measurement noise

This paper describes a novel approach towards optimal tuning of distributed order fractional PID controller parameters. A distributed order fractional PID controller (DPID) is approximated by a compound fractional controller with multiple fractional differintegrators connected in parallel. Orders of these differintegrators have been equally spaced, with the first one being the classical integrator of order 1, and the last one being the classical differentiator of order 1. A classical noise cancellation filter is considered as a part of controllers structure. The controller parameters, being the gains of all differintegrators, have been tuned. Proposed tuning procedure maximizes gain of classical integrator term under constraints on Maximum sensitivity, Ms, Maximum sensitivity to measurement noise, Mn and Maximum complementary sensitivity, Mp. Results are presented via a number of numerical simulations.

Fractional order PD control of Furuta pendulum: D-decomposition approach

This paper deals with stability problem of inverted pendulum controlled by a fractional order PD controller. D-decomposition method for determining stability region in controller parameters space is hereby presented. The D-decomposition problem for linear systems is extended for linear fractional systems and for the case of nonlinear parameters dependence. Some comparisons of fractional and integer order PID controllers are given based on simulation results.

Some electromechanical systems and analogies of mem-systems integer and fractional order

This paper will provide some an applications of memristors and mem-systems with a particular focus on electromechanical systems and analogies that holds great promise for advanced modeling and control of complex objects and processes. Also, we present the connection between fractional order differ integral operators and behavior of the mem-systems. Finally, several potential applications of electromechanical analogies of integer and fractional order are discussed.

Stabilization of Inverted Pendulum by Fractional Order PD Controller with Experimental Validation: D-decomposition Approach

This paper deals with the stability problem of two types of inverted pendulum controlled by a fractional order PD controller. Rotational inverted pendulum and cart inverted pendulum are under-actuated mechanical systems with two degrees of freedom and one control input. Detailed mathematical models of both pendulums are derived using the Rodriguez method. Fractional order PD controller is introduced for inverted pendulum stabilization. Stability regions in control parameters space are calculated using the D-decomposition approach, based on which tuning of the fractional order controller can be carried out. Numerical simulations and experimental realization are given to demonstrate the effectiveness of the proposed method.

A fast closed-loop process dynamics characterization.

Stable, integrating and unstable processes, including dead-time, are analyzed in the loop with a known PI/PID controller. The ultimate gain and frequency of an unknown process G(p)(s), and the angle of tangent to the Nyquist curve G(p)(iω) at the ultimate frequency, are determined from the estimated Laplace transform of the set-point step response of amplitude r0. Gain G(p)(0) is determined from the measurements of the control variable and known r0. These estimates define a control relevant model G(m)(s), making possible the use of the previously determined and memorized look-up tables to obtain PID controller guaranteeing desired maximum sensitivity and desired sensitivity to measurement noise. Simulation and experimental results, from a laboratory thermal plant, are used to demonstrate the effectiveness and merits of the proposed method.

A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas

A novel class of quasi-polynomials orthogonal with respect to the fractional integration operator has been developed in this paper. The related Gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed. By allowing the commensurate order of quasi-polynomials to vary independently of the integration order, a family of fractional quadrature formulas has been developed for each fractional integration order, including novel quadrature formulas for numerical approximation of classical, integer order integrals. A distinct feature of the proposed quadratures is high computational efficiency and flexibility, as will be demonstrated in the paper. As auxiliary results, the paper also presents methods for Lagrangian and Hermitean quasi-polynomial interpolation and Hermitean fractional quadratures. The development is illustrated by numerical examples.