Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

Abstract

In this paper, a robust fractional-order PID (FOPID) controller design method for fractional-order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D-decomposition method. Then, the optimal fractional orders and of FOPID controller are achieved at the biggest three-dimensional PSR, which means the best robustness. Given the optimal and , the other FOPID controller parameters k p , k i , k d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI).The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm. KEYWORDSdelay system, FOPID control, fractional-order, positive stability region (PSR), robust analysis INTRODUCTIONFractional calculus is an extension of traditional calculus. The integral and differential orders in fractional calculus are not limited to integers, they can be any arbitrary real number or even complex number. 1 The concepts and mathematical analysis methods of fractional calculus are more complex than traditional calculus. Strictly speaking, most of the physical phenomena and processes can be described more precisely by fractional-order models. 2,3 Early fractional calculus related research works concentrated in the mathematics fields. With the rapid development of science and technology, the fractional calculus related engineering studies have also shown their great potential, for example, bioengineering, 4 signal process, 5 control engineering, 6-11 and applied physics. 12 Among these fractional calculus related fields, the development of fractional-order control technologies is particularly fast. Fractional-order control algorithm has more tuning knobs compared with conventional control algorithm. Therefore, it provides more probabilities in improving system robustness, transient control performance, and disturbance suppression ability. One of the most remarkable fractional-order control methodologies is fractional-order PID (FOPID or PI D ) control, which was proposed by Podlubny. 13 The PID controller has been overwhelmingly applied and was reported to be applied in more than 90% industrial control loops. 14 The FOPID controller is an extension of traditional PID controller with extra two parameters, ie, integral order and differential order. Hence, extra flexibility in meeting different Int J Robust Nonlinear Control. 2019;29:5195-5212.wileyonlinelibrary.com/journal/rnc

General type industrial temperature system control based on fuzzy fractional-order PID controller

“…The Fractional-order PID (FOPID or PI λ D μ ) type controller is an extension of PID controller which has two extra tuning parameters, namely integral order λ and differential order μ [6,7]. It also has the similar advantages with the PID controller, such as a simple structure.…”

Robust observer‐based‐controller design for uncertain fractional‐order time‐varying‐delay systems

“…The problem of stability analysis, control and observer designs for this kind of systems have attracted the attention of many researchers. Recently, fractional‐order systems (FOS) with time‐varying‐delays or constant delays have been subject to some works and studies (see for example References 9‐17).…”

“…On the other hand, the main goal of the feedback controller design is to stabilize unstable systems, 11,18‐23 or to improve the stability in the presence of transient phenomenon which does not disappear fastly. From a practical point of view, the knowledge of all the pseudo‐state vector is not possible in all cases in the practice for different reasons (technical, financial, ).…”

Design, Implementation, and Validation of Robust Fractional-Order PD Controller for Wheeled Mobile Robot Trajectory Tracking

“…e CRONE controller which can compensate the disturbance caused by parameter uncertainties and load changes has been successfully used in vehicle suspension [26] and path tracking [27]. A series of robust FOPID controllers tuned based on the flat phase property have been used in motion control [28,29]. Moreover, in [30], an FO controller is used in the trajectory tracking task of a differential drive WMR.…”