On the selection of tuning methodology of FOPID controllers for the control of higher order processes

Abstract

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modelling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.

Introduction

Modelling of process plants for control analysis and design often give rise to higher order models in order to capture delicate dynamic behaviours of the process, with higher accuracy [1], [2], [3]. It has been shown by Saha et al. [4] that a nonlinear process dynamics under shift in operating point can be nicely captured using system identification techniques with several higher order process models and then a generalized varying gain model. Tuning of suitable controllers for these higher order processes are a bit challenging. It is well known that among various types of industrial controllers, PID dominates most of the process control applications due to its simple structure, easy tuning and robustness [5], [6]. In recent past, FOPID or $P{I}^{\lambda }{D}^{\mu }$ controllers have been proposed by Podlubny [7] which are capable of enhancing the closed loop performance of a system over a simple integer order PID structure [8], [9]. In fact, the true potential of a FOPID controller greatly depends on its tuning methodology and often the performance may degrade severely, with contradictory design specifications to be met by the FO controllers. The present work attempts to show the inherent advantages and limitations of different tuning strategies, while designing FOPID controllers for higher order processes with specified time/frequency response.

In [5], [6], it has been shown that a reduced order model is required for a higher order plant before its tuning with a PID controller using classical tuning rules. Classical model reduction techniques for PID tuning mostly used First Order Plus Time Delay (FOPTD) and Second Order Plus Time Delay (SOPTD) templates, which are enhanced in this paper, with the introduction of new templates known as Non-Integer Order Plus Time Delay (NIOPTD) having flexible order elements. This allows robust iso-damped tuning of FOPID controllers without compromising the accuracy of the reduced order models. In other words, with the introduction of NIOPTD templates, robustness of a FOPID controller can be increased by the significant reduction in modelling error. In conventional frequency domain model reduction the suboptimal approach, proposed by Xue et al. [10] by minimizing the $H_2$ norm between the reduced order and the higher order process are popular among research communities and is also capable of extracting the delays in a model which finds great scope of application especially in building reduced order process models.

For process control applications, FO controllers have been classified in four categories in [11] among which Podlubny’s $P{I}^{\lambda }{D}^{\mu }$ or FOPID [7] and Oustaloup’s CRONE controller [12] and its three generations [13], [14], [15] deserve special merit. Other FO controllers like the FO lead-lag compensator [16] and FO phase shaper [17], [18], [19], [4] are also becoming popular in recent robust process control applications. Several tuning strategies have been proposed by many contemporary researchers to tune FOPID controllers in both frequency domain and time domain. It has been found that the frequency domain design technique requires a reduced order template of the original higher order process. The time domain tuning techniques, on the other hand, do not necessarily require a reduced order model and hence the higher order process model is sufficient to find out the controller parameters by an optimization technique with some time domain performance indices as the design criteria. In present day, most of the industrial controllers are tuned with a few set of design specifications, either in time domain (e.g. error index, rise time, percentage of overshoot, settling time, overshoot–undershoot ratio etc.) or frequency domain (e.g. gain margin, phase margin, cross-over frequencies, maximum sensitivity and complementary sensitivity magnitudes etc.) [5], [6]. Hence, a single tuning methodology cannot satisfy all of the above design criteria i.e. simultaneously satisfying time and frequency domain performance specifications. Indeed, such contradictory design criteria may often give unsatisfactory, even unstable closed loop response due to over-specification. Thus, a FOPID controller, as argued above, satisfying few set of time domain specifications may not have sufficient robustness against system parameter uncertainties in frequency domain analysis and vice versa. Thus, it is clear, that every tuning strategy possesses its own inherent strength and weakness. The present work tries to focus on those characteristics of some well-established tuning approaches and their extensions for FOPID controllers in a comparative manner.

This paper also proposes a robust frequency domain tuning strategy FOPID controllers using highly accurate NIOPTD-II template for open loop stable, minimum/non-minimum phase higher order processes. The proposed technique uses a simultaneous nonlinear equation solving based robust tuning of FOPID controllers, which requires lesser computational load unlike a constrained nonlinear optimization used by the contemporary researchers for iso-damped tuning of FOPIDs. Further, it is seen that time domain optimal tuning method for FOPIDs, as in recent literature, do not always guarantee the closed loop stability of the process. An extension of FOPID tuning strategy is proposed in this paper to guarantee the closed loop stability and also to select the most suitable integral performance index for optimal time domain tuning.

The rest of the paper is organized as follows. Tuning methodologies for FOPID controllers, proposed by contemporary researchers are outlined in Section 2. In Section 3, the robust frequency domain tuning of FOPID controller is discussed along with the proposal for new model reduction templates and simulations with a test-bench of higher order processes. Section 4 discusses about the time domain optimal FOPID design by minimizing a chosen time domain integral performance index. In Section 5, the design performances of the robust and optimal FOPID controllers are compared from various perspectives of control system analysis. The paper ends with the contributions of the present work as the conclusion in Section 6, followed by the references.