A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

doi:10.1016/j.cnsns.2012.01.007

Communications in Nonlinear Science and Numerical Simulation

Volume 17, Issue 9, September 2012, Pages 3628-3642

https://doi.org/10.1016/j.cnsns.2012.01.007 Get rights and content

Abstract

A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI^λD^μ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI^λD^μ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as “M-curve”. This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller’s effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.

Highlights

► LQR based PID controller design is proposed with guaranteed dominant pole placement. ► Conformal mapping transforms the FOPID zeros from the complex w-plane to s-plane. ► Variation in order of zeros shows different root locus with fixed closed loop poles. ► Cost of control & controller effort get reduced in optimum fractional pole placement. ► Proposed “M-curve” reduces FOPID zeros as simple PID zeros for ease of realization.

Introduction

In most process control applications, dominant pole placement tuning is a popular technique with second order approximations for sluggish or oscillatory processes [1], [2]. It is shown in Wang et al. [3] that guaranteed dominant pole placement can be done with PID controllers if real part of the resulting non-dominant closed loop poles is at least 3–5 times larger than that of the dominant closed loop poles. He et al. [4] tried to tune PID controllers using LQR based technique taking pole placement into consideration while selecting the weighting matrices for the optimal quadratic regulator. The technique, proposed in [4] ensures pole placement with PID controllers while also minimizing the state deviations and controller effort for doing so [5]. The concept of LQR based PI/PID tuning was primarily derived for first order systems with PI controllers in [4]. Then it was extended for second order systems with PID controllers as well; in order to cancel the open loop system pole with one real controller zero [4]. This reduces the formulation to a simple PI controller design for handling first order systems. This approach fails to give satisfactory results especially for systems having highly oscillatory open loop dynamics i.e. complex conjugate open loop poles. The present methodology developed in this paper, primarily tries to focus on the unresolved issues addressed in [3], [4] e.g.

(a)
Condition for guaranteed dominant pole placement.
(b)
Extension of LQR based PID tuning for highly oscillatory and sluggish processes with the same pole placement technique.
(d)
Comparison of the control cost and also initial controller efforts from the regulator design point of view.

FOPID or PI^λD^μ controllers [6] recently have become popular in process control as it has two extra parameters for tuning i.e. the integro-differential orders; giving us extra freedom of tuning. Therefore, FOPID controller is expected to give better performance over conventional PID controllers [6], [7]. While conformal mapping for FOPID is done with s to w plane-transform, the transfer function of a FOPID controller [6], [7] places two fractional zeros and one fractional pole at the origin in the complex w-plane; with differ-integral fractional orders of the controller are equal. The solution of fractional order systems can run into multiple Riemann Sheets [7]. Let us assume that the domain of existence of the controller zeros and poles are limited to only the primary Riemann-sheet, so as to ensure that the closed loop system is not hyper-damped or ultra-damped which does not come into dominant dynamics [7], [8], [9], [10], [11]; then position of the two fractional order zeros in complex s-plane can always be replaced by two integer order zeros. This is because guaranteed pole placement is feasible with a PID controller anywhere in the negative real part of the s-plane. With this approach, our focus is to represent the approximated PID controller gains in terms of FOPID controller parameters which reduce the complicacy of FO controller realization; while also preserving the time response of the controlled system.

In this paper, it is observed that with the decrease in the order of a FOPID controller, the position of the zeros shifts towards greater damping regions and then move very fast towards smaller damping and ultimately goes towards instability. While the obtained trajectory of the controller zeros or the dominant closed loop poles takes a certain pattern and we termed as “M-curve”. The proposed methodology first assumes a LQR based PID controller with desired location of the dominant pole placement. Also, it is always feasible to place the dominant closed loop poles at the same location of the primary Riemann sheet with a FOPID controller. Now with variation in the FO controller orders, the equivalent integer order PID controller with fractional zeros in the same location gives different set of controller gains. Upon approximation with integer order PID, the controller does not remain optimal but is suboptimal. This conformal mapping based sub-optimal controller design possesses certain strengths over the LQR based optimal PID controller, which is illustrated and elucidated with credible and exhaustive numerical simulations.

The objective of this paper is to put forward a novel methodology that tunes a PID controller with an LQR based dominant pole placement method at a lower damping than the desired one in the first stage; and then considering FOPID controller zeros at the same location for pole-placement. Thereafter the order of FO controller is decreased so as to obtain the approximated integer order suboptimal PID gains that forces the closed loop poles to move towards greater (desired) damping. Simulation results are given to justify that the proposed two-stage tuning of a PID controller significantly reduces the control signal vis-à-vis a single stage LQR based PID controller to achieve the same desired closed loop damping that is percentage overshoot.

The rest of the paper is organized as follows. Section 2 proposes LQR based guaranteed dominant pole placement of integer order PID controller with highly oscillatory processes too [12] and discusses about the inverse optimal control costs involved in pole placement problem. Section 3 introduces a new fractional order approach of PID controller tuning. Simulation studies are carried out in Section 4 for three different class of second order processes to show the effect of variation in orders of FO controller and its effect on the integer order approximation of the fractional order controller (zeros and dominant closed loop poles). The advantage of two-stage sub-optimal tuning of PID controllers over the conventional LQR based optimal tuning is dealt in Section 5. The paper ends with the conclusion as Section 6, followed by the references.

Section snippets

Criteria for guaranteed dominant pole placement

In this section, a brief idea is presented regarding the accuracy of guaranteed pole placement with PID controllers. Let us consider that a second order process G_p (with sluggish “S” shaped or oscillatory open loop dynamics) needs to be controlled with an integer order PID controller C(s) of the form (1). $C (s) = K_{p} + \frac{K_{i}}{s} + K_{d} s = \frac{K_{d} s^{2} + K_{p} s + K_{i}}{s}$ Here, the process is characterized by the open loop transfer function $G_{p} (s) = \frac{K}{s^{2} + 2 ζ^{ol} ω_{n}^{ol} s + {(ω_{n}^{ol})}^{2}}$ Then, the closed loop transfer function becomes $G_{cl} = \frac{G_{p} C}{1 + G_{p} C} = K (K_{d} s^{2} + K_{p} s + K_{i})$

Fractional zero placement approach for FOPID controllers and conformal mapping based approximation in complex s ↔ w plane

Let us consider a FOPID or PI^λD^μ controller (37) with integral order (λ) and the derivative order (μ) set to the same value (q), i.e. λ = μ = q similar to that in [16], [17]. Therefore, the transfer function of the FOPID controller becomes $\tilde{C} (s) = K_{p} + \frac{K_{i}}{s^{λ}} + K_{d} s^{μ} = K_{p} + \frac{K_{i}}{s^{q}} + K_{d} s^{q} = \frac{K_{d} s^{2 q} + K_{p} s^{q} + K_{i}}{s^{q}}$ First we shall transform the controller (37) to w-plane, by putting variable transformation as s^q = w. This is called conformal mapping [7], and the transfer function is now in w-plane. The controller is transformed as;

Effect of variation in controller’s differ-integral orders on the closed loop control performance

The closed loop performance of a well tuned FOPID controller gets heavily affected by slight change in its differ-integral orders [7]. The variation in the order of the FOPID controller (37) in the pre defined way leads to an advanced tuning strategy for the approximated PID controller with gains (50) which is described in the next section for three different classes of second order systems.

Two stage suboptimal tuning and its advantages over conventional LQR based tuning

In order to show the effectiveness of the proposed two stage sub-optimal PID controller tuning methodology, the basic design steps are given as follows.

Step 1:
For a given second order system of the form (2) with specified ζ^ol and $ω_{n}^{ol}$ , calculate the PID controller gains via pole placement for a low ζ^cl using Eq. (5).
Step 2:
Location of the PID controller zeros can be intuitively modified using the approximated fractional order PID controller approach (50) by gradually decreasing the fractional order q, so as

Conclusion

A novel two stage sub-optimal PID controller tuning methodology is proposed in this paper, with the help of fractional order pole-zero placement approach. The methodology uses an LQR based optimal PID tuning approach with dominant pole placement in the first stage. Then considering the same closed loop pole location, achieved with a FOPID, the integro-differential orders of the FO controller is decreased in the second level of tuning to achieve the desired damping for the approximated integer

Acknowledgements

This work has been supported by the Board of Research in Nuclear Sciences (BRNS) of the Department of Atomic Energy, Govt. of India, sanction No. 2006/34/34-BRNS dated March 2007.

References (26)

Karl J. Astrom et al.
PID controller: theory, design, and tuning
(1995)
Yu Zhang et al.
Dominant pole placement for multi-loop control systems
Automatica
(2002)
Qing-Guo Wang et al.
Guaranteed dominant pole placement with PID controllers
J Process Control
(2009)
Jian-Bo He et al.
PI/PID controller tuning via LQR approach
Chem Eng Sci
(2000)
Brian D.O. Anderson et al.
Optimal control: linear quadratic methods
(1989)
Igor Podlubny
Fractional-order systems and PI^λD^μ-controllers
IEEE Trans Autom Control
(1999)
Shantanu Das
Functional fractional calculus
(2011)
Tom T. Hartley et al.
Dynamics and control of initialized fractional-order systems
Nonlinear Dyn
(2002)
Vinagre BM, Monje CA, Calderon AJ. Fractional order systems and fractional order control actions, Lecture 3 IEEE CDC...
Farshad Merrikh-Bayat et al.
Extension of the root-locus method to a certain class of fractional-order systems
ISA Trans
(2009)

F. Merrikh-Bayat et al.

Method for designing PI^λD^μ stabilizers for minimum-phase fractional-order systems

IET Control Theor Appl

(2010)

G.M. Malwatkar et al.

Tuning PID controllers for higher-order oscillatory systems with improved performance

ISA Trans

(2009)

Rames C. Panda et al.

PID tuning rules for SOPDT systems: review and some new results

ISA Trans

(2004)

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A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

Abstract

Highlights

Introduction

Section snippets

Criteria for guaranteed dominant pole placement

Fractional zero placement approach for FOPID controllers and conformal mapping based approximation in complex s ↔ w plane

Effect of variation in controller’s differ-integral orders on the closed loop control performance

Two stage suboptimal tuning and its advantages over conventional LQR based tuning

Conclusion

Acknowledgements

PID controller: theory, design, and tuning

Dominant pole placement for multi-loop control systems

Automatica

Guaranteed dominant pole placement with PID controllers

J Process Control

PI/PID controller tuning via LQR approach

Chem Eng Sci

Optimal control: linear quadratic methods

Fractional-order systems and PIλDμ-controllers

IEEE Trans Autom Control

Functional fractional calculus

Dynamics and control of initialized fractional-order systems

Nonlinear Dyn

Extension of the root-locus method to a certain class of fractional-order systems

ISA Trans

Method for designing PIλDμ stabilizers for minimum-phase fractional-order systems

IET Control Theor Appl

Tuning PID controllers for higher-order oscillatory systems with improved performance

ISA Trans

PID tuning rules for SOPDT systems: review and some new results

ISA Trans

Fractional-order systems and PI^λD^μ-controllers

Method for designing PI^λD^μ stabilizers for minimum-phase fractional-order systems