**1. Introduction**

Several processes encountered in various fields of engineering exhibit an inherently unstable behaviour coupled with time delays. To approximate the open loop dynamics of such systems for the purpose of designing controllers, many of these processes can be satisfactorily described by unstable transfer function models. The most widely used models of this type is the unstable first order plus dead-time (UFOPDT) and the unstable second order plus dead-time (USOPDT) transfer function models, which take into account dead times that might appear in the model, due to measurement delay or due to the approximation of higher order dynamics of the process, by a simple transfer function model.

Research on tuning methods of two or three-term controllers for unstable dead-time processes has been very active in the last 20 years. The most widely used feedback schemes for the control of such processes are the Proportional-Integral-Differential (PID) controller with set-point filter (Jung et al, 1999; Lee et al, 2000), the Pseudo-Derivative Feedback (PDF) or I-PD controller (Paraskevopoulos et al, 2004), and the Proportional plus Proportional– Integral–Derivative (P-PID) controller (Jacob & Chidambaram, 1996; Park et al, 1998). These control schemes are identical in practice, provided that the parameters of the controllers and of the pre-filters needed in some cases are selected appropriately. Controller tuning for unstable dead-time processes has been performed according to several methods, the most popular of them being various modifications of the Ziegler-Nichols method (De Paor & O' Malley, 1989; Venkatashankar & Chidambaram, 1994; Ho & Xu, 1998), several variations of the direct synthesis tuning method (Jung et al, 1999; Prashanti & Chidambaram, 2000; Paraskevopoulos et al, 2004; Padma Sree & Chidambaram, 2004), the ultimate cycle method (Poulin & Pomerleau, 1997), the pole placement method (Clement & Chidambaram, 1997), the method based on the minimization of various integral criteria, the Internal Model Control (IMC) tuning method (Rotstein & Lewin, 1991; Lee et al, 2000; Yang et al, 2002; Tan et al, 2003), the optimization method (Jhunjhunwala & Chidambaram, 2001; Visioli, 2001), the two degrees of freedom method (Huang & Chen, 1997; Liu et al, 2005; Shamsuzzoha et al, 2007), etc. (see the work (O'Dwyer, 2009), and the references cited therein). Moreover, due to the wide practical acceptance of the gain and phase margins (GPM) in characterizing system robustness, some tuning methods for unstable dead-time models, based on the satisfaction of GPM specifications, have also been reported (Ho & Xu, 1998; Fung et al, 1998;

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 53

1 1 ( ) <sup>1</sup> *I D*

, , 2

*G sG s s F PDF F PID I D* , , ( ) 

controller in its parallel form. In the case of the series PID controller, the pre-filter *GF,PID* is used in order to cancel out all or some of the zeros introduced by the controller and to smoothen the set-point step response of the closed loop system. The pre-filters *GF,P-PID* and *GF,PDF* are the equivalent pre-filters of the corresponding control schemes. Note that, the pre-filter *GF,PDF* can be used only when the reference input is a known and differentiable signal. Therefore, is seldom used in real practice. From Fig. 1, one can easily recognize that in the case of regulatory control the three control schemes are identical when the controller parameters are chosen as suggested by (1), even if there are no pre-filters used. Moreover, one can also see that the stability properties of the closed loop system are not affected, in any case, by the respective pre-filter used, which is applied here, only to filter the set point and to prevent excessive overshoot in closed-loop responses to set-point changes, which are common in the case of unstable time-delay systems (Jacob & Chidambaram, 1996). Thus, the loop transfer functions obtained for the above three

Fig. 1. Equivalent three-term controller schemes with appropriate pre-filters: (a) The series

D P sK K

**PDF-Controller** 

**+** \_ +

+\_ <sup>s</sup>

PID controller, (b) The PDF (or I-PD) controller, and (c) The P-PID controller.

**G**F,PID**(s) +** \_ +

i <sup>1</sup> 1 sk <sup>s</sup> 

c,i GF,P-PID(s) k

series form Pre-Filter

**P-PID Controller** Pre-Filter

KI **R(s)**

Pre-Filter

GF,PDF(s)

**PID-Controller** 

I D

I ( s 1)( 1) <sup>K</sup> <sup>s</sup> 

**R(s) <sup>+</sup> GP(s)**

**+**\_ +

d c

*F P PID F PID*

*G sG*

where *KC*

 , *<sup>I</sup>* and

*D*

alternative control schemes are identical.

 

*s s*

 1 1

**E(s) Y(s)**

C

**E(s) Y(s)**

**E(s) Y(s)**

+\_

**L(s)**

**+**

are the three controller parameters of the conventional PID

**GP(s)**

**U(s)**

**L(s)**

**<sup>+</sup> GP(s) U(s)**

 

(2)

**(a)**

**(b)**

**(c)**

*id i s s*

Wang & Cai, 2002; Lee & Teng, 2002; Paraskevopoulos et al, 2006). The vast majority of the tuning methods mentioned above refer to the design of controllers for UFOPDT models and less attention has been devoted to USOPDT models (Lee et al, 2000; Rao & Chidambaram, 2006). Usually these models are further simplified to second order ones without delay, or to UFOPDT models, in order to design controllers for this type of processes. However, this simplification is not possible when the time delay of the system and/or the stable dynamics (stable time constant) are significant.

The aim of this work is to present a variety of innovative tuning rules for designing PIDlike controllers for USOPDT processes. These tuning rules are obtained by imposing various specifications on the closed-loop system, such as the appropriate assignment of its dominant poles, the satisfaction of several time response criteria (like the fastest settling time and the minimization of the integral of squared error), as well as the simultaneous satisfaction of stability margins specifications. In particular, the development of the proposed tuning methods relying on the assignment of dominant poles as well as on time response criteria is performed on the basis of the fact that (under appropriate selection of the derivative term), the delayed open loop response of a 3rd order system, with poles equal to the three dominant poles of the closed loop system, is identical to the closed loop step response of an USOPDT system. Simple numerical algorithms are, then, used to obtain the solution of the tuning problem. To reduce the computational effort and to obtain the controller settings in terms of the process parameters (a fact that permits online tuning), the obtained solution is further approximated by analytical functions of these parameters. Moreover, in the case of the method that relies on the satisfaction of stability margin specifications, the controller parameters are obtained using iterative algorithms, whose solutions, in a particular case, are further approximated quite accurately by analytic functions of the process parameters. The obtained approximate solutions have been obtained using appropriate curve-fitting optimization techniques. Furthermore, the admissible values of the stability robustness specifications for a particular process are also given in analytic forms. Finally, the tuning rules proposed in this work, are applied to the control of an experimental magnetic levitation system that exhibits highly nonlinear unstable behaviour. The experimental results obtained clearly illustrate the practical efficiency of the proposed tuning methods.
