**2. PID-like controller structures for USOPDT processes**

The three main feedback configurations applied in the extant literature in order to control unstable processes with time delay are depicted in Fig. 1 (see Jacob & Chidambaram, 1996; Park et al, 1998, Paraskevopoulos et al, 2004). As it can easily verified, the loop transfer functions obtained by these control schemes are identical, provided that the following relations hold

$$
\overline{K}'\_{\mathbb{C}} = \overline{K}\_{\mathbb{C}} \left(\overline{\tau}\_{I} + \overline{\tau}\_{D}\right) / \ \overline{\tau}\_{I} = \left(1 + \overline{k}\_{c}\right) \overline{k}\_{c,i} = \overline{K}\_{P}
$$

$$
\overline{\tau}'\_{I} = \overline{\tau}\_{I} + \overline{\tau}\_{D} = \overline{\tau}\_{i} (1 + \overline{k}\_{c}) / \ \overline{k}\_{c} = \overline{K}\_{P} / \ \overline{K}\_{I} \tag{1}
$$

$$
\overline{\tau}'\_{D} = \overline{\tau}\_{D} \overline{\tau}\_{I} / \left(\overline{\tau}\_{I} + \overline{\tau}\_{D}\right) = \overline{\tau}\_{d} \overline{k}\_{c} / \left(1 + \overline{k}\_{c}\right) = \overline{K}\_{D} / \ \overline{K}\_{P}
$$

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

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

(stable time constant) are significant.

efficiency of the proposed tuning methods.

relations hold

**2. PID-like controller structures for USOPDT processes** 

 

 

The three main feedback configurations applied in the extant literature in order to control unstable processes with time delay are depicted in Fig. 1 (see Jacob & Chidambaram, 1996; Park et al, 1998, Paraskevopoulos et al, 2004). As it can easily verified, the loop transfer functions obtained by these control schemes are identical, provided that the following

> , ( )/ (1 ) *K K C C I D I c ci P*

*kk K*

(1 )/ / *k kKK* (1)

 

 *D DI I D dc c D P* /( ) /(1 ) / *k k KK*

 *IIDi c c P I*

$$\begin{aligned} \mathbf{G}\_{F,P-PID}(\mathbf{s}) &= \mathbf{G}\_{F,PID} \frac{\left(\overline{\tau}\_{l}\mathbf{s} + \mathbf{1}\right)\left(\overline{\tau}\_{D}\mathbf{s} + \mathbf{1}\right)}{\overline{\tau}\_{i}\overline{\tau}\_{d}\mathbf{s}^{2} + \overline{\tau}\_{i}\mathbf{s} + \mathbf{1}} \\\\ \mathbf{G}\_{F,PDF}(\mathbf{s}) &= \mathbf{G}\_{F,PID}\left(\overline{\tau}\_{l}\mathbf{s} + \mathbf{1}\right)\left(\overline{\tau}\_{D}\mathbf{s} + \mathbf{1}\right) \end{aligned} \tag{2}$$

where *KC* , *<sup>I</sup>* and *D* are the three controller parameters of the conventional PID 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 alternative control schemes are identical.

Fig. 1. Equivalent three-term controller schemes with appropriate pre-filters: (a) The series PID controller, (b) The PDF (or I-PD) controller, and (c) The P-PID controller.

PID-Like Controller Tuning for Second-Order Unstable Dead-Time Processes 55

It is not difficult to recognize that the Nyquist plot of the *GL(ŝ)* has tow crossover points with the real axis, which determine the critical (or crossover) frequencies *wmin* and *wmax*, and the critical gains *KC,min=1/AL(wmin)* and *KC,max=1/AL(wmax).* These crossover frequencies are

when the values of the *atan* function are assigned in the range *(-π/2, π/2)*. Having computed *wmin* and *wmax*, one can determine the acceptable values for the controller gain *KC*, for which the closed-loop system is stable. In particular *KC,min<KC<KC,max*, where, with subscript "*M*"

> , 2 2 1 1

We next define the increasing gain margin *GMinc*, the decreasing gain margin *GMdec* and the

Obviously for the closed loop system to be stable *GMinc* and *GMdec* should be grater than one. Note that, the largest the values of *GMprod*, the more robust the system becomes with respect to the gain uncertainty, if the controller gain *KC* is appropriately selected. Furthermore, the phase margin of the closed loop system is defined by *PM=φL(wG)+π*, where *wG* is the frequency at which *AL(wG)=1*. From (7), one can easily conclude that *wG* is given by the

22 6 22 2 222 4 2 2 2 2 2 2 0

In order to obtain the maximum phase margin for given *d*, *τS*, *τI* and *τD*, the controller gain

1 1

*Ip p Sp*

*ww w*

1 1

where *wp* is the frequency at which the argument of the loop transfer function is maximized.

11 1 1

 

From (6), one can easily conclude that *wp* is given by the solution of *dφL/dω*

 

 

2 22 22 22 <sup>1</sup> <sup>0</sup>

 

*p p Dp S p*

*wwww*

positive real root. Substituting *wp* in (6), the respective maximum argument *φL(wp)* is

*Ip Dp*

*w w*

 

*D S*

<sup>2</sup> <sup>2</sup>

 

2 2

 

 *SG S C D G C D G C K KK* (12)

*IM M SM*

*ww w*

1 1

 

*C M*

*K*

gain margin product of the closed-loop system as follows

*-π/2-dwC+atan(wC)+atan(τΙwC)+atan(τDwC)-atan(τSwC)=0* (8)

*GMinc=KC,max/KC , GMdec=KC/KC,min* (10)

*GMprod= GMincGMdec=KC,max/KC,min* (11)

(9)

(13)

(14)

*<sup>2</sup>*, with only one acceptable

*w=wp =0*, or

2 2

 

 

*w w*

*IM DM*

obtained as the solutions of the equation *φL(wC)=-π*, or equivalently, of the equation

used for either "min" or "max"

maximum real root of the equation

*KC* should be selected as

equivalently of the equation

calculated.

 

*d*

 

 

*C*

that results in a fourth order linear equation with respect to *wp*

*K*

 

 


Table 1. Normalized vs. original system parameters.

In the sequel, our focus of interest is the design of PID-like controllers when applied to control USOPDT process, with the following transfer function model

$$G\_P(\mathbf{s}) = \frac{\overline{K} \exp\left(-\overline{d}\mathbf{s}\right)}{(\overline{\tau}\_{\rm S}\mathbf{s} + 1)(\overline{\tau}\_{\rm U}\mathbf{s} - 1)}\tag{3}$$

where *K* , *d* , *<sup>S</sup>* and *<sup>U</sup>* are the process gain, the time delay and the stable and unstable time constants, respectively. In order to simplify the analysis and in order to facilitate comparisons, all system and controller parameters are normalized with respect to *<sup>U</sup>* and

*K* . Thus, the original process and controller parameters are replaced with the dimensionless parameters shown in Table 1.

Observe now that, the loop transfer function of an USOPDT system in connection with a PID-like controller, is given by

$$G\_L(\hat{s}) = \frac{K\_\mathbb{C} \left(\tau\_I \hat{s} + 1\right) \left(\tau\_D \hat{s} + 1\right) \exp\left(-d\hat{s}\right)}{\tau\_I \hat{s} \left(\tau\_S \hat{s} + 1\right) \left(\hat{s} - 1\right)}\tag{4}$$

while, using the pre-filter *GF, PID(ŝ)=(τΙŝ+1)-1*, the closed-loop transfer function becomes

$$G\_{\rm CL}(\hat{s}) = \frac{K\_{\rm C}(\tau\_{\rm D}\hat{s} + 1)\exp(-d\hat{s})}{\tau\_{I}\hat{s}(\tau\_{\rm S}\hat{s} + 1)(\hat{s} - 1) + K\_{\rm C}(\tau\_{I}\hat{s} + 1)(\tau\_{\rm D}\hat{s} + 1)\exp(-d\hat{s})}\tag{5}$$

Relations (2) and (5) are next elaborated for the derivation of the tuning methods proposed in this work.
