Analytical Design of PI Controllers for First Order plus Time Delay Systems

This study proposes an analytical design method of a Proportional Integral controller for the stability and performance of first order plus time delay systems. The method proposed in the study achieves general computation equations for such systems. Inspired from Bode’s ideal loop, gain crossover frequency and phase margin specifications are considered for the system. Then, these specifications are used to obtain the parameters of the proportional integral controller. Analytically derived formulas by the proposed method are tested with existing plants in the literature and the results are illustrated graphically. It is shown that the tuning method satisfies desired gain crossover frequency and phase margin specifications.


Introduction
A large number of industrial processes are approximately modeled by first order plus time delay (FOPTD) transfer functions.FOPTD models are widely used in electronics [1], automation [2], thermal [3] and chemical processes [4] etc. Thus, control of these plants is a challenging area of research.From this idea, researchers are motivated on better design methods or alternative controller ideas [5][6][7].
Proportional Integral Derivative (PID) controllers are one of the leading industrial controllers for decades.There can be found numerous studies related to control of FOPTD plants by using PID controllers in the literature [8][9][10].Formed with the deficiency of the derivative operator, Proportional Integral (PI) controllers are also utilized in so many processes.For instance, Onat et al. presented a PI tuning method for FOPTD plants in [11].A comparative study of cascaded PI-PD controllers applied on a coupled tank system can be found in [12].Miao et al. optimized PI parameters for different performance criterias in [13].
This paper proposes a method to tune PI controllers for the stability and performance of FOPTD plants.Phase crossover frequency and phase margin specifications based on Bode's ideal transfer function are determined and parameters of the controller are analytically derived considering these specifications.The method gives generalized parameters of the PI controller for FOPTD plants.Efficiency of the proposed equations are tested with existing plants in the literature and the results are illustratively given.
Organization of this paper is as follows.Section 2 gives remindful information about PI controllers and FOPTD plans.Section 3 presents the computation process of the PI controller.Illustrative examples clarify the process in section 4 and section 5 has the concluding remarks.

PI Controllers and FOPTD Plants
This section presents the transfer functions of a PI controller and a FOPTD plant.For instance, general components of a Bode diagram is also briefly reminded.Following equation denotes the general representation of a FOPTD plant.
where, K is the gain, T is the time constant and L is the delay.Similarly, transfer function of a PI controller is given as follows. () Thus, the system can be shown with the following equation.
Figure 1 shows the closed loop scheme of the system implemeted in this paper.where, () Rs is the input signal and () Ys is the output signal.() Ps is the transfer function of the FOPTD plant in Eq. 1 and () Cs is the PI controller in Eq. 2. Figure 2 shows an instance Bode diagram of an open loop system.It would be useful to describe the components of a Bode diagram.The frequency value that the gain curve crosses the 0dB line is called as the gain crossover frequency and denoted as c  is this paper.Difference of the phase value with the -180 degrees line at the gain crossover frequency is the phase margin and denoted as m  .Phase margin shows the tolerance of the open loop system phase to ensure the stability of the closed loop system.Now, the desired gain and phase specifications can be given.

Design Specifications of PI Controller for FOPTD Plant
In order to analyse the system in the frequency domain, substitution of Then, frequency response of the FOPTD plant in Eq.1 can be written as, Likewise, frequency response of the PI controller is, () Magnitude and phase of the FOPTD plant are obtain in the following way.

()
1 ( ) Similarly, magnitude and phase of the PI controller are, () Therefore, magnitude and phase of the system can be written as follows. ( Assuming that the gain crossover frequency is c  and the phase margin is m  , following gain and phase specifications are desired to be satisfied. () Considering Eq. 11 and Eq. 13, gain specification of the system can be rewritten as, ) Similarly, considering Eq. 12 and Eq. 14 phase margin specification of the system is, ( ) Together solution of Eq. 15 and Eq.16 leads to the following parameters of the PI controller.
( ) Parameters of the PI controller in Eq. 17 and Eq.18 can be used to obtain the PI controller to satisfy given gain and phase margin specifications in this paper.It would be clarifying to explain the results on illustrative examples.

Illustrative Examples
This section gives two examples to clarify the given procedure.
Example 1: Consider the following FOPTD plant provided from [14].Replacing the unknown variables in Eq. 17 and Eq. 18, following PI controller is obtained. .Thus, the proposed method is successfully implemented.

Figure 3. Bode diagram of the system
We can also check the stability of the system with the step response of the closed loop given in Figure 4.The method can also be tested by varying phase margin values.  1 are given in Figure 5.    Similarly, stability of the systems with 1 () Ps and the 7 controllers can be checked with the step responses in Figure 6.From this example, efficiency of the proposed method is clearly shown.It would be advantageous to apply the proposed method in another example.
Example 2: Consider the following FOPTD plant [11].with increment steps of 5 o .Table 2 shows the parameters of the PI controller obtained for this case.Ps and the 7 controllers in Table 2.
Thus, the method is proved illustratively.

Conclusion
This paper proposes a design scheme of proportional integral controllers for performance and stability of first order plus time delay plants.Gain and phase specifications for the system are inspired from Bode's ideal loop.The method analytically obtains general computation equations for the mentioned systems.Illustrative examples are given with two existing plants from the literature and the results are shown graphically.It is investigated that proposed method is successfully implemented.

Figure 1 .
Figure 1.Block diagram of the closed loop system.
G s C s P s = is illustrated in Figure3.It is clearly seen in the figure that the phase crossover frequency is tuned to be 10

Figure 4 .
Figure 4. Step response of the closed loop system in Example 1.

Figure 5 .
Figure 5. Bode diagrams of the systems with1 () Ps and the 7 controllers listed in Table1.

Figure 6 .
Figure 6.Step responses of the systems with

Figure 7 .
Figure 7. Bode diagrams of the system with 2 () Ps and the 7 controllers in Table 2.

Figure 8 .
Figure 8. Step responses of the systems with 2 ()Ps and the 7 controllers in Table2.

Table 1 . Parameters of the PI controller found for
Table 1 lists the parameters of the PI controller found for [30 , 60 ] Ps and the 7 controllers listed in Table