Analytical Design of PI Controller for First Order Transfer Function Plus Time Delay: Stability Triangle Approach

In this study, proportional-integral (PI) controller design with a geometric approach for first order time-delayed systems is presented. This method can be expressed as an improved version of the weighted geometric center method. The method is based on calculating the center of gravity of a stability triangle selected inside the stability boundary locus (SBL). Stability boundary is determined along the <inline-formula> <tex-math notation="LaTeX">$(k_{p},k_{i})$ </tex-math></inline-formula> axes with the SBL. A stability triangle is formed with the two boundary points of the SBL intersecting the <inline-formula> <tex-math notation="LaTeX">$k_{p}$ </tex-math></inline-formula> axis for <inline-formula> <tex-math notation="LaTeX">$k_{i} =0$ </tex-math></inline-formula> and the point corresponding to the weighted geometric center value of the angular frequency. The center of gravity of the determined stability triangle gives the optimum PI control gains. Also, an analytical solution method for the PI controller design is presented. The proposed method (stability triangle) is examined on numerical examples. The time response performances of the controller calculated with the proposed method and the controllers calculated using the weighted geometric center method were compared. In addition, the comparisons of the PI controller determined by the proposed method and the different controllers selected in the neighborhood of this point are included. Comparisons with the studies in the literature including the PI controller design of the time-delayed first-order open-loop unstable transfer function are presented. As a result, it has been seen that the proposed method determines the parameters that provide optimum system performance in the tested region.


I. INTRODUCTION
The presence of time delay is an important issue in system modeling and control [1], [2], [3]. Studies on the analysis and design of time-delayed control systems are increasing day by day under the leadership of rapid development in software and hardware technology. Time delay, which is an important problem of control applications, online data flow from sensors at different points of the system, calculation of control signal, sending data and signals to the actuator, etc. occur in such cases. Stability analysis and control of delayed systems, robotic surgery and teleoperation, coordination of unmanned vehicles, active vibration suppression on flexible The associate editor coordinating the review of this manuscript and approving it for publication was Valentina E. Balas . robot arms in satellite systems are studied in control problems. The dynamics of many systems (electromechanical, thermal, fluid, economic, biological, etc.) that we encounter in real life and their response to the signal stimulating the system are obtained from the solution of the differential equation that characterizes the system [4]. However, in dynamic modeling the existing delays are often overlooked, but many physical events have time delays and even very small amounts of delay in the system can cause huge changes in the stability of the system. For this reason, the time delays in the relevant system should be carefully considered to obtain the most realistic open and closed loop system models. Also, time delays are known as the source of oscillation and instability [5]. Such structures are encountered quite frequently in pneumatic systems, long transmission lines, nuclear reactors, hydraulic systems, engines, and production processes [6], [7]. Many studies have concluded that the time delay will reduce the phase margin in the control system and cause a decrease in relative stability. The presence of time delay in systems creates a great difficulty in providing the desired control performance [8].
It is possible to come across many studies in literature, especially for first-order time-delay control systems. A complex fractional order PID controller is presented by Abdulwahhab [22]. Enwerem et al. present a PID controller tuning strategy for first-order plus time delay processes [23]. Gerov and Jovanovic propose a method for designing PI controller is based on the pole assignment method [24]. Şenol and Demiroğlu propose an analytical design method of a proportional integral controller for first order plus time delay systems [25]. A study presents a fractional PI controller design method satisfying both time-domain and frequency requirements by Hmed et al. [26]. A fractional order proportional integral controller is designed for controlling the level of a spherical tank which is modeled as a first order plus dead time system by Gopinath [27]. Jung et al. present a tuning method for the PI controller settings of unstable first-order-plus-time-delay processes [28]. Izci et al. present feed forward compensated PI controller design for optimum control of the air-fuel ratio (AFR) system [29]. A Matlab graphical interface to teach basic control concepts based on a set of PI tuning rules are presented for stable and unstable first order lag plus time delay models by Ruz et al. [30]. Also Ruz et al. present an interactive tool which, provides a set of tuning rules for both open-loop stable and unstable first order plus time delay processes [31].
In order to improve the response of a system, different controller parameters may always be available, and the methods mentioned above may not always give good results. All these tuning methods may respond differently in different control systems. In order to select the controller parameters that can provide the best control performance, many studies have been carried out on the calculation of all parameters of PI and PID controllers, which can make the system stable [32], [33]. Some of these methods can be summarized as follows. In [34], a study based on Nyquist criterion is presented for the calculation of all PID controllers that make a system stable.
In [35], a parameter space approximation study using the singular frequency approximation is discussed. A method for calculating stability zones by the SBL method is mentioned in [36] and [37]. SBL analysis is a graphical method used to obtain all the controller parameters that stabilize a closedloop system. In some studies, the weighted geometric center method was used to determine the optimum point within the SBL [38], [39].
Due to the high instability tendencies of time-delayed systems, the practical and easy determination of suitable controller parameters has motivated this study. In this study, the stability triangle method, which is formed within the SBL, is proposed to determine the PI controller parameters for the time-delayed first-order transfer functions. With this method, three points are determined within the stability boundary. With these three points, the SBL is reduced to a triangular area. The center of gravity of the stability triangle specifies the parameters of the PI controller. Also, an analytical solution method is presented for a first order time delayed systems. The proposed method has been tested in the simulation environment with the weighted geometric center method and different controller parameters selected in the neighborhood of the determined point.
The remaining parts of this study are organized as follows: in section II, the SBL equations of a control system which includes a time-delayed transfer function, and a PI controller are derived; in section III, the weighted geometric center method is introduced; in section IV, the methodology of the stability triangle method is explained. The tests of the proposed method are presented in section V with numerical examples. The results are given in section VI.

II. CALCULATION OF SBL FOR PI CONTROLLER
In this section, the SBL equations for the closed-loop control system with PI controller given in Figure 1 are presented. Where, C(s) denotes a PI controller with k p and k i parameters as defined in Equation (1), and G(s) denotes a time-delayed transfer function as defined in Equation (2).
In Equation (2), N (s) and D(s) represent the numerator and denominator polynomials of G p (s), respectively. The characteristic equation of the closed loop system in Figure 1 is obtained as Equation (3).
70378 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Equation (4) is obtained by transforming s = jω in G p (s). Thus, the numerator and denominator of G p (s) are decomposed into odd (N o ) and even (N e ) parts. To simplify Equation (4), the expression (−ω 2 ) will be omitted from the following equations.
In Equation (5), R (ω) denotes the real part of the characteristic equation and I (ω) denotes the imaginary part of the characteristic equation. The real and imaginary parts of the characteristic equation are obtained as Equations (6) and (7), respectively.
When the real and imaginary parts of the characteristic equation are set to zero, Equations (8) and (9) are obtained.
From the solution of Equations (10) and (11), k p and k i are calculated as Equations (18) and (19).
It follows that the SBL can be formed in the (k p , k i ) plane when the denominator expression of Equations (20) and (21) is When SBL is obtained, all controller parameters that make the closed-loop system stable or unstable can be tested. Integral gain parameter of the PI controller k i = 0 on the SBL divides the parameter plane into stable and unstable. The SBL is obtained by changing the angular frequency ω from 0 to ∞. However, the controllers can operate in the ω ∈ [0, ω c ] frequency range. Where, ω c denotes the frequency at which the phase of G(s) is −180 • . This is called the critical frequency and is expressed as Equation (22). Equation (23) is obtained by the Trigonometric transformation made in Equation (22). The critical frequency ω c is obtained by graphical method from the intersection point of tan(ωθ) and f (ω) curves.
Example 1: Let's plot the SBL in the time-delayed transfer function given in Equation (24).
Equation (25) is written to calculate the critical frequency. The critical frequency ω c is determined from the graphical solution of Equation (25) given in Figure 2. As it can be seen from Figure 2, the critical frequency ω c is determined as 3.10 rad/s from the intersection point of the curves.
For Equation (24) the SBL is plotted using Equations (20) and (21). The stable region to be constrained for the selection of appropriate PI controller parameters is obtained by plotting the SBL in the range of ω ∈ [0, 3.10] rad/s as in Figure 3.

III. WEIGHTED GEOMETRICAL CENTER METHOD
In this section, the determination of PI controller parameters using the weighted geometric center method is explained through Example 2.
Example 2: Let us consider the time-delayed transfer function given in Equation (26).   The characteristic equation of the closed loop system is calculated as Equation (27).
Functions k P and k i determining the SBL are calculated as Equation (28) and (29), respectively.
The critical frequency ω c of G(s) is obtained from the solution of Equation (29).
The graphical solution for Equation (29) is given in Figure (4). The critical frequency ω c of G(s) is determined as 2.03 rad/s from the intersection point of the tan(ω) and f (ω) curves.
Using Equation (28) for ω ∈ [0.01, 2.03] rad/s, SBL is obtained as Figure 5. In Figure 5, each point represents a (k p , k i ) pair and is located at different distances for each ω value. For example, the points are closely spaced at small values of ω. The points are tightened at large values of ω. The SBL terminates at the real root limit k i = 0 for the appropriate controller parameters.  Closed stability limits consist of n points depending on the angular frequency step ω . Thus, using the SBL and the real root boundary line, the weighted geometric center is obtained as in Equation (30). Where, k pwgc and k iwgc parameters are coordinate of the weighted geometric center.
In Equation (30), small ω values give better results than large ω values. Let's determine the weighted geometric center point (PI controller gains) for two different angular frequency steps of ω = 0.05 and ω = 0.01. Size of the angular frequency array n is computed as 41 for ω = 0.05. Thus, 41 (k p , k i ) pairs are formed, and the weighted geometric center is calculated as in Equation (31).
In the same way, size of the angular frequency array n is computed as 203 for ω = 0.01. Thus, 203 (k p , k i ) pairs are 70380 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   formed, and the weighted geometric center is calculated as in Equation (32).
By using PI controllers calculated with the weighted geometric center method for ω = 0.05 and ω = 0.01, the unit step responses of the control system are plotted as in Figure 6.
Here it can be seen that a better unit step response is obtained for smaller ω .

IV. DETERMINATION OF STABILITY TRIANGLE AND PI CONTROLLER PARAMETERS
In this section, the stability triangle approach, which adds innovation to the study, is proposed. It introduces a novel analytical PI controller design for time-delayed first-order transfer functions. The general form of a time-delayed firstorder transfer function model is expressed as Equation (33). For this general model, functions k p (ω) and k i (ω) are calculated parametrically as Equations (34) and (35). The region that limits the PI controller gains is calculated in the range of ω ∈ [0, ω c ] and is obtained as in Figure 7. The SBL shows the limits of all stable controller gains for the PI controller.
The critical frequency ω c can be solved graphically using Equation (36). However, in order to obtain an analytical solution, the approximation of the tan(α) function defined in the range of 0 < α < π was calculated as Equation (38) using the least-square curve fitting method.
In Figure 7, the corner coordinates of the stability triangle are defined as P1, P2, and P3 as in Equation (40).
At point P1, ω 0 specifies the initial angular frequency. At point P2, ω m indicates the average angular frequency and is calculated using the weighted geometric center method in the defined frequency range. At point P3, ω c is the critical frequency. The size of the angular frequency array n is calculated by Equation (41). The ω was preferred as 0.01 in VOLUME 11, 2023 A. Yuce: Analytical Design of PI Controller for First Order Transfer Function Plus Time Delay Step responses for C wgc and C st . the calculations. The first term of the frequency sequence is ω 1 , the increase amount is ω , and thus the average angular frequency ω m is calculated as in Equation (42).
In Equation (42), when the initial frequency is selected as ω 1 = ω ≈ 0, the average angular frequency is calculated approximately as in Equation (43).
Finally, in the G(s) system model, the center of gravity of the stability triangle within the boundaries of the SBL is determined as in Equation (47). This point is the parameters of the PI controller that controls the system optimally.

V. NUMERICAL EXAMPLES FOR THE PERFORMANCE TESTS
In this section, two examples are considered to test the success of the stability triangle method. In the first example, a circle is determined in the neighborhood of the center of gravity C st (k p , k i ) of the stability triangle. The unit step response performances of randomly selected controller parameters from this circle were examined according to the point determined by the center of gravity method. In the second example, an open loop unstable first-order time-delayed system model, which has been examined in the articles in the literature [24], [40], [41], is discussed. Example 3: For performance tests, let's consider the firstorder time-delayed transfer function given in Equation (48).
Using Equation (39), ω c is computed as 4.899 rad/s. For the angular frequency step range ω = 0.01, the SBL in the range of ω ∈ [0.01, ω c ] is obtained as in Figure 8. In order to form the stability triangle, the point P1 is obtained as Equation (49).
Using the critical frequency ω c = 4.899 rad/s, the mean angular frequency ω m is calculated as in Equation (50). Thus, the point P2 is obtained as in Equation (51).
The center of gravity of the stability triangle is calculated as in Equation (53). This point indicates the optimum controller parameter within the SBL. Thus, the PI controllers determined by the proposed method (stability triangle) and the weighted geometrical center method are obtained as in Equation (55) and (56), respectively.
The unit step responses of the closed loop system for controllers tuned using the stability triangle and weighted geometric center methods are given in Figure 9. The purpose of the controllers is to minimize their transient response performance. The most important of these performances are rise time, settling time and overshoot rate. The performance Step responses for the controllers which are selected around the center of gravity.  data of the unit step responses calculated with both controllers are given in Table 1 comparatively.
In the stability triangle method, all of the performance criteria are more successful than the weighted geometrical center method. Let us now examine the time response performance of the system at randomly selected points by drawing a

FIGURE 13.
Step responses plotted using the PI controllers, which is determined the different methods. circle around the C st point. A circle with radius r = 0.05 unit and center C st in the stability triangle is drawn as seen in Figure 10. The time response performance analysis is performed for eight different points equally spaced on this circle. PI controller parameters of the determined points are given in Table 2.
The unit step responses of the PI controllers given in Table 2 and the PI controller calculated using the stability triangle method in the closed loop system are plotted in Figure 11. Here, it is observed that C st (s) remains in the center of the other eight unit step response and this point gives optimum performance. In addition, the time response performances of these eight unit step responses are given in Table 3. The performance values and maximum and minimum performance differences of each controller are shown in Figure 12 in detail. The bar chart shows that C st (s) has the smallest size. These data show that the PI controller tuned using the stability triangle method exhibits optimum control performance.   Table 4.
The unit step responses calculated with the PI controllers given in Table 4 are plotted as Figure 13. The performance criteria of the unit step responses of the methods are given in Table 5. Normalized performance criteria and averages of performance data for each method are presented in Figure 14.
70384 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. According to these data, it is observed that the performance of the proposed method is quite close to the method of Majhi.
The a, b and θ values is changed by 10% to investigate the robustness of the given unstable first-order time-delayed transfer function. The unit step responses of the transfer function in the new situation are obtained as in Figure 15. The performance criteria of the unit step answers are given in Table 6. Likewise, normalized performance criteria and average performance data for each method are presented in Figure 16. In the new case, it is understood that the performance of the proposed method is better than other methods and shows robust control performance.

VI. CONCLUSION
In this study, a simple and practical PI controller tuning method is proposed for first-order time-delayed systems. The proposed method is based on the calculation of all stable PI controllers in the k p , k i plane using the stability boundary curve method. Then, the center of gravity of the geometric region, which is obtained in the SBL and called the stability triangle, is determined. This point specifies the optimum PI controller parameters. The method presented in this study provides an analytical solution for the determination of PI controller parameters. With the presented equations, it is very practical to calculate the optimum PI controller gains using the coefficients and time delay values of a first-order time-delayed system without drawing the SBL. It has been tested that the method gives reliable results with the numerical examples given. It is understood from Figure 12 that C st (s) performs optimally according to the controllers given in Table 3. In addition, accurate results are obtained in unstable first-order time-delayed systems and proposed method is compared with the studies in the literature. The proposed method provides an alternative analytical solution method for PI controller designs of first order time-delayed transfer functions. It is thought that the proposed method can also be used in PID, FOPI, FOPID controllers. In future studies, analytical controller tuning techniques can be investigated in PID, FOPI and FOPID controller design using the stability triangle method.