An Improved Analytical Tuning Rule of a Robust PID Controller for Integrating Systems with Time Delay Based on the Multiple Dominant Pole-Placement Method

An improved analytical tuning rule of a Proportional-Integral-Derivative (PID) controller for integrating systems with time delay is proposed using the direct synthesis method and multiple dominant pole-placement approach. Different from the traditional multiple dominant pole-placement method, the desired characteristic equation is obtained by placing the third-order dominant poles at −1/λ and placing the second-order non-dominant poles at −5/λ (λ is the tuning parameter). According to root locus theory, the third-order dominant poles and the second-order non-dominant poles are nearly symmetrically located at the two sides of the fifth-order dominant poles. This makes the third-order dominant poles closer to the imaginary axis than the fifth-order dominant poles, which means that, possibly, better performances can be achieved. Analytical formulas of a PID controller with a lead-lag filter are derived. Simple tuning rules are also given to achieve the desired robustness, which is measured by the maximum sensitivity (Ms) value. The proposed method can achieve better performances and maintain better performances when there exist parameters’ perturbation compared with other methods. Simulations for various integrating processes as well as the nonlinear continuous stirred tank reactor (CSTR) model illustrate the applicability and effectiveness of the proposed method.


Introduction
Recently, integrating systems have become a study hot topic in terms of automatic control. These types of systems have at least one pole at the origin, which is regarded as the characterization of the open-loop instability [1]. Integrating systems do not have a self-regulating nature. This implies that bounded inputs may lead to unbounded outputs in time domain. This phenomenon makes controlling such processes a difficult task. The type of integral system depends on the number of poles at the origin and the position of other poles in the denominator of closed-loop transfer function [2]. Generally speaking, the following integral systems can be divided into two types: a pure integral plus time delay system (PIPTD), a double integral plus time delay system (DIPTD), a stable first-order plus integral time delay system (FOPTDI) and an unstable first-order integral time delay system (UFOPTD) [3]. Time delay integrated systems exist in many industrial processes, such as a jacketed continuous stirred tank reactor [4], distillation column level and composition control loop [5], boiler drum [6,7], paper drum dryer drum [8], liquid storage tank and bioreactor [9].

Performance Indices
The maximum value of the sensitivity function amplitude ratio (Ms) is considered as a measurement of the robustness and stability of the proposed PID controller. The sensitivity function of the closed-loop control system for the proposed method is written as: Generally speaking, the process model uncertainty has a direct impact on the performance of the closed-loop system. The types of uncertainties mentioned here include process gain uncertainty, time constant and time delay uncertainties of the process model. The sensitivity function quantitatively reflects the sensitivity of the closed-loop transfer function to the uncertainty of process model parameters. The larger the value of MS, the more susceptible the controller is to parameter uncertainty. Conversely, the smaller the MS value, the better the stability of the controller to process parameter uncertainty. For the stable system, a range of 1.2 to 2 of Ms indicates good robustness. For integral systems considered as a subset of unstable systems, MS is equal to or greater than 2 for robust control. With regard to the proposed PID controller, Ms can be calculated as:

Theoretical Statements
The multiple dominant pole placement method is a frequently used tuning method of PID controllers for an integrating process. In general, most authors [30][31][32][33][34] use this method to obtain controller parameters by placing two main complex conjugate poles at desired positions. The dominance of a pole requires that the ratio of the real part of any other pole to the real part of the dominant pole is m (the value varies from 3 to 5), and there is no zero nearby. Therefore, all non-dominant poles are located on the left side of the line s = −ma, where a is the real part of the dominant pole. The core problem of pole preserving assignment is to determine PID parameters so that the closed-loop poles Symmetry 2020, 12, 1449 4 of 24 are located in the desired region except for the main poles. This method makes it possible to determine adjustable PID parameters without overshoot of step response [37].
The multi-dominant pole placement method proposed in this paper is based on the assumption that the performance requirements of closed-loop control in time domain are transformed into multiple dominant poles and non-dominant poles, and there are no zeros nearby. The distance from the non-dominant poles to the origin is five times the distance from the dominant pole to the origin. Simultaneously, all the poles are real and stable and their values of multiplicity are equal to the number of the adjustable parameters of the designed PID controller.
Ch. Anil [37] has placed fifth-order dominant poles in the same location of −1/λ (λ is the tuning parameter) to constitute the closed-loop transfer function. PID parameters are derived on a constraint condition that Ms equals to 2 or exceeds 2. To satisfy the robustness specification, λ should be selected to reach the desired Ms. If the combination of the dominant pole and the non-dominant pole is used to replace the fifth order main pole, the dominant poles and the non-dominant poles must be neatly symmetrically located on both sides of the fifth order main pole to achieve the same Ms. Therefore, the dominant poles are closer to the origin than the fifth-order dominant. Figure 1a,b give the locations of the 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles with their corresponding non-dominant poles and fifth-order dominant poles for a DIPTD system and an FOPTDI system with Ms = 2 (second-order poles denote two poles located at the same place, 3rd-order poles denote three poles located at the same place, and so forth). In Figure 1, 'x' denotes one or more poles at the same location, that is to say, 'x' only represents the location of the poles instead of the number. Pi (i = 1, 2, 3, 4, 5) denotes the ith-order dominant poles. It can be seen apparently that the dominant poles and the non-dominant poles are located at both sides of fifth-order dominant poles. The multi-dominant pole placement method proposed in this paper is based on the assumption that the performance requirements of closed-loop control in time domain are transformed into multiple dominant poles and non-dominant poles, and there are no zeros nearby. The distance from the non-dominant poles to the origin is five times the distance from the dominant pole to the origin. Simultaneously, all the poles are real and stable and their values of multiplicity are equal to the number of the adjustable parameters of the designed PID controller.
Ch. Anil [37] has placed fifth-order dominant poles in the same location of −1/λ (λ is the tuning parameter) to constitute the closed-loop transfer function. PID parameters are derived on a constraint condition that Ms equals to 2 or exceeds 2. To satisfy the robustness specification, λ should be selected to reach the desired Ms. If the combination of the dominant pole and the non-dominant pole is used to replace the fifth order main pole, the dominant poles and the non-dominant poles must be neatly symmetrically located on both sides of the fifth order main pole to achieve the same Ms. Therefore, the dominant poles are closer to the origin than the fifth-order dominant. Figure 1a,b give the locations of the 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles with their corresponding non-dominant poles and fifth-order dominant poles for a DIPTD system and an FOPTDI system with Ms = 2 (second-order poles denote two poles located at the same place, 3rd-order poles denote three poles located at the same place, and so forth). In Figure 1, 'x' denotes one or more poles at the same location, that is to say, 'x' only represents the location of the poles instead of the number. Pi (i = 1, 2, 3, 4, 5) denotes the ith-order dominant poles. It can be seen apparently that the dominant poles and the non-dominant poles are located at both sides of fifthorder dominant poles.
(a) (b) Figure 1. Locations of 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles with their corresponding non-dominant poles and fifth-order dominant poles for a double integral plus time delay system (DIPTD) system (a) and a first-order plus integral time delay system (FOPTDI) system (b) with Ms = 2.
Generally, a smaller distance from poles to the origin results better servo and regulatory performance on the condition that there is only a single pole in a close-loop transfer function or all poles are at the same location. Figure 2 gives the dominance of the 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles for a DIPTD system and an FOPTDI system with Ms = 2. It can be verified that the higher order of the dominant poles yields stronger dominance. In order to achieve better set point tracking performance and robustness, a trade-off between the distance from the multiple dominant poles to the origin and their dominance should be considered. Locations of 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles with their corresponding non-dominant poles and fifth-order dominant poles for a double integral plus time delay system (DIPTD) system (a) and a first-order plus integral time delay system (FOPTDI) system (b) with Ms = 2.
Generally, a smaller distance from poles to the origin results better servo and regulatory performance on the condition that there is only a single pole in a close-loop transfer function or all poles are at the same location. Figure 2 gives the dominance of the 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles for a DIPTD system and an FOPTDI system with Ms = 2. It can be verified that the higher order of the dominant poles yields stronger dominance. In order to achieve better Step response of the closed-loop systems consist of 1st-order, 2nd-order, 3rd-order, 4thorder dominant poles with their corresponding non-dominant poles (solid line) and the step response of the systems consist of only dominant poles (dotted line) for a DIPTD system (a) and an FOPTDI system (b) with Ms = 2.
After numerous simulations, the third-order dominant poles are selected to achieve the best setpoint tracking performance and robustness. The correctness and universal applicability of the theory are tested and verified in a large number of simulation experiments. The simulation results are similar to those shown in Figure 3.

The Process Model
The dynamic process of integral plus time delay process is modeled by the following general transfer function:  Step response of the closed-loop systems consist of 1st-order, 2nd-order, 3rd-order, 4th-order dominant poles with their corresponding non-dominant poles (solid line) and the step response of the systems consist of only dominant poles (dotted line) for a DIPTD system (a) and an FOPTDI system (b) with Ms = 2.
After numerous simulations, the third-order dominant poles are selected to achieve the best setpoint tracking performance and robustness. The correctness and universal applicability of the theory are tested and verified in a large number of simulation experiments. The simulation results are similar to those shown in Figure 3. After numerous simulations, the third-order dominant poles are selected to achieve the best setpoint tracking performance and robustness. The correctness and universal applicability of the theory are tested and verified in a large number of simulation experiments. The simulation results are similar to those shown in Figure 3.

The Process Model
The dynamic process of integral plus time delay process is modeled by the following general transfer function:

The Process Model
The dynamic process of integral plus time delay process is modeled by the following general transfer function: Case1: If τ = 0, c = 1, P = 0, denote PIPTD systems with the transfer function Symmetry 2020, 12, 1449 6 of 24 Case2: If τ = 1, c = 0, P = 0, denote DIPTD systems with the transfer function Case3: If c = 1, P = 0, denote stable FOPTDI systems with the transfer function Case4: If c = 1, P > 0 or P < 0, denote stable FOPTDI systems with a positive/negative zero, and the transfer function is expressed as:

Controller Design
The PID controller with a lead-lag filter is given by Equation (8), where k c is the process gain, τ i is the integral time constant and τ d is the derivative time constant, α and β are viewed as the filter time constant.
The characteristic equation can be calculated by: Substituting Equations (3) and (8) into (9) yields: By means of the first order Pade's approximation for the time delay term, i.e., e −Ls ≈ (1 − 0.5Ls)/(1 + 0.5Ls), Equation (10) can be simplified as: Simplifying and rearranging Equation (11) to extract the common factor of s 5 , s 4 , s 3 , s 2 and s, an equation can be obtained as follows: Symmetry 2020, 12, 1449 7 of 24 The expected characteristic equation can be written as: Expanding the Equation (13) yields the equation as follows: Equating the corresponding coefficients of s 5 , s 4 , s 3 , s 2 and s in Equation (12) with that of Equation (14), the following five equations can be obtained as: It can be deduced logically from the above equation that the adjusting parameter λ is directly related to the controller parameters. By selecting diverse λ for different Ms, simultaneous equations of Equations (15)- (19) can be solved by Matlab.

Condition for PIPTD System
For PIPTD systems, the lead-lag filter does not show the effect of improving performance. Hence, the conventional PID controller without a lead-lag filter is designed for this system. There are three adjustable PID parameters. The desired characteristic equation is obtained by dealing with the placement of three poles. In order to obtain better performance, the ratio of the distance between the non-dominate pole and the dominate poles to the origin is set to 3. Simultaneously, a combination of second-order dominant poles and a first-order non-dominant pole is chosen to achieve the trade-off between the distance from the multiple dominant poles to the origin and their dominance. The controller parameters can be obtained with the third order characteristic equation.
The characteristic equation for PIPTD processes can be written by: The expected characteristic equation is chosen as: Symmetry 2020, 12, 1449 8 of 24 By comparing the corresponding coefficients of Equations (20) and (21), PID parameters can be derived from the following three equations.

Selection of the Tuning Parameter and the Tuning Rules
The primary requirement for selecting the tuning parameter λ is to guarantee that all the controller parameters are positive. Furthermore, the adjustment method of λ should make the final controller have good nominal value and robustness (in terms of Ms). After a great deal of simulation experiments and analysis, it was observed that the initial value of λ can be selected as equal to 0.8L, where L is the time delay of the process plant. If good performance is not achieved with this value of λ, then gradually increase the adjustment parameters from this value until a good nominal value and robustness are reached. The range of λ that guarantees good nominal and robustness is L to 3L.
When a specific value of λ is taken into Equations (15)- (19) and Equations (22)-(24), the controller parameters can be obtained with the help of Matlab. Subsequently, Ms can be calculated with these known controller parameters. For different values of λ, obtain the Ms and PID settings, and draw the relationship between Ms and λ. According to this figure, the equations of Ms and λ are fitted by the curve fitting tool in Matlab. Taking the desired Ms into this equation, the tuning parameter of λ can be easily calculated. Finally, all the controller parameters can be obtained. Figure 4 gives the variation of Ms with λ/L for PIPTD systems and DIPTD systems. Tables 1 and 2 give the tuning rules for PIPTD, DIPTD and FOPTDI (Ms = 2) systems. By comparing the corresponding coefficients of Equations (20) and (21), PID parameters can be derived from the following three equations.

Selection of the Tuning Parameter and the Tuning Rules
The primary requirement for selecting the tuning parameter λ is to guarantee that all the controller parameters are positive. Furthermore, the adjustment method of λ should make the final controller have good nominal value and robustness (in terms of Ms). After a great deal of simulation experiments and analysis, it was observed that the initial value of λ can be selected as equal to 0.8L, where L is the time delay of the process plant. If good performance is not achieved with this value of λ, then gradually increase the adjustment parameters from this value until a good nominal value and robustness are reached. The range of λ that guarantees good nominal and robustness is L to 3L.
When a specific value of λ is taken into Equations (15)- (19) and Equations (22)-(24), the controller parameters can be obtained with the help of Matlab. Subsequently, Ms can be calculated with these known controller parameters. For different values of λ, obtain the Ms and PID settings, and draw the relationship between Ms and λ. According to this figure, the equations of Ms and λ are fitted by the curve fitting tool in Matlab. Taking the desired Ms into this equation, the tuning parameter of λ can be easily calculated. Finally, all the controller parameters can be obtained. Figure  4 gives the variation of Ms with λ/L for PIPTD systems and DIPTD systems. Tables 1 and 2 give the tuning rules for PIPTD, DIPTD and FOPTDI (Ms = 2) systems.    Table 1. Proportional-Integral-Derivative (PID) tuning rules for IPTD and DIPTD. Table 2. PID tuning rules for FOPTDI (Ms = 2).

Robustness, Input Usage and Performance Index
The proposed PID controller is designed not only to ensure the stability of the closed-loop system, but also achieve robustness and performance for servo and regulation problems. In order to illustrate the effectiveness of the proposed analytical method, integral performance index and robustness are adopted in this paper. Moreover, overshoot (OS) is considered as an index of performance evaluation to Section 3.1.

Integral Performance Index
To measure the performance of the controller, the integral performance index is considered. For single variable systems, there are four frequently encountered error integral rules: the Integral Square Error (ISE), Integral Time Square Error (ITSE), Integral Absolute Error (IAE) and Integral Time Absolute Error (ITAE). In the present work, comparison is carried out in terms of IAE with the PID controllers in other methods. A small value of the indexes mentioned above means fast setpoint tracking and interference suppression. The integral performance index IAE is given by:

Total Variation Index
Total Variation (TV) is a performance index to measure the smoothness of the process input or the controller output. It is the sum of the differences between the current process input and previous process input over a period of time which equals to settling time of the systems. A smaller value of TV indicates that the controlled output of the controller is smoother than other methods. When the manipulated variable u(t) of the closed loop system is discretized as a sequence (u1, u2, u3, ..., ui, ...), TV is expressed as:

Simulation and Analysis Studies
In this section, the proposed method are compared with methods reported recently by Ch. Anil and R. Padma Sree (CH) [35,37], Q.B. Jin and Q. Liu (JL) [27], M. Viteckova and A. Vitecek (V.V) [33], M. Ajmeri and A. Ali (A.A) [36], Shamsuzzoha [29] and Lee et al. (Lee) [26]. Various classes of integrating plustime delay models and a non-linear model are considered to illustrate the effectiveness and applicability of the proposed method. The tuning parameter λ is selected to give the same Ms as given by other methods to make sure the performance comparison is carried out for the same robustness level.

Remark 1.
The set point weighting is implemented in the PID controller designed in this paper to improve the servo performance. For fair comparison, the set point weight parameter is set to be ε= 0.4 (the same as chosen in Ch. Anil and R. Padma Sree [37]) in the controllers of the proposed method for case 1, case 2, case 3 and case 5. When there is a negative zero (1 + Ps) in the transfer function of process such as case 4 (P is positive), a first order lag filter of 1/(1 + Ps) is added along with the original PID filter (αs + 1)/(βs + 1). This is implemented on the integrating stable first order plus time delay system with a negative zero.
PID settings for 6 cases of various methods are displayed in Table 3. For simulation, step reference and disturbance are considered to illustrate the superiority of the proposed method in the simulation. Table 4 concludes the performance indices of the step response in terms of IAE, TV and OS (maximum overshoot) for all cases. Table 5 concludes the performance indices of step responses considering parameter perturbations. From Table 4, it can be proved that the proposed method almost has the smallest or the second smallest IAE, TV and OS compared with other methods, which implies that the proposed method has almost the best set point performance and the smoothest controller output. From Table 5, it can be seen that the proposed method can maintain the best performances compared with other methods when there exist parameter perturbations. In conclusion, compared with other methods, this method has good set-point tracking and interference suppression performance, and has better robustness.

PIPTD System
The simulation is conducted for a specific pure integrating time delay process model  Table 3, it can be verified that the controller reported by M. Ajmeri and A. Ali [36] has better servo performance than other controllers owing to the PD controller and the 2-DOF structure, but it has a large overshoot and large value of TV at the same time. The proposed controller has the smallest OS and TV and the IAE is close to the methods designed by Ch. Anil and R. Padma Sree [37] and M. Viteckova and A. Vitecek [33]. For the regulatory problem, the method of Ch. Anil and R. Padma Sree [37] has the smallest IAE and the IAE of the proposed method is close to Ch. Anil and R. Padma Sree [37]. The proposed method has the smallest OS and has the second smallest TV value.
To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is, G r = 0.05e −7s s . Figures 7 and 8 give the closed loop responses and controller outputs of different methods. From Figures 7 and 8 and Table 5, it can be seen that the proposed method has the smallest IAE, TV and OS for servo problem. For the regulation problem, the proposed method has the smallest IAE and OS, and the TV is slightly bigger than M. Viteckova and A. Vitecek (V.V) [33].
value of TV at the same time. The proposed controller has the smallest OS and TV and the IAE is close to the methods designed by Ch. Anil and R. Padma Sree [37] and M. Viteckova and A. Vitecek [33]. For the regulatory problem, the method of Ch. Anil and R. Padma Sree [37] has the smallest IAE and the IAE of the proposed method is close to Ch. Anil and R. Padma Sree [37]. The proposed method has the smallest OS and has the second smallest TV value. To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is,  Table 5, it can be seen that the proposed method has the smallest IAE, TV and OS for servo problem. For the regulation problem, the proposed method has the smallest IAE and OS, and the TV is slightly bigger than M. Viteckova and A. Vitecek (V.V) [33]. close to the methods designed by Ch. Anil and R. Padma Sree [37] and M. Viteckova and A. Vitecek [33]. For the regulatory problem, the method of Ch. Anil and R. Padma Sree [37] has the smallest IAE and the IAE of the proposed method is close to Ch. Anil and R. Padma Sree [37]. The proposed method has the smallest OS and has the second smallest TV value. To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is,  Table 5, it can be seen that the proposed method has the smallest IAE, TV and OS for servo problem. For the regulation problem, the proposed method has the smallest IAE and OS, and the TV is slightly bigger than M. Viteckova and A. Vitecek (V.V) [33].

DIPTD System
The DIPTD system  Table 4, it can be seen that proposed method has the second smallest IAE, TV and OS for servo problem. Though the method in M. Ajmeri and A. Ali (A.A) [36] has the smallest IAE value, but the OS and TV are much bigger than the proposed method. For the regulation problem, the proposed method has the smallest IAE and OS.

DIPTD System
The DIPTD system G p (s) = e −s s 2 is considered in this section. The proposed method is compared with the methods presented by Ch. Anil  Servo and regulatory step responses of various methods are illustrated in Figure 9a,b. Figure 10 gives the controller outputs. From Figures 9a and 10a and Table 4, it can be seen that proposed method has the second smallest IAE, TV and OS for servo problem. Though the method in M. Ajmeri and A. Ali (A.A) [36] has the smallest IAE value, but the OS and TV are much bigger than the proposed method. For the regulation problem, the proposed method has the smallest IAE and OS.  To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is   To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is . Figures 11 and 12 give the closed loop responses and controller outputs of different methods. From Figures 11 and 12 and Table  5, To solve servo problem, the proposed method has the smallest IAE. The value of TV and OS are To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is G r = e −1.4s s 2 . Figures 11 and 12 give the closed loop responses and controller outputs of different methods. From Figures 11 and 12 and Table 5, To solve servo problem, the proposed method has the smallest IAE. The value of TV and OS are close to the methods in Ch. Anil and R. Padma Sree (CH) [37] and M. Viteckova and A. Vitecek (V.V) [33]. For the regulation problem, the proposed method has the smallest IAE and OS. To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 40% perturbation is added to time delay, that is . Figures 11 and 12 give the closed loop responses and controller outputs of different methods. From Figures 11 and 12 and Table  5, To solve servo problem, the proposed method has the smallest IAE. The value of TV and OS are close to the methods in Ch. Anil and R. Padma Sree (CH) [37] and M. Viteckova and A. Vitecek (V.V) [33]. For the regulation problem, the proposed method has the smallest IAE and OS.

Stable FOPTDI System
A stable FOPTDI system 0.2 ( ) (4 1)  Figure 13. Figure 14 gives the controller outputs of different methods. From Figure 13, it can be seen that the proposed method has the smallest OS for both servo and regulation problems. Though the method in M. Ajmeri and A. Ali [36] has the smallest IAE, it has a large overshoot and large value of TV at the same time. For the regulation problem, as far as IAE and OS are concerned, the proposed method has almost the best performance.

Stable FOPTDI System
is considered in this part. The proposed method is compared with the methods proposed by Ch. Anil and R. Padma Sree [37], Q.B. Jin and Q. Liu [27] and M. Ajmeri and A. Ali [36]. For the proposed method, the tuning parameter λ is chosen as a value of 2.184 to satisfy the desired Ms. The dominant poles and the non-dominant poles are placed at −0.4579 and −2.2895, respectively. Servo and regulatory step responses of various methods are shown in Figure 13. Figure 14 gives the controller outputs of different methods. From Figure 13, it can be seen that the proposed method has the smallest OS for both servo and regulation problems. Though the method in M. Ajmeri and A. Ali [36] has the smallest IAE, it has a large overshoot and large value of TV at the same time.
For the regulation problem, as far as IAE and OS are concerned, the proposed method has almost the best performance. has the smallest OS for both servo and regulation problems. Though the method in M. Ajmeri and A. Ali [36] has the smallest IAE, it has a large overshoot and large value of TV at the same time. For the regulation problem, as far as IAE and OS are concerned, the proposed method has almost the best performance.  To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 20% perturbations of all parameters are added, that is, . Figures 15 and 16 give the closed loop responses and controller outputs of different methods. From Figures 15 and 16 and Table 5, it can be seen that the proposed method has minimal OS for both servo problems and adjustment problems. For servo problems, the values of IAE and TV are close to other methods. For regulation problem, the proposed method has the second smallest IAE, which is very close to the method in Ch. Anil and R. Padma Sree (CH) [37]. To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 20% perturbations of all parameters are added, that is, G r = 0.24e −1.2s s(4.8s + 1) . Figures 15 and 16 give the closed loop responses and controller outputs of different methods. From Figures 15 and 16 and Table 5, it can be seen that the proposed method has minimal OS for both servo problems and adjustment problems. For servo problems, the values of IAE and TV are close to other methods. For regulation problem, the proposed method has the second smallest IAE, which is very close to the method in Ch. Anil and R. Padma Sree (CH) [37].  Table 5, it can be seen that the proposed method has minimal OS for both servo problems and adjustment problems. For servo problems, the values of IAE and TV are close to other methods. For regulation problem, the proposed method has the second smallest IAE, which is very close to the method in Ch. Anil and R. Padma Sree (CH) [37].
. Figure 15. Servo step responses and regulatory step responses of case 3 with parameter mismatch (all parameters +20%).

Stable FOPTD System with a Negative Zero
Consider a stable FOPTD system with a negative zero given by proposed method is compared with the method proposed by Ch. Anil and R. Padma Sree [37]. The tuning parameter λ of the proposed method is adjusted to 1.67 in a way that Ms equals to 2.35 as that is given in [37]. The dominant poles and the non-dominant poles are placed at −0.5988 and −2.994, respectively. Servo and regulatory step responses and the controller outputs are shown in Figure 17. From Figure 17 and Table 4, it can be seen that the servo and regulation performances of the proposed method are better than [37] in terms of IAE, TV and OS.

Stable FOPTD System with a Negative Zero
Consider a stable FOPTD system with a negative zero given by G p (s) = (1 + 10s)e −s s(2s + 1) . The proposed method is compared with the method proposed by Ch. Anil and R. Padma Sree [37]. The tuning parameter λ of the proposed method is adjusted to 1.67 in a way that Ms equals to 2.35 as that is given in [37]. The dominant poles and the non-dominant poles are placed at −0.5988 and −2.994, respectively. Servo and regulatory step responses and the controller outputs are shown in Figure 17. From Figure 17 and Table 4, it can be seen that the servo and regulation performances of the proposed method are better than [37] in terms of IAE, TV and OS.
is given in [37]. The dominant poles and the non-dominant poles are placed at −0.5988 and −2.994, respectively.
Servo and regulatory step responses and the controller outputs are shown in Figure 17. From Figure 17 and Table 4, it can be seen that the servo and regulation performances of the proposed method are better than [37] in terms of IAE, TV and OS. To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 20% perturbations of all parameters are added, that is, . Figure 18 gives the closed loop responses and controller outputs of different methods. From Figure 18 and Table 5, it can be seen that the proposed method has smaller IAE, TV and OS for both servo problem and regulation problem. To test the robustness of the proposed method, parameter perturbations are considered.
In simulation, 20% perturbations of all parameters are added, that is, G r = (1+12s)e −1.2s s(2.4s+1) . Figure 18 gives the closed loop responses and controller outputs of different methods. From Figure 18 and Table 5, it can be seen that the proposed method has smaller IAE, TV and OS for both servo problem and regulation problem.

High-Order System
To illustrate that the proposed method can also apply to high-order systems, a plant Servo and regulatory step responses and controller outputs are detailed in Figure 19. From Figure 19 and Table 4, it can be observed that the proposed method has the smallest OS and the IAE and TV performances are close to the methods in Ch. Anil and R. Padma Sree [37] and Q.B. Jin and Q. Liu [27].

High-Order System
To illustrate that the proposed method can also apply to high-order systems, a plant G p (s) = 0.5(1 − 0.5s)e −0.7s s(0.4s + 1)(0.1s + 1)(0.5s + 1) is considered in this section. The process model is reduced by the reduction technique to an FOPTDI model given by G p (s) = 0.5183e −1.2799s s(1.1609s + 1) . The proposed method is compared with the methods proposed by Ch. Anil and R. Padma Sree [37] and Q.B. Jin and Q. Liu [27]. The tuning parameter λ is adjusted to 2.235 to match Ms = 2.81. The dominant poles and the non-dominant poles are placed at −0.4474 and −2.237, respectively. Servo and regulatory step responses and controller outputs are detailed in Figure 19. From Figure 19 and Table 4, it can be observed that the proposed method has the smallest OS and the IAE and TV performances are close to the methods in Ch. Anil and R. Padma Sree [37] and Q.B. Jin and Q. Liu [27]. [27]. The tuning parameter λ is adjusted to 2.235 to match Ms = 2.81. The dominant poles and the nondominant poles are placed at −0.4474 and −2.237, respectively.
Servo and regulatory step responses and controller outputs are detailed in Figure 19. From Figure 19 and Table 4, it can be observed that the proposed method has the smallest OS and the IAE and TV performances are close to the methods in Ch. Anil and R. Padma Sree [37] and Q.B. Jin and Q. Liu [27]. . Figure 20 gives the closed loop responses and controller outputs of different methods. From Figure  20 and Table 5, it can be seen that the proposed method has the smallest IAE and OS for the servo problem and the TV value is close to other methods. For the regulation problem, the proposed method has the smallest IAE, TV and OS. Squar up the robustness of the proposed method, parameter perturbations are considered.
In simulation, 20% perturbations of all parameters are added, that is, G r = 0.6(1 − 0.6s)e −0.84s s(0.48s + 1)(0.12s + 1)(0.6s + 1) . Figure 20 gives the closed loop responses and controller outputs of different methods. From Figure 20 and Table 5, it can be seen that the proposed method has the smallest IAE and OS for the servo problem and the TV value is close to other methods. For the regulation problem, the proposed method has the smallest IAE, TV and OS.

Non-Linear Model
A Continuous Stirred Tank Reactor (CSTR) [35] is considered with model equations expressed as:

Non-Linear Model
A Continuous Stirred Tank Reactor (CSTR) [35] is considered with model equations expressed as: Table 6 shows the values of the process parameters. The above nonlinear equation is linearized near the unstable operating point C As = 10 K mol/m 3 , C Bs = 5.62 K mol/m 3 , T s = 590 K. The transfer function model about the operating temperature to the feed concentration is expressed as: Equation (31) can be simplified as After the pole zero cancellation, Equation (32) can be reduced to: T(s) C AF (s) = 0.07803 0.0805 × 0.0039(12.4224s + 1)(256.4103s − 1) In Equation (33), the unstable time constant with a value of 256.4103 in the denominator is considered into process gain because it is very large. Subsequently, a transfer function of the stable FOPTDI system is obtained as follows: Considering the 1 s measurement time delay in Equation (34), the transfer function model is expressed as: A conventional PID controller is designed for the transfer function model of Equation (33) by selecting a tuning parameter with a value of 2.5 to match the same Ms as Ch. Anil and R. Padma Sree [35]. Lee et al. [26] proposed a simple analytical PID controller using a modified IMC method, and the PID parameters are obtained by assuming the transfer function model to be an FOPTDI plant. The tuning parameter λ is tuned to 2.5 to match Ms = 2. The dominant poles and the non-dominant poles are placed at −0.4 and −2, respectively.
Servo and regulatory responses and controller outputs of CSTR for the linear model in case 6 ( T(s) C AF (s) = 0.9693e −s s(12.4224s + 1) ) given in Figure 21. From Figure 21 and Table 7, it can be seen that the proposed method has the smallest IAE and OS for servo problem. For the regulation problem, the proposed method has the smallest IAE, TV and OS.
poles are placed at −0.4 and −2, respectively. Servo and regulatory responses and controller outputs of CSTR for the linear model in case 6 (  Table 7, it can be seen that the proposed method has the smallest IAE and OS for servo problem. For the regulation problem, the proposed method has the smallest IAE, TV and OS.    To test the robustness of the proposed method, parameter perturbations are considered. In simulation, 20% perturbations of all parameters are added, that is, G r = 1.1632e −1.2s s(14.9069s + 1) . Figure 22 gives the closed loop responses and controller outputs of different methods. From Figure 22 and Table 7, it can be seen that the proposed method has the smallest IAE and OS for servo problem and the TV value is close to other methods. For regulation problem, the proposed method has the smallest IAE, TV and OS. . Figure 22 gives the closed loop responses and controller outputs of different methods. From Figure 22 and Table 7, it can be seen that the proposed method has the smallest IAE and OS for servo problem and the TV value is close to other methods. For regulation problem, the proposed method has the smallest IAE, TV and OS.
(a) (b) To explain the effectiveness of the method proposed in this article for nonlinear plant, simulations for the non-linear model are given. Figure 23 gives the servo responses and controller outputs of the non-linear model in case 6 for a step change in operating temperature from 590 K to 595 K. Figure 24 gives the regulatory responses and controller outputs of non-linear model in case 6 for a load change of feed concentration from 15.61 to 20 K mol/m 3 . Table 8 concludes the performance comparison of the CSTR nonlinear model in terms of IAE, TV and OS. From Table 7, it can be proved that the proposed method has the best performances for both servo and regulation problems. To explain the effectiveness of the method proposed in this article for nonlinear plant, simulations for the non-linear model are given. Figure 23 gives the servo responses and controller outputs of the non-linear model in case 6 for a step change in operating temperature from 590 K to 595 K. Figure 24 gives the regulatory responses and controller outputs of non-linear model in case 6 for a load change of feed concentration from 15.61 to 20 K mol/m 3 . Table 8 concludes the performance comparison of the CSTR nonlinear model in terms of IAE, TV and OS. From Table 7, it can be proved that the proposed method has the best performances for both servo and regulation problems.
To explain the effectiveness of the method proposed in this article for nonlinear plant, simulations for the non-linear model are given. Figure 23 gives the servo responses and controller outputs of the non-linear model in case 6 for a step change in operating temperature from 590 K to 595 K. Figure 24 gives the regulatory responses and controller outputs of non-linear model in case 6 for a load change of feed concentration from 15.61 to 20 K mol/m 3 . Table 8 concludes the performance comparison of the CSTR nonlinear model in terms of IAE, TV and OS. From Table 7, it can be proved that the proposed method has the best performances for both servo and regulation problems.   From the above simulations, it can be concluded that the proposed method is superior to other methods in terms of IAE values and overshoot for both linear model and non-linear model of CSTR.

Conclusions
In this paper, a novel analytical tuning rule of a PID controller is proposed for integrating systems with time delay using the direct synthesis method and modified multiple dominant poleplacement method. A combination of third-order dominant poles and second-order non-dominant poles is adopted to achieve better performances. When using the analytical formula, the PID parameters can be directly calculated. According to the adjustment parameters and Ms, the adjustment rules are given to meet the required robustness and performance requirements.
Compared with other methods, the designed controller is found to be more effective in terms of IAE, TV and OS for both servo and regulation problems from simulation results. The controller used in this paper is a conventional PID control scheme. In the future, a differential forward PID control scheme or other advanced control schemes, such as a parallel control system and cascade control structure, can be considered to further improve the performance and robustness.  From the above simulations, it can be concluded that the proposed method is superior to other methods in terms of IAE values and overshoot for both linear model and non-linear model of CSTR.

Conclusions
In this paper, a novel analytical tuning rule of a PID controller is proposed for integrating systems with time delay using the direct synthesis method and modified multiple dominant pole-placement method. A combination of third-order dominant poles and second-order non-dominant poles is adopted to achieve better performances. When using the analytical formula, the PID parameters can be directly calculated. According to the adjustment parameters and Ms, the adjustment rules are given to meet the required robustness and performance requirements.
Compared with other methods, the designed controller is found to be more effective in terms of IAE, TV and OS for both servo and regulation problems from simulation results. The controller used in this paper is a conventional PID control scheme. In the future, a differential forward PID control scheme or other advanced control schemes, such as a parallel control system and cascade control structure, can be considered to further improve the performance and robustness.