# Performance Evaluation of Composite CRONE Controller Strategy for an Air Pressure System

Authors: V. Velmurugan; N. N. Praboo; K. Ramash Kumar
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

This article explores and determines the robust performance of a CRONE controller for an air pressure system (APS). Generally, the CRONE controller is a robust noninteger order controller to achieve linear performance for a highly nonlinear system. In this regard, the noninteger order controller can be designed using the transfer function of an APS. Whereas, the transfer function of the system is obtained by the worst-case modelling technique. The servo performance of the controller is analyzed in terms of error and time indices performances for three different operating points. Finally, the simulation results are compared with the conventional controller. According to the performance characteristics, it is visualized that the composite CRONE (CC) controller gives superior performance than the first generation CRONE (FGC) controller and conventional controller for the selected nonlinear system.


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## Body

## 1. Introduction

CRONE controller has attracted more attention of the research community in process control, who are specifically growing on the fractional order controller strategy system. It can be utilized in many sectors due to its effective robust controller performance. Even though many controllers are available in the controller community, it constantly provides profound effective performance based on the process.Pressure is the foremost thing in every process application like boilers, vessels, atomic reactors, and steam drums. Hence, it needs to be controlled in an effective manner due to the fact that it is closely tied to individual human safety and the resources of the specific application. As a result of this research, an air pressure system (APS) for investigation was proposed in this article.In this regard, Manabe [1] proposed to people who have worked in the domain of noninteger that the behaviour of the noninteger order controller is dependent on the calculation of a constant phase margin around the gain crossover frequency ωgc [2]. Furthermore, Alain Oustaloup has introduced newly fractional order controller as a CRONE controller in the year of 1991. The term “CRONE controller” is an acronym by “Commande Robuste d’Ordre Non Entire,” which refers to different types of frequency domain approach-based noninteger order controllers. In reality, there are three types of CRONE controllers: first generation CRONE (FGC) controllers, second generation CRONE (SGC) controllers, and third generation CRONE (TGC) controllers. The first generation of CRONE controller provides a constant phase of the plant around ωgc [3]. But unlike FGC, the SGC control eliminates directly the phase margin variation since the plant reparametrization and is possible to choose the expected ωgc within the asymptotic frequency range of the plant [4]. Whereas, the TGC controller has additional advanced features such as pole and zero misplacement [5]. As a primary strategy, the FGC controller has been proposed to rectify the steam plant parameter uncertainty using a frequency domain approach [6]. The CRONE controller has been effectively utilized to obtain robust performance for different process systems in terms of time and frequency domain characteristics [7–9]. The APS system has been briefly discussed in terms of time and error indices performance using a noninteger order controller. But unfortunately, the tuning of the CRONE controller has been failed drastically due to higher values of IAE and ISE as well as rise and settling time performances [10]. Real-time APS performance has been compared between CRONE and conventional controllers and tuning methods have been found to be unsuccessful based on the observation of acquired values [11]. The two distinct controllers, namely, CRONE and H∞ controller have been adopted in the wind conversion system and addressed problems such as turbine speed, harmonics, mechanical vibrations, and load variations. Finally, both controllers have been compared and evaluated the performance on the speed of the turbine and attenuation of mechanical vibration of the system [12].The air pressure in the tank has been controlled and studied by the CRONE controller and it implemented into embedded connected in the hydraulic process system [13]. The pressure control application has been investigated and examined by the conventional proportional integral derivative (PID) controller under various plant uncertainties [14]. The air compressor is a key device, which plays a vital role in many significant departments. Generally, the air compressor unit is the time dependent nonlinear system. The intelligent controller such as the fuzzy logic controller has been considered to ensure the smooth operation with the achievement of effective efficiency and rectify the crisis of air pressure in air compressor [15]. The comparative analysis has been done between PID and FGC controllers in frequency and time domain characteristics for an electrical vehicle to ensure passenger comfort. The detailed study of PID and CRONE controller of robust performance for perturbed plants was recorded [16]. The brief discussion of the tuning methodology of fractional proportional integral (PI) or proportional derivative (PD) controller with the process model and auto-tuning without process model strategy was outlined and has exhibited comparative results [17].The automatic tuning of the PID controller has been proposed for the determination of optimum controller performance and the tuning can be obtained by calculating critical frequency and critical gain for complicated adaptive regulator [18, 19]. The air pressure control supply in the heating, ventilation, and air-conditioning (HVAC) system has been implemented by the adaptive neuro fuzzy (ANF) controller and validated with classical PID controller performance [20]. Moreover, nowadays, the CRONE controller has been implemented in various sectors like automobile [21], thermal application [22], robotics [23], and industrial applications [24].Based on the literature reviewed previously, this article proposes the composite CRONE controller, which is a possible combination of an FGC and a conventional controller in both a parallel and a series configuration. When it gets this different combination of structures, the controller provides two different performances. Among the getting distinct performance, the parallel configuration works more effectively than the series combination. Hence, this article proposes a parallel structure for FGC and ZN-PI controllers.The remaining part of the manuscript is structured as follows: Section2 discusses the material and methods of the procedure. The process of determining the transfer function is revealed in Section 3. The design of controllers is briefly explained in Section 4. Section 5 provides the detailed implementation procedures of the controllers. Respectively, Section 6 presents simulation results. Section 7 presents the conclusions of the proposed system.


## 2. Materials and Methods

### 2.1. Air Pressure System (APS)

Pressure is an important physical parameter that is involved in many process control applications such as boiler drum water level control, atomic reactor, steam boilers, domestic water tanks, and flow transmitter lines. Hence, it is required to measure and control the pressure in order to ensure the safety of the user and equipment.Figure1 depicts a simplified block representation of the APS. The reservoir tank’s storage pressure must be managed and regulated. Its components include pressure transmitter, air filter regulator (AFR), control valve, and pressure reservoir tank. The compressor generates the air source, which is regulated by an air filter regulator in the range of 0–75 psi. The air development is saved in a pressure reservoir tank for later use. The tank pressure is measured using a piezoelectric transmitter. Thus, the reservoir tank pressure is compared to the reference value to determine the error signal. Applying measured error input to the controller determines controller output in terms of electrical value at the span of 4–20 mA. An electric to pressure (E/P) converter is then used to transform the controller output into a pressure signal. The pneumatic control valve is activated by the pressure signal and regulates the pressure inside the reservoir tank.Figure 1
Block diagram of APS [10, 11].


### 2.2. Experimental Setup of APS

Figure2 depicts the real-time experimental arrangement of the APS. The process tank, I/P converter, pressure transmitter, air filter regulator, V-MAT01 card, V/I and I/V converter, pressure transmitter, and pressure gauges comprise the system. The compressor generates air, which is then regulated by an air filter regulator. Through the air to open (ATO) control valve, the regulated air enters the process tank and the compressed air in the process tank is measured by the piezoelectric sensor. The measured current signal is turned into a voltage signal using an I/V converter, and the appropriate digital signal is found using the V-MAT01 card’s analog to digital converter (ADC). Thus, the signal is applied to a controller in the personal computer to manipulate the controller output. Finally, after employing the digital to analog converter (DAC), the pneumatic output is obtained utilizing an I/P converter. A pneumatic control valve uses the pneumatic signal to regulate the air pressure in the process tank.Figure 2
Experimental setup of APS.


## 2.1. Air Pressure System (APS)

Pressure is an important physical parameter that is involved in many process control applications such as boiler drum water level control, atomic reactor, steam boilers, domestic water tanks, and flow transmitter lines. Hence, it is required to measure and control the pressure in order to ensure the safety of the user and equipment.Figure1 depicts a simplified block representation of the APS. The reservoir tank’s storage pressure must be managed and regulated. Its components include pressure transmitter, air filter regulator (AFR), control valve, and pressure reservoir tank. The compressor generates the air source, which is regulated by an air filter regulator in the range of 0–75 psi. The air development is saved in a pressure reservoir tank for later use. The tank pressure is measured using a piezoelectric transmitter. Thus, the reservoir tank pressure is compared to the reference value to determine the error signal. Applying measured error input to the controller determines controller output in terms of electrical value at the span of 4–20 mA. An electric to pressure (E/P) converter is then used to transform the controller output into a pressure signal. The pneumatic control valve is activated by the pressure signal and regulates the pressure inside the reservoir tank.Figure 1
Block diagram of APS [10, 11].


## 2.2. Experimental Setup of APS

Figure2 depicts the real-time experimental arrangement of the APS. The process tank, I/P converter, pressure transmitter, air filter regulator, V-MAT01 card, V/I and I/V converter, pressure transmitter, and pressure gauges comprise the system. The compressor generates air, which is then regulated by an air filter regulator. Through the air to open (ATO) control valve, the regulated air enters the process tank and the compressed air in the process tank is measured by the piezoelectric sensor. The measured current signal is turned into a voltage signal using an I/V converter, and the appropriate digital signal is found using the V-MAT01 card’s analog to digital converter (ADC). Thus, the signal is applied to a controller in the personal computer to manipulate the controller output. Finally, after employing the digital to analog converter (DAC), the pneumatic output is obtained utilizing an I/P converter. A pneumatic control valve uses the pneumatic signal to regulate the air pressure in the process tank.Figure 2
Experimental setup of APS.


## 3. Process Model Identification, APS

### 3.1. Worst-Case Modelling Strategy

Air pressure in the process tank is maintained at the desired level in the real-time environment for a variety of operating points, including 30%, 40%, 50%, and 60%. Figures3 and 4 show the effect of applying 10% of the step input to each operating point and observing the change in pressure value with respect to time until reaching a new steady state. Using the observed data process gain (Kp), time delay (td) and time constant (τp) are measured and the values are summarized in Table 1.Figure 3
+10% positive step changes in open loop responses.Figure 4
−10% negative step changes in open loop responses.Table 1
Determination of the worst-case model parameters.
Operating region %KpτptdKp(max)τp(min)td(max)40–500.96706.30.9668740–300.768750–601.1646.31.1646.350–400.8706.2Model recognised in a broad region of 30–60%1.1647It is clear that a larger process gain (Kp), a smaller time constant (tp), and a longer delay (td) were taken into account from the tabulation for the identification of worst-case parameters [25] and used for the design of controllers.As a result of the APS, the worst-case model transfer function is obtained as(1)Gs=1.164s+1e−7s.


## 3.1. Worst-Case Modelling Strategy

Air pressure in the process tank is maintained at the desired level in the real-time environment for a variety of operating points, including 30%, 40%, 50%, and 60%. Figures3 and 4 show the effect of applying 10% of the step input to each operating point and observing the change in pressure value with respect to time until reaching a new steady state. Using the observed data process gain (Kp), time delay (td) and time constant (τp) are measured and the values are summarized in Table 1.Figure 3
+10% positive step changes in open loop responses.Figure 4
−10% negative step changes in open loop responses.Table 1
Determination of the worst-case model parameters.
Operating region %KpτptdKp(max)τp(min)td(max)40–500.96706.30.9668740–300.768750–601.1646.31.1646.350–400.8706.2Model recognised in a broad region of 30–60%1.1647It is clear that a larger process gain (Kp), a smaller time constant (tp), and a longer delay (td) were taken into account from the tabulation for the identification of worst-case parameters [25] and used for the design of controllers.As a result of the APS, the worst-case model transfer function is obtained as(1)Gs=1.164s+1e−7s.


## 4. Controller Strategies

### 4.1. FGC Controller Design Procedure

Figure5 shows the generalized closed-loop configuration of the FGC controller methodology [3].Figure 5
Closed-loop configuration of FGC controller.The FGC controller technique is appropriate when the response is asymptotic and the desired open loop gain crossover frequencyωcg is located within a band of the frequency range between ωA and ωB. The nominal and perturbed plant phase values remain constant in this frequency range, and changes in gain are the only factors affecting the plant’s uncertainty. The transfer function of the FGC controller CF(s) is identified within the band of the frequency range between ωA and ωB close by the ωcg for “n”th order integrodifferentiator, and it is expressed as(2)CFs=C0sn,withn,C0ϵ♣,where “n” correlates the fractional order open loop integrodifferentiator and C0 is the FGC controller gain.Figure6 depicts the FGC controller’s Bode plot. As seen in the figure, the controller ensures that the phase value (n/2) around the ωcg [2] remains constant throughout the frequency range of ωA and ωB.Figure 6
Bode plot of the FGC controller.Even if the plant corner frequency or plant gain changed in this constant phase controllerCF(s), the controller’s phase margin could not. Therefore, the frequency band (ωA, ωB) needs to be within ωcg. By utilizing corner frequencies ωl and ωh, the transfer function of the FGC controller CF(s) can also be characterized using a band-limited form of the transfer function and is then defined as follows:(3)CFs=C01+s/ωl1+s/ωhn,ωl<ωAandωh<ωB.In order to elude steady-state error and attain the control effort, the band-restricted integrator with ordernI and the low-pass filter with order nF must be cascaded in addition to the band-limited noninteger order transfer function FGC controller.(4)CFs=C0ωIs+1nI1+s/ωl1+s/ωhn11+s/ωFnF.To suppress the amplification of high-frequency noise and input disturbances, it is necessary to satisfy the conditions of the controllerωI < ωl < ωcg > ωh > ωF and the order of the integrator nI and nF = 1 accordingly.


### 4.2. Composite CRONE Controller

After a few decades, the researchers concentrated on the development of new controllers, which outperform the existing controllers. Meanwhile, a new controller is proposed in this research work and is named as composite CRONE controller. The composite CRONE controller is a combination of the FGC controller and conventional controller.A systematic endeavour was made to use CRONE controllers in parallel with a conventional controller. It is observed that the CRONE controllerCFGC(s) with conventional controller CCONV(s) in a peculiar arrangement of summed output parallel form (Figure 7) outperforms all other controllers based on its performance. The same has been tested and shows superior performance. This combination is named as composite CRONE controller.Figure 7
Peculiar arrangement of the CC controller.In Figure8, the composite CRONE controller closed-loop architecture is depicted.Figure 8
Closed-loop structure of the proposed CC controller.As a result, the CC controller transfer function is defined as the combination of a rationalized FGC controller and a conventional controller, and it is written as(5)CCCs=CFGCs+CCONVs.


### 4.3. Conventional Controller

Ziegler and Nichols (ZN), engineers at a control hardware company in the United States, introduced the most well-known tuning method in 1942 [14]. They developed a closed-loop tuning mechanism based on their knowledge of transients from diverse processes, and it is still in use in some capacity today.Because the controller remains in the loop as an active controller in automated mode, the method is described as a closed-loop method. Using the ZN tuning method, PID controllers can be automatically tuned online. The conventional controller is responsible for monitoring the system or plant with regard to set points and process variables, regardless of the system’s or plant’s nature.The Ziegler–Nichols tuning approach [18] was the main effort to establish a realistic approach for dealing with the PID controller tuning. The ZN-PI controller is made using the APS worst-case model. It states that the controller transfer function is written as(6)Cs=KC+KCKIs,where(7)KC=0.9τKP×td,KI=1τI=13.33×td.


## 4.1. FGC Controller Design Procedure

Figure5 shows the generalized closed-loop configuration of the FGC controller methodology [3].Figure 5
Closed-loop configuration of FGC controller.The FGC controller technique is appropriate when the response is asymptotic and the desired open loop gain crossover frequencyωcg is located within a band of the frequency range between ωA and ωB. The nominal and perturbed plant phase values remain constant in this frequency range, and changes in gain are the only factors affecting the plant’s uncertainty. The transfer function of the FGC controller CF(s) is identified within the band of the frequency range between ωA and ωB close by the ωcg for “n”th order integrodifferentiator, and it is expressed as(2)CFs=C0sn,withn,C0ϵ♣,where “n” correlates the fractional order open loop integrodifferentiator and C0 is the FGC controller gain.Figure6 depicts the FGC controller’s Bode plot. As seen in the figure, the controller ensures that the phase value (n/2) around the ωcg [2] remains constant throughout the frequency range of ωA and ωB.Figure 6
Bode plot of the FGC controller.Even if the plant corner frequency or plant gain changed in this constant phase controllerCF(s), the controller’s phase margin could not. Therefore, the frequency band (ωA, ωB) needs to be within ωcg. By utilizing corner frequencies ωl and ωh, the transfer function of the FGC controller CF(s) can also be characterized using a band-limited form of the transfer function and is then defined as follows:(3)CFs=C01+s/ωl1+s/ωhn,ωl<ωAandωh<ωB.In order to elude steady-state error and attain the control effort, the band-restricted integrator with ordernI and the low-pass filter with order nF must be cascaded in addition to the band-limited noninteger order transfer function FGC controller.(4)CFs=C0ωIs+1nI1+s/ωl1+s/ωhn11+s/ωFnF.To suppress the amplification of high-frequency noise and input disturbances, it is necessary to satisfy the conditions of the controllerωI < ωl < ωcg > ωh > ωF and the order of the integrator nI and nF = 1 accordingly.


## 4.2. Composite CRONE Controller

After a few decades, the researchers concentrated on the development of new controllers, which outperform the existing controllers. Meanwhile, a new controller is proposed in this research work and is named as composite CRONE controller. The composite CRONE controller is a combination of the FGC controller and conventional controller.A systematic endeavour was made to use CRONE controllers in parallel with a conventional controller. It is observed that the CRONE controllerCFGC(s) with conventional controller CCONV(s) in a peculiar arrangement of summed output parallel form (Figure 7) outperforms all other controllers based on its performance. The same has been tested and shows superior performance. This combination is named as composite CRONE controller.Figure 7
Peculiar arrangement of the CC controller.In Figure8, the composite CRONE controller closed-loop architecture is depicted.Figure 8
Closed-loop structure of the proposed CC controller.As a result, the CC controller transfer function is defined as the combination of a rationalized FGC controller and a conventional controller, and it is written as(5)CCCs=CFGCs+CCONVs.


## 4.3. Conventional Controller

Ziegler and Nichols (ZN), engineers at a control hardware company in the United States, introduced the most well-known tuning method in 1942 [14]. They developed a closed-loop tuning mechanism based on their knowledge of transients from diverse processes, and it is still in use in some capacity today.Because the controller remains in the loop as an active controller in automated mode, the method is described as a closed-loop method. Using the ZN tuning method, PID controllers can be automatically tuned online. The conventional controller is responsible for monitoring the system or plant with regard to set points and process variables, regardless of the system’s or plant’s nature.The Ziegler–Nichols tuning approach [18] was the main effort to establish a realistic approach for dealing with the PID controller tuning. The ZN-PI controller is made using the APS worst-case model. It states that the controller transfer function is written as(6)Cs=KC+KCKIs,where(7)KC=0.9τKP×td,KI=1τI=13.33×td.


## 5. Implementations of the Controller

The CRONE CSD toolbox [25, 26] plays a vital role in this article, and it is mainly used to implement CRONE controllers for an APS. The primary stage of the design procedure of the CRONE controller is to define the plant model with different perturbations and user-defined CRONE controller design specifications. The change in plant gain is considered as a perturbation of the plant model.The CRONE research group created and maintains the CRONE control system design (CSD) toolbox. It enables the user to configure essential requirements such as plant parameter and its perturbations, frequency domain and time domain specifications, open loop fixed parameters and open loop adjusted parameters, and soon.As a result, the information from the APS plant model is kept in the CRONE CSD (CRONE control system design) toolbox for future usage. Its plant disturbances are also taken into account. Because a broader range of disturbance is anticipated to integrate robustness to the controller, the plant gain is perturbed to a range of 20% and is in the range of 0.88 < 1.1 < 1.32.The user-defined CRONE controller design specifications include open loop gain crossover frequencyωcg, needed phase margin Pm, and soon. Table 2 shows the user-defined settings for the system under consideration in this work.Table 2
CRONE controller settings.
Open loop gain crossover frequency-ωcg3ωA/ωl-ratio10Phase margin,Pm56.3921ωh/ωB-ratio1Integral order,ni1ωAB/ωcg-ratio10Low-pass filter order,nf1ωcg/ωi-ratio30Rational approximation cell no.,N5ωf/ωcg-ratio30Fractional effect width,ωA/ωB ratio0.8783All three controllers are presented in Table3 after using the design methodologies discussed in Section 4 and with the support of the CRONE CSD toolbox [26].Table 3
Controller design parameters.
ControllersStructure of the controllerZN-PICZN−PIs=6.857+0.293/sFGCCFRs=1.427Xs+0.1s+0.6482s+1.5349s+3.4735/ss+0.5372s+1.2056s+2.4940Xs+8.417s+16.382/s+6.3741s+13.2s+90CCFGC-ZN-PICFR−ZN−PIs=1.427Xs+0.1s+0.6482s+1.5349s+3.4735/ss+0.5372s+1.2056s+2.4940Xs+8.417s+16.382/s+6.3741s+13.2s+90+6.857+0.293/s


## 6. Result and Discussion

### 6.1. Set Point Tracking Performance

The effectiveness of the controller is evaluated using error indices like as integral absolute error (IAE) and integral square error (ISE), in addition to time domain parameters such as settling time (ts) and rise time (tr).The set point tracking method is used to determine the performance of controllers such as the ZN-PI, FGC, and CC. The simulations are recorded in MAT lab for three different operating pressures such as 40%,50%, and 60% of the air pressure system and different step changes of −5%, +5%, −10%, and +10% are applied at each operating pressure, respectively. The set point tracking response for the above-discussed controllers are validated and the recorded the simulated responses are shown in Figures9–17. The performances of these figures are catalogued in Tables 4 and 5.Figure 9
Simulated servo responses at 40% operating pressure using FGC.Figure 10
Simulated servo responses at 50% operating pressure using FGC.Figure 11
Simulated servo responses at 60% operating pressure using FGC.Figure 12
Simulated servo responses at 40% operating pressure using ZN-PI.Figure 13
Simulated servo responses at 50% operating pressure using ZN-PI.Figure 14
Simulated servo responses at 60% operating pressure using ZN-PI.Figure 15
Simulated servo responses at 40% operating pressure using CC.Figure 16
Simulated servo responses at 50% operating pressure using CC.Figure 17
Simulated servo responses at 60% operating pressure using CC.Table 4
Set point tracking performance of the controllers in terms of ISE and IAE.
Step input changeZN-PIFGCCCISEIAEISEIAEISEIAE40%operating pressure+10%492.143.99370.827.09225.516+5%470.839.66354.424.38215.514.4−5%471.836.99354.424.25215.514.33−10%49439.99370.826.96225.515.9350%operating pressure+10%753.352.88567.132.51344.919.2+5%731.948.62550.829.8334.917.61−5%732.746.02550.829.74334.917.57−10%754.849.81567.132.45344.919.1760%operating pressure+10%107261.68807.137.92490.722.4+5%105157.54790.735.22480.820.8−5%105255.4790.735.19480.820.79−10%107459.51807.137.89490.722.39Table 5
Set point tracking performance of the controllers in terms ofts and tr.
Step input changeZN-PIFGCCCtstrtstrtstr40%operating pressure+10%302526201510+5%25232318107−5%24232218106−10%30252520161050%operating pressure+10%302627201510+5%26222317107−5%25232318106−10%29252620161060%operating pressure+10%312628201711+5%26232416107−5%27232317107−10%302527201510Tables2 and 3 show the servo performance in terms of error indices and time domain performances of several controllers such as ZN-PI, FGC, and CC for an air pressure system at three distinct operating pressures with variable step changes. Table 4 provides information about the results obtained through simulations like error indices performance of the set point tracking for various step changes at different operating pressures of an air pressure system; it can be seen from Table 2 that the CC controller provides the ISE value of 225.5 significantly. Whereas, the FGC controller and ZN-PI controllers give 370.8 and 492.1, respectively, for +10% of step changes on 40% of operating pressure mode. Similarly, in the measurement of IAE for −5% step change on 60% of operating pressure, the CC controller yields 16 optimistically. Besides the FGC and conventional controllers dispenses 27.09 and 43.99 accordingly.Hence, the CC controller offers superior performance, which means it gives the least value of ISE and IAE and occupies the first position. Decently, the FGC controller stands second comparatively than the other controllers. The ZN-PI controller is pushed into the last position due to its poor performance in terms of both ISE and IAE value. Likewise, the same types of comparison and validation have been made for all other operating pressures for different step input changes.Table5 deliberates the time domain performances of an air pressure system for the three different operating pressures with different step input changes of a process system. It is validated from Table 3 that the CC controller output settles at 15 seconds for +10% of step input changes of 40% operating pressure. Meanwhile, the FGC and ZN-PI controller settles at 26 seconds and 30 seconds, respectively. Similarly, in the view of rise time, for the CC controller is 10 seconds. Meanwhile, for the FGC and conventional controller is 20 and 26 seconds periodically. Therefore, the CC controller gives extremely good performances and stands in first place by providing lesser values of ts and tr, respectively. The FGC and ZN-PI controllers are giving second and third-level performance when compared with other controller strategies. Finally, the ZN-PI controller stands the last position due to its higher value of ts and tr.Figures18 and 19 make it abundantly evident how much better the proposed CC controller’s robust performance is than that of the other controller utilized for comparison.Figure 18
Graphical representations of ISE and IAE performances.Figure 19
Graphical representation of settling time and rise time performances.The graphical representation of performances criteria of the error indices and time domain performances for all the controller strategies for three different operating pressure with different step changes are shown in Figures18 and 19.


## 6.1. Set Point Tracking Performance

The effectiveness of the controller is evaluated using error indices like as integral absolute error (IAE) and integral square error (ISE), in addition to time domain parameters such as settling time (ts) and rise time (tr).The set point tracking method is used to determine the performance of controllers such as the ZN-PI, FGC, and CC. The simulations are recorded in MAT lab for three different operating pressures such as 40%,50%, and 60% of the air pressure system and different step changes of −5%, +5%, −10%, and +10% are applied at each operating pressure, respectively. The set point tracking response for the above-discussed controllers are validated and the recorded the simulated responses are shown in Figures9–17. The performances of these figures are catalogued in Tables 4 and 5.Figure 9
Simulated servo responses at 40% operating pressure using FGC.Figure 10
Simulated servo responses at 50% operating pressure using FGC.Figure 11
Simulated servo responses at 60% operating pressure using FGC.Figure 12
Simulated servo responses at 40% operating pressure using ZN-PI.Figure 13
Simulated servo responses at 50% operating pressure using ZN-PI.Figure 14
Simulated servo responses at 60% operating pressure using ZN-PI.Figure 15
Simulated servo responses at 40% operating pressure using CC.Figure 16
Simulated servo responses at 50% operating pressure using CC.Figure 17
Simulated servo responses at 60% operating pressure using CC.Table 4
Set point tracking performance of the controllers in terms of ISE and IAE.
Step input changeZN-PIFGCCCISEIAEISEIAEISEIAE40%operating pressure+10%492.143.99370.827.09225.516+5%470.839.66354.424.38215.514.4−5%471.836.99354.424.25215.514.33−10%49439.99370.826.96225.515.9350%operating pressure+10%753.352.88567.132.51344.919.2+5%731.948.62550.829.8334.917.61−5%732.746.02550.829.74334.917.57−10%754.849.81567.132.45344.919.1760%operating pressure+10%107261.68807.137.92490.722.4+5%105157.54790.735.22480.820.8−5%105255.4790.735.19480.820.79−10%107459.51807.137.89490.722.39Table 5
Set point tracking performance of the controllers in terms ofts and tr.
Step input changeZN-PIFGCCCtstrtstrtstr40%operating pressure+10%302526201510+5%25232318107−5%24232218106−10%30252520161050%operating pressure+10%302627201510+5%26222317107−5%25232318106−10%29252620161060%operating pressure+10%312628201711+5%26232416107−5%27232317107−10%302527201510Tables2 and 3 show the servo performance in terms of error indices and time domain performances of several controllers such as ZN-PI, FGC, and CC for an air pressure system at three distinct operating pressures with variable step changes. Table 4 provides information about the results obtained through simulations like error indices performance of the set point tracking for various step changes at different operating pressures of an air pressure system; it can be seen from Table 2 that the CC controller provides the ISE value of 225.5 significantly. Whereas, the FGC controller and ZN-PI controllers give 370.8 and 492.1, respectively, for +10% of step changes on 40% of operating pressure mode. Similarly, in the measurement of IAE for −5% step change on 60% of operating pressure, the CC controller yields 16 optimistically. Besides the FGC and conventional controllers dispenses 27.09 and 43.99 accordingly.Hence, the CC controller offers superior performance, which means it gives the least value of ISE and IAE and occupies the first position. Decently, the FGC controller stands second comparatively than the other controllers. The ZN-PI controller is pushed into the last position due to its poor performance in terms of both ISE and IAE value. Likewise, the same types of comparison and validation have been made for all other operating pressures for different step input changes.Table5 deliberates the time domain performances of an air pressure system for the three different operating pressures with different step input changes of a process system. It is validated from Table 3 that the CC controller output settles at 15 seconds for +10% of step input changes of 40% operating pressure. Meanwhile, the FGC and ZN-PI controller settles at 26 seconds and 30 seconds, respectively. Similarly, in the view of rise time, for the CC controller is 10 seconds. Meanwhile, for the FGC and conventional controller is 20 and 26 seconds periodically. Therefore, the CC controller gives extremely good performances and stands in first place by providing lesser values of ts and tr, respectively. The FGC and ZN-PI controllers are giving second and third-level performance when compared with other controller strategies. Finally, the ZN-PI controller stands the last position due to its higher value of ts and tr.Figures18 and 19 make it abundantly evident how much better the proposed CC controller’s robust performance is than that of the other controller utilized for comparison.Figure 18
Graphical representations of ISE and IAE performances.Figure 19
Graphical representation of settling time and rise time performances.The graphical representation of performances criteria of the error indices and time domain performances for all the controller strategies for three different operating pressure with different step changes are shown in Figures18 and 19.


## 7. Conclusions

The presented article has proposed two distinct noninteger order controllers with a conventional controller. These controllers are designed using the transfer function of the air pressure system, which is analytically derived based on the worst-case modelling technique. The comparison has been made between the CRONE controller, composite controller, and conventional controller. It examines the robust performance validation of the controller in terms of the error and time indices. The set point tracking simulation results have been obtained and validated in the MAT lab platform with help of the CSD tool box. Hence, it is seen from the result of validation that the composite CRONE controller provides excellent performance than the FGC controller, and similarly, FGC was better than the conventional controller.



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*Source: 9933623-2023-02-03.xml*