# A New Design Method for PI-PD Control of Unstable Fractional-Order System with Time Delay

Authors: Min Zheng; Tao Huang; Guangfeng Zhang
Journal: Complexity
 (2019)

---

## Abstract

In this paper, a practical PI-PD controller parameter tuning method is proposed, which uses the incenter of the triangle and the Fermat point of the convex polygon to optimize the PI-PD controller. Combined with the stability boundary locus method, the PI-PD controller parameters that can ensure stability for the unstable fractional-order system with time delay are obtained. Firstly, the parameters of the inner-loop PD controller are determined by the centre coordinates of the CSR in thekd−kf plane. Secondly, a new graphical method is used to calculate the parameters of the PI controller, in which Fermat points in the CSR of (kp−ki) plane are selected. Furthermore, the method is extended to uncertain systems, and the PI-PD controller parameters are obtained by using the proposed method through common stable region of all stable regions. The proposed graphical method not only ensures the stability of the closed-loop system but also avoids the complicated optimization calculations. The superior control performance of this method is illustrated by simulation.


---

## Body

## 1. Introduction

Proportional-integral-derivative (PID) controller has been widely used in industrial control systems for decades because of its simple structure and convenient implementation [1–6]. As the control plant becomes more and more complex, the performance of the controller is required to be higher and higher and the controllability of the system becomes difficult to solve. By using Schade’s fixed point theorem, the controllability problem of nonlinear fractional integro-differential dynamical systems is transformed into the fixed point problem in [7]. The integral-order PID (IOPID) controller has limitations in control integration, instability, and delay process, and it often leads to large step response, large overshoot, and large impact, especially for unstable complex systems with time delay, and it is difficult to obtain good closed-loop performance [8–12]. Researchers have proposed a series of controller design schemes, including integer-order PID (IOPID) controller, sliding mode controller (SMC) to fractional-order PID (FOPID) controller, fractional-order sliding mode controller and other complex controllers, and so on [12–19]. The computational cost of these controllers increases geometrically with the change of controller structure complexity, but the performance improvement is not satisfactory. For this reason, researchers put forward a PI-PD controller structure by changing the structure of PID controllers in [8, 9, 20]. The controller converts the unstable open-loop plant into the stable open-loop plant through the PD controller of the internal loop circuit, ensuring that the open-loop poles of the resonance and integration processes are in the proper position. The PI controller of the outer loop controls the inner loop of the system as a whole. Many studies have shown that the PI-PD structure has superior closed-loop performance for an unstable system with time delay [20]. The IOPID controller has three adjustable parameters, while the PI-PD controller has four adjustable parameters. Therefore, the PI-PD controller has advantages over the IOPID controller in terms of control structure and control freedom [21, 22]. The traditional controller has a good control effect on the precise model with certain parameters. However, in most cases, the exact model of the actual system could not be obtained [23]. Therefore, an excellent controller design must meet the requirements of robustness, stability, control performance, and the ability to overcome the uncertainty of model parameters [24, 25].In practical applications, PID controller parameter tuning methods are divided into three categories: the iterative optimization method based on intelligent optimization [26, 27]; the setting formula method for specific system [3, 4, 28]; and a parameter selection method [29–34] based on the stable region. The stable boundary locus (SBL) method has been widely studied, which provides an effective solution for some specific systems.In recent years, PI-PD controllers have been used extensively in systems with unstable processes and time delay. Kaya and Atherton [28] studied the first-order time-delay unstable system of a PI-PD controller by using a simple regulation formula tuning method, taking integral of squared error (ISE) and integral of squared time weighted error (ISTE) as performance indicators. Tan [29] and Srivastava and Pandit [30] proposed a graphic method based on SBL by using gain phase tester to achieve the specified gain and phase margin and extended this method to the systems with parameter uncertainty by Kharitonov theorem. For the systems with uncertain time-varying delays, Shariati proposed a neutral system approach to stabilize and synthesize H∞ PI-PD controllers. A new bounded real lemma was given for neutral systems with efficiency in designing the H∞ controllers with both state delay and state derivative delay coefficients depending on the controller parameters [35]. Onat et al. [31, 32] proposed the concept of the weight geometric centre (WGC) method based on stable domain, which was applied to the parameter design of a PI controller. Ozyetkin [36] extended the WGC method to the fractional-order PI-PD. All the sample points that make up the stable locus boundary would be used to calculate the coordinate of WGC point. However, it has the large calculation cost which is not conducive to the application of real-time system control. In addition, under fractional-order control, WGC points do not always fall into the stable region. Onat [34] simplified the WGC method and constructed a convex stabilization region with only three points to ensure that the WGC is in the stable region.However, the above literature only considered integer-order systems. On the other hand, there is little research on the PI-PD controller design through these graphical parameter selection methods. Inspired by the above analysis, the main contribution of this paper is to solve the parameters of the general fractional-order time-delay system PI-PD controller by the graphical method and generalize it to the parameter uncertainty system. In this paper, a new method to tune the parameters of PI-PD is proposed by using the incenter point, Fermat point, and geometry method in convex stable region (CSR). The main design process includes the following steps: firstly, for the open-loop uncertain system, the internal loop PD controller is designed. The stability boundary locus on thekd−kf plane is obtained by using the internal loop closed-loop characteristic polynomial. Then, the incenter coordinates of the CSR are computed. Secondly, embedding PD controller parameters into transfer function, desired stability boundary locus of Pl controller is computed by the same procedure. Then, a quadrilateral Fermat point in CSR of the outer loop is obtained which is the parameter of the PI controller. This method only needs simple geometric calculation to obtain the parameters of the PI-PD controller, which is solved in the stable region to ensure the closed-loop stability. It uses several special points of the stable region and fewer points describing the coordinates than the WGC method, ensuring that the calculated points are situated in the stable region and the PI-PD controller has superior robustness. The simulation example shows that the method presented in this paper has a satisfactory control effect.The structure of this paper is organized as follows. In Section2, the problem statements of the controlled plant and controller are given, and then the proposed method is presented and extended to uncertain systems. Simulative and experimental application examples are considered to illustrate the effectiveness of the proposed method in Section 3. Section 4 gives conclusion.


## 2. Main Results

### 2.1. Problem Statement

The generalized PI-PD negative feedback system shown in Figure1, where r is the set point and y is the negative feedback output. The generalized fractional transfer function Gps is written as follows:(1)Gps=NpsDpse−τs,where τ is the time delay and Nps and Dps are the numerator and the denominator of the transfer function, respectively. They are defined with the forms:(2)Nps=b0sβ0+b1sβ1+⋯+bn−1sβn−1+bnsβn=∑i=0nbisβi,Dps=a0sα0+a1sα1+⋯+an−1sαn+ansαn=∑i=0naisαi,where aj,bi,αj,βi,i,j=0,1,…,n are real numbers and αn>αn−1>⋯>α2>α1>α0≥0,βn>βn−1>⋯>β2>β1>β0≥0,αn>βn. The internal loop PD controller CPDs and external loop PI controller CPIs are defined as(3)CPDs=NPDsDPDs=kf+kdsTs+1,(4)CPIs=NPIsDPIs=kp+kis,where T=0.01, kf and kd are proportional and differential gains of the internal loop PD controller, and kp and ki are proportional and integral gains of the outer-loop PI controller, respectively.Figure 1
The PI-PD control system for generalized fractional plant.The main purpose is to solve the parameters of the generalized fractional-order time-delay system (1) with PI-PD controller (3) and (4) by the graphical method.


### 2.2. The Proposed Method

The PI-PD controller has four adjustable parameters (kf,kd,kp,ki) that need to be determined. The internal PD controller is used to change the poles of the plant transfer function Gps to more desirable locations for control by the outer-loop PI controller.Substitutingsα=ωαcosαπ/2+jsinαπ/2 into (1) and dividing the numerator and denominator into a real part and an imaginary part, one can write(5)NpjωDpjω=NPEjω+jNPOjωDPEjω+jDPOjω.SimplifyNPEjω,NPOjω,DPEjω,andDPOjω as NPE,NPO,DPE,andDPO, where NPE,NPO are the real part and the imaginary part and DPE,DPO are the real part and the imaginary part, respectively. The expressions are given as(6)NPE=∑i=0nbiωβicosβi2π;NPO=∑i=0nbiωβisinβi2π,DPE=∑i=0naiωαicosαi2π;DPO=∑i=0naiωαicosαi2π.As shown in Figure1, the internal loop negative feedback transfer function is(7)ϕ1s=Gps1+CPDsGps.The closed-loop characteristic polynomial of the internal loop negative feedback transfer function is written as(8)ΔPDs;kf,kd=1+CPDsGps=s+100Dps+kf+100kds+100kfNpse−τs.Substitutes=jω and e−jτω=cosτω−jsinτω into inner-loop transfer function (8) and divide it into real part and imaginary part as follows:(9)ΔPDjω;kf,kd=100DPE+jDPO+ωjDPE−DPO+m1NPE+m2NPO+jm1NPO+m2NPEcosτω+m1NPO+m2NPE−jm1NPE+m2NPOsinτω=ReΔ,PD+jImΔ,PD=0,where m1=100kf and m2=kf+100kdω.Taking the real part and the imaginary part of the characteristic polynomial as zero, they can be simplified into the following form:(10)ReΔ,PD=A1kf+B1kd+C1=0,ImΔ,PD=A2kf+B2kd+C2=0,where A1=100NPE+ωNPOcosτω+100NPO+ωNPEsinτω; A2=100NPO+ωNPEcosτω−100NPE+ωNPOsinτω; B1=100ωNPOcosτω+100ωNPEsinτω; B2=100ωNPEcosτω−100ωNPOsinτω; C1=100DPE−ωDPO; and C2=100DPO−ωDPE.Solving equation (10), we can get the expression of kf and kd and draw the stable boundary curve on the kd−kf plane when ω∈0,∞. The expressions kf and kd are given (11)kd=A1C2−A2C1B1A2−A1B2,kf=B2C1−B1C2B1A2−A1B2.According to the obtained SBL, the vertices and corner points of the stable region are devoted to describe the CSR. In a generalkd−kf plane, the CSR is designed as a triangle with one vertex and two corner points, whose coordinates are kd1,kf1,kd2,kf2,andkd3,kf3. In order to ensure that the obtained point is in the stable region, here we propose the concept of the incenter of the CSR, whose coordinate expression is as follows:(12)H=kd1kd2kd3kf1kf2kf3abca+b+c,where a,b,andc are the modules of the three-sided vectors. The vector H obtained in (12) is the inner-loop PD controller parameter kd;kf.Then, the inner-loop negative feedback system has the following form:(13)ϕ1s=NsDs=Gps1+CPDsGps=s+100Npse−τss+100Dps+Npse−τskf+100kds+100kf.Simplify the inner-loop transfer function (13) and split the numerator and denominator into real parts and imaginary parts, respectively:(14)NjωDjω=NEjω+jNOjωDEjω+jDOjω.SimplifyNEjω,NOjω,DEjω,andDOjω as NE,NO,DE,andDO, where NE,NO are the real part and imaginary part of numerator and DE,DO are the real part and imaginary part of the denominator, respectively. Their expressions are as follows:(15)NE=NPEcosτω+NPOsinτωn1+100+NPEsinτω−NPOcosτωn2,NO=NPEcosτω+NPOsinτωn2+NPOcosτω−NPEsinτωn1+100,DE=A1kf+B1kd+C1;DO=A2kf+B2kd+C2.Regarding the inner loop as a whole, the transfer function of the system is written as(16)ϕ2s=CPIsϕ1s1+CPIsϕ1s.Therefore, the closed-loop characteristic equation of the outer loop is determined as follows:(17)ΔPIs;kp,ki=1+CPIsGps=sDs+kps+kdNs.Substitute (4) and (12) into the closed-loop characteristic equation (16) and divide it into real part and imaginary parts:(18)ΔPIjω;kp,ki=ReΔ,PI+jImΔ,PI=0.Simplify the characteristic equation into a binary equations with unknown parameterskp,ki:(19)ReΔ,PI=A1¯kp+B1¯ki+C1¯=0,ImΔ,PI=A2¯kp+B2¯ki+C2¯=0,where A1¯=−ωNO; A2¯=ωNE; B1¯=NE; B2¯=NO; C1¯=−ωDO; and C2¯=ωDE.The parameters of SBL and the stability region of the outer-loop PI controller on thekp−ki plane are obtained by solving equation (19). And a SBL is drawn on the kp−ki plane:(20)kp=B2¯C1¯−B1¯C2¯B1¯A2¯−A1¯B2¯,ki=A1¯C2¯−A2¯C1¯B1¯A2¯−A1¯B2¯.For obtaining the outer-loop PI controller parameters, this paper introduces a Fermat point concept that has the smallest sum of distances to the vertices of the polygon. In order to optimize calculation of Fermat point, a trapezoid is designed at first. The four vertices include two corner points and two points on the SBL which are 1/2 of the longitudinal distance between the cusp and corner point. Fermat point is the diagonal intersection point.The proposed method includes the following five steps:Step 1: according to equation (11), the relationship between the parameters kd,kf of the inner-loop PD controller could be obtained. And the SBL could be drawn on the kd−kf plane.Step 2: by selecting the vertices and corner points of the CSR, the incenter of the convex CSR is computed according to equation (12). Its coordinates Hkd,kf are the parameters of the inner-loop PD controller.Step 3: calculating the transfer function of the inner loop based on the obtained parameterskd,kf.Step 4: according to equation (20), the relation of the outer-loop PI controller parameters kp,ki could be obtained. The SBL is drawn on the kp−ki plane.Step 5: determining the corner point and the boundary point of the SBL; the intersection point of the trapezoid diagonal is the parameter of the outer-loop PI controller.Remark 1.
For fractional-order time-delay systems, the traditional method of selecting stable regions is usually determined by manually selecting a large number of test points in all the regions divided by the stable boundary trajectories, which will lead to time consumption. Inspired by [37], this paper gives a simple parameter tuning method of FOPID controller, which avoids time-consuming stability test.


### 2.3. Method Extension of Uncertain Systems

Due to modelling errors and other factors, it is an important issue for the analysis and design of the uncertain parameter control systems with time delay in control theory. In this article, the design method is extended to the fractional-order system with time delay and parameter uncertainty. Multiple transfer functions are attained in the condition of uncertain parameters. Consider the transfer function with parameter uncertainty for the control system shown in Figure1. It is defined as follows:(21)Gps=∑j=0nbjsβj∑i=0maisαie−τs,where ai∈a¯i,a¯i,bj∈b¯j,b¯j,αi∈α¯i,α¯i,βj∈β¯j,β¯j, i=0,1,…,m and j=0,1,…,n. According to the number of uncertain parameters in the general transfer function, there are m2×n2 Kharitonov plants in total.For the uncertain parameter system with time delay, all steps of the method put forward in paper are individually processed for each transfer function in terms of the Kharitonov theorem. Then, the common stable domain of all transfer functions and its convex stable domain are obtained. Finally, the incenter of the inner loop and the Fermat points of outer loop of the common convex stability region are obtained, and then a robust PI-PD controller is designed.The proposed method extension of the time-delay uncertain system could be classified into the following steps:Step 1: for all submodels, the inner-loop stable boundary loci are calculated separately, and the common stable region is determined graphically on thekd−kf plane.Step 2: then, the CCSR is determined according to the corner points, intersection points, and individual tip points. The incenter of the CCSR is computed by applying (12). The coordinate of the incenter is the parameter kd,kf of the PD controller in the internal loop.Step 3: by substituting the valueskd and kf into the inner-loop system, the transfer functions of the outer loop are obtained for all the submodels simplified based on the Kharitonov theorem.Step 4: the common stable region of the outer-loop PI controller on thekp−ki plane is determined based on all closed-loop transfer functions.Step 5: looking for corner points, tip points, and bipartite points of the common stable area, the CCSR based on these points is obtained. The Fermat point of the CCSR is given by the graphical method, which is parameterkp,ki of the outer-loop PI controller.Remark 2.
The method proposed in this paper is an empirical parameter adjustment method, which is only suitable for systems that can clearly distinguish stable regions in images. For uncertain systems, it is only applicable to the case with common stable region. Otherwise, further research is needed to find other control methods.


## 2.1. Problem Statement

The generalized PI-PD negative feedback system shown in Figure1, where r is the set point and y is the negative feedback output. The generalized fractional transfer function Gps is written as follows:(1)Gps=NpsDpse−τs,where τ is the time delay and Nps and Dps are the numerator and the denominator of the transfer function, respectively. They are defined with the forms:(2)Nps=b0sβ0+b1sβ1+⋯+bn−1sβn−1+bnsβn=∑i=0nbisβi,Dps=a0sα0+a1sα1+⋯+an−1sαn+ansαn=∑i=0naisαi,where aj,bi,αj,βi,i,j=0,1,…,n are real numbers and αn>αn−1>⋯>α2>α1>α0≥0,βn>βn−1>⋯>β2>β1>β0≥0,αn>βn. The internal loop PD controller CPDs and external loop PI controller CPIs are defined as(3)CPDs=NPDsDPDs=kf+kdsTs+1,(4)CPIs=NPIsDPIs=kp+kis,where T=0.01, kf and kd are proportional and differential gains of the internal loop PD controller, and kp and ki are proportional and integral gains of the outer-loop PI controller, respectively.Figure 1
The PI-PD control system for generalized fractional plant.The main purpose is to solve the parameters of the generalized fractional-order time-delay system (1) with PI-PD controller (3) and (4) by the graphical method.


## 2.2. The Proposed Method

The PI-PD controller has four adjustable parameters (kf,kd,kp,ki) that need to be determined. The internal PD controller is used to change the poles of the plant transfer function Gps to more desirable locations for control by the outer-loop PI controller.Substitutingsα=ωαcosαπ/2+jsinαπ/2 into (1) and dividing the numerator and denominator into a real part and an imaginary part, one can write(5)NpjωDpjω=NPEjω+jNPOjωDPEjω+jDPOjω.SimplifyNPEjω,NPOjω,DPEjω,andDPOjω as NPE,NPO,DPE,andDPO, where NPE,NPO are the real part and the imaginary part and DPE,DPO are the real part and the imaginary part, respectively. The expressions are given as(6)NPE=∑i=0nbiωβicosβi2π;NPO=∑i=0nbiωβisinβi2π,DPE=∑i=0naiωαicosαi2π;DPO=∑i=0naiωαicosαi2π.As shown in Figure1, the internal loop negative feedback transfer function is(7)ϕ1s=Gps1+CPDsGps.The closed-loop characteristic polynomial of the internal loop negative feedback transfer function is written as(8)ΔPDs;kf,kd=1+CPDsGps=s+100Dps+kf+100kds+100kfNpse−τs.Substitutes=jω and e−jτω=cosτω−jsinτω into inner-loop transfer function (8) and divide it into real part and imaginary part as follows:(9)ΔPDjω;kf,kd=100DPE+jDPO+ωjDPE−DPO+m1NPE+m2NPO+jm1NPO+m2NPEcosτω+m1NPO+m2NPE−jm1NPE+m2NPOsinτω=ReΔ,PD+jImΔ,PD=0,where m1=100kf and m2=kf+100kdω.Taking the real part and the imaginary part of the characteristic polynomial as zero, they can be simplified into the following form:(10)ReΔ,PD=A1kf+B1kd+C1=0,ImΔ,PD=A2kf+B2kd+C2=0,where A1=100NPE+ωNPOcosτω+100NPO+ωNPEsinτω; A2=100NPO+ωNPEcosτω−100NPE+ωNPOsinτω; B1=100ωNPOcosτω+100ωNPEsinτω; B2=100ωNPEcosτω−100ωNPOsinτω; C1=100DPE−ωDPO; and C2=100DPO−ωDPE.Solving equation (10), we can get the expression of kf and kd and draw the stable boundary curve on the kd−kf plane when ω∈0,∞. The expressions kf and kd are given (11)kd=A1C2−A2C1B1A2−A1B2,kf=B2C1−B1C2B1A2−A1B2.According to the obtained SBL, the vertices and corner points of the stable region are devoted to describe the CSR. In a generalkd−kf plane, the CSR is designed as a triangle with one vertex and two corner points, whose coordinates are kd1,kf1,kd2,kf2,andkd3,kf3. In order to ensure that the obtained point is in the stable region, here we propose the concept of the incenter of the CSR, whose coordinate expression is as follows:(12)H=kd1kd2kd3kf1kf2kf3abca+b+c,where a,b,andc are the modules of the three-sided vectors. The vector H obtained in (12) is the inner-loop PD controller parameter kd;kf.Then, the inner-loop negative feedback system has the following form:(13)ϕ1s=NsDs=Gps1+CPDsGps=s+100Npse−τss+100Dps+Npse−τskf+100kds+100kf.Simplify the inner-loop transfer function (13) and split the numerator and denominator into real parts and imaginary parts, respectively:(14)NjωDjω=NEjω+jNOjωDEjω+jDOjω.SimplifyNEjω,NOjω,DEjω,andDOjω as NE,NO,DE,andDO, where NE,NO are the real part and imaginary part of numerator and DE,DO are the real part and imaginary part of the denominator, respectively. Their expressions are as follows:(15)NE=NPEcosτω+NPOsinτωn1+100+NPEsinτω−NPOcosτωn2,NO=NPEcosτω+NPOsinτωn2+NPOcosτω−NPEsinτωn1+100,DE=A1kf+B1kd+C1;DO=A2kf+B2kd+C2.Regarding the inner loop as a whole, the transfer function of the system is written as(16)ϕ2s=CPIsϕ1s1+CPIsϕ1s.Therefore, the closed-loop characteristic equation of the outer loop is determined as follows:(17)ΔPIs;kp,ki=1+CPIsGps=sDs+kps+kdNs.Substitute (4) and (12) into the closed-loop characteristic equation (16) and divide it into real part and imaginary parts:(18)ΔPIjω;kp,ki=ReΔ,PI+jImΔ,PI=0.Simplify the characteristic equation into a binary equations with unknown parameterskp,ki:(19)ReΔ,PI=A1¯kp+B1¯ki+C1¯=0,ImΔ,PI=A2¯kp+B2¯ki+C2¯=0,where A1¯=−ωNO; A2¯=ωNE; B1¯=NE; B2¯=NO; C1¯=−ωDO; and C2¯=ωDE.The parameters of SBL and the stability region of the outer-loop PI controller on thekp−ki plane are obtained by solving equation (19). And a SBL is drawn on the kp−ki plane:(20)kp=B2¯C1¯−B1¯C2¯B1¯A2¯−A1¯B2¯,ki=A1¯C2¯−A2¯C1¯B1¯A2¯−A1¯B2¯.For obtaining the outer-loop PI controller parameters, this paper introduces a Fermat point concept that has the smallest sum of distances to the vertices of the polygon. In order to optimize calculation of Fermat point, a trapezoid is designed at first. The four vertices include two corner points and two points on the SBL which are 1/2 of the longitudinal distance between the cusp and corner point. Fermat point is the diagonal intersection point.The proposed method includes the following five steps:Step 1: according to equation (11), the relationship between the parameters kd,kf of the inner-loop PD controller could be obtained. And the SBL could be drawn on the kd−kf plane.Step 2: by selecting the vertices and corner points of the CSR, the incenter of the convex CSR is computed according to equation (12). Its coordinates Hkd,kf are the parameters of the inner-loop PD controller.Step 3: calculating the transfer function of the inner loop based on the obtained parameterskd,kf.Step 4: according to equation (20), the relation of the outer-loop PI controller parameters kp,ki could be obtained. The SBL is drawn on the kp−ki plane.Step 5: determining the corner point and the boundary point of the SBL; the intersection point of the trapezoid diagonal is the parameter of the outer-loop PI controller.Remark 1.
For fractional-order time-delay systems, the traditional method of selecting stable regions is usually determined by manually selecting a large number of test points in all the regions divided by the stable boundary trajectories, which will lead to time consumption. Inspired by [37], this paper gives a simple parameter tuning method of FOPID controller, which avoids time-consuming stability test.


## 2.3. Method Extension of Uncertain Systems

Due to modelling errors and other factors, it is an important issue for the analysis and design of the uncertain parameter control systems with time delay in control theory. In this article, the design method is extended to the fractional-order system with time delay and parameter uncertainty. Multiple transfer functions are attained in the condition of uncertain parameters. Consider the transfer function with parameter uncertainty for the control system shown in Figure1. It is defined as follows:(21)Gps=∑j=0nbjsβj∑i=0maisαie−τs,where ai∈a¯i,a¯i,bj∈b¯j,b¯j,αi∈α¯i,α¯i,βj∈β¯j,β¯j, i=0,1,…,m and j=0,1,…,n. According to the number of uncertain parameters in the general transfer function, there are m2×n2 Kharitonov plants in total.For the uncertain parameter system with time delay, all steps of the method put forward in paper are individually processed for each transfer function in terms of the Kharitonov theorem. Then, the common stable domain of all transfer functions and its convex stable domain are obtained. Finally, the incenter of the inner loop and the Fermat points of outer loop of the common convex stability region are obtained, and then a robust PI-PD controller is designed.The proposed method extension of the time-delay uncertain system could be classified into the following steps:Step 1: for all submodels, the inner-loop stable boundary loci are calculated separately, and the common stable region is determined graphically on thekd−kf plane.Step 2: then, the CCSR is determined according to the corner points, intersection points, and individual tip points. The incenter of the CCSR is computed by applying (12). The coordinate of the incenter is the parameter kd,kf of the PD controller in the internal loop.Step 3: by substituting the valueskd and kf into the inner-loop system, the transfer functions of the outer loop are obtained for all the submodels simplified based on the Kharitonov theorem.Step 4: the common stable region of the outer-loop PI controller on thekp−ki plane is determined based on all closed-loop transfer functions.Step 5: looking for corner points, tip points, and bipartite points of the common stable area, the CCSR based on these points is obtained. The Fermat point of the CCSR is given by the graphical method, which is parameterkp,ki of the outer-loop PI controller.Remark 2.
The method proposed in this paper is an empirical parameter adjustment method, which is only suitable for systems that can clearly distinguish stable regions in images. For uncertain systems, it is only applicable to the case with common stable region. Otherwise, further research is needed to find other control methods.


## 3. Simulation

Several simulation examples show the effectiveness of the proposed method. It includes the following plants: two unstable time-delay systems of order 1 and order 2, respectively, the time-delay unstable fractional plant with uncertain parameters, and the two fractional-order plants.


### 3.1. Example A

Consider a first-order unstable transfer function with time delay which was studied by Tan [29], Onat [34], Kaya [38], and Visioli [39]. It has the following form:(22)Gps=44s−1e−2s.According to the proposed steps, the stable region of the inner-loop system is obtained. The SBL is drawn in thekd−kf plane. And the convex region of the stable region is demonstrated in Figure 2. It shows that the convex stability region consists of one vertex V and two corner points C1 and C2. The convex stability region of the PD controller can be described as a triangle by these points. According to the coordinates of the above points, the incenter (Inc) of the PD controller can be calculated by applying equation (12), and its coordinate is (0.461, 0.469). The inner-loop PD controller parameters are kd=0.461andkf=0.499.Figure 2
Convex stability region and incenter of PD controller in example A.Substituting the PD controller parameters into the inner loop, the system transfer function is obtained by using (20). For the outer loop, PI controller parameters’ stability region is drawn in the kp−ki plane in Figure 3. Figure 3 also demonstrates the CSR. In this step, two corner points C3 and C4 and two points E1 and E2 on the SBL with a longitudinal distance of 1/2 from the corner point to the cusp point are given as shown in Figure 3. These points describe the convex region as a trapezoid. Then, the coordinates of Fermat points (Fps) are obtained as (0.06475, 0.04573), and the parameters of the inner-loop PI controller are kp=0.06459andki=0.04564. Thus, parameters of the PI-PD controller are kd=0.461,kf=0.499,kp=0.06459,andki=0.04564.Figure 3
Convex stability region and Fermat point of PI controller in example A.In Figure4, the unit step response of the method put forward in this paper is compared with the methods put forward by Onat [34], Tan [29], Kaya [38], and Visioli [39]. Table 1 gives the parameters of the various methods. The settling time, overshoot, rise time, peak time, and IAE values are used as evaluation index. Table 2 gives the evaluation values of this proposed method, other controller designs, and optimization methods. In light of the evaluation criteria, the tuning method proposed in this paper exhibits the superior performance. This method has the lowest settling time and the smallest overshoot. The response speed of the PID controller is fast, but other performances are not ideal. In this example, the PI-PD controller proved to be superior to PID controller, and the parameter tuning method is the best in comparison.Figure 4
Unit step responses comparison of various methods for example A.Table 1
Controller parameters comparison of various methods in example A (other data cited from [34]).
Controller
Proposed PI-PD
Onat PI-PD
Tan PI-PD
Kaya PI-PD
Visioli PID
Parameters
k
d
=
0.461
k
f
=
0.499
k
p
=
0.06459
k
i
=
0.04564
k
d
=
0.3412
k
f
=
0.439
k
p
=
0.107
k
i
=
0.0393
k
d
=
0.20
k
f
=
0.413
k
p
=
0.07
k
i
=
0.03
k
p
=
0.6217
k
i
=
0.0722
k
d
=
0.5352
k
p
=
0.652
k
i
=
0.0789
k
d
=
0.6309Table 2
Performance metrics comparison of various methods in example A (other data cited from [34]).
Controller
Settling time (s)
Overshoot (%)
Rise time (s)
Peak time (s)
IAE
Proposed PI-PD
11.81
1.35
5.80
15.68
5.63
Onat PI-PD
16.04
2.13
3.77
8.63
4.87
Tan PI-PD
20.26
6.24
4.64
10.62
5.89
Kaya PI-PD
17.70
174.16
0.45
5.01
8.57
Visioli PID
24.15
191.65
0.32
4.00
9.18


### 3.2. Example B

View a second-order unstable transfer function which was studied by Kaya and Atherton [28]:(23)Gps=12s2+s−1e−0.3s.The stability region of the inner-loop PD controller is shown in Figure5. A vertex V and two corner points C1 and C2 in Figure 5 construct a CSR. Inc stands for convex stable region, and Inc can be calculated as (5.51, 4.8667), and then the inner-loop PD controller parameters are kd=5.51andkf=4.8667.Figure 5
Convex stability region and incenter of PD controller in example B.By substituting the PD controller parameters (kd=5.51,kf=4.8667) into the inner loop, the transfer function can be obtained by using (12). Figure 6 gives the stable region and its convex region of the outer ring PI controller in the kp−ki plane. C3 and C4 are the corner points and E1 and E2 are the bisection points. The diagonal intersection coordinate of the quadrilateral is obtained by the graphic method, and the coordinate values of Fp are kp=3.77andki=4.75. Through the above analysis and calculation, the PI-PD controller parameters calculated by the proposed method are determined. They are kd=5.51, kf=4.8667, kp=3.77, and ki=4.75.Figure 6
Convex stability region and Fermat point of PI controller in example B.Figure7 gives the comparison between the PI-PD controller optimized by the parameter tuning method put forward in this paper and the controller put forward by Kaya and Atherton [28]. The unit step responses of these controllers are illustrated in Figure 7. Controller parameters are shown in Table 3. Table 4 gives the performance indexes of the several studies in this example. Obviously, the proposed method has lowest overshoot and peak time. And it also has a small IAE in this example. It is obvious that the tuning method put forward in this paper is better than the others.Figure 7
Step responses comparison of various methods for example B.Table 3
Controller parameters comparison of various methods in example B (other data cited from [34]).
Controller
Proposed PI-PD
Onat PI-PD
Kaya PI-PD
Atherton PID
Parameters
k
d
=
5.51
k
f
=
4.8667
k
p
=
3.77
k
i
=
4.75
k
d
=
5.46
k
f
=
5.69
k
p
=
3
k
i
=
5.3
k
d
=
0.535
k
f
=
3
k
p
=
1.398
k
i
=
0.793
k
p
=
1.398
k
i
=
0.793
k
d
=
0.535Table 4
Performance metrics comparison of various methods in example B (other data cited from [34]).
Controller
Settling time (s)
Overshoot (%)
Rise time (s)
Peak time (s)
IAE
Proposed PI-PD
11.81
1.35
5.80
15.68
5.63
Onat PI-PD
16.04
2.13
3.77
8.63
4.87
Tan PI-PD
20.26
6.24
4.64
10.62
5.89
Kaya PI-PD
17.70
174.16
0.45
5.01
8.57
Visioli PID
24.15
191.65
0.32
4.00
9.18


### 3.3. Example C

This example considers a time-delay unstable transfer function with uncertain parameters. The transfer function is written as(24)Gps=1sa−be−0.2s,where a∈0.8,1.2 and b∈0.5,1.5. Four boundary transfer functions are taken into account based on Kharitonov’s theorem which are Gp1s,Gp2s,Gp3s, and Gp4s. The plants are given in (25)–(28), respectively:(25)Gp1s=1s1.2−0.5e−0.2s,(26)Gp2s=1s1.2−1.5e−0.2s,(27)Gp3s=1s0.8−1.5e−0.2s,(28)Gp4s=1s0.8−0.5e−0.2s.Above all, stability boundary loci of edge plants in the inner loop are drawn in thekd−kf plane in Figure 8. According to the stable boundary loci obtained above, the common stability region is obtained as shown as the shadow in Figure 8. Figure 9 enlarges CSR in Figure 8 and gives CCSR composed of corner points C1 and C2 and vertex V. C1 is the intersection of two stable boundary loci. The parameters of the inner-loop PD controller are the values of Inc coordinates with kd=0.0314andkf=2.205.Figure 8
Stability boundary loci of inner-loop PD controller for uncertain systems in example C.Figure 9
Common convex stable region and incenter of PD controller in example C.Furthermore, for outer loop, stability boundary loci of edge plants are drawn in Figure10, and the shadow is the CSR. Figure 11 enlarges CSR in Figure 10 and gives CCSR as a trapezoid composed of corner points C3 and C4 and the bisection points E1 and E2 of the common stable region. Applying the method proposed in the paper, the common convex region is designed as a trapezoid and Fermat point is the diagonal intersection of the trapezoid. Therefore, the values of Fp coordinate in Figure 11 are the parameters of the outer-loop PI controller with kp=1.518andkf=1.644. Through the above two steps, the PI-PD controller of the system is obtained, and its parameters are kd=0.0314,kf=2.205,kp=1.518,andkf=1.644.Figure 10
Stability boundary loci of outer-loop PI controller for uncertain systems in example C.Figure 11
Common convex stable region and Fermat point of the PI controller in example C.Finally, for the parameters of the uncertain system, the cell step responses of the four boundary plants are obtained in Figure12. For the uncertain system, the controller has strong robustness.Figure 12
Step response comparison of various methods for example C.


### 3.4. Example D

Consider the following transfer function, which has an order of 0.8:(29)Gps=1s0.8−1e−0.2s.In this example, if the point atω=0 is used as a corner point, the CSR is as shown in Figure 13. The convex region contains a lot of unstable regions, and the WGC point and the incenter of the convex region are outside of the stable region. Therefore, as shown in Figure 14, a new point C1 is defined as the left corner point. The height of C1 is equal to the original corner point. And C1 is closer to the stable region. C2 is another corner point of the new convex stable region, and V is the vertex of the stable region.Figure 13
General corner point of inner-loop PD controller in example D.Figure 14
New corner point of inner-loop PD controller in example D.For inner loop, Inc coordinates are obtained askd=−0.041andkf=1.55 and centroid coordinates are obtained as kd=0.00947andkf=2.5633.And for the outer loop, using the WGC method, the convex stability region is given as Figure15. C3, C4, and C5 are the corner points of the stable region, and V is the vertex of the stable region. The centroid PI controller parameters are obtained as kp=0.1072andki=2.805. Applying the method put forward in this paper, the convex stability region of PI controller is given in Figure 16. The convex stable region is a trapezoid composed of the corner points C6 and C7 and the bisection points E1 and E2 of the common stable region. The outer-loop PI controller parameters are computed as Fp with kp=1.527andki=2.515. Accordingly, the PI-PD controller optimized by using the WGC method and the method proposed in this paper is determined. The parameters of the WGC method are kd=0.00947,kf=2.5633, and kp=0.1072,ki=2.805, and the parameters of the proposed method are kd=−0.041,kf=1.55, and kp=1.527,ki=2.515. The unit step responses are given in Figure 17. Compared with Onat’s method, the method proposed in this paper can achieve faster response.Figure 15
Convex stability region and centroid of the PI controller in example D.Figure 16
Convex stability region and Fermat point of PI controller in example D.Figure 17
Step responses in example D.


### 3.5. Example E

Consider the case of order 1.2 of fractional transfer function in example 3:(30)Gps=1s1.2−1e−0.2s.According to the method proposed in this paper, the corner points are selected. The stable region and its regular convex stability region of the inner loop and outer loop are shown in Figures18 and 19, respectively. C1, C2, C3, and C4 are the corner points, V is the vertex point, and E1 and E2 are the bisection points. The inner PD controller parameters are the coordinates of Inc with kd=0.799andkf=1.823, and the outer PI controller parameters are the coordinates of Fp with kp=4.976andki=7.75.Figure 18
Convex stability region designed by regular point in example E.Figure 19
Regular convex stability region and Fermat point of PI controller in example E.Due to the inward extension of corner points, the following attempts were made in this paper to select inflection points on the boundary to replace corner points. The stable region and the convex region are as shown in Figures20 and 21. C5, C6, C7, and C8 are the corner points, V is the vertex point, and E3 and E4 are the bisection points. kd,kfandkp,ki are the coordinates of Inc and Fp, respectively. The parameters of this improved PI-PD controller are kd=0.533,kf=2.941,kp=3.488,andki=5.363. The unit step responses are compared as shown in Figure 22. The two methods show the same settling time. However, the improved method is better than the former one in the field of overshot and IAE capabilities.Figure 20
Changed convex stability region and incenter of PD controller in example E.Figure 21
Changed convex stability region and Fermat point of PI controller in example E.Figure 22
Step response for example E.


## 3.1. Example A

Consider a first-order unstable transfer function with time delay which was studied by Tan [29], Onat [34], Kaya [38], and Visioli [39]. It has the following form:(22)Gps=44s−1e−2s.According to the proposed steps, the stable region of the inner-loop system is obtained. The SBL is drawn in thekd−kf plane. And the convex region of the stable region is demonstrated in Figure 2. It shows that the convex stability region consists of one vertex V and two corner points C1 and C2. The convex stability region of the PD controller can be described as a triangle by these points. According to the coordinates of the above points, the incenter (Inc) of the PD controller can be calculated by applying equation (12), and its coordinate is (0.461, 0.469). The inner-loop PD controller parameters are kd=0.461andkf=0.499.Figure 2
Convex stability region and incenter of PD controller in example A.Substituting the PD controller parameters into the inner loop, the system transfer function is obtained by using (20). For the outer loop, PI controller parameters’ stability region is drawn in the kp−ki plane in Figure 3. Figure 3 also demonstrates the CSR. In this step, two corner points C3 and C4 and two points E1 and E2 on the SBL with a longitudinal distance of 1/2 from the corner point to the cusp point are given as shown in Figure 3. These points describe the convex region as a trapezoid. Then, the coordinates of Fermat points (Fps) are obtained as (0.06475, 0.04573), and the parameters of the inner-loop PI controller are kp=0.06459andki=0.04564. Thus, parameters of the PI-PD controller are kd=0.461,kf=0.499,kp=0.06459,andki=0.04564.Figure 3
Convex stability region and Fermat point of PI controller in example A.In Figure4, the unit step response of the method put forward in this paper is compared with the methods put forward by Onat [34], Tan [29], Kaya [38], and Visioli [39]. Table 1 gives the parameters of the various methods. The settling time, overshoot, rise time, peak time, and IAE values are used as evaluation index. Table 2 gives the evaluation values of this proposed method, other controller designs, and optimization methods. In light of the evaluation criteria, the tuning method proposed in this paper exhibits the superior performance. This method has the lowest settling time and the smallest overshoot. The response speed of the PID controller is fast, but other performances are not ideal. In this example, the PI-PD controller proved to be superior to PID controller, and the parameter tuning method is the best in comparison.Figure 4
Unit step responses comparison of various methods for example A.Table 1
Controller parameters comparison of various methods in example A (other data cited from [34]).
Controller
Proposed PI-PD
Onat PI-PD
Tan PI-PD
Kaya PI-PD
Visioli PID
Parameters
k
d
=
0.461
k
f
=
0.499
k
p
=
0.06459
k
i
=
0.04564
k
d
=
0.3412
k
f
=
0.439
k
p
=
0.107
k
i
=
0.0393
k
d
=
0.20
k
f
=
0.413
k
p
=
0.07
k
i
=
0.03
k
p
=
0.6217
k
i
=
0.0722
k
d
=
0.5352
k
p
=
0.652
k
i
=
0.0789
k
d
=
0.6309Table 2
Performance metrics comparison of various methods in example A (other data cited from [34]).
Controller
Settling time (s)
Overshoot (%)
Rise time (s)
Peak time (s)
IAE
Proposed PI-PD
11.81
1.35
5.80
15.68
5.63
Onat PI-PD
16.04
2.13
3.77
8.63
4.87
Tan PI-PD
20.26
6.24
4.64
10.62
5.89
Kaya PI-PD
17.70
174.16
0.45
5.01
8.57
Visioli PID
24.15
191.65
0.32
4.00
9.18


## 3.2. Example B

View a second-order unstable transfer function which was studied by Kaya and Atherton [28]:(23)Gps=12s2+s−1e−0.3s.The stability region of the inner-loop PD controller is shown in Figure5. A vertex V and two corner points C1 and C2 in Figure 5 construct a CSR. Inc stands for convex stable region, and Inc can be calculated as (5.51, 4.8667), and then the inner-loop PD controller parameters are kd=5.51andkf=4.8667.Figure 5
Convex stability region and incenter of PD controller in example B.By substituting the PD controller parameters (kd=5.51,kf=4.8667) into the inner loop, the transfer function can be obtained by using (12). Figure 6 gives the stable region and its convex region of the outer ring PI controller in the kp−ki plane. C3 and C4 are the corner points and E1 and E2 are the bisection points. The diagonal intersection coordinate of the quadrilateral is obtained by the graphic method, and the coordinate values of Fp are kp=3.77andki=4.75. Through the above analysis and calculation, the PI-PD controller parameters calculated by the proposed method are determined. They are kd=5.51, kf=4.8667, kp=3.77, and ki=4.75.Figure 6
Convex stability region and Fermat point of PI controller in example B.Figure7 gives the comparison between the PI-PD controller optimized by the parameter tuning method put forward in this paper and the controller put forward by Kaya and Atherton [28]. The unit step responses of these controllers are illustrated in Figure 7. Controller parameters are shown in Table 3. Table 4 gives the performance indexes of the several studies in this example. Obviously, the proposed method has lowest overshoot and peak time. And it also has a small IAE in this example. It is obvious that the tuning method put forward in this paper is better than the others.Figure 7
Step responses comparison of various methods for example B.Table 3
Controller parameters comparison of various methods in example B (other data cited from [34]).
Controller
Proposed PI-PD
Onat PI-PD
Kaya PI-PD
Atherton PID
Parameters
k
d
=
5.51
k
f
=
4.8667
k
p
=
3.77
k
i
=
4.75
k
d
=
5.46
k
f
=
5.69
k
p
=
3
k
i
=
5.3
k
d
=
0.535
k
f
=
3
k
p
=
1.398
k
i
=
0.793
k
p
=
1.398
k
i
=
0.793
k
d
=
0.535Table 4
Performance metrics comparison of various methods in example B (other data cited from [34]).
Controller
Settling time (s)
Overshoot (%)
Rise time (s)
Peak time (s)
IAE
Proposed PI-PD
11.81
1.35
5.80
15.68
5.63
Onat PI-PD
16.04
2.13
3.77
8.63
4.87
Tan PI-PD
20.26
6.24
4.64
10.62
5.89
Kaya PI-PD
17.70
174.16
0.45
5.01
8.57
Visioli PID
24.15
191.65
0.32
4.00
9.18


## 3.3. Example C

This example considers a time-delay unstable transfer function with uncertain parameters. The transfer function is written as(24)Gps=1sa−be−0.2s,where a∈0.8,1.2 and b∈0.5,1.5. Four boundary transfer functions are taken into account based on Kharitonov’s theorem which are Gp1s,Gp2s,Gp3s, and Gp4s. The plants are given in (25)–(28), respectively:(25)Gp1s=1s1.2−0.5e−0.2s,(26)Gp2s=1s1.2−1.5e−0.2s,(27)Gp3s=1s0.8−1.5e−0.2s,(28)Gp4s=1s0.8−0.5e−0.2s.Above all, stability boundary loci of edge plants in the inner loop are drawn in thekd−kf plane in Figure 8. According to the stable boundary loci obtained above, the common stability region is obtained as shown as the shadow in Figure 8. Figure 9 enlarges CSR in Figure 8 and gives CCSR composed of corner points C1 and C2 and vertex V. C1 is the intersection of two stable boundary loci. The parameters of the inner-loop PD controller are the values of Inc coordinates with kd=0.0314andkf=2.205.Figure 8
Stability boundary loci of inner-loop PD controller for uncertain systems in example C.Figure 9
Common convex stable region and incenter of PD controller in example C.Furthermore, for outer loop, stability boundary loci of edge plants are drawn in Figure10, and the shadow is the CSR. Figure 11 enlarges CSR in Figure 10 and gives CCSR as a trapezoid composed of corner points C3 and C4 and the bisection points E1 and E2 of the common stable region. Applying the method proposed in the paper, the common convex region is designed as a trapezoid and Fermat point is the diagonal intersection of the trapezoid. Therefore, the values of Fp coordinate in Figure 11 are the parameters of the outer-loop PI controller with kp=1.518andkf=1.644. Through the above two steps, the PI-PD controller of the system is obtained, and its parameters are kd=0.0314,kf=2.205,kp=1.518,andkf=1.644.Figure 10
Stability boundary loci of outer-loop PI controller for uncertain systems in example C.Figure 11
Common convex stable region and Fermat point of the PI controller in example C.Finally, for the parameters of the uncertain system, the cell step responses of the four boundary plants are obtained in Figure12. For the uncertain system, the controller has strong robustness.Figure 12
Step response comparison of various methods for example C.


## 3.4. Example D

Consider the following transfer function, which has an order of 0.8:(29)Gps=1s0.8−1e−0.2s.In this example, if the point atω=0 is used as a corner point, the CSR is as shown in Figure 13. The convex region contains a lot of unstable regions, and the WGC point and the incenter of the convex region are outside of the stable region. Therefore, as shown in Figure 14, a new point C1 is defined as the left corner point. The height of C1 is equal to the original corner point. And C1 is closer to the stable region. C2 is another corner point of the new convex stable region, and V is the vertex of the stable region.Figure 13
General corner point of inner-loop PD controller in example D.Figure 14
New corner point of inner-loop PD controller in example D.For inner loop, Inc coordinates are obtained askd=−0.041andkf=1.55 and centroid coordinates are obtained as kd=0.00947andkf=2.5633.And for the outer loop, using the WGC method, the convex stability region is given as Figure15. C3, C4, and C5 are the corner points of the stable region, and V is the vertex of the stable region. The centroid PI controller parameters are obtained as kp=0.1072andki=2.805. Applying the method put forward in this paper, the convex stability region of PI controller is given in Figure 16. The convex stable region is a trapezoid composed of the corner points C6 and C7 and the bisection points E1 and E2 of the common stable region. The outer-loop PI controller parameters are computed as Fp with kp=1.527andki=2.515. Accordingly, the PI-PD controller optimized by using the WGC method and the method proposed in this paper is determined. The parameters of the WGC method are kd=0.00947,kf=2.5633, and kp=0.1072,ki=2.805, and the parameters of the proposed method are kd=−0.041,kf=1.55, and kp=1.527,ki=2.515. The unit step responses are given in Figure 17. Compared with Onat’s method, the method proposed in this paper can achieve faster response.Figure 15
Convex stability region and centroid of the PI controller in example D.Figure 16
Convex stability region and Fermat point of PI controller in example D.Figure 17
Step responses in example D.


## 3.5. Example E

Consider the case of order 1.2 of fractional transfer function in example 3:(30)Gps=1s1.2−1e−0.2s.According to the method proposed in this paper, the corner points are selected. The stable region and its regular convex stability region of the inner loop and outer loop are shown in Figures18 and 19, respectively. C1, C2, C3, and C4 are the corner points, V is the vertex point, and E1 and E2 are the bisection points. The inner PD controller parameters are the coordinates of Inc with kd=0.799andkf=1.823, and the outer PI controller parameters are the coordinates of Fp with kp=4.976andki=7.75.Figure 18
Convex stability region designed by regular point in example E.Figure 19
Regular convex stability region and Fermat point of PI controller in example E.Due to the inward extension of corner points, the following attempts were made in this paper to select inflection points on the boundary to replace corner points. The stable region and the convex region are as shown in Figures20 and 21. C5, C6, C7, and C8 are the corner points, V is the vertex point, and E3 and E4 are the bisection points. kd,kfandkp,ki are the coordinates of Inc and Fp, respectively. The parameters of this improved PI-PD controller are kd=0.533,kf=2.941,kp=3.488,andki=5.363. The unit step responses are compared as shown in Figure 22. The two methods show the same settling time. However, the improved method is better than the former one in the field of overshot and IAE capabilities.Figure 20
Changed convex stability region and incenter of PD controller in example E.Figure 21
Changed convex stability region and Fermat point of PI controller in example E.Figure 22
Step response for example E.


## 4. Conclusions

In this paper, a PI-PD control parameter graphical tuning method for unstable fractional time-delay systems is proposed by using the concept of incenter and Fermat points of CSRs. The method obtains the inner-loop stable boundary locus and the outer-loop stable boundary locus by establishing the characteristic equations. CSR is designed, and the incenter of the inner-loop CSR and Fermat point of outer-loop convex stable region are obtained, respectively. This method uses several special points on the stable boundary locus to determine the convex stable region. The method has the advantages of convenient calculation and easy implementation in engineering. This method is popularized to the uncertain system by applying Kharitonov’s theorem. Simulation examples verify the effectiveness of the method put forward in the paper. In future studies, the proposed method can also be extended to the PIλ-PDμ controller.



---

*Source: 3253497-2019-10-30.xml*