# Stabilizing and Robust Fractional PID Controller Synthesis for Uncertain First-Order plus Time-Delay Systems

Authors: Aymen Rhouma; Sami Hafsi; Kaouther Laabidi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, by using the nonintegerPIλDμ controllers, we conduct an investigation into the subject of robust stability area of time-delay interval process. Our method is based on setting up of the noninteger interval closed-loop characteristic equation using the inferior and superior bounds of uncertain parameters into several vertexes. We have combined the composition of the value set of vertex with the zero exclusion principle to analyse the stability of the uncertain process. A generalized version of the Hermite–Biehler theorem, applicable to fractional quasipolynomials, is exploited to determine the stability region of each vertex. The robust stability region of the noninteger regulator can be given by the crossing of the stability area of all the vertex characteristic noninteger quasipolynomials. By using the value set method and zero exclusion theory, the effectiveness of the stability region can be tested. Also, we propose a suitable procedure to determine the whole of stabilizing parameters for an interval process. An explicative example is given to point out the advantage and reliability of the approach.


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## Body

## 1. Introduction

The PID controllers are the best controllers applied in industrial applications due to their simplicity and their remarkable effectiveness of implementation [1–3]. Based on a little process knowledge, various simple and effective setting methods have been developed. As we know, control system is an active field of research, good results are forever required in order to achieve prescribed levels of performance. Meanwhile, alternative form PID regulators have been suggested to enhance the closed-loop response, for typical examples, IMC-PID [4–7] and Smith PID controller [8, 9].The mathematical modeling and simulation of systems and processes based on the description of their physical laws leads to differential equations of a fractional order [10–12]. In fact, large efforts of academic and industrial research have been concentrated on modeling and fractional-order control, primarily in the fields of tuning techniques and stabilization approaches. The earliest contributions in this field are the three generations of CRONE controllers [13]. The authors of [13] proved the advanced performance of the CRONE regulators over the traditional PID controllers. Lately, Podlubny has suggested a new version of the PID regulator named fractional-order PID controller (PIλDμ). This kind of controllers has received more and more attention. Thanks to the two extra freedom degree λ and μ, the PIλDμ has showed a better response and a strong robustness, in correspondence with the standard PID, when applied for both integer-order and noninteger-order process [14, 15].Many researchers find the key to the problem of stability, for example, Nyquist and Routh–Hurwitz stability criterions. Unfortunately, there are several problems to extend these results for noninteger systems. Recently, a few design approaches and tuning methods forPIλ, PDμ, and PIλDμ controllers have been presented from the generalizations of the classical rules.In [15], the authors designed a PIλ controller to fulfill three design specifications: robustness to system gain, phase margin variations, and time constant variations.Also, an optimal noninteger-order regulator based on specified performances has been given in [16]. Many authors use the three stability boundaries, Real Root Boundary (RRB), Complex Root Boundary (CRB), and Infinite Root Boundary (IRB), to establish the stability area [17]. This approach yields the specific formula obverse to these limits in terms of noninteger-order controller parameters [18]. More recently, an effective method is proposed to compute the stability area of noninteger-order PIλDμ for an arbitrarily given noninteger-order time-delay system [19].For practical reasons, the authors of [20, 21] presented an implementation of the fractional-order integrator capable to realize a PIλDμ controller.We often face time delays in industrial systems and several problems in process control engineering. When dealing with a system involving time delays, controlled by aPIλDμ controller, it is essential to be mindful that the closed-loop characteristic equation is not a polynomial but a quasipolynomial function. The problem will be more complicated when the mathematical representation of the dynamic system suffers from uncertainties. These uncertainties are due to measurement fault, industrial tolerances, nonlinearities, and so on. Maybe the most famous result about interval systems is Kharitonov’s theorem [22, 23]. This theorem proves that the interval polynomial set is Hurwitz stable if the four Kharitonov polynomials are Hurwitz stable.The D-decomposition procedure has been widely employed to establish the stability region of fractional-order controllers manually, although this method presents some little complicated problems that still remain to be resolved [17, 18, 24]. For this method, a point in every area split by the boundaries with respect to the singular frequencies has to be picked to quantify the process stability in the sense of the Nyquist rule. If the number of the fragmented areas is big, the technique of computing stability region will be needless. Consequently, the stability region of the PIλDμ controller in this case cannot be carried out by this method.In this paper, an efficient procedure is proposed to compute the stability area of noninteger regulator of time-delay system. The motivation here is to resolve the case whereλ+μ=2, and this case cannot be achieved by other methods. The results also expose many forms of stabilizing areas for the PIλDμ regulators. It should be noted that, for the uncertain first-order process with time delay, the stability study was not well elaborated and it is still a renewed research area. Indeed there are not various research works that deal with the problem of uncertain parameters of time-delay systems with noninteger PID regulators.Motivated by the above discussion, a robust stabilization method ofPIλDμ regulators for first-order process with time delay is given in this work. The objective is to determine the stability area of the PIλDμ controllers in the (Kp,Ki,Kd) plane for the uncertain system.This paper is organised as follows: in Section2, we present Hermite–Biehler and Pontryagin theorems which will be useful to compute stability region of time-delay system. A method for the setting up of the value set is given and a new procedure is offered to handle the stability area of uncertain noninteger process in Section 3. In Section 4, an example of time-delay plant controlled by a noninteger regulator is given. Finally, Section 5 is reserved for the conclusion and some remarks.


## 2. Stabilization of Uncertain First-Order Time-Delay System

A SISO noninteger control plant is shown in Figure1.Figure 1
Closed-loop noninteger system.In Figure1, r is the reference input, u is the control law, y is the output signal, and Gs and Cs are the transfer functions, respectively, of the time-delay process and the noninteger PID controller.Our objective in this work is to compute the set ofKp,Ki,Kd parameters where the closed-loop process stays stable, where Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain, and the orders λ and μ may assume real noninteger positives values. The output of the overall noninteger control process is given by(1)ys=CsGs1+CsGsrs.The complex closed-loop characteristic equation of this system is given as(2)Qjω=qrω+jqiω,where qrω and qiω expose the real and the imaginary part, respectively, of Qjω.The stabilization of this kind of systems is relatively complicated, since the number of roots is infinite when time delays are introduced. In this work, we use the generalization Hermite–Biehler theorem for quasipolynomials [1, 25]. This theorem is useful to find the fractional-order PID regulator parameters that stabilize a given time-delay systems.Hence, the characteristic equation of the closed-loop system is(3)Qs=ds+e−sLinis,where ds and nis for i=1,2,…,m are quasipolynomials with real coefficients:(4)degds=n,degnis<n,fori=1,2,…,m.Li for i=1,2,…,m represent time delays with order L1<L2<⋯<Lm.From equation (1), we can define the fractional characteristic expression of the process in Figure 1 as(5)Qs=sλ1+Ts+KKpsλ+Ki+Kdsμ+λe−Ls.We can rewrite the quasipolynomialQs as follows:(6)Qs=sλ1+TseLs+KKpsλ+Ki+Kdsμ+λ.In order to guarantee thatQs admits a nonzero principal term, for this, it is necessary that the coefficients of the higher degrees of s and es are different from zero, we must suppose that degds=λ+1>degns=λ+μ. This assumption implies that μ<1; in this work, we suppose that λ+μ=2, where λ=2/3 and μ=1/3.Then, forz=Ls, Qs becomes(7)Qz=zLλ1+TLzez+KKpzLλ+Ki+KdzLμ+λ.By replacingz by jω, the noninteger quasipolynomial (7) is given by(8)Qjω=jωLλ1+TLjωejω+KKpjωLλ+KKi+KKdjωLμ+λ.


### 2.1. Determination ofKp Range

#### 2.1.1. Hermite–Biehler Theorem [1, 25]

LetQjω be in the following form:(9)Qjω=qrω+jqiω,where qrω and qiω, respectively, represent the real and the imaginary part of Qjω.Qjω is stable if and only if(1)
qrω and qiω have only simple real roots and these are interlaced.(2)
qi′ωqrω−qiωqr′ω>0, for some ω∈−∞;+∞,whereqr′ω and qi′ω, respectively, represent the derivatives of qrω and qω.A key step in Hermite–Biehler theorem to examine stability of the process is to assure thatqrω and qiω have just real roots. This characteristic can be tested due to Pontryagin theorem [26, 27].


#### 2.1.2. Pontryagin Theorem

M and N indicate the uppermost degrees of s and es, respectively, in Qjω, and η is a suit constant such that the coefficients of terms of uppermost degree in qrω and qiω do not become zero at ω=η. To fulfill that the equations qrω=0 and qiω=0 have just real roots, it is necessary and sufficient that(10)−2lπ+η≤ω≤2lπ+η;l=1,2,3,….qrω and qiω have exactly 4lN+M roots, with large l.⋄ This theorem allows us to specify the suitable value of η so as to get the widest interval of Kp parameter to make sure that qrω and qiω have just real roots. According to H. B theorem, it is useless to verify if roots of the imaginary and real parts are all real. Since the interlacing ownership between the roots of qrω and qiω is confirmed, the other part has also just only real roots.Forjωλ=ωλcosπ/2λ+jsinπ/2λ, qrω and qiω can be expressed by(11)qrω=ωLλcosπ2λKKp+cosω−TLωsinω−ωLλsinπ2λsinω+TLωcosω+KKi+KKdωLμ+λcosπ2λ+μ,(12)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω+KKdωLμ+λsinπ2λ+μ.Clearly, thePIλDμ gains Kp and Kd appear in the imaginary part (12), while the real part (11) is affected by all controller parameters. In order to examine the stability of closed-loop system, we require to compute all controller parameter (Kp,Ki,Kd). As a first step, based on P. theorem, we compute the range of the proportional gain Kp which assures real roots for qiω with an appropriate η value as explained in the previous theorem. In [28], the authors proposed an important method to carry out the stability area for the case λ+μ=2, which cannot be determined by an other method for the stabilization of the noninteger-order PIλDμ regulator. In such a case, it should be noted that we have cited this logical exigency λ+μ=2 on the stabilizing noninteger-order regulator which is well defined due to the above assumption: degds=λ+1>degns=λ+μ. As a consequence of this requirement, the term sinπ/2λ+μ is equal to zero and the imaginary part is as follows:(13)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω.It is remarkable to note that function (13) depends on only one unknown fractional-order PID parameter. Therefore, as stated by H. B theorem, we use the P. theorem to determine the range of Kp parameter which ensures that the real and imaginary parts have only real roots for a given range of η.where Nr represents the number of roots for qi∗ω and Kp1 and Kp2 represent the first interval limits of Kp arbitrary chooses. The values of Kp ensuring real roots for qiω are given by Algorithm 1, as shown in Figure 2. It is possible to note that the projection η=0.8 corresponds to the largest range for Kp.Algorithm 1:Compute Kp range for several η values.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1.(2)
Count the roots of the equationqiω=0 for ω∈−2π+η:2π+η.(3)
IfNr=4lN+M then go to 4.Else incrementKp=Kp+step and go to 5.(4)
SaveKp and increment Kp=Kp+step.(5)
IfKp<Kp2, go to 2, else give Pontryagin interval,Figure 2
Different values ofKp for η∈0:6.2.From this figure, we choose an appropriate value ofη=1.9 which fulfills the condition that η would not vanish both real and imaginary parts; this value corresponds to −0.9<Kp<1.6. After determining the Kp range, we present an elegant procedure in order to compute the set Ki,Kd values.


### 2.2. Stability Area inKi,Kd Plan

In this section, we compute the wholeKi and Kd gains of the PIλDμ regulator that make system stable. In fact, by taking into account the interlacing property between the roots of the two parts of Qjω and using the lemma given in [29], we calculate qrω with the zeros of qiω.Lemma 1.
ForKp range given by Algorithm 1, the property of interlacing between the roots of qiz and qrz is handled by(14)−1jqrzj>0j=0,1,2,…,qrz0>0qrz1<0qrz2>0qrz3<0⋮⟶Ki>0,±Kd<−Lz1μBz1+mz1Ki,±Kd>−Lz2μBz2+mz2Ki,±Kd<−Lz3μBz3+mz3Ki,⋮,where(15)Bzj=cosπ2λKp+1kcoszj−TLzjsinzj−1Ksinπ2λsinzj+TLzjcoszj,mzj=Lzjλ.
The borderlines of these areas are shown in Figure3, which are determined by straight lines with the following equations:(16)Kd=−LωjμBωj+mωjKi.
The shaded area of these four lines gives the stability region in the (Ki,Kd) plane with Kp=0.4.
In view of the above lemma, we show this second algorithm to calculate the ensemble of stabilizing (Ki,Kd) parameters for a fixed Kp parameter. (Algorithm 2)Figure 3
Stability region forKp=0.4.Algorithm 2:Stability region for a system with fixed parameters.
(1)
ComputeKp interval using Algorithm 1.(2)
Select aKp value from the already determined range.(3)
Get the rootsωjj=1…4 from equation (11).(4)
Compute the parametersBωjj=1…4 and mωjj=1…4.(5)
Delimit theKi,Kd area using equation (16).(6)
IfKp<Kp2 return to Step 2.


## 2.1. Determination ofKp Range

### 2.1.1. Hermite–Biehler Theorem [1, 25]

LetQjω be in the following form:(9)Qjω=qrω+jqiω,where qrω and qiω, respectively, represent the real and the imaginary part of Qjω.Qjω is stable if and only if(1)
qrω and qiω have only simple real roots and these are interlaced.(2)
qi′ωqrω−qiωqr′ω>0, for some ω∈−∞;+∞,whereqr′ω and qi′ω, respectively, represent the derivatives of qrω and qω.A key step in Hermite–Biehler theorem to examine stability of the process is to assure thatqrω and qiω have just real roots. This characteristic can be tested due to Pontryagin theorem [26, 27].


### 2.1.2. Pontryagin Theorem

M and N indicate the uppermost degrees of s and es, respectively, in Qjω, and η is a suit constant such that the coefficients of terms of uppermost degree in qrω and qiω do not become zero at ω=η. To fulfill that the equations qrω=0 and qiω=0 have just real roots, it is necessary and sufficient that(10)−2lπ+η≤ω≤2lπ+η;l=1,2,3,….qrω and qiω have exactly 4lN+M roots, with large l.⋄ This theorem allows us to specify the suitable value of η so as to get the widest interval of Kp parameter to make sure that qrω and qiω have just real roots. According to H. B theorem, it is useless to verify if roots of the imaginary and real parts are all real. Since the interlacing ownership between the roots of qrω and qiω is confirmed, the other part has also just only real roots.Forjωλ=ωλcosπ/2λ+jsinπ/2λ, qrω and qiω can be expressed by(11)qrω=ωLλcosπ2λKKp+cosω−TLωsinω−ωLλsinπ2λsinω+TLωcosω+KKi+KKdωLμ+λcosπ2λ+μ,(12)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω+KKdωLμ+λsinπ2λ+μ.Clearly, thePIλDμ gains Kp and Kd appear in the imaginary part (12), while the real part (11) is affected by all controller parameters. In order to examine the stability of closed-loop system, we require to compute all controller parameter (Kp,Ki,Kd). As a first step, based on P. theorem, we compute the range of the proportional gain Kp which assures real roots for qiω with an appropriate η value as explained in the previous theorem. In [28], the authors proposed an important method to carry out the stability area for the case λ+μ=2, which cannot be determined by an other method for the stabilization of the noninteger-order PIλDμ regulator. In such a case, it should be noted that we have cited this logical exigency λ+μ=2 on the stabilizing noninteger-order regulator which is well defined due to the above assumption: degds=λ+1>degns=λ+μ. As a consequence of this requirement, the term sinπ/2λ+μ is equal to zero and the imaginary part is as follows:(13)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω.It is remarkable to note that function (13) depends on only one unknown fractional-order PID parameter. Therefore, as stated by H. B theorem, we use the P. theorem to determine the range of Kp parameter which ensures that the real and imaginary parts have only real roots for a given range of η.where Nr represents the number of roots for qi∗ω and Kp1 and Kp2 represent the first interval limits of Kp arbitrary chooses. The values of Kp ensuring real roots for qiω are given by Algorithm 1, as shown in Figure 2. It is possible to note that the projection η=0.8 corresponds to the largest range for Kp.Algorithm 1:Compute Kp range for several η values.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1.(2)
Count the roots of the equationqiω=0 for ω∈−2π+η:2π+η.(3)
IfNr=4lN+M then go to 4.Else incrementKp=Kp+step and go to 5.(4)
SaveKp and increment Kp=Kp+step.(5)
IfKp<Kp2, go to 2, else give Pontryagin interval,Figure 2
Different values ofKp for η∈0:6.2.From this figure, we choose an appropriate value ofη=1.9 which fulfills the condition that η would not vanish both real and imaginary parts; this value corresponds to −0.9<Kp<1.6. After determining the Kp range, we present an elegant procedure in order to compute the set Ki,Kd values.


## 2.1.1. Hermite–Biehler Theorem [1, 25]

LetQjω be in the following form:(9)Qjω=qrω+jqiω,where qrω and qiω, respectively, represent the real and the imaginary part of Qjω.Qjω is stable if and only if(1)
qrω and qiω have only simple real roots and these are interlaced.(2)
qi′ωqrω−qiωqr′ω>0, for some ω∈−∞;+∞,whereqr′ω and qi′ω, respectively, represent the derivatives of qrω and qω.A key step in Hermite–Biehler theorem to examine stability of the process is to assure thatqrω and qiω have just real roots. This characteristic can be tested due to Pontryagin theorem [26, 27].


## 2.1.2. Pontryagin Theorem

M and N indicate the uppermost degrees of s and es, respectively, in Qjω, and η is a suit constant such that the coefficients of terms of uppermost degree in qrω and qiω do not become zero at ω=η. To fulfill that the equations qrω=0 and qiω=0 have just real roots, it is necessary and sufficient that(10)−2lπ+η≤ω≤2lπ+η;l=1,2,3,….qrω and qiω have exactly 4lN+M roots, with large l.⋄ This theorem allows us to specify the suitable value of η so as to get the widest interval of Kp parameter to make sure that qrω and qiω have just real roots. According to H. B theorem, it is useless to verify if roots of the imaginary and real parts are all real. Since the interlacing ownership between the roots of qrω and qiω is confirmed, the other part has also just only real roots.Forjωλ=ωλcosπ/2λ+jsinπ/2λ, qrω and qiω can be expressed by(11)qrω=ωLλcosπ2λKKp+cosω−TLωsinω−ωLλsinπ2λsinω+TLωcosω+KKi+KKdωLμ+λcosπ2λ+μ,(12)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω+KKdωLμ+λsinπ2λ+μ.Clearly, thePIλDμ gains Kp and Kd appear in the imaginary part (12), while the real part (11) is affected by all controller parameters. In order to examine the stability of closed-loop system, we require to compute all controller parameter (Kp,Ki,Kd). As a first step, based on P. theorem, we compute the range of the proportional gain Kp which assures real roots for qiω with an appropriate η value as explained in the previous theorem. In [28], the authors proposed an important method to carry out the stability area for the case λ+μ=2, which cannot be determined by an other method for the stabilization of the noninteger-order PIλDμ regulator. In such a case, it should be noted that we have cited this logical exigency λ+μ=2 on the stabilizing noninteger-order regulator which is well defined due to the above assumption: degds=λ+1>degns=λ+μ. As a consequence of this requirement, the term sinπ/2λ+μ is equal to zero and the imaginary part is as follows:(13)qiω=ωLλcosπ2λsinω+TLωcosω+ωLλsinπ2λKKp+cosω−TLωsinω.It is remarkable to note that function (13) depends on only one unknown fractional-order PID parameter. Therefore, as stated by H. B theorem, we use the P. theorem to determine the range of Kp parameter which ensures that the real and imaginary parts have only real roots for a given range of η.where Nr represents the number of roots for qi∗ω and Kp1 and Kp2 represent the first interval limits of Kp arbitrary chooses. The values of Kp ensuring real roots for qiω are given by Algorithm 1, as shown in Figure 2. It is possible to note that the projection η=0.8 corresponds to the largest range for Kp.Algorithm 1:Compute Kp range for several η values.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1.(2)
Count the roots of the equationqiω=0 for ω∈−2π+η:2π+η.(3)
IfNr=4lN+M then go to 4.Else incrementKp=Kp+step and go to 5.(4)
SaveKp and increment Kp=Kp+step.(5)
IfKp<Kp2, go to 2, else give Pontryagin interval,Figure 2
Different values ofKp for η∈0:6.2.From this figure, we choose an appropriate value ofη=1.9 which fulfills the condition that η would not vanish both real and imaginary parts; this value corresponds to −0.9<Kp<1.6. After determining the Kp range, we present an elegant procedure in order to compute the set Ki,Kd values.


## 2.2. Stability Area inKi,Kd Plan

In this section, we compute the wholeKi and Kd gains of the PIλDμ regulator that make system stable. In fact, by taking into account the interlacing property between the roots of the two parts of Qjω and using the lemma given in [29], we calculate qrω with the zeros of qiω.Lemma 1.
ForKp range given by Algorithm 1, the property of interlacing between the roots of qiz and qrz is handled by(14)−1jqrzj>0j=0,1,2,…,qrz0>0qrz1<0qrz2>0qrz3<0⋮⟶Ki>0,±Kd<−Lz1μBz1+mz1Ki,±Kd>−Lz2μBz2+mz2Ki,±Kd<−Lz3μBz3+mz3Ki,⋮,where(15)Bzj=cosπ2λKp+1kcoszj−TLzjsinzj−1Ksinπ2λsinzj+TLzjcoszj,mzj=Lzjλ.
The borderlines of these areas are shown in Figure3, which are determined by straight lines with the following equations:(16)Kd=−LωjμBωj+mωjKi.
The shaded area of these four lines gives the stability region in the (Ki,Kd) plane with Kp=0.4.
In view of the above lemma, we show this second algorithm to calculate the ensemble of stabilizing (Ki,Kd) parameters for a fixed Kp parameter. (Algorithm 2)Figure 3
Stability region forKp=0.4.Algorithm 2:Stability region for a system with fixed parameters.
(1)
ComputeKp interval using Algorithm 1.(2)
Select aKp value from the already determined range.(3)
Get the rootsωjj=1…4 from equation (11).(4)
Compute the parametersBωjj=1…4 and mωjj=1…4.(5)
Delimit theKi,Kd area using equation (16).(6)
IfKp<Kp2 return to Step 2.


## 3. Generalisation of the Hermite–Biehler Criterion Stability for Uncertain Noninteger-Order Quasipolynomial

We exploit the studies presented previously to discuss the stability of the uncertain noninteger-order time-delay systems. For that, let us consider the uncertain first-order system as follows:(17)Gs=K1+Tse−Ls,where the intervals of parameters are(18)K∈K¯;K¯,T∈T¯;T¯,L∈L¯;L¯.The objective of this article is to compute the robust stability region of the interval time-delay system using the noninteger-orderPIλDμ regulator. Based on the argument value interval concept, we construct the geometrical techniques of complex analysis. In fact, the uncertain fractional-order first-order system is dissociated into diverse vertexes of uncertain parameters. Based on these vertexes of parameters, we construct the set of values of characteristic quasipolynomials. The method presented in Section 2 is used to determine the stability area of each vertex of the closed-loop characteristic quasipolynomial. The stability region of the interval system is given by the intersection of these stability areas which is called robust stability region.Using the lower and upper bounds, the interval time-delay systemGs given by equation (17) can be decomposed into 23=8 vertex fractional-order quasipolynomials and 3∗22=12 exposed edges. Therefore, by taking a point (Ki,Kd) from the stability region in Figure 3, given for fixed Kp=0.4, we can write all the eight vertex fractional quasipolynomials of Qs,p in the following pattern:(19)q1s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q2s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q3s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q4s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q5s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q6s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q7s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ,q8s,p=sλ1+T¯seL¯s+K¯Kpsλ+Ki+Kdsμ+λ.For example, we remark from equation (19) that q1s,p and q2s,p have the same structure except the value of the parameter T. Hence, one of the exposed edges can be presented by the following expression:(20)1−τq1s+τq2s,τ=0,1.In the same way, we build the other exposed edges and we determine the ensemble which contains3∗22=12 exposed edges as(21)QEs=eq1s,q2s,eq1s,q3s,eq1s,q5s,eq2s,q4s,eq2s,q8s,eq3s,q7s,eq3s,q4s,eq4s,q8s,eq5s,q6s,eq5s,q7s,eq6s,q8s,eq7s,q8s.For further survey, the lector can refer to [30] and to the extensive results on such fractional uncertain quasipolynomials given in [26]. In this context, we give the following theorems.Theorem 1.
(see [29, 31]). The uncertain parameters are linearly dependent to each other for single delay systems and fixed $\omega$. Then, by mapping the vertices $q_i(s, q)$and the exposed edges $Q_E(s∗)$, we obtain a convex polygon in the complex plane (Figure 4). From the complex plane geometry, one can say that the boundary of the value set of the fractional quasipolynomial $Q(j\omega∗, q)$ can be obtained from the images of the exposed edges(22)∂Qs∗,q⊂QEs∗,where ∂ indicates the limit and QE is given in (21)Theorem 2.
(see [29]). The noninteger-order uncertain quasipolynomial Qjω∗,q is robust stable if QEjω∗ is stable.
In this work, we chooseT=1.5,2.5, L=3,4, and K=0.8,1.2 and we depict the value sets QEjω,p. Therefore, by taking 21 points within the uncertain parameters T=1.5:0.05:2.5 and L=1:0.05:2 and 5 points within the uncertain parameter K=0.8:0.05:1.2, the images of 21×21×5=2205 quasipolynomials at ω=4 rad/s are shown in Figure 4. Also, the images and the exposed edges at ω=4 rad/s are shown in the same figure by using a solid line. We can also see that the value set is limited by the set of exposed edges.Figure 4
The images of 2205 quasipolynomials and the exposed edges of equation (19).Theorem 3.
(see [29]). Qs,q is a stable quasipolynomial family if there exists at least one point s∗0∈∂Qs∗,q such that 0∉Qs∗0,q for all frequencies.
According to Theorem2, it is possible to ensure the robust stability of the uncertain quasipolynomials given in (19). Consequently, we can ensure that the entire fractional quasipolynomial family Qs;Kp,Ki,Kd is a stable fractional quasipolynomial family if it has at least one stable vertex fractional quasipolynomial Qjs;Kp,Ki,Kd and the value set of QEs,p does not include the origin for all frequencies.
It is clear from Figure5 that the origin is not included in the value sets for 0≤ω≤4. Therefore, based on Theorem 3, the fractional-order controlled system is robust stable. Let us now summarize the simple and efficient Algorithm 3.Figure 5
The value sets of interval polynomial of equation (19) for 0≤ω≤4.Algorithm 3:Stability of uncertain time-delay system.
(1)
DefineQs;Kp,Ki,Kd by equation (19).(2)
Produce exposed edges by equation (20).(3)
Compose the set of exposed edges,QEs;p, equation (21).(4)
Using Algorithm1, the allowable ranges of Kp of each vertex (19) is found.(5)
Determine the intersection of all theKp intervals.(6)
Using Algorithm2, we compute the (Ki,Kd) stabilizing values for the entire family.(7)
Plot the robust stability region for every vertex in theKp,Ki,Kd plan.


## 4. Illustrate Example

We consider the time-delay first-order system, whereK=0.8,1.2,T=1.5:2.5, and L=1:2. Based on the closed-loop characteristic equation given from expression (1), we determine the vertex fractional quasipolynomials as follows:(23)q1s,p=sλ1+1.5ses+0.8Kpsλ+Ki+Kdsμ+λ,q2s,p=sλ1+2.5ses+0.8Kpsλ+Ki+Kdsμ+λ,q3s,p=sλ1+2.5ses+1.2Kpsλ+Ki+Kdsμ+λ,q4s,p=sλ1+1.5ses+1.2Kpsλ+Ki+Kdsμ+λ,q5s,p=sλ1+1.5se2s+0.8Kpsλ+Ki+Kdsμ+λ,q6s,p=sλ1+2.5se2s+0.8Kpsλ+Ki+Kdsμ+λ,q7s,p=sλ1+2.5se2s+1.2Kpsλ+Ki+Kdsμ+λ,q8s,p=sλ1+1.5se2s+1.2Kpsλ+Ki+Kdsμ+λ.The main idea is to discuss the stability of each closed-loop vertex characteristic quasipolynomial. The intersection stability area gives the robust stability area of the fractional-order interval plant. By applying the first algorithm to every vertex fractional quasipolynomials in equation (23), we obtain the Kp ranges for each η value as shown in Figures 6(a), 6(b), 7(a), and 7(b).Figure 6
Kp vs. η as computed by applying Pontryagin theorem: (a)K¯,T¯,L¯; (b)K¯,T¯,L¯.
(a)(b)Figure 7
Kp vs. η as computed by applying Pontryagin theorem: (a)K¯,T¯,L¯; (b)K¯,T¯,L¯.
(a)(b)Forη=0.9, Kp=−0.9:1.7 is the intersection range of the eight Kp ranges given for each qj=1…8s,p. For each value of Kp inside this common region, the imaginary part qiω has five roots in −2π+η≤ω≤2π+η. Outside this nonempty intersection range of Kp, there are no stabilizing PI4/3D2/3 controllers.By scanning over allKp∈−0.9,1.7, we can yield the ensemble of Kp,Ki,Kd values that make the closed-loop system stable. The robust stability region can be visualized in the three-dimensional plot Kp,Ki,Kd for λ=2/3 and μ=1/3, as shown in Figure 8. It is clear that the set of linear inequalities in terms of (Ki,Kd) and this admissible set values have a special structure for each proportional gain, for example, (i)
ForKp=−0.1, the intersection area of the stability regions in the Ki,Kd plane is a trapezoid as shown in Figures 9(a) and 9(b)(ii)
ForKp=0.4, the intersection area of the stability regions in the Ki,Kd plane is a quadrilateral as shown in Figures 10(a) and 10(a)(iii)
ForKp=0.9, the intersection area of the stability regions in the Ki,Kd plane is a polygon as shown in Figures 11(a) and 11(b).Figure 8
The global stability region for thePI2/3D1/3 controller.Figure 9
(a) The boundary lines of the stabilizing set in the (Ki,Kd) space for Kp=−0.1. (b) Stability region in the (Ki,Kd) plane for Kp=−0.1.
(a)(b)Figure 10
(a) The boundaries lines of the stabilizing set in the (Ki,Kd) space for Kp=0.4. (b) Stability region in the (Ki,Kd) plane for Kp=0.4.
(a)(b)Figure 11
(a) The boundaries lines of the stabilizing set in the (Ki,Kd) space for Kp=0.9. (b) Stability region in the (Ki,Kd) plane for Kp=0.9.
(a)(b)


## 5. Conclusion

A new procedure based on generalisation version of Hermite–Biehler theorem and Pontryagin theorem has been presented in this work to compute the robust stability region of aPIλDμ regulator for an interval system with time delay. Using the lower and upper limits, the system with coefficient interval uncertainties is decomposed to various vertex plants. The stability area of each vertex characteristic quasipolynomial is given by the three algorithms which are based on Hermite–Biehler and Pontryagin theorems and zero exclusion principle. The common stability region of all vertex characteristic quasipolynomial defines the robust stability region.



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*Source: 9940634-2021-05-24.xml*