# Generic Noninteger Order Controller for Time-Delay Systems

Authors: Sami Hafsi; Sadem Ghrab; Kaouther Laabidi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper focuses on the problem of fractional controllerPI stabilization for a first-order time-delay systems. For this reason, we utilize the Hermite–Biehler and Pontryagin theorems to compute the complete set of the stabilizing PIλ parameters. The widespread industrial utilization of PID controllers and the potentiality of their noninteger order representation justify a timely interest in PIλ tuning techniques. Step responses are calculated through Kp,Ki,lambda parameters inside and outside stability region to prove the method efficiency.


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## Body

## 1. Introduction

Time delay usually appears in many real-time engineering systems in the state, the measurements, or the control input [1, 2]. Delays are involved in challenging areas of robotics and real-time systems [3], which are generally sources of instability. To eliminate this instability, researchers introduced several approaches of control among the conventional PID controller [4, 5]. These controllers are the most used in industrial purposes due to their simplicity, easy designing method, and their setting methods [6]. Many tuning methods have been proposed in the literature to tune P, PI, and PID controllers [6].After that, Podlubny has proposed a generalization of the classicPI and PID controllers defined as PIλ and PIλDμ where the order integrator λ and the order differentiator μ assumed real noninteger values. Also, he proved that these types of fractional controllers are the best for dynamics systems control [7, 8]. Using PIλDμ allows one to choose besides the parameters of the classical PID (Kp, Ki, and Kd) the orders of integration λ and derivation μ [9]. In most cases, first-order plus dead time (FOPDT) can be used with success to represent the process dynamics of single-input single-output (SISO) systems [10, 11]. Fractional-order PID controllers have become an interest research topic in control engineering. It has been evolving along with the progress of the technology, and nowadays it is very often implemented in digital form rather than with pneumatic or electrical components. Also, fractional controllers are introduced as a solution to solve the problem of instability of time-delay systems, which give more performance for many types of systems [8]. Caponetto developed a new procedure to compute the parameters of fractional-order controller PIλDμ designed to stabilize a first-order time delay systems [12]. Recently, Hafsi et al. has focused on the stabilization of first order with time-delay systems with a fractional controller of the form PIλ [13]. After that, an extension of this work has been developed in [14] for a fractional-order system with delay.In this paper, we develop analytical characterization of the stabilizing set of fractional controllers for first-order systems with time delay. Based on an extension of the Hermite–Biehler theorem and the Pontryagin theorem, the necessary and sufficient conditions for the stability of SISO first-order systems with known time delay are derived. The algorithms for computing the stabilizing sets ofPIλ controller are presented on the basis of these conditions. These algorithms simplify the tedious calculations of the intervals of the parameters Kp and Ki presented in papers [13, 14]. The entire setKp,Ki,λ is then found by sweeping the fractional order λ over appropriate ranges.This paper is structured as follows. Section2 is an introductory step that presents the usual definitions of fractional calculus and also presents some advantages that characterize fractional control actions. Section 3 introduces the problem of the stabilization of time-delay systems and presents the theory of the analysis of systems with time-lags which are based on the Hermite–Biehler theorem. Section 4 is attempted to establish a fractional analysis theory for first-order systems with time-delay systems controlled via a PIλ regulator. Finally, we present the results of the simulations in Section 5.


## 2. Fractional-Order Calculus: Preliminary Study

Noninteger calculus can be considered as the generalization of classical calculus to fractional orders of integration or differentiation. The definitions of Riemann–Liouville (RL), Caputo, and Grûnwald–Letnikov are the most well-known definitions used in fractional calculus. According to Grunwald–Letnikov, fractional integration and differentiation orders are defined as follows [15]:(1)aDtλvt=limh⟶0h−λ∑j=0t/h−1jλ,jTvt−jh,where t represents the integer part of t, h is the step size, λj is a binomial coefficient, and the subscripts a and t denote the low limits related to the operation of fractional differentiation. Caputo’s definition of the fractional derivative function vt of order is defined as follows [8]:(2)aDtλvt=1Γλ−n∫atdnvτ/dτn1−τa−n+1dτ,−1<λ<n.The RL fractional integral or derivative of order is defined as follows [8]:(3)Datλvt=1Γ−λ!∫at1−x−λ−1vxdx,siλ<0,vt,siλ=0,DnaDtλ−nvx,siλ>0.The Gamma functionΓ is defined by(4)Γz=∫0∞ettz−1dt.


## 3. Fractional-Order Control Actions

Noninteger order control is the use of fractional calculus to obtain a control system with fractional actions more efficiently than conventional actions. However, the system is modeled in a classical manner or fractional one. From a certain point of view, fractional-order control presented an evolution analogous to that of classical control, following two parallel paths depending on the starting point: the time domain or the frequency domain. In this section, we will look at control actions of typeKsλ where λ∈−1,1.λ=−1: integral actionλ=0: proportional actionλ=1: derivative actionThe basic control actions traditionally considered will be special cases [8], in which in the closed-loop system shown in Figure 1, r is the reference input, u is the control signal, and y is the output.Figure 1
Block diagram of the unity feedback control system.


### 3.1. Complex Plane Analysis

In the complex plane, the roots of the system given in (1) are located according to the following equation [8]:(5)1+KsλGs=0.The magnitude and the phase are given by(6)K=1sλGs,argsλGs=2k+1π,k=0,±1,±2,…,or s=sejθ, such that θ is the argument of s.


### 3.2. Fractional-Order Integral Action

Integral action slows down the system, eliminates steady-state error, and decreases relative stability [8]. In the frequency domain, the integral action increases the slopes of the amplitude curve by a coefficient −20dB/dec and decrements the phase plots by a coefficient π/2rad. In the complex plane, the integral action makes it possible to move the root locus of the system on the right half-plane [8]. On the other hand, the fractional integral control action is analyzed in the frequency domain as follows. The magnitude and the phase curves are, respectively, given by [8](7)20logsλGss=jω=20logGjω+20λlogω.The phase plot is given by(8)argsλGs=argGs+λπ2.Accordingly, if we chooseλ in the interval −1;0,The magnitude curve is shifted with an interval value−20dB/dec0dB/decThe phase plot is delayed by an angle with respect to the interval−π/2rad,0


### 3.3. Fractional-Order Derivative Action

The derivative control action increases the relative stability but unfortunately it makes the system more sensitive to high frequency noise [8]. In the complex plane, it allows to move the locus of the roots of the system on the left half-plane. In the frequency domain, the derivative control action produces a constant phase advance of π/2rad and increases the slopes of the magnitude curves with a value equal to 20dB/dec [16]. By following a procedure similar to that of the integral action and from equations (7) and (8), we note that with the presence of the fractional derivative action [8],The magnitude curve is shifted with an interval value0dB/dec;20dB/decThe phase plot is delayed by an angle with respect to the interval0;π/2rad


## 3.1. Complex Plane Analysis

In the complex plane, the roots of the system given in (1) are located according to the following equation [8]:(5)1+KsλGs=0.The magnitude and the phase are given by(6)K=1sλGs,argsλGs=2k+1π,k=0,±1,±2,…,or s=sejθ, such that θ is the argument of s.


## 3.2. Fractional-Order Integral Action

Integral action slows down the system, eliminates steady-state error, and decreases relative stability [8]. In the frequency domain, the integral action increases the slopes of the amplitude curve by a coefficient −20dB/dec and decrements the phase plots by a coefficient π/2rad. In the complex plane, the integral action makes it possible to move the root locus of the system on the right half-plane [8]. On the other hand, the fractional integral control action is analyzed in the frequency domain as follows. The magnitude and the phase curves are, respectively, given by [8](7)20logsλGss=jω=20logGjω+20λlogω.The phase plot is given by(8)argsλGs=argGs+λπ2.Accordingly, if we chooseλ in the interval −1;0,The magnitude curve is shifted with an interval value−20dB/dec0dB/decThe phase plot is delayed by an angle with respect to the interval−π/2rad,0


## 3.3. Fractional-Order Derivative Action

The derivative control action increases the relative stability but unfortunately it makes the system more sensitive to high frequency noise [8]. In the complex plane, it allows to move the locus of the roots of the system on the left half-plane. In the frequency domain, the derivative control action produces a constant phase advance of π/2rad and increases the slopes of the magnitude curves with a value equal to 20dB/dec [16]. By following a procedure similar to that of the integral action and from equations (7) and (8), we note that with the presence of the fractional derivative action [8],The magnitude curve is shifted with an interval value0dB/dec;20dB/decThe phase plot is delayed by an angle with respect to the interval0;π/2rad


## 4. Problem Characterization

Many industrial processes can be characterized by a first order with time-delay transfer function:(9)Gs=KTs+1e−Ls,where K represents the steady-state gain, L represents the time delay, and T represents the time constant for the plant. These three parameters are supposed to be positive. In this part, we consider the problem of stabilizing the first-order plus time-delay system with a noninteger regulator PIλ.(10)Cs=Kp+Kisλ.The theory of fractional analysis, which is to be developed in this part, was based on the Hermite–Biehler theorem and was intended for first-order time-delay systems. Generally, this kind of system is characterized by the following relation:(11)Hs=bs+e−L1sm1s+e−L2sm2s+⋯+e−Lnsmns,where bs and mjs for j=1,2,…n are polynomials with real coefficients and L1,L2,…,Ln represent the delays. This type of equations, including s and es as variables, is called a quasi-polynomial and is used to characterize the closed-loop system when the transfer function involves time delay. This analysis is based on a new version of the Hermite–Biehler theorem for Hurwitz quasi-polynomials while respecting the following assumptions:(12)A1:0<L1<L2<…<Ln,A2:degbs=m,etdegmjs<m.


### 4.1. Hermite–Biehler Theorem

χjω=χrω+jχiω where χiω and χrω are, respectively, the imaginary and the real part of the quasi-polynomial χjω [17, 18]. Within A1 and A2, χjω is stable if and only if(1)
χrω and χiω have only simple real roots, and these are interlaced(2)
dχiω/ω∗χrω−dχrω/ω∗χiω>0, for some ω∈−∞;+∞⋄


### 4.2. Pontryagin Theorem

LetM and N indicate the highest powers of s and es, respectively, in χjω [17, 18]. ς is a suitable constant such that the coefficients of terms of highest degree in χrω and χiω do not vanish at ω=ς. As well, it is necessary and sufficient that in the intervals(13)−2lπ+ς≤ω≤2lπ+ςl=1,2,3,…,the equations χrω=0 and χiω=0 have exactly 4lN+M roots, starting with sufficiently large l.


## 4.1. Hermite–Biehler Theorem

χjω=χrω+jχiω where χiω and χrω are, respectively, the imaginary and the real part of the quasi-polynomial χjω [17, 18]. Within A1 and A2, χjω is stable if and only if(1)
χrω and χiω have only simple real roots, and these are interlaced(2)
dχiω/ω∗χrω−dχrω/ω∗χiω>0, for some ω∈−∞;+∞⋄


## 4.2. Pontryagin Theorem

LetM and N indicate the highest powers of s and es, respectively, in χjω [17, 18]. ς is a suitable constant such that the coefficients of terms of highest degree in χrω and χiω do not vanish at ω=ς. As well, it is necessary and sufficient that in the intervals(13)−2lπ+ς≤ω≤2lπ+ςl=1,2,3,…,the equations χrω=0 and χiω=0 have exactly 4lN+M roots, starting with sufficiently large l.


## 5. Stability Region for PI Fractional Controller

According to the Hermite–Biehler theorem, the first condition ensures that the real and imaginary parts of the characteristic equation have only simple real roots and these are interlaced. The first part of this theorem is checked by the Pontryagin theorem.The characteristic equation of the closed-loop system in Figure1 is as follows:(14)Hs=KKi+KKpsλe−Ls+1+Tssλ.Consider that the system to be controlled in this section is the first-order time-delay system presented by expression (9). Thus, our objective is to study the stability of the system, so we must analytically calculate all the parameters of the controller Kp,Ki, and λ of the transfer function of the noninteger order controller given by expression (10). The real and imaginary parts of the characteristic relation become(15)Hr∗ω=KKi+KKp+cosω−TLωsinωRejλωλ1Lλ−sinω+TLωcosωImjλωλ1Lλsignω,Hi∗ω=KKp+cosω−TLωsinωImjλωλ1Lλ∗signω−sinω+TLωcosωRejλωλ1Lλ.


### 5.1.Kp Range and Pontryagin Condition

Clearly here,Hr∗ω is function on the two variables Kp and Ki while the imaginary part Hi∗ω depends only on the parameter Kp. For this reason, let us now consider the system where the parameters are fixed like L=4, K=1, and T=2 and the fractional integral action λ=0.5. To ensure the stability of the closed-loop system, we first implement the Pontryagin theorem to determine the proportional gain Kp which ensures the real roots of the imaginary part Hi∗ω. This step allows us to define the best value of ς which gives the largest range of parameter Kp for−2lπ+ς≤ω≤2lπ+ς,l=1,2,3,…. This theoretical development of these results is quite involved in the following algorithm.Here,Nr represents the number of roots for Hi∗ω and Kp1 and Kp2 represent the first interval limits of Kp arbitrary choose. The values of Kp ensuring real roots for Hi∗ω are given by Algorithm 1 as shown in Figure 2. Note that the projection ς=0.8 corresponds to the largest range for Kp.Algorithm 1:Kp vs. ς is calculated by considering the Pontryagin condition to the PIλ regulator.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1(2)
Count the roots of the equationHi∗ω=0 for ω∈−2π+ς:2π+ς(3)
IfNr=4lN+M, then go to 4; else increment Kp=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
Evolution ofKp for a PI1/2 regulator.From Figure3, it is remarkable that, for some values inside this larger range of −1<Kp<2.5, the Pontryagin condition is respected. We can verify that for Kp=−0.5, Kp=0, and Kp=1, Hi∗ω has exactly five roots while for Kp=3, it has only one root. Therefore, out of this range, Hi∗ω does not have exactly five roots.Figure 3
Trend ofHi∗ω for different values of Ki for the PI1/2 regulator.Since the value ofς is chosen, we calculate the range of the proportional gain Kp for different values of λ in the interval 0.2;1.8 as shown in Figure 4.Figure 4
Kp vs. λ as computed by applying the Pontryagin theorem to the PI1/2 controller.


### 5.2. Solution of the Hermite–Beihler Conditions and Set of Stability Region

The next phase consists of calculating the intervals ofKp and Ki which satisfy the interlacing condition between the roots of the imaginary and real parts. From there, we remind that Kp and Ki affect the real part Hr∗ω. However, it is enough to ensure that the interlacing property is respected between the roots of the real and imaginary parts of the characteristic equation, which also means that the real part has only simple real roots.In fact, the real part of the characteristic equation can be written as follows:(16)Hr∗ω=KKi+Kppω+qω,where(17)pω=Rejλωλ1Lλ,(18)qω=ωλ1/LλKRejλcosω−TLωsinω−Imjλsinω+TLωcosωsignω.We now evaluateHr∗ω at the roots of the imaginary part Hi∗ω. For ωj=0, using (15), we obtain(19)Hr∗0=KKi.Forω=ωj≠0, Hr∗ωj can take the following form:(20)Hr∗ωj=KKi+pωjKp+qωj.Interlacing the roots ofHr∗ωj and Hi∗ωj is equivalent to Hr∗0>0 since Ki>0, Hr∗ω1<0, Hr∗ω2>0, Hr∗ω3<0, and so on.Now, let us explore the conditions that allow us to verify whether the roots ofHr∗ωj and Hi∗ωj are interlaced. In fact, we compute the values of the real part Hr∗ωj in the zeros of the imaginary part Hi∗ωj. Therefore, the interlacing property ensures that(21)Hr∗0>0⟶KKi>0,which implies(22)Ki>0,for ω=ωj≠0,(23)−1jHr∗ωj>0,(24)−1jpωjKp+−1jqωj>−1j+1Ki.Letaj be a complex parameter equal to(25)aj=−1jpωjKp+−1jqωj.From equations (22)–(24), it is possible to develop the system of inequalities as follows:(26)Hr∗0>0Hr∗ω1<0Hr∗ω2>0Hr∗ω3<0Hr∗ω4>0⟶Ki>0Ki<a1Ki>a2Ki<a3Ki>a4.The approach developed previously is summarized in Algorithm2.Algorithm 2:Stability region for the system with fixed parameters.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1.(2)
Compute the roots forHr∗ω=0.(3)
Determine the values ofKi.(4)
IncrementKp=Kp+step; If Kp<Kp2, then go to 2; else give the region Kp;Ki.We note that,Kp1 and Kp2 represent the terminals of the interval Kp determined by the first algorithm. By applying Algorithm 2, we can find the stability region of the system Gs=1/2s+1e−4s as shown in Figure 5. Note that for Kp values outside this range, there is no stabilizing PIλ=0.5 controller.Figure 5
Stability region forλ=0.5.By sweeping over all−1.4<Kp<1.5, we can generate the set of values Kp;Ki;λ for which the closed-loop system is stable. The region of global stability can then be visualized in a 3D plot as shown in Figure 6.Figure 6
Global Stability region.


## 5.1.Kp Range and Pontryagin Condition

Clearly here,Hr∗ω is function on the two variables Kp and Ki while the imaginary part Hi∗ω depends only on the parameter Kp. For this reason, let us now consider the system where the parameters are fixed like L=4, K=1, and T=2 and the fractional integral action λ=0.5. To ensure the stability of the closed-loop system, we first implement the Pontryagin theorem to determine the proportional gain Kp which ensures the real roots of the imaginary part Hi∗ω. This step allows us to define the best value of ς which gives the largest range of parameter Kp for−2lπ+ς≤ω≤2lπ+ς,l=1,2,3,…. This theoretical development of these results is quite involved in the following algorithm.Here,Nr represents the number of roots for Hi∗ω and Kp1 and Kp2 represent the first interval limits of Kp arbitrary choose. The values of Kp ensuring real roots for Hi∗ω are given by Algorithm 1 as shown in Figure 2. Note that the projection ς=0.8 corresponds to the largest range for Kp.Algorithm 1:Kp vs. ς is calculated by considering the Pontryagin condition to the PIλ regulator.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1(2)
Count the roots of the equationHi∗ω=0 for ω∈−2π+ς:2π+ς(3)
IfNr=4lN+M, then go to 4; else increment Kp=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
Evolution ofKp for a PI1/2 regulator.From Figure3, it is remarkable that, for some values inside this larger range of −1<Kp<2.5, the Pontryagin condition is respected. We can verify that for Kp=−0.5, Kp=0, and Kp=1, Hi∗ω has exactly five roots while for Kp=3, it has only one root. Therefore, out of this range, Hi∗ω does not have exactly five roots.Figure 3
Trend ofHi∗ω for different values of Ki for the PI1/2 regulator.Since the value ofς is chosen, we calculate the range of the proportional gain Kp for different values of λ in the interval 0.2;1.8 as shown in Figure 4.Figure 4
Kp vs. λ as computed by applying the Pontryagin theorem to the PI1/2 controller.


## 5.2. Solution of the Hermite–Beihler Conditions and Set of Stability Region

The next phase consists of calculating the intervals ofKp and Ki which satisfy the interlacing condition between the roots of the imaginary and real parts. From there, we remind that Kp and Ki affect the real part Hr∗ω. However, it is enough to ensure that the interlacing property is respected between the roots of the real and imaginary parts of the characteristic equation, which also means that the real part has only simple real roots.In fact, the real part of the characteristic equation can be written as follows:(16)Hr∗ω=KKi+Kppω+qω,where(17)pω=Rejλωλ1Lλ,(18)qω=ωλ1/LλKRejλcosω−TLωsinω−Imjλsinω+TLωcosωsignω.We now evaluateHr∗ω at the roots of the imaginary part Hi∗ω. For ωj=0, using (15), we obtain(19)Hr∗0=KKi.Forω=ωj≠0, Hr∗ωj can take the following form:(20)Hr∗ωj=KKi+pωjKp+qωj.Interlacing the roots ofHr∗ωj and Hi∗ωj is equivalent to Hr∗0>0 since Ki>0, Hr∗ω1<0, Hr∗ω2>0, Hr∗ω3<0, and so on.Now, let us explore the conditions that allow us to verify whether the roots ofHr∗ωj and Hi∗ωj are interlaced. In fact, we compute the values of the real part Hr∗ωj in the zeros of the imaginary part Hi∗ωj. Therefore, the interlacing property ensures that(21)Hr∗0>0⟶KKi>0,which implies(22)Ki>0,for ω=ωj≠0,(23)−1jHr∗ωj>0,(24)−1jpωjKp+−1jqωj>−1j+1Ki.Letaj be a complex parameter equal to(25)aj=−1jpωjKp+−1jqωj.From equations (22)–(24), it is possible to develop the system of inequalities as follows:(26)Hr∗0>0Hr∗ω1<0Hr∗ω2>0Hr∗ω3<0Hr∗ω4>0⟶Ki>0Ki<a1Ki>a2Ki<a3Ki>a4.The approach developed previously is summarized in Algorithm2.Algorithm 2:Stability region for the system with fixed parameters.
(1)
Initializestep=0.1; Nr=0; Kp=Kp1.(2)
Compute the roots forHr∗ω=0.(3)
Determine the values ofKi.(4)
IncrementKp=Kp+step; If Kp<Kp2, then go to 2; else give the region Kp;Ki.We note that,Kp1 and Kp2 represent the terminals of the interval Kp determined by the first algorithm. By applying Algorithm 2, we can find the stability region of the system Gs=1/2s+1e−4s as shown in Figure 5. Note that for Kp values outside this range, there is no stabilizing PIλ=0.5 controller.Figure 5
Stability region forλ=0.5.By sweeping over all−1.4<Kp<1.5, we can generate the set of values Kp;Ki;λ for which the closed-loop system is stable. The region of global stability can then be visualized in a 3D plot as shown in Figure 6.Figure 6
Global Stability region.


## 6. Simulation Results

In order to examine the accuracy of the stability region and the efficiency of our tuning method, step responses are calculated on either side of the stability region. Figures7 and 8 illustrate the calculation of the step responses with a value of the pair Kp,Ki chosen within the interlimit of the stability region. On the other hand, Figure 9 shows the step responses calculated with a value of the pair Kp,Ki choosing from the outer limit of the stability region. It is clear that the closed-loop system has a convergent stable dynamic when the controller parameters choose within the internal limit of the stability region, and it has a divergent unstable dynamic when the controller parameters choose from the outer limit of the stability region.Figure 7
Responses to step for varyingλ>1.Figure 8
Responses to step for varyingλ<1.Figure 9
Responses to step for varyingλ=1.3.We can also observe that for lowλ values, this gives high static error values; otherwise, that is, λ greater than 1, the static error is zero. It is interesting to note that the variation of the references is essential to evaluate the performances of the method developed in the presence of time delay.


## 7. Conclusion

In this work, we studied the stabilization problem of first-order plus time-delay systems with fractional controller. The quasi-polynomials which model the delayed systems are characterized by an infinite number of roots which makes their study more complicated. In this case, a fractional analysis theory based on the Hermite–Biehler and Pontryagin theorems makes it possible to define a stability region for this kind of systems. The interval ofKp was calculated by applying Pontryagin’s theorem. According to these values, the roots of the imaginary part function are all simple and real. According to the interlacing property, for each value of Kp, there will be an interval of values of Ki which represent the region of stability established for this fractional controller.



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