International Conference on Fractional Differentiation and its Applications, [601919]

International Conference on Fractional Differentiation and its Applications,
Novi Sad, Serbia, July 18 – 20, 2016
1

Lag and Lead Fractional -Order Compensator s Design

Reyad El -Khazali

ECE Department
Khalifa University
Sharjah -UAE
e-mail: [anonimizat]

Abstract : This paper introduces a new class of fractional -order lag and lead compensators. Both
types of compensators are described by biquadratic transfer functions that are uniquely identi fied
in terms of their fractional orders, thus, reducing the number of parameters that need to be
determined. The proposed design technique utilize s the frequency response of the unco mpensated
system to improve its gain and phase margins . It is straightforward and the proposed lag and lead
compensators enjoy flat phase response to provide more robustness. Moreover, the goals of the
design are achieved from the first attempt at desired cross over frequencies. The proposed design
algorithms are demonstrated via numerical examples .

Key words : Fractional -order calculus , controller design, Lag compensator, Lead
compensator , biquadratic.

1. Introduction

Fractional calculus is more than 300 years old. It is a generalization of the
“integer -order” calculus, where most dynamic systems that exhibit non -Newtonian
behavior or have some type of hereditary effect are best described by fraction al-order
differentiation or integration . For example, motion of bodies through viscous medium,
propagation of sound waves , the diffusion of a heat flow through a semi -infinite solid ,
and t he voltage -current relationship of a semi -infinite lossy t ransmissio n line obey
fractional -order dynamics [1, 2, 5 ].
Fractional calculus is widely used in many applications such as in circuit theory,
control applications, system modeling, and fractional -order capacitors and inductors [ 8].
The application of fractional -order controllers supersedes their integer -order counterparts
[3,13 ]. Fractional -order PID controllers , (or 𝑃𝐼𝜆𝐷𝜇), for example represent weighted
forms of lag and lead controllers. They enjoy two additional parameters to tune; i.e., the
order of the integral dynamics , 𝜆, and the order of its differential action, 𝜇, which
provides more control over their dynamic changes .
The interest here is to introduce and a new straightforward design method of new
class of lag and lead compensators. T he purpose of the lag and/or lead compensator
design is to improve the transient and steady -state system response b y improving the
system’s phase and gain margin s. In addition, one could choose t he gain cross over
frequency to reshape the system’s bandwidth and its speed response [11].

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
2 The phase margin defines the relative stability and the desired transient respo nse
characteristics . The speed of system response is inversely proportional to the system
bandwidth. The lead compensator is used to adjust the system’s Bode phase curve in
order to reduce the steady state error and to achieve a required phase margin at a specified
frequency. The fractional -order lag compensator is used to drop the magnitude curve
down to reshape the system’s transient response [12]. It can be considered as a
controllable predictive compensator.
The integer -order s ingle -stage lag or lead c ompensators can provide up to 55°-60°
phase change, which sometimes may not be sufficient and does not provide robustness
due to their narrow band response. M ultiple -stage, or cascaded compensators, can be used
to shift the phase diagram by a sufficient am ount usually cannot be accomplished by
single -stage ones. Moreover, precise cross over frequency or exact phase margins cannot
be achieved easily by using the classical forms of single -stage or multiple -stage
compensators [OGATA]. It may need several attem pts to tune several parameters before
reaching the goal.
Therefore, a new class of fractional -order biquadratic lag and lead compensators
are introduced in this work to overcome the design limitations of the classical integer –
order compensators. They provi de flat phase and completely characterized by only two
parameters ; the gain and the order of the compensator. Similar biquadratic forms were
used to design 𝑃𝐼𝜆𝐷𝜇 controllers, and to realize fractional -order capacitors and inductors
[13, 14, 15]. The phase diagrams of such compensators are completely controllable and
can be centered at any desired frequency to control the cross over frequency of the
system. In some cases, these compensators are uniquely determined by their fractional
order.
The paper is organized as follows: next section introduces preliminaries and
fundamentals . Section 3 outlines the design pro cedure of the lag and lead compensators,
which includes simple examples to clarify the design steps. Finally, section 5 summarizes
the main findings of this work.
.
2. Preliminaries and Background
Fractional calculus is a generalization of the differentia l-order (integral -order)
operators; denoted by 𝐷𝛼≡ 𝐷𝑡𝛼
𝑎 (𝐼𝑡−𝛼
𝑎 ), respectively, where a and t are the limit s of
operation. T he Riemann -Liouville definition of fractional -order derivative of order α is
considered and defined by [1, 7, 16]:
𝐷𝑡𝛼𝑦(𝑡)=1
𝛤(𝑛−𝛼)𝑑𝑛
𝑑𝑡𝑛∫𝑦(𝜏)
(𝑡−𝜏)𝛼−𝑛+1𝑑𝜏𝑡
𝑎 𝑎 ; 𝑛−1<𝛼<𝑛 (1)
The Laplace transform of (1) is given by [2]
ℒ{𝐷𝑡𝛼
𝑎 𝑦(𝑡)}=𝑠𝛼𝑌(𝑠)−∑ 𝑠𝛼−𝑘−1𝑦(𝑘)(0) 𝑁
𝑘=0 ; 𝑛−1<𝛼<𝑛 (2)
In addition, i f the signal, y(t), is initially at rest, then £{𝐷𝑡𝛼
𝑎 𝑓(𝑡)}=𝑠𝛼𝐹(𝑠). Similarly,
the Laplace transform of the fractional -order integral of y(t); i.e., 𝐼𝑡−𝛼𝑦(𝑡)0 , is given by:
ℒ{𝐼𝑡−𝛼𝑦(𝑡)0 }=𝑠−𝛼𝑌(𝑠) (3)

2.1. System Model
A typical open -loop transfer function of a fractional -order system can be
described by the following transfer function :

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
3 𝐺(𝑠)=𝑏𝑚𝑠𝜈𝑚+𝑏𝑚−1𝑠𝜈𝑚−1+⋯+𝑏1𝑠𝜈1+𝑏𝑜𝑠𝜈𝑜
𝑎𝑛𝑠𝛼𝑛+𝑎𝑛−1𝑠𝛼𝑛−1+⋯𝑎1𝑠𝛼1+𝑎𝑜𝑠𝛼𝑜 (4)
where , without loss of generality, 0≤𝛼𝑜<𝛼1<𝛼2… <𝛼𝑛, and 0≤𝜈𝑜<𝜈1<
𝜈2… <𝜈𝑚.

2.2 New Lag and Lead Compensators
The classical class of 1st-order lag or le ad compensator s have three
unknown parameters to design ; the value of the pole, the zero, and the gain of the
compensator. If one wish to design 2nd-order classical lag or lead compensators to
achieve robustness , a total of fiv e parameters have to be deter mined , which
increase s the comple xity of the design. Moreover, the design requirements may
not be achieved from the first attempt [11].
In this work, a new form of biquadratic lag or lead compensators
comprises a special class of 2nd-order filters that can only be identified by two
parameters instead of five; i.e., the gain and order of the compensator s. Thus,
reducing design complexity and ensuring system robustness.
The proposed class of lag or lead compensators are given by:
𝐺𝑐(𝑠,𝛼)=𝐾𝑐𝑎2𝑠2+𝑎1𝑠+𝑎𝑜
𝑎𝑜𝑠2+𝑎1𝑠+𝑎2 (5)

where 0<𝛼≤1 is the order of the compensator, and where ,

a2=(𝛼𝛼+2𝛼+1)
𝑎o=(𝛼𝛼−2𝛼+1)
𝑎1=(a2−ao)𝑡𝑎𝑛((2−𝛼)𝜋
4) (6)
The biquadratic transfer functions (5) were thoroughly investigated in [ 14]
and similar forms were also used to design fractional -order PID controllers [13]
Observe that the parameters of the compensators given by (6) depend only on its
order, 𝛼. Therefore, the design will be accomplished by finding the gain and order of the
compensator . In some cases, only the order of t he compensator will be sufficient to
complete the design, which represents a significant improvement over the classical
methods of lag or lead compensator s design .
Observe that a proper selection of 𝑎𝑜,𝑎1, and 𝑎2 approximate s a band -limited
fractional -order differential (integral) operator, which can be used to design fractional –
order lag (𝑃𝐼−𝛼) or fractional -order lead ( 𝑃𝐷𝛼) compensators [13].
In orde r to change the operating point (or the centre frequency) of the control ler,
one can shift the cross -over frequency of the compensator (5 ) from 𝜔=1 rad/s to any
desired frequency at 𝜔𝑐 rad/s; i.e.,
𝐺𝑐(s
ωc,α)=Kc𝑎2(s/ωc)2+𝑎1(s/ωc)+𝑎o
𝑎o(s/ωc)2+𝑎1(s/ωc)+𝑎2 (7)
when |𝐺𝑐(𝜔)|=𝐾𝑐 at 𝜔=𝜔𝑐, and its phase contribution is equal to:
𝜑𝐺𝑐(𝜔/ωc)=tan−1(𝑎1
𝑎𝑜−𝑎2)−tan−1(𝑎1
𝑎2−𝑎𝑜) (8)

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
4 From (6 ), and for 0<𝛼≤1, 𝑎2>𝑎o then the phase angle of equation (8 ) is given by :
𝜑𝐺𝑐(𝜔𝑐)=tan−1(𝑎1
𝑎o−𝑎2)−tan−1(𝑎1
𝑎2−𝑎o)=𝜋−2tan−1(𝑎1
𝑎2−𝑎o)>0 (9)
which represents a fractional -order lead compensator centered at 𝜔𝑐.
Consequently, due to the biquadratic structure of (7 ), its reciprocal yields a
lagging phase contribut ion; i.e.,
𝐺𝑐(s
ωc,α)=Kc𝑎𝑜(s/ωc)2+𝑎1(s/ωc)+𝑎2
𝑎2(s/ωc)2+𝑎1(s/ωc)+𝑎𝑜 (10)
Since 𝑎o<𝑎2, then (10 ) implies :
𝜑𝐺𝑐(𝜔𝑐)=tan−1(𝑎1
𝑎2−𝑎𝑜)−tan−1(𝑎1
𝑎𝑜−𝑎2)=−𝜋+2tan−1(𝑎1
𝑎2−𝑎0) <0 (11)
which describes a fractional -order lag compensator .

(a) (b)
Figure 1 . Bode diagram of 𝐺𝑐(𝑠,𝛼) of a) lag, and b) lead compensators .
Figure 1 shows the bode diagrams of (10) and (7 ), respectively, and reassures
their lagging and leading nature . For example, when 𝐾𝑐=1,𝜔𝑐=1, figure 1 shows the
bode diagrams of fractional -order lag and lead compensators of orders 𝛼=0.5,and 𝛼=
0.9, respectively,. Clearly, the phase contribution of this class of compensators is
completely controllable by their fractional orders.
To appreciate the robustness and the simplicity of the proposed class of
compensators, figure 2 shows the bode diagrams of two lead compensators that provide a
leading phase of 𝜋/4 at 𝜔𝑐=1 rad/s. The first compensator is a classical 1st-order
compensator described by the following transfer function [13,14,10]:
𝐺1(𝑠)=2.414 𝑠 +1
0.4142 𝑠+1 (12)
while the second one is a 2nd-order compensator defined by (7 ) and given by the
followi ng biquadratic transfer function :
𝐺𝑐=2.37152.7071 𝑠2 + 4.8284 𝑠 + 0.7071
0.7071 𝑠2 + 4.8284 𝑠 + 2.7071; 𝐾𝑐=2.3715 , 𝛼=0.5 (13)
-40-2002040Magnitude (dB)

10-310-210-1100101102103-90-81-450Phase (deg)Bode Diagram
Frequency (rad/sec) = 0.5
 = 0.9
-40-2002040Magnitude (dB)
10-310-210-11001011021030458190Phase (deg)
Bode Diagram
Frequency (rad/sec) = 0.5
 = 0.9

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
5
Figure 2 . Bode diagrams of (12) and (13 ).

Obviously the 1st-order compensator (12) provides a maximum phase angle
over a narrow frequency band, which does not provide robustness to the controlled
plant. Comparing a 1st-order compensator with a 2nd-order one is not accurate as one
may claim. Hence, to justify the comparison between similar struct ures, one may
cascade the 1st-order compensator (12 ) by another 1st-order one centred at 𝜔=10
rad/s with a slight gain adjustment to yield;
𝐺1(𝑠)=0.4127(2.414 𝑠 +1)(0.2414 𝑠+1)
(0.4142 𝑠+1)(0.04142 𝑠+1) (14)
Figure 3 display s the bode diagram of (13) and (14 ). Clearly, the bandwidth of
the two com pensators are almost similar , however, the cascaded first -order compensators
do not provide the exact phase contribution at the desired gain crossover frequency ,
which requires many trials and errors to the achieve the desire design goals. This adds
another reason to adopt the biquadratic structure described by (7) and (10 ) as will be
verified by the design procedure of the proposed lag and lead compensators [10, 6].

Figure 3 . Bode diagrams of the compensators defined by (1 3) and (1 4).

The following section out lines a systematic procedure to design the lag
and lead compensators.

-505101520Magnitude (dB)

10-210-110010110203060Phase (deg)Bode Diagram
Frequency (rad/sec)Integer-order lead compensator
Fractional-order lead compensator
-20-10010203040Magnitude (dB)
10-210-11001011021030306090Phase (deg)
Bode Diagram
Frequency (rad/sec)Biquadratic Lead Compensator
Casvaded Lead Compensator

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
6 3. Compensators Design

3.1 Fractional -order Lag Compensator Design
Lag compensators are used to improve the steady -state response of
system s [13]. It can be considered as a fractiona l-order predictive controller ,
which provides a negative phase margin , and shifts the crossover frequency of the
controlled system to a lower value . Thus, the controlled system becomes slower
due to a decrease in its bandwidth . Notice that t he phase differ ence between the
controlled and the uncontrolled systems determine the order of the compensator.
Consider the lag compensator given by (10 ). As mentioned earlier, it is
sufficient to design this class of compensators by solely selecting the gain , 𝐾𝑐, and
the fractional order ,𝛼. When 𝛼=1, equation (6 ) yields 𝑎𝑜=0,𝑎1=𝑎2=4 and
the lag compensator reduces to 𝐺𝑐(𝑠
𝜔𝑐,1)=𝐾𝑐𝜔𝑐
𝑠. In this case , equation (10 )
defines a pure integrator.

3.1.1 Design Procedure
The d esign procedure of the lag compensator defined by (10 ) can be
summarized by the following steps:

Step 1 : Plot the bode diagram of th e open -loop system using MATLAB and
determine its phase margin, 𝜑𝑚𝑜, gain margin, 𝑔𝑚𝑜, and the correspond ing gain
and phase crossover frequencies; 𝜔𝑐𝑔𝑜,𝑎𝑛𝑑 𝜔𝑐𝑝𝑜, respectively.
Step2: Specify a desired phase margin , Φ𝑚. Sketch a horizontal line that is Φ𝑚
degrees away from the -180° line, which intersects the phase diagram at a
frequency denote it as 𝜔𝑐.
Step 3: Determine the centre frequency of the compensator t o be one frequency
decade below 𝜔𝑐, i.e.,
𝜔𝑐𝑝𝑛=𝜔𝑔𝑛
10 rad/s (15)
Step 4: Determine the phase of the uncompensated system at 𝜔𝑐𝑝𝑛 and define it by
𝜙𝑐𝑝𝑛. Calculate the phase lag needed, ( 𝜙𝑐𝑝𝑛−Φ𝑚)>0, to reduce the phase to
the desired value.
Step 5 : Calculate the order of the lag compensator from:

𝛼=𝜙𝑐𝑝𝑛−Φ𝑚
π/2 (16)
Step 6: Determine the gain margin of the uncompensated system at 𝜔𝑐𝑝𝑛 , and
denote it by 𝑔𝑐𝑝𝑛dB.
Step 7: Calculate th e gain of the compensator that reduces the gain margin of the
uncompensated system to 0 dB at 𝜔𝑐𝑝𝑛 from :

𝐾𝑐=10(−𝑔𝑐𝑝𝑛/20) (17)

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
7 Step 8 : Sketch the bode diagram of the controlled s ystem to verify the design
requirements . If the design requirements are not met, slightly shift the corner
frequency of the compensator, 𝜔𝑐𝑝𝑛, to the left and repeat your design steps.

Example 1 : Consider the following transfer function of an o pen-loop system :
𝐺(𝑠)=300
𝑠(𝑠2+6𝑠+600 )
It is desired to design a lag compensator such that the controlled system has a phase
margin of 55𝑜 and a gain margin greater than 10 dB.

1) Initially plot the bode diagram of the uncompensated system as shown in Figure 4.
The phase and gain margins of the uncontrolled system are 𝑔𝑚𝑜=1.58 dB at 𝜔𝑔=
7.5 rad/s and 𝜙𝑠𝑜=16.1𝑜 at 𝜔𝑝=6.93 rad/s, respectively, which do not m eet the
design requirements.
2) Sketch a horizontal line that intersects the phase diagram. Determine t he new
crossover frequency that corresponds to a phase margin of 55𝑜, which is equal to
𝜔𝑐=5.25 rad/s. Hence, the new phase crossover frequency of the compensator,
which will be its centre frequency, is chosen one decade below 𝜔𝑐; i.e., 𝜔𝑐𝑝𝑛=
0.525 rad/s .
3) The new phase of the uncompensated system at 𝜔𝑐𝑛=0.525 rad/s is almost equal
to 𝜙𝑐𝑝𝑛=−95𝑜. Hence, from (16 ), 𝛼=−95−(−125 )
90=0.333 .
4) The gain margin of the uncompensated system at 𝜔𝑐𝑝𝑛=0.525 is equal to 𝑔𝑐𝑛=
−20 dB. From (17 ), the gain of the compensator 𝐾𝑐=10(−20/20) =0.1.
5) Finally , using (6) and (10 ) for 𝐾𝑐=0.1 and 𝛼=0.333 yield the following transfer
function of the desired lag compensator :
𝐺𝑐(𝑠
0.525,0.333 )=0.11.027 𝑠2+3.483 𝑠+1.156
2.36𝑠2+43.483 𝑠+0.5031 (18)

-100-50050100
Magnitude (dB)
10-210-1100101102-270-225-180-135-90-450Phase (deg)Bode Diagram
Gm = 27.3 dB (at 7.19 rad/sec) , Pm = 57.1 deg (at 0.507 rad/sec)
Frequency (rad/sec)Uncompensated SystemCompensated System
Lag Compensator

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
8 Figure 4 . Bode diagram of the system with and without compensator.

Clearly the phase margin of the compensated system is 57.1°>55°at 𝜔𝑐𝑝𝑛=
0.507 rad/s and the gain margin is 27.3 dB>10 dB𝜔𝑐𝑔𝑛=7.19 rad/s . The large value
of the gain margin permits a gain increase to the compensated systems in order to adjust
its transient response . Figure 5 shows the step response of the uncompensated and
compensated systems for two values of compensator gains; 𝐾𝑐=0.1 and 𝐾𝑐=1.

Figure 5 . Step response of the uncompensated and compensated systems.

3.2. Lead Compensator Design
Lead compensator s are used to improve the transient response of the system. The
lead compensator s, defined by (7 ), represent a class of fractional -order controllers, 𝑃𝐷𝛼,
and provide a leading phase 0<𝜑𝐺𝑐(𝜔/ωc)≤𝛼𝜋/2. If a leading phase of more than 𝜋/2
is needed, additional compensators may be cascaded at the same corner frequency.
Let G(𝑗𝜔) be the freque ncy response of an uncompensated system. Let φ𝑐𝑝𝑜
and 𝑔𝑐𝑔𝑜 be the ph ase margin and gain margin of the uncon trolled system at the
corresponding cross over frequency, 𝜔𝑐𝑝𝑜, and 𝜔𝑐𝑔𝑜, respectively. Let t he characteristic
equation of the controlled system be described by :
𝑇(𝑗𝜔)=1+𝐺𝑐(𝑗𝜔/𝜔𝑐,𝛼)G(𝑗𝜔)=0. (19)
The goal is to design a fractional -order lead compensator, or a 𝑃𝐷𝛼 controller, to
achieve a phase margin, Φ𝑚, at 𝜔𝑐𝑝𝑜, and a desired gain margin, 𝑔𝑚 at 𝜔𝑐𝑔𝑜. This can be
accomplished by, simultaneously, maintain ing:
𝐴𝑟𝑔 (𝐺𝑐(𝑗𝜔
𝜔𝑐𝑝𝑜,𝛼)G(𝑗𝜔
𝜔𝑐𝑝𝑜))=𝛼𝜋
2+𝜃𝑐𝑝𝑜=−𝜋+Φm (20)
and
20𝑙𝑜𝑔|𝐺𝑐(𝑗𝜔
𝜔𝑐𝑔𝑜,𝛼)G(𝑗𝜔/𝜔𝑐𝑔𝑜)|=𝒈𝒎 (21)
0 5 10 1500.511.5
Step Response
Time (sec)AmplitudeUncompensated System
Compensated system with Kc = 1
Compensated System with Kc = 0.1

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
9 The order of the lead compensator is then evaluated from (20) and given by:
𝛼=Φ𝑚−(𝜋+𝜃𝑐𝑝𝑜)
𝜋/2 (22)
while the gain of the compensator is determined from (21) and given by:
𝐾𝑐=10𝑔𝑚
|G(jω/𝜔𝑐𝑔𝑜)| (23)
In general, the lead compensator is much easier to design than the lag
compensator since adding an additional phase would migrate the phase cross over
frequency, 𝜔𝑐𝑝𝑜, to the right, which improves the system ga in margin. Hence, the first
and most important step of the design is to determine order of the lead compensato r.
To provide comprehensive approach to the lead compensator s design , the design
procedure is divided into the following two methods :
A. First Metho d: 𝜔𝑐=𝜔𝑐𝑝𝑜.
In this case, t he design steps are straightforward since they depend on finding the
order of the compensators. The design steps are summarized as follows:
Step 1 : Calculate the order of the compensator from (2 2).
Step 2 : Set the gain of the compensator to unity, i.e., 𝐾𝑐=1.
Step 3 : Determine the compensator parameters and transfer function from (6) and (7 ).
Step 4: Check your design and slightly tune 𝐾𝑐 to achieve the design requirements. If the
design requirements are not met for this case, one has to adopt the following second
method of design:
B. Second Method : 𝜔𝑐≠𝜔𝑐𝑝𝑜
This method will be considered for highly unstable systems, where placing the
centre frequency of the compensator at 𝜔𝑐=𝜔𝑐𝑝𝑜 yields a phase increase >90°. In this
case more than one compensator would be needed to achieve the design requirements.
Therefore, if one needs to obtain a gain margin, 𝑔𝑚, by using a single compensator , the
centre frequency has to be careful ly selected at a frequency 𝜔𝑐≠𝜔𝑐𝑔𝑝. In this case , the
gain and phase margins will be simultaneously achieved. The following steps summarize
the second method of lead compensator design:
Step 1: Plot the bode diagram of the uncompensated syst em. Sketch a horizontal line that
is 𝒈𝒎 dB away from the 0 dB line.
Step 2 : Determine the frequency at which this line intersects t he gain diagram and denote
it by 𝜔𝑐𝑝𝑛. This frequency will be considered as the new phase crossover frequency of
the compensator. The cross over frequency of the compensated system will be in the
neighborhood of 𝜔𝑐𝑝𝑛.
Step 3 : Determine the phase of the uncompensated system at 𝜔𝑐𝑝𝑛and define it by 𝜑𝑐𝑝𝑛.
The required leading phase to be provided by the compensator is then equal to (−180 −
𝜑𝑐𝑝𝑛 +Φ𝑚). The order of the compensator is then determined from:

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
10 𝛼=Φ𝑚−(𝜋+𝜑𝑐𝑝𝑛)
𝜋/2 (24)
Step 4: Determine the gain of the compensator , 𝐾𝑐, that shift s the gain diagram to cross
the 0 dB line at 𝜔𝑐𝑝𝑛. Let the amount of the gain at 𝜔𝑐𝑝𝑛be 𝒈𝒎 dB.
Since |𝐺𝑐(𝑗𝜔𝑐𝑝𝑛
𝜔𝑐𝑝𝑛,𝛼)|=Kc, then from (23 ):
𝐾𝑐=10−𝒈𝒎/𝟐𝟎 (25)
Step 5: The transfer function of the lead compensator is then determined from (24), (25),
(6) and (7 ).
Step 6 : Plot the bode diagram of the compensated system and check your design. If the
design requirements are not met, adjust the gain and the order of the compensator from
(24) and (25), respectively , and repeat step 5 if necessary .
Remark 1 : For the case when 𝛼>1, and if the seco nd method did not achieve the design
requirements, one may cascade several fractional order lead compensators of a total order
equals to 𝛼.
Example 2 : Consider the following transfer function of an open -loop system:
𝐺(𝑠)=200
𝑠3+6𝑠2+6𝑠+10 (26)
It is desired to design a lead compensator such that the controlled system has a phase
margin of 45𝑜 and a gain margin greater than 10 dB .
1) From figure 6, 𝑔𝑚𝑜=−17.7 dB at 𝜔𝑐𝑔𝑜=2.45 rad/s, and 𝜑𝑚𝑜=−36.7° at
𝜔𝑐𝑝𝑜=5.33 rad/s. Obviously, the system is unstable.

Figure 6. Bode diagram of the uncompensated system.
2) Since the required 10 dB gain margin is not very large, the first method of the lead
compensator design will be used . Now, set 𝜔𝑐=𝜔𝑐𝑝𝑜=5.33 rad/sec and 𝐾𝑐=1.
-80-60-40-2002040Magnitude (dB)
10-210-1100101102-270-225-180-135-90-450
Phase (deg)Bode Diagram
Gm = -17.7 dB (at 2.45 rad/sec) , Pm = -36.7 deg (at 5.33 rad/sec)
Frequency (rad/sec)Uncompensated system

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
11 3) The phase of the uncompensated system at 𝜔𝑐=𝜔𝑐𝑝𝑜=5.33 rad/s is equal to
𝜑𝑐𝑝𝑜=−216 .7°; (corresponds to a phase margin of -36.7°. Hence, from (25), 𝛼=
(45°−(180°+(−216 .7°))
90° =81.7°
90°=0.9078 .
4) From (6), (7 ), and for 𝛼=0.9078 and 𝐾𝑐=1, the transfer function of the lead
compensator centered at 𝜔𝑐=5.33 rad/s is given by:
𝐺𝑐(𝑠)= 3.731 𝑠2+22.38 𝑠+2.851
0.1004 𝑠2 + 22.38 𝑠+ 106
5) The bode diagram of both the compensated and uncompensated systems are shown in
figure 7. Clearly, the design requirements are perfectly met from the first attempt .
6) Figure 8 shows the step response of bot h systems, which justifies the compensator
design.

Figure 7 . Bode diagram of the compensated and the compensated system.

Figure 8 . Step response of the compensated and compensated systems.
.
4. Conclusions
New design techni ques of new class of fractional -order lag and lead
compensators are introduced. The biquadratic structure of these compensators provides
-200-150-100-50050Magnitude (dB)
10-310-210-1100101102103104-270-180-90090
Phase (deg)Bode Diagram
Gm = 30 dB (at 32.6 rad/sec) , Pm = 45 deg (at 5.33 rad/sec)
Frequency (rad/sec)Uncompensated System
Compensated System
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2-1.5-1-0.500.511.52
Step Response
Time (sec)Amplitude
Compensated System
Uncompensated System

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
12 flat phase response that adds robustness to the controlled system. There are only two
unknown parameters to design; the gain and the order of the compensator. The design
methods are straightforward and, in most cases, depend on finding the fractional -order of
the compensator. The order of such class of compensators is accurately determined from
the phase information of the uncontrolled system.
The case when the order of the compensator is greater than one, which
corresponds to phase changes of less than −90° (more than 90°) for lag (lead)
compensators, respectively, two or more compensators may be cascaded to obtain
minimum phase controllers. The lag -lead controller design is left for future work.

5. References

[1] K.B. Oldham and J. Spanier, The Fractional Calculus , 1974, Academic Press.
[2] Podlubny, I., Fractional differential equations , Academic Press, New York, 1999.
[3] Podlubny I., “Fractional -Order Systems and 𝑃𝐼𝜆𝐷𝜇 controllers,” IEEE Trans.
Automatic Control , vol. 44(1), pp. 208 -214.
[4] J Sabatier, OP Agrawal, JAT Machado, Advances in fractional calculus , Spring er 4
(9).
[5] C. A. Monje , YangQuan Chen , B. M. Vinagre , Dingyu Xue , V. Feliu –
Batlle , Fractional -order Systems and Controls: Fundamentals and Applications ,
Springer London, Nov. 2014 .
[6] YangQuan Chen, Ivo Petras, and Dingyu Xue, “Fractional -order Control -A tutorial,”
ACC Conference, St. Louis, MO, USA, June 10 -12, 2009
[7] R. Caponetto, J. A. T. Machado, and J. J. Trujillo, Theory and Applications of
Fractional Order Systems, Hindawi P ub. Corp. , Math. Prob. in Engineering , Vol.
2014, Article ID 596195 .
[8] Ortigueira MD (2011), Fractional calculus for scientists and engineers , vol 84.
Springer, Berlin
[9] C. Zhao , D. Xue , and Y., “A fractional order PID tuning algorithm for a class of
fractional order plants, ” Proc. Of the IEEE Int. Conf . On Mechatronics and
Automation, 2005 IEEE Int. Conf. (ICMA), Niagra -Canada, pp. 216 -221.
[10] M. Salehtavazoei ; M. Tavakoli -Kakhki , “Compensation by fractional -order phase –
lead/lag compensators, ” IET Control Theory & Applications , Vol. 8, No. 5, 2014.
[11] R. C.Dorf , and R. H. Bishop, Modern Control Systems , AddisonWesley, 12th –
edition, Jan. 2016
[12] K. Ogata, Modern Control Engineering , Prentice Hall, Upper SaddleRiver, NJ, 4th-
edition, 2002.
[13] El-Khazali, R., “Fractional -Order PIλDµ Controller Design,” J. of Computers &
Mathematics with Applications ,” Vol. 66, No. 5, pp. 639-646, 2013.
[14] El-Khazali, R., “On the Biquadratic Approximation of Fractional -Order L aplacian
Operators, ” Analog Integrated Circuits and Signal Processing , March 2015, Volume
82, Issue 3, pp 503 -517.

Reyad El -Khazali , Lag and Lead Fractional -Order Compensators Design
13 [15] El-Khazali, R. and Tawalbe h, N., “Realization of fractional -order capacitors and
Inductors, ” 5th-IFAC Symposium on Fractional Diff. and its Applications, Nanjing,
Chin a, 14-16/5/2012.
[16] MD Ortigueira, JAT Machado, JJ Trujillo, BM Vinagre , “Fractional signals and
systems ,” Signal Processing 91 (3), 349 .