lachhab2013.pdf

Fractional Order PID Controller (FOPID)-Toolbox
Nabil Lachhab1, Ferdinand Svaricek1, Frank Wobbe2 and Heiko Rabba2
Abstract— This paper presents a fractional order PID con-
troller (FOPID)-Toolbox to design robust fractional PID con-
trollers achieving a desired crossover frequency and a desired
phase margin. A novel approach based on nonsmooth optimiza-
tion techniques is used. Two types of controllers are considered,
the (PID)n
and PIα
Dβ
controllers. The requirements to be
fulfilled by the controller are expressed in terms of a desired
open-loop response. Loop shaping configuration is used to
synthesize the controller. To optimize the fractional orders an
optimization algorithm based on the steepest descent method
is used. Simulation results show the benefit of our method.
I. INTRODUCTION
There is no doubt that the PID controller is one of the
most used controller type in control-loops. The design and
tuning of such a controller is well studied and still an active
field of research, see [1]. A generalization of this type of
controller is given by the fractional PIα
Dβ
controller and
was introduced by [2], at first. Due to the additional fractional
order α and β this controller, when well tuned, outperfoms
the classical PID controller. In [3] a method to tune the
PIα
Dβ
controller is presented. It is based on solving a
set of nonlinear equations. In [4] and [5] a tuning rule for
the PDβ
controller with application to motion systems is
given. Genetic algorithms are used in [6] to design a PIα
Dβ
controller.
Another class of fractional controllers is proposed by
[7], namely (PI)n
and (PID)n
controllers. This class of
controllers is more appropriate to ensure robustness of the
closed-loop to static gain variations with a conventional
CRONE template. Based on output feedback techniques,
the controller is derived using the Lyapunov stability condi-
tion expressed in terms of LMIs. The time domain constraints
are introduced using the equality of moments between the
closed-loop system and its fractional reference model.
Unlike in the PID controller case, the number of pub-
lications which deals with the design and tuning of the
fractional PID controller is still small. Therefore, there
is a need to explore new tuning methods. The goal of
this work is to develop a systematic tool to optimize the
parameters of the fractional order controller (PID)n
and
PIα
Dβ
. Our approach is based on the recently developed
nonsmooth optimization techniques by [8]. The requirements
to be satisfied by the controller are expressed in terms of a
1N. Lachhab and 1F. Svaricek are with Department of Aerospace Engi-
neering, Group of Control Engineering, University of the Federal Armed
Forces Munich, 85577 Neubiberg, Germany {nabil.lachhab,
ferdinand.svaricek}@unibw.de
2F. Wobbe and 2H.Rabba are with the Department of Powertrain
Mechatronics Development Gasoline Engines, IAV GmbH, 38518 Gifhorn,
Germany {frank.wobbe, Heiko.Rabba}@IAV.de
desired open-loop response. The loop shaping approach is
used to formulate the problem in the H∞ framework.
This paper is organized as following, in section 2 we give
the problem formulation for the PIα
Dβ
and the (PID)n
controllers such as the solution of the related optimization
problem. In section 3 we present the FOPID-Toolbox for
Matlab. Numerical examples and simulation results are given
in section 4.
II. PROBLEM FORMULATION
The approach proposed in this paper deals with the de-
sign of fractional PID controllers in the form (PID)n
or
PIα
Dβ
. It is based on the work [8] in which the authors
proposed an algorithm to solve the fixed structure H∞
problem without using the Lyapunov stability in LMI form
to avoid the related high number of decision variables. The
motivation of this work is to extend this technique to cope
with fractional order controllers.
A. Fractional PIα
Dβ
controller
The fractional controller in the form PIα
Dβ
is a gen-
eralization of the PID controller for fractional orders and
was introduced by [2], at first. In this work an approach is
proposed to optimize the parameters of this controller such
that some requirements are fulfilled. For this purpose the
feedback control-loop configuration, see Fig. 1, is consid-
ered. This configuration is well known as the loop shaping
configuration [9], with Ld(s) as the desired open-loop trans-
fer function which includes the requirements to be fulfilled
by the controller K(s). The signal r denotes the reference
Fig. 1. Loop shaping configuration
signal, e = r − y is the error signal, nw and ew are the
exogenuous input and output, respectively. G(s) is a SISO
2013 European Control Conference (ECC)
July 17-19, 2013, Zürich, Switzerland.
978-3-033-03962-9/©2013 EUCA 3694

LTI-system with static gain variations. The transfer function
from (r, nw) to (y, ew) is denoted by T(r,nw)→(y,ew)(s).
We are interested in designing a fractional controller of
the form
PIα
Dβ
= KP +
KI
sα
+ KDsβ
(1)
achieving a given phase margin φm and a crossover fre-
quency ωc. The controller should also be robust in the
presence of static gain variations which is given by

d (arg(L (jω)))
dω

ω=ωc
= 0 (2)
and means that the phase plot is flat around ωc. These spec-
ifications are translated into a desired open-loop response
Ld(s) =
ωf
sv
, (3)
with
v = 2 −
φm
90
and ωf = ωv
c . (4)
Equation (3) defines a fractional integrator. To approximate
the fractional order v in the whole frequency range, a
high order transfer function is needed. For a band limited
implementation of (3), the CRONE approximation method
given in [10]
N
Y
i=1
1 + s
ω
0
i
1 + s
ωi
, ω
0
i, ωi R (5)
with
ωi
ωi
0 = α,
ω
0
i+1
ω
0
i
= η, n =
log(α)
log(αη)
(6)
is used. The order N should be choosen depending on
the bandwidth in which the approximation is valid. A very
important point to be mentioned here is that due to the
inverse of Ld(s), see Fig. 1, the CRONE approximation is a
very suitable method to be used. Filter (5) is bi-proper with
stable poles and zeros.
Fig. 2. Desired open-loop response
An example of a desired open-loop response using the
fractional integrator
Ld(s) =
1
s1.5
(7)
is presented in Fig. 2. The approximation is valid in a
specified bandwidth Bw around the crossover frequency
ωc = 1 rad/s. The phase margin for (7) is φ = 45o
.
The constant phase enforces the closed-loop to be robust
against static gain variations. To express the phase margin
specification in term of the overshoot of the related closed
loop system see [11].
Before proceeding to the problem definition the following
notation
kG(s)k∞ := max
ω
σ̄(G(jω)) (8)
is introduced to denote the H∞ norm of the transfer function
G(s). σ̄(G) is the maximal singular value of G. For SISO
systems this norm is the maximum gain over all frequencies.
In the MIMO case it is the peak value of the maximum
singular value over all frequencies. With the help of this
norm and using the configuration in Fig. 1 our optimization
problem is formulated as follows
min
K∈Ω
T(r,nw)→(y,ew)(K) ∞
(9)
with K ∈ Ω is a structural constraint on the controller. In
our case this constraint is represented by fractional PID
controllers in the form PIα
Dβ
. Without the restriction K ∈
Ω, problem (9) falls into the scope of convex optimization
and can be solved efficiently. For example to solve (9) one
can first define a generalized plant (10) consisting of the
plant G(s), the filter Ld(s) and 1/(Ld(s))
P :



ẋ = A x + B1 w + B2 u
z = C1 x + D11 w + D12 u
e = C2 x + D21 w + D22 u
(10)
and a controller K
K :

ẋK = AK xK + BK e
u = CK xK + DK e .
(11)
With the help of the bounded real lemma and the projection
lemma [12], problem (9) can be transformed into the LMI
form and then be solved efficiently using LMI-Solvers. The
obtained controller is of full order which means that the size
of Ak is equal to the size of A. As mentioned in [8], adding
the constraint K ∈ Ω changes the whole situation. Problem
(9) can not be converted into the LMI form or any other
convex program. To solve this problem other algorithmic
methodologies are required. The authors in [8] have proposed
a new nonsmooth optimization technique to solve the H∞
problem under structural constraints on the controller. In
the scope of this work, the set Ω consists of fractional
PID controllers which includes an additional constraint, the
fractional order.
3695

Now considering the fractional controller (1) and using
the approximation (5) for the fractional order α and β, this
controller is equivalent to
K̃(s) = Kp + KIFI(s) + KDFD(s) (12)
with FI(s) and FD(s) used for the approximation of α and
β, respectively. Substituting (12) in (9) and considering the
case of a fixed value αk and βk problem (9) reduces to
min
K̃
T(r,nw)→(y,ew)(K̃, αk, βk)
∞
. (13)
At this point we want to clarify the difference between (9)
and (13). In (9) the fractional orders of the PIα
Dβ
controller
are variables of the H∞ minimization problem and so a
posteriori known. Unlike in (13) they are known a priori. In
the second case the optimal values of α and β are computed
afterwards using an outer loop. Problem (13) can be seen
as computing a static output feedback controller for the new
plant G̃(s) as shown in Fig. 3. This is a typical application
of the method proposed in [8] which is implemented in the
Matlab function hinfstruct.
Fig. 3. Loop shaping configuration
The main idea in this work is to replace problem (9) by
problem (13) and then optimize over the fractional orders
α and β. Formally the considered optimization problem is
expressed as follows
min
α,β

min
K̃
T(r,nw)→(y,ew)(K̃, α, β)
∞

(14)
and solved for α and β using the following algorithm
1) Initialize α0 and β0
2) Compute the gradient dk with respect to αk and βk
3) If kdkk2 smaller then a value t STOP, else update αk
and βk and go back to step (2).
To update αk and βk in step (3) we used the steepest descent
algorithm with a line search. The computation of the gradient
of the H∞ norm in (14) with respect to α and β was
performed numerically. The values of KP , KI and KD are
computed by substituting the obtained values of αk and βk
in (13).
B. Fractional (PID)n
controller
This kind of controller was presented in [7] and [13] and a
method to tune the controller using the equality of moments
between the closed-loop system and its fractional reference
model is given. Contrary to [7] the fractional order n is also
a tuning parameter to get an additional degree of freedom
in designing the controller. The requirements to be fulfilled
Fig. 4. Reference model configuration
by the (PID)n
controller are the same used in the PIα
Dβ
controller case. To formulate the optimization problem, the
configuration Fig. 4 is used. Formally speaking we want to
solve the following problem
min
K∈Ω
Tr→ef
(K) ∞
(15)
with the set of fractional controllers Ω in the form
(PID)n
=
1
sn
(KP +
KI
s
+ KDs) . (16)
To render the PID controller in (16) realizable, the deriva-
tive term KDs is replaced by KD
s
τs+1 and will be denoted
with Df . As mentioned in [7] it becomes necessary to define
a new open-loop function given by
L̃(s) =
L(s)
τs + 1
. (17)
The filter describing the desired closed-loop response, see
Fig. 4, becomes
F(s) =
L̃(s)
1 + L̃(s)
. (18)
To solve problem (15) the same idea as in the PIα
Dβ
con-
troller case is adopted here. Using the order approximation
(5) the controller (16) is equivalent to
K̃(s) = Fn(s)(KP +
KI
s
+ KD
s
τs + 1
), (19)
with Fn(s) the integer approximation of the order n. Sub-
stituting (19) in (15) and considering the case of a known
value nk, problem (15) reduces to
min
K̃
Tr→ef
(K̃, nk)
∞
. (20)
Solving problem (20) can be seen as computing a PID
controller for the augmented system G̃(s) = G(s)Fn(s).
The optimal value nk that minimizes the H∞ norm in (20)
is computed using the approach proposed to optimize the
PIα
Dβ
controller.
3696

III. FOPID-TOOLBOX
As mentioned in [8], the problem of designing low order
controllers is not convex. This means that the used algorithm
does not necessarily lead to the global minimum. The authors
in [8] deal with this difficulty by starting the algorithm with
different initial sets of parameters and then pick up the one
with the lowest value of the H∞ norm. As our approach is
based on this method, it is also a non convex one. Because
of this fact we give the user the opportunity to analyse
the relevant open and closed-loop plot of the resulting
transfer function. Moreover with our FOPID-Toolbox the
obtained controller parameters can be retuned if necessary.
We also implemented several approximation methods for the
fractional order (continuous and discrete). The main features
of the FOPID-Toolbox are summarized as following
• Compute a fractional PID controller in the form
PIα
Dβ
or (PID)n
achieving robust performance in
the presence of static gain variation
• Analyse the following plots
– Open- and closed-loop Bode plot
– Nyquist and Nichols plots
– Step Response
• Tuning of the computed parameters
• Analyse and compare several approximation methods
for the fractional orders
The discretization methods implemented in the toolbox are
based on the work [14] in which the authors proposed about
28 methods. The FOPID-Toolbox contains seven methods to
approximate the fractional orders α, β and n. For a detailled
review about existing approximation methods we recommend
the reader to view the work [14].
To compute a controller with the help of the FOPID-
Toolbox, Matlab Robust Control Toolbox (2011) or
higher is required. The user has only to define the plant
to be controlled, the desired phase margin and the desired
crossover frequency. These specifications are translated into a
desired open-loop response and then the related optimization
problem is solved using the proposed approach.
IV. EXAMPLES
In this section we present several examples to show the
benefit of using the FOPID-Toolbox to design fractional
order PID controllers. We give also a comparison between
the (PID)n
and PIα
Dβ
controller.
A. Example 1
The plant for the first example is borrowed from the
work [13], in which the authors considered the design of a
(PIDf )n
controller based on the method of moments. The
plant (21) describes a time delay second order system
P(s) =
G
2s2 + 3s + 1
e−0.2s
. (21)
The parameter G is uncertain and varies in the region [0.5 2].
The goal is to optimize a PIα
Dβ
controller that fulfills the
following requirements
• Phase margin φm = 51◦
, approximately 24% overshoot
• Crossover frequency ωc = 0.5 rad/s
• Flat phase at ωc.
The FOPID-Toolbox translates these requirements into a
desired response Ld(s) using (3). Due to the time-delay of
the plant, it is necessary to define a new desired response of
the open-loop transfer function
L̃(s) =
ωf
sv
e−T s
. (22)
To approximate the time-delay e−T s
, the Pade method is
used. After minimizing the objective function (9) in the
parameters (KP , KI, KD, α, β), we get the following con-
troller
K(s) = 1.65 +
1.9
s1.14
+ 0.18
s1.17
0.05s + 1
. (23)
The step response with the obtained controller is shown
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(sec)
Amplitude
Step response
G=1
G=0.5
G=2
Fig. 5. Step response for different static gain variations (PIαDβ)
in Fig. 5. The overshoot is approximately constant for all
values of G. The overshoot for the minimal and maximal
value of G is 19% and 33% respectively. The performance
of the PIα
Dβ
controller is good. For the seek of comparison
we compute for the same plant with the same requirements
a (PID)n
controller. As in the previous case a new desired
reference model is defined
T(s) =
L̃(s)
1 + L̃(s)
(24)
with
L̃(s) =
ωf
(τs + 1)sv
e−0.2s
. (25)
After minimizing (15) in the parameters (KP , KI, KD, n),
we get the following controller
K(s) =
1
s0.37
(1.1 +
0.78
s
+ 0.37
s
0.005s + 1
). (26)
The step response with the controller (26) is shown in
Fig. 6. The overshoot is nearly constant for the nominal,
minimal and maximal value of G. The robustness to gain
variations is achieved with this controller. Comparing now
the step response Fig. 5 and Fig. 6 it becomes clear that
3697

0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
Time(sec)
Amplitude
Step response
G=1
G=0.5
G=2
Fig. 6. Step response for different static gain variations (PIDf )n
the (PIDf )n
controller outperforms the PIα
Dβ
controller.
With Ld denoting the desired open loop response Fig. 7
shows that the design requirements are not fully satisfied
by the PIα
Dβ
controller. Fig. 8 shows that the (PID)n
−50
0
50
Magnitude
(dB)
10
−2
10
−1
10
0
10
1
−225
−180
−135
−90
Phase
(deg)
Bode Diagram
Frequency (rad/s)
G=1
G=0.5
G=2
Ld
Fig. 7. Bode plot PIαDβ
controller provides a very good fit of Ld. Moreover, we give
−50
0
50
Magnitude
(dB)
10
−2
10
−1
10
0
10
1
−225
−180
−135
−90
Phase
(deg)
Bode Diagram
Frequency (rad/s)
G=1
G=0.5
G=2
Ld
Fig. 8. Bode plot (PID)n
also a comparison between the controllers (26) and (23) and
the controllers in the work [13]. The results are shown in
Table I. The FOPID-(PIDf )n
controller gives a slightly
better performance then the (PIDf )n
controller in [13]. The
performance of the FOPID-PIα
Dβ
controller is adequate
TABLE I
OVERSHOOTS IN (%)
G 0.5 1 2
FOPID-(PIDf )n 24% 24.4% 28.9%
[13] (PIDf )n 22% 24% 29.5%
FOPID-(PIαDβ) 19% 25% 33%
[13] (PIDf ) 8.75% 24.5% 47.5%
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
Time(sec)
Amplitude
Step response
G=1
G=0.8
G=1.2
Fig. 9. Step response for different static gain variations (PD)n
and the PIDf controller failed to ensure robustness in the
presence of static gain variations.
B. Example 2
The plant for the second example is taken from the work
[5], in which the authors designed a PDα
controller using
a tuning rule based on a set of nonlinear equations. The
following plant
P(s) =
G
s(0.4s + 1)
(27)
describes a simplified motion control system. The parameter
G is uncertain with 0.8 as a minimal value and 1.2 as a
maximal value. The controller should fulfill the following
requirements
• Phase margin φm = 70◦
• Crossover frequency ωc = 10 rad/s
These requirements are translated using (4) into a desired
open-loop response
Ld(s) =
16.68
s1.22
. (28)
After minimizing the objective function (9) in the parameters
(KP , KD, β), we get the fractional controller
K(s) =
1
s0.21
(16.36 + 6.42
s
0.005s + 1
). (29)
The step response with the controller (29) is presented in
Fig. 9. The overshoot of the three step responses is constant.
The robustness to static gain variations is achieved by this
controller. The performance of this controller is approxi-
mately the same as with the PDα
in [5]. Both controllers
satisfy the design requirements.
3698

C. Example 3
With this example we want to show that our approach
is also valid for higher order systems. The plant considered
here is a fourth order model
P(s) =
G
(s + 10)(s + 2)(s + 1)(s + 0.5)
(30)
with G an uncertain parameter varying in the region [0.5 1.5].
The requirements to be satisfied by the controller are
• Phase margin φm = 60
• Crossover frequency ωc = 0.5 rad/s
We are interested in designing a PIα
Dβ
controller. After
translating the requirements in the desired open-loop re-
sponse
Ld(s) =
0.39
s1.33
(31)
the H∞ norm (9) is minimized using the proposed method.
The step response with the obtained controller
K(s) = 1.55 +
0.74
s1.0665
+ 1.34
s0.93
0.001s + 1
(32)
is shown in Fig. 10. Clearly the controller ensures robustness
for static gain variations. The overshoot of the three step
0 5 10 15 20 25 30 35 40 45 50
0
0.2
0.4
0.6
0.8
1
1.2
Time(sec)
Amplitude
Step response
G=1
G=0.5
G=1.5
Fig. 10. Step response for different static gain variations (PIαDβ)
responses is nearly constant. This is due to the flat phase
around the crossover frequency 0.5 rad/s, see Fig. 11.
−50
0
50
Magnitude
(dB)
10
−3
10
−2
10
−1
10
0
10
1
−225
−180
−135
−90
Phase
(deg)
Bode Diagram
Frequency (rad/s)
G=1
G=0.5
G=1.5
Flat
phase
Fig. 11. Open-loop bode plot PIαDβ
V. CONCLUSIONS
In this work a new tuning method for fractional PID
controllers in the form PIα
Dβ
or (PID)n
is presented. It
is based on the recently developed nonsmooth optimization
techniques and a steepest descent algorithm. Moreover, based
on our tuning method a FOPID-Toolbox for Matlab is
presented. The user provides a desired phase margin φm
and a crossover frequency ωc. These specifications are then
internally translated to a desired open-loop response. A frac-
tional controller is computed that best fits this desired open-
loop response in the frequency domain. Several numerical
examples have shown that the proposed method provides
a robust controller satisfying the prespecified requirements.
Future works will be for example to extend the toolbox
to cope with general uncertainty and to extend the results
for MIMO systems. Another point will be to implement
the controllers on a real plant and to explore automotive
applications.
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