Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
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Atul NarkhedeReview, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
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Atul NarkhedeI.J.
Intelligent Systems and Applications, 2016, 7, 73-96
Published Online July 2016 in MECS (http://www.mecs-press.org/)
DOI: 10.5815/ijisa.2016.07.08
Review, Design, Optimization and Stability
Analysis of Fractional-Order PID Controller
Ammar SOUKKOU, M.C. BELHOUR
Faculty of Sciences and Technology, Department of Electronics, Jijel University,
P.O. Box 98, Ouled Aissa, Jijel 18000, Algeria.
E-mail: soukkou.amr@gmail.co m; belhour.mohammed.cherif@g mail.co m
Salah LEULMI
Department of Electric Power Engineering, Skikda Electric Power Systems Laboratory,
University of August 20th, 1955, Skikda, Algeria.
E-mail: salah.leulmi@yahoo.fr
Abstract—This paper will establish the importance and generally, do not work well fo r nonlinear co mp lex and
significance of studying the fractional-order control of vague systems that have no precise mathematical models.
nonlinear dynamical systems. The foundation and the However, a class of newly mod ified-type of Oo-PID
sources related to this research scope is going to be set. controller based on advanced and intelligent approaches
Then, the paper incorporates a brief overview on how this has been designed and simulated for this purpose.
study is performed and p resent the organization of this Recently, fract ional-order calculus, included in
study. The present work investigates the effectiveness of advanced techniques, has received an increasing interest
the physical-fractional and biological-genetic operators to due to the fact that fractional operators are defined by
develop an Optimal Form of Fract ional-order PID natural phenomena [1].
Controller (O2Fo-PIDC). The newly developed Fo-PIDC Fractional derivatives and integrals operators
with optimal structure and parameters can, also, imp rove provide an excellent instrument fo r
the performances required in the modeling and control of c Dt f t ,
modern manufacturing-industrial process (MIP). The the description of memory and hereditary properties of
synthesis methodology of the proposed O2Fo-PIDC can various materials and processes.
be viewed as a mu lt i-level design approach. The The theoretical and practical interest of these operators
hierarchical Multiobject ive genetic algorith m (M GA ), is nowadays, well established. Its applicability to science
adopted in this work, can be visualized as a comb ination and engineering can be considered as an emerging new
of structural and parametric genes of a controller topic. It includes some areas, such as biology and
orchestrated in a h ierarch ical fashion. Then, it is applied chemical engineering (irrigation canal control [2], low
to select an optimal structure and knowledge base of the pressure flowing water network [3], networked control
developed fractional controller to satisfy the various system [4]), control and simulat ion of photovoltaic-wind
design specification contradictories (simplicity, accuracy, energy systems [5, 6], hydraulic turbine regulation [7],
stability and robustness). flight control [8], fractional-order chaotic systems
modeling and control [9, 10], Modeling of Econo mic
Index Terms—Optimal fractional-order controllers, Order Quantity [11], control of a servo systems [12, 13],
BIBO stability analysis, Multiobjective genetic algorith m, realization of higher-order filters [14], robotic control
CE150 Helicopter model. [15], astronomy [16], .. etc.
The fractional-order Controllers (FoC) were introduced
firstly by Prof. Oustaloup [17]. He developed the so -
I. INT RODUCT ION called CRONE controller, French abbreviation of
“commande Robuste d‟ordre non entier”. The idea of
Systems and control theory has evolved as an using FoC for the dynamic system control is well
important confluence between the engineering and addressed in [18, 19]. More recently, Podlubny proposed
mathematics discip lines. In general, it relates to the a generalizat ion of the Oo -PIDC, as a natural extension to
dynamical system analysis and behaviour and to the
the Fo-PIDC or a PI D controller, involving an
various techniques in which these systems can be
integrator of
order and a differentiator of
influenced to obtain some desired results.
Actually, a wide variety of mathemat ical techniques order. There are many variants of FoC that have been
are used in the field of control system design. Moreover, used in different applicat ions depending on the
many control, modelling and optimizat ion problems can application specific requirements: CRONE controller and
be recast as a learning problem and be solved with its three generations [18], Fo-PIDC [20], non-integer
appropriate mathematical tools. It has been shown that integral [21] and tilted proportional and integral
famous ordinary-order PID controllers (Oo-PIDC),
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
74 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
compensator [22]. Other FoC like fractional-order lead- application in engineering field. The present work
lag compensator [23] and fract ional-order phase shaper investigates the effectiveness of the physical and
are also becoming popular in recent robust process biological-based operators (fractional and genetic
control applications [24]. The most co mmon is the family approaches) to develop an optimal form (structure with
of the Fo-PID controllers. optimal dimension)) of fractional-order proportional-
The application of fractional-order calculus has been integral-derivative controller (O2Fo-PIDC) with optimal
significantly increased and the FoC are, attractively, parameters (knowledge base) and able to improve the
becoming a major topic of control. Ho wever, designing performances required in the modelling and control of
high-performance and cost effective controllers is a very industrial process. In this context, issues related to the
complex task which could be, practically, impossible to design of the proposed controller can be classified into
optimally acco mplish by some conventional trial an d mathematical configuration and parametric optimizat ion
error methods considering different design criterion in categories.
time and frequency-domain.
There are many methods to construct and design such Characterizat ion 'mathe matical configuration' of
FoC which can be regarded as analytical and learn ing the control law under develop ment by a judicious
problems of a desired performance of the controlled choice of d iscretization techniques of derivative
c Dt f t , ,
systems
and integral
The first step is to define a mathemat ical structure c It f t c Dt f t , , operato
of the controller based on the application of
approximation (discretization) methods of the rs involved in modeling of the control signals. In
addition, it is indispensable to take into account
fractional operators c Dt , (resp. s ) the type of the association (parallel or serial) of
based on the analysis field (time or frequency- various proportionality, derivative and integral
domain). actions. It depends on the design field, time-
The second step is to identify the gains domain (control of nonlinear dynamical
„parameters‟ of the predefined structure. After an continuous or discrete systems) or frequency-
extensive literature search on methods of tuning domain (control of linear continuous systems). In
and optimization of FoC, especially the Fo-PIDC, this phase, the constraint of computing time (long
we arrived to classify these methods, according to memo ry effect) related to an analytical model of
the fractional systems will be treated with prudence
and is also the seeking tools to imp rove the
Manual methods (trial-and-error) [25]; tradeoff between simp licity and accuracy
Analytical methods (phase and gain marg in- specifications.
based specifications [3], flat phase [26], The learn ing process involves the development or
dominant roots-pole placement [27, 28], internal the implementation of the learning and
model control (IMC) [29]); optimization algorithms of relevant parameters of
Optimization-based methods (F-MIGO fractional-order control system under develop ment.
Algorith m [23], Monje‟s methods [3], simplex This transaction is based upon the factors of
search min imization [3], metaheuristic accuracy and stability conditions of the control
algorith ms (GA [4, 30], PSO [31], DE [4], SA loop and the used optimizat ion algorith ms
[32], ABCA [12], IWO [33], FOA [34], CSA (conventional, advanced or intelligent-based
[15]) and least squares optimization method approaches).
[35]);
Tuning rule for p lants (with an S-shaped step The paper is organized as follows. Section 2 provides
response - with a critical gain) [29, 36]; fundamentals of fract ional calcu lus and its properties in
Auto tuning [3, 21]. time and frequency-domain. A presentation of the Fo-
PIDC, their description and properties are also included.
Based on the previous classification, we may notice A survey of various design and optimization
that metaheuristic algorith ms „evolutionary algorith ms approaches of Fo-PIDC is considered. The aim o f Section
(EA)‟, especially GA , PSO and DE, have the majority of 3 is to design an optimal and simp lified form o f Fo-PIDC
references in the design of the predefined (characterized) by considering various contradictory objective functions
FoC. The frequency-domain analysis is the dominant case to control dynamical systems, such as simplicity,
of these applications , where the problem is how to efficiency (precision) and stability. The design deal
approximate the integrator-derivator operator s and
with time-do main analysis of the O2Fo-PIDC adapted to
nonlinear dynamical systems. In section 4, some
how to optimize the associated parameters of the simu lation results to illustrate the effectiveness and
controller under development? robustness of the proposed control system are displayed.
This research is aimed for appreciat ion of the fractional Finally, we present some conclusions and the
calculus, coupling fractional calculus –intelligent design contributions developed in this research.
and optimization approaches Thus, it is made as an
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 75
f
n
C D f t 1
c t
II. FRACT IONAL CALCULUS: A N OVERVIEW t
d ,
n c t 1 n
Fractional calculus, capable of representing natural (4)
phenomena in a more general way and do not
approximate the processes by considering that the order
n 1 n
of the governing differentials are integers only [23, 37], is
a branch of mathemat ical analysis. It studies the As shown by the RL and GL defin itions, the fractional-
possibility of tacking real nu mbers power of the order derivatives are global operators having a memory
differential and integration operator. The generalized of all past events. This property is used to model
differ-integrator may be put forward as [38] hereditary and memory effects in most materials and
systems [13]. For nu merical calcu lation of fract ional-
order derivative, we can use the relation (2) derived fro m
c Dt f t c It f t Dt c f t the GL definition. Th is approach is based on the fact that
for a wide class of functions, GL (2), RL (3) and
d f t dt , 0 Caputo‟s (4) are equivalent [10]. The fract ional-order
d f t
(1)
derivative and integral can also be defined in the
1, 0
d t c
transformation-do main. It is shown that the Laplace
t f t d
c , 0 transform of a fractional derivative of a signal f t is
given by
where represents the real-order of the differ-
integral, t is the parameter for wh ich the differ-integral is
L Dt f t s L f t
taken and c is the lower limit, generally c 0 for the F s
causal systems. Several alternative definitions of the (5)
n 1
fractional-order integrals and derivatives exist. The three
s k 0 Dt k 1 f t
most common known defin itions for fractional
k 1 t 0
derivatives-integrals are Grünwald-Letnikov (GL)
Initialization function
definit ion, Riemann–Liouville (RL) defin ition and
Caputo (C) definition, given, in time-domain, as follows
Considering null in itial conditions, the last expression
(5) is reduced to the following s uitable form:
Grünwald-Letnikov definition
GL D f t lim h N 1 1 j f t j h
c t
L Dt f t s F s (6)
h 0 j 0 j
(2) The Laplace t ransform reveals to be a valuable tool for
the analysis and design of fractional-order control
t c systems for reasons of analysis and synthesis simplicity
where N is the upper limit of the co mputational [13]. It is noted that some MATLAB-tools of the
h
fractional-order dynamic system modeling, control and
universe, where means the integer part, h is a grid size filtering can be found in [39, 40]. Nu merical methods for
simu lating and discretization fractional-order systems are
and is a binomial coefficients. given in detail in [10] and [41].
j
A. Approximation and numerical implementation
Riemann–Liouville definit ion given by the ‘discretization’ of fractional operators: An overview
generalized form The ideal d igital fractional-o rder differentiator-
is a d iscrete-time whose output y n
integrator 0 Dt
RL D f t 1 d
nt f
c t 1 n
d , is uniformly sampled version of the th -order derivative
n dt
c t (3)
of f t . Henceforth, we assume that 0 , we write
n 1 n
y n 0 Dt f t (7)
where is the Euler Gamma function. t nh
Caputo definit ion, which so metimes called smooth In general, if a function f t is appro ximated by a
fractional derivative, defined by
grid function, f n h , where h is the period of
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
76 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
discretizat ion (step size of calculation or sampling time). rational appro ximat ion of functions [45, 46].
The appro ximation for its fractional derivative order These techniques are based on the approximat ion
can be expressed as [42] of an irrational function, G s , by a rational one
defined by the quotient of two polynomials in the
d f t
lim h f t
variable s.
yh nh
dt h 0
b. Methods based on the explicit recursive location of
(8) poles and zeros of the rational approximation.
h w 1 f h n h
Many iterative techniques exist for realization of
the fractional elements in continuous time. The
most used are Oustaloup [18], Charef [47],
Carlson [48] and Matsuda [49].
where f n h f n h f n h h ,
1 is c. Methods based on frequency domain identificat ion,
t nh that is, on finding a rational integer-order system
the shift operator and w 1 is a so-called generating whose frequency response fits that of the
fractional-order operator [45]. Generally, linear
function (GF). This GF and its expansion determine both least squares and metaheuristic optimizat ion
the form of the approximation and the coefficients [43]. techniques are proposed to solve the frequency-
Fractional derivative and integral operators are often domain identification problem [50, 51].
difficult to find analytically and the simulat ion of
Step 3: Replace the last approximation of s and
fractional systems is complicated due to their long
memo ry behaviour as shown by Oustaloup [18]. A
evaluate the inverse of Laplace operator
number of techniques are available for appro ximat ing
fractional derivatives and integrals. These latter can be L1 L 0 Dt f t in (5) to obtain the
classified into three main groups based on the field under
study, temporal (t-p lane), frequency (s-plane) and time- expression to simulate the fract ional derivator and
integrator of signal f t .
discrete (z-plane) analysis as illustrated in figure 1.
1
t-plane 3. Methods based on the approximat ion of a
fractional model by a rational discrete-time one
z Laplace (discrete-time analysis). The z-transform is the
Transform Transform most used algorithm. The main steps to compute
the general formula of the output are:
z-plane s-plane
3 2
1
Step 1: Discretizat ion of the operator s by s z
s z
Transform and then the approximation of a GF of the form
Fig.1. Simulation-discretization tools of fractional operators.
z 1
corresponding to s
. The most
used s-to-z transform s
are [52]: Euler
1. Methods based on the computation of analytical
expression of derivative and integral signal (backward - forward), Tustin, Al-A laoui, Simpson,
(temporal-analysis). The GL (2) and the Adams - Al-Alaoui – Simpson, Al-Alaoui – SKG and Al-
Bashforth-Moulton (ABM) algorith ms based on Alaoui – Schneider.
the Caputo‟s definition are the most used methods. Step 2: To obtain the coefficients of the approximat ion
2. Methods based on the approximat ion of a
fractional model by a rat ional continuous-time one
(frequency-analysis). The Laplace transform is the
equations for fractional calcu lus z
1
, we
can perform the most used algorithms:
most used algorithm. The main steps to compute
the general formula of the output are:
The direct power series expansion (PSE) of the
Euler operator [53, 54] (FIR filter structure);
Step 1: Compute the Laplace transform from Eq. (5).
The continuous fractional expansion (CFE) of the
Step 2: Evaluate s s N s D s by using the Tustin operator [53, 54] (IIR filter structure); or
The numerical integration-based method [55, 56].
most used s
approximat ion and can be
classified into the three following categories [44]:
Step 3: Evaluate the inverse operator Z 1 Z 0 Dt f t
to obtain the approximat ion of 0 Dt f t .
a. Methods based on mathemat ical techniques for
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 77
Fro m a control and signal processing perspective, the K D K P Td T ) actions based on the error signal,
GL and ABM approaches seem to be the most useful and given in continuous and discrete-time as
intuitive, particularly , for a discrete-time imp lementation.
Moreover, in the analysis and design of control systems,
I-action D-action
we usually adopt the Laplace transform method. As a P-action
result, fractional-order controller synthesis is preferably K P e t K I Dt1e t K D Dt1e t
x t
carried out in the frequency domain. Th is is the reason u x
x nT K e nT K T n e j T
why the most design methods proposed, so far, for FoC P I j 0
are based on using frequency-response information.
B. Fractional-order PID controller
K D e nT e n 1T T
(9)
The motivation on using the digital Fo-PIDC was that
Oo-PIDC belongs to the dominating industrial controllers. can be generalized to a fract ional controller involving an
Therefore, there is a continuous effort to imp rove their
quality and robustness. The traditional Oo-PIDC wh ich integrator of order and a differentiator of order
involves proportional (with gain K P ) plus integral (with [19, 20]. Figure 2 indicates the schematic
g ain K I K P T Ti ) p lu s d eriv at iv e (w it h g ain diagram of the Fo-PIDC loop.
DPL Discretization
Design layer
techniques
LPL Learning
algorithms
Digital Fo-PID
Controller
IPL
KP Dt0
yr t et +
+ u t
KI Dt Actuator
+ +
-
y t KD
Dt
Plant
Sensor
Implementation layer
Fig.2. Fo-PIDC loop design.
The mathemat ical model of the Fo-PIDC described in The first is to define an analytical structure of the
continuous and discrete-time is given by controller by applying the methods of
discretizat ion and approximation of fract ional-
K P e t
K I Dt e t K D Dt e t
order operators.
u (10) The second consists of determining the knowledge
K P e nT K I e nT K e nT
nT D nT base of the controller structure, previously
identified, by application of learning and
optimization algorithms.
To analyze and simulate co mplex dynamical systems, The last phase is the verification of the coherence
variable-o rder fract ional is proposed by Hu Sheng et al. of the proposed algorithm by real-time execution.
[57] as a generalization of the constant-order
PI D controller, where , in (10) are rep laced The frequency-domain analysis is the most used in the
design of the FoC, where the continuous transfer function
by t , t . of the Fo-PIDC is obtained through Laplace transform or
As shown in Fig. 2, it consists of two layers: a discrete z-transfer function.
computational layer (design) and the decision layer
(operational). Laplace transform of t ransfer function of system
Three main stages are necessary for the with input E s (error signal) and controller‟s
implementation of the control loop:
output U s , as
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
78 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
U s or frequency-domain).
Gc s K P K I s K D s
E s
Develop ment or imp lementation of the learning
(11) approaches of relevant parameters of control
, 0 system under develop ment (tuning and
optimization).
Discrete transfer function given by the following
λ λ
expression:
λ>1
PI (0, 1) (1, 1)
PID
λ=1
PI PID
Gc z K P K I z 1 K D z 1 PID
(12) P µ
(0, 0) (1, 0)
µ
, 0
PD P PD
µ=1
µ>1
(a) (b)
where
s z 1 denotes the discrete operator,
PID
µ
PD
PID
expressed as function of the complex variable z or the 1
λ
shift operator z 1 . The fractional d ifferentiator-
P
PI
0
integrator s is substituted by its discrete-time -1 (c)
equivalent z 1 . Fig.3. Generalization of the Fo-PIDC
Fro m figure 3(a), it can be remarked that the Oo-PIDC
can only be switched in four conditions: {P, PI, PD, PID}.
It restricts the controller performance. Relatively, III. CONT ROLLER DESIGN
depending on the value of orders , ( shaded area This section should address the most significant issues
represented in figure 3 (b )), we get an infinite nu mber of schematized in figure 2 by codes into circles DPL, LPL
choices for the controller‟s type, defined continuously on and IPL, respectively.
the , plane, i.e., The Fo-PIDC expands the Oo-
The main question in DPL is pronounced as: Is it
PIDC fro m point to plane, thereby it adds the flexibility possible to analytically design the Fo-PIDC with
to controller design and allows us to control our real the best achievable performances, such as
world p rocesses more accurately [38, 58]. In [59], the simplicity and accuracy?
authors applied the P I D controller to enhance the The aim o f LPL: It is how this Fo-PIDC can be
performance of the force feedback control system, where tuned for quantitative performances, such as
the pair Ki , i , , represents the control gain and
accuracy, stability and robustness?
The problem of IPL is pronounced as: What a
the non-integer order. Thus, the design procedure performance limit does the Fo-PIDC have?
involves the parameters of a Fo C with three terms
K e t , K D e t , K D e t associated with A. Problem formulation and adopted tools
the P, I and D-actions, respectively. This last form The fractional calcu lus has received an increasing
expands the Oo-PIDC fro m point to , , -plane as interest due to the fact that fractional operators are
defined by natural phenomena [1]. In the imp lementation
shown in figure 3 (c). of the fractional controllers, especially Fo-PIDC fin ite
Fo-PID controllers have received a considerable order approximat ion is required since fract ional elements
attention in the last years both from academic and have infinite o rder. Th is difficu lty is caused by the
industrial point of view. However, due to the complexity mathematical nature of the fractional order operators
of the fractional-order systems, these control design which are defined by convolution and require „unlimited
techniques available for the ubiquitous fractional-order memory‟. Du ring the synthesis of the proposed control
systems suffer fro m a lack of d irect systematic law, the required performances are articulated around
approaches based on the fair comparison with the various contradictory objective functions , such as:
traditional integer-order controllers [58], [60, 61].
Searches in the field of fractional control systems are Simp le and optimal analytical structure and
concentrated around two areas parameters;
Efficiency and precision;
Develop ment of tools for calculat ing (d iscretizing) Stability and robustness against disturbances;
fractional operators Dt , (resp. s
) Implementation ability in real-time.
based on the used analysis field under study (time
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 79
In this context, issues related to the design of the Use a series of discrete samp ling point
proposed controller can be classified into mathemat ical t n T , n 0,1, 2, to instead the recursive
modeling „characterization‟ o f the control law under
algorith m, the steady-state value of Oo-PIDC process is
development „analytical structure‟ and identification
obtained as
„learn ing‟ of the knowledge base associated to the
predefined analytical model of controller „Paramet ric
optimization‟. u n T u n 1 T K0 e n T
(17)
B. Mathematical background: An overview K1 e n 1 T K 2 e n 2 T
The Oo-PIDC algorithm is one of the most commonly
used control algorithms in industry. It‟s considered as a As shown in (17), the calcu lus of the u n T is,
mathematically, based on the u n 1 T , e n T ,
generic closed loop feedback mechanism. The Oo-PIDC
output is computed in continuous time as illustrated in (9).
For a small sample increment T 0 , (9) can be turned e n 1 T and e n 2 T . In the software
into a difference equation by discretization [62]. It can be
predicted that (9) needs a large quantity of memory space implementation, the incremental algorith m (17) can avoid
„RAM‟ to store the results of the computation, i.e, to accumulat ion of all past errors e n and can realize
compute the sum n , all past errors smooth switching from manual to automatic operation,
j 0
compared with the position algorithm [62]. It is observed
e 0 , e 1 , , e n , have to be stored. that in (17), the efficiency and the flexib ility of the
This algorith m is called the „position algorithm‟ [62]. operating environment can be improved than (9) [63].
To avoid the redundancy and to derive the recursive Many tuning methods are presented in literatures during
algorith m, we can apply the integer derivative operator in this last decade that are based on a few structures of the
(9), as given by process dynamics.
As mentioned above, the conventional Oo-PIDC (9)
du t dt K P de t dt K I e t
can be generalized to a Fo C involving an integrator of
(13) order and a differentiator of order as
K D d dt de t dt
given in (10). Applied the derivative of the expression
(10), we get
In order to take better advantage of discrete form of the
data, (13) has been approximated by the following 1 1 e t
t u t K P Dt e t K I Dt e t K D Dt
D1
1
difference equations based on the first-order Eu ler
approximation approach (18)
dx t dt lim x t x t T T (14) In discrete-time control systems, it is required that the
T 0 data are in form of the samp led data. Thus, it needs to
discretize the fractional derivative or integral operation to
realize the discrete-time fractional-order control. In
d 2 x t dt 2 d dt dx t dt general, the GL defin ition given in (2) is the most suitable
T 0
lim x t x t T T x t T x t 2 T T T method for the realization of discrete control algorith ms.
(15) In (2), 1 j represents the binomial
j
Replace (14) and (15) in (13), we obtain the coefficients c j
, j 0,1, 2, , calculated accord ing
approximate expression of the Oo-PIDC in time-do main
as to the relation
u t u t T K 0 e t K1 e t T
K2 e t 2 T 0, j 0
c j 1, j 0 (19)
and,
K
0 KP KI T KD T
1
(16) 1 1 c , j 0
j j 1
K1 K P 2 K D T
1
Replacing the analytical exp ressions of Dt1 x t ,
K 2 K D T
1
1
Dt x t computed fro m (2) and Dt1x t fro m (14) in
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
80 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
(18), we can obtain the appro ximate expression of the Fo- Each microprocessor-based system has its own
PIDC in time-domain as minimum value of the sample period;
Then, it is necessary to perform all the
u t u t T K P e t e t T computations required by the control law between
two samples.
t T
(20)
K I T d j K D T q j e t j T Due to this last reason, it is very impo rtant to obtain
j 0 good approximat ions with a min imal set of parameters.
On the other hand, when the number of parameters in the
The coefficient d j and q j are co mputed from (19), approximation increases, it increases the amount of the
required memory too. Fro m Eq. (22), concluding remarks
respectively, and given by:
can be pronounced.
1, j 0, The sampling time T determines the rate at which
d j 2
the feedback gains are updated. Faster sampling
1 d j 1, j 1, 2,... rate T allow higher adaptation rate to be
j
(21) used, which in turn leads to a better tracking
1, j 0, performance. Therefore, it is difficu lt to
q j 1 2 q
implement the algorithm in real-time because of
j 1, j 1, 2,...
j the existence of the term RL e in (22).
Cu mulat ive values represent a constraint for the
Using series of discrete sampling point implementation of the algorith m in real-time. The
t n T , n 0,1, 2, instead of the recursive problem of running time in the phase of design and
simulation is, also, a problem to report (alert).
algorith m, the final expression of Fo-PIDC obtained after In general, when the sampling rate is slow (large
simplification of (20) is given by the generalized form T), the effects of sampling and discretization are
more pronounced and the control system
u n T u n 1 T performance is degraded. Therefore, in pract ical
K0 e n T K1 e n 1 T K 2 e n 2 T implementation, it is highly desirable to increase
the sampling rate as much as possible by
ubase nT
optimizing the real-time control program o r using
K3 e n 3 T Kn e 0 a multiprocessor parallel computing system.
RL e
and, C. Elaboration of the proposed Fo-PID model
In practical applications of the Fo-PIDC (22), real-t ime
K0 K P K I T K D T
realization problems arise due to linearly growing
processed samples (known as „the growing calculat ion
K1 K P K I T 1 K D T 1 tail‟) and fin ite microprocessor memory. The prag matic
solution is to adapt the digital control algorith m to the
1
K 2 K I T 1 K D T dynamic characteristics of the process under study. In
2 2
order to
n
RL e K j e n j T and
j 3 Reduce the amount of historical data;
K j K I T d j K D T q j
Improve the accuracy of the numerical solution;
Avoid the redundancy;
(22) Then, derive the recursive algorithm in its optimal
form,
This expression is similar (in form) to that given by the
relation (17) with the addition of a correction term a simplified analytical form „structure‟ of the Fo-PIDC
RL e which represents an infinite sequence of (22) is proposed. The main idea is developed by
SOUKKOU et al. in [64]. It consists to make the
linear regulators.
correction term RL e in (22) adjustable (manually
In the case of controller implementation, it is necessary
to take into account some important considerations [45]. or automatically) accord ing to the physical characteristics
First of all, the value of T, the step when dealing with of the system to be controlled and technological
numerical evaluation is limited by the characteristics of performances of the monitoring PLC (storage space -
the microprocessor-based system, used for the controller speed) used to imp lement the control algorith m. Eq. (22)
implementation, in two ways can, also, be rewritten in the following manner
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 81
u n T u n 1 T window limiting to some fixed time spam and T is the
K0 e n T K1 e n 1 T K 2 e n 2 T „transformed t ime‟ samp ling period. is the Euler
ui, j nT , i 0; j 0,1,2 Gamma function given by 1 k k !, k .
K3 e n 3 T K 4 e n 4 T K 5 e n 5 T Brian P. Sprouse et al. [66] introduce an adaptive time-
step method for co mputing the contribution of the
ui, j nT , i 1; j 3,4,5 memo ry effect associated with the h istory of a system,
K6 e n 6 T K7 e n 7 T K8 e n 8 T where fractional methods must be taken into account. Zhe
Gao et al. [67] proposed an alternative d iscretizat io n
ui, j nT , i 2; j 6,7,8 algorith m based on the Haar wavelet method and reduces
the store space of the history data, just calculating the
fractional order integral with five sampled data
(23)
f n T , f n 1 T , f n 2 T , f n 3 T
Co mpared with (17), the expression (23) can be seen as
a comb ination of a large set of linear PID regulators,
f n 4 T and the initial value f 0 .
temporarily shifted with moderate The present work represents a newly powerful and
gains Ki , i 0,1, , n . In (23), each i th iteration
simp le approach to provide a reasonable tradeoff between
computational overhead, storage space and numerical
requires the re-calculat ion and summat ion of every accuracy. Rearranging (23), the discrete Fo-PIDC law can
previous time point convolved with Ki . This becomes be rewritten as:
increasingly cu mbersome for large t imes, which requires
significant nu mbers of co mputations and memory storage u n T u n 1 T u n T
requirements. and
Applying some ideas as, for instance, short memory
nbr_subs-1 3i 2
K j e n j T (24)
principle [19], we can reduce the computational cost of
time-do main methods. Short memo ry principle means u n T
taking into account the function behavior only in the i 0 j 3i
“recent past”. The results obtained by these methods are ui, j nT
more reliab le than those determined using the frequency-
based approximat ion [10]. This would then result in e
approximating (22) by truncating backwards summation,
only taking into account times on the interval t , L T The expression (24) represents an association of a
instead of 0, t , where L is defined as the memory collection of nbr_subs linear-PID subsystems as
length. summarized by figure 4. The fo llowing three-input
Accord ing to the short memo ry p rincip le [19], the variables
length of the system memory can be substantially reduced
in the nu merical algorithm to get reliab le results. Thus, N e n 3 i T , e n 3 i 1 T , e n 3 i 2 T , i 0,1, 2,
i n ( 2 ) b e c o m e s N N t min t L h , L h .
Obviously, for this simplification, we pay a penalty in the are used by the discrete-time subsystem PID control in
form of some inaccuracy. If f t M , we can easily
incremental form, where e n T y r n T y n T ,
estab lish the fo llo wing est imate fo r determin ing the
memo ry length L , prov id ing the requ ired accu racy y r n T and y n T are a set-point/reference signal
1 for process output, and the process output at sampling
L M 1 a n d t h e u p p e r l i m it o f time n.T, respectively.
The zero-o rder co mponent of the combination (similar
1
s u mmat io n N is N M T 1 q
. Th e to Oo-PID (17)), ui 0, j 1,2,3 n T , is required since
it is the fundamental engine of the control law as showed
in figure 4. e is the number of terms e n j T
upper limit of summation N in (2) can be calculated fro m
the inho mogeneous samp ling algorith m p ro posed by
Fu jio Ikeda in [65] as N L T 1 , where
not taken into effect during the association (formu lation)
of elementary controllers ui, j .
x means t runcat ion to x , L is the calcu lat ion
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
82 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
of Fo-PID Controller
-3
Subsyk
Switching logic
z
-3
z Subsy2
-3
z Subsy1
y r nT e nT u nT
Subsy0 ++ ++ ++
+
-
y nT
Fig.4. Equivalent model of the developed FoC.
We can consider that the control variation o f the newly implement the algorithm (25).
model developed by (24) will be div ided into two parts For slow systems where T , needs to be, preferably,
(B) and (A). Part (B) is the basic element of the control a large number of subsystems with respect to the amount
law. Whereas, the additive portion (A) is adjusted to of memo ry reserved for the application. Indeed, for faster
obtain more performances. The control variation (24) can systems where T , we need the minimu m nu mber of
be written in iterative form as: selected subsystems must meet a co mpro mise between
efficiency and precision of the system to be controlled.
N Fro m the incremental algorith m of Fo-PID (25), various
u n T ubase n T ui, j n T architectures have been investigated to determine wh ich
(B) i 1 one will p rovide the best results in terms of
(A) computational speed and accuracy.
3i 2 (25)
ui, j n T K j e n j T
Full Form of Fractional-order PID controller
(F2Fo-PID) well suited to slow systems ( T ). It
j 3i
requires large space memo ry (RAM) and
K j K I T d j K D T q j significant time- response.
Extended Form o f Oo-PIDC o r just Reduced Form
of Fractional-order PID controller (EFOo-PIDC
where N nbr_subs-1 is the upper limit of
or R2Fo-PIDC), where RL e 0 is well
summation (A). The number of linear regulators
suited to very fast systems ( T ). It requires a
ui, j n T , j 3 i, 3 i 1, 3 i 1 , i 1, 2, , N short storage space and much reduced time-
in Eq. (25) depends on the number of samples response.
N (temporal universe of operation of the process under Optimal Form of Fractional-order PID (O2Fo-
study). The maximu m nu mber of subsystems in (25) is PIDC) that represents the F2Fo-PIDC with an
approximated by optimal nu mber of EFOo -PID subsystems. The
value of nbr_subs in (25) is selected and adjusted,
nbr_subs N 3 (26) by application of learning algorith ms to search the
optimal value, to achieve the control design
where N 3 int N 3 is a means the integer part of
specifications with respect to the functional
characteristics of the plant to be controlled. The
N 3 . The high order (higher dimension) of the last ith O2Fo-PID model represented a trade-off between
term in (25), i.e, j 3 i 2, N be defined by:
simp licity and efficiency. Moreover, it requires a
reduced storage space and optimal time-response.
orderi N 2 3 (27) The R2Fo -PID and O2Fo-PID approaches make the
simu lation and implementation of Fo-PIDC much easier
and enables a smooth transaction for industry to take
In th is wo rk, the cho ice of nu mber o f subsystems
advantages of this new approach. Further, the
ui, j nT depends on the des ired accu racy and the effectiveness of the proposed controllers remains to be
specifications of the system to be controlled (fast, very approved through application examp les in d ifferent
fas t , s lo w, et c.) and th e mo n it o rin g PLC u s ed t o industrial and scientific disciplines.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 83
D. Stability analysis of the proposed Fo-PID controller: Controller
An overview e1 f e1
u1 f(.) y1
The study of the stability o f fractional-order systems
can be carried out by studying the solutions of the
differential equations that characterize the m, linear or g e2 e2
y2 g(.) u2
nonlinear. Every study of stability of linear t ime invariant
fractional-order (LTI-Fo) system is, generally, based on Process
frequency-domain analysis. An alternative way is the
study of the transfer function of the system [23]. There Fig.5. A typical closed-loop control system.
are even some attempts to develop polynomial techniques,
either Routh or Jury type, to analy ze their stability [10], Suppose that the incremental control formu la of
[68, 69]. Saeed Balochian et al. [70] present the fractional control law (25) is used. By defining
stabilization problem of a LTI-Fo switched system by a
single Lyapunov function whose derivative is negative e1 n T e n T , e2 n T u n T
and bounded by a quadratic function within the activation
u1 n T y n T , u2 n T u n 1 T (29)
of each subsystem. The switching law is extracted based r
on the variable structure control with a sliding sector.
Stability of the fractional-order nonlinear system is u n T f e n T , y n T g u n T
very comp lex and is different fro m the fract ional-order
linear systems. The main difference is that for a nonlinear It‟s easy to see that we obtain an equivalent closed -
system, it is necessary to investigate steady states having loop control system as shown in figure 5, we have fro m
two characteristics, such as limit cycle and equilibriu m (28)
point. For nonlinear systems (may have several
equilibriu m points), there are many definit ions of stability
(asymptotic, global, orbital, …etc.) [10]. Stability of
u n T y r n T e n T g u n T
1
e1 n T g e2 n T
fractional-order nonlinear dynamic systems is studied in
[71] by using Lyapunov direct method with the (30)
introduction of Mittag-Leffler stability and generalized u2 n T e2 n T f e1 n T
Mittag-Leffler stability notions. Actually, in the field of
fractional-order control, many stability methods are based
on the original Matignon‟s stability theorem and the Based on the general form of the incremental O2Fo -
classic Lyapunov stability theory. Recent examp les PID form (25), we have
include:
N
Fractional-order linear matrix inequalities method u n T ubase n T ui, j n T
[72, 73], i 1
The robust interval check method [74]; N
Fractional-order Lyapunov inequality method [71]. ubase n T ui, j n T
i 1
In this subsection, by using the small gain theorem
(SGT) [75] (see appendix), we will find the generalized K0 e n T (31)
sufficient BIBO stability conditions of the O2Fo -PID 1
control system (25). Figure 5, simplified form of figure 2
well suited to the analysis of the BIBO stability, shows a + K1 e n 1 T K 2 e n 2 T
general nonlinear control system in a block diagram form.
The subsystem f is a fractional controller model and g is a N
nonlinear dynamic system to be controlled. It is clear + ui, j n T
from figure 5 that: i 1
y2 t y2 t
where
e1 t u1 t g e2 t u1 t e1 t g e2 t i 1, , N ; j 3, 4,5 , 6,7,8 , , n 2, n 1, n 3
or (28)
e2 t u2 t f e1 t u2 t e2 t f e1 t
and
y1 t y1 t
1 K P K I T K D T (32)
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
84 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
and 1
K P K I T K D T g 1 and
y n T g e2 n T g u n T (37)
(33)
2
g u n T
g
which is of the form
Applying the SGT, we can obtain the sufficient conditions
2 g (34)
for the BIBO stability for the linear Oo-PIDC system,
where 1 and 1 in (37).
The operator norm g is the gain of the given
It can be remarked that the stability conditions for the
nonlinear system g , usually defined as [76] controllers O2Fo-PID, R2Fo-PID and F2Fo-PID are
similar.
g u1 n g u2 n The zero -order co mponent of the
g sup (35) combination, ui 0, j 1,2,3 n T , is required in study
u1 u2 , n 0 u1 n u2 n
of stability analysis, since it is the fundamental engine of
the control law. The other terms of the comb ination may
The norm is the gain of the given nonlinear system affected, either locally or globally, on the precision,
over a set of admissible control signals that have any robustness and the possibility of the real-t ime
mean ingful function norms and u1 n and u2 n are implementation of the control law (25).
any two of the control signals in the set.
So, the sufficient conditions for the O2Fo-PIDC
system to be BIBO stable if the parameters of the O2Fo - IV. OPT IMIZAT ION OF T HE O2FO-PID CONT ROLLER
PIDC satisfy the inequality 1 2 1 where 1 and 2 Multiobjective optimization (M O) process is applied
are defined in (32) and (34), respectively. which simultaneously minimizes n objective functions
Theorem: Sufficient conditions of the O2Fo-PIDC system J which are functions of decision variables
(25) shown in figure 2 to be globally BIBO stable are bounded by some equality and inequality constraints
(constraints could be linear or nonlinear).
i. The given nonlinear process has a bounded norm, A MO problem can be formulated as follows
i.e., the nonlinear system has a finite gain
g and Objectives
T
J J1 , ..., J k
ii. The knowledge base of the O2Fo-PIDC
Minimizes
KP , KI , KD,, ,T satisfies the condition
Subject to
KP KI T KD T g 1.
Functional constraints
Conditions: g j 0, j 1, l (38)
hi 0, i 1, m
1
K K T K T g 1 and
P I D
Parametric constraints
(36)
r r r , r 1, n
2
g
where is defined as the decision vector,
together provide the BIBO stability criteria for the O2Fo- J k as the objective function vector.
PIDC design for a given bounded system. By noting that
g g j , j 1,2,..., l and h hi , i 1, 2,..., m
K P 0, K I 0, K D 0, T 0 , the result for the
stability of O2Fo-PIDC system will be: are the vector of equality and inequality constraints,
respectively. r and r are the lower and upper bounds
in the decision space for r parameter. The minimizat ion
problem (38) can be further represented as
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 85
Minimizes G (39) GA , are rapidly beco ming the most used methods of
choice for some intractable systems [79-81].
A. Genetic optimization of the developed FoC
where G incorporates the objective function J
GA are global parallel search and optimizat ion
and the constraint functions g and h , techniques based around Darwin ian principles, working
respectively. There are different ways to convert (38) into on a population of potential solutions (chromosomes).
(39), i.e., the transformation of the optimization problem Every individual in the population represents a particular
(38) to an unconstrained problem is made through many solution to the problem under study, often exp ressed in
methods. The weighted sum (or min-max) of objectives- binary (or real) code. The population is evolved over a
constraints methods are the two popular approaches [77]. series of generation (based on selection, crossover and
In order to incorporate constraints into the optimizat ion mutation operators) to produce better solutions to th e
problem, the sequential unconstrained minimizat ion problem.
technique is used. Here, the pseudo objective function is Figure 6 shows the structure of the control design and
formulated as optimization process for a nonlinear dynamical system
that contains three main blocks
G , rP G rP (40)
Optimization layer characterized by the MGA;
Control layer representing the closed loop O2Fo-
G is a co mbination of objectives in (38) (sum or
PIDC strategy;
Dynamical system to be controlled.
max of objectives) and is termed the penalty
Once the „optimal‟ structure of the controller is
function that introduces the constraints in (38) into the identified as given in (25), the optimization of the O2Fo-
cost function. rP denotes the scalar penalty mu ltiplier PID model is to find the „best‟ parameters, i.e., an
optimal knowledge base, which can be represented as an
and is typically increased as the optimizat ion goes on to extremum problem of optimization index (39).
put more and mo re emphasis on avoiding constraint
violations [78]. Metaheuristic algorith ms, that include
Optimisation Layer
Controller Performances y t
Designer :
MGA
Criteria yr t
vt
Knowledge
Disturbances
base
KP KI KD
λ µ
Plant modeled by
y t
I
u t
Actuators
I
x t f t , xt , u t
O2Fo-PID
controller
yr t y t g t , xt , u t
Decision Layer
y m t Sensors
x1, ..., xn T
bt
Measurement noise
Low power signal High power signal
Fig.6. Elements of the control loop.
There are many methods to design such fractional Using arith metic map and specified probability do
controllers which can be regarded as some optimizat ion crossover of parents to form new offsprings -
problems of certain performance measures of the Adaptation of genetic operators to appropriate
controlled systems. coding mode-.
The controller design based on GA can be considered Using chaotic map and specified probability
as a MO search procedure over a large objective- mutate chro mosomes to form new offspring‟s. The
parameter space. Figure 7 su mmarizes the evolution chaotic mutation operator [82] is used to solve the
process of GA, used in this work, to design the optimal problem of maintaining the population diversity of
FoC able to realize the desired objectives . GA in the learning process.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
86 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
The concept of elite strategy is adopted, where the elites - Improvement of the basic algorithm -.
best individuals in a population are regarded as
Encoding &
Initial population using
initialization
Start
random generator
Process
g:=0 Knowledge
base
Set population
of individuals
K u t y t
Controller
y ref t Proces
s
g := g + 1 Parent Selection
Evaluation Process
Evolution Process
New Population
Crossover
Knowledge base of
K
Mutation the controller
operator F Fitness function
Elitism Strategy
individual
Yes Optimal
No Are optimization
Best
knowledge
Loop criteria met ? base
Fig.7. A general cycle of the GA and the interaction between GA–closed loop control systems.
The parameters to be optimized are obviously the Small maximu m error with small total squared
O2Fo-PIDC gains, non-integer integrator-differentiator error J1 ;
orders and the number of PID subsys tems. So, the
Reduced control effort to be applied to the process
optimized parameter set may be such that
J2 ;
T Reduced number of linear-PID subsystems J3 ;
Kb Gc Sc ,
G K
Stability analysis conditions g st .
c P KI KD ,
T (41)
Sc nbr_subs T The objectives-constraints for a MIM O system with m
outputs are given by their expressions:
Therefore, three different possibilities exist in general
1 N 1 T
J1 x k x k Q k x k
Optimize gains only Gc ; M k 0
Optimize order only Sc ;
Optimize all parameters Kb
M
1
k
max xT k Q k x k (42)
. Q k 1 k n n
In order to emp loy the GA to optimize the knowledge
base of the FoC, we establish the fitness function
according to the objectives specified by the designer.
N 1
J2 u k M u k R k u k
Thus, the controller design based on GA can be
1 T
considered as a M O search procedure over a large (43)
k 0
objective-parameter space.
R k
2 k mm
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 87
3
J nbr_subs T S i
sub sub (44)
The expression (48) is a compro mise between better
accuracy, a reduced consumption of energy control (more
S i security), optimal structure and satisfied stability analysis
3 i mm
conditions. In the rest of the algorith m, the performance
criterion to minimize, will be selected and adapted to the
case of MIMO systems. The additional constraints are the
1 T
g g st H i g st
T
lower K T and upper K limits of the
m b b
H i 3 i mm controller parameters. The optimizat ion procedure may
be pronounced as follows: Find the knowledge base
g st K Pi K Ii T i K Di T i g 1 0
i K T (41) that
b
i 1, , m
Minimizes G
(45)
M N T denotes the integer number of co mputing
Subject to (49)
steps N is the running time and T is the sampling
period. x k xr k x k is the tracking error at
sampling time k. Il m is the identity matrix o f The fitness function is a measure of how well a set of
dimension l m and n is the nu mber of state variables. candidate coefficients meet the design specifications. A
fitness value used by the MGA can then defined as
sub is a vector of linear-PID subsystems. The weighting
factors j 0, j 1, 2,3 , selected to provide a
Fitness (50)
compromise between design specifications. G
By using the interior penalty function (IPF) method,
the stability constraints are included. The transfo rmed
unconstrained problem via the IPF method, used in this
work, is given by [78] where , 2 and is a s mall positive constant
used to avoid the numerical erro r of div iding by zero.
n Four majo r operations are necessary for correctly
Minimizes G wi Ji
, rg , rh (46) handling the execution of the optimization algorithm:
j 1
Determine the O2Fo-PID knowledge base to be
where
, rg , rh is the IPF, expressed as
optimized;
Determine an optimization objective;
Constraint equations:
Bounds for the elements of the design vector.
l
, rg , rh rh hk 2 MGA based learning provide an alternative way to
k 1
(47) learn fo r the O2Fo -PID knowledge base Kb (41). The
m
1
rg adaptation of these parameters continues until the overall
j 1 g j number of generations is satisfied.
B. Chromosome structure
In the expression (46), n 3 and hk 0, k . The key to put a genetic search for the O2Fo-PIDC
The function G in (46), mainly, depends on the into practice is that all design variables to be optimized
(41) are encoded as a finite length string, called
design specifications required by the user. The weights chromosome.
wi , rg 0 can, therefore, be used in control system The real-valued (or mixed) genes representation is used
design as design parameters to trade-off between d ifferent in this work. Global structure of the chromosome, in the
performance specificat ions. Fro m (46), the local and case of MIMO systems, and the corresponding
global minima can be calcu lated if the reg ion of configurations is illustrated in figure 8.
realizability of is convex [77]. It is usually assumed Every chro mosome which encodes the knowledge base
that of the O2Fo-PIDC can rep resent a solution of the
problem, that is, a O2Fo-PID optimal knowledge base. To
wi , rg , rh 1
simp lify the genes coding mode, i.e., avoidance of the
(48) mixed coding (real-integer) and the genetic operators
i adapted for each mode, the real-coding is used with the
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
88 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
restriction
the control vector, respectively. v t q represents the
nbr_subs ctrl
state of an external signal generator (dynamic of
disturbances), x0 is the in itial state vector,
ctrl f : C
(51) n m n is a nonlinear relationship
ctrl 1, N 3 between the state variables of the controlled system, the
generated control signals and interference (disturbances)
where ctrl models the number of subsystems, signals, respectively. The output or observation
y t q is generated via an output map g .
nbr_subs , included in modeling of O2Fo-PIDC
control law under development of (25) and x represents A. Control of a nonlinear helicopter process
the integer part of x . The CE150 helicopter model offered by Hu musoft Ltd
[83] for the theoretical study and practical investigation
Linear model of basic and advanced control engineering principles. The
parameters system consists of a body, carrying two propellers driven
by DC motors and a massive support. The body has two
λ µ ctrl KP KI KD AT degrees of freedom. Both body position angles
(horizontal and vertical) are influenced by the rotation of
Configuration TA propellers. The axes of a body rotation are perpendicular.
factors Both body position angles, i.e. azimuth angle in
horizontal and elevation angle in vert ical p lane are
Knowledge base
influenced by the rotating propellers, simultaneously. The
AT DC motors for driv ing propellers are controlled
AT
proportionally to the output signal of the computer. All
Controller i
inputs and outputs variables are coupled. The user of the
simu lator co mmunicates with the system via the data-
(a) (b) processing interface. The schematic diagram of the
Fig.8. Chromosome structure of O2Fo-PID controller for (a) SISO helicopter model is shown in figure 9.
systems. (b) MIMO systems.
In the next section, the developed algorithm is applied
to stabilize the CE150 helicopter model. The process is
modelled by a strongly nonlinear MIMO system. u2
u1
V. SIMULAT ION 𝚽
The proposed design controller is applicable to a
specific class of nonlinear system that can be described
by the differential equation of form
dx t
f t, x t , u t , v t Fig.9. Helicopter simulator configuration.
dt
y t g t , x t , u t , v t (52) The helicopter model is a multivariable dynamical
system with t wo man ipulated inputs and two measured
dv t
s v t outputs. The system is essentially nonlinear, naturally
dt unstable with significant cross coupling. The
x o x , t 0, t f mathematical model of the helicopter is given by the
0 following differential equations system [84]
where x t n and u t m are the state vector and
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 89
x1 t x2 t
x2 t 0.8764 x2 t sin x1 t 3.4325 x4 t cos x1 t u1 t
0.4211 x2 t 0.0035 x52 t
46.35 x6 2 t 0.8076 x5 t x6 t 0.0259 x5 t
2.9749 x6 t
x3 t x4 t
x4 t 21.4010 x4 t 31.8841 x8 t 14.2029 x8 t
2 (53)
21.7150 x9 t 1.4010 u1 t
x t 6.6667 x t 2.7778 x t 2 u t
5 5 6 1
x t 4 x t
6 5
x7 t 8 x7 t 4 x8 t 2 u2 t
x8 t 4 x7 t
x9 t 1.3333 x9 t 0.0625 u1 t
where, x1 t t , x2 t d t dt , x3 t t , where r t and r t are the desired values (set-
x4 t d t dt and x5 t x9 t are state variables points) of the considered variables. Figure 10 su mmarizes
representing the two DC motors and the coupling effects. the block diagram of the feedback control by using the
Assuming that the helicopter model (53) is a rigid body developed O2Fo-PIDC.
with t wo degrees of freedo m, the helicopter simulator In our applicat ion, the helicopter model is dicret ized by
CE150 can be viewed as an interconnexion of two using the fourth-order Runge-Kutta method with
subsystems. The elevation subsystem characterized by the sampling time chosen to be 0.1sec and a time horizon
input/output variables u1 t , t and the azimuth of 50 sec . The control object ive is to stabilize the
subsystem characterized by the input/output T
variables u2 t , t , where, t is the elevation system states t t around a set-point
angle (pitch angle), t is the azimuth angle (yaw r t r t 1 rad 1 rad under the control
T
angle), u1 t is the main motor voltage and u2 t is the T
actions u1 k u2 k . The in itial conditions , used in
tail motor voltage. The decentralized O2Fo-PIDC
strategy is adopted in this application, where the simu lation of the process, are taken
as x t 0 0
mu ltivariable process is treated as two separate single T
0 .
variables process, namely:
u1 t O2Fo-PID1 r t , t and
u2 t O2Fo-PID 2 r t , t
(54)
u1 t x1 t t
Digital controller
Elevation
O2Fo_PID(1) subsystem x2 t
r t
r t u2 t
x9 t
O2Fo_PID(2) Azimuth
subsystem
x3 t t
Helicopter simulator
Measurement noises
Chaotic added disturbances
Fig.10. Block diagram of the feedback control of the Helicopter model using two O2Fo -PIDC.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
90 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
The disturbances used to test a particular control performances than the R2Fo-PIDC, F2Fo -PIDC and the
strategy play a critical evaluation role. Thus, to carry out Oo-PIDC with similar optimization algorithm.
a comp lete and unbiased evaluation, it is necessary to The simu lation results at the end of the execution of
define a series of d isturbances and to subject each control the algorithm developed in this work are shown
strategy to all disturbances. schematically in figures 13 and 14.
Chaotic wave perturbations [83] on the system states Graphical co mparative results between O2Fo-PIDC
and the controller signals are added in order to tes t the and the Oo-PIDC, F2Fo-PIDC and R2Fo-PIDC are
efficiency and robustness of the controller. mentioned in these figures. Figures 13 (a)-(b) illustrate
the evolution of the output variables and their
B. Simulation results
corresponding reference trajectories.
The adapted chromosome structure with real-coding of It has been demonstrated that the developed O2Fo-
genes of the MGA for this application has the same form PIDC presents better performances then the others
of the figure 8 (a), but doubled. A part for O2Fo -PID(1) controllers with similar optimizat ion algorith m (tuning
and the other for O2Fo -PID(2). The M GA characteristics and stability conditions). The outputs of the developed
are summarized in table 1. controllers are illustrated by the figures 14 (a)-(b).
Figures 11 (a)-(d), show, respectively, the evolution of
the control parameters K Pi , K Ii , K Di ,i , i i 1, 2 of
0.08
0.07
the controllers O2Fo-PID(1) and O2Fo-PID(2) during the
optimization process. Around 500 optimization iterat ion 0.06
K
P1
steps (generation number), the number o f linear -PID K
D1
Control gains 0.05 K
subsystems (51) and the objective function (49) are I1
minimized as shown in figures 12 (a) and (b), 0.04
respectively. It can be remarked that the nu mber of linear
0.03
PID subsystems is minimized to 80% of its maximu m
value. 0.02
For generalizat ion, a comparison of the evaluation
values obtained at the end of the algorith m execution 0.01
between the proposed O2Fo-PIDC (25), the classical Oo- 0
0 50 100 150 200 250 300 350 400 450 500
PIDC, F2Fo-PIDC and R2Fo-PIDC using the same Generation
optimization procedure (49) is summarized in Table 2. (a)
0.08
T able 1. Specifications of the MGA.
0.07
Characteristic Value
0.06
Population Size 50
Control gains
0.05
Max_gen 500
Coding chromosome Real 0.04
Gain factors K Pi , K Ii , K Di i 1, 2
-6
[10 , 0.1] 0.03 KP2
Control factors i , i i 1, 2 [10-4, 1.0]
0.02
K
D2
KI2
Number of subsystems ctrl1 2 [1.0, 166.0]
0.01
Selection process Tournament
Arithmetic Crossover Pc = 0.8 0
0 50 100 150 200 250 300 350 400 450 500
Generation
Chaotic Mutation Pm = 0.02
(b)
As shown in nu merical values in Table 2 (ro ws and
columns have been ticked), it has been demonstrated that
the newly developed O2Fo-PIDC presents better
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 91
T able 2. O2Fo-PIDC, R2Fo-PIDC, F2Fo-PIDC and Oo-PIDC performances.
Controller
O2Fo-PID R2Fo-PID F2Fo-PID Oo-PID
Parameter O2Fo-PID(1) O2Fo-PID (2) R2Fo-PID(1) R2Fo -PID(2) F2Fo-PID(1) F2Fo -PID(2) Oo-PID(1) Oo -PID(2)
KP 0.026086 0.067830 0.012292 0.060458 0.017621 0.075926 0.010315 0.079464
KD 0.006076 0.072876 0.007261 0.038959 0.017838 0.077685 0.016022 0.007824
KI 0.078126 0.047305 0.074661 0.063871 0.079673 0.023698 0.078145 0.008156
0.613409 0.555782 0.613581 0.247233 0.991755 0.851211 1 1
0.027912 0.493221 0.011761 0.171282 0.855631 0.352196 1 1
nbr_subs 1base+8 1base+8 1base+0 1base+0 1base+166 1base+166 1base+0 1base+0
ISE 41.683407 14.456318 43.583710 15.754932 49.608883 14.828958 50.623753 24.129436
IAE 56.091587 21.582956 57.496647 23.204674 66.638168 24.532221 66.733536 40.239185
ITSE 102.022896 12.010715 109.308533 14.237223 144.248703 13.459677 149.341995 38.848053
ITAE 186.917908 52.184265 193.163345 43.827129 277.177338 79.772552 261.660492 168.433258
1 60
Lambda 1 nbr-subs 1
0.9 Mu 1 nbr-subs 2
50
0.8
Configuration factors
Number of subsytems
0.7
40
0.6
0.5 30
Optimal
Structure
0.4
20
0.3
0.2
10
0.1
0 0
0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500
Generation Generation
(c) (a)
1 0.52
Lambda 2 With Elitism Strategy
0.9 Mu 2 0.5 Without Elitism Strategy
0.8 0.48
Configuration factors
0.7 0.46
Best objective
0.6
0.44
0.5
0.42
0.4 Best
0.4
0.3 Objective
0.38
0.2
0.36
0.1
0.34
0 0 50 100 150 200 250 300 350 400 450 500
0 50 100 150 200 250 300 350 400 450 500
Generation
Generation
(d) (b)
Fig.11. Convergence of the O2Fo-PIDC parameters: (a) and (b) Gains Fig.12. Convergence of the (a) number of subsystems for the O2Fo-
of the O2Fo-PIDC(1) and O2Fo-PIDC(2) controllers. (c) and (d) Order PIDC(1) and O2Fo-PIDC(2). (b) Objective function.
of integral and derivative operators.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
92 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
1.4 0.7
Response O2Fo-PID
R2Fo-PID
1.2 Setpoint 0.6
F2Fo-PID
Oo-PID
1
Elevation angle [rad]
0.5
Control signal u1
0.8 0.4
0.6 0.3
O2Fo-PID(1)
0.4 0.2
0.2 0.1
0 0
0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50
Time [sec] Time [sec]
1.4
(a)
O2Fo-PID
R2Fo-PID 0.7
1.2 O2Fo-PID
F2Fo-PID R2Fo-PID
Oo-PID 0.6
F2Fo-PID
1
Elevation angle [rad]
Oo-PID
0.5
Control signal u2
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0 5 10 15 20 25 30 35 40 45 50
Time [sec] 0
0 5 10 15 20 25 30 35 40 45 50
(a) Time [sec]
1.2 (b)
1 Fig.14. Control signal: (a) u1 k . (b) u2 k .
0.8
Azimuth angle [rad]
Response Simu lation results show the good performance of the
0.6
Setpoint developed algorithm and confirm the effectiveness of the
O2Fo-PID(2) control law in the tracking of the desired trajectories.
0.4 Finally, the control inputs to the plant are adequately
constrained within their saturation limits. The stability of
0.2
the overall closed loop system is also preserved
independently from the uncertainties.
0
-0.2
0 5 10 15 20 25 30 35 40 45 50
Time [sec] VI. CONCLUSIONS
1.2
Progress has been made in coupling advanced
1
modeling and control methods in modern manufacturing
industry. This paper has considered a new alternative for
0.8 O2Fo-PID the synthesis of a simple, accurate, stable and robust FoC
Azimuth angle [rad]
R2Fo-PID
F2Fo-PID
controller with an optimal structure and parameters.
0.6
Oo-PID It has been demonstrated that the parameters
optimization of fractional-order controller based on M GA
0.4
is highly effective. According to optimization target, the
0.2
newly proposed method can search the best global
solution for O2Fo-PIDC parameters and guarantee the
0 objective solution space in defined search space.
The results showed the methodologies proposed in this
-0.2
0 5 10 15 20 25 30 35 40 45 50 study could achieve the desired optimization goal. The
Time [sec] FoC can yield high precision trajectory control results.
(b) Simu lation results demonstrate that the proposed O2Fo-
Fig.13. System states using O2Fo-PIDC, R2Fo-PIDC, F2Fo-PIDC and PIDC offers encouraging advantages and has better
Oo-PIDC: (a) Elevation angle. (b) Azimuth angle. performances.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller 93
establishes that the closed loop system is finite gain
A PPENDIX stable.
Proof. It follows from
BIBO ST ABILIT Y AND SMALL GAIN THEOREM
e1 u1 g e2
Theorem 1: BIBO stability [74]. For a given b, there
exists m0 b such that u t b g u t m0 b That
for all t 0, . Thus, a bounded input to the nonlinear
e1 u1 g e2
system is assumed to produce a bounded output.
u1 2 2 e2
Theorem 2: Small Gain Theorem [74]. Consider the
interconnected nonlinear feedback system of figure 5 with Similarly, we have
inputs u1 t , u2 t and outputs y1 t , y2 t , which
is described by the relation (28). Suppose that both e2 u2 1 2 e1
subsystems f and g are causal and let
Combining these two inequalities, we obtain
1 f the gain of f and 2 g the
e1 1 2 e1 u1 2 u2 2 2 1
gain of g . Also, suppose that there exists
constants 1, 2 , 1 0 , and 2 0 so that or, using the fact
f e1 t 1 1 e1 t 1 2 1 ,
1
g e2 t 2 2 e2 t
e1 1 1 2 u1 2 u2 2 2 1
The rest of the theorem follows immediately.
then, the closed loop system is also finite gain stable from
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87-92.
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
96 Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller
evolution, an alternative approach to evolutionary Ph.D at the Electronics department, University of Jijel, Algeria.
algorithm‟‟, in „„Modern heuristic optimization His current research interests include the fractional-order
techniques : Theory and applications to power systems’’, systems applied on the photovoltaic control systems, Artificial
IEEE Press Editorial Board Kwang Y. Lee, M ohamed A. Intelligence and advanced approaches in control engineering.
EL-Sharkawi, Chapter 9, pp. 171-187, 2008.
[78] Rolf Isermann, M arco M ünchhof, Identification of
dynamic systems: An introduction with applications, S alah Leulmi, born in 1951 in Algeria, ,
Springer-Verlag Berlin Heidelberg, 2011. received the State Engineer Degree in
[79] Shaminder Singh, Jasmeen Gill,"Temporal Weather Electric Power Systems Engineering from
Prediction using Back Propagation based Genetic the Algiers National Polytechnic School in
Algorithm Technique", IJISA, vol.6, no.12, pp.55-61, 1976, Algeria, a M aster Degree of
2014. DOI: 10.5815/ijisa.2014.12.08 Engineering from RPI, Troy, NY, USA in
[80] Osama I. Hassanein,Ayman A. Aly,Ahmed A. Abo- Electric Power Systems Engineering in 1978
Ismail,"Parameter Tuning via Genetic Algorithm of Fuzzy and a Ph.D. in Electrical Engineer ing from
Controller for Fire Tube Boiler", IJISA, vol.4, no.4, pp.9- ISU, Ames, IA, USA in 1983. He is the author of around 50
18, 2012. publications in journals & proceedings. He was the Head
[81] M olly M ehra, M .L. Jayalal, A. John Arul, S. Rajeswari, K. "Director" of the University of August 20th, 1955, Skikda,
K. Kuriakose, S.A.V. Satya M urty,"Study on Different Algeria. From 1992 to 2010, he was the President of the
Crossover M echanisms of Genetic Algorithm for Test Scientific Council of the Faculty of Science & Technology at
Interval Optimization for Nuclear Power Plants", IJISA, the same University. He is a Professor since 1988 up to now.
vol.6, no.1, pp.20-28, 2014. DOI: Prof. S. Leulmi is, also, a referee of 4 Algerian Journals & some
10.5815/ijisa.2014.01.03 Proceedings & one overseas society "WSEAS" for Proceedings
[82] A. Soukkou, A. Khellaf, S. Leulmi, M . Grimes, “Control & Journals. Since 1992 up to now, he is the President of the
of dynamical systems: An intelligent approach,‟‟ NSC of the Equivalency Degrees, since 1992 to 2015.
International Journal of Control, Automation, and
Systems, vol. 6, no. 4, August 2008, pp. 583-595
[83] Humusoft, CE 150 helicopter model: User's manual,
Humusoft, Prague 2002. How to cite this paper: Ammar SOUKKOU, M.C. BELHOUR,
[84] H. Boubertakh, M . Tadjine, Pierre-Yves Glorennec, Salim Salah LEULM I, "Review, Design, Optimization and Stability
Labiod, „„Tuning fuzzy PD and PI controllers using Analysis of Fractional-Order PID Controller", International
reinforcement learning,‟‟ ISA Transactions, vol. 49, 2010, Journal of Intelligent Systems and Applications (IJISA), Vol.8,
pp. 543-551. No.7, pp.73-96, 2016. DOI: 10.5815/ijisa.2016.07.08
[85] A. Soukkou, S. Leulmi, A. Khellaf, „„Intelligent nonlinear
optimal controller of a biotechnological process,‟‟
Archives of Control Sciences, vol. 19, no. 2, 2009, pp.
217-240.
Authors’ Profiles
Ammar S OUKKOU received his Diploma
in Engineer (BsC), the M agister (M sC)
degree and the Doctorate (PhD) in
Engineering Control (2008), Electronics
Department, Setif University, Algeria. Since
2000 to 2005, he held different positions
involved in industrial field and education.
Since 2005, he has been an Assistant
Professor at the Electronics department, University of Jijel,
Algeria. He is the author of more than 25 publications in
international journals & proceedings. His current rechearch
interrests include intelligent and advanced modeling and control
of Biotechnological and renewable energy processes,
Fractional-order field, Advanced optimization techniques,
Artificial Intelligence and advanced approaches in control
engineering.
M.C. Belhour was born in Jijel - ALGERIA
at the 7th June, 1989. received his Diploma
of License in electronics from the
department of electronics, Jijel University,
Algeria, 2010 and M aster degree in
Electronics and System Analysis from the
Department of Electronics, Jijel University,
Algeria, 2012. He is currently working on a
Copyright © 2016 MECS I.J. Intelligent Systems and Applications, 2016, 7, 73-96
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