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Neural adaptive quantized output-feedback control-based synchronization of uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities
Author’s Accepted Manuscript Neural Adaptive quantized output-feedback control- based synchronization of uncertain timedelay incommensurate fractional-order chaotic systems with input nonlinearities Farouk Zouari, Abdesselem Boulkroune, Asier Ibeas www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)31437-0 http://dx.doi.org/10.1016/j.neucom.2016.11.036 NEUCOM17792
To appear in: Neurocomputing Received date: 13 July 2016 Revised date: 27 September 2016 Accepted date: 19 November 2016 Cite this article as: Farouk Zouari, Abdesselem Boulkroune and Asier Ibeas, Neural Adaptive quantized output-feedback control- based synchronization of uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.11.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Neural Adaptive quantized output-feedback control- based synchronization of uncertain time-delay incommensurate fractionalorder chaotic systems with input nonlinearities Farouk Zouari1*, Abdesselem Boulkroune2, Asier Ibeas3,4 1
Laboratoire de Recherche en Automatique (LARA), École Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, BP. 37, Le Belvédère, 1002 Tunis, Tunisie 2 LAJ, University of Jijel, BP. 98, Ouled-Aissa, 18000 Jijel, Algeria 3
Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, 08193- Bellaterra, Barcelona, Spain 4 Departa ento de Ingenier a, acultad de iencias Naturales e Ingenier a, ni ersidad de ogot orge Tadeo o ano, treet, No - , od , ogot , D.C. 110311, Colombia *[email protected] [email protected] [email protected]
Abstract This research is concerned with the problem of generalized function projective synchronization of nonlinear uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities. The considered problem is challenging owing to the presence of unmeasured master-slave system states, external dynamical disturbances, unknown nonlinear system functions, unknown time-varying delays, quantized outputs, unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct fractional-orders. Under some mild assumptions and using
aputo’s definitions for
fractional-order integrals and derivatives, the design procedure of the proposed neural adaptive controller consists of a number of steps to solve the generalized function projective synchronization problem. First, smooth functions and the mean value theorem are utilized to overcome the difficulties from actuator nonlinearities and distributed time-varying delays, respectively. Then, a simple linear observer is established to estimate the unknown synchronization error variables. In addition, a Nussbaum function is incorporated to cope with the unknown control direction and a neural network is adopted to tackle the unknown nonlinear functions. The combination of the frequency distributed model, the Razumikhin Lemma, the neural network parameterization, the Lyapunov method and the
arbalat’s le
a is
employed to perform the stability proof of the closed-loop system and to derive the adaption laws. The major advantages of this research are that: (1) the Strictly Positive Real (SPR) condition on the estimation error dynamics is not required, (2) the 1
considered class of master-slave systems is relatively large, (3) all signals in the resulting closed-loop systems are semi-globally uniformly ultimately bounded and the synchronization errors semi-globally converge to zero. Finally, numerical examples are presented to illustrate the performance of the proposed synchronization scheme. Keywords Generalized function projective synchronization, uncertain time-delay chaotic systems, incommensurate fractional-order systems, input nonlinearities, Nussbaum function,
Razumikhin Lemma, Frequency distributed model, adaptive quantized
output-feedback control.
1. Introduction Nowadays, a great deal of interest has been paid to the synchronization of noninteger-order chaotic systems due to their broad applications in various branches of science and engineering like bioengineering [1], chemical science [1-4], continuum mechanics [5], nonlinear control systems [1-33], electrical networks [20], financial systems [21], robotics [22], modeling brain activities [1] and many others. Generally, the objective of the synchronization of two or more dynamical systems is that those systems are coupled or one among them somehow traces the motion of the others [122, 24-27, 29-41]. Recently, approaches about synchronization between the drive (master) and response (slave) systems of two identical or non-identical systems with different initial conditions, have been proposed in various works, for instance, see [17-45] and references therein. As stated in [1-45], there exist many categories of synchronization
such
as
identical
or
complete
synchronization,
partial
synchronization, generalized projective synchronization, generalized synchronization, Q-S synchronization, phase synchronization, hybrid projective synchronization, anticipated and lag synchronization, amplitude envelope synchronization, modified projective synchronization, function projective synchronization, generalized function projective synchronization, and so on. It should be pointed out that the generalized projective synchronization, anti-phase synchronization, generalized synchronization, projective synchronization and complete synchronization are special cases of the generalized function projective synchronization [20-45]. In some works, several control approaches have been developed for the chaotic synchronization phenomena, such as linear and nonlinear feedback control, active control, adaptive (direct or 2
indirect) control, time delay feedback approach, Sampled-data feedback control, adaptive open-plus-closed-loop control, fractional PID control, robust control, sliding mode control, Backstepping control, data-driven control, among others [1-10, 2389]. It is also well known that the singularity problem and chattering phenomena can appear while implementing some control approaches [87-94]. From the control theory point of view, a chaotic system is a dynamical nonlinear deterministic system governed by one or more control parameters and characterized by an unusual sensitivity to initial conditions, a fractal structure, a non-periodic behavior and the existence of a positive Lyapunov exponent [1-15]. As pointed out in [1, 3-35], Lyapunov exponent criteria can be employed to identify a chaotic behavior. In the literature, many systems can produce chaotic behaviors such as, Lü system, Chen system, Sprott systems, Li system, Cai system, Tigan system, Wei–Yang system, fractional-order
hua’s circuit, Sundarapandian systems, fractional-order
Arneodo system, Pehlivan system, fractional-order Lorenz system, fractional-order Rössler system, Sampath–Vaidyanathan system, fractional-order Lü system, uncertain fractional-order Duffing–Holmes chaotic time-delay systems, to name but a few [120]. Generally, the mathematical models of chaotic systems can be described by differential or difference equations in such a way that the present states of systems are expressed in terms of the past ones [1, 3-25, 27-57]. Most of the above-mentioned works have focused on the simple chaotic systems with affine structures. However, there are various nonlinear nonaffine systems in the control literature, such as biochemical processes, chemical reactors, and so on [26, 56, 75, 87, 88]. It should be noted that affine systems are a very special case of nonaffine systems [26, 75, 87, 88].Thanks to the considerable efforts devoted by researchers, significant control techniques have been proposed for nonaffine systems [56, 87]. In general, there exist five approaches used to cope with the no affinity problem, namely: (1) the local inversion of the Takagi-Sugeno fuzzy affine model, (2) the mean value theorem, (3) the implicit function theorem, (4) the differentiation of the original system equation, (5) Taylor series expansion [1-26, 87, 88]. In addition, most of these above-mentioned approaches have been sometimes employed to determine the control direction (i.e., the so-called sign of the control gain) [26-56]. It should be noted that for facilitating the design of adaptive controls for nonlinear nonaffine systems, the sign of the control
3
gain has been assumed to be known (strictly negative or strictly positive) in some works [57-87]. From a practical point-of-view, the assumption that the sign of the control gain is known a priori, is more an exception than a rule because there are various systems with unknown control directions such as biochemical and biophysical processes, robotics, electrical systems, and so on [1-4, 6, 9, 12, 23, 43, 87]. It must be pointed out that the lack of knowledge of the sign of the control gain can complicate the adaptive control design [2, 12, 4, 26, 33, 39, 74, 78-80, 84, 88]. The available approaches for coping with the unknown control problem, can be classified into five groups, such as [79, 84, 88]: (1) an approach based on a hysteresis-type function, (2) an approach based on a monitoring function, (3) an approach based on a Nussbaumtype function, (4) an approach based on directly estimating unknown parameters, (5) an approach based on Nussbaum and hysteresis-dead zone-type functions.
For
example in [78,79, 83, 84, 88], synchronization schemes based on some of the previous methods have been developed for a class of uncertain fractional-order chaotic systems with unknown control direction. In some of the above-mentioned researches, the models of studied systems are assumed to be known or partially known. In real applications, many systems are harshly nonlinear and contain uncertainties due to the presence of variable changes, unknown parameters, disturbances, unmodeled dynamics, and different conditions [1-70, 88, 89]. In the literature [3, 5, 70-88], iterative data-driven controller tuning techniques like
data-driven reinforcement
learning control, frequency-domain tuning, iterative feedback tuning, iterative regression
tuning,
correlation-based
tuning,
data-driven
predictive
control,
simultaneous perturbation stochastic approximation, model-free adaptive control, adaptive online iterative feedback tuning, model-free control, and model-free or datadriven iterative learning control, have been developed for handling the structured uncertainties. Moreover, based on the universal approximation property [2-4, 8, 14, 33, 39, 49-58, 62, 74, 78-88], various adaptive intelligent controllers like neural network and fuzzy-based adaptive controllers have been introduced in the synchronization schemes. The principal idea of such control methods is to employ neural networks or fuzzy logic systems for estimating the uncertain nonlinear functions in dynamic systems [2-4, 62, 74, 78-88]. It is well known that some neural 4
network-based control techniques are iterative data-driven control techniques [2-4, 8, 14, 15, 33, 88]. For coping with approximation errors and external disturbances, neural adaptive controllers have been augmented by robust compensators in some real applications [7, 9, 37, 57, 76, 77, 88]. Nevertheless, the effect of actuator (input) and sensor nonlinearities on designing and implementing synchronization control systems is omitted in some of the aforementioned works. In recent years, actuator and sensor nonlinearities are ubiquitous in physical systems and devices such as biological optics, electro-magnetism, mechanical actuators, and electronic relay circuits due to their excellent characteristics [26, 39, 46-60, 67, 74, 84, 88]. In engineering practice, the output of the actuators and the input of the sensors are unknown signals because the actuators and the sensors have been usually employed to actuate the systems as an input and to measure the system output, respectively [26, 39, 46-60, 67]. It is well known that actuator and sensor nonlinearities like dead-zone, saturation, backlash, quantization, and hysteresis, could guide to poor performances or even instability of the synchronization control systems [46-60, 67, 74, 84, 88]. Moreover, it is also noted that the quantizer is commonly employed for transforming a real-valued signal into piecewise continuous [46-52, 67]. The studies about actuator nonlinearities (backlash-like hysteresis, dead zone and saturation) and quantized output problems have attracted a great deal of attention, for example, see [26, 39, 46-60, 67, 74, 84] and the references therein. The works in [67, 67, 74, 84, 88] have solved the problems of backlash-like hysteresis, dead zone and unknown asymmetric saturation actuators by using smooth functions. Besides, approaches based on static and dynamic quantizers have been developed to cope with the quantized feedback control problems [46-52]. The major limitation of most of the above mentioned results is that all the state variables of the controlled systems are directly measured or accessible. In most practical situations, the state variables of the systems are immeasurable or difficult to measure [28, 51, 54, 58, 67, 75-77, 79, 80, 83, 85, 88]. In these cases, the observer-based controls (or output-feedback controls) must be used to get the desired performance [75-77, 79, 80, 83, 85, 88]. In general, there exist three approaches of observer-based control designs for uncertain nonlinear systems, such as [75, 79, 80, 83, 85, 88]: (1) an approach based on the Strictly Positive Real (SPR) condition, (2) an approach based on the separation principle paradigm, (3) an approach based on the 5
non-separation principle. In [80, 83, 85, 88], based on the separation approach, the controller and the high-gain observer that provides state estimates have been constructed separately in order to make the closed loop system stable. It must be mentioned that pole-placement algorithms, Riccati equation-based algorithms, and Lyapunov equation-based approaches are the available techniques for the design of high-gain observers [75-77, 79, 80, 83, 88]. Furthermore in [53, 54, 58, 67, 75-77, 79, 80, 83, 85, 88], based on the non-separation principle, Implicit Function Theorem, Mean Value Theorem and neural network parametrization, an output feedback control scheme has been developed for a class of non-affine nonlinear systems, so that the design of state observer has been coupled with the control system design. On the other hand, based on the SPR condition of the observation error dynamics (i.e. the estimation error dynamics), direct and indirect adaptive control schemes have been proposed for uncertain nonlinear perturbed systems [75-77, 79, 80, 83, 85, 88]. As stated in [80, 83, 85, 88], the SPR condition (or a suitable observation error filtering) can be employed in the Lyapunov stability analysis for designing the adaptation laws on the basis of the Meyer-Kalman-Yakubovich (MKY) Lemma and available measurements (i.e. the output observation error and the output tracking error) [75-77, 79, 80, 88]. It should be noted that the SPR condition is a very restrictive condition and it is not satisfied by various systems [75, 79, 80, 88]. Moreover, in [75, 76, 79, 80, 88], it has been proven that the control system can be augmented by a stable filter to satisfy a SPR condition of a transfer function of the observation error dynamics [75, 88]. It must be pointed out that some of the aforementioned studies do not cope with state time-delays. Delays usually appear in many industrial control systems, such as underwater vehicles, chemical processes, communication networks, rolling mill systems, biological systems, metallurgical processing systems, and so on [5, 13, 15, 21, 34, 41, 44, 45, 52, 53, 60-62, 84, 86-89]. In fact, the presence of delays usually causes poor performance, exhibition of undesirable inaccuracies, and even instability of the system under control [60-62, 84, 86-89]. The Lyapounv-Razumikhin and the Lyapunov-Krasovskii functionals are approaches usually used for controller design and stability analysis of time-delay systems [52, 53, 60-62, 75, 84, 86-89]. Unlike the Lyapunov-Krasovskii techniques in which the derivative of the time-varying delay terms should be strictly less than unity, the Lyapunov-Razumikhin technique has been 6
applied to various time-delay systems, so that the stability analysis and the control design can be relatively straightforward and the time-varying delays only satisfy the boundedness [5, 13, 15, 21, 34, 41, 88]. Moreover, the classical Lyapunov-Krasovskii functional, which is always composed by integral and quadratic functions, has been extended to fractional-order systems [87, 89]. In various works, the LyapunovRazumikhin functional has been generally built with quadratic functions in order to construct the control and to determine the (asymptotic, uniform, global asymptotic) stability of fractional functional differential systems with time-varying, constant, and distributed time varying delays [87-89]. In [88, 89], stability conditions of fractionalorder nonlinear systems have been obtained based on the Razumikhin technique, Lyapunov functions and Caputo derivatives. In the above-mentioned researches, stochastic, pure-feedback, strict-feedback and nonlinear systems have been studied and it has been proven that many real systems could be described by fractional-order differential equations, such as [1-23]: electrical circuits, electromagnetic waves, regular variation in thermodynamics, viscoelastic systems, dielectric polarization, biological and financial systems, heat conduction in a semi-infinite slab, robotics, bioengineering, and so on. In several branches of engineering and science, the fractional calculus provides more precise models of the physical systems than the regular calculus does. It is well known that integer order systems is a very special case of fractional (non-integer) order systems [6-30, 33, 43, 59, 63-66, 69-73, 76, 77, 81, 82, 87-91, 94-97]. That is to say, fractional calculus is an excellent mathematical tool for an exact description of memory and heredity features of many materials and processes [1-23, 33, 43, 59, 88, 91]. In [94-99], an interpretation of fractional operators in the time domain has been presented by using the physical realization of fractional operators, construction of a Cantor set, linear filters and four concepts of fractal geometry. It has been shown that fractional operators can be classified into filters with partial memory, which lies between two extreme categories of filters with no memory and those with complete memory [9599]. In [97, 98], it has been deduced that fractional operators may be used for modeling systems with partial dissipation [96-99]. In addition, it has been shown that the fractional-order of a fractional integral represents the indication of the preserved or remaining energy of a signal existing in such system [95-97]. However, it has been established in [96, 97] that the fractional-order of a differentiator represents the rate of 7
a portion of the lost energy. On the other hand, it has been found that the Grünwald– Letnikov, Hadamard, Weyl, Riesz, Riemann– iou ille, and
aputo’s definitions are
the most popular definitions for fractional-order integrals and derivatives [59, 63-66, 69-73, 76, 94-99]. As stated in [1-45, 76, 77, 81, 82, 87-93], the most classical stability tools of integer order systems cannot be extended or applied directly to that of the fractional-order differential systems. Recently, stability conditions for a class of fractional-order nonlinear systems has been developed based on Mittag-Leffler function, Laplace transform, Caputo derivatives and the generalized Gronwall inequality [8, 63-65, 88]. Besides, Lyapunov-Razumikhin and Mittag-Leffler stability theorems for fractional differential systems with delay have been proposed in [63-65]. Additionally, with the help of the frequency distributed model and indirect Lyapunov methods, adaptive controllers have been constructed and some stability conditions for the synchronization of fractional-order systems have been obtained [76, 77, 81, 82, 88]. In comparison with many research works on integer order systems, a few works focused on the synchronization of multi-input multi-output (MIMO) fractional-order systems because the specificity of MIMO systems and the difficulties with the extension of the approaches employed for integer order systems to fractional ones [145, 87-89]. So far, to the best of authors’ knowledge, the problem of the generalized function projective synchronization of MIMO uncertain nonlinear time-delay incommensurate
fractional-order
chaotic
systems
with
unknown
actuator
nonlinearities (including backlash-like hysteresis, asymmetric saturation actuators and dead-zone) has not been studied before in the literature. Motivated by the preceding discussion, a novel neural adaptive quantized outputfeedback control is proposed in this research for achieving an appropriate generalized function projective synchronization of nonlinear uncertain MIMO time-delay incommensurate
fractional-order
nonlinearities. The main
chaotic
systems
with
unknown
actuator
difficulties of this research are how to cope with
unmeasured master-slave system states, external dynamical disturbances, unknown nonlinear system functions, unknown distributed time-varying delays, unknown timevarying delays, quantized outputs, an unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct fractional-orders. In order to solve the generalized function projective synchronization problem, the design of the proposed neural adaptive 8
controller follows a number of steps under some mild assumptions and using aputo’s definitions for fractional-order integrals and derivatives. First, smooth functions and the mean value theorem are introduced to avoid the difficulties from distributed timevarying delays and unknown actuator nonlinearities, respectively. Then, a simple observer is incorporated for estimating the unknown synchronization error variables. Additionally, a neural network is used to neutralize the uncertain nonlinear dynamics and a Nussbaum function is incorporated to cope with the unknown control direction. A combination of the Razumikhin Lemma, the neural network parameterization, the frequency distributed model, the Lyapunov method and the
arbalat’s le
a is
adopted for deriving all the parameter adaption laws and for performing the stability proof of the closed-loop system. The main contributions of this research lie in the following: (1) Unlike in [1-4, 6, 8, 9, 12, 14, 26, 27, 29, 33, 34, 37], the proposed design approach does not require any prior knowledge of the sign of control gains nor any information of the bound of actuator nonlinearities. (2) In contrast to the adaptive controls [1-18, 22, 24, 25, 27, 28, 29, 39-41, 45], the proposed neural adaptive controller is very simple and without the singularity problem, so that the number of adjustable variables is reduced. (3) Unlike the previous research works [80, 83, 85, 88], the SPR (Strictly Positive Real) condition on the estimation error dynamics is neglected. (4) Compared to the previous works [1-15, 20, 23, 28-30, 33, 43, 59, 63-66, 68-73, 83, 87, 89], the stability of the corresponding closed-loop system is analyzed in this manuscript without determining the fractional derivatives of the Lyapunov functions, (5) Unlike the closely related research works [2, 4, 26, 33, 74], the synchronization errors semi-globally converge to zero and all signals of the closed-loop system are semi-globally uniformly ultimately bounded. The remainder of this paper is organized as follows. In Section 2, the problem statement is presented. In Section 3, the neural adaptive quantized output-feedback control is designed and analyzed while in Section 4, simulation examples are given to confirm the effectiveness, the feasibility and the validity of the proposed design. Finally, conclusions are drawn in section 5. Notations. D
represents the order Caputo differential operator. q . denotes
the quantization function. . is the floor function. . is a nonlinear input 9
function. .
indicates the Euclidean norm. The superscript T denotes the matrix
transpose operation. I n represents the identity matrix of size n . min . denotes the minimum eigenvalue of corresponding matrices. . denotes the absolute value function. 0n m is the zero matrix of size n m .
diag . denotes a block diagonal
matrix. t denotes the time index. sgn . represents the sign function. set of all n m real matrices.
n
n m
denotes the
represents the n-dimensional Euclidean space. L 2 is
the space of square-summable functions. L is the space of bounded functions.
2. Problem formulation Let us consider two nonlinear uncertain MIMO time-delay incommensurate fractional-order chaotic systems (master and slave systems), expressed as follows
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 Master: x Mi t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, , n i , i 1, p
(1)
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t d i , j t t t 3,i , j t f 3,i , j x S , d , j 1, , ni 1, t 0 2 ,i D x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t d i ,ni t t f Slave : x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,ni t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 S i 2 ,i i mi 1 y Si t x Si ,1 t , for i 1, , p
10
(2)
where
the
for i 1,
fractional
incommensurate
0 1,i 1
orders
0 2 ,i 1 ,
and
, p are known constants.
x M t x M 1T t , x M 2T t , x S t x S 1T t , x S 2T t ,
, x MpT t T
, x SpT t T
n
n
and
are the states of the master
and slave, respectively, with
x Mi t x Mi ,1 t , x Mi ,2 t ,
x Si t x Si ,1 t , x Si ,2 t ,
and n1 n 2
, x Mi ,ni t
, x Si ,ni t T
T
ni
,
, for i 1,
ni
,p
np n .
y M t y M 1 t , y M 2 t ,
, y Mp t T
and y S t y S 1 t , y S 2 t ,
p
, y Sp t T
p
are the outputs of the master
and slave, respectively.
d t d 1T t , d 2T t ,
, d pT t T
n
is the external bounded disturbance
vector for the slave system with d i t d i ,1 t , d i ,2 t , for i 1,
, d i ,ni t
T
ni
,
,p .
The initial condition functions for the slave 1,i t
ni
, and 2 ,i t
ni
i 1,
,p ,
are unknown, smooth and bounded t 0 . For k 1,
, 4 , j 1,
, ni , i 1,
, p , k ,i , j t are unknown time-varying delays
satisfying 0 k ,i , j t * with the constant * being unknown and strictly positive.
11
For j 1,
, ni , i 1,
, p , f 1,i , j x M t , x M t 4 ,i , j t
,
f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t , f 3,i , j x S t , t and f 4,i x S t , i t
are unknown smooth functions.
u t u1T t , u 2T t ,
, u pT t
system with u i t u i ,1 t , u i ,2 t , and m1 m 2
T
m
,
is the control input of the slave
, u i ,mi t
T
mi
, for i 1,
,p
mp m .
The output of the actuator nonlinearity t 1 t , 2 t ,
, p t T
p
(including backlash-like hysteresis, asymmetric saturation actuators and dead-zone) is described by the following equation [39, 52-60, 67, 74, 84, 86, 88, 90-92] u u min , if i t min 1 0 2 u min i t 1 2 , if 1 i t 1 2 i t u i t 0 , if 1 i t 3 u i t 3 4 ,if 3 i t max 3 4 u u max , if i t max 3 0 4 T T d i t d i u i t T u t t d i u i t , for i 1, , p 5 6 i i i 7 dt dt dt
where u min , l , 2 , 3 , 4 and u max are unknown strictly positive constants.
5 , 6 and 7 are unknown constants satisfying 6 7 .
12
(3)
diag 1 , 2 ,
, p
and i 0 , for i 1,
mp
is a known constant matrix so that i
mi
,p .
Remark 1. In contrast to [1, 3, 4, 6-9, 12-14, 25-27, 29, 37, 39], the master and slave systems (1) and (2) presented in this paper are partially dissimilar, with inhomogeneous
1,1 1,2 for i 1,
(or
incommensurate)
1,p and 2 ,1 2 ,2 p
, p and
i 1
1,i
fractional-orders
(i.e.
2 , p ) and with different orders (i.e., 1,i 2 ,i ,
p
2 ,i ). In addition, these master and slave systems with i 1
uncertain dynamics and external dynamic disturbances can be used to describe many physical (practical) systems, such as:
fractional-order Lorenz system, chemical
processes, aircraft wing rock, Duffing chaotic system, communication networks, fractional-order Lü system, induction servo-motor drive, fractional-order Chen system, fractional-order unified chaotic system, underwater vehicles, and so on [2145, 59]. Remark 2. As already discussed in [1-15, 20, 23, 28-30, 33, 43, 59, 68-73, 88], the Hadamard, Riemann–Liouville, Riesz, Grünwald–Letnikov, Weyl, and Caputo’s definitions are the most widely used definitions for fractional-order integrals and derivatives. Therefore, the Caputo fractional operator is employed in this paper because it is more consistent than other ones, for the following reasons [68-73]: the Caputo derivative of a constant is zero, the initial conditions for the fractional-order differential equations in Caputo sense have the same form as for the integer-order differential equations, the Caputo operator allows the avoidance of hyper-singular improper integral and mass balance error. Remark 3. It is easy to see that the actuator nonlinearity (3) has non-smooth nonlinear characteristics and sharp corners when i t u min and i t u max , for i 1,
, p . Consequently, many techniques cannot be directly applied for
constructing the adaptive controllers [53-59, 67].To handle such a problem, smooth functions are usually employed for approximating the system (3).
13
Remark 4. It must be emphasized that commensurate fractional-order chaotic systems are a special case of incommensurate fractional-order chaotic systems [1-5, 23, 28-30, 33, 43, 59, 68-73, 88]. In this research, the following Assumptions 1-4 are necessary. Assumption 1. Only the quantized outputs of the master and slave systems described by the following equation (4) are available for measurement. y Mi t q1,i y Mi t 1,i t , 1,i t y Si t q 2 ,i y Si t t 2 ,i t , for i 1, , p 2 ,i T q y t q y t , q y t , , q1, p y Mp t 1 M 1,1 M1 1, 2 M2 T q y t q y t , q y t , , q 2 , p y Sp t 2 S 2 ,1 S1 2 ,2 S2
(4)
p
p
where the smooth functions 1,i t and 2,i t are strictly positive and bounded such that
d is bounded, t 0 , for i 1, t
0
1,i
2 ,i
,p .
Assumption 2. There exist unknown class k functions k,i, j . , k 1, j 1,
, ni , i 1,
, p and unknown positive constants 1 , 2 , such that
14
,7 ,
p p ni f x t , x t t x t f 4 ,i x S t , 0 M 4 ,i , j M 1,i , j M i 1 j 1 i 1 p ni f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t x S t i 1 j 1 p ni p ni 1,i, j x M t 2 ,i, j x S t i 1 j 1 i 1 j 1 p ni 3 ,i, j x S t 1,i , j t i 1 j 1 p ni p ni 4 ,i, j t 2 ,i , j t 7 ,i, j x M t 4 ,i , j t 1 i 1 j 1 i 1 j 1 p ni p ni p ni f x t , t x t 5 ,i, j S 6 ,i, j t 2 3 ,i , j S i 1 j 1 i 1 j 1 i 1 j 1
(5)
Assumption 3. f 4 ,1 x S t , 1 t f 4 , p x S t , p t is nonThe matrix g 1 x S t , t diag , , 1 t p t
singular and either strictly negative-definite or strictly positive-definite so that there exist two unknown positive constants 3 and 4 , satisfying
0 3
f 4 ,i x S t , i t i t
4 , for i 1,
,p
(6)
Assumption 4. There exists an unknown strictly positive constant 5 , so that
p
p
i 1
i 1
x M t d t 1,i t 2 ,i t 7
4
i 1
i 1
* i i n p m p
p
i 1
i 1
i u max u min D
2 ,i
(7) x Mi t
5 np
15
, t 0
Remark 5.
Note that Assumption 1 is satisfied by many modern engineering
systems such as electric power grids, intelligent transportation systems, groups of unmanned (aerial, ground, and underwater) vehicles, networked control systems, to name but a few [46-52, 67]. According to [46-52, 67], it can be deduced that the
quantizer q . in this paper is dynamic. Indeed, Assumption 2 is not conservative and is crucial for employing the Razumikhin Lemma in the control design process [53, 6066]. Besides, Assumption 3 is not restrictive and it can be seen as a controllability condition, e.g. see [1-4, 6, 8, 11, 12, 26, 27, 29, 33, 34, 37, 39, 53, 67, 78]. In fact, from this last assumption, one can see that the control-gains matrix g 1 x S t , t is always regular and the control direction is unknown. Furthermore, Assumption 4 has been commonly employed in the literature [1, 56, 58, 62]. As pointed out in [33], the external disturbances and the states of the master system (in general, the states of uncontrolled chaotic systems) are always bounded. Remark 6.
Generally, a class k function . is continuous, strictly increasing
n and unbounded [53, 60-66, 88]. Then, 0 0 and ai i 1
for i 1,
n nai , ai 0 i 1
,n .
The objective of this research is to determine an adaptive neural controller to achieve the generalized function projective synchronization between the master system (1) and the slave one (2) under Assumptions 1-4, while all the signals in the derived closed-loop systems are bounded and the corresponding synchronization errors converge to zero.
Therefore, let us define the synchronization error variables between the two systems as follows
16
i t q 2 ,i y Si t i t q1,i y Mi t t x t t x ,p Si i Mi t , for i 1, di , p t p t 1 t , 2 t , T T T , dpT t d t d 1 t , d 2 t ,
where the functions i t , i 1, p
D
2 ,i
p
Mi
i 1
n
, p , are known, smooth and bounded, such that 5
t x t t i
i 1
(8)
i
i 1
6 with 6 being an unknown strictly
i
positive constant.
Remark 7.
In the synchronization literature, the scaling factors, the states of the
master system and their fractional derivatives are already assumed to be bounded [110, 26, 33]. From the above research works [63, 95, 96], one can deduce that D
2 ,i
t x t 12 t D i
Mi
i
2 ,i
x Mi t x Mi t D
2 ,i
i t , for i 1,
,p .
Remark 8. It is pointed out that the synchronization errors i t , i 1,
, p are
available for measurement according to Assumption 1. Besides, the synchronization error variables di t , i 1,
, p are immeasurable.
Remark 9. It is worth noting that the generalized function projective synchronization
problem of different incommensurate fractional-order chaotic systems is turned into [1-23, 26, 33]: a generalized function projective synchronization problem of different commensurate fractional-order chaotic systems if the fractional-orders
1,1 1,2
1,p and 2 ,1 2 ,2
2, p ,
a generalized function projective synchronization problem of integer-order chaotic systems, if 1,1 1,2
1,p 1 and 2 ,1 2 ,2
17
2, p 1 ,
a projective function synchronization problem if the scaling factors
1 t 2 t
p t ,
a complete synchronization problem if 1 t 2 t
p t 1 ,
an anti-phase synchronization problem if 1 t 2 t a chaos control problem, if 1 t 2 t
p t 1 ,
p t 0 .
On the other side, the quantization errors are given by
e i t y Si t q 2 ,i y Si t
(9)
y Mi t q1,i y Mi t i t , for i 1, with e t e1 t , e 2 t ,
, e p t T
p
,p .
Remark 10. As stated in [46-52, 67], the quantization error e i t is bounded, so p p p that e i t 1,i t 2 ,i t 6 . i 1 i 1 i 1
3. Controller Design and Stability Analysis
Under Assumptions 1-4, in order to solve the generalized projective synchronization
problem, we will use: smooth functions for describing the actuator nonlinearity (3), the mean value theorem to overcome the problems of unknown distributed time-varying delays and to transform the slave system (2) into an affine-like form in which the control input appears in a linear fashion, an adaptive observer to estimate the unmeasured synchronization error variables, a neural network to approximate the unknown nonlinear functions, a Nussbaum-type function to cope with the unknown control direction,
18
the Razumikhin Lemma, the frequency distributed model, the Lyapunov method and the Barbalat’s le
a to derive the adaptation control laws and to
perform the stability proof of the closed-loop system.
3.1 Approximation of unknown actuator nonlinearities Based on the previous research works [39, 52-59, 60, 67, 74, 84, 86, 88, 90-92] and by utilizing smooth functions, the unknown nonlinear input (3) can be approximated as follows i t 6 i T u i t 1,i t t t t t i 2 ,i i i t u i t 0 ,i i t 0 ,i t , for i 1,
(10)
,p
with
1,i t i 0 6 i T u i 0 exp i T u i t i T u i 0 5 sgn i T u i t
exp 5 sgn i T u i t
u 0 7 6 exp 5 sgn i T u i t d i T u i t
,
T
i
i
, if i t 0 , t 2 4 , if i t 0
2 ,i
4 3 , if 3 i t t t i T u i t , if 1 i t 3 , 2 1 , if i t 1
u u u u 0 ,i i t =u M erf i t , u M max min max min sgn i t , 2u M 2 2
i
i
i
2 erf i t 2u M i
2u M i 0
i t
exp 2 d
and 0 ,i t u i t 0 ,i i t , for i 1,
19
,p .
Remark 11. It is worthy to note that the functions t , 1,i t , 2,i t ,
0 ,i i t , u M , erf
2u M i
i
i t and 0,i t are bounded, for i 1,
,p
and satisfy p
p
t u 0 ,i
i 1
i 1
p
p
i 1
i 1
p
1,i t t
Mi
i 1
2 ,i t 0 ,i i t
erf i t 7 2u M i 1 i p
,
with the constant 7 being unknown and strictly positive.
Moreover, based on the mean value theorem [53, 67, 88], we can write
t f x , d t 3 ,i , j t 3 ,i , j S 3 ,i , j t f 3 ,i , j x S t 5 ,i , j t , t 5 ,i , j t , f 4 ,i x S t , i t f 4 ,i x S t , 0 g 1,i x S t , t i t , 1 ,i i t 1,i i t u i t 1,i t , for i 1, , p , j 1, , n i
(11)
with 0 5 ,i , j t 3,i , j t * ,
g 1,i x S t , 1,i t
f 4 ,i x S t , i t
i t
i t 1 ,i t
0 1,i 1 , 1,i i t t 6 i T
1,i t 2 ,i t
0 ,i i t i t
0 ,i i t i t
, 1,i t 1,i i t ,
0 ,i i t i t
i t 2 ,i t
,
t 1,i t 0 ,i t i t 2 ,i t
20
, i t 2 ,i t
0
0 ,i i t i t 2 ,i
1 , i t 2 ,i t
t 2 ,i i t
and 0 2 ,i 1 for j 1,
, ni , i 1,
,p .
Remark 12. The difficulty from the distributed time-varying delays can be processed by using the mean value theorem for integrals [53, 67, 88], as done in (11). Remark 13. It is worth mentioning that unlike the Taylor series expansion, which is
only valid locally at specified points, the Mean Value Theorem is valid globally [26, 53, 56, 67, 74, 78, 88]. Another important difference between the Taylor series expansion and the Mean Value Theorem is that by means of the Taylor series expansion, a smooth function would be expressed around one point with high-order terms (high-order approximating error), whereas, with help of the Mean Value Theorem, this function would be approximated between two points without highorder terms [56, 67, 74, 75, 88]. Consequently, by using (11), the slave system (2) can be rewritten as
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t 3 ,i , j t f 3 ,i , j x S t 5 ,i , j t , t 5 ,i , j t d i , j t , j 1, , n i 1, t 0 D 2 ,i x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t 3 ,i ,ni t f 3 ,i ,ni x S t 5 ,i ,ni t , t 5 ,i ,ni t Slave : f 4 ,i x S t , 0 g 2 ,i t u i t d i ,ni t 2 ,i t , t 0 x Si t 2 ,i t , u i t 0m i 1 , t 0 y t x t , for i 1, , p Si ,1 Si
21
(12)
where 2 ,i t 1,i t g 1,i x S t , 1,i t
and g 2 ,i t g 1,i x S t , 1,i t 1,i i t are bounded
for i 1,
,p .
3.3. Adaptive observer for the unmeasured synchronization error variables To estimate the unmeasured synchronization error variables, the following observer is considered
D 2 ,i ˆ di t Ai bi k ci ˆ di t k oi i t , ˆ i t c i T ˆ di t , ˆ i t i t i t , for i 1, , p where ˆ di t
ni
and ˆ i t
respectively, for i 1,
For i 1,
T
, p , k ci
Ai bi k ci
n i n i
are the estimates of di t and i t ,
,p .
0 ni 11 I ni 1 Ai 0 1 n i and c i 1, 01 ni 1
(13)
ni ni
, i 1,
ni
1ni
T
, bi 01 ni 1 , 1
,p .
and k oi
ni
and Ai k oi c i T
are chosen such that the matrices n i n i
are strictly Hurwitz.
On the other hand, one has ˆ t ˆ 1 t , ˆ 2 t ,
ˆ d t ˆ d 1T t , ˆ d 2T t ,
and t 1 t , 2 t ,
ni
, ˆ dpT t
, p t
T
T
p
22
n
.
, ˆ p t
T
p
,
Remark 14. Compared with the existing observers in [49, 51, 54, 76, 77], the proposed observer (13) has a simple linear form which is not related to control input and intelligent systems (neural networks or fuzzy logic systems).
According to [78-80, 88], k ci and k oi , i 1,
, p can be given in the following forms
T k C 1 , C 2 2 , , C nnii 1ni ni 1 ni 1 oi , for i 1, k ci C nnii 2 ni , C nnii 1 2 ni 1 , , C n1i 2
with C nji 1
and
2
ni ! , for j 1, j! n i j !
, ni , i 1,
,p
(14)
,p .
are known strictly positive constants.
ˆ t , for i 1, Let the observation errors be defined as di t di t di
, p . Then,
by using (8), (9) and (13), the dynamics of the observation errors can be written as D 2 ,i di t Ai k oi c i T di t bi g 2 ,i t u i t F0 ,i t T i t c i di t e i t , for i 1 , p
with
23
(15)
d i ,1 t d i , 2 t F0 ,i t k e t A i di t oi i d t i ,n i
f 2 ,i ,1 x S t , x S t 1,i ,1 t , t 2 ,i ,1 t f 2 ,i , 2 x S t , x S t 1,i , 2 t , t 2 ,i , 2 t f 2 ,i ,n x S t , x S t 1,i ,n t , t 2 ,i ,n t i i i
t t
3 ,i ,1 t f 3 ,i ,1 x S t 5 ,i ,1 t , t 5 ,i ,1 3 ,i , 2 t f 3 ,i , 2 x S t 5 ,i , 2 t , t 5 ,i , 2 3 ,i ,n t f 3 ,i ,n x S t 5 ,i ,n t , t 5 ,i ,n t i i i i 0 0 D 2 ,i i 0 0 ˆ f 4 ,i x S t , 0 2 ,i t k ci di t
t x Mi t , for i 1
,p.
From (13) and (15), one can get D
2 ,i
i t Ad i i t bd i g 2 ,i t u i t F1,i t , for i 1, , p
k e t with F1,i t oi i F t 0 ,i ˆ t i t di t di
for i 1,
2 ni
2 ni
Ai bi k ci , Ad i 0 n n i i
, bd i 01ni , bi T
T
,p .
24
2 ni
(16)
Ai k oi ci T k oi c i T
and cd i 01ni , c i T
2 n i 2 ni
T
,
2 ni
,
By the aid of the frequency distributed model [76, 77, 81, 82, 88], the dynamic system (16) can be rewritten as Z i ,t Z i ,t Ad i i t bd i g 2 ,i t u i t F1,i t t i t 0 i Z i ,t d , for i 1, , p sin 2 ,i 2 ,i with i and Z i ,t 2ni , for i 1,
(17)
,p .
Remark 15. It is noteworthy that thanks to auxiliary times and frequency domain functions, the fractional-order systems are transformed into its continuous frequency distributed equivalent models (equation (17)), in order to simplify the stability analysis [76, 77, 81, 82, 88].
Furthermore, since Adi is stable, thus, there exist constant symmetric positive definite matrices Pi Pi T 0 and Qi Qi T 0 , such that [3, 75-82, 88]
Adi T Pi Pi Adi Qi , for
i 1,
(18)
,p
3.3. Neural network for approximating uncertain functions By utilizing the Razumikhin lemma [53, 60-66, 88], it has been proven that there exists a constant 8 1 , such that x S t 1,i , j t 8 x S t x S t 5 ,i , j t 8 x S t , for j 1, , n i , i 1, , p
(19)
Remark 16. The Razumikhin condition (19) has been established in [53, 60-66, 88]
and will be employed in the procedure of adaptive controller design later. Indeed, from (3), (7) and (8), the following inequality holds
25
ni
t t x t d t p
2 ,i , j
i 1 j 1
M
p
p
p
i 1
i 1
i 1
k oi e i t 2 ,i t i t Ai x Mi t p
D
2 ,i
i 1
(20)
ni
t x t t t p
i
Mi
4 ,i , j
i 1 j 1
9
p p 8 where 9 2 2np i 2 k ci 2 k oi i 1 i 1 i
2
is an unknown strictly positive
constant.
Now, by using (5), (16), (19) and (20), one has p ni 7 p F1,i t 1 k,i, j i 1 j 1 k 1 i 1 p g 2 ,i t u i t 10 i 1 p 2 ,i t 10 i 1
t
with t 1T t , 2T t ,
10
10
t 10
(21)
, pT t T
2n
2
p ni 7 and 10 2 2 9 k,i, j 2 9 . i 1 j 1 k 1
Let us denote
p
t P T
F2 t
p ni 7 1 k,i, j t 9 9 t 2 2 t t i 1 j 1 k 1 i
i 1
i
9 (22)
p
9 i T t Pi bd i i t i 1
t 2 t 2
26
where the smooth function t is known, strictly positive and bounded such that
d is bounded, t 0 . t
0
By using (21) and (22), it is easy to obtain that p
i 1
i T t Pi F1,i t
p
g 2 ,i t u i t i T t Pi bd i i t i 1
(23)
t t F2 t
Thanks to the universal approximation theorem [2-5, 8, 14, 15, 53, 56, 61, 78-80, 88], the unknown function F2 t can be approximated, on the sufficiently large
compact set t
t M
2n
2n
, by a three-layer neural network as
follows F2 t w *T t
(24)
where t ˆ d T t , T t T
np
is the input of the neural network. The constant
M is unknown and strictly positive.
, is unknown and bounded over the
sufficiently large compact set , i.e., * , with the constant * being unknown and strictly positive. t
, is the neural activation function with
1 being the
number of the neural network nodes. Then, the ideal constant neural network weight vector w *
, can be defined as
follows [3, 5, 8, 14, 15, 88]
w * arg min sup F2 t w T t w* t
27
(25)
In general, w * , is unknown, constant and bounded. Besides, t is known, and bounded [3, 5, 8, 14, 15, 53, 56, 61]. Remark 17.
Similar to [3, 5, 8, 14, 15, 53, 56, 61, 88], the neural network
w T t used in this research for approximating the unknown function F2 t has a simple structure, consisting of an input layer, a single output layer with one neuron, and a single hidden layer with
neurons. As stated in [5, 8, 14, 15, 53, 56,
61], the hidden-layer activation function may be a radial basis function, a logistic sigmoid function, or a hyperbolic tangent function. Furthermore, the output-layer activation function is a linear function. To alleviate the burden and complexity of computation, only the weight vector between the hidden layer and the output layer is unknown and must be estimated in controller design, while the weight matrix between the hidden layer and the input layer is randomly initialized and fixed. As pointed out in [3, 5, 8, 14, 15, 53, 56, 61], radial basis function (RBF) neural networks are generally used in practical control engineering thanks to their satisfactory approximation properties and simple architectures.
Over the sufficiently large compact set , there exist an unknown constants 11 and * such that
w
t 1 2
0 F2 t w * * *
2
*
(26)
10
* * t 2
11 with t 11 and * w * * k 2 . k 1
And t t t 1 is a known function.
Remark 18. Similar to [53, 61, 75], the Neural Network parameterization (26) will be used in the design procedure of the controller in order to reduce the burden of
28
computation and complexity. On the other hand, the neural network can be employed to avoid the global Lipschitz assumption for unknown nonlinear functions [75].
3.4. Adaptive control law To achieve synchronization between the master (1) and the slave systems (2), the adaptive control law is constructed as follows u t N t ˆ t 1 t t 2 ˆ t 1 t t t ˆ t 2 t ˆ t 1 t t 2 t 1 t 2 2 t 2 t t
(27)
where N t cos t exp 2 t is a Nussbaum function [78,79, 83, 84, 2 88]. The constant is known and strictly positive. ˆ t is the estimate of the unknown positive constant * .
Remark 19. Similar to [53, 61, 75, 88], it should be selected ˆ 0 0 in such a way that the inequality ˆ t 0 holds t 0 .
Remark 20. As pointed out in [78, 79, 83, 84, 88], commonly used Nussbaum functions are : 2 t cos t , 2 t sin t , t cos and cos t exp 2 t . 2 29
t
3.5. Stability analysis In this stage, we summarize the main results of the proposed controller by the following theorem.
Theorem 1. Let us consider the master-slave systems (1) and (2), satisfying Assumptions 1–4. Then, for any bounded initial conditions, the adaptive observer (13) and the control law (27) guarantee the following properties: All signals in the closed-loop system are semi-globally uniformly ultimately bounded, i.e., ˆ t , t , u t , t and t L . The synchronization error variables semi-globally converge to zero, i.e., d t 0 and t 0 , as t .
Proof of Theorem 1 Now, let us consider the following Lyapunov function candidate p
V t V i t i 1
with V i t
1 2 t 2
(28)
1 i Z i T , t Pi Z i , t d , for i 1, 0 2
,p
and t * ˆ t .
From (17), it is easy to show that p
i Z i T ,t Pi Z i ,t d 0 i 1
0
(29)
Based on Rayleigh inequality [93] and by utilizing the equations ((18), (23) and (29)), the time derivative of V t along the solutions of (17) can be expressed in the following form
30
V t
min Q 2
t t F2 t
t t
(30)
t F2 t T t g 2 t u t with Q diag Q1 ,
, Qp
pp
and g 2 t diag g 2 ,1 t , g 2 ,2 t ,
, , , g 2 , p t
p m
.
Using the neural network parameterization (26) and the control law (27), (30) becomes
min Q t t * t t * 2 2 t t t t ˆ t t
V t
1
(31)
N t ˆ t 1 t T t g 2 t t
According to [83, 85-88], one can conclude that 2 T t g 2 t t t t 2 * * * t t 1 t t t t ˆ t t t 1 2 t t 1 *2 2 2 where t is a bounded function satisfying t 0, t 0 .
(32)
By substituting (32) into (31) leads to V t
min Q 2
t 1 N t t t
(33)
1 2 t t *2 2 *
31
After integrating (33) over 0, t , one obtain V t
min Q t 2 * 1 *2 t d V 0 2 0 d 2 0 2
1 N d t
0
(34)
Similar to the discussion in [83, 85-88] and according to (34), one can easily check Q t 1 t that V t , min d , 2 * *2 d , V 0 0 2 2 0 and
1 N d
are bounded t 0, .
t
0
This fact implies that t L2 .
From (16), since the functions f 1,i , j x M t , x M t 4 ,i , j t
,
f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t , f 3,i , j x S t , t and f 4,i x S t , i t
are smooth functions,
and the variables d i t , e i t , D bounded, then, one has D
2,i
2 ,i
t x t , t , i
Mi
i t L , for j 1,
Finally, since t L2 L and D that lim t 0
2,i
2 ,i
, ni , i 1,
i t L for i 1,
urther ore, by using arbalat’s le
t
,
i t
are
,p .
, p , one can deduce
a [78, 83, 85-88], it
follows that the synchronization error variables converge to zero, i.e., lim d t 0 , lim ˆd t 0 and lim t 0 because d t 2 t , t
t
t
p
p
i 1
ˆd t 2 t , t 1,i t 2,i t 2 6 2 t i 1
p
p
i 1
i 1
and lim 1,i t 2,i t 0 . t
The proof is completed here. 32
Remark
21.
Since
the
well-known
Leibniz
and
chain
rules
D f t g t D f t g t f t D g t
and D f g t
f x D g t are not satisfied for fractional derivatives, x x g t
the classical stability tools for integer order systems cannot be directly applicable to non-integer ones [1-33, 43, 59, 63-66, 69-73, 76, 77, 81, 82, 87-95]. Up to now, the stability analysis of fractional-order systems has attracted wide attention, (see, e.g. [63-66, 69-73, 76, 77, 81, 82, 87-91]). For example in [63-66, 69-73], stability theorems for nonlinear fractional-order systems have been obtained by using Volterratype Lyapunov functions and Caputo fractional derivative. In [63-66], some stability analyses for fractional-order nonlinear systems have been performed based on MittagLeffler function, Laplace transform, and the generalized Gronwall inequality. In [5, 13, 76, 77, 81, 82, 87-89], the stability of fractional-order systems with delays has been studied by utilizing Caputo fractional derivative, Lyapunov approaches, Razumikhin-type stability theorems and Lyapunov-Krasovskii approaches.
Remark 22. Similar to most research works on neural network and fuzzy-based adaptive control schemes [2-5, 8, 14, 15, 33, 39, 49, 50, 53, 56, 58, 61, 74, 78-80, 8385, 87-89, 92, 93], the stability results obtained in this work are semi-global which means that the stability results are valid as much as the master-slave system states remain within some compact sets. 3.6. Practical implementation
To summarize the above analysis, Figure 1 demonstrates the overall scheme of the proposed synchronization in this research, which is consisting of an observer (13) and an adaptive control law (27). To further illustrate the implementation of the proposed controller, Figure 2 presents an algorithmic description based on a flowchart. It follows from Figures 1-2, that the proposed controller is running on-line without any offline-learning phase similar to the previous controllers in [1-22, 26, 33, 78-89].
33
From the previous discussions, the design algorithm of the proposed synchronization scheme is presented step-by-step as follows: Step 1: Choose initial conditions ˆ d 0 , ˆ 0 , 0 , a neural activation function
t , appropriate function t , strictly positive constants 1 ,
2
,
and a learning gain used in the observer (13) and the control law (27). Step 2: Compute t and obtain t by using (13). Step 3: Calculate the control input u t according to (27). Step 4: Use the control input u t for the master-slave systems (1) and (2) under Assumption 1-4. Step 5: Measure the quantized outputs of the master and slave systems q1 y M t and q 2 y S t .
Step 6: Go to Step 2.
34
t
ˆ d t
Proposed neural adaptive control law Equation (27)
u t
x S t
Actuator nonlinearity Equation (3)
Master system (1)
y M t
t
Slave system (2), satisfying Assumptions 1-4
q 2 y S t Quantizer Equation (4)
q1 y M t
y S t
d t Synchronization error variables
x M t
Equation (8)
t +
Observer (13)
ˆ d t
To estimate the unmeasured synchronization error variables
Figure 1. Proposed synchronization scheme
35
-
ˆ t
t
Start t 0
Set initial condition and parameters Go to next sample interval Calculate t and obtain
t by using (13) Determine the control input u t according to (27) Use the control input u t for the master-slave systems (1) and (2) under Assumptions 1-4 Measure the quantized outputs of the master and slave systems q1 y M t and q 2 y S t
Figure 2. Flowchart of implementation of the proposed controller
36
3.7. Comparison to available methods
In comparison to some available control schemes, the most important advantages of our work are that: unlike the closely related works [1-4, 6, 8, 9, 12, 14, 26, 27, 29, 33, 34, 37], the problems of the actuator nonlinearities, the quantized output and the unmeasured master-slave states of nonlinear systems are resolved, there is no filtering of the neural activation functions and the SPR condition imposed in previous works [80, 83, 85, 88] is avoided. In addition, the SPRLyapunov design approach and the SPR-filter approach are not used, the requirement that the sign of control gains are known or satisfied the matching condition [1-4, 6, 8, 9, 12, 14, 26] is removed,
unlike the previous works [2, 4, 26, 33, 74], the synchronization errors semiglobally converge to zero, the considered class of slave systems is relatively large in the sense that the unknown nonlinear functions are not affine and could depend on the full state vector, the time-varying delays and the actuator nonlinearities, unlike in [1-12, 14-22, 24-27, 29-41, 43, 55, 89], unlike in the existing literature [1-18, 22, 24, 25, 27, 28, 29, 39-41, 45], the proposed control has a simple structure in such a manner that the number of tuning variables is reduced.
4. Simulation examples
In this section, two examples are used to evaluate the performance of the proposed projective function synchronization scheme. 37
Example 1: (an academic MIMO fractional-order chaotic system) Let us consider two different uncertain MIMO fractional-order Duffing–Holmes chaotic time-delay systems (master and slave systems) as follows [89]: Master system
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 x Mi t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, 2 , i 1, 2
(35)
Slave system
D 2 ,i x Si ,1 t f 2 ,i ,1 x S t , x S t 1,i ,1 t , t 2 ,i ,1 t d i ,1 t t t 0 t 3,i ,1t f 3,i ,1 x S , d , D 2 ,i x t f Si , 2 2 ,i , 2 x S t , x S t 1,i , 2 t , t 2 ,i , 2 t d i , 2 t t f x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,2 t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 2 ,i i Si y Si t x Si ,1 t , for i 1, 2
(36)
where: 1,1 0.98 , 1,2 0.94 , 2 ,1 0.94 incommensurate orders, 38
and 2 ,2 0.98 are the fractional
x M t x M 1,1 t , x M 1,2 t , x M 2 ,1 t , x M 2 ,2 t T
4
and x S t x S 1,1 t , x S 1,2T t , x S 2 ,1 t , x S 2 ,2 t T
4
are the states
of the master and slave systems, respectively ,
y M t y M 1 t , y M 2 t T
2
and y S t y S 1 t , y S 2 t T
2
are the outputs of the master and slave systems, respectively,
d 1,1 t d 1,2 t 0.01sin t and d 2 ,1 t d 2 ,2 t 0.01cos t are the external bounded disturbances,
1,1 t 1,2 t 2 ,1 t 0, 0 T
2
and 2 ,2 t 1, 1 T
2
are the initial
condition functions,
1,1,1 t 2 ,1,1 t 3,1,1 t 4 ,1,1 t 0 ,
1,2 ,1 t 2 ,2 ,1 t 3,2 ,1 t 4 ,2 ,1 t 0 ,
1,1,2 t 2 ,1,2 t 3,1,2 t 4 ,1,2 t 0.01 0.02sin t , and 1,2 ,2 t 2 ,2 ,2 t 3,2 ,2 t 4 ,2 ,2 t 0.01 0.01sin t , are timevarying delays,
f 1,1,1 x M t , x M t 4 ,1,1 t 2.5x M 1,2 t ,
f 1,1,2 x M t , x M t 4 ,1,2 t
3
1 x M 1,1 t 0.01x M 1,2 2 t 4 ,1,2 t 2.5 0.1x M 1,2 t 0.01x M 1,1 t 4 ,1,2 t
1 x M 1,1 t 0.01x M 1,12 t 4 ,1,2 t 25cos 1.29 t 2.5
39
,
f 1,2 ,1 x M t , x M t 4 ,2 ,1 t 2.7x M 2 ,2 t ,
f 1,2 ,2 x M t , x M t 4 ,2 ,2 t
3
1 x M 2 ,1 t 0.01x M 2 ,2 2 t 4 ,2 ,2 t 2.7 0.1x M 2 ,2 t 0.01x M 1,1 t 4 ,2 ,2 t
,
1 x M 2 ,1 t 0.01x M 2 ,12 t 4 ,2 ,2 t 25cos 1.28t 2.7
f 2 ,1,1 x S t , x S t 1,1,1 t , t 2 ,1,1 t 2.8x S 1,2 t , f 3,1,1 x S t , t 0.001 t 0.002x S 1,2 t ,
f 2 ,1,2 x S t , x S t 1,1,2 t , t 2 ,1,2 t
3
1 x S 1,1 t 0.01x S 1,2 2 t 1,1,2 t 2.8 0.1x S 1,2 t 0.01x S 1,1 t 1,1,2 t
1 0.001 t 2 ,1,2 t x S 1,1 t 2.8 2 0.01x S 1,1 t 1,1,2 t 24cos 1.5t
,
f 3,1,2 x S t , t 0.001 t 0.001x S 1,2 t ,
f 4 ,1 x S t , t tanh 0.001 x S t 1 t , 2
f 2 ,2 ,1 x S t , x S t 1,2 ,1 t , t 2 ,2 ,1 t 2.7x S 2 ,2 t , f 3,2 ,1 x S t , t 0.001 t 0.003x S 2 ,2 t ,
f 2 ,2 ,2 x S t , x S t 1,2 ,2 t , t 2 ,2 ,2 t
3
1 x S 2 ,1 t 0.01x S 2 ,2 2 t 1,2 ,2 t 2.7 0.1x S 2 ,2 t 0.01x S 1,1 t 1,2 ,2 t
1 0.002 t 2 ,2 ,2 t x S 2 ,1 t 2.7 2 0.01x S 2 ,1 t 1,2 ,2 t 24cos 1.9 t
,
f 3,2 ,2 x S t , t 0.002 t 0.001 x S t ,
and f 4 ,2 x S t , t tanh 0.002 x S t 2 t are smooth functions.
u t u1 t , u 2 t T
2
2
is the control input of the slave system,
40
t 1 t , 2 t T
2
is the output of the actuator nonlinearity defined
in (3) with u min 10 , l 1 , 2 1.6 , 3 3 , 4 1.8 , 5 0.3 , 6 1 ,
7 0.1, u max 12 and 1 2 1 .
Figures 3 and 4 show the two-dimensional projections of the phase portrait of master system on x M 1,1 t , x M 1,2 t and x M 2 ,1 t , x M 2 ,2 t -planes, respectively.
Besides, Figures 5 and 6 describe the two-dimensional projections of the phase portrait of slave system without control input on
x t , S 2 ,1
x t , S 1,1
x S 1,2 t and
x S 2 ,2 t -planes, respectively
Figure 3. Two-dimensional projection of the phase portrait of master system on x M 1,1 t , x M 1,2 t -plane
41
Figure 4. Two-dimensional projection of the phase portrait of master system on x M 2 ,1 t , x M 2 ,2 t -plane
Figure 5. Two-dimensional projection of the phase portrait of slave system without control input on x S 1,1 t , x S 1,2 t -plane
42
Figure 6. Two-dimensional projection of the phase portrait of slave system without control input on x S 2 ,1 t , x S 2 ,2 t -plane
Our objective consists in designing a neural adaptive control achieving a generalized function projective synchronization between the two different fractional-order chaotic systems (35) and (36), such that: the
master
and
slave
systems
satisfy
Assumptions
1-4
with
1,1 t 1,2 t 2 ,1 t 2 ,2 t exp 0.4t , the scaling factors are given by 1 t sin t and 2 t cos t . According to section 3, the design of the proposed neural adaptive controller is realized based on the following four steps: Step 1: Choose the design parameters employed in the observer (13) and the control
law
(27)
as
follows
1
10 ,
2
8,
100
and t exp 2t . Step 2: Determine the vector t and t by using (13). Step 3: : For approximating the nonlinear unknown functions, select a radial basis function (RBF) neural network w *T t containing 40 nodes with 43
centers evenly distributed in the interval t 40, 40 40, 40 40, 40 40, 40 40, 40 40, 40
and all the widths being equal to 70 . Step 4: Calculate the control input u t by using (27).
The computer simulation results obtained by applying the proposed controller, are presented in Figures 7-12 with the initial conditions ˆ 0 0 , 0 1.2 T and ˆ d 0 0, 0, 0 , 0 .
From Figures 7 and 8, it can be clearly seen that the signals q 2 ,1 y S 1 t and q 2 ,2 y S 2 t
satisfactorily
track
the
signals
1 t q1,1 y M 1 t
and 2 t q1,2 y M 2 t , respectively. Figures 9 and 10 exhibit the control input
u t and the output of the actuator nonlinearities t . Figures 11 and 12 show that the Euclidean norms of synchronization error vectors t and ˆ d t quickly converge to zero.
As one can see, the proposed neural controller is able to synchronize successfully two time-delay incommensurate fractional-order chaotic systems with unknown actuator nonlinearities, i.e., a fast synchronization can be carried out. In addition, it is easy to confirm the robustness of the proposed neural adaptive control scheme with respect to uncertain dynamics, disturbances, and unknown actuator nonlinearities, by these simulation results (Figures 7-12).
44
Figure 7. Signals q 2 ,1 y S 1 t and 1 t q1,1 y M 1 t
Figure 8. Signals q 2 ,2 y S 2 t and 2 t q1,2 y M 2 t
45
Figure 9. Output of the actuator nonlinearity 1 t and control input u1 t
Figure 10. Output of the actuator nonlinearity 2 t and control input u 2 t
Figure 11. Euclidean norm of synchronization error vector t 46
Figure 12. Euclidean norm of synchronization error vector ˆ d t
Example 2: (a physical MIMO fractional-order chaotic system)
Consider two MIMO time-delay chaotic satellite systems (master and slave systems) described by [39]:
Master system
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 x Mi , j t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, , n i , i 1, , 3
47
(37)
Slave system
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t d i , j t t f x S , d , j 1, , n i 1, t 0 t 3 ,i , j t 3 ,i , j 2 ,i D x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t d i ,ni t t f x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,ni t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 S i 2 ,i i m i 1 y Si t x Si ,1 t , for i 1, , p
(38)
where: n1 2 , n 2 3 , n3 4 and p 3 are constants, 1,1 2 ,1
1 , 2
1,2 2 ,2
1 3
1,3 2 ,3
and
1 4
are
the
fractional
incommensurate orders, x M t x M 1,1 t , x M 1,2 t , x M 2 ,1 t , x M 2 ,2 t , x M 2 ,3 t , x M 3,1 t , x M 3,2 t , x M 3,3 t , x M 3,4 t
9
and x S t x S 1,1 t , x S 1,2 t , x S 2,1 t , x S 2,2 t , x S 2 ,3 t , x S 3,1 t , x S 3,2 t , x S 3,3 t , x S 3,4 t
9
T
T
are the states of the master and slave systems, respectively ,
y M t y M 1 t , y M 2 t , y M 3 t T
3
and y S t y S 1 t , y S 2 t , y S 3 t T
3
are the outputs of the master and slave systems, respectively,
48
1,1 t 10, 0 T
2 ,1 t 2, 0 T
2
, 1,2 t 3, 0, 0 T
2
, 2 ,2 t 1, 0, 0 T
and 2 ,3 t 5, 0, 0, 0 T
4
3
, 1,3 t 6, 0, 0, 0 T
4
3
are the initial condition functions,
1,1,1 t 2 ,1,1 t 3,1,1 t 4 ,1,1 t 0.01sin t ,
1,2 ,1 t 2 ,2 ,1 t 3,2 ,1 t 4 ,2 ,1 t 0.02cos t , t
1,2 ,2 t 2 ,2 ,2 t 3,2 ,2 t 4 ,2 ,2 t 0.01sin , 2 t
1,3,1 t 2 ,3,1 t 3 ,3 ,1 t 4 ,3 ,1 t 0.01sin , 4
1,3,2 t 2 ,3,2 t 3,3,2 t 4 ,3,2 t 0.001exp t , t
1,3,3 t 2 ,3,3 t 3 ,3 ,3 t 4 ,3 ,3 t 0.002sin , 8
1,1,2 t 2 ,1,2 t 3,1,2 t 4 ,1,2 t 0.01 0.02sin t , t
1,2 ,3 t 2 ,2 ,3 t 3,2 ,3 t 4 ,2 ,3 t 0.02 0.01sin 2 and 1,3,4 t 2 ,3,4 t 3,3,4 t 4 ,3,4 t 0.01 0.01sin t , are timevarying delays,
49
,
1 d 1,1 t 2exp 100 x S t 1,1,1 t t 2 ,1,1 t 1 exp t 3 ,1 ,1 t 100 x S
2
t
d
100 x S t d 1,2 t 2exp 1 x S t 1,1,2 t 2 t 2 ,1,2 t 100 x S t 2 exp d t 3 ,1 ,2 t 1 2 x S x S 1,1 t x S 2 ,1 t 0.4x S 3 ,1 t
0.01sin t ,
0.02sin t ,
d 2 ,1 t 0.01cos t 2exp x S t 1,2 ,1 t 2exp t 2 ,2 ,1 t 2
t
t 3 ,2 ,1 t
exp x
S
exp d
d
,
d 2 ,2 t 0.02cos t 2exp x S t 1,2 ,2 t 2exp t 2 ,2 ,2 t 2
t
t 3 ,2 ,2 t
exp 2 10 x
S
d 2 ,3 t 0.03cos t 2exp 4 t 2 ,2 ,3 t 3 x S t 1,2 ,3 t 2
t
t 3 ,2 ,3 t
exp 4 3 x
S
,
d
,
0.1 x S 1,1 t 0.46 x S 2 ,1 t 2x S 3 ,1 t 2
2 t d 3 ,1 t 0.02cos 2exp 1 x S t 1,3 ,1 t t 2 ,3 ,1 t 2 2 exp t 3 ,3 ,1 t 1 x S
2
t
50
d
,
4 t d 3 ,2 t 0.03cos 2exp 1 x S t 1 , 3 , 2 t t 2 , 3 , 2 t 2 4 2 exp t 3 ,3 ,2 t 1 x S t
,
,
d
5 t d 3 ,3 t 0.04cos 2exp 2 x S t 1 , 3 , 3 t t 2 , 3 , 3 t 2 t 5 2 exp t 3 ,3 ,3 t 2 x S
d
and
7 t d 3 ,4 t 0.05cos 2exp 5 x S t 1,3 ,4 t t 2 ,3 ,4 t 2 7 exp t 3 ,3 ,3 t 5 x S
2
t
d
2 1 0.87 x S 1,1 t 3x S 2 ,1 t 7x S 3 ,1 t 4
are
the external bounded disturbances,
f 1,1,1 x M t , x M t 4 ,1,1 t x M 1,2 t ,
1 6 f 1,1,2 x M t , x M t 4 ,1,2 t x M 2 ,1 t x M 3,1 t 0.4x M 1,1 t x M 3,1 t , 3 6
f 1,2 ,1 x M t , x M t 4 ,2 ,1 t x M 2 ,2 t ,
f 1,2 ,2 x M t , x M t 4 ,2 ,2 t x M 2 ,3 t , f 1,2 ,3 x M t , x M t 4 ,2 ,3 t x M 1,1 t x M 3,1 t 0.175x M 2 ,1 t ,
f 1,3,1 x M t , x M t 4 ,3,1 t x M 3,2 t , f 1,3,2 x M t , x M t 4 ,3,2 t x M 3,3 t , f 1,3,3 x M t , x M t 4 ,3,3 t x M 3,4 t , f 1,3,4 x M t , x M t 4 ,3,4 t x M 1,1 t x M 2 ,1 t 6x M 1,1 t 0.4x M 3,1 t , 51
f 2 ,1,1 x S t , x S t 1,1,1 t , t 2 ,1,1 t
1 x S 1,2 t exp 100 x S t 1,1,1 t t 2 ,1,1 t
1 f 3 ,1,1 x S t , t exp 100 t x S t
f 2 ,1,2 x S t ,x S t 1,1,2 t , t 2 ,1,2 t
tanh 0.001 x S t
2
,
100 x S t exp 1 x S t 1,1,2 t 2 t 2 ,1,2 t
100 x S t f 3 ,1,2 x S t , t exp 1 2 t x S t
,
1 f 4 ,1 x S t , t x S 2 ,1 t x S 3 ,1 t 0.4x S 1,1 t 3
2 6 x S 3 ,1 t tanh 0.001 x S t 1 t 6
,
f 2 ,2 ,1 x S t , x S t 1,2 ,1 t , t 2 ,2 ,1 t x S 2 ,2 t exp x S t 1,2 ,1 t
exp t 2 ,2 ,1 t
f 3,2 ,1 x S t , t exp x S t exp t ,
f 2 ,2 ,2 x S t ,x S t 1,2 ,2 t , t 2 ,2 ,2 t x S 2 ,3 t exp x S t 1,2 ,2 t
exp t 2 ,2 ,2 t
f 3,2 ,2 x S t , t exp 2 t 10 x S t ,
f 2 ,2 ,3 x S t , x S t 1,2 ,3 t , t 2 ,2 ,3 t
x S 1,1 t x S 3 ,1 t 0.175x S 2 ,1 t
,
exp 4 t 2 ,2 ,3 t 3 x S t 1,2 ,3 t
f 3,2 ,3 x S t , t exp 4 t 3 x S t , f 4 ,2 x S t , t x S 1,1 t x S 2 ,1 t 0.175x S 2 ,1 t
tanh 0.004 x S t 2 t 2
52
,
,
,
,
,
f 2 ,3 ,1 x S t , x S t 1,3 ,1 t , t 2 ,3 ,1 t
2 x S 3 ,2 t exp 1 x S t 1,3 ,1 t t 2 ,3 ,1 t
2 f 3 ,3 ,1 x S t , t exp 1 t x S t
f 2 ,3 ,2 x S t , x S t 1,3 , 2 t , t 2 ,3, 2 t
,
4 x S 3 ,3 t exp 1 x S t 1,3 ,2 t t 2 ,3 ,2 t
4 f 3 ,3 ,2 x S t , t exp 1 t x S t
f 2 ,3 ,3 x S t , x S t 1,3 ,3 t , t 2 ,3 ,3 t
5 f 3 ,3 ,3 x S t , t exp 2 t x S t f 2 ,3 ,4 x S t , x S t 1,3 ,4 t , t 2 ,3 ,4 t
,
7 exp 5 x S t 1 , 3 , 4 t t 2 , 3 , 4 t
tanh 0.004 x S t
2
7 f 3 ,3 ,4 x S t , t exp 5 t x S t
,
and f 4 ,3 x S t , t x S 1,1 t x S 2 ,1 t 0.175x S 2 ,1 t
tanh 0.004 x S t 3 t 2
are smooth functions. 53
,
,
5 x S 3 ,4 t exp 2 x S t 1 , 3 , 3 t t 2 , 3 , 3 t
,
,
,
u t u1 t , u 2 t , u 3 t T
t 1 t , 2 t , 3 t T
3
3
is the control input of the slave system,
is the output of the actuator nonlinearity
defined in (3) with u min 5 , l 1 , 2 1.6 , 3 3 , 4 1.8 , 5 0.3 ,
6 1 , 7 0.1 , u max 6 and 1 2 1 .
Our main objective consists in determining a neural adaptive control achieving a generalized function projective synchronization between the two MIMO time-delay chaotic satellite systems (37) and (38), such that: the
master
and
slave
systems
satisfy
Assumptions
1-4
with
1,1 t exp 0.2t , 1,2 t 1,3 t exp 0.22t , 2 ,1 t exp 0.25t and 2 ,2 t 2 ,3 t exp 0.26t , the scaling factors are given by 1 t 1 sin t , 2 t cos t sin t and 3 t 1 sin t .
According to section 3, the design of the proposed neural adaptive controller is carried out based on the following four steps: Step 1: Select the design parameters employed in the observer (13) and the control
law
(27)
as
follows
1
12 ,
2
10 ,
200
and t exp 2t . Step 2: Determine the vector t and t by using (13). Step 3: For approximating the nonlinear unknown functions, select a radial basis function (RBF) neural network w *T t containing 45 nodes with centers evenly distributed in the interval 54
t 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50
and all the widths being equal to 80 . Step 4: Calculate the control input u t by using (27).
The Simulation results obtained by applying the proposed controller, are given in Figures
13-20
with
the
initial
conditions
ˆ 0 0
,
0 1.07
T and ˆ d 0 0, 0, 0, 0, 0 , 0 , 0 , 0 , 0 .
From Figures 13-15, it is clear that the signals q 2 ,1 y S 1 t , q 2 ,2 y S 2 t and
q 2 ,3 y S 3 t satisfactorily track the signals 1 t q1,1 y M 1 t , 2 t q1,2 y M 2 t , and 3 t q1,3 y M 3 t respectively. Figures 16-18 depict the control input u t and the output of the actuator nonlinearities t . Figures 19 and 20 demonstrate that the Euclidean norms of synchronization error vectors t
and ˆ d t
quickly
converge to zero.
As seen in these figures, the proposed neural controller is capable to synchronize successfully two MIMO time-delay chaotic satellite systems with unknown actuator nonlinearities, i.e., a fast synchronization can be carried out. Furthermore, it is easy to confirm the robustness of the proposed neural adaptive control scheme with respect to uncertain dynamics, disturbances, and unknown actuator nonlinearities, by these simulation results (Figures 13-20). In summary, we can conclude that the performance of the proposed controller is validated through this physical system (two MIMO time-delay chaotic satellite systems).
55
Figure 13. Signals q 2 ,1 y S 1 t and 1 t q1,1 y M 1 t
Figure 14. Signals q 2 ,2 y S 2 t and 2 t q1,2 y M 2 t
Figure 15. Signals q 2 ,3 y S 3 t and 3 t q1,3 y M 3 t 56
Figure 16. Output of the actuator nonlinearity 1 t and control input u1 t
Figure 17. Output of the actuator nonlinearity 2 t and control input u 2 t
Figure 18. Output of the actuator nonlinearity 3 t and control input u 3 t 57
Figure 19. Euclidean norm of synchronization error vector t
Figure 20. Euclidean norm of synchronization error vector ˆ d t
5. Conclusions This paper has addressed the generalized function projective synchronization of MIMO uncertain time-delay incommensurate fractional-order chaotic systems with unknown input nonlinearities including backlash-like hysteresis, dead-zone and asymmetric saturation actuators. The main technical difficulties in this research have come from the existence of unknown nonlinear system functions, unmeasured masterslave system states, unknown distributed time-varying delays, unknown time-varying delays, quantized output, unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct 58
fractional-orders. A novel combination of a linear observer, a neural network, a Nussbaum function, the mean value theorem, the Lyapunov-Razumikhin method and a frequency-distributed model, has been employed to cope with these difficulties under some suitable assumptions. The main advantages of the proposed control design are summarized as follows: (1) the number of the adjustable parameters is reduced, (2) all signals of the closed-loop system remain semi-globally uniformly ultimately bounded and the synchronization errors semi-globally converge to zero. The effectiveness of the proposed controller has been demonstrated throughout simulation examples. In the future, we will study the fuzzy adaptive synchronization problem for a class of incommensurate uncertain non-integer-order chaotic systems with input nonlinearities and faults based on the results of this research work.
6. Acknowledgments The authors would like to express their sincere gratitude and appreciation to the anonymous Reviewers, Associate-Editors and Editors for their insightful suggestions and comments, and for their efforts and time spent in helping us to improve the presentation and quality of the present research work. The third author is grateful to the Spanish Ministry of Economy and Competitiveness (MINECO) for its financial support through grant no. DPI2013-47825-C3-1-R. 7. Conflict of interest statement Authors declare no conflict of interests. REFERENCES [1] S.K. Agrawal, M. Srivastava, S. Das, “Synchronization of fractional order chaotic systems using active control method,” Chaos, Solitons & Fractals, Volume 45, Issue 6, pp. 737–752, 2012. [2] A. Bouzeriba, A. Boulkroune , T. Bouden, S. Vaidyanathan, “Fuzzy Adaptive Synchronization of Incommensurate Fractional-Order Chaotic Systems,” Advances and Applications in Chaotic Systems, Volume 636, pp. 363-378, 2016.
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Farouk ZOUARI was born in Tunis, Tunisia, on August 27, 1980. He received his Engineer degree in Electrical Engineering, his magister degree in Automatic and Signal Processing, and his PhD degree in Electrical Engineering from the "National Engineering School of Tunis, University of Tunis El Manar, Tunisia, in 2004, 2005 and 2014, respectively. He is currently a researcher at aboratoire de Recherche en
uto atique (
R ), École Nationale d’Ingénieurs de
Tunis, Université de Tunis El Manar. His current research interests include singular systems, neural control theory, nonlinear control, and intelligent adaptive control.
Abdesselem Boulkroune received his Engineering degree from Setif University in 1995, his Master grade from the military polytechnic school (EMP) of Algiers in 2002, his PhD degree in Automatic from the national polytechnic school (ENP) of Algiers in 2009, in Algeria. In 2003, he joined the automatic control department at Jijel University, in Algeria, where he is currently a professor. His research interests are in nonlinear control and adaptive control.
70
Asier Ibeas was born in Bilbao, Spain, on July 7, 1977. He received his MSc degree in Applied Physics and his PhD degree in Automatic Control from the University of the Basque Country, Spain, in 2000 and 2006, respectively. He is currently Associate Professor of Control Systems at Autonomous University of Barcelona, Spain. His research interests include time-delayed systems, robust adaptive control, applications of artificial intelligence to control systems design and nonconventional applications of control such as to epidemic systems, supply chain management and financial systems, fields where he has published more than 130 contributions in international journals and conferences.
71
PII: DOI: Reference:
S0925-2312(16)31437-0 http://dx.doi.org/10.1016/j.neucom.2016.11.036 NEUCOM17792
To appear in: Neurocomputing Received date: 13 July 2016 Revised date: 27 September 2016 Accepted date: 19 November 2016 Cite this article as: Farouk Zouari, Abdesselem Boulkroune and Asier Ibeas, Neural Adaptive quantized output-feedback control- based synchronization of uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.11.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Neural Adaptive quantized output-feedback control- based synchronization of uncertain time-delay incommensurate fractionalorder chaotic systems with input nonlinearities Farouk Zouari1*, Abdesselem Boulkroune2, Asier Ibeas3,4 1
Laboratoire de Recherche en Automatique (LARA), École Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, BP. 37, Le Belvédère, 1002 Tunis, Tunisie 2 LAJ, University of Jijel, BP. 98, Ouled-Aissa, 18000 Jijel, Algeria 3
Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, 08193- Bellaterra, Barcelona, Spain 4 Departa ento de Ingenier a, acultad de iencias Naturales e Ingenier a, ni ersidad de ogot orge Tadeo o ano, treet, No - , od , ogot , D.C. 110311, Colombia *[email protected] [email protected] [email protected]
Abstract This research is concerned with the problem of generalized function projective synchronization of nonlinear uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities. The considered problem is challenging owing to the presence of unmeasured master-slave system states, external dynamical disturbances, unknown nonlinear system functions, unknown time-varying delays, quantized outputs, unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct fractional-orders. Under some mild assumptions and using
aputo’s definitions for
fractional-order integrals and derivatives, the design procedure of the proposed neural adaptive controller consists of a number of steps to solve the generalized function projective synchronization problem. First, smooth functions and the mean value theorem are utilized to overcome the difficulties from actuator nonlinearities and distributed time-varying delays, respectively. Then, a simple linear observer is established to estimate the unknown synchronization error variables. In addition, a Nussbaum function is incorporated to cope with the unknown control direction and a neural network is adopted to tackle the unknown nonlinear functions. The combination of the frequency distributed model, the Razumikhin Lemma, the neural network parameterization, the Lyapunov method and the
arbalat’s le
a is
employed to perform the stability proof of the closed-loop system and to derive the adaption laws. The major advantages of this research are that: (1) the Strictly Positive Real (SPR) condition on the estimation error dynamics is not required, (2) the 1
considered class of master-slave systems is relatively large, (3) all signals in the resulting closed-loop systems are semi-globally uniformly ultimately bounded and the synchronization errors semi-globally converge to zero. Finally, numerical examples are presented to illustrate the performance of the proposed synchronization scheme. Keywords Generalized function projective synchronization, uncertain time-delay chaotic systems, incommensurate fractional-order systems, input nonlinearities, Nussbaum function,
Razumikhin Lemma, Frequency distributed model, adaptive quantized
output-feedback control.
1. Introduction Nowadays, a great deal of interest has been paid to the synchronization of noninteger-order chaotic systems due to their broad applications in various branches of science and engineering like bioengineering [1], chemical science [1-4], continuum mechanics [5], nonlinear control systems [1-33], electrical networks [20], financial systems [21], robotics [22], modeling brain activities [1] and many others. Generally, the objective of the synchronization of two or more dynamical systems is that those systems are coupled or one among them somehow traces the motion of the others [122, 24-27, 29-41]. Recently, approaches about synchronization between the drive (master) and response (slave) systems of two identical or non-identical systems with different initial conditions, have been proposed in various works, for instance, see [17-45] and references therein. As stated in [1-45], there exist many categories of synchronization
such
as
identical
or
complete
synchronization,
partial
synchronization, generalized projective synchronization, generalized synchronization, Q-S synchronization, phase synchronization, hybrid projective synchronization, anticipated and lag synchronization, amplitude envelope synchronization, modified projective synchronization, function projective synchronization, generalized function projective synchronization, and so on. It should be pointed out that the generalized projective synchronization, anti-phase synchronization, generalized synchronization, projective synchronization and complete synchronization are special cases of the generalized function projective synchronization [20-45]. In some works, several control approaches have been developed for the chaotic synchronization phenomena, such as linear and nonlinear feedback control, active control, adaptive (direct or 2
indirect) control, time delay feedback approach, Sampled-data feedback control, adaptive open-plus-closed-loop control, fractional PID control, robust control, sliding mode control, Backstepping control, data-driven control, among others [1-10, 2389]. It is also well known that the singularity problem and chattering phenomena can appear while implementing some control approaches [87-94]. From the control theory point of view, a chaotic system is a dynamical nonlinear deterministic system governed by one or more control parameters and characterized by an unusual sensitivity to initial conditions, a fractal structure, a non-periodic behavior and the existence of a positive Lyapunov exponent [1-15]. As pointed out in [1, 3-35], Lyapunov exponent criteria can be employed to identify a chaotic behavior. In the literature, many systems can produce chaotic behaviors such as, Lü system, Chen system, Sprott systems, Li system, Cai system, Tigan system, Wei–Yang system, fractional-order
hua’s circuit, Sundarapandian systems, fractional-order
Arneodo system, Pehlivan system, fractional-order Lorenz system, fractional-order Rössler system, Sampath–Vaidyanathan system, fractional-order Lü system, uncertain fractional-order Duffing–Holmes chaotic time-delay systems, to name but a few [120]. Generally, the mathematical models of chaotic systems can be described by differential or difference equations in such a way that the present states of systems are expressed in terms of the past ones [1, 3-25, 27-57]. Most of the above-mentioned works have focused on the simple chaotic systems with affine structures. However, there are various nonlinear nonaffine systems in the control literature, such as biochemical processes, chemical reactors, and so on [26, 56, 75, 87, 88]. It should be noted that affine systems are a very special case of nonaffine systems [26, 75, 87, 88].Thanks to the considerable efforts devoted by researchers, significant control techniques have been proposed for nonaffine systems [56, 87]. In general, there exist five approaches used to cope with the no affinity problem, namely: (1) the local inversion of the Takagi-Sugeno fuzzy affine model, (2) the mean value theorem, (3) the implicit function theorem, (4) the differentiation of the original system equation, (5) Taylor series expansion [1-26, 87, 88]. In addition, most of these above-mentioned approaches have been sometimes employed to determine the control direction (i.e., the so-called sign of the control gain) [26-56]. It should be noted that for facilitating the design of adaptive controls for nonlinear nonaffine systems, the sign of the control
3
gain has been assumed to be known (strictly negative or strictly positive) in some works [57-87]. From a practical point-of-view, the assumption that the sign of the control gain is known a priori, is more an exception than a rule because there are various systems with unknown control directions such as biochemical and biophysical processes, robotics, electrical systems, and so on [1-4, 6, 9, 12, 23, 43, 87]. It must be pointed out that the lack of knowledge of the sign of the control gain can complicate the adaptive control design [2, 12, 4, 26, 33, 39, 74, 78-80, 84, 88]. The available approaches for coping with the unknown control problem, can be classified into five groups, such as [79, 84, 88]: (1) an approach based on a hysteresis-type function, (2) an approach based on a monitoring function, (3) an approach based on a Nussbaumtype function, (4) an approach based on directly estimating unknown parameters, (5) an approach based on Nussbaum and hysteresis-dead zone-type functions.
For
example in [78,79, 83, 84, 88], synchronization schemes based on some of the previous methods have been developed for a class of uncertain fractional-order chaotic systems with unknown control direction. In some of the above-mentioned researches, the models of studied systems are assumed to be known or partially known. In real applications, many systems are harshly nonlinear and contain uncertainties due to the presence of variable changes, unknown parameters, disturbances, unmodeled dynamics, and different conditions [1-70, 88, 89]. In the literature [3, 5, 70-88], iterative data-driven controller tuning techniques like
data-driven reinforcement
learning control, frequency-domain tuning, iterative feedback tuning, iterative regression
tuning,
correlation-based
tuning,
data-driven
predictive
control,
simultaneous perturbation stochastic approximation, model-free adaptive control, adaptive online iterative feedback tuning, model-free control, and model-free or datadriven iterative learning control, have been developed for handling the structured uncertainties. Moreover, based on the universal approximation property [2-4, 8, 14, 33, 39, 49-58, 62, 74, 78-88], various adaptive intelligent controllers like neural network and fuzzy-based adaptive controllers have been introduced in the synchronization schemes. The principal idea of such control methods is to employ neural networks or fuzzy logic systems for estimating the uncertain nonlinear functions in dynamic systems [2-4, 62, 74, 78-88]. It is well known that some neural 4
network-based control techniques are iterative data-driven control techniques [2-4, 8, 14, 15, 33, 88]. For coping with approximation errors and external disturbances, neural adaptive controllers have been augmented by robust compensators in some real applications [7, 9, 37, 57, 76, 77, 88]. Nevertheless, the effect of actuator (input) and sensor nonlinearities on designing and implementing synchronization control systems is omitted in some of the aforementioned works. In recent years, actuator and sensor nonlinearities are ubiquitous in physical systems and devices such as biological optics, electro-magnetism, mechanical actuators, and electronic relay circuits due to their excellent characteristics [26, 39, 46-60, 67, 74, 84, 88]. In engineering practice, the output of the actuators and the input of the sensors are unknown signals because the actuators and the sensors have been usually employed to actuate the systems as an input and to measure the system output, respectively [26, 39, 46-60, 67]. It is well known that actuator and sensor nonlinearities like dead-zone, saturation, backlash, quantization, and hysteresis, could guide to poor performances or even instability of the synchronization control systems [46-60, 67, 74, 84, 88]. Moreover, it is also noted that the quantizer is commonly employed for transforming a real-valued signal into piecewise continuous [46-52, 67]. The studies about actuator nonlinearities (backlash-like hysteresis, dead zone and saturation) and quantized output problems have attracted a great deal of attention, for example, see [26, 39, 46-60, 67, 74, 84] and the references therein. The works in [67, 67, 74, 84, 88] have solved the problems of backlash-like hysteresis, dead zone and unknown asymmetric saturation actuators by using smooth functions. Besides, approaches based on static and dynamic quantizers have been developed to cope with the quantized feedback control problems [46-52]. The major limitation of most of the above mentioned results is that all the state variables of the controlled systems are directly measured or accessible. In most practical situations, the state variables of the systems are immeasurable or difficult to measure [28, 51, 54, 58, 67, 75-77, 79, 80, 83, 85, 88]. In these cases, the observer-based controls (or output-feedback controls) must be used to get the desired performance [75-77, 79, 80, 83, 85, 88]. In general, there exist three approaches of observer-based control designs for uncertain nonlinear systems, such as [75, 79, 80, 83, 85, 88]: (1) an approach based on the Strictly Positive Real (SPR) condition, (2) an approach based on the separation principle paradigm, (3) an approach based on the 5
non-separation principle. In [80, 83, 85, 88], based on the separation approach, the controller and the high-gain observer that provides state estimates have been constructed separately in order to make the closed loop system stable. It must be mentioned that pole-placement algorithms, Riccati equation-based algorithms, and Lyapunov equation-based approaches are the available techniques for the design of high-gain observers [75-77, 79, 80, 83, 88]. Furthermore in [53, 54, 58, 67, 75-77, 79, 80, 83, 85, 88], based on the non-separation principle, Implicit Function Theorem, Mean Value Theorem and neural network parametrization, an output feedback control scheme has been developed for a class of non-affine nonlinear systems, so that the design of state observer has been coupled with the control system design. On the other hand, based on the SPR condition of the observation error dynamics (i.e. the estimation error dynamics), direct and indirect adaptive control schemes have been proposed for uncertain nonlinear perturbed systems [75-77, 79, 80, 83, 85, 88]. As stated in [80, 83, 85, 88], the SPR condition (or a suitable observation error filtering) can be employed in the Lyapunov stability analysis for designing the adaptation laws on the basis of the Meyer-Kalman-Yakubovich (MKY) Lemma and available measurements (i.e. the output observation error and the output tracking error) [75-77, 79, 80, 88]. It should be noted that the SPR condition is a very restrictive condition and it is not satisfied by various systems [75, 79, 80, 88]. Moreover, in [75, 76, 79, 80, 88], it has been proven that the control system can be augmented by a stable filter to satisfy a SPR condition of a transfer function of the observation error dynamics [75, 88]. It must be pointed out that some of the aforementioned studies do not cope with state time-delays. Delays usually appear in many industrial control systems, such as underwater vehicles, chemical processes, communication networks, rolling mill systems, biological systems, metallurgical processing systems, and so on [5, 13, 15, 21, 34, 41, 44, 45, 52, 53, 60-62, 84, 86-89]. In fact, the presence of delays usually causes poor performance, exhibition of undesirable inaccuracies, and even instability of the system under control [60-62, 84, 86-89]. The Lyapounv-Razumikhin and the Lyapunov-Krasovskii functionals are approaches usually used for controller design and stability analysis of time-delay systems [52, 53, 60-62, 75, 84, 86-89]. Unlike the Lyapunov-Krasovskii techniques in which the derivative of the time-varying delay terms should be strictly less than unity, the Lyapunov-Razumikhin technique has been 6
applied to various time-delay systems, so that the stability analysis and the control design can be relatively straightforward and the time-varying delays only satisfy the boundedness [5, 13, 15, 21, 34, 41, 88]. Moreover, the classical Lyapunov-Krasovskii functional, which is always composed by integral and quadratic functions, has been extended to fractional-order systems [87, 89]. In various works, the LyapunovRazumikhin functional has been generally built with quadratic functions in order to construct the control and to determine the (asymptotic, uniform, global asymptotic) stability of fractional functional differential systems with time-varying, constant, and distributed time varying delays [87-89]. In [88, 89], stability conditions of fractionalorder nonlinear systems have been obtained based on the Razumikhin technique, Lyapunov functions and Caputo derivatives. In the above-mentioned researches, stochastic, pure-feedback, strict-feedback and nonlinear systems have been studied and it has been proven that many real systems could be described by fractional-order differential equations, such as [1-23]: electrical circuits, electromagnetic waves, regular variation in thermodynamics, viscoelastic systems, dielectric polarization, biological and financial systems, heat conduction in a semi-infinite slab, robotics, bioengineering, and so on. In several branches of engineering and science, the fractional calculus provides more precise models of the physical systems than the regular calculus does. It is well known that integer order systems is a very special case of fractional (non-integer) order systems [6-30, 33, 43, 59, 63-66, 69-73, 76, 77, 81, 82, 87-91, 94-97]. That is to say, fractional calculus is an excellent mathematical tool for an exact description of memory and heredity features of many materials and processes [1-23, 33, 43, 59, 88, 91]. In [94-99], an interpretation of fractional operators in the time domain has been presented by using the physical realization of fractional operators, construction of a Cantor set, linear filters and four concepts of fractal geometry. It has been shown that fractional operators can be classified into filters with partial memory, which lies between two extreme categories of filters with no memory and those with complete memory [9599]. In [97, 98], it has been deduced that fractional operators may be used for modeling systems with partial dissipation [96-99]. In addition, it has been shown that the fractional-order of a fractional integral represents the indication of the preserved or remaining energy of a signal existing in such system [95-97]. However, it has been established in [96, 97] that the fractional-order of a differentiator represents the rate of 7
a portion of the lost energy. On the other hand, it has been found that the Grünwald– Letnikov, Hadamard, Weyl, Riesz, Riemann– iou ille, and
aputo’s definitions are
the most popular definitions for fractional-order integrals and derivatives [59, 63-66, 69-73, 76, 94-99]. As stated in [1-45, 76, 77, 81, 82, 87-93], the most classical stability tools of integer order systems cannot be extended or applied directly to that of the fractional-order differential systems. Recently, stability conditions for a class of fractional-order nonlinear systems has been developed based on Mittag-Leffler function, Laplace transform, Caputo derivatives and the generalized Gronwall inequality [8, 63-65, 88]. Besides, Lyapunov-Razumikhin and Mittag-Leffler stability theorems for fractional differential systems with delay have been proposed in [63-65]. Additionally, with the help of the frequency distributed model and indirect Lyapunov methods, adaptive controllers have been constructed and some stability conditions for the synchronization of fractional-order systems have been obtained [76, 77, 81, 82, 88]. In comparison with many research works on integer order systems, a few works focused on the synchronization of multi-input multi-output (MIMO) fractional-order systems because the specificity of MIMO systems and the difficulties with the extension of the approaches employed for integer order systems to fractional ones [145, 87-89]. So far, to the best of authors’ knowledge, the problem of the generalized function projective synchronization of MIMO uncertain nonlinear time-delay incommensurate
fractional-order
chaotic
systems
with
unknown
actuator
nonlinearities (including backlash-like hysteresis, asymmetric saturation actuators and dead-zone) has not been studied before in the literature. Motivated by the preceding discussion, a novel neural adaptive quantized outputfeedback control is proposed in this research for achieving an appropriate generalized function projective synchronization of nonlinear uncertain MIMO time-delay incommensurate
fractional-order
nonlinearities. The main
chaotic
systems
with
unknown
actuator
difficulties of this research are how to cope with
unmeasured master-slave system states, external dynamical disturbances, unknown nonlinear system functions, unknown distributed time-varying delays, unknown timevarying delays, quantized outputs, an unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct fractional-orders. In order to solve the generalized function projective synchronization problem, the design of the proposed neural adaptive 8
controller follows a number of steps under some mild assumptions and using aputo’s definitions for fractional-order integrals and derivatives. First, smooth functions and the mean value theorem are introduced to avoid the difficulties from distributed timevarying delays and unknown actuator nonlinearities, respectively. Then, a simple observer is incorporated for estimating the unknown synchronization error variables. Additionally, a neural network is used to neutralize the uncertain nonlinear dynamics and a Nussbaum function is incorporated to cope with the unknown control direction. A combination of the Razumikhin Lemma, the neural network parameterization, the frequency distributed model, the Lyapunov method and the
arbalat’s le
a is
adopted for deriving all the parameter adaption laws and for performing the stability proof of the closed-loop system. The main contributions of this research lie in the following: (1) Unlike in [1-4, 6, 8, 9, 12, 14, 26, 27, 29, 33, 34, 37], the proposed design approach does not require any prior knowledge of the sign of control gains nor any information of the bound of actuator nonlinearities. (2) In contrast to the adaptive controls [1-18, 22, 24, 25, 27, 28, 29, 39-41, 45], the proposed neural adaptive controller is very simple and without the singularity problem, so that the number of adjustable variables is reduced. (3) Unlike the previous research works [80, 83, 85, 88], the SPR (Strictly Positive Real) condition on the estimation error dynamics is neglected. (4) Compared to the previous works [1-15, 20, 23, 28-30, 33, 43, 59, 63-66, 68-73, 83, 87, 89], the stability of the corresponding closed-loop system is analyzed in this manuscript without determining the fractional derivatives of the Lyapunov functions, (5) Unlike the closely related research works [2, 4, 26, 33, 74], the synchronization errors semi-globally converge to zero and all signals of the closed-loop system are semi-globally uniformly ultimately bounded. The remainder of this paper is organized as follows. In Section 2, the problem statement is presented. In Section 3, the neural adaptive quantized output-feedback control is designed and analyzed while in Section 4, simulation examples are given to confirm the effectiveness, the feasibility and the validity of the proposed design. Finally, conclusions are drawn in section 5. Notations. D
represents the order Caputo differential operator. q . denotes
the quantization function. . is the floor function. . is a nonlinear input 9
function. .
indicates the Euclidean norm. The superscript T denotes the matrix
transpose operation. I n represents the identity matrix of size n . min . denotes the minimum eigenvalue of corresponding matrices. . denotes the absolute value function. 0n m is the zero matrix of size n m .
diag . denotes a block diagonal
matrix. t denotes the time index. sgn . represents the sign function. set of all n m real matrices.
n
n m
denotes the
represents the n-dimensional Euclidean space. L 2 is
the space of square-summable functions. L is the space of bounded functions.
2. Problem formulation Let us consider two nonlinear uncertain MIMO time-delay incommensurate fractional-order chaotic systems (master and slave systems), expressed as follows
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 Master: x Mi t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, , n i , i 1, p
(1)
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t d i , j t t t 3,i , j t f 3,i , j x S , d , j 1, , ni 1, t 0 2 ,i D x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t d i ,ni t t f Slave : x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,ni t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 S i 2 ,i i mi 1 y Si t x Si ,1 t , for i 1, , p
10
(2)
where
the
for i 1,
fractional
incommensurate
0 1,i 1
orders
0 2 ,i 1 ,
and
, p are known constants.
x M t x M 1T t , x M 2T t , x S t x S 1T t , x S 2T t ,
, x MpT t T
, x SpT t T
n
n
and
are the states of the master
and slave, respectively, with
x Mi t x Mi ,1 t , x Mi ,2 t ,
x Si t x Si ,1 t , x Si ,2 t ,
and n1 n 2
, x Mi ,ni t
, x Si ,ni t T
T
ni
,
, for i 1,
ni
,p
np n .
y M t y M 1 t , y M 2 t ,
, y Mp t T
and y S t y S 1 t , y S 2 t ,
p
, y Sp t T
p
are the outputs of the master
and slave, respectively.
d t d 1T t , d 2T t ,
, d pT t T
n
is the external bounded disturbance
vector for the slave system with d i t d i ,1 t , d i ,2 t , for i 1,
, d i ,ni t
T
ni
,
,p .
The initial condition functions for the slave 1,i t
ni
, and 2 ,i t
ni
i 1,
,p ,
are unknown, smooth and bounded t 0 . For k 1,
, 4 , j 1,
, ni , i 1,
, p , k ,i , j t are unknown time-varying delays
satisfying 0 k ,i , j t * with the constant * being unknown and strictly positive.
11
For j 1,
, ni , i 1,
, p , f 1,i , j x M t , x M t 4 ,i , j t
,
f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t , f 3,i , j x S t , t and f 4,i x S t , i t
are unknown smooth functions.
u t u1T t , u 2T t ,
, u pT t
system with u i t u i ,1 t , u i ,2 t , and m1 m 2
T
m
,
is the control input of the slave
, u i ,mi t
T
mi
, for i 1,
,p
mp m .
The output of the actuator nonlinearity t 1 t , 2 t ,
, p t T
p
(including backlash-like hysteresis, asymmetric saturation actuators and dead-zone) is described by the following equation [39, 52-60, 67, 74, 84, 86, 88, 90-92] u u min , if i t min 1 0 2 u min i t 1 2 , if 1 i t 1 2 i t u i t 0 , if 1 i t 3 u i t 3 4 ,if 3 i t max 3 4 u u max , if i t max 3 0 4 T T d i t d i u i t T u t t d i u i t , for i 1, , p 5 6 i i i 7 dt dt dt
where u min , l , 2 , 3 , 4 and u max are unknown strictly positive constants.
5 , 6 and 7 are unknown constants satisfying 6 7 .
12
(3)
diag 1 , 2 ,
, p
and i 0 , for i 1,
mp
is a known constant matrix so that i
mi
,p .
Remark 1. In contrast to [1, 3, 4, 6-9, 12-14, 25-27, 29, 37, 39], the master and slave systems (1) and (2) presented in this paper are partially dissimilar, with inhomogeneous
1,1 1,2 for i 1,
(or
incommensurate)
1,p and 2 ,1 2 ,2 p
, p and
i 1
1,i
fractional-orders
(i.e.
2 , p ) and with different orders (i.e., 1,i 2 ,i ,
p
2 ,i ). In addition, these master and slave systems with i 1
uncertain dynamics and external dynamic disturbances can be used to describe many physical (practical) systems, such as:
fractional-order Lorenz system, chemical
processes, aircraft wing rock, Duffing chaotic system, communication networks, fractional-order Lü system, induction servo-motor drive, fractional-order Chen system, fractional-order unified chaotic system, underwater vehicles, and so on [2145, 59]. Remark 2. As already discussed in [1-15, 20, 23, 28-30, 33, 43, 59, 68-73, 88], the Hadamard, Riemann–Liouville, Riesz, Grünwald–Letnikov, Weyl, and Caputo’s definitions are the most widely used definitions for fractional-order integrals and derivatives. Therefore, the Caputo fractional operator is employed in this paper because it is more consistent than other ones, for the following reasons [68-73]: the Caputo derivative of a constant is zero, the initial conditions for the fractional-order differential equations in Caputo sense have the same form as for the integer-order differential equations, the Caputo operator allows the avoidance of hyper-singular improper integral and mass balance error. Remark 3. It is easy to see that the actuator nonlinearity (3) has non-smooth nonlinear characteristics and sharp corners when i t u min and i t u max , for i 1,
, p . Consequently, many techniques cannot be directly applied for
constructing the adaptive controllers [53-59, 67].To handle such a problem, smooth functions are usually employed for approximating the system (3).
13
Remark 4. It must be emphasized that commensurate fractional-order chaotic systems are a special case of incommensurate fractional-order chaotic systems [1-5, 23, 28-30, 33, 43, 59, 68-73, 88]. In this research, the following Assumptions 1-4 are necessary. Assumption 1. Only the quantized outputs of the master and slave systems described by the following equation (4) are available for measurement. y Mi t q1,i y Mi t 1,i t , 1,i t y Si t q 2 ,i y Si t t 2 ,i t , for i 1, , p 2 ,i T q y t q y t , q y t , , q1, p y Mp t 1 M 1,1 M1 1, 2 M2 T q y t q y t , q y t , , q 2 , p y Sp t 2 S 2 ,1 S1 2 ,2 S2
(4)
p
p
where the smooth functions 1,i t and 2,i t are strictly positive and bounded such that
d is bounded, t 0 , for i 1, t
0
1,i
2 ,i
,p .
Assumption 2. There exist unknown class k functions k,i, j . , k 1, j 1,
, ni , i 1,
, p and unknown positive constants 1 , 2 , such that
14
,7 ,
p p ni f x t , x t t x t f 4 ,i x S t , 0 M 4 ,i , j M 1,i , j M i 1 j 1 i 1 p ni f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t x S t i 1 j 1 p ni p ni 1,i, j x M t 2 ,i, j x S t i 1 j 1 i 1 j 1 p ni 3 ,i, j x S t 1,i , j t i 1 j 1 p ni p ni 4 ,i, j t 2 ,i , j t 7 ,i, j x M t 4 ,i , j t 1 i 1 j 1 i 1 j 1 p ni p ni p ni f x t , t x t 5 ,i, j S 6 ,i, j t 2 3 ,i , j S i 1 j 1 i 1 j 1 i 1 j 1
(5)
Assumption 3. f 4 ,1 x S t , 1 t f 4 , p x S t , p t is nonThe matrix g 1 x S t , t diag , , 1 t p t
singular and either strictly negative-definite or strictly positive-definite so that there exist two unknown positive constants 3 and 4 , satisfying
0 3
f 4 ,i x S t , i t i t
4 , for i 1,
,p
(6)
Assumption 4. There exists an unknown strictly positive constant 5 , so that
p
p
i 1
i 1
x M t d t 1,i t 2 ,i t 7
4
i 1
i 1
* i i n p m p
p
i 1
i 1
i u max u min D
2 ,i
(7) x Mi t
5 np
15
, t 0
Remark 5.
Note that Assumption 1 is satisfied by many modern engineering
systems such as electric power grids, intelligent transportation systems, groups of unmanned (aerial, ground, and underwater) vehicles, networked control systems, to name but a few [46-52, 67]. According to [46-52, 67], it can be deduced that the
quantizer q . in this paper is dynamic. Indeed, Assumption 2 is not conservative and is crucial for employing the Razumikhin Lemma in the control design process [53, 6066]. Besides, Assumption 3 is not restrictive and it can be seen as a controllability condition, e.g. see [1-4, 6, 8, 11, 12, 26, 27, 29, 33, 34, 37, 39, 53, 67, 78]. In fact, from this last assumption, one can see that the control-gains matrix g 1 x S t , t is always regular and the control direction is unknown. Furthermore, Assumption 4 has been commonly employed in the literature [1, 56, 58, 62]. As pointed out in [33], the external disturbances and the states of the master system (in general, the states of uncontrolled chaotic systems) are always bounded. Remark 6.
Generally, a class k function . is continuous, strictly increasing
n and unbounded [53, 60-66, 88]. Then, 0 0 and ai i 1
for i 1,
n nai , ai 0 i 1
,n .
The objective of this research is to determine an adaptive neural controller to achieve the generalized function projective synchronization between the master system (1) and the slave one (2) under Assumptions 1-4, while all the signals in the derived closed-loop systems are bounded and the corresponding synchronization errors converge to zero.
Therefore, let us define the synchronization error variables between the two systems as follows
16
i t q 2 ,i y Si t i t q1,i y Mi t t x t t x ,p Si i Mi t , for i 1, di , p t p t 1 t , 2 t , T T T , dpT t d t d 1 t , d 2 t ,
where the functions i t , i 1, p
D
2 ,i
p
Mi
i 1
n
, p , are known, smooth and bounded, such that 5
t x t t i
i 1
(8)
i
i 1
6 with 6 being an unknown strictly
i
positive constant.
Remark 7.
In the synchronization literature, the scaling factors, the states of the
master system and their fractional derivatives are already assumed to be bounded [110, 26, 33]. From the above research works [63, 95, 96], one can deduce that D
2 ,i
t x t 12 t D i
Mi
i
2 ,i
x Mi t x Mi t D
2 ,i
i t , for i 1,
,p .
Remark 8. It is pointed out that the synchronization errors i t , i 1,
, p are
available for measurement according to Assumption 1. Besides, the synchronization error variables di t , i 1,
, p are immeasurable.
Remark 9. It is worth noting that the generalized function projective synchronization
problem of different incommensurate fractional-order chaotic systems is turned into [1-23, 26, 33]: a generalized function projective synchronization problem of different commensurate fractional-order chaotic systems if the fractional-orders
1,1 1,2
1,p and 2 ,1 2 ,2
2, p ,
a generalized function projective synchronization problem of integer-order chaotic systems, if 1,1 1,2
1,p 1 and 2 ,1 2 ,2
17
2, p 1 ,
a projective function synchronization problem if the scaling factors
1 t 2 t
p t ,
a complete synchronization problem if 1 t 2 t
p t 1 ,
an anti-phase synchronization problem if 1 t 2 t a chaos control problem, if 1 t 2 t
p t 1 ,
p t 0 .
On the other side, the quantization errors are given by
e i t y Si t q 2 ,i y Si t
(9)
y Mi t q1,i y Mi t i t , for i 1, with e t e1 t , e 2 t ,
, e p t T
p
,p .
Remark 10. As stated in [46-52, 67], the quantization error e i t is bounded, so p p p that e i t 1,i t 2 ,i t 6 . i 1 i 1 i 1
3. Controller Design and Stability Analysis
Under Assumptions 1-4, in order to solve the generalized projective synchronization
problem, we will use: smooth functions for describing the actuator nonlinearity (3), the mean value theorem to overcome the problems of unknown distributed time-varying delays and to transform the slave system (2) into an affine-like form in which the control input appears in a linear fashion, an adaptive observer to estimate the unmeasured synchronization error variables, a neural network to approximate the unknown nonlinear functions, a Nussbaum-type function to cope with the unknown control direction,
18
the Razumikhin Lemma, the frequency distributed model, the Lyapunov method and the Barbalat’s le
a to derive the adaptation control laws and to
perform the stability proof of the closed-loop system.
3.1 Approximation of unknown actuator nonlinearities Based on the previous research works [39, 52-59, 60, 67, 74, 84, 86, 88, 90-92] and by utilizing smooth functions, the unknown nonlinear input (3) can be approximated as follows i t 6 i T u i t 1,i t t t t t i 2 ,i i i t u i t 0 ,i i t 0 ,i t , for i 1,
(10)
,p
with
1,i t i 0 6 i T u i 0 exp i T u i t i T u i 0 5 sgn i T u i t
exp 5 sgn i T u i t
u 0 7 6 exp 5 sgn i T u i t d i T u i t
,
T
i
i
, if i t 0 , t 2 4 , if i t 0
2 ,i
4 3 , if 3 i t t t i T u i t , if 1 i t 3 , 2 1 , if i t 1
u u u u 0 ,i i t =u M erf i t , u M max min max min sgn i t , 2u M 2 2
i
i
i
2 erf i t 2u M i
2u M i 0
i t
exp 2 d
and 0 ,i t u i t 0 ,i i t , for i 1,
19
,p .
Remark 11. It is worthy to note that the functions t , 1,i t , 2,i t ,
0 ,i i t , u M , erf
2u M i
i
i t and 0,i t are bounded, for i 1,
,p
and satisfy p
p
t u 0 ,i
i 1
i 1
p
p
i 1
i 1
p
1,i t t
Mi
i 1
2 ,i t 0 ,i i t
erf i t 7 2u M i 1 i p
,
with the constant 7 being unknown and strictly positive.
Moreover, based on the mean value theorem [53, 67, 88], we can write
t f x , d t 3 ,i , j t 3 ,i , j S 3 ,i , j t f 3 ,i , j x S t 5 ,i , j t , t 5 ,i , j t , f 4 ,i x S t , i t f 4 ,i x S t , 0 g 1,i x S t , t i t , 1 ,i i t 1,i i t u i t 1,i t , for i 1, , p , j 1, , n i
(11)
with 0 5 ,i , j t 3,i , j t * ,
g 1,i x S t , 1,i t
f 4 ,i x S t , i t
i t
i t 1 ,i t
0 1,i 1 , 1,i i t t 6 i T
1,i t 2 ,i t
0 ,i i t i t
0 ,i i t i t
, 1,i t 1,i i t ,
0 ,i i t i t
i t 2 ,i t
,
t 1,i t 0 ,i t i t 2 ,i t
20
, i t 2 ,i t
0
0 ,i i t i t 2 ,i
1 , i t 2 ,i t
t 2 ,i i t
and 0 2 ,i 1 for j 1,
, ni , i 1,
,p .
Remark 12. The difficulty from the distributed time-varying delays can be processed by using the mean value theorem for integrals [53, 67, 88], as done in (11). Remark 13. It is worth mentioning that unlike the Taylor series expansion, which is
only valid locally at specified points, the Mean Value Theorem is valid globally [26, 53, 56, 67, 74, 78, 88]. Another important difference between the Taylor series expansion and the Mean Value Theorem is that by means of the Taylor series expansion, a smooth function would be expressed around one point with high-order terms (high-order approximating error), whereas, with help of the Mean Value Theorem, this function would be approximated between two points without highorder terms [56, 67, 74, 75, 88]. Consequently, by using (11), the slave system (2) can be rewritten as
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t 3 ,i , j t f 3 ,i , j x S t 5 ,i , j t , t 5 ,i , j t d i , j t , j 1, , n i 1, t 0 D 2 ,i x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t 3 ,i ,ni t f 3 ,i ,ni x S t 5 ,i ,ni t , t 5 ,i ,ni t Slave : f 4 ,i x S t , 0 g 2 ,i t u i t d i ,ni t 2 ,i t , t 0 x Si t 2 ,i t , u i t 0m i 1 , t 0 y t x t , for i 1, , p Si ,1 Si
21
(12)
where 2 ,i t 1,i t g 1,i x S t , 1,i t
and g 2 ,i t g 1,i x S t , 1,i t 1,i i t are bounded
for i 1,
,p .
3.3. Adaptive observer for the unmeasured synchronization error variables To estimate the unmeasured synchronization error variables, the following observer is considered
D 2 ,i ˆ di t Ai bi k ci ˆ di t k oi i t , ˆ i t c i T ˆ di t , ˆ i t i t i t , for i 1, , p where ˆ di t
ni
and ˆ i t
respectively, for i 1,
For i 1,
T
, p , k ci
Ai bi k ci
n i n i
are the estimates of di t and i t ,
,p .
0 ni 11 I ni 1 Ai 0 1 n i and c i 1, 01 ni 1
(13)
ni ni
, i 1,
ni
1ni
T
, bi 01 ni 1 , 1
,p .
and k oi
ni
and Ai k oi c i T
are chosen such that the matrices n i n i
are strictly Hurwitz.
On the other hand, one has ˆ t ˆ 1 t , ˆ 2 t ,
ˆ d t ˆ d 1T t , ˆ d 2T t ,
and t 1 t , 2 t ,
ni
, ˆ dpT t
, p t
T
T
p
22
n
.
, ˆ p t
T
p
,
Remark 14. Compared with the existing observers in [49, 51, 54, 76, 77], the proposed observer (13) has a simple linear form which is not related to control input and intelligent systems (neural networks or fuzzy logic systems).
According to [78-80, 88], k ci and k oi , i 1,
, p can be given in the following forms
T k C 1 , C 2 2 , , C nnii 1ni ni 1 ni 1 oi , for i 1, k ci C nnii 2 ni , C nnii 1 2 ni 1 , , C n1i 2
with C nji 1
and
2
ni ! , for j 1, j! n i j !
, ni , i 1,
,p
(14)
,p .
are known strictly positive constants.
ˆ t , for i 1, Let the observation errors be defined as di t di t di
, p . Then,
by using (8), (9) and (13), the dynamics of the observation errors can be written as D 2 ,i di t Ai k oi c i T di t bi g 2 ,i t u i t F0 ,i t T i t c i di t e i t , for i 1 , p
with
23
(15)
d i ,1 t d i , 2 t F0 ,i t k e t A i di t oi i d t i ,n i
f 2 ,i ,1 x S t , x S t 1,i ,1 t , t 2 ,i ,1 t f 2 ,i , 2 x S t , x S t 1,i , 2 t , t 2 ,i , 2 t f 2 ,i ,n x S t , x S t 1,i ,n t , t 2 ,i ,n t i i i
t t
3 ,i ,1 t f 3 ,i ,1 x S t 5 ,i ,1 t , t 5 ,i ,1 3 ,i , 2 t f 3 ,i , 2 x S t 5 ,i , 2 t , t 5 ,i , 2 3 ,i ,n t f 3 ,i ,n x S t 5 ,i ,n t , t 5 ,i ,n t i i i i 0 0 D 2 ,i i 0 0 ˆ f 4 ,i x S t , 0 2 ,i t k ci di t
t x Mi t , for i 1
,p.
From (13) and (15), one can get D
2 ,i
i t Ad i i t bd i g 2 ,i t u i t F1,i t , for i 1, , p
k e t with F1,i t oi i F t 0 ,i ˆ t i t di t di
for i 1,
2 ni
2 ni
Ai bi k ci , Ad i 0 n n i i
, bd i 01ni , bi T
T
,p .
24
2 ni
(16)
Ai k oi ci T k oi c i T
and cd i 01ni , c i T
2 n i 2 ni
T
,
2 ni
,
By the aid of the frequency distributed model [76, 77, 81, 82, 88], the dynamic system (16) can be rewritten as Z i ,t Z i ,t Ad i i t bd i g 2 ,i t u i t F1,i t t i t 0 i Z i ,t d , for i 1, , p sin 2 ,i 2 ,i with i and Z i ,t 2ni , for i 1,
(17)
,p .
Remark 15. It is noteworthy that thanks to auxiliary times and frequency domain functions, the fractional-order systems are transformed into its continuous frequency distributed equivalent models (equation (17)), in order to simplify the stability analysis [76, 77, 81, 82, 88].
Furthermore, since Adi is stable, thus, there exist constant symmetric positive definite matrices Pi Pi T 0 and Qi Qi T 0 , such that [3, 75-82, 88]
Adi T Pi Pi Adi Qi , for
i 1,
(18)
,p
3.3. Neural network for approximating uncertain functions By utilizing the Razumikhin lemma [53, 60-66, 88], it has been proven that there exists a constant 8 1 , such that x S t 1,i , j t 8 x S t x S t 5 ,i , j t 8 x S t , for j 1, , n i , i 1, , p
(19)
Remark 16. The Razumikhin condition (19) has been established in [53, 60-66, 88]
and will be employed in the procedure of adaptive controller design later. Indeed, from (3), (7) and (8), the following inequality holds
25
ni
t t x t d t p
2 ,i , j
i 1 j 1
M
p
p
p
i 1
i 1
i 1
k oi e i t 2 ,i t i t Ai x Mi t p
D
2 ,i
i 1
(20)
ni
t x t t t p
i
Mi
4 ,i , j
i 1 j 1
9
p p 8 where 9 2 2np i 2 k ci 2 k oi i 1 i 1 i
2
is an unknown strictly positive
constant.
Now, by using (5), (16), (19) and (20), one has p ni 7 p F1,i t 1 k,i, j i 1 j 1 k 1 i 1 p g 2 ,i t u i t 10 i 1 p 2 ,i t 10 i 1
t
with t 1T t , 2T t ,
10
10
t 10
(21)
, pT t T
2n
2
p ni 7 and 10 2 2 9 k,i, j 2 9 . i 1 j 1 k 1
Let us denote
p
t P T
F2 t
p ni 7 1 k,i, j t 9 9 t 2 2 t t i 1 j 1 k 1 i
i 1
i
9 (22)
p
9 i T t Pi bd i i t i 1
t 2 t 2
26
where the smooth function t is known, strictly positive and bounded such that
d is bounded, t 0 . t
0
By using (21) and (22), it is easy to obtain that p
i 1
i T t Pi F1,i t
p
g 2 ,i t u i t i T t Pi bd i i t i 1
(23)
t t F2 t
Thanks to the universal approximation theorem [2-5, 8, 14, 15, 53, 56, 61, 78-80, 88], the unknown function F2 t can be approximated, on the sufficiently large
compact set t
t M
2n
2n
, by a three-layer neural network as
follows F2 t w *T t
(24)
where t ˆ d T t , T t T
np
is the input of the neural network. The constant
M is unknown and strictly positive.
, is unknown and bounded over the
sufficiently large compact set , i.e., * , with the constant * being unknown and strictly positive. t
, is the neural activation function with
1 being the
number of the neural network nodes. Then, the ideal constant neural network weight vector w *
, can be defined as
follows [3, 5, 8, 14, 15, 88]
w * arg min sup F2 t w T t w* t
27
(25)
In general, w * , is unknown, constant and bounded. Besides, t is known, and bounded [3, 5, 8, 14, 15, 53, 56, 61]. Remark 17.
Similar to [3, 5, 8, 14, 15, 53, 56, 61, 88], the neural network
w T t used in this research for approximating the unknown function F2 t has a simple structure, consisting of an input layer, a single output layer with one neuron, and a single hidden layer with
neurons. As stated in [5, 8, 14, 15, 53, 56,
61], the hidden-layer activation function may be a radial basis function, a logistic sigmoid function, or a hyperbolic tangent function. Furthermore, the output-layer activation function is a linear function. To alleviate the burden and complexity of computation, only the weight vector between the hidden layer and the output layer is unknown and must be estimated in controller design, while the weight matrix between the hidden layer and the input layer is randomly initialized and fixed. As pointed out in [3, 5, 8, 14, 15, 53, 56, 61], radial basis function (RBF) neural networks are generally used in practical control engineering thanks to their satisfactory approximation properties and simple architectures.
Over the sufficiently large compact set , there exist an unknown constants 11 and * such that
w
t 1 2
0 F2 t w * * *
2
*
(26)
10
* * t 2
11 with t 11 and * w * * k 2 . k 1
And t t t 1 is a known function.
Remark 18. Similar to [53, 61, 75], the Neural Network parameterization (26) will be used in the design procedure of the controller in order to reduce the burden of
28
computation and complexity. On the other hand, the neural network can be employed to avoid the global Lipschitz assumption for unknown nonlinear functions [75].
3.4. Adaptive control law To achieve synchronization between the master (1) and the slave systems (2), the adaptive control law is constructed as follows u t N t ˆ t 1 t t 2 ˆ t 1 t t t ˆ t 2 t ˆ t 1 t t 2 t 1 t 2 2 t 2 t t
(27)
where N t cos t exp 2 t is a Nussbaum function [78,79, 83, 84, 2 88]. The constant is known and strictly positive. ˆ t is the estimate of the unknown positive constant * .
Remark 19. Similar to [53, 61, 75, 88], it should be selected ˆ 0 0 in such a way that the inequality ˆ t 0 holds t 0 .
Remark 20. As pointed out in [78, 79, 83, 84, 88], commonly used Nussbaum functions are : 2 t cos t , 2 t sin t , t cos and cos t exp 2 t . 2 29
t
3.5. Stability analysis In this stage, we summarize the main results of the proposed controller by the following theorem.
Theorem 1. Let us consider the master-slave systems (1) and (2), satisfying Assumptions 1–4. Then, for any bounded initial conditions, the adaptive observer (13) and the control law (27) guarantee the following properties: All signals in the closed-loop system are semi-globally uniformly ultimately bounded, i.e., ˆ t , t , u t , t and t L . The synchronization error variables semi-globally converge to zero, i.e., d t 0 and t 0 , as t .
Proof of Theorem 1 Now, let us consider the following Lyapunov function candidate p
V t V i t i 1
with V i t
1 2 t 2
(28)
1 i Z i T , t Pi Z i , t d , for i 1, 0 2
,p
and t * ˆ t .
From (17), it is easy to show that p
i Z i T ,t Pi Z i ,t d 0 i 1
0
(29)
Based on Rayleigh inequality [93] and by utilizing the equations ((18), (23) and (29)), the time derivative of V t along the solutions of (17) can be expressed in the following form
30
V t
min Q 2
t t F2 t
t t
(30)
t F2 t T t g 2 t u t with Q diag Q1 ,
, Qp
pp
and g 2 t diag g 2 ,1 t , g 2 ,2 t ,
, , , g 2 , p t
p m
.
Using the neural network parameterization (26) and the control law (27), (30) becomes
min Q t t * t t * 2 2 t t t t ˆ t t
V t
1
(31)
N t ˆ t 1 t T t g 2 t t
According to [83, 85-88], one can conclude that 2 T t g 2 t t t t 2 * * * t t 1 t t t t ˆ t t t 1 2 t t 1 *2 2 2 where t is a bounded function satisfying t 0, t 0 .
(32)
By substituting (32) into (31) leads to V t
min Q 2
t 1 N t t t
(33)
1 2 t t *2 2 *
31
After integrating (33) over 0, t , one obtain V t
min Q t 2 * 1 *2 t d V 0 2 0 d 2 0 2
1 N d t
0
(34)
Similar to the discussion in [83, 85-88] and according to (34), one can easily check Q t 1 t that V t , min d , 2 * *2 d , V 0 0 2 2 0 and
1 N d
are bounded t 0, .
t
0
This fact implies that t L2 .
From (16), since the functions f 1,i , j x M t , x M t 4 ,i , j t
,
f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t , f 3,i , j x S t , t and f 4,i x S t , i t
are smooth functions,
and the variables d i t , e i t , D bounded, then, one has D
2,i
2 ,i
t x t , t , i
Mi
i t L , for j 1,
Finally, since t L2 L and D that lim t 0
2,i
2 ,i
, ni , i 1,
i t L for i 1,
urther ore, by using arbalat’s le
t
,
i t
are
,p .
, p , one can deduce
a [78, 83, 85-88], it
follows that the synchronization error variables converge to zero, i.e., lim d t 0 , lim ˆd t 0 and lim t 0 because d t 2 t , t
t
t
p
p
i 1
ˆd t 2 t , t 1,i t 2,i t 2 6 2 t i 1
p
p
i 1
i 1
and lim 1,i t 2,i t 0 . t
The proof is completed here. 32
Remark
21.
Since
the
well-known
Leibniz
and
chain
rules
D f t g t D f t g t f t D g t
and D f g t
f x D g t are not satisfied for fractional derivatives, x x g t
the classical stability tools for integer order systems cannot be directly applicable to non-integer ones [1-33, 43, 59, 63-66, 69-73, 76, 77, 81, 82, 87-95]. Up to now, the stability analysis of fractional-order systems has attracted wide attention, (see, e.g. [63-66, 69-73, 76, 77, 81, 82, 87-91]). For example in [63-66, 69-73], stability theorems for nonlinear fractional-order systems have been obtained by using Volterratype Lyapunov functions and Caputo fractional derivative. In [63-66], some stability analyses for fractional-order nonlinear systems have been performed based on MittagLeffler function, Laplace transform, and the generalized Gronwall inequality. In [5, 13, 76, 77, 81, 82, 87-89], the stability of fractional-order systems with delays has been studied by utilizing Caputo fractional derivative, Lyapunov approaches, Razumikhin-type stability theorems and Lyapunov-Krasovskii approaches.
Remark 22. Similar to most research works on neural network and fuzzy-based adaptive control schemes [2-5, 8, 14, 15, 33, 39, 49, 50, 53, 56, 58, 61, 74, 78-80, 8385, 87-89, 92, 93], the stability results obtained in this work are semi-global which means that the stability results are valid as much as the master-slave system states remain within some compact sets. 3.6. Practical implementation
To summarize the above analysis, Figure 1 demonstrates the overall scheme of the proposed synchronization in this research, which is consisting of an observer (13) and an adaptive control law (27). To further illustrate the implementation of the proposed controller, Figure 2 presents an algorithmic description based on a flowchart. It follows from Figures 1-2, that the proposed controller is running on-line without any offline-learning phase similar to the previous controllers in [1-22, 26, 33, 78-89].
33
From the previous discussions, the design algorithm of the proposed synchronization scheme is presented step-by-step as follows: Step 1: Choose initial conditions ˆ d 0 , ˆ 0 , 0 , a neural activation function
t , appropriate function t , strictly positive constants 1 ,
2
,
and a learning gain used in the observer (13) and the control law (27). Step 2: Compute t and obtain t by using (13). Step 3: Calculate the control input u t according to (27). Step 4: Use the control input u t for the master-slave systems (1) and (2) under Assumption 1-4. Step 5: Measure the quantized outputs of the master and slave systems q1 y M t and q 2 y S t .
Step 6: Go to Step 2.
34
t
ˆ d t
Proposed neural adaptive control law Equation (27)
u t
x S t
Actuator nonlinearity Equation (3)
Master system (1)
y M t
t
Slave system (2), satisfying Assumptions 1-4
q 2 y S t Quantizer Equation (4)
q1 y M t
y S t
d t Synchronization error variables
x M t
Equation (8)
t +
Observer (13)
ˆ d t
To estimate the unmeasured synchronization error variables
Figure 1. Proposed synchronization scheme
35
-
ˆ t
t
Start t 0
Set initial condition and parameters Go to next sample interval Calculate t and obtain
t by using (13) Determine the control input u t according to (27) Use the control input u t for the master-slave systems (1) and (2) under Assumptions 1-4 Measure the quantized outputs of the master and slave systems q1 y M t and q 2 y S t
Figure 2. Flowchart of implementation of the proposed controller
36
3.7. Comparison to available methods
In comparison to some available control schemes, the most important advantages of our work are that: unlike the closely related works [1-4, 6, 8, 9, 12, 14, 26, 27, 29, 33, 34, 37], the problems of the actuator nonlinearities, the quantized output and the unmeasured master-slave states of nonlinear systems are resolved, there is no filtering of the neural activation functions and the SPR condition imposed in previous works [80, 83, 85, 88] is avoided. In addition, the SPRLyapunov design approach and the SPR-filter approach are not used, the requirement that the sign of control gains are known or satisfied the matching condition [1-4, 6, 8, 9, 12, 14, 26] is removed,
unlike the previous works [2, 4, 26, 33, 74], the synchronization errors semiglobally converge to zero, the considered class of slave systems is relatively large in the sense that the unknown nonlinear functions are not affine and could depend on the full state vector, the time-varying delays and the actuator nonlinearities, unlike in [1-12, 14-22, 24-27, 29-41, 43, 55, 89], unlike in the existing literature [1-18, 22, 24, 25, 27, 28, 29, 39-41, 45], the proposed control has a simple structure in such a manner that the number of tuning variables is reduced.
4. Simulation examples
In this section, two examples are used to evaluate the performance of the proposed projective function synchronization scheme. 37
Example 1: (an academic MIMO fractional-order chaotic system) Let us consider two different uncertain MIMO fractional-order Duffing–Holmes chaotic time-delay systems (master and slave systems) as follows [89]: Master system
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 x Mi t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, 2 , i 1, 2
(35)
Slave system
D 2 ,i x Si ,1 t f 2 ,i ,1 x S t , x S t 1,i ,1 t , t 2 ,i ,1 t d i ,1 t t t 0 t 3,i ,1t f 3,i ,1 x S , d , D 2 ,i x t f Si , 2 2 ,i , 2 x S t , x S t 1,i , 2 t , t 2 ,i , 2 t d i , 2 t t f x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,2 t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 2 ,i i Si y Si t x Si ,1 t , for i 1, 2
(36)
where: 1,1 0.98 , 1,2 0.94 , 2 ,1 0.94 incommensurate orders, 38
and 2 ,2 0.98 are the fractional
x M t x M 1,1 t , x M 1,2 t , x M 2 ,1 t , x M 2 ,2 t T
4
and x S t x S 1,1 t , x S 1,2T t , x S 2 ,1 t , x S 2 ,2 t T
4
are the states
of the master and slave systems, respectively ,
y M t y M 1 t , y M 2 t T
2
and y S t y S 1 t , y S 2 t T
2
are the outputs of the master and slave systems, respectively,
d 1,1 t d 1,2 t 0.01sin t and d 2 ,1 t d 2 ,2 t 0.01cos t are the external bounded disturbances,
1,1 t 1,2 t 2 ,1 t 0, 0 T
2
and 2 ,2 t 1, 1 T
2
are the initial
condition functions,
1,1,1 t 2 ,1,1 t 3,1,1 t 4 ,1,1 t 0 ,
1,2 ,1 t 2 ,2 ,1 t 3,2 ,1 t 4 ,2 ,1 t 0 ,
1,1,2 t 2 ,1,2 t 3,1,2 t 4 ,1,2 t 0.01 0.02sin t , and 1,2 ,2 t 2 ,2 ,2 t 3,2 ,2 t 4 ,2 ,2 t 0.01 0.01sin t , are timevarying delays,
f 1,1,1 x M t , x M t 4 ,1,1 t 2.5x M 1,2 t ,
f 1,1,2 x M t , x M t 4 ,1,2 t
3
1 x M 1,1 t 0.01x M 1,2 2 t 4 ,1,2 t 2.5 0.1x M 1,2 t 0.01x M 1,1 t 4 ,1,2 t
1 x M 1,1 t 0.01x M 1,12 t 4 ,1,2 t 25cos 1.29 t 2.5
39
,
f 1,2 ,1 x M t , x M t 4 ,2 ,1 t 2.7x M 2 ,2 t ,
f 1,2 ,2 x M t , x M t 4 ,2 ,2 t
3
1 x M 2 ,1 t 0.01x M 2 ,2 2 t 4 ,2 ,2 t 2.7 0.1x M 2 ,2 t 0.01x M 1,1 t 4 ,2 ,2 t
,
1 x M 2 ,1 t 0.01x M 2 ,12 t 4 ,2 ,2 t 25cos 1.28t 2.7
f 2 ,1,1 x S t , x S t 1,1,1 t , t 2 ,1,1 t 2.8x S 1,2 t , f 3,1,1 x S t , t 0.001 t 0.002x S 1,2 t ,
f 2 ,1,2 x S t , x S t 1,1,2 t , t 2 ,1,2 t
3
1 x S 1,1 t 0.01x S 1,2 2 t 1,1,2 t 2.8 0.1x S 1,2 t 0.01x S 1,1 t 1,1,2 t
1 0.001 t 2 ,1,2 t x S 1,1 t 2.8 2 0.01x S 1,1 t 1,1,2 t 24cos 1.5t
,
f 3,1,2 x S t , t 0.001 t 0.001x S 1,2 t ,
f 4 ,1 x S t , t tanh 0.001 x S t 1 t , 2
f 2 ,2 ,1 x S t , x S t 1,2 ,1 t , t 2 ,2 ,1 t 2.7x S 2 ,2 t , f 3,2 ,1 x S t , t 0.001 t 0.003x S 2 ,2 t ,
f 2 ,2 ,2 x S t , x S t 1,2 ,2 t , t 2 ,2 ,2 t
3
1 x S 2 ,1 t 0.01x S 2 ,2 2 t 1,2 ,2 t 2.7 0.1x S 2 ,2 t 0.01x S 1,1 t 1,2 ,2 t
1 0.002 t 2 ,2 ,2 t x S 2 ,1 t 2.7 2 0.01x S 2 ,1 t 1,2 ,2 t 24cos 1.9 t
,
f 3,2 ,2 x S t , t 0.002 t 0.001 x S t ,
and f 4 ,2 x S t , t tanh 0.002 x S t 2 t are smooth functions.
u t u1 t , u 2 t T
2
2
is the control input of the slave system,
40
t 1 t , 2 t T
2
is the output of the actuator nonlinearity defined
in (3) with u min 10 , l 1 , 2 1.6 , 3 3 , 4 1.8 , 5 0.3 , 6 1 ,
7 0.1, u max 12 and 1 2 1 .
Figures 3 and 4 show the two-dimensional projections of the phase portrait of master system on x M 1,1 t , x M 1,2 t and x M 2 ,1 t , x M 2 ,2 t -planes, respectively.
Besides, Figures 5 and 6 describe the two-dimensional projections of the phase portrait of slave system without control input on
x t , S 2 ,1
x t , S 1,1
x S 1,2 t and
x S 2 ,2 t -planes, respectively
Figure 3. Two-dimensional projection of the phase portrait of master system on x M 1,1 t , x M 1,2 t -plane
41
Figure 4. Two-dimensional projection of the phase portrait of master system on x M 2 ,1 t , x M 2 ,2 t -plane
Figure 5. Two-dimensional projection of the phase portrait of slave system without control input on x S 1,1 t , x S 1,2 t -plane
42
Figure 6. Two-dimensional projection of the phase portrait of slave system without control input on x S 2 ,1 t , x S 2 ,2 t -plane
Our objective consists in designing a neural adaptive control achieving a generalized function projective synchronization between the two different fractional-order chaotic systems (35) and (36), such that: the
master
and
slave
systems
satisfy
Assumptions
1-4
with
1,1 t 1,2 t 2 ,1 t 2 ,2 t exp 0.4t , the scaling factors are given by 1 t sin t and 2 t cos t . According to section 3, the design of the proposed neural adaptive controller is realized based on the following four steps: Step 1: Choose the design parameters employed in the observer (13) and the control
law
(27)
as
follows
1
10 ,
2
8,
100
and t exp 2t . Step 2: Determine the vector t and t by using (13). Step 3: : For approximating the nonlinear unknown functions, select a radial basis function (RBF) neural network w *T t containing 40 nodes with 43
centers evenly distributed in the interval t 40, 40 40, 40 40, 40 40, 40 40, 40 40, 40
and all the widths being equal to 70 . Step 4: Calculate the control input u t by using (27).
The computer simulation results obtained by applying the proposed controller, are presented in Figures 7-12 with the initial conditions ˆ 0 0 , 0 1.2 T and ˆ d 0 0, 0, 0 , 0 .
From Figures 7 and 8, it can be clearly seen that the signals q 2 ,1 y S 1 t and q 2 ,2 y S 2 t
satisfactorily
track
the
signals
1 t q1,1 y M 1 t
and 2 t q1,2 y M 2 t , respectively. Figures 9 and 10 exhibit the control input
u t and the output of the actuator nonlinearities t . Figures 11 and 12 show that the Euclidean norms of synchronization error vectors t and ˆ d t quickly converge to zero.
As one can see, the proposed neural controller is able to synchronize successfully two time-delay incommensurate fractional-order chaotic systems with unknown actuator nonlinearities, i.e., a fast synchronization can be carried out. In addition, it is easy to confirm the robustness of the proposed neural adaptive control scheme with respect to uncertain dynamics, disturbances, and unknown actuator nonlinearities, by these simulation results (Figures 7-12).
44
Figure 7. Signals q 2 ,1 y S 1 t and 1 t q1,1 y M 1 t
Figure 8. Signals q 2 ,2 y S 2 t and 2 t q1,2 y M 2 t
45
Figure 9. Output of the actuator nonlinearity 1 t and control input u1 t
Figure 10. Output of the actuator nonlinearity 2 t and control input u 2 t
Figure 11. Euclidean norm of synchronization error vector t 46
Figure 12. Euclidean norm of synchronization error vector ˆ d t
Example 2: (a physical MIMO fractional-order chaotic system)
Consider two MIMO time-delay chaotic satellite systems (master and slave systems) described by [39]:
Master system
D 1,i x Mi , j t f 1,i , j x M t , x M t 4 ,i , j t , t 0 x Mi , j t 1,i t , t 0 y Mi t x Mi ,1 t , for j 1, , n i , i 1, , 3
47
(37)
Slave system
D 2 ,i x Si , j t f 2 ,i , j x S t , x S t 1,i , j t , t 2 ,i , j t d i , j t t f x S , d , j 1, , n i 1, t 0 t 3 ,i , j t 3 ,i , j 2 ,i D x Si ,ni t f 2 ,i ,ni x S t , x S t 1,i ,ni t , t 2 ,i ,ni t d i ,ni t t f x S , d f 4 ,i x S t , i t , t 0 t 3 ,i ,ni t 3 ,i ,ni i t u i t , x t t , u t 0 , t 0 S i 2 ,i i m i 1 y Si t x Si ,1 t , for i 1, , p
(38)
where: n1 2 , n 2 3 , n3 4 and p 3 are constants, 1,1 2 ,1
1 , 2
1,2 2 ,2
1 3
1,3 2 ,3
and
1 4
are
the
fractional
incommensurate orders, x M t x M 1,1 t , x M 1,2 t , x M 2 ,1 t , x M 2 ,2 t , x M 2 ,3 t , x M 3,1 t , x M 3,2 t , x M 3,3 t , x M 3,4 t
9
and x S t x S 1,1 t , x S 1,2 t , x S 2,1 t , x S 2,2 t , x S 2 ,3 t , x S 3,1 t , x S 3,2 t , x S 3,3 t , x S 3,4 t
9
T
T
are the states of the master and slave systems, respectively ,
y M t y M 1 t , y M 2 t , y M 3 t T
3
and y S t y S 1 t , y S 2 t , y S 3 t T
3
are the outputs of the master and slave systems, respectively,
48
1,1 t 10, 0 T
2 ,1 t 2, 0 T
2
, 1,2 t 3, 0, 0 T
2
, 2 ,2 t 1, 0, 0 T
and 2 ,3 t 5, 0, 0, 0 T
4
3
, 1,3 t 6, 0, 0, 0 T
4
3
are the initial condition functions,
1,1,1 t 2 ,1,1 t 3,1,1 t 4 ,1,1 t 0.01sin t ,
1,2 ,1 t 2 ,2 ,1 t 3,2 ,1 t 4 ,2 ,1 t 0.02cos t , t
1,2 ,2 t 2 ,2 ,2 t 3,2 ,2 t 4 ,2 ,2 t 0.01sin , 2 t
1,3,1 t 2 ,3,1 t 3 ,3 ,1 t 4 ,3 ,1 t 0.01sin , 4
1,3,2 t 2 ,3,2 t 3,3,2 t 4 ,3,2 t 0.001exp t , t
1,3,3 t 2 ,3,3 t 3 ,3 ,3 t 4 ,3 ,3 t 0.002sin , 8
1,1,2 t 2 ,1,2 t 3,1,2 t 4 ,1,2 t 0.01 0.02sin t , t
1,2 ,3 t 2 ,2 ,3 t 3,2 ,3 t 4 ,2 ,3 t 0.02 0.01sin 2 and 1,3,4 t 2 ,3,4 t 3,3,4 t 4 ,3,4 t 0.01 0.01sin t , are timevarying delays,
49
,
1 d 1,1 t 2exp 100 x S t 1,1,1 t t 2 ,1,1 t 1 exp t 3 ,1 ,1 t 100 x S
2
t
d
100 x S t d 1,2 t 2exp 1 x S t 1,1,2 t 2 t 2 ,1,2 t 100 x S t 2 exp d t 3 ,1 ,2 t 1 2 x S x S 1,1 t x S 2 ,1 t 0.4x S 3 ,1 t
0.01sin t ,
0.02sin t ,
d 2 ,1 t 0.01cos t 2exp x S t 1,2 ,1 t 2exp t 2 ,2 ,1 t 2
t
t 3 ,2 ,1 t
exp x
S
exp d
d
,
d 2 ,2 t 0.02cos t 2exp x S t 1,2 ,2 t 2exp t 2 ,2 ,2 t 2
t
t 3 ,2 ,2 t
exp 2 10 x
S
d 2 ,3 t 0.03cos t 2exp 4 t 2 ,2 ,3 t 3 x S t 1,2 ,3 t 2
t
t 3 ,2 ,3 t
exp 4 3 x
S
,
d
,
0.1 x S 1,1 t 0.46 x S 2 ,1 t 2x S 3 ,1 t 2
2 t d 3 ,1 t 0.02cos 2exp 1 x S t 1,3 ,1 t t 2 ,3 ,1 t 2 2 exp t 3 ,3 ,1 t 1 x S
2
t
50
d
,
4 t d 3 ,2 t 0.03cos 2exp 1 x S t 1 , 3 , 2 t t 2 , 3 , 2 t 2 4 2 exp t 3 ,3 ,2 t 1 x S t
,
,
d
5 t d 3 ,3 t 0.04cos 2exp 2 x S t 1 , 3 , 3 t t 2 , 3 , 3 t 2 t 5 2 exp t 3 ,3 ,3 t 2 x S
d
and
7 t d 3 ,4 t 0.05cos 2exp 5 x S t 1,3 ,4 t t 2 ,3 ,4 t 2 7 exp t 3 ,3 ,3 t 5 x S
2
t
d
2 1 0.87 x S 1,1 t 3x S 2 ,1 t 7x S 3 ,1 t 4
are
the external bounded disturbances,
f 1,1,1 x M t , x M t 4 ,1,1 t x M 1,2 t ,
1 6 f 1,1,2 x M t , x M t 4 ,1,2 t x M 2 ,1 t x M 3,1 t 0.4x M 1,1 t x M 3,1 t , 3 6
f 1,2 ,1 x M t , x M t 4 ,2 ,1 t x M 2 ,2 t ,
f 1,2 ,2 x M t , x M t 4 ,2 ,2 t x M 2 ,3 t , f 1,2 ,3 x M t , x M t 4 ,2 ,3 t x M 1,1 t x M 3,1 t 0.175x M 2 ,1 t ,
f 1,3,1 x M t , x M t 4 ,3,1 t x M 3,2 t , f 1,3,2 x M t , x M t 4 ,3,2 t x M 3,3 t , f 1,3,3 x M t , x M t 4 ,3,3 t x M 3,4 t , f 1,3,4 x M t , x M t 4 ,3,4 t x M 1,1 t x M 2 ,1 t 6x M 1,1 t 0.4x M 3,1 t , 51
f 2 ,1,1 x S t , x S t 1,1,1 t , t 2 ,1,1 t
1 x S 1,2 t exp 100 x S t 1,1,1 t t 2 ,1,1 t
1 f 3 ,1,1 x S t , t exp 100 t x S t
f 2 ,1,2 x S t ,x S t 1,1,2 t , t 2 ,1,2 t
tanh 0.001 x S t
2
,
100 x S t exp 1 x S t 1,1,2 t 2 t 2 ,1,2 t
100 x S t f 3 ,1,2 x S t , t exp 1 2 t x S t
,
1 f 4 ,1 x S t , t x S 2 ,1 t x S 3 ,1 t 0.4x S 1,1 t 3
2 6 x S 3 ,1 t tanh 0.001 x S t 1 t 6
,
f 2 ,2 ,1 x S t , x S t 1,2 ,1 t , t 2 ,2 ,1 t x S 2 ,2 t exp x S t 1,2 ,1 t
exp t 2 ,2 ,1 t
f 3,2 ,1 x S t , t exp x S t exp t ,
f 2 ,2 ,2 x S t ,x S t 1,2 ,2 t , t 2 ,2 ,2 t x S 2 ,3 t exp x S t 1,2 ,2 t
exp t 2 ,2 ,2 t
f 3,2 ,2 x S t , t exp 2 t 10 x S t ,
f 2 ,2 ,3 x S t , x S t 1,2 ,3 t , t 2 ,2 ,3 t
x S 1,1 t x S 3 ,1 t 0.175x S 2 ,1 t
,
exp 4 t 2 ,2 ,3 t 3 x S t 1,2 ,3 t
f 3,2 ,3 x S t , t exp 4 t 3 x S t , f 4 ,2 x S t , t x S 1,1 t x S 2 ,1 t 0.175x S 2 ,1 t
tanh 0.004 x S t 2 t 2
52
,
,
,
,
,
f 2 ,3 ,1 x S t , x S t 1,3 ,1 t , t 2 ,3 ,1 t
2 x S 3 ,2 t exp 1 x S t 1,3 ,1 t t 2 ,3 ,1 t
2 f 3 ,3 ,1 x S t , t exp 1 t x S t
f 2 ,3 ,2 x S t , x S t 1,3 , 2 t , t 2 ,3, 2 t
,
4 x S 3 ,3 t exp 1 x S t 1,3 ,2 t t 2 ,3 ,2 t
4 f 3 ,3 ,2 x S t , t exp 1 t x S t
f 2 ,3 ,3 x S t , x S t 1,3 ,3 t , t 2 ,3 ,3 t
5 f 3 ,3 ,3 x S t , t exp 2 t x S t f 2 ,3 ,4 x S t , x S t 1,3 ,4 t , t 2 ,3 ,4 t
,
7 exp 5 x S t 1 , 3 , 4 t t 2 , 3 , 4 t
tanh 0.004 x S t
2
7 f 3 ,3 ,4 x S t , t exp 5 t x S t
,
and f 4 ,3 x S t , t x S 1,1 t x S 2 ,1 t 0.175x S 2 ,1 t
tanh 0.004 x S t 3 t 2
are smooth functions. 53
,
,
5 x S 3 ,4 t exp 2 x S t 1 , 3 , 3 t t 2 , 3 , 3 t
,
,
,
u t u1 t , u 2 t , u 3 t T
t 1 t , 2 t , 3 t T
3
3
is the control input of the slave system,
is the output of the actuator nonlinearity
defined in (3) with u min 5 , l 1 , 2 1.6 , 3 3 , 4 1.8 , 5 0.3 ,
6 1 , 7 0.1 , u max 6 and 1 2 1 .
Our main objective consists in determining a neural adaptive control achieving a generalized function projective synchronization between the two MIMO time-delay chaotic satellite systems (37) and (38), such that: the
master
and
slave
systems
satisfy
Assumptions
1-4
with
1,1 t exp 0.2t , 1,2 t 1,3 t exp 0.22t , 2 ,1 t exp 0.25t and 2 ,2 t 2 ,3 t exp 0.26t , the scaling factors are given by 1 t 1 sin t , 2 t cos t sin t and 3 t 1 sin t .
According to section 3, the design of the proposed neural adaptive controller is carried out based on the following four steps: Step 1: Select the design parameters employed in the observer (13) and the control
law
(27)
as
follows
1
12 ,
2
10 ,
200
and t exp 2t . Step 2: Determine the vector t and t by using (13). Step 3: For approximating the nonlinear unknown functions, select a radial basis function (RBF) neural network w *T t containing 45 nodes with centers evenly distributed in the interval 54
t 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50 50 , 50
and all the widths being equal to 80 . Step 4: Calculate the control input u t by using (27).
The Simulation results obtained by applying the proposed controller, are given in Figures
13-20
with
the
initial
conditions
ˆ 0 0
,
0 1.07
T and ˆ d 0 0, 0, 0, 0, 0 , 0 , 0 , 0 , 0 .
From Figures 13-15, it is clear that the signals q 2 ,1 y S 1 t , q 2 ,2 y S 2 t and
q 2 ,3 y S 3 t satisfactorily track the signals 1 t q1,1 y M 1 t , 2 t q1,2 y M 2 t , and 3 t q1,3 y M 3 t respectively. Figures 16-18 depict the control input u t and the output of the actuator nonlinearities t . Figures 19 and 20 demonstrate that the Euclidean norms of synchronization error vectors t
and ˆ d t
quickly
converge to zero.
As seen in these figures, the proposed neural controller is capable to synchronize successfully two MIMO time-delay chaotic satellite systems with unknown actuator nonlinearities, i.e., a fast synchronization can be carried out. Furthermore, it is easy to confirm the robustness of the proposed neural adaptive control scheme with respect to uncertain dynamics, disturbances, and unknown actuator nonlinearities, by these simulation results (Figures 13-20). In summary, we can conclude that the performance of the proposed controller is validated through this physical system (two MIMO time-delay chaotic satellite systems).
55
Figure 13. Signals q 2 ,1 y S 1 t and 1 t q1,1 y M 1 t
Figure 14. Signals q 2 ,2 y S 2 t and 2 t q1,2 y M 2 t
Figure 15. Signals q 2 ,3 y S 3 t and 3 t q1,3 y M 3 t 56
Figure 16. Output of the actuator nonlinearity 1 t and control input u1 t
Figure 17. Output of the actuator nonlinearity 2 t and control input u 2 t
Figure 18. Output of the actuator nonlinearity 3 t and control input u 3 t 57
Figure 19. Euclidean norm of synchronization error vector t
Figure 20. Euclidean norm of synchronization error vector ˆ d t
5. Conclusions This paper has addressed the generalized function projective synchronization of MIMO uncertain time-delay incommensurate fractional-order chaotic systems with unknown input nonlinearities including backlash-like hysteresis, dead-zone and asymmetric saturation actuators. The main technical difficulties in this research have come from the existence of unknown nonlinear system functions, unmeasured masterslave system states, unknown distributed time-varying delays, unknown time-varying delays, quantized output, unknown control direction, unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct 58
fractional-orders. A novel combination of a linear observer, a neural network, a Nussbaum function, the mean value theorem, the Lyapunov-Razumikhin method and a frequency-distributed model, has been employed to cope with these difficulties under some suitable assumptions. The main advantages of the proposed control design are summarized as follows: (1) the number of the adjustable parameters is reduced, (2) all signals of the closed-loop system remain semi-globally uniformly ultimately bounded and the synchronization errors semi-globally converge to zero. The effectiveness of the proposed controller has been demonstrated throughout simulation examples. In the future, we will study the fuzzy adaptive synchronization problem for a class of incommensurate uncertain non-integer-order chaotic systems with input nonlinearities and faults based on the results of this research work.
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Farouk ZOUARI was born in Tunis, Tunisia, on August 27, 1980. He received his Engineer degree in Electrical Engineering, his magister degree in Automatic and Signal Processing, and his PhD degree in Electrical Engineering from the "National Engineering School of Tunis, University of Tunis El Manar, Tunisia, in 2004, 2005 and 2014, respectively. He is currently a researcher at aboratoire de Recherche en
uto atique (
R ), École Nationale d’Ingénieurs de
Tunis, Université de Tunis El Manar. His current research interests include singular systems, neural control theory, nonlinear control, and intelligent adaptive control.
Abdesselem Boulkroune received his Engineering degree from Setif University in 1995, his Master grade from the military polytechnic school (EMP) of Algiers in 2002, his PhD degree in Automatic from the national polytechnic school (ENP) of Algiers in 2009, in Algeria. In 2003, he joined the automatic control department at Jijel University, in Algeria, where he is currently a professor. His research interests are in nonlinear control and adaptive control.
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Asier Ibeas was born in Bilbao, Spain, on July 7, 1977. He received his MSc degree in Applied Physics and his PhD degree in Automatic Control from the University of the Basque Country, Spain, in 2000 and 2006, respectively. He is currently Associate Professor of Control Systems at Autonomous University of Barcelona, Spain. His research interests include time-delayed systems, robust adaptive control, applications of artificial intelligence to control systems design and nonconventional applications of control such as to epidemic systems, supply chain management and financial systems, fields where he has published more than 130 contributions in international journals and conferences.
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