A unified approach for design of indirect adaptive output-feedback fuzzy controller
Abdesselem Boulkroune* Department of Automatic, University of Jijel, Jijel 18000, Algeria E-mail: [email protected] *Corresponding author
Mohamed TadjineDepartment of the Electric Engineering, ENP, Algiers, Algeria E-mail: [email protected]
Mohammed M’Saad and Mondher Farza GREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN, 6, Bd Maréchal Juin, 14050 Caen Cedex, France E-mail: [email protected] E-mail: [email protected]
Abstract: In this paper, an indirect adaptive fuzzy output-feedback control based on the observer is presented for Single-Input Single-Output (SISO) uncertain non-linear systems. On the basis of the estimation of the tracking error, and without resorting to the famous Strictly Positive Real condition or the filtering of the observation error, a Proportional-Integral law for updating the adjustable parameters is proposed. Then, a unified observer is used to estimate the tracking error. Indeed, the corrective term of the proposed observer involves a well-defined design function which is shown to be satisfied by the commonly used High-Gain (HG)-based observers, namely, for the usual HG observers and the Sliding Modes observers together with their implementable versions. The Lyapunov synthesis approach is used to guarantee a Uniformly Ultimately Bounded property of the observation and tracking errors, as well as of all other signals in a closed-loop system. The viability and the efficiency of the obtained fundamental results are clearly illustrated through a numerical simulation involving the usual benchmark example of the fuzzy control community.
Reference to this paper should be made as follows: Boulkroune, A., Tadjine, M., Saad, M.M. and Farza, M. (2008) ‘A unified approach for design of indirect adaptive output-feedback fuzzy controller’, Int. J. Intelligent Systems Technologies and Applications, Vol. 5, Nos. 1/2, pp.83–103.
84 A. Boulkroune et al.
Biographical notes: Abdesselem Boulkroune received the Engineering degree from the Setif University and the Master grade from the Military Polytechnic School of Algiers (EMP), Algeria, in 1995 and 2002, respectively. In 2003, he joined the Automatic Department at the Jijel University, Algeria, where he is currently an Assistant Professor. His research interests are non-linear control, adaptive control, observer and fuzzy systems.
Mohamed Tadjine received his Engineering degree from the National Polytechnic School of Algiers (ENP), Algeria in 1990 and a PhD degree in Automatic Control from the National Polytechnic Institute of Grenoble (INPG), France, in 1994. From 1995 to 1997, he was a Researcher at the Automatic Systems Laboratory, Amiens, France. Since 1997, Professor Tadjine has been with the Department of Electric Engineering, ENP, Algeria. His research interests are in robust and non-linear control.
Mohammed M’Saad was educated at the Mohammadia School of Engineering in Rabat (Morocco) where he held an Assistant Professor position in 1978. He held a Research Position at the Laboratoire d’Electronique et d’Etude des Systèmes in Rabat where he prepared his Doctor Engineering degree in Process Control. In November 1982, he joined the Laboratoire d’Automatique de Grenoble to work on theory and applications of adaptive control. He received his Doctorat d’Etat-es-Sciences Physiques from the Institut National Polytechnique de Grenoble in April 1987. In March 1988, he held a Research Position at the the Centre National de Recherche Scientifique. In September 1996, he held a Professor position at the Ecole Nationale Supérieure d’ingénieurs de Caen where he is the Head of the Control Group at the GREYC. His main research areas are adaptive control theory, system identification and observation, advanced control methodologies and applications, computer aided control engineering.
Mondher Farza received his degrees of Engineering and MSc in Computer Sciences and Applied Mathematics from the ENSEEIHT Toulouse in 1988, his PhD in control sciences from the INPG Grenoble in 1992. From 1992 to 1994 and 1994 to 1997, he worked at the Laboratoire d’Automatique de Grenoble and at Laboratoire d’Automatique et de Génie des Procédés in Lyon (LAGEP), respectively. In September 1997, he joined the the Control Group at the GREYC and the University of Caen as an Associated Professor. His reasearch interests are in non-linear control and systems and applications.
1 Introduction
Fuzzy Systems (FSs) have been successfully applied to many control problems because they do not need an accurate mathematical model of the system under control and they can cooperate with human expert knowledge. It is also known that FSs as well as Neural Networks (NNs) can approximate uniformly any non-linear continuous function over a compact set (Hornik, 1989; Wang and Mendel, 1992; Wang, 1994). Wang and Mendel (1992) proved that the FS is a universal approximator and the output of FS can be represented by a linear combination of the so-called Fuzzy Basis Functions (FBFs). Later, Castro (1995) relaxed the restrictions on the components of the FSs and proved that the general FSs also have approximation abilities. Based on this property, several adaptive fuzzy control schemes have been developed for a class of uncertain non-linear systems
Unified approach for indirect adaptive output-feedback fuzzy controller design 85
(Sue and Stepanenko, 1994; Wang, 1994; Spooner and Passino, 1996; Chang, 2001). The stability of the closed-loop system, in such schemes, is established according to Lyapunov’s theory. Compared with conventional adaptive control schemes, the key advantage of these adaptive fuzzy control schemes is that there is no need for a linear parameterisation condition. To cope with the approximation errors and the external disturbances, the adaptive fuzzy controllers are augmented by a robust control term that can be a supervisory control (Wang, 1994), Sliding Mode (SM) control (Sue and Stepanenko, 1994; Spooner and Passino, 1996), and/or H control (Chang, 2001). A key assumption in these methods is that all the states of the plant are available for measurement. But this measurement requirement is more an exception than a rule in the engineering practice. This is the rational behind the potential interest for the observer-based controllers.
Based on state/or error observer, direct or indirect adaptive fuzzy control schemes were developed in (Leu, Lee and Wang, 1999; Li and Tong, 2003; Wang et al., 2003; Tong, Li and Wang, 2004; Leu, Wang and Lee, 2005). These schemes require Strictly Positive Real (SPR) condition on the observation error dynamics so that they can use Meyer-Kalman-Yakubovich (MKY) lemma. The original observation error dynamics, which are not SPR in general, are augmented by a low-pass filter designed to satisfy the SPR condition of a transfer function associated with the Lyapunov stability analysis. However, these schemes result in the filtering of FBFs which makes the dynamic order of the system controller/observer very large.
Others schemes of the adaptive fuzzy controllers based on observer were developed in Tong and Li (2002, 2003), Park and Kim (2004) and Park et al. (2005). In these schemes, no SPR condition is need for the stability proof. In Park and Park (2003) and Park et al. (2005), the output observation error is filtered and the state variables of the filter are used to design the underlying update law as well as the robust control term. However, in these control schemes, one can see a kind of redundancy: on one hand, an observer is designed to evaluate the system states which will be employed in FBF construction, on the other hand, a chain of integrators is designed to estimate the filter states which will be used in the design of the robust control term and the parameters update. This chain of integrators is quite similar to an observer. Moreover, these control schemes require a restrictive assumption on a priori boundedness of the control. In Tong and Li (2002) and Tong and Li (2003), the authors developed direct and indirect adaptive fuzzy controllers based on a High-Gain (HG) observer for a class of SISO and Multi-Input Multi-Output (MIMO) non-linear systems, respectively. Although this kind of adaptive fuzzy control schemes can assure the stability of the closed-loop system and achieve a prescribed tracking performance, the HG observer sometimes exhibits a peaking phenomenon in the transient behaviour due to the HG. To overcome this kind of problem, the solutions are proposed in (Khalil, 1995; Seshagiri and Khalil, 2000; Ge and Zhang, 2003).
Unlike the above contributions, in this paper, an indirect fuzzy adaptive controller based on any observer (HG observer, SM (like) observer, etc.) is designed. There are three main contributions to emphasise. First, there are still no reports of the fuzzy (or NNs) adaptive control with SM (like) observers independently on the nature of the system to control, namely, the MIMO or SISO unknown non-linear dynamical systems. In the literature, the design methods of the observer-based-control system are not general and the states are commonly estimated by using a relatively simple observer: a linear error observer (Leu, Lee and Wang, 1999; Li and Tong, 2003; Wang et al., 2003; Tong,
86 A. Boulkroune et al.
Li and Wang, 2004; Park and Kim, 2004; Leu, Wang and Lee, 2005), a HG observer (Seshagiri and Khalil, 2000; Tong and Li, 2002; Ge and Zhang, 2003) and an adaptive observer (Tong and Li, 2003; Park et al., 2005). Secondly, the controller implementation is carried out without resorting to the famous SPR condition, as in Leu, Lee and Wang (1999), Li and Tong (2003), Wang et al. (2003), Tong, Li and Wang (2004) and Leu, Wang and Lee (2005), or the output observation error filtering, as in Park and Park (2003) and Park et al. (2005). In this present work, a new update PI law is designed based on the tracking error estimate. The proportional part involves a well-defined design function derived from the Lyapunov stability analysis. Thirdly, compared with the previous indirect adaptive output-feedback fuzzy control schemes, the suggested approach has the following advantages:
The proposed scheme is more cost-effective than the previous indirect methods (Leu, Lee and Wang, 1999; Tong, Li and Wang, 2004; Park et al., 2005). To control a SISO system y(n) = f(x) + g(x)u + d, the previous indirect methods would require two FS to implement the controller (one FS to estimate f(x), other to estimate g(x)),whereas the proposed method requires only one FS.
In the previous indirect methods (Leu, Lee and Wang, 1999; Tong, Li and Wang, 2004; Park et al., 2005), extra care should be taken to ensure that ˆ (.)g does not equal zero during the adaptation to avoid the singularity of the feedback linearisation. However, in the proposed method, no extra care is need.
In contrast to (Park et al., 2005) where the control input is supposed bounded in advance and before stability analysis, this kind of restrictive assumption will not take place here.
The rest of the paper is organised as follows, following the introduction, the problem to be considered is formulated in Section 2. In Section 3, the FS is reviewed briefly. A unified approach for design of the indirect adaptive output-feedback fuzzy controller is given in Section 4. Simulation results of the inverted pendulum are presented in Section 5, followed by a conclusion given in Section 6.
2 Notation and problem statement
2.1 Notation and preliminaries
Let R denote the real numbers, Rn, the real n-vectors and Rn m, the real m n matrices. One
defines the norm of a vector nx R as 2 21 nx x x and the norm of a matrix
n mA R as Tmax max2 ( ) ( )A A A A , max (.) and min (.) are largest and smallest
eigenvalues of a matrix and max ( )A is the maximum singular value. The absolute value is noted by . .
2.2 Problem statement
Consider the nth order non-linear dynamical system of the form (Wang, 1994):
Unified approach for indirect adaptive output-feedback fuzzy controller design 87
( ) ( 1) ( 1)( , , , ) ( , , , ) ,n n nx f x x x g x x x u d y x (1)
or equivalently of the form
[ ( ) ( ) ],x Ax B f x g x u d y C x (2)
where
T
0 1 0 0 0 10 0 1 0 0 0
, , . 0 0 0 1 0 00 0 0 0 1 0
A B C
u R is the input, y R is the output, f and g are unknown but continuous functions, d is the external bounded disturbances and ( 1) T T
1 2[ , , , ] [ , , , ]n nnx x x x x x x R is the
state vector where not all xi are assumed to be available for measurement. Only the system output y is assumed to be measurable.
Let ym be a bounded reference signal, e = ym – y the output tracking error and x the estimate of x. Denote
( 1) ( 1)1 2[ , ,..., ] , [ , ,..., ] [ , ,..., ] ,n T n T T
m m m nm my y y y e y x e e e e e e
Tn
Tnm
eeeeeexye ]ˆ,...,ˆ,ˆ[]ˆ,...,ˆ,ˆ[ˆˆ 21)1( , .~ eee
In order for Equation (2) to be controllable, it is required that g(x) 0 for x in certain controllability region x Rn. Without loss of generality, one makes the following assumption.
Assumption 1. In Slotine and Li (1991) and Wang et al. (2003), it is assumed that 0 ( ) Hg x g for xx , where Hg is a positive constant.
The control problem consists in determining the control input u to steer the state variables of the system close to the reference signals, while ensuring all involved signals in the closed-loop remain bounded. Since the non-linear functions ( ( ) and ( ))f x g x are not known, and the system states are not available, a FS will be used to approximate the unknown non-linearities, and an observer will be designed to estimate the system states.
3 Description of the used fuzzy logic systems
The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine and a defuzzifier as shown in Figure 1. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input vector T
1 2[ , , , ] MM R to an output f R .
The ith fuzzy rule is written as ( )
1 1ˆ:if and ...and is , then isi i i i
M MR is A A f f (3)
88 A. Boulkroune et al.
where 1 2, , ,and i i iMA A A are fuzzy variables and f i is the fuzzy singleton for the output in
the ith rule. By using the singleton fuzzifier, product inference, and centre-average defuzzifier, the output of the FS can be expressed as follows:
11 T
11
( )ˆ ( ) ( )
( )
ij
ij
r MijAji
r MjAji
ff (4)
where ( )ij
jA is the membership function value of the fuzzy variable j, r is the number
of fuzzy rules, T = [f 1, f 2, …, f r] is the adjustable parameter vector (composed of consequent parameters), and T = [ 1, 2, …, r], where
1
11
( )( )
( )
ij
ij
MjAji
r MjAji
(5)
is the FBF (i.e. the normalised firing strength). In this paper, it is assumed that there
exists always at least one active rule, 1
1
i.e. ( ) 0.ij
rM
jAji
Figure 1 The basic configuration of a fuzzy logic system
The FS Equation (4) is the most frequently used one in the control application. Following the universal approximation results (Wang and Mendel, 1992; Wang, 1994), the FS in Equation (4) is able to approximate any non-linear smooth function f on a compact operating space to any degree of accuracy. In this paper, it is assumed that the structure of the FS and the membership function parameters are properly specified in advance by the designer. But, the consequent parameters, i.e. must be calculated by learning algorithms.
4 Fuzzy adaptive controller and observer design
Following the idea in Park and Kim (2004), the design of the proposed controller is based on the implicit function theorem.
Unified approach for indirect adaptive output-feedback fuzzy controller design 89
After adding and subtracting the terms bu to Equation (1), the nth derivative of the tracking error can be written as follows.
( ) ( ) ( ) ( ) ( )[ ( ) ( ( ) ) ] ( , )n n n n nm m me x y f x g x b u y bu d x u y bu d (6)
where
( , ) ( ) ( ( ) )x u f x g x b u (7)
and b is a positive design constant which must satisfy the following condition (please refer to Jiang et al. (2001), Jiang and Wu (2002), Park and Kim (2004) and Jiang, Wu and Wen (2004)):
( ) , .2 x
g xb x (8)
The state space realisation of the system (6) can be given as follows ( ) [ ( , ) ],nme Ae B x u y bu d e Ce (9)
where m
e y x is the tracking error vector.
From Equation (9), the control input can be determined as
a1
[ ]u u vb
(10)
where ua is an adaptive term designed to cancel the non-linearities (x, u) by utilising the fuzzy logic system and ( )T n
c mv K e y is a linear controller used to stabilise linearised
dynamics. 1 2 T[ , , , ]nc c c cK k k k is the feedback gain vector, chosen such that the
characteristic polynomial of TcA BK is Hurwitz because (A, B) is controllable.
If ( , )x u , in Equation (9) is perfectly cancelled by a fuzzy term *au , i.e.
*a a( , )u x u u , and in addition 0d , one can show that ( ) T 0n
ce K e . Thus, econverges to zero as t .
(x, u) can be estimated by other methods, namely:
1 A perturbation observer (Jiang et al., 2001; Jiang and Wu, 2002; Jiang, Wu and Wen, 2004).
2 Other universal approximators: NN, Radial basis function RBF and Wavelet.
4.1 Fuzzy approximation of ( , )x u
The FS Equation (4) will be employed to approximate (x, u). However, the inputs to the FS are x and u (where u =[–ua + v]/b), since (.) is a function of them. In this formulation, the FS is to be a recurrent FS because the output of the FS ua is directly a feedback into the FS to produce the control input u. Figure 2 illustrates this situation. However, if one uses the recurrent FS, a fixed-point problem must be solved at every time instant and this imposes computational burden (Park and Kim, 2004). To avoid this problem, one applies the implicit function theorem (Khalil, 1995) to guarantee that *
au which satisfies
90 A. Boulkroune et al.
* * *a a a( , , ) ( , ( ) / ) 0h x v u x u v b u (11)
a function of x and v only, which means that the feedback path in Figure 2 is not need. Therefore, a static FS can be employed to approximate (x, u). The following lemma is introduced to show that *
au which satisfies Equation (11) is a function of x and v only.
Figure 2 Recurrent FS
Lemma 1. Park and Kim (2004): if the constant b satisfies condition (8), then there exists a set x Rn and unique *
au which is a function of x and v such that *a ( , )u x v satisfies
Equation (11) for all (x, v) x R.
Proof of lemma 1. The process for proof is similar to that in Park and Kim (2004). First, one shows that the solution *
au to Equation (11) exists. The sufficient condition
for the existence of *au is that the mapping (.) is a contraction over the entire input
domain, i.e. the following inequality (Park and Kim, 2004) must hold:
*a
1u
(12)
which can be rewritten as
* *
* * *a a
( ( ) ( ( ) ) ) 1 ( )( ( ) ) 1 1f x g x b u u g xg x b
b bu u u (13)
where * *a( ) /u u v b . One can easily observe that the last inequality holds if
condition (8) is always satisfied. Secondly, one shows that the function *
a(.)h u is non-singular. Differentiating the
left-hand side of Equation (11) with respect to *au yields
* * * * *a a a a* * *
a a a*
** *
a
( , , ) { ( , ( ) / ) } { ( ) ( ( ) ) }
1 ( ){ ( ) ( ( ) ) } 1 { ( ) } 1
h x v u x u v b u f x g x b u uu u u
u g xf x g x b u g x b
b bu u
(14)
which is non-zero because 0 < g(x). Thus, according to the implicit function theorem (Khalil, 1995), there exists an unique solution *
a ( , )u x v satisfying Equation (11) for all ( , ) xx v R .
Unified approach for indirect adaptive output-feedback fuzzy controller design 91
Lemma 1 makes it possible to employ a static FS rather than a recurrent FS to approximate *
au which enables the controller to avoid solving a fixed problem at every time instant. In this paper, *
a ( , )u x v is identified by a static FS which is well-known as a universal approximator (Wang and Mendel, 1992; Wang, 1994). Then, the input vector to the FS is T T[ , ]x v whose dimension is M = n + 1.
From Equation (4), the fuzzy adaptive control term au can be described as
Ta ( )u (15)
The function *a ( )u which satisfies Equation (11) can be optimally approximated
according to the universal approximation theorem (Wang and Mendel, 1992; Wang, 1994) such that
* * *Ta a( ) ( , ) ( ) ( ) ( )u u (16)
where ( ) is the fuzzy approximation error and * is the optimal parameter vector
which is defined as follows
* *a aarg min sup ( , ) ( )
x Ru u (17)
where denotes the set of suitable bounds on . Note that the optimal parameter vector * is an artificial constant quantity introduced only for analysis purpose and its value is
not need when implementing the controller. However, one needs to following assumption for the optimal parameter vector (Wang and Mendel, 1992; Wang, 1994).
Assumption 2. The optimal parameter vector satisfies
* M (18)
where M is unknown positive constant.According to the universal approximation theorem (Wang and Mendel, 1992; Wang,
1994) for the fuzzy logic systems, there exists a positive constant 0c such that the following inequality holds for all ( ) ( x R):
0( ) .c (19)
Since the input vector to FS ( ) is not available, it will be replaced by its estimated
vector T Tˆ ˆ ˆ[ , ]x v . Thereafter, one will design a unified observer to estimate the
tracking error vector e . Thus, the tracking error estimation e will be used to calculate the system states estimation (by using
mˆ ˆx y e ) and the linear term v , as follows:
( ) Tˆ .nm cv y K e (20)
Then, the output of the FS ua can be described as
92 A. Boulkroune et al.
Ta ˆ ˆ( , ) ( ).u (21)
Now, one needs the following lemma in the stability analysis.
Lemma 2. (Park and Kim, 2004): If Assumption 2 holds, then there exists positive constants 1c and 2c such that
*1 2( , ) ( , ) .x u x u c c (22)
For all ( ) ( x R) where * *a( ) /u u v b and * is the parameter error
vector.
Proof of lemma 2. The process for proof is similar to that in Park and Kim (2004). If b is selected so that condition (8) is always satisfied and using Equations (7) and (13), one has
* * *( , ) ( , ) ( )x u x u g x b u u b u u (23)
By using Equations (10), (18)–(21), the inequality Equation (23) can be further written as
* * * Ta a a a
*T1 2
ˆ ˆ( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) ( )
ˆ( ( ) ( )) ( ) .
bx u x u u v u v u u
b
c c
Denoting 1 t ˆsup ( )c and 2 t 0ˆ( sup ( ) ( ) )c M c yields Equation (22). Now, let us define the observation error vector as ˆ ˆe e e x x . Substituting the
proposed control law, i.e. T ( ) Tˆ ˆ[ ( ) ] /nm cu y K e b into Equation (9), and using
Equations (11) and (16), one gets the tracking error dynamics as T T T
1ˆ( ) [ ( ) ],c ce A BK e B K e w e Ce (24)
where * is the parameter error vector and * *T
1 ˆ( ) [ ( , ) ( , )] [ ( ) ( )] .w x u x u d (25)
Assume that P is a positive definite solution of the following matrix equation: Tc cA P PA Q (26)
where Tc cA A BK and Q is an arbitrary positive definite symmetric matrix.
4.2 Design of the adaptive laws
By using the estimate of the tracking error vector e , one can propose a unified update PI law as follows
T2 1 1 ˆˆ PB ( )e (27)
Unified approach for indirect adaptive output-feedback fuzzy controller design 93
where 1 2, , 0 are design constants and rR is a design function which has to satisfy the following condition:
TT ˆ ˆ[ PB ( )] 0.e (28)
Later, one will give some examples for such function. The adaptation law Equation (27) is augmented by two terms for different purposes: a -modification term (Ioannou and Kokotovic, 1984) which can ensure the parameters boundedness and a proportional term which can contribute to the stability of the closed-loop system. Indeed, thereafter, one
will see that the design of this term according to condition (28) makes possible to have an important negative term in the Lyapunov’s function derivative. Such proportional term can replace the robust control term which is generally added at the adaptive fuzzy control in order to overcome the fuzzy approximation error and the external disturbances.
4.3 Design of a unified observer
In order to solve the problem of the unavailable states, one proposes a unified error observer in this subsection.
Before giving the observer, one introduces the following interesting notations:
a Let be the diagonal matrix defined by
11 1
diag 1, , , n (29)
where > 0 is a real number.
b Let S be the unique solution of the following algebraic Lyapunov equation (Farza and M’Saad, 2004; Farza, M’Saad and Sekher, 2005)
T T .S A S SA C C (30) It can be shown that the solution of Equation (30) is symmetric positive definite.
c 1[ ,..., ] nn R , set , T
1 1( ) [ ( ),0,...,0] nK k R be a vector of
smooth or non-smooth functions satisfying:
T T T 1
1: ( ) .2
nR K C C (31)
According to dynamics of the Equation (24), one can propose the following unified observer to estimate the tracking error vector:
T 1 1ˆ ˆ ˆ ˆ[ ] ( ),ce A BK e S K e e Ce (32)
Subtracting Equation (32) from Equation (24), one gets the following observation error dynamics
T 11ˆ[ ( ) ] ( ),e Ae B w SK e e Ce (33)
where w1 is given by Equation (25). To make easy the stability analysis, let us define a state transformation as
94 A. Boulkroune et al.
.z e (34)
One can easily show that z has the following useful properties:
1(i) nz e z (35)
(ii)C z z Ce e (36)
Since 1A A and ( ) ( )K z K e , Equation (33) can be written in terms of z as follows:
1 T1ˆ( ) [ ( ) ], .z Az S K z B w z C z (37)
Lemma 3. If Assumption 2 is satisfied, there exist positive constants 1 3,c c and 4c such that
1 1 3(a) w c c (38)
T1 4 3ˆ(b) ( ) w c c (39)
For all ( ) ( x R).
Proof of lemma 3.
(a) Using Equation (25), 1w can be bounded as follows:
* *T1 ˆ( ) ( , ) ( , ) [ ( ) ( )] ( ) .w x u x u d t (40)
Since (.) and ( )d t L , by using Equations (18) and (19) and lemma 2, one gets
*1 0 1 2 t t 1 3ˆsup ( ) ( ) sup ( )w c c c d t c c (41)
Denoting 3 0 2 t tˆsup ( ) ( ) sup ( )c c c M d t yields Equation (38).
(b) Since (.) L and using Equation (41), one has
T T1 1 4 3ˆ ˆ( ) ( )w w c c
where 4 t 1ˆsup ( ) .c c
4.4 Stability analysis
The following results conclude the stability of the closed-loop systems (24) and (37).
Theorem. Consider the system (1) with its unified observer Equation (32), and its controller T ( ) Tˆ ˆ[ ( ) ] /n
m cu y K e b . Let the parameter vector be adjusted by the
update law Equation (27) and Assumptions (1) and (2) be true. Then, one has the following property:
Unified approach for indirect adaptive output-feedback fuzzy controller design 95
1 All involved signals are Uniformly Ultimately Bounded (UUB), i.e. ˆ, , ,e e e and u L .
Proof of theorem. Consider the following Lyapunov function candidate
1 2V V V (42)
where 2 21/ n , and
T1V z S z (43)
and
T T2 2 2
1
1 1 ( ) ( ).2 2
V e Pe (44)
Differentiating V1 along of the solution (37) and using Equation (30), one obtains T T T TT T
1 1ˆ2 [ ( ) 0.5 ] 2 [ ( ) ].V z S z z K z z C Cz z S B w (45)
If K(z) is selected so that condition (31) is always satisfied, Equation (45) becomes 2 T T
1 min 1ˆ( ) 2 [ ( ) ].V S z z S B w (46)
Due to special form of the matrix and of the vector B, it can be shown easily that ,S B SB and using lemma 3, (46) can be arranged as follows
2 T T1 min 1
2 Tmin 4 3
ˆ( ) 2 ( )
( ) 2
V S z z SB w
S z z SB c c (47)
Denoting 5 4c c SB and 6 3c c SB , Equation (47) becomes
21 min 4 3
2min 5 6
( ) 2 2
( ) 2 2
V S z c SB z c SB z
S z c z c z (48)
Differentiating V2 with to respect the time along of the solution (24) and using Equation (26), one gets
T T T T T2 1 2 2
1
1 1ˆ ( ) ( ) ( )2 cV e Qe e PB K e w (49)
where * .Using Equations (27), (28) and (38), (49) becomes
96 A. Boulkroune et al.
2 1
22 min 7 8
9 10
1 ˆ ˆˆ ( ) ( )2
1ˆˆ( ( )) ( )2
T T TT T T Tc
TT
T
V e Qe e PB K e w e PB
e PB Q e c e c e e
c e c e
(50)
where 7 t ˆPB sup ( )c , T8 PB cc K , 9 1 PBc c and 10 3 PBc c .
From Equation (50), the choice of according to Equation (28) allows to introduce a negative term which can be important, if 2 is selected large.
Since 22T *2 , Equation (50) can be rewritten as follows
222 *2 min 7 8 9 10
1 ( )2 2 2
V Q e c e c e e c e c e (51)
Using e z and from Equations (48) and (51), the time derivative of Equation (42) can be bounded by
2min 11 6
2min 8 9 10
22 *
( ) 2
1 ( )2
2 2
V S z c z c z
Q e c e z c e c e (52)
where 11 5 72c c c .By using the following inequalities:
2 221111 1
14c
c z z
2 26 2 6
2
12 c z z c
22 23 8
83
14
cc e z z e
2 2299 4
44c
c e e
2 210 5 10
54c e e c
where 1 2, , 3 , 4 and 5 0, one has:
Unified approach for indirect adaptive output-feedback fuzzy controller design 97
Choosing min/ ( )S , min Q( )Q and , one can guarantee that V is negative as long as e is outside the compact set e defined as
min/ .
( )e e eS
(54)
According to a standard Lyapunov theorem (Ioannou and Sun, 1996), one concludes that e is bounded and will converge to e . Moreover, the radius of the set can be made
arbitrary small if is chosen to be sufficiently large. Similarly, the signal e is bounded and will converge to e defined as
min
2/( )e
Qe e
Q (55)
whose radius can be made also arbitrary small if min(Q) is selected to be sufficiently large. The parameter error vector is also bounded and converges to which is defined as
2/ (56)
Because of Assumption 2, i.e. * L , the boundedness of can guarantee that of .Since ,e e L and e e e , then e L . Finally, observe that ˆ( ) , , ( )n
my , and
e L . Hence, u L . This ends the proof of theorem. To summarise, Figure 3 shows the overall scheme of the indirect adaptive fuzzy
control based on any observer, proposed in this paper.
Remark 1. The adaptive law (27) can be rewritten as follows:
T1 1 2
0
ˆˆ( ( PB) ( ))d .t
e (57)
Using the particular expression of the design function satisfying condition (28), the robust PI update law can be derived:
98 A. Boulkroune et al.
1 with a proportional non-smooth term: Tˆ ˆsign( ( PB) ( ))e , where sign
denotes the usual sign function. Although this function satisfies condition (28), it cannot be used due to the discontinuity of the sign function. In order to overcome this problem, one shall use continuous functions which have similar properties that those of sign function.
2 With a proportional smooth term: Tˆ ˆ( PB) ( )e , Tˆ ˆSat( ( PB) ( ))e ,Tˆ ˆtanh( ( ( PB) ( )))k e , Tˆ ˆArctan( ( ( PB) ( )))k e , etc. where Sat denotes
the usual saturation function, tanh the hyperbolic tangent function, Arctan the inverse tangent function, with , 0k are real numbers. Recall that the term is introduced in the adaptive law to improve the robustness in the presence of the fuzzy approximation error and the external disturbances.
Figure 3 Over all scheme of the proposed indirect adaptive fuzzy control based on observer
Remark 2.
1 By giving the particular expressions to ( )K e satisfying condition (31), HG observers and SM observers can be derived. Table 1 summarises the observers obtained according to the choice of ( )K e where 0 0, and 0k p are real numbers.
2 Table 2 summarises the second type of SM observers derived. It is very easy to see that the expressions given in Table 2 satisfy condition (31) for relatively high value of l (e.g. the observer with T
SM ( ) ( sign( ))K e lC C e satisfies condition (31), if
t0.5supl e ). This condition imposed on the gain l is used in the literature (i.e. in the convergence proof of the SM observers, e.g. Barbot, Boukhobza and Djemai (1996), Xiong and Saif (2001) and Jiang and Wu (2002)).
Unified approach for indirect adaptive output-feedback fuzzy controller design 99
Table 1 HG observers and SM observers (first type), obtained according to the choice of ( )K e
)~(eK Observer derived
HG ( ) ( )TK e C C e High-Gain observer (Seshagiri and Khalil, 2000; Tong and Li, 2002)
SM( ) ( ) ( sign ( ))T TK e C C e lC C e Non-smooth Sliding Mode observer, (Jiang and Wu, 2002; Chao et al., 2005)
SM 0( ) ( ) ( Tan ( ))T TK e C C e lC C h k e Smooth Sliding Mode observer (Filipescu, Dugard and Dion, 2003)
SM 0( ) ( ) ( arctan( ))T TK e C C e lC C k e Smooth Sliding Mode observer
SM ( ) ( ) ( Sat( ))T TK e C C e lC C e Smooth Sliding Mode observer
SM( ) ( ) ( /( ))T ToK e C C e lC C e C e Smooth Sliding Mode observer
SM ( ) ( ) sign( )PT TK e C C e l Ce C C eNon-smooth Sliding Mode observer
Table 2 SM observers (second type) obtained according to the choice of )~(eK
( )K e Observer derived
SM( ) ( sign( ))TK e lC C e Non-smooth Sliding Mode observer, (Barbot, Boukhobza and Djemai, 1996)
SM ( ) ( tanh( ))ToK e lC C k e Smooth Sliding Mode observer
SM ( ) ( arctan( ))ToK e lC C k e Smooth Sliding Mode observer
SM ( ) ( Sat( ))TK e lC C e Smooth Sliding Mode observer
SM ( ) ( /( ))ToK e lC C e C e Smooth Sliding Mode observer
0.5SM ( ) ( ) sign( )TK e l Ce C C e Non-smooth Sliding Mode observer (Cao,
2000)
5 Simulation
The control of an inverted pendulum has been a long history because it is a good test bed for new control schemes. Almost all kinds of control strategies have been applied to this system. Since the process is both non-linear and unstable, it gives rise to a number of interesting problems. Its unstable nature makes it a good process for testing real-time controllers, since if the timing fails, the pendulum is likely to fall down. The inverted pendulum can be simulating some systems, e.g. rocket.
Let x1 = be the angle of the pendulum with respect to the vertical line and 2x .The dynamic equations of such system are given by (Wang, 1994):
100 A. Boulkroune et al.
1 1 1
2 2 2
0 1 0 , [1 0]
0 0 1x x x
f gu d yx x x
(58)
where 21 2 2 1 1 1 1( , ) ( sin cos ( ) sin ) /( cos (4 / 3) ( ))f x x mLx x x M m g x mL x L M m and
21 2 1 1( , ) ( cos ) /( cos (4/ 3) ( ))g x x x mL x L M m
and g is the acceleration due gravity, M is the mass of the cart, m is the mass of the pole, L is the half-length of pole and u is the applied force.
From Table 2, for the indirect adaptive fuzzy controller (10), one designs the following SM observer:
T 21 2 0 2 0ˆ ˆ ˆ ˆ2 arctan( ), arctan( ).ce e l k e e K e l k e (59)
It is assumed that the external disturbances d( )t is a square wave having an amplitude 1with a period of 2 (s). The control objective is to maintain the system to track the desired angle trajectory, sin( )my t , under the condition that only the system output y is measurable. The system parameters are given as M =1 kg, m = 0.1 kg, L =0.5 m, g = 9.8 m/s2. The design parameters are selected as = 2, 1 = 100, 2 = 50, b = 1, l = 0.16, k0 = 20, T [144, 24]cK , diag[5,5]Q and 90 .
The initial values are chosen as, Tˆ (0) [0, 0]x , T (0) [0.5,0]x , and (0) 0 (i.e. no a priori information on the fuzzy parameters). The following membership functions for ˆ j ,
1, 2j (i.e. for 1 2ˆ ˆ,x x ), are given as
1 2 321 1ˆ ˆ ˆ ˆ( ) ; ( ) exp( ( ) ); ( ) .
ˆ ˆ1 exp( 3( 1)) 1 exp(3( 1))j j jj j j jA A A
j j
For ˆ j , j = 3 (i.e. for v ), one chooses the following membership functions:
1 2 321 1ˆ ˆ ˆ ˆ( ) ; ( ) exp( ( ) ); ( ) .
ˆ ˆ1 exp( 0.2( 15)) 1 exp(0.2( 15))j j jj j j jA A A
j j
From Remark 1, the following law is selected for the adaptation of the parameters:
Figure 4 illustrates the simulation results of the indirect fuzzy adaptive controller based on SM observer. Figure 4(a) shows that the output x1 = y (dot line) tracks its reference ym
effectively. Similarly, in Figure 4(b), one clearly observes that the state x2 (dot line) follows its reference my with precision, even in the presence of disturbances and unknown dynamics. Also, it can be seen that after a very short period of peaking shown in Figure 4(c), the tracking error estimations (ê1, solid line, ê2, dot line) converge fast to zero. Figure 4(d) indicates the history of the control input u and its boundedness. The
norm of the fuzzy parameter estimates T is given in Figure 4(f) to illustrate the boundedness of fuzzy parameter estimates. Finally, Figure 4(e) clearly shows that the estimated function by the FS ˆ au (dot line) follows the true function
Unified approach for indirect adaptive output-feedback fuzzy controller design 101
(where ( , ) ( ) ( ( ) ) )x u f x g x b u . Recall that the expression of the true function Equation (7), which is used for simulation purposes, is not known by the controller.
Therefore, the results of the simulation show good transient performance and the tracking error is very small with all signals in the closed-loop system being bounded, even though there were external disturbances.
Figure 4 Simulation results (see online version for colours)
6 Conclusions
In this paper, a unified approach for designing of the indirect adaptive output-feedback fuzzy controller for SISO uncertain non-linear systems has been presented. In designing the controller, neither differentiation of the system output nor exact knowledge of non-linearities of the non-linear system are required. Indeed, a unified observer has been designed to estimate the tracking error vector and an adaptive FS has been adopted to approximate the unknown dynamics. On the basis of the tracking error estimate, and without resorting to the famous SPR condition or the filtering of the observation error, a PI law for updating the adjustable parameters has been proposed. The proposed controller, which does not require an offline learning phase, has several advantages which were explained in Section 1. The performances of the proposed controller have been demonstrated in a realistic simulation framework involving a non-linear system.
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