Accepted Manuscript Title: A new Robust Observer-Based Adaptive Type-2 Fuzzy Control for a class of Nonlinear Systems Author: Ardashir Mohammadzadeh Farzad Hashemzadeh PII: DOI: Reference:
S1568-4946(15)00476-7 http://dx.doi.org/doi:10.1016/j.asoc.2015.07.036 ASOC 3111
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
25-1-2015 10-7-2015 27-7-2015
Please cite this article as: Ardashir Mohammadzadeh, Farzad Hashemzadeh, A new Robust Observer-Based Adaptive Type-2 Fuzzy Control for a class of Nonlinear Systems, Applied Soft Computing Journal (2015), http://dx.doi.org/10.1016/j.asoc.2015.07.036 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 proof before it is published in its final 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.
Highlights 1) To cope with the problem of the curse of dimensionality, new type-2 3-dimensional
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membership function is presented. 2) To decrease the computational cost in the type-reduction part, improved and simplified
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type-2 fuzzy neural network is presented.
3) It’s assumed that the all states of the system are unmeasurable, and a robust observer is
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designed.
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4) The effect of external disturbance and approximation errors and state estimation errors are eliminated using the new proposed adaptive compensator.
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5) The stability and zero convergence of the tracking errors is investigated using Lyapunov
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te
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and Barbalat’s theorems.
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*Manuscript
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A new Robust Observer-Based Adaptive Type-2 Fuzzy Control for a class of Nonlinear Systems Ardashir Mohammadzadeha , Farzad Hashemzadeha,b a
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Control Engineering Department, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran b Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada
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Abstract
In this paper, a novel robust observer-based adaptive controller is presented
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using a proposed Simplified Type-2 Fuzzy Neural Network (ST2FNN) and a new three dimensional type-2 membership function is presented. Proposed
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controller can be applied to the control of high-order nonlinear systems and
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adaptation of the consequent parameters and stability analysis are carried out using Lyapunov Theorem. Moreover, a new adaptive compensator is
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presented to eliminate the effect of the external disturbance, unknown nonlinear functions approximation errors and sate estimation errors. In the proposed scheme, using the Lyapunov and barbalat’s theorem it is shown that the system is stable and the tracking error of the system converges to zero asymptotically. The proposed method is simulated on a flexible joint robot, two-link robot manipulator and inverted double pendulums system. Simulation results confirm that in contrast to other robust techniques, our proposed method is simple, give better performance in the presence of noise, Email addresses:
[email protected] (Ardashir Mohammadzadeh),
[email protected] (Farzad Hashemzadeh)
Preprint submitted to Applied Soft Computing
July 9, 2015
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external disturbance and uncertainties, and has less computational cost.
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Keywords: Simplified Interval Type-2 Fuzzy Neural Network, Indirect Adaptive Control, Approximation Error, Adaptive Compensator, robust,
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observer
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1. Introduction
Over the past decade, adaptive control of uncertain nonlinear systems
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has attracted growing interest. To cope with uncertainties many adaptive fuzzy control methods have been presented based on approximation capabil5
ity of fuzzy neural networks [1-16]. Adaptive fuzzy control methods include
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direct and indirect schemes. The direct one uses a fuzzy system to estimate an unknown ideal controller and in the indirect one, fuzzy systems are em-
d
ployed to estimate unknown nonlinear functions. One of the most limitations
10
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of indirect scheme is the effects of external disturbances, function approximation and state estimation errors. To solve this problem some techniques
Ac ce p
are presented in literatures. In [1, 2] upper bound of optimal approximation error(AE) has been estimated and in [3, 4, 5, 6] a variable structure term has been added to the control signal where bounds of unknown functions must be known that is difficult in practice. In [7] based on the bounds of
15
fuzzy approximation parameters and back-stepping technique the robustness of the adaptive fuzzy controller to external disturbance has been improved. In [8, 9] sliding-mode approach has been proposed in which the bounds of unknown functions must be known. Instead of estimation of the unknown functions directly, a Takagi–Sugeno fuzzy system has been employed to ap-
20
proximate a so-called virtual linearized system in [10], which include external
2
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disturbance. In [2] by obtaining the upper bound of optimal AE the stability
ip t
of the closed loop system has been studied. Recently H ∞ techniques [11, 12] have also been used to deal with robust
25
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stabilization in fuzzy control problems. In [13, 14, 15] based on small gain
theorem a way of eliminating for external disturbance effect has been de-
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scribed. A common drawback of aforementioned works [11-15] is that many parameters need to be tuned. Furthermore the sufficient stabilization con-
an
dition for the existence of a positive definite matrix, is difficult to achieve. Furthermore some controller based on intelligent computing have been pre30
sented. For instance in [16] PSO-GA optimization method has been used to
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minimize the steady state error of a plant’s response. In [17], a controller by using generalized type-2 fuzzy system has been proposed to control of a
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mobile robot. In [18] a type-2 fuzzy system has been employed to calculate
35
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the feedback gain such that the steady-state error be reduced. The results has been compared with conventional PID controller.
Ac ce p
Another main problem in the design of controller, based on approximation property of fuzzy systems, is the problem of “curse of dimensionality” and “explosion of complexity” . With the rise of the input variables of a fuzzy system, the number of rules increases exponentially. To handle this problem,
40
the hierarchical fuzzy systems in [19] has been introduced in which the num-
ber of rules is linearly depending to the number of input variables. In this paper a new simplified type-2 fuzzy neural network by using 3-dimentsinal membership functions is presented to overcome the problem of exponential growth of the number of fuzzy rules. Dynamic surface control has been devel-
45
oped in [20, 21, 22] to cope with this problem in the back-stepping approach
3
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procedure.
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Type-2 fuzzy system can model uncertainties better than type-1 [23] and the development of type-2 fuzzy systems is important in the performance
50
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improvement [24]. In the past few years interval type-2 fuzzy system has been widely studied [25, 26, 27, 28, 29]. Adaptive dynamic surface control
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has been studied in [29]. In this paper the uncertainties have been modeled by T2FNNs and closed-loop stability has been investigated based on
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Lyaponuv theorem. Adaptive type-2 fuzzy PI control has been proposed in [30] for class of uncertain nonlinear time delay systems. Backstepping 55
control by using type-2 fuzzy systems has been designed in [31] and has been
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applied on flexible air-breathing hypersonic vehicle. In [32] an adaptive type2 fuzzy control has been presented, in which a H∞ compensator has been
d
added to attenuate the effect of fuzzy approximation error. A review on
60
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the applications of type-2 fuzzy systems in pattern recognition and intelligent control has been presented in [33, 34]. To design optimal type-2 fuzzy
Ac ce p
systems several techniques have been presented for tuning free parameters. For instance recursive orthogonal least-squares algorithm has been proposed in [35] to design type-2 fuzzy controller for prediction of the transfer bar surface temperature in an industrial hot strip mill facility. Multi-objective
65
genetic optimization of type-2 fuzzy controllers has been investigated in [36]. Steepest-descent method, which is referred to as recursive back-propagation mechanism, has been proposed in [37] to adjust the parameters of proposed type-2 fuzzy system. A review on the techniques used in the optimization of type-2 fuzzy controllers has been considered in [38].
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Type-2 fuzzy systems have more computational cost in the type-reduction
4
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part and to solve this problem and reduce the computational burden, some
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simplified typ-2 fuzzy systems have been proposed. For instance in [39, 40], the q coefficient has been used to adjust the proportion of the upper and lower
75
cr
bounds in the type-reduction part instead of Karnik–Mendel (KM) iterative procedure. In [41] original KM algorithm has been improved to increase its
us
speed. Some alternative type-reduction algorithms [42, 43, 44] have been presented which are faster than KM algorithm. The direct defuzzification
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for type-2 fuzzy systems has been proposed in [45] and has been compared with type-reduction. In [46] genetic algorithm has been proposed for type80
reduction. A useful review on type-reduction of type-2 fuzzy sets has been
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presented in [47].
As mentioned above, in the design of adaptive fuzzy controllers, we have
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some challenges such as great computational cost, the effects of approxima-
85
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tion error, external disturbances and the effect of state estimation errors on the stability of closed-loop system. In this paper a novel robust observer-
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based indirect adaptive type-2 fuzzy controller is presented for a class of uncertain nonlinear systems, in which a new adaptive compensator is designed to eliminate of the approximation error, external disturbance effect and state estimation errors. Furthermore simplified type-2 FNN with less
90
computational cost in type-reduction part is proposed. By using proposed 3-dimensional type-2 membership functions, the problem of the “curse of dimensionality” is solved and the proposed controller can be applied to high order systems. Most contribution of this paper are as follows: • To cope with the problem of the curse of dimensionality, new type-2
95
3-dimensional membership function is presented. 5
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• To decrease the computational cost in the type-reduction part, im-
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proved and simplified type-2 fuzzy neural network is presented. • It’s assumed that the all states of the system are unmeasurable, and a
• The effect of external disturbance and approximation errors and state
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100
cr
robust observer is designed.
estimation errors are eliminated using the new proposed adaptive com-
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pensator.
• The stability and zero convergence of the tracking errors is investigated
105
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using Lyapunov and Barbalat’s theorems.
The remainder of this paper is organized as follows: problem formulation
d
and preliminaries are given in Section.2. State observer scheme is presented
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in Section.3 and proposed simplified T2FNN is illustrated in Section.4. Stability analysis and derivation of adaption laws are presented in Section.5.
110
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Simulation studies to demonstrate the performances of the proposed method are provided in Section.6 and Section.7 gives the conclusions of the advocated design methodology.
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2. SYSTEM DESCRIPTION AND PRELIMINARIES
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The following class of single input single output (SISO) nonlinear systems is considered:
cr
x˙ 1 = x2
(1)
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x˙ 2 = x3 .. .
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an
x˙ n = f (x) + g(x)u + d ¯ ¯ y = x1 iT h (n−1) is a state vector and xi , i = where x = [x1 , x2 , ..., xn ]T = x1 , x˙ 1 ..., x1 ¯ 2, ..., n are assumed to be unmeasurable, vector functions f (.) and g(.) are unknown but bounded, u ∈ R is the control signal, y ∈ R is the output of the system and d is the bounded unknown external disturbance. The reference
d
signal and tracking error vectors are defined as follows:
Ac ce p
te
h iT (n−1) yr = yr , y˙ r , ..., yr ¯ T e = x − yr = e, e, ˙ ..., e(n−1) ¯ ¯ ¯ h iT eˆ = xˆ − yr = eˆ, eˆ˙ , ..., eˆ(n−1) ¯ ¯ ¯
(2)
where xˆ and eˆ are the estimate of x and e. The equation (1), can be rewritten ¯ ¯ ¯ ¯ in the state space form xˆ = Ax + B [f (x) + g(x)u + d] , ¯ ¯ ¯ ¯ y = CT x ¯
(3)
7
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··· ··· ···
0
0 1 0 0 0 0 .. . · · · · · · , B = . , C = .. 0 0 0 1 0 0 1 0
ip t
···
0
(4)
cr
···
0 1 0 0 0 0 1 0 A = ··· ··· ··· ··· 0 0 0 0 0 0 0 0
us
in which,
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Assumption1: we assume 0 < g(x) < ∞, so (3) is controllable in certain ¯ n controllability region Uc ⊂ R , [48, 49]. If the system functions f (x) and g(x) ¯ ¯ are known and the disturbance d(t) = 0, based on the well-known feedback
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linearization theory, the ideal controller u∗ can be chosen as follows: (5)
d
u∗ = g −1 (x) −f (x) + yr(n) − KcT e ¯ ¯ ¯
where Kc = [kc1 , kc2 , ..., kcn ]T ∈ Rn are chosen such that polynomial sn +
te
Ac ce p
115
kcn sn−1 + · · · + kc1 be Hurwitz stable [50]. Since the state vector x is not ¯ available to measure and the functions f (x) and g(x) are unknown, the ideal ¯ ¯ controller (5) cannot be implementable. To cope these problems following steps are considered in this paper. • The unknown nonlinear functions are estimated using a proposed adaptive ST2FNN.
120
• A state observer is designed. • A new compensator is designed to eliminate the effect of the external disturbance and the approximation error.
8
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ip t cr us
3. STATE OBSERVER SCHEME
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Fig. 1: Control Block Diagram
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Based on the previously mentioned problems and to solve them, (5) is rewritten as follows:
(6)
te
d
h i u = (ˆ g (xˆ) + εsign (ˆ g (xˆ)))−1 −fˆ(xˆ) + yr(n) − KcT eˆ + us ¯ ¯ ¯ ¯
Ac ce p
where xˆ, gˆ(xˆ), fˆ(xˆ) and eˆ are the estimations of x, g(x), f (x) and e respec¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ tively. The ε is a small positive constant and using the term εsign (ˆ g (xˆ)) in ¯ (6), solve the singularity problem of u and the sign (ˆ g (xˆ)) is a typical signum ¯ defined as follows: 1 gˆ(xˆ) ≥ 0 ¯ sign (ˆ g (xˆ)) = (7) 0 ¯ gˆ(xˆ) < 0 ¯
The us is the adaptive term which has been considered to eliminate the effect of the approximation error and external disturbance and it can guarantee the robustness of the controller. The block diagram of the proposed scheme is shown in Fig.1. By adding and subtracting (ˆ g (xˆ) + εsign (ˆ g (xˆ))) u, (3) ¯ ¯ 9
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becomes
ip t
x˙ = Ax + B {f (x) + [g(x) − gˆ(xˆ) − εsign (ˆ g (xˆ))] u ¯ ¯ ¯ ¯ ¯ ¯ + [ˆ g (xˆ) + εsign (ˆ g (xˆ))] u + d} ¯ ¯ y = CT x ¯ Applying (6) to (8) gives
cr
(8)
M
an
us
e˙ = Ae − BKcT eˆ ¯ ¯ ¯ n o (9) +B f (x) − fˆ(xˆ) + [g(x) − gˆ(xˆ) − εsign (ˆ g (xˆ))] u + us + d , ¯ ¯ ¯ ¯ ¯ e1 = C T e ¯ where e1 = yr − y = yr − x1 . From (9), an observer to estimate the vector e ¯ is obtained as follows [48, 51]:
(10)
te
d
e˙ = Ae − BKcT eˆ ¯ ¯ n ¯ o +B f (x) − fˆ(xˆ) + [g(x) − gˆ(xˆ) − εsign (ˆ g (xˆ))] u + us + d ¯ ¯ ¯ ¯ ¯ T e1 = C e ¯
where, Ko = [kon , kon−1 , ..., ko1 ]T ∈ Rn is the observer gain vector that is
Ac ce p
chosen such that the characteristic polynomial A − Ko C T is Hurwitz. Subtracting (10) from (9) gives the dynamics of the observation error as follows:
e˜˙ = A − Ko C T e˜ ¯ ¯ n o +B f (x) − fˆ(xˆ) + [g(x) − gˆ(xˆ) − εsign (ˆ g (xˆ))] u + us + d , ¯ ¯ ¯ ¯ ¯ T e˜1 = C e˜ ¯
(11)
4. PROPOSED TYPE-2 FUZZY NEURAL NETWORK 125
In this part an interval type-2 FNN is employed where the new proposed 3-dimensional gaussian Membership Functions (MF) with mean m and stan10
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ip t cr
0.8
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0.6 0.4 0.2
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Membership Degree
1
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0 1 0
−1
d
−1
0 −0.5 x1
te
x2
1 0.5
Ac ce p
Fig. 2: Proposed interval type-2 fuzzy set dard deviation σ ∈ [¯ σ, σ] is used which is shown in Fig.2. The T2FNN struc¯ ture is shown in Fig.3. The output of the proposed ST2FNN is computed as follows:
M embership Layer: In this layer the firing degree of upper and lower
MF is obtained as below:
130
! j 2 j 2 m m 1 (x − ) + (x − ) p q k k µ ¯jk (xp , xq ) = exp − j 2 2 σ ¯k ! (12) 2 j j 2 m m 1 (x − ) + (x − ) p q j k k µk (xp , xq ) = exp − 2 2 ¯ σkj ¯ j j in which µ ¯k and µk are the upper and lower firing degrees of j-th MF for the ¯ 11
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k-th input set that is contain inputs p and q. The mjk is the center of j-th
ip t
MF for the k-th input set and, σ ¯kj and σkj are the upper and lower widths of ¯ j-th MF for k-th input set. It must be noted that in the T2FNN flowchart
cr
as given in Fig.3, we consider N membership functions for each input set.
Rule Layer: each node in this layer represent a fuzzy rule which compute
Z¯ j =
n Y
µ ¯pkk
j
Z = ¯
µpkk k=1 ¯
(13)
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in which Z¯ j and Z j are the upper and lower firing degrees of j-th rule respec¯ pk pk tively. µ ¯k and µk are the upper and lower firing degree of pk -th MF for the ¯ k-th input set respectively and n is the number of inputs set.
and yl defines as follows:
M L M P P P Z j wrj + Z¯ j wrj Z¯ j wlj + Z j wlj ¯ j=1 j=1 j=R+1 j=L+1 ¯ yr = , y = l M M R L P P P P Zj + Z¯ j Z¯ j + Zj ¯ j=1 j=1 j=R+1 j=L+1 ¯
te
R P
d
T ype − reduction Layer: Based on the center of sets type reduction yr
(14)
Ac ce p
135
n Y
an
k=1
us
the upper and lower firing degrees [52].
in which R and L are obtained from KM iterative algorithm [52]. wrj and wlj are the consequent parameters in the j-th upper and lower rules respectively. Z¯ j and Z j are the upper and lower firing degree of j-th rule respectively ¯ and M is the number of rules. To increase the controller speed and reduce computational cost, motivated by [39], we propose following type-reduction
12
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in which qr and ql are the adaptive parameters.
cr
yr =
M M M P P P Z¯ j wrj Z j wrj + (1 − qr ) ql Z¯ j wlj + (1 − ql ) Z j wlj ¯ j=1 j=1 j=1 j=1 ¯ , y = l M M M M P P P P qr Z j + (1 − qr ) Z¯ j ql Z¯ j + (1 − ql ) Zj j=1 ¯ j=1 j=1 j=1 ¯ (15) M P
ip t
qr
us
If the number of MFs for each input set be N , then M will be M = N 2 in which n is the number of input set. In the conventional center of sets type
an
reduction, in each sample time, KM iterative algorithm must be repeated that impose great computational cost. In our proposed method only two parameters qr and ql must be updated. We obtain adaptation laws for these
Ac ce p
te
d
M
parameters as follows (see equations (22), (23) and (31). ! ! M M M M M M P P P P P P Z¯ j Zj − Z¯ j wrj Z¯ j wrj − Z j wrj − Z¯ j ¯ ¯ j=1 j=1 j=1 j=1 j=1 j=1 q˙r = 12 γf e˜T Po B ! #2 " ¯ M M M P P P Zj − Z¯ j qr + Z¯ j j=1 ¯ j=1 j=1 ! ! M M M M M M P P P P P P Zj Z¯ j wlj − Z j wlj − Z j wlj Z¯ j − Zj j=1 ¯ j=1 j=1 ¯ j=1 ¯ j=1 j=1 ¯ q˙l = 21 γf e˜T Po B " ! #2 ¯ M M M P P P Z¯ j − Z j ql + Zj j=1 j=1 ¯ j=1 ¯ (16) Output Layer: The defuzzified crisp output defined as the average of yr
and yl .
y=
yr + yl 2
(17)
The unknown nonlinear functions f (x) and g(x) in (5) are estimated using ¯ ¯ the proposed ST2FNN respectively. Optimal tunable parameters of fˆ(xˆ) and ¯
13
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i h ˆ ∗ = arg min supx∈Ux f (xˆ θf ) − f (x) ¯ ¯ θg∗ = arg min supx∈Ux gˆ(xˆ θg∗ ) − g(x) ¯ ¯
θf∗
ip t
gˆ(xˆ) are defined as follows: ¯ (18)
M
an
us
cr
where fˆ(xˆ) and gˆ(xˆ) are the estimation of f (x) and g(x), and Ux is the range ¯ ¯ ¯ ¯ of x . According to the Taylor expansion series: " # ˆ(xˆ|θf ) ∗ 2 ∗ δ f T ∗ ∗ ˆ ˆ ¯ f (xˆ θf ) − f (xˆ θf ) = θf − θf + o θf − θf ¯ ¯ δθf (19) 2 T δˆ g (xˆ|θg ) ¯ gˆ∗ (xˆ θg∗ ) − gˆ(xˆ θg ) = θg∗ − θg + o θg∗ − θg ¯ ¯ δθg 2 2 ∗ ∗ in which o θf − θf and o θg − θg are the higher- order terms. Then
140
Ac ce p
te
d
fˆ(xˆ θf ) = θfT ζf (xˆ) ¯ ¯ (20) T gˆ(xˆ θg ) = θg ζg (xˆ) ¯ ¯ ˆ g (xˆ|θg ) ˆ|θf ) ∆ δˆ ∆ δ f (x ¯ ¯ T and ζg (xˆ) = . The θf and θg are trainable where ζf (xˆ) = ¯ ¯ δθf δθg T parameters that are consist of wr , wl , qr and ql . It must be noted that the
center and width of membership functions are not trained (are fixed), but are chosen to cover all range of its input. From (15) and (17), it is possible δy δy to calculate and as: T δ wr δ wl T δy 1 = δwr T 2
qr Z + (1 − qr )Z¯ δy 1 ql Z¯ + (1 − ql )Z ¯ ¯M , = T M M M P P P P j δw 2 l qr Z j + (1 − qr ) Z¯ j ql Z¯ j + (1 − ql ) Z j=1 ¯ j=1 j=1 j=1 ¯ (21)
14
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δy , we rewrite (15) as follows: δqr ! M M M P P P Z¯ j wrj Z¯ j wrj qr + Z j wrj − j=1 j=1 j=1 ¯ ! M M M P P P Zj − Z¯ j qr + Z¯ j j=1 ¯ j=1 j=1
⇒
1 j=1 δy = δqr 2
cr us
M P Z¯ j
! ! M M M M P P P P Zj − Z¯ j Z¯ j wrj Z¯ j wrj − Z j wrj − j=1 ¯ j=1 j=1 j=1 j=1 ¯ #2 " ! M M M P P P Z¯ j Z¯ j qr + Zj − ¯ j=1 j=1 j=1 (22) M P
an
yr =
ip t
To calculate
Ac ce p
te
d
M
δy similar to (22) is computed as follows: δ ql ! ! M M M M M M P P P P P P Zj Z j wlj Z¯ j − Z¯ j wlj − Z j wlj − Zj j=1 ¯ j=1 ¯ j=1 j=1 j=1 ¯ 1 j=1 ¯ δy (23) = " ! #2 δql 2 M M M P ¯j P j P j Z − Z ql + Z j=1 j=1 ¯ j=1 ¯ Note that derivative equations are calculated at the operating point. 5. STABILITY ANALYSIS In this part the stability and zero convergence of the tracking errors is
investigated using lyapunov and barbalat’s theorems. By adding and subtracting fˆ∗ (xˆ) and gˆ∗ (xˆ)u into (11), it is possible to see that ¯ ¯ n ˙e˜ = A − Ko C T e˜ + B f (x) − fˆ∗ (xˆ) + [g(x) − gˆ∗ (xˆ) − εsign (ˆ g (xˆ))] u ¯ ¯ ¯ ¯ ¯ ¯ ¯ o ∗ ∗ ˆ ˆ + [ˆ g (xˆ) − gˆ(xˆ)] u + f (xˆ) − f (xˆ) + us + d ¯ ¯ ¯ ¯ e˜1 = C T e˜ ¯ (24) 15
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ip t cr us an
Ac ce p
Input set 2
te
d
M
Input set 1
Members hip Layer
Rule Layer
Type Reduction Layer
Output Layer
Fig. 3: Proposed T2FNN structure
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cr
ip t
h ∗ i From (19) and (20), [ˆ g ∗ (xˆ) − gˆ(xˆ)] and fˆ (xˆ) − fˆ(xˆ) are rewritten as fol¯ ¯ ¯ ¯ lows: h ∗ i T fˆ (xˆ) − fˆ(xˆ) = θf∗ − θf ζf = θ˜fT ζf ¯ ¯ (25) T ∗ ∗ T ˜ [ˆ g (xˆ) − gˆ(xˆ)] = θg − θg ζg = θg ζg ¯ ¯ Let’s define approximation errors Ef and Eg as:
an
us
∗ ∆ Ef = f (x) − fˆ (xˆ) ¯ ¯ (26) ∆ Eg = g(x) − gˆ∗ (xˆ) ¯ ¯ It is assumed that f (x) and g(x) are bounded functions, and so Ef and Eg ¯ ¯ in (26) are bounded and upper bounds E¯f and E¯g can be defined. Using (25)
te
d
M
and (26), the equations (24) become e˜˙ = A − Ko C T e˜ ¯ ¯ o (27) +B {Ef + [Eg − εsign (ˆ g (xˆ))] u +θ˜gT ζg u + θ˜fT ζf + us + d ¯ e˜1 = C T e˜ ¯ The Lyapunov design scheme is used to prove the stability and robustness of
Ac ce p
the system. Consider the following Lyapunov function: V (t) = 12 eˆT Pc eˆ + 12 e˜T Po e˜ + 2γ1f θ˜fT θ˜f + 2γ1g θ˜gT θ˜g ¯ ¯ ¯ 2 ¯ 2 2 + 2γ1ˆ E¯ f − Eˆ¯ f + 2γ1ˆ E¯ g − Eˆ¯ g + 2γ1 ˆ d¯ − dˆ¯ ¯ E f
¯g E
(28)
d¯
ˆ¯ are estimations of E¯ , E¯ and d¯ respectively. E¯ , E¯ and where Eˆ¯ f , Eˆ¯ f and d, f g f g
¯ are the upper bounds of Ef , Eg and d, respectively. θ˜ = θ∗ − θ and θ˜ = d, g f f f ˆ ˆ ∗ θ − θ , γ , γ , γ and γ , are the adaptation rate of E¯ , E¯ , θ and θ g
g
ˆ ¯g E
ˆ ¯f E
f
f
g
f
f
g
respectively. Pc and Po are n × n positive definite symmetric matrixes which satisfy the Lyaponuv equations Ac T Pc + Pc Ac = −Qc Ao T Po + Po Ao = −Qo
(29)
17
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where Ac = A − BKcT and Ao = A − Ko C T . Qc and Qo are the arbitrary
ip t
n × n positive definite matrixes. Using (27), time derivative of V becomes
us
cr
V˙ (t) = 21 eˆT Ac T Pc + Pc Ac eˆ + 12 e˜T Ac T Po + Po Ac e˜ + eˆT Pc Ko e˜1 ¯ ¯ ¯ ¯ ¯ o T T T ˜ ˜ +e˜ Po B {Ef + [Eg − εsign (ˆ g (xˆ))] u +θg ζg u + θf ζf + us + d ¯ ¯ ˙ ˙ − γ1f θ˜fT θ˙f − γ1g θ˜gT θ˙g − γ 1ˆ E¯ f − Eˆ¯ f Eˆ¯ f − γ 1ˆ E¯ g − Eˆ¯ g Eˆ¯ g ¯ ¯g E E f ˙ − γ1ˆ d¯ − dˆ¯ dˆ¯ d¯
(30)
an
Considering the terms θ˜fT ζf e˜T Po B − γ1f θ˜fT θ˙f and θ˜gT ζg e˜T Po Bu − γ1g θ˜gT θ˙g , to ¯ ¯ simplify (29) the adaptation laws of θf and θg , can be obtained as follows:
M
∆ θ˙f = γf e˜T Po Bζf ¯ ∆ θ˙g = γg e˜T Po Bζg u ¯
te
d
Note that e˜T Po B is scalar. From (29) and (31), V˙ (t) in (30), becomes ¯ V˙ (t) = − 12 eˆT Qc eˆ − 12 e˜T Qo e˜ + eˆT Pc Ko e˜1 ¯ ¯ ¯ ¯ ¯ T +e˜ Po B {Ef + [Eg − εsign (ˆ g (xˆ))] u +us + d} ¯ ˙ ˙ ¯ ˙ − γ 1ˆ E¯ f − E¯ˆ f E¯ˆ f − γ 1ˆ E¯ g − E¯ˆ g Eˆ¯ g − γ1ˆ d¯ − dˆ¯ dˆ¯
Ac ce p
145
(31)
¯ E f
¯g E
d¯
≤
− 21 eˆT Qc eˆ − 12 e˜T Qo e˜ + eˆT Pc Ko e˜1 ¯ T ¯ ¯ ¯ T ¯ + e˜ Po B E¯ f + e˜ Po B E¯ g |u| ¯ ¯ −e˜T Po Bεsign (ˆ g (xˆ)) u + e˜T Po B us + e˜T Po B d¯ ¯ ¯ ¯˙ ¯ ˙ ˙ ˆ 1 1 1 ˆ ˆ ˆ ˆ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ − γ ˆ E f − E f E f − γ ˆ E g − E g E g − γ ˆ d − d dˆ¯ ¯ E f
¯g E
(32)
d¯
where E¯f , E¯g and d¯ are the upper bounds of Ef , Eg and d respectively. By adding and subtracting e˜T Po B Eˆ¯ f + e˜T Po B Eˆ¯ g |u| + e˜T Po B dˆ into (32), ¯ ¯ ¯ 18
Page 19 of 44
ip t
we get
˙ Eˆ¯ g
1
γEˆ¯
an
us
cr
V˙ (t) ≤ − 12 eˆT Qc eˆ − 12 e˜T Qo e˜ + eˆT Pc Ko e˜1 ¯ ¯ ¯ ¯ ¯ ˙ T 1 ˆ ˆ ˆ + E¯ f − E¯ f e˜ Po B − γ ˆ E¯ f + E¯ g − E¯ g e˜T Po B |u| − ¯ ¯ E¯ f ˙ + d¯ − dˆ¯ e˜T Po B − γ1ˆ dˆ¯ ¯ d ¯ T ˆ T ¯ + e˜ Po B E f + e˜ Po B E¯ˆ g |u| ¯ ¯ T −e˜ Po Bεsign (ˆ g (xˆ)) u + e˜T Po B us + e˜T Po B dˆ¯ ¯ ¯ ¯ ¯
g
(33)
To simplify (33), the adaptation laws of E¯ˆ f , E¯ˆ f and d¯ˆ can be define as:
(34)
d
M
˙ ∆ Eˆ¯ f = γEˆ¯ f e˜T Po B ¯T ˙ ∆ ˆ ¯ E g = γEˆ¯ g e˜ Po B |u| ¯ ˙∆ dˆ¯ = γdˆ¯ e˜T Po B ¯
te
Using adaptation laws (34), equation (33), can be rewritten as follows:
Ac ce p
V˙ (t) ≤ − 21 eˆT Qc eˆ − 12 e˜T Qo e˜ + eˆT Pc Ko e˜1 T ¯ ¯ ˆ ¯ T ¯ ¯ ˆ + e˜ Po B E¯ f + e˜ Po B E¯ g |u| ¯ ¯ −e˜T Po Bεsign (ˆ g (xˆ)) u + e˜T Po B us + e˜T Po B dˆ¯ ¯ ¯ ¯ ¯
(35)
Considering (35), it is possible to design us as ) ( T e ˆ P K e ˜ c o 1 + εsign (ˆ g (xˆ)) u us = −sign e˜T Po B Eˆ¯ f + Eˆ¯ g |u| + dˆ¯ + ¯T ¯ ¯ e˜ Po B + ε ¯ (36) From (35) and (36), it can be easily concluded that 1 1 V˙ (t) ≤ − eˆT Qc eˆ − e˜T Qo e˜ ≤ 0 ¯ 2¯ ¯ 2¯
(37)
19
Page 20 of 44
ip t
t
Z
V˙ (τ )dτ = V (t) − V (0)
(38)
cr
0
us
Since V is positive definite and non-increasing function then Z t V˙ (τ )dτ = V (0) − V (t) ≤ V (0) < ∞ − 0
1 1 − e˜T Qo e˜ ≤ − λmin (Qo ) ke˜k2 ¯ ¯ 2¯ 2
M
1 1 − eˆT Qc eˆ ≤ − λmin (Qc ) keˆk2 , ¯ ¯ 2¯ 2
an
Using the following facts:
(39)
(40)
respectively, then
d
where λmin (Qo ) and λmin (Qc ) are the minimum eigenvalues of Qo and Qc
te
Rt 2 2 λ (Q ) k e ˆ (τ )k + λ (Q ) k e ˜ (τ )k dτ < ∞ min c min o 0q ¯ ¯ Rt keˆ(τ )k2 + ke˜(τ )k2 dτ < ∞ ⇒ 0 q ¯ ¯ R t keˆ(τ )k2 dτ < ∞ q 0 ¯ ⇒ R t ke˜(τ )k2 dτ < ∞ 0 ¯ 1 2
Ac ce p
150
By using Barbalat’s lemma, to show lim eˆ = 0 and lim e˜ = 0, it must be t→∞ ¯ t→∞ ¯ shown that eˆ ∈ `2 , e˜ ∈ `2 and eˆ˙ , e˜˙ are bounded. Note that ¯ ¯ ¯ ¯
(41)
Therefore eˆ ∈ `2 , e˜ ∈ `2 . Considering (10) and (27), and assuming control ¯ ¯ signal and approximation errors Ef and Eg are bounded, it can be seen that
eˆ˙ ∈ `∞ , e˜˙ ∈ `∞ . Using barbalat’s lemma it can be easily concluded that ¯ ¯ lim eˆ(t) = 0 ¯ lim e˜(t) = 0 t→∞ ¯ t→∞
(42)
Remark 1: for soft switching, it is better to replace the sign function in (36) with tanh. 20
Page 21 of 44
Remark 2: in order to avoid parameters drift, the adaptation laws of pamodified as follows:
ip t
rameters in (31) and (34), can be γ e˜T P Bζ or f¯ o f θ˙f = or 0 if or
an
us
cr
if θf < θf < θ¯f ¯ θf = θf and γf e˜T Po Bζf > 0 ¯ ¯ (43) θf = θ¯f and γf e˜T Po Bζf < 0 ¯ θf = θf and γf e˜T Po Bζf < 0 ¯ ¯ θf = θ¯f and γf e˜T Po Bζf > 0 ¯ ¯ where θf and θf are the upper and lower bounds of θf . By defining upper ¯ ˆ ¯E ˆ and d, ¯ˆ the modified and lower bounds for θ and maximum values for ¯E, g
f
g
(44)
Ac ce p
te
d
M
adaptation laws of these parameters would be similar to (43) as: γg e˜T Po Bζg u if θg < θg < θ¯g ¯ ¯ or θ = θ and γg e˜T Po Bζg > 0 g g ¯ ¯ θ˙g = or θg = θ¯g and γg e˜T Po Bζg < 0 ¯ T 0 if θ = θ and γ e ˜ P Bζ < 0 g g o g ¯ ¯g or θg = θ¯g and γg e˜T Po Bζg > 0 ¯ T ˆ ˆ ¯ ¯ γEˆ¯ e˜ Po B if E f < M ax(E f ) ˙ f ¯ Eˆ¯ f = 0 if Eˆ¯ f > M ax(Eˆ¯ f ) T ˆ¯ < M ax(Eˆ¯ ) e ˜ P B |u| if E γ ˆ o g g ¯g ˙ E ¯ Eˆ¯ g = ˆ ˆ ¯ ¯ 0 if E g > M ax(E g ) T ˆ¯ < M ax(d) ˆ¯ γ e ˜ P B if d o ¯ ˙ dˆ ¯ dˆ¯ = ˆ¯ 0 if dˆ¯ > M ax(d)
(45)
6. SIMULATIONS In this section three illustrative examples are given to domonstrate the effectiveness of the proposed controller. Example1. In this example, the 21
Page 22 of 44
proposed method is simulated on a flexible joint robot. Using Euler-Lagrange
1
2
cr
2
ip t
equations, the analytical model of the flexible joint robot is derived as [53]: I q¨ + M gL sin(q ) + K(q − q ) = 0 1 1 2 1 (46) J q¨ − K(q − q ) = u
x˙ 1 = x2 x˙ 2 = − MIgL sin(x1 ) − K J
x˙ 4 =
(x1 − x3 )
(47)
an
x˙ 3 = x4
K I
us
The state space representation of the system is obtained as:
(x1 − x3 ) + J1 u
space representation as follows:
d
z˙ 1 = z2
z˙ 3 = z4
cos(z1 ) +
Ac ce p
M gL I
te
z˙ 2 = z3
z˙ 4 = −
M
This system is feedback linearizable and it is posible to change the state
K I
+
K J
z3 +
(48) M gL I
z22 −
K J
sin(z1 ) +
K u IJ
in which,
z1 = x1
z2 = x2
z3 = − MIgL sin(x1 ) −
K I
z4 = − MIgL x2 cos(x1 ) −
(49) (x1 − x3 ) K I
(x2 − x4 )
We rewrite (48), as follows:
z˙ i = zi+1 , i = 1, 2, 3 z˙ 4 = f (z) + u + d h iT z = z1 z2 z3 z4
(50)
22
Page 23 of 44
The numerical values of the parameters considered in the simulation studies
ip t
are: g = 9.8m/s2 , M = 1kg, K = 1N/m, I = 1Kgm2 and L = 1m. The d is external disturbance that is considered to be white noise with zero mean and variance 0.1. We design controller step by step as follows:
cr
155
1. The feedback and observer gains Kc and Ko are chosen such that all
us
roots of the polynimial s4 + kc4 s3 + kc3 s2 + kc2 s + kc1 and s4 + ko1 s3 + ko2 s2 + ko3 s + ko4 are in −10 and −20 respectively.
an
2. By considering Qc and Qo in (29), as what are coming below, after
Ac ce p
te
d
M
solving (29), the positive definite matrix Pc and Po are: 1 0 0 0 0 1 0 0 −3 Qc = Qo = 10 0 0 1 0 0 0 0 1 1.3881 0.6549 0.0955 0 0.6549 0.3297 0.0542 0.0003 Pc = 0.0955 0.0542 0.0120 0.0001 0 0.0003 0.0001 0 313.5117 −0.0005 −0.5047 0.0005 −0.0005 0.5047 −0.0005 −0.0040 Po = −0.5047 −0.0005 0.0040 −0.0005 0.0005 −0.0040 −0.0005 0.0001
160
(51)
10 1 3. Given that y = z1 = x1 and yr (s) = s+10 , solve (10) to obtain zˆ = s h iT zˆ1 zˆ2 zˆ3 zˆ4 . 4. A proposed ST2FNN fˆ(ˆ z ), is used to estimate f (z) in (50). We consider
three MF as shown in figure 2 for input sets (ˆ z 1 , zˆ2 ) and (ˆ z 3 , zˆ4 ) with 23
Page 24 of 44
2 y1 r
0 −1
5
10
15
20 Time(s)
25
30
2
35
40
cr
−2 0
ip t
1
Error
us
1 0
−2 0
5
10
an
−1
15
20 Time(s)
25
30
35
40
M
Fig. 4: The tracking response when the proposed controller is applied to the flexible joint robot, in example 1 h
i −1 0 1 , upper widths 0.8 and lower widths 0.4 .
d
centers
te
5. We chose adaptation rates γEˆ¯ f = 0.5, γdˆ¯ = 0.5, γθf = 0.1
165
Ac ce p
6. The compensator control signal is obtained as (6) in which, ( ) T e ˜ P K e ˜ c o 1 us = − tanh e˜T Po B Eˆ¯ f + dˆ¯ + T ¯ ¯ e˜ Po B + 0.001 ¯
(52)
Simulation results are shown in Fig.4 and Fig.5, where Fig.4 gives the tracking response and tracking error. Fig.5 depicts the control signal. It can be seen that tracking error converged to small neighborhood around zero, expeditiously. Simulation results confirm that our proposed controller can guarantee the stability and robustness of the closed loop system. Example2: In this example, the proposed controller is applied to a twolink robot manipulator as shown in Fig.6. The dynamic equation of the
24
Page 25 of 44
ip t
Control signal
cr
500
0
0
5
10
15 Time(s)
us
−500
20
25
Ac ce p
te
d
M
an
Fig. 5: Control signal, in example 1
Fig. 6: A two link robot manipulator
25
Page 26 of 44
ip t
system is as follows [54]. q˙ τ −hq˙2 −h (q˙2 + q˙2 ) q¨ H H12 1 = 1 1 + 11 q˙2 τ2 hq˙1 0 q¨2 H21 H22
us
H11 = α1 + 2α3 cos q2 + 2α4 sin q2 a
cr
with
(53)
H12 = H21 = α2 + α3 cos q2 + α4 sin q2
an
H22 = α2
h = α3 sin q2 − α4 cos q2
(54)
M
2 2 α1 = I1 + m1 lc1 + Ie + me lce + me l12 2 α2 = Ie + me lce
α3 = me l1 lce cos δe
d
α4 = me l1 lce sin δe
Ac ce p
x˙ 11 = x12
te
The state space model of the system can be written as follows:
x˙ 12 = f1 (x11 , x12 , x21 , x22 , u2 ) + g1 (x11 , x12 , x21 , x22 ) u1 (55)
x˙ 21 = x22
x˙ 22 = f2 (x11 , x12 , x21 , x22 , u1 ) + g2 (x11 , x12 , x21 , x22 ) u2 y1 = x12 , y2 = x22
where
x11 = q1 , x12 = q˙1 , x21 = q2 , x22 = q˙2 , u1 = τ1 , u2 = τ2 −1 f1 H11 H12 −hx22 −h (x22 + x12 ) x12 = f2 H21 H22 hx12 0 x22 g1 =
H22 H22 H11 −H12 H21
, g2 =
(56)
H11 H22 H11 −H12 H21
26
Page 27 of 44
Simulation parameters are m1 = 1 , l1 = 1, me = 2, δe = 30◦ , I1 = 0.12, lc1 =
ip t
0.5, Ie = 0.25, lce = 0.6 Reference signal is considered to be yr = sin(t). According to the design procedure, f1 , f2 , g1 and g2 in (55), are estimated
using the proposed ST2FNN fˆ1 , fˆ2 , gˆ1 and gˆ2 . Note that functions fˆ1 and
us
cr
fˆ2 have five inputs. We consider three proposed MF for input sets (ˆ x11 , xˆ12 ) h i and (ˆ x21 , xˆ22 ) as shown in figure 2, with centers −1 0 1 , upper widths 0.8 and lower widths 0.4 , and three Gaussian MFs as shown in Fig.7, for
an
inputs u1 and u2 . The controllers u1 and u2 are designed as illustrated in
(57)
d
M
example 1. The controller parameters are chosen as follows: h iT 1 0 3.3462 0.003 , Pc = , Kc = 169 26 Qc = 0 1 0.003 0.0193 h iT 1 0 10.0062 −0.5 , Po = , Ko = 80 1600 Qo = 0 1 −0.5 0.0313 f
Tracking performance and tracking errors are shown in Fig.13, and the con-
Ac ce p
170
te
γEˆ¯ = 0.1, γdˆ¯ = 0.1, γθf = 10, γθg = 10, γwˆ¯g = 0.1, ε = 0.001
trol signal is given in Fig.14. The state estimation results are depicted in Fig.15 and Fig.11. It can be seen from figures 8-10 that the closed loop system is asymptotically stable and confirm that the proposed controller performs very well.
Example3: In this example two inverted pendulums that are connected
by a moving spring and mounted on two carts are considered [7] which its configuration is shown in Fig.12. State space representation of the system
27
Page 28 of 44
0.4 0.2 0 −20
−15
−10
−5
0 input
5
10
15
ip t
0.6
20
cr
Membership Degree
1 0.8
Fig. 7: Gaussian membership functions, which considered for 5-th input in
us
example 2
Our result
an
2
0
−2 0
2
4
6
y1 y2 r
8
10
M
Time(s) Our result
1
−1 0
2
4
d
0
y
1
y2
6
8
10
te
Time(s)
Fig. 8: Tracking performance and tracking error, when proposed controller
Ac ce p
has been applied to a two link robot manipulator in example 2
200
u1
0
−200 0
2
4 Time(s)
200
u2
0
−200
0
2
4 Time(s)
Fig. 9: Control signals, when proposed controller has been applied to a two-link robot manipulator in example 2
28
Page 29 of 44
estimation of x11
2
0 0
0.5
1
1.5 Time(s)
2
5
2.5 3 estimation of x12 x12
0.5
1
1.5 Time(s)
2
3
x21
0.5
1
1.5 Time(s)
10
us
0 2
2.5 estimation of x
3
22
x22
0.5
an
0 −10 0
2.5
estimation of x21
2
−2 0
cr
0 −5 0
ip t
x11
1
1
1.5 Time(s)
2
2.5
3
M
Fig. 10: States estimation performance, when proposed observer has been
0.5 0 0.5
Ac ce p
−0.5 0
te
d
applied to a two link robot manipulator in example 2
1
e11
1.5 Time(s)
2
2.5
10
3 e12
0
−10 0
0.5
1
1.5 Time(s)
2
2.5
3 e21
0.5 0
−0.5 0
0.5
1
1.5 Time(s)
2
2.5
10
3 e22
0
−10 0
0.5
1
1.5 Time(s)
2
2.5
3
Fig. 11: State estimation errors, when proposed observer has been applied to a two link robot manipulator in example 2
29
Page 30 of 44
ip t cr us
an
Fig. 12: Two inverted pendulum on carts can be written as follows [7]:
M
x˙ 11 = x12
x˙ 12 = f1 (x11 , x12 , x21 , x22 ) + g1 u1 x˙ 21 = x22
(58)
d
x˙ 22 = f2 (x11 , x12 , x21 , x22 ) + g2 u2
te
y1 = x12 , y2 = x22
Ac ce p
with
g m 2 x −M x12 cl 11 k[a(t)−cl] (−a(t)x11 cml2
f1 (x11 , x12 , x21 , x22 ) =
sin(x11 )+ + a(t)x21 − x1 + x2 )
g m 2 x −M x22 sin(x21 )+ cl 21 k[a(t)−cl] (−a(t)x21 + a(t)x11 + cml2
f2 (x11 , x12 , x21 , x22 ) =
g1 = g2 =
(59) x1 − x2 )
1 cml2
a(t) = sin(wt),
x1 = sin(w1 t), x2 = sin(w2 t) + L
Simulation parameters are M = m = 1 0, l = 1, c = 0.5, w = 5, w1 = 2, w1 = 3, L = 2. Reference signal is considered to be yr = 0. According to the design proce30
Page 31 of 44
dure, it is assumed that f1 (x11 , x12 , x21 , x22 ) and f2 (x11 , x12 , x21 , x22 ) in (58),
ip t
are unknown and are estimated using the proposed ST2FNN. The initial
cr
membership functions for input sets (ˆ x11 , xˆ12 ) and (ˆ x21 , xˆ22 ) are considered h i similar to the previous examples with centers −1 0 1 , upper widths
175
f1
= 0.05, γEˆ¯
f2
= 0.05, γθf 1 = 5, γθf 2 = 5, ε = 0.001
M
γEˆ¯
an
us
0.8 and lower width 0.4 . The controller parameters are chosen as follows: h iT 1 0 25.01 0 Qc = , Pc = , Kc = 10000 200 0 1 0 0.0025 h iT 1 0 137.5 −0.5 , Po = , Ko = 1100 302500 Qo = 0 1 −0.5 0.0023 (60)
Tracking performance is shown in Fig.13. Control signals are given in Fig.14
d
and state estimation results are depicted in Fig.15 and Fig.16. From figures
satisfactory.
te
12-14, it can be seen that the performance of the proposed controller is
180
Ac ce p
Robust method based on back-stepping technique in [7] is applied to the control of the inverted double pendulums system of example 3. It is valuable to note that the tunable parameters in our proposed method are less than that of [7] and the tracking errors is significantly less. Remark 3: For comparison, the results with using type-1 and type-2
fuzzy systems are given in Table.1, where ui , i = 1, 2 are the control signals,
185
ei = yi −r i = 1, 2 are the tracking errors and eij = xij − xˆij , i, j = 1, 2 are the
state estimation errors, T is final time and ts = 0.001 is the sample time. As it can be seen the performance by using type-2 fuzzy sets is better than type1 counterpart. Furthermore the number MFs to achieve good performance 31
Page 32 of 44
ip t
Our result 0.5
y1 y2
0
−0.5 0
0.5
1
1.5
2
2.5 Time(s) The result of [7]
3
3.5
0.5
4
4.5
5
1
1.5
2
2.5 Time(s)
3
3.5
4
y2
4.5
5
an
0.5
us
y1
0
−0.5 0
cr
r
Fig. 13: Tracking performance, when proposed controller has been applied
Ac ce p
1000
te
d
M
to the inverted double pendulums system
u1
0
−1000 0
0.5 Time(s) Control signal
1
1000
u2
0
−1000 0
0.5 Time(s)
1
Fig. 14: Controls signals, when proposed controller has been applied to the inverted double pendulums system
32
Page 33 of 44
estimation of x11
0.5
−0.5 0
0.5 Time(s)
1
estimation of x12 x12
10
cr
0 −10 0
ip t
x11
0
0.5 Time(s)
1
estimation of x21
0.5
−0.5 0
0.5 Time(s)
10
us
x21
0
1
estimation of x22 x22
an
0 −10 0
0.5 Time(s)
1
M
Fig. 15: States estimation performance, when proposed controller has been
0.5 0
Ac ce p
−0.5 0
te
d
applied to the inverted double pendulums system
e11
0.5 Time(s)
1 e
10
12
0
−10 0
0.5 Time(s)
1 e21
0.5 0
−0.5 0
0.5 Time(s)
10
1 e22
0
−10 0
0.5 Time(s)
1
Fig. 16: States estimation errors, when proposed controller has been applied to the inverted double pendulums system
33
Page 34 of 44
in the case of type-2 is less than type-1. As mentioned before, a higher level
ip t
of uncertainty can be modelled by type-2 fuzzy sets.
Type-1
Type-2
Type-1
e1 2 (t)
8.9897
6.5844
us
e2 2 (t)
10.4152
9.1876
eˆ11 2 (t)
0.6245
eˆ12 2 (t)
536.1854
eˆ21 2 (t)
10.2652
3.9912
3.9108
3.9633
3.9456
0.0980
6.5231
4.7794
515.1561
908.7340
803.7724
14.1028
2.5452
0.8003
53.8202
363.0379
468.0687
t=1
s
TP /ts t=1
t=1
s
TP /ts
M
s
TP /ts
Type-2
an
s
TP /ts
Example 3
te
Example 2
cr
Table 1: The comparison of results, with using type-1 and type-2 fuzzy sets
s
TP /ts t=1
s
TP /ts
96.8741
Ac ce p
t=1
eˆ22 2 (t)
d
t=1
s
TP /ts
u1 2 (t)
30825
18574
14102
13940
u2 2 (t)
906.1204
977.7816
18181
18342
3
2
5
2
t=1
s
TP /ts t=1
Num Of MFs
190
7. CONCLUSION In this research an observer-based indirect adaptive fuzzy control strategy proposed for a class of uncertain high order nonlinear systems in which 34
Page 35 of 44
a new simplified type-2 FNN is employed for online estimation of nonlinear functions of the system. All trainable parameters of ST2FNN are adjusted
ip t
195
base on the adaptation laws which are derived from Lyapunov stability anal-
cr
ysis. In this scheme a new adaptive compensator proposed to eliminate the approximation error, external disturbance effect and state estimation errors.
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The effectiveness of the proposed method are verified by simulations on three different systems. In cotrast to other robust techniques, it is confirmed that
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the proposed method is simple, has less computational cost and give better performance in the presence of noise, external disturbance and uncertainties.
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