Fuzzy Sets and Systems 143 (2004) 211–232
www.elsevier.com/locate/fss
Observer-based indirect adaptive fuzzy-neural tracking
control for nonlinear SISO systems using VSS and
H
∞approaches
Tsung-Chih Lin
a;∗, Chi-Hsu Wang
b, Han-Leih Liu
baDepartment of Electronic Engineering, Feng-Chia University, 100 Wenhwa Road, Seatwen, Taichung, Taiwan 407, China
bDepartment of Electrical and Control Engineering, Chiao-Tung University, Hsinchu, Taiwan Received 21 November 2001; received in revised form 18 February 2003; accepted 8 April 2003
Abstract
Fuzzy control is a model free approach, i.e., it does not require a mathematical model of the system under control. An observer-based indirect adaptive fuzzy neural tracking control equipped with VSS and H∞ control algorithms is developed for nonlinear SISO systems involving plant uncertainties and external
disturbances. Three important control methods, i.e., adaptive fuzzy neural control scheme, VSS control design and H∞ tracking theory, are combined to solve the robust nonlinear output tracking problem. A modi5ed
algebraic Riccati-like equation must be solved to compensate the e6ect of the approximation error via adaptive fuzzy neural system on the H∞ control. The overall adaptive scheme guarantees the stability of the resulting
closed-loop system in the sense that all the states and signals are uniformly bounded and arbitrary small attenuation level of the external disturbance on the tracking error can be achieved. The simulation results con5rm the validity and performance of the advocated design methodology.
c
2003 Elsevier B.V. All rights reserved.
Keywords: State observer; Indirect adaptive control; FNNVSS,
1. Introduction
The fuzzy controllers provide a systematic and e9cient framework to incorporate linguistic fuzzy information from human expert [10,15,21]. Furthermore, fuzzy control is a model free approach,
∗Corresponding author.
E-mail address: tclin@fcu.edu.tw(T.-C. Lin).
0165-0114/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-0114(03)00167-2
i.e., it does not require a mathematical model of the system under control. Hence fuzzy control has found more extensive applications for a wide variety of industrial systems and consumer prod-ucts. Control engineers are now facing more and more complex systems, and the mathematical models of these systems are more and more di9cult to obtain. Thus, in control engineering, model free approaches become more important. There are some model free approaches in conven-tional control, such as nonlinear adaptive control and PID control. Fuzzy control gives another model free approach [9,1,5,22,23]. In the meantime, variable structure systems (VSS)
con-trol [6,7] design technique has been developed as a popular robust strategy to treat
un-certain systems with external disturbance, quickly varying parameters and unmodeled dynamics.
Fuzzy systems are structured numerical estimators. They start from highly formalized insights about the structure of categories found in the real world and then articulate fuzzy IF-THEN rules as a kind of expert knowledge. Also, they combine fuzzy sets with fuzzy rules to produce overall complex nonlinear behavior. We have witnessed a rapid growth in the use of fuzzy system in a wide variety of consumer products and industrial systems.
The adaptive control for feedback linearizable nonlinear systems is an approach to nonlinear con-trol design that has attracted a great deal of interest in the nonlinear concon-trol community for at least a quarter of a century. The nonlinear adaptive problem is transformed into a linear
adap-tive control problem by feedback linearization [12,13,19,26,20]. Therefore, the linear adaptive
con-trol methodologies can be applied to acquire the desired performance. More recently, an important adaptive fuzzy control system has been proposed to incorporate with the expert information
sys-tematically and the stability is guaranteed by universal approximation theorem [2,3,16,17,21,24,25].
An adaptive fuzzy controller is constructed from adaptive fuzzy systems. An adaptive fuzzy sys-tem is de5ned as a fuzzy logic syssys-tem equipped with a training algorithm, where the fuzzy logic system is constructed from a set of fuzzy IF-THEN rules using fuzzy logic principles, and the training algorithm adjusts the parameters of the fuzzy logic system based on training data. The adaptive fuzzy controllers are classi5ed into direct and indirect adaptive fuzzy controller
cate-gories [3,14,25,4]. More speci5cally, direct adaptive fuzzy controllers use fuzzy logic system as
controllers; therefore, linguistic fuzzy control rules can be directly incorporated into the controllers. On the other hand, indirect adaptive fuzzy controllers use fuzzy logic systems to model the plant and construct the controllers assuming that the fuzzy logic systems represent the true plant; there-fore, fuzzy IF-THEN rules describing the plant can be directly incorporated into the indirect adap-tive fuzzy controller. In this paper, we develop the observer-based indirect adapadap-tive fuzzy-neural
tracking control for nonlinear SISO systems by using VSS and H∞ approaches under the
con-straint that only the system output is available for measurement. The proposed design method
attempts to combine the attenuation technique via H∞ tracking design scheme, fuzzy logic
uni-versal approximation theorem and adaptive control algorithm for the robust tracking design of the nonlinear systems with a large uncertainty or unknown variation in the plant parameters and structures.
This paper is organized as follows. First, the problem formulation is presented in Section 2. A
brief description of fuzzy-neural networks is then made in Section 3. VSS indirect adaptive H∞
tracking control design is given in Section 4. Simulation examples to illustrate the performance of
the proposed method is provided in Section 5. Section 6 gives the conclusions of the advocated
2. Problem formulation
Consider the nth-order nonlinear dynamical system of the form ˙x1= x2 ˙x2= x: 3 : : : ˙xn= f(x1; x2; : : : ; xn) + g(x1; x2; : : : xn)u + d y = x1 (1)
or equivalently the form
x(n)= f(x;x; : : : ; x: (n−1)) + g(x;x; : : : ; x: (n−1))u + d; y = x; (2)
where f and g are unknown but bounded functions, u ∈ R and y ∈ R are the control input and output of the system, respectively, and d is the external bounded disturbance. We can rewrite (2) in state space representation ˙x = Ax + B[f(x) + g(x)u + d]; y = CTx; (3) where A = 0 1 0 0 · · · 0 0 0 0 1 0 · · · 0 0 · · · · 0 0 0 0 · · · 0 1 0 0 0 0 · · · 0 0 ; B = 0 0 ... 0 1 ; C = 1 0 ... 0 0 (4)
and x = [x1; x2; : : : ; xn]T= [x;x; : : : ; x: (n−1)]T∈ Rn is a state vector where not all xi are assumed to be
available for measurement. Only the system output y is assumed to be measurable. In order for
(2) to be controllable, it is required that g(x) = 0 for x in certain controllability region Uc ⊂ Rn.
Without loss of generality, we assume 0¡g(x)¡∞ for x ∈ Uc. The control object is to force the
system output y to follow a given bounded reference signal yr, under the constraint that all signals
involved must be bounded.
To begin with, the reference signal vector yr, the tracking error vector e will be de5ned as
yr= [yr; ˙yr; : : : ; y(nr −1)]T ∈ Rn;
e = x − yr= [e; ˙e; : : : ; e(n−1)]T∈ Rn;
ˆe = ˆx − yr= [ ˆe; ˙ˆe; : : : ; ˆe(n−1)]T ∈ Rn;
If the functions f(x) and g(x) are known and the system is free of external disturbance d, then we can choose the controller u∗ to cancel the nonlinearity and design controller. Specially, let kc= [kc
1; k2c; : : : ; knc]T∈ Rn to be chosen such that all roots of the polynomial p(s) = sn+ kncsn−1 + · · · + kc
1 are in the open left half-plane and control law of the certainty equivalent controller is
obtained as [17]
u∗ = g(x)1 [−f(x) + y(n)
r − kTce]: (5)
Substituting (5) into (2), we obtain the closed-loop system governed by e(n)+ kc
ne(n−1)+ · · · + k1ce = 0;
where the main objective of the control is limt→∞ e(t) = 0. However, f(x) and g(x) are unknown, the ideal controller (5) cannot be implemented and not all system states x can be measured. We have to design an observer to estimate the state vector x in the following context.
2.1. State observer scheme
Replacing the functions f(x), g(x) and e in (5) by the estimation functions f(ˆx), g(ˆx) and ˆe, the
control law (5) is rewritten as
u = ˆg( ˆx)1 [− ˆf( ˆx) + y(n)
r − kTcˆe]: (6)
Applying (6) to (3) and after some simple manipulations, we can obtain the error equation ˙e = Ae − BkT
cˆe + B{f(x) − ˆf( ˆx) + (g(x) − ˆg( ˆx))u + d};
e1= CTe; (7)
where e1= yr− y = yr− x1 denotes the output tracking error. Therefore, the tracking problem can
be converted into the regulation problem to design a state observer for estimating the state vector e
in (7) in order to regulate e1 to zero.
From (7), the following is an observer that estimates the state vector e in (7)
˙ˆe = Aˆe − BkT
cˆe + ko(e1− ˆe1);
ˆe1= CTˆe; (8)
where kT
o= [kno; kno−1; : : : ; k1o] is the state observer gain vector.
The observation errors are de5ned as ˜e = e − ˆe and ˜e1= e1− ˆe1. Subtracting (8) from (7), we can
obtain the output error dynamics
˙˜e = (A − koCT) ˜e + B{f(x) − ˆf( ˆx) + (g(x) − ˆg( ˆx))u + d};
where = A − koCT= −ko n 1 0 0 · · · 0 0 −ko n−1 0 1 0 · · · 0 0 ... ... ... ... ... ... ... −ko 2 0 0 0 · · · 0 1 −ko 1 0 0 0 · · · 0 0 :
Since (C; ) pair is observable, the observer gain vector ko can be chosen such that the characteristic polynomial of is strictly Hurwitz (i.e., the roots of the closed-loop system are in the open left half-plane) and we know that there exists a positive de5nite symmetric n × n matrix P which satis5es the Lyapunov equation
TP + P = −Q; (10)
where Q is an arbitrary n × n positive de5nite matrix. Let us rewrite (8) as
˙ˆe = ˆAˆe + koCT˜e; (11)
where ˆA = A − BkT
c is a strictly Hurwitz matrix. Therefore, there exists a positive de5nite symmetric
n × n matrix ˆP which satis5es the Lyapunov equation ˆATˆP + ˆP ˆA = − ˆQ;
(12) where ˆQ is an arbitrary n × n positive de5nite matrix. Let Vˆe=12ˆeTˆP ˆe, then by using (11) and (12),
we have
˙Vˆe=12 ˙ˆeTˆP ˆe + 12ˆeTˆP ˙ˆe = 12{ ˆA ˆe + koCT˜e}TˆP ˆe + 12ˆeTˆP{ ˆA ˆe + koCT˜e}
= −1
2ˆeT ˆQ ˆe + ˆeTˆPkoCT˜e: (13)
Since ˆQ and ko are determined by the designer, we can choose ˆQ and ko, such that ˙Vˆe60. Hence,
Vˆe is a bounded function and there exists a constant value LVˆe, such that Vˆe6 LVˆe.
3. The Takagi–Sugeno FNN system [23]
Fuzzy logic systems address the imprecision of the input and output variables directly by de5ning them with fuzzy numbers (and fuzzy sets) that can be expressed in linguistic terms (e.g., slow, medium and fast). The basic con5guration of the Takagi–Sugeno (T–S) FNN system includes a fuzzy rule base, which consists of a collection of fuzzy IF-THEN rules in the following form:
R(l): IF x
1 is F1l; and · · · ; and xn is Fnl;
where Fl
i are fuzzy sets and Tl = [ql0; ql1; : : : ; qnl] is a vector of the adjustable factors of the
con-sequence part of the fuzzy rule. Also yl is a linguistic variable, and a fuzzy inference engine to
combine the fuzzy IF-THEN rules in the fuzzy rule base into a mapping from an input linguistic vec-tor xT= [x1; x2; : : : ; xn] ∈ Rn to an output variable y ∈ R. Let M be the number of the fuzzy IF-THEN
rules. The output of the fuzzy logic systems with central average defuzzi5er, product inference and singleton fuzzi5er can be expressed as
1y(x) = M l=1 l· yl M l=1 l = M l=1l· Tl[1xT] M l=1 l ; (15) where !Fl
i(xi) is the membership function value of the fuzzy variable xi and
l=n i=1!Fl
i(xi) is the
truth value of the lth implication. The actual membership functions Fl
i in Eq. (14) are normally the
bell-shaped functions with parameters to be de5ned to suit di6erent applications. Eq. (15) can
be rewritten as
y(x) = T"(x); (16)
where T= [ T
1 T2· · · TM] is an adjustable parameter vector and "T(x) = ["1(x), "2(x); : : : ; "M(x)] is a
fuzzy basis function vector de5ned as "l(x) = l[1 xT]
M l=1 l
: (17)
When the inputs are fed into the T–S FNN, the truth value l of the lth implication is computed.
Applying the common defuzzi5cation strategy, the output of the neural network expressed as (15)
is pumped out. The overall con5guration of the T–S FNN is shown in Fig. 1.
Based on the universal approximation theorem [2,24], the above fuzzy logic system is capable of
uniformly approximating any well-de5ned nonlinear function over a compact set Uc to any degree of
accuracy. Also it is straightforward to show that a multi-output system can always be approximated by a group of single-output approximation systems.
4. VSS Adaptive H∞tracking control design
To begin with, our task is to use fuzzy neural network to approximate the nonlinear functions f(x), g(x) and develop the adaptive laws to adjust the parameters of fuzzy neural networks to attenuate the approximation errors and external disturbance.
First, from (16) the fuzzy systems ˆf(ˆx; f) and ˆg(ˆx; g) can be described as
ˆf( ˆx; f) = "T( ˆx) f
and
ˆg( ˆx; g) = "T( ˆx) g;
where f = [ f1; : : : ; fn]T and g = [ g1; : : : ; gn]T also "(x) = ["1; : : : ; "n]T. The universal fuzzy
system ˆf(ˆx; f) with input vector ˆx ∈ Uˆx for some compact set Uˆx∈ Rn is proposed here to
approxi-mate the uncertain function f(x), where f is a parameter vector to be tuned. Similar, the universal fuzzy system ˆg(ˆx; g) is de5ned here to approximate the uncertain functions g(x), where g is a parameter vector to be tuned.
Next, in order for the linearly parameterized fuzzy model is employed in the approximation proce-dure of the uncertain dynamics, the membership functions !fl
i(xi) and !gli(xi) for 16i6n, 16l6M
should be speci5ed beforehand in this paper. By universal approximate theorem [2,25], there exist
optimal approximation parameters ∗f and ∗g such f(x) and g(x) can be approximated as close as
possible. By using the adaptive laws, these optimal parameters are arti5cial quantities required only for analytical purpose as in much previous adaptive fuzzy research.
In order to guarantee the parameters f and g within the given constraint regions # f and
# g for all t¿0, respectively, we use the parameter projection algorithm [3,14,16,17]. All constraint
regions # f and # g are assumed to be convex. First, let us describe the convex hypercube of # f
as
#0f= { f|afi6 fi6 bfi; 1 6 i 6 n}
and
# f= { f|afi− &f6 fi6 bfi+ &f; 1 6 i 6 n};
where all values afi, bfi and &f¿0 are speci5ed by the designer, also de5ne (f= "BTPe with
algorithm with respect to # f will be given as [7,27] ˙ fi= r1L*fi if fi¿ bfi and *fi¿ 0; r1*ˆfi if fi¡ afi and *fi¡ 0; r1*fi otherwise; 1 6 i 6 n; (18) where *fi is the ith component of (f and r1 denotes the adaptive gain, also
L*fi = 1 +bfi&− fi f *fi and ˆ *fi = 1 + fi&− afi f *fi:
Next, the convex hypercube of # g is given by
#0g= { g|agi 6 gi6 bgi; 1 6 i 6 n}
and
# g= { g|agi− &g6 gi 6 bgi+ &g; 1 6 i 6 n};
where all values agi, bgi and &g¿0 are speci5ed by the designer too, also de5ne (g= "BTPeu. The
smooth projection algorithm with respect to # g will be given as
˙ gi= r2L*gi if gi¿ bgi and *gi¿ 0; r2*ˆgi if gi¡ agi and *gi¡ 0; r2*gi otherwise; 1 6 i 6 n; (19) where *gi is the ith component of (g and r2 denotes the adaptive gain, also
L*gi = 1 +bgi&− gi g *gi and ˆ *gi = 1 + gi&− agi g *gi:
The minimum approximation errors are de5ned as Mf( ˆx) = f(x) − ˆf( ˆx; ∗f)
and
In order to derive the control laws, we need the following assumptions hold for all ˆx ∈ Uˆx, f∈ # f
and g∈ #gf.
A1. There exists an upper bound ,f¿0, such that |Mf(ˆx)| 6 ,f:
A2. There exists an upper bound 0¡,g¡1, such that |Mg(ˆx) ˆg−1(ˆx; g)| 6 ,g:
A3. There exists a 5nite constant Bd¿0, such that
∞
0 d
2(t) dt 6 B
d; i:e:; d(t) ∈ L2[0; ∞):
Theorem 1. For nonlinear SISO system (2), let the assumptions A1–A3 be true. If the VSS
adaptive FNN control is chosen as u = ˆg( ˆx; 1 g) − ˆf( ˆx; f) + y(n) r − kTcˆe + us+ uh (20) where ˙ f in (18), ˙ g in (19) with b2i− &2¿0 for all 16i6m2, and the VSS controller us and the
robust H∞ controller uh are
us = −1 − ,Be( ˆx) gsgn(B
TP ˜e); (21)
uh= −2r1 BTP ˜e; (22)
where Be(ˆx) = ,f+ ,g| − ˆf(ˆx; f) + y(n)r − kTcˆe|, r is a positive scalar value and P = PT¿0 is the
solution of the following Riccati-like equation: TP + P + Q + PB 1 /2− 1 − ,g r BTP = 0 (23)
then the H∞ tracking can be achieved for a prescribed attenuation level /.
Proof. Let us reconsider the output error dynamic equation (9) and take into account the minimum
approximation errors, the error dynamic equation (9) can be rewritten as
˙˜e = ˜e + B{−"T˜
f− "T˜ gu + Mf( ˆx) + Mg( ˆx)u + us+ uh+ d}; (24)
where ˜ f= f− ∗f, ˜ g= g− ∗g, us is the VSS controller and uh is the robust H∞ controller.
Consider the Lyapunov-like function candidate
V = 1 2 ˜eTP ˜e + 1 2r1 ˜ T f˜ f+ 2r1 2 ˜ T g˜ g: (25)
Di6erentiating (25) with respect to time, and using the control law (20) we obtain ˙V = 12{ ˜e + B(1 + Mg( ˆx) ˆg−1( ˆx; g))(us+ uh) − B"T˜ f− B"T˜ gu + BMf( ˆx)
+ B Mg( ˆx) ˆg−1( ˆx; g)(− ˆf( ˆx; f) + y(n)
r − kTcˆe) + Bd}TP ˜e +12 ˜eTP{ ˜e + B(1 + Mg( ˆx) ˆg−1( ˆx; g)(us+ uh) − B"T˜ f− B"T˜ gu + BMf( ˆx) + B Mg( ˆx) ˆg−1( ˆx; g)(− ˆf( ˆx; f) + y(n) r − kTcˆe) + Bd} +r1 1 ˙˜ T f˜ f+r1 2 ˙˜ T g˜ g =1
2 ˜eT(TP + P) ˜e + uh(1 + Mg( ˆx) ˆg−1( ˆx; g))BTP ˜e + dBTP ˜e + us(1 + Mg( ˆx) ˆg−1( ˆx; g)BTP ˜e − ˜ Tf"BTP ˜e + r1 1 ˙˜ T f˜ f− ˜ Tg"BTP ˜eu + 1 r2 ˙˜ T g˜ g+ (Mf( ˆx) + Mg( ˆx) ˆg−1( ˆx; g)(− ˆf( ˆx; f) + y(n)r − kTcˆe)BTP ˜e: (26)
By substituting the robust H∞ controller uh in (22) and completing the square, we get
˙V = 12 ˜eT TP + P + PB 1 /2 − 1 + Mg( ˆx)g−1( ˆx; g) r BTP ˜e −12 1 /BTP ˜e − /d 2 +12/2d2+ u s(1 + Mg( ˆx) ˆg−1( ˆx; g))BTP ˜e − ˜ Tf"BTP ˜e + 1 r1 ˙˜ T f˜ f− ˜ Tg"BTP ˜eu +r1 2 ˙˜ T g˜ g + (Mf( ˆx) + Mg( ˆx) ˆg−1( ˆx; g)(− ˆf( ˆx; f) + yr(n)− kTcˆe))BTP ˜e: (27)
In fact, from adaptive law in (16) we can get (1=r1) ˙˜ T
f˜ f− ˜ Tf"BTP ˜e60 and f(t) ∈ # f for all t¿0
if f(0) ∈ #0f [7]. Also, from the adaptive law in (19) we obtain (1=r2) ˙˜
T
g˜ g− ˜ Tg"BTP ˜eu60 and
g(t) ∈ # g for all t¿0 if g(0) ∈ #0g. Next, from the assumptions A1–A2 and the VSS controller
us in (21), we obtain
us(1 + Mg( ˆx) ˆg−1( ˆx; g))BTP ˜e + (Mf( ˆx) + Mg( ˆx) ˆg−1( ˆx; g)(− ˆf( ˆx; f) + yr(n)− kTcˆe))BTP ˜e 6 −Be(1 + Mg( ˆx) ˆg1 − ,−1( ˆx; f))
g |B
TP ˜e| + B
From above results and Eq. (28), let us consider the Riccati-like equation (23), then the Eq. (27) can be rewritten as ˙V 6 12 ˜eT TP + P + PB 1 /2 − 1 + Mg( ˆx)g−1( ˆx; g) r BTP ˜e + 12/2d2 612 ˜eTTP + P + PB 1 /2− 1 − ,g r BTP ˜e +1 2/2d2 6 −1 2 ˜eTQ ˜e + 1 2/2d2: (29)
Integrating both sides of Eq. (29) from t = 0 to T and after simple manipulations yields V (T) +12 T 0 ˜e(t) 2 Qdt 6 V (0) +/ 2 2 T 0d 2(t) dt ∀0 6 T ¡ ∞: (30)
Considering the Lyapunov-like function V (t) in (25) and the assumption A3, it yields ˜eT(t)P ˜e(t) 6
2V (0) + /2Bd for all t¿0, therefore the compact set Ux can be constructed as
ˆx(t) ∈ Uˆx, ˆx| ˜e(t) 6 2V (0) + /2Bd 2min(P) 1=2 ; yr(t) ∈ #r; ∀t ¿ 0 ; (31) where yr= [yr; yr; : : : ; y(nr −1)]T.
This demonstrates all states and signals involved of the closed loop system are bounded. Further-more, the H∞ performance can be achieved from Eq. (30), i.e.,
T 0 ˜e(t) 2 Qdt 6 2V (0) + /2 T 0 d 2(t) dt; ∀0 6 T ¡ ∞: (32)
From the smooth projection algorithm (19), we know that agi− &g6 gi6bgi+ &g for 16i6n and
all values agi, bgi and &g¿0 can be arbitrary chosen by the designer. If agl≡ min16i6n(agi− &g)
and bgu≡ max16i6n(bgi+ &g), then it can be obtained that agl6 ˆg(ˆx; g)6bgu from
agl6 n i=1 agl"i 6 n i=1 (agi− &g)"i6 ˆg( ˆx; g) 6 n i=1 (bgi+ &g)"i6 n i=1 bgu"i= bgu; (33)
since "i(ˆx) ∈ (0; 1] and ni=1"i(ˆx) = 1. Therefore, ˆg(ˆx; g) is invertible if agi and &g can be chosen
suitably such that agl¿0. This completes the proof.
Remark 1. In comparison with previous work [3,8] the VSS adaptive FNN controller developed
above can be implemented, i.e., the fuzzy system ˆg(ˆx; g) can be guaranteed to be invertible and in
Remark 2. The upper bound value ,g de5ned in A2 is supposed to be a state-dependent bound
and can be independently chosen as ,g(ˆx)63g(ˆx)=agl, where 3g(ˆx) = max |Mg(ˆx)|. It implies that the
maximal perturbation of g(ˆx), i.e. |Mg(ˆx)|, should be less than max |g(ˆx; g)|.
Remark 3. Our advocated design methodology combines three important control techniques, i.e., adaptive fuzzy neural network control scheme, VSS control design and H∞ tracking control theory. Two adaptive neural systems, i.e., ˆf(ˆx; f) and ˆg(ˆx; g) equipped with update laws (18) and (19), are constructed to model the unknown systems f(x) and g(x), respectively. The VSS controller
us is required to e6ectively eliminate the e6ect of the approximation errors from the universal
approximation property. The robust H∞ controller uh can be applied such that the e6ect of the
external disturbance on the tracking error can be attenuated to any prescribed level.
To summarize the above analysis, the design algorithm for an observer-based indirect adaptive
fuzzy neural tracking control equipped with VSS and H∞ control is proposed as follows:
Design procedure
Step 1: Specify the feedback and observer gain vector kc and ko such that the characteristic
matrices A − BkT
c and A − koCT are strictly Hurwitz matrices, respectively.
Step 2: Specify a positive de5nite n × n matrix Q and solve the Lyapunov equation (10) to obtain
a positive de5nite symmetric n × n matrix P.
Step 3: Solve the state equation (8) to obtain estimate state vector ˆx = ˆe + yr.
Step 4: Specify the design parameters, based on the practical constraints.
Step 5: De5ne the membership function !Fl
i(ˆx) for i = 1; 2; : : : ; n and compute the fuzzy basis
functions "(ˆx). Then the fuzzy logic control systems ˆf(ˆx; f) and ˆg(ˆx; g) can be constructed as
ˆf(ˆx; f) = "T( ˆx) f
and
ˆg( ˆx; g) = "T( ˆx) g:
Step 6: Obtain the control and apply to the plant, then compute the adaptive laws (18) and (19)
to adjust the parameter vector f and g.
5. The illustrative examples
In this section, we will apply our observer-based indirect adaptive FNN controller using VSS and
H∞ for two cases. The 5rst example is to let the inverted pendulum to track a sine-wave trajectory.
The second example is to let the output of mass–spring–damper system to track a sine-wave trajectory as well.
Example 1. Consider the inverted pendulum system as shown in Fig. 2. Let x1= be the angle of the pendulum with respect to the vertical line.
u mgv sin(x1) x1 mc l x2
Fig. 2. The inverted pendulum system.
The dynamic equations of the inverted pendulum system [3,25,8,20] and [11] are ˙x1 ˙x2 = 0 1 0 0 x1 x2 + 0 1 (f + gu + d); y = [1 0] x1 x2 ; (34) where
f = gv sin x1− (mlx22cos x1 sin x1)=mc+ m
l 4 3− m cos2x 1 mc+ m ; g = cos x1 mc+m l 4 3 − m cos2 x 1 mc+ m
and gv= 9:8 m=s2 is the acceleration due to gravity, mc is the mass of the cart, l is the
half-length of the pole, m is the mass of the pole and u is the control input. In this example, we assume that mc= 1 kg, m = 0:1 kg, l = 0:5 m. The control object is to control the state x1 of the
system to track the reference trajectory yr(t) = 0:5 sin(t). Also the external disturbance d is assumed
0:2 sin(2t) exp(−0:1t). The choices of r’s and h are to improve the convergence rate of the closed-loop system controlled by our proposed controller.
According to the design procedure, the design is given in the following steps: Step 1: The observer and feedback gain vectors are chosen as kT
o= [89 184], and kTc= [4 4],
respectively.
Step 2: We select Q in (10) as 6
0 06
, then after solving (10), the positive de5nite symmetric
2 × 2 matrix P in (10) is −0:504180:9945 −0:504190:2956 .
Step 3: Solve (8) to obtain ˆx.
Step 4: We select r1= 152:0302, r2= 88:4132, &f= 10, &g= 0:1, agi= 1:3, bgi= 1:5 and / =
[0:50:05 0:01]. Also we choose ,f= 0:1, ,g= 0:2, Bd= 20, and ˆQ in (12) is chosen as 4025 2530
and ˆA = 0
−4 −41
in (12). Therefore the positive de5nite symmetric 2 × 2 matrix ˆP in (12) can be
solved as 15
5 55
0 0 .0 5 0 .1 0 .1 5 0 .2 -0 .1 5 -0 .1 -0 .0 5 0 0 .0 5 0 .1 0 .1 5 0 .2 t (sec) x1, = 0 .5 x1, = 0 .0 5 x1, = 0 .0 1 1 ˆx , = 0 .5 1 ˆx, = 0 .0 5 1 ˆx, = 0 .0 1 rad
Fig. 3. Trajectories of the state x1 and the estimated state ˆx1.
Step 5: The following membership functions for ˆxi, i = 1; 2 are selected as
!Fi 1(ˆxi) = {1 + exp[10( ˆxi+ 0:4)]} −1; ! Fi 2( ˆxi) = exp − 2( ˆxi+ 0:2)2 ; !Fi 4( ˆxi) = exp(−2 ˆx 2 i); !Fi 4( ˆxi) = exp − 2( ˆxi− 0:2)2 ; !Fi 5( ˆxi) = {1 + exp[10( ˆxi− 0:4)]} −1:
To cover whole cases, we apply (25) fuzzy rules.
Step 6: Obtain control input and compute the adaptive laws (18), (19).
The trajectories of the state x1 and the estimated state ˆx1 for three di6erent levels of attenuation,
i.e., / = 0:01, 0.05 and 0.5, are shown in Fig. 3 and it shows that the estimated state ˆx1 takes
very short time to catch up the system state x1. For di6erent levels of attenuation, the tracking
performances are also very good as shown in Fig. 4, where yr is the reference trajectory.
Also, the generalized velocity ˙ (t) trajectories for di6erent levels of attenuation and reference
are shown in Fig. 5. Under the di6erent prescribed attenuation levels, the integral of the error
T
0 e(t) 2dt are indicated in Fig. 6. Therefore, the simulation result shows that the desired H∞
attenuation requirement can be achieved. Furthermore the e6ects due to plant uncertainties and external disturbances can be e9ciently diminished by proposed observer-based VSS indirect adaptive
FNN H∞ tracking controller.
The trajectories of the control input for di6erent prescribed attenuation levels are shown in
Fig. 7(a)–(c), i.e., / = 0:01, 0.05 and 0.5.
Example 2. Consider the mass–spring–damper system as described in Fig.8, with system parameters as body mass M (kg), spring coe9cient K (N/m), friction coe9cient B (N/m/s) and applied torque
input u (N). The equation of motion for the system can be expressed as [27]
0 5 1 0 1 5 2 0 -0 .1 5 -0 .1 -0 .0 5 0 0 .0 0 .1 0 .1 0 .2 0 .2 yr = 0 .5 = 0 .0 5 = 0 .0 1 t (sec) (t ) ρ ρ ρ
Fig. 4. The output trajectory for di6erent levels of attenuation and reference trajectory.
0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 r y t (sec) (t ) ρ=0.5 =0.05 ρ=0.01 ρ . .
Fig. 5. The velocity trajectory ˙ (t) for di6erent levels of attenuation and reference trajectory.
where fK(x) denotes the spring force due to K, fB(x) is the friction force from and fC(x) is the
coulombfriction force.
Let x1= y; x2= ˙x1 and x = [x1; x2]T, the state space representation of the system can be
described as ˙x1= x2;
˙x2= M1 (−fK(x) − fB(x) − fC(x)) + M1 u +M1 d: (36)
This mass–spring–damper system su6ers from plant uncertainties, unmodeled force and external disturbances. The nominal parameters of the system are given by M0= 1, K0= 2 and B0= 2. The
0 5 1 0 1 5 2 0 0 2 4 6 8 1 0 1 2 1 4 t (sec) ρ=0.5 ρ= 0 . 0 5 = 0 .0 1 dt t e T 2 0 ) ( ρ ∫
Fig. 6. The integral of the error 0Te(t)2dt for di6erent prescribed attenuation levels.
0 5 10 15 20 -4 -2 0 2 4 6 u (t) u (t) u (t) t (sec) 0 5 10 15 20 -50 0 50 100 t (sec) 0 5 10 15 20 -100 0 100 200 300 t (sec) (a) (b) (c)
Fig. 7. The trajectories of the control input for di6erent prescribed attenuation levels.
Also the nonlinear spring force and friction force are assumed to be fK(x) = K0y + MKy3 and
fB(x) = B0˙y + MB ˙y2. In addition, suppose there is a coulombfriction force fC(x) = 0:01sgn( ˙y).
Therefore, state space representation can be rewritten as ˙x1 ˙x2 = 0 1 0 0 x1 x2 + 0 1 (f(x) + g(x)u + d1); y = [1 0] x1 x2 ;
fB B K f K M y u -
Fig. 8. The mass–spring–damper system.
where f(x) = M 1 0+ MM (−fK(x) − fB(x) − fC(x)); (37) g(x) = M 1 0+ MM (38) and d1= M 1 0+ MM d: (39)
By substituting all parameters into Eq. (37) and (38), we get f(x) = 1 + 0:1 sin(x1 1)(−2x1− 0:5x 3 1− 2x2− 0:5x22− 0:01sgn(x2)); g(x) = 1 + 0:1 sin(x1 1):
We also have to determine the bounds fU, gU and g
L as follows: |f(x1; x2)| 6 1 |1 + 0:1 sin(x1)| (0:5|x3 1| + 2|x1| + 2 + 0:01|sgn(x2)|) 6 1:5 × (0:5|x3 1| + 2|x1| + 2:01) = fU(x1; x2) ≈ fU(ˆx1; ˆx2); |g(x1; x2)| 6 1:5 = gU(x1; x2) ≈ gU( ˆx1; ˆx2); (40) |g(x1; x2)| ¿ 0:9 = gL(x1; x2) ≈ gL( ˆx1; ˆx2): (41)
The control object is to control the state x1 of the system to track the reference trajectory yr(t) = 0:5
Table 1
Three cases of the initial states
Cases Initial states
Case1 x(0) = [0:25 0]T ˆx(0) = [−0:25 0]T Case 2 x(0) = [0:15 0:15]T ˆx(0) = [−0:15 − 0:15]T Case 3 x(0) = [−0:15 − 0:05]T ˆx(0) = [0:15 0:25]T
b e 0:2 sin(2t) exp(−0:1t). The choices of 7’s and h are to improve the convergence rate of the closed-loop system controlled by our proposed controller.
According to the design procedure, the design is given in the following steps: Step 1: The observer and feedback gain vectors are chosen as kT
o= [89 184], and kTc= [4 4],
respectively.
Step 2: We select Q in (12) as 6
0 06
, then after solving (10), the positive de5nite symmetric
2 × 2 matrix P in (10) is 0:9945
−0:50418 −0:504190:2956
.
Step 3: Solve (8) to obtain ˆx.
Step 4: We select r1= 500, r2= 1:0, &f= 10, &g= 0:1, agi= 1:3, bgi= 1:5 and / = 0:5. Also we
choose ,f= 0:1, ,g= 0:2, Bd= 20, and ˆQ in (12) is chosen as 4025 2530 and ˆA =
0
−4 −41
in (12).
Therefore the positive de5nite symmetric 2 × 2 matrix ˆP in (12) can be solved as 15
5 55
. The H∞
gain r = 0:7/2 and the step size is chosen as h = 0:001667.
Step 5: The following membership functions for ˆxi, i = 1; 2 are selected as
!Fi 1( ˆxi) = 1=(1 + exp(10( ˆxi+ 1))); !F2i( ˆxi) = exp(−2( ˆxi+ 0:5) 2); !Fi 3( ˆxi) = exp(−2 ˆx 2 i); !Fi 4( ˆxi) = exp(−2( ˆxi− 0:5) 2); !Fi 5( ˆxi) = 1=(1 + exp(−10( ˆxi− 1))):
To cover whole cases, we apply 25 fuzzy rules.
Step 6: Obtain control input and compute the adaptive laws (18) and (19).
According to the initial states, three cases are simulated as shown in Table 1.
The trajectories of the state x1 and the estimated state ˆx1 for three di6erent initial states are shown
in Fig. 9 and it shows that the estimated state ˆx1 takes very short time to catch up the system state
x1. For di6erent initial states, the tracking performances are also very good as shown in Fig. 10,
where yr is the reference trajectory.
Also, the generalized velocity ˙y(t) and yr(t) trajectories for three cases are shown in Fig. 11.
Under the di6erent prescribed attenuation levels, the integral of the error 0T e(t) 2dt are indicated
in Fig. 12. Therefore, the simulation result shows that the desired H∞ attenuation requirement can
be achieved. Furthermore the e6ects due to plant uncertainties and external disturbances can be
0 0.1 0.2 0.3 0.4 0.5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 t (sec) Case 1 Case 2 Case 3
Fig. 9. The trajectories the states x1 (solid line) and ˆx1 (dash line) of 3 cases (time: 0–0:483 s).
yr 0 5 10 15 20 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 y of case 1 t (sec) y of case 3 y of case 2 y, yr (rad)
Fig. 10. The output trajectory for three cases and reference trajectory.
The trajectories of the control input of three di6erent initial states are shown in Fig. 13. 6. Conclusions
An indirect adaptive FNN controller with observer design by using VSS and H∞ control
al-gorithms is developed for nonlinear SISO systems involving plant uncertainties and external distur-bances, in which only the system output can be measured. Based on the Lyapunov synthesis approach, the free parameters of the adaptive FNN controller can be tuned on-line by the observer-based output
0 5 10 15 20 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 y of case 1 y of case 2 y of case 3 r y (rad/se c) t (sec) r y y , . . . . . .
Fig. 11. The generalized velocity ˙y(t) and yr(t) trajectories for three cases.
0 5 1 0 1 5 2 0 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 t (sec) Case 1 Case 2 Case 3 dt t e T 2 0 ) ( ∫
Fig. 12. The integral of the error 0Te(t)2dt for three cases.
feedback control and the adaptive laws. Also the robust nonlinear output tracking requirement can be achieved by three control design techniques, adaptive fuzzy neural control scheme, VSS control design and H∞ tracking theory. Simulation results show that the overall observer-based adaptive FNN control scheme guarantees stability of the resulting closed-loop system in the sense that all the states and signals are uniformly bounded and H∞ tracking performance can be achieved.
0 5 10 15 20 -5 0 5 10 15 20 t(sec) u (t) u (t) u (t) 0 5 10 15 20 -5 0 5 10 15 t(sec) 0 5 10 15 20 -50 0 50 100 t(sec) Case 1 Case 2 Case 3
Fig. 13. Trajectories of the control input for three cases.
Acknowledgements
The authors are grateful to reviewers for their insightful comments and suggestions. References
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