1. No category

Automatica 35 (1999) 741—747
Technical Communique
A direct adaptive controller for dynamic systems with
a class of nonlinear parameterizations
S. S. Ge*, C. C. Hang, T. Zhang
Department of Electrical Engineering, National University of Singapore, Singapore 119260, Singapore
Received 2 March 1998; revised 30 June 1998; received in final form 23 October 1998
Abstract
In this note, the adaptive control problem is considered for a class of nonlinearly parametrized systems. By introducing a novel kind
of Lyapunov functions, a direct adaptive controller is developed for achieving asymptotic tracking control. The transient performance
of the resulting closed-loop system can be guaranteed by suitably choosing the Lyapunov function to construct the controller. The
effectiveness of the proposed scheme is illustrated with two examples. 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Nonlinear system; Adaptive control; Lyapunov stability; Nonlinear parameterization; Transient performance
1. Introduction
Adaptive control of nonlinear systems has been an
active research area and many good theoretical results
have been obtained in the literature (Sastry and Isidori
1989; Kanellakopoulos et al. 1991; Krstic et al. 1995;
Johansen and Ioannou, 1996; Marino and Tomei, 1995)
and the references therein. Most available adaptive controllers deal with control problem of systems with nonlinearities being linear in the unknown parameters. In
practice, however, nonlinear parameterization is very
common in many physical plants. Adaptive control for
nonlinearly parametrized systems is an interesting and
challenging problem in control community. Marino and
Tomei (1993) studied the global output feedback control for systems containing nonlinear parameterizations,
which is designed using high-gain adaptation and applicable to set-point regulation problem. For a class of firstorder nonlinearly parametrized models similar to those
arising in fermentation processes, Boskovic (1995) provided an interesting adaptive control scheme with three
unknown parameters (two of them do not enter linearly).
The key points of this design method lie in the appropriate parameterization of the plant and the suitable choice
* Corresponding author. Tel.: 00 65 874 6821; fax: 00 65 779 1103;
e-mail: [email protected].
This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Editor Peter Dorato.
of a Lyapunov function with a cubic term for developing
the stable adaptive controller.
In this paper, we deal with the state-feedback adaptive
tracking control problem for nonlinear systems with
a class of nonlinear parameterizations. A novel kind of
Lyapunov functions is developed to construct a
Lyapunov-based controller and parameter updating law.
It is shown that the globally asymptotic tracking is
achieved with guaranteed control performance. This
paper is organized as follows. Section 2 shows the
control problem and the definition of weighted control
Lyapunov function (WCLF). Section 3 presents the
direct adaptive controller and stability analysis of the
closed-loop system. Tracking performance of the adaptive system is discussed in Section 4. Two examples are
given in Section 5 to show the effectiveness of the controller proposed. Section 6 contains the conclusion.
2. Problem statement
Let us consider the nonlinear systems given in the
following form
xR "x , i"1, 2,2, n!1
G
G>
1
xR "
[ f (x)#g(x)u],
L b(x)
y"x ,
0005-1098/99/$—see front matter 1999 Elsevier Science Ltd. All rights reserved
PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 1 5 - 5
(1)
742
S.S. Ge et al./Automatica 35 (1999) 741—747
where x"[x x 2 x ]231L, u31, y31 are the
L
state variables, system input and output, respectively;
g(x) is a known continuous function; functions
f (x), b(x)3C can be expressed as
f (x)"h2w (x)#f (x),
D
b(x)"h2w (x)#b (x)
@
(2)
where h31N is a vector of unknown constant parameters, w (x)31N and w (x)31N are known regressor
D
@
vectors, functions f (x), b (x)3C are known. The con
trol objective is to find a controller u such that output
y follows a given reference signal y .
B
Clearly, the unknown parameter vector h enters into
system (1) nonlinearly. Many practical systems, such as
pendulum plants (Cannon 1967, Balestrino et al. 1984)
and fermentation processes (Boskovic, 1995), can be described by system (1) and possess such a kind of nonlinear
parameterizations. In this paper, the following assumption is made.
Assumption 1. g(x)/b(x)O0, ∀x31L and its sign is
known.
The above assumption implies that the continuous
function g(x)/b(x) is strictly either positive or negative.
From now onward, without losing generality, we shall
assume g(x)'0 and b(x)'0 for all x31L. Define
vectors x , e and a filtered error e as
B
Q
x "[y yR 2 yL\]2, e"x!x "[e e 2 e ]2,
B
B B
B
B
L
d
L\
e " #j
e "["2 1]e,
Q
dt
(3)
where constant j'0 and
""[jL\ (n!1)jL\ 2 (n!1)j]2.
Remark 2.1. It has been shown in the reference
(Slotine and Li, 1991) that definition (2) has the following properties: (i) the equation e "0 defines a
Q
time-varying hyperplan in 1L on which the tracking
error e converges to zero asymptotically, (ii) if the
magnitude of e is bounded, the error vector e(t)
Q
is also bounded, and (iii) a state representation
of Eq. (3) can be expressed as fQ "A f#b e with
Q
QQ
f"[e e 2 e ]2, n52, A a stable matrix depend L\
Q
ing on j and b "[0 0 2 0 1]2.
Q
From Eqs. (1) and (3), the time derivative of e can be
Q
written as
1
eR "
[ f (x)#g(x)u]#l,
Q b(x)
(4)
where l"!yL#[0 "2]e. Let b (x)"b(x)a(x) with
B
?
the smooth function a(x): 1LP1 to be specified later. It
>
is can be seen from Eq. (3) that x "e #yL\!
L
Q
B
["2 0]e. For the ease of discussion, we shall denote
b (xN , e #l )"b (x) with xN "[x x 2 x ]2 and
?
Q
?
L\
l "yL\!["2 0]e.
B
Definition 2.1. For a bounded reference vector x , a scalar
B
function
e
pb? (xN , p#l) dp
Q
(5)
»"
C
is called a weighted control Lyapunov function (WCLF)
for system (1), if there exist a smooth function a(x) and
a control input u such that » satisfies:
C
1. » is positive definite in the filtered error e ,
C
Q
2. » is radially unbounded with respect to e , i.e.,
C
Q
» PR as "e "PR, and
C
Q
3. »Q (0, ∀e O0
C
Q
In addition, a(x) is called a weighting function (WF).
3. Adaptive controller design
In this section, we first show that for system (1) satisfying Assumption 1, there indeed exists a WF a(x) and
a control input u such that » defined in Eq. (5) is
C
a WCLF. Then, we construct an adaptive controller
using this WCLF for achieving asymptotic tracking
control.
As b(x)'0 is linear in the unknown constant
parameters, a smooth function a(x) can be found such
that » satisfies conditions 1 and 2 in Definition 2.1.
C
For example, if b(x)"exp(!x ) (h #x) with constant
L L
h '0, then we may choose a(x)"exp(x ) which
L
leads to
e
p[h#(p#l)] dp
Q
»"
C
e
" Q [(e # l )# l#2h ].
4 Q Clearly, the above function is positive definite and
radially unbounded with respect to e . Taking the time
Q
derivative of » given in Eq. (5), we have
C
»Q "b (x)e eR
C
?
QQ
p e
Q
#
*b (xN , p#l )
*b (xN , p#l )
?
xNQ # ?
lR dp.
*xN
*l
(6)
743
S.S. Ge et al./Automatica 35 (1999) 741—747
Because *b (xN , p#l )/*l "*b (xN , p#l )/*p and l"
?
?
!lR , it follows that
e
Q
p
"!l
e
Q
e
Q
"!l pb (xN , p#l ) ! b (xN , p#l ) dp
?
?
"!le b (x)#l
Q ?
e
Q
b?(xN , p#l) dp.
Substituting the above equation into Eq. (6) and using
Eq. (4), we obtain
b (x)
»Q " ? [ f (x)#g(x) u]e
C b(x)
Q
e
Q
#
»Q "e a(x) [h2w(z)#g(x) u#h(z)],
C
Q
(7)
where
1
w(z)"w (x)#
D
e a(x)
Q
p
e
Q
*wN (xN , p#l )
@
xNQ #lwN (xN , p#l ) dp, (8)
@
*xN
1
h(z)"f (x)#
e a(x)
Q
p
;
e
Q
(10)
*bM (xN , p#l )
xNQ #lbM (xN , p#l ) dp,
*xN
»Q "!ke!hI 2w(z)a(x) e .
(11)
C
Q
Q
The system stability is not clear at this stage because the
last term in Eq. (11) is indefinite and contains unknown hI .
To remove such an uncertainty, parameter adaptive
tuning is introduced for hK . For constructing an adaptive
law, we augment » as follows
C
(12)
»"» # (hI 2!\hI )
C with gain matrix !"!2'0. The time derivative of
» along Eq. (11) is
»Q "!ke#hI 2 [!w(z)a(x) e #!\hQK ].
(13)
Q
Q
To eliminate hI from »Q , the adaptive law can be chosen as
*b (xN , p#l )
xNQ #lb (xN , p#l ) dp.
p ?
?
*xN
Noting the expressions in Eq. (2), we have
;
where hª is the estimate of h. Define a parameter estimate
error hI "hK !h and substitute Eq. (10) into Eq. (7), we
obtain
*b (xN , p#l )
dp
p ?
*p
1
ke
u"
! Q !hK 2w(z)!h(z) ,
g(x)
a(x)
*b (xN , p#l )
?
lR dp
*l
e
Q
case of unknown parameter h, we employ its certaintyequivalence controller as
(9)
z"[x2 x2 yL]231L>, wN (xN , p#l )"w (xN , p#l )
B B
@
@
a(xN , p#l )31N
and
bM (xN , p#l )"b (xN , p#l )
a(xN , p#l )31. It can be checked that
lwN (xN , l )
,
lim w(z)"w (x)# @
D
a(x)
e P0
Q
lbM (xN , l )
.
lim h(z)"f (x)# a(x)
e P0
Q
Hence, both w(z) and h(z) are well defined. If the
parameter vector h is available, a possible controller
is u*"g\(x) [!k(e /a(x))!h2w(z)!h(z)] with design
Q
parameter k'0. For this controller, Eq. (7) becomes
»Q "!ke(0, ∀e O0. According to Definition 2.1, we
C
Q
Q
conclude that » is a WCLF and e P0 as tPR. In the
C
Q
hKQ "!w(z)a(x)e
Q
which leads to
(14)
»Q "!ke40.
(15)
Q
Since function b (x)3C, Eq. (5) shows that » is a
?
C
C function of x and x . This guarantees that » (0)3¸
B
C
for any bounded initial values x(0) and x (0). IntegraB
ting Eq. (15), we have ke (q) dq4»(0)(R and
Q
04»(t)4»(0). This implies that e 3¸ 5¸ and hK (t) is
Q
bounded. Consequently, u and eR are also bounded. Since
Q
e 3¸ 5¸ and eR 3¸ , we conclude lim
e "0 by
Q
Q
R Q
Barbalat’s lemma (Popov, 1973). It follows from
Remark 2.1 that x3¸ and the tracking error converges
to zero asymptotically. The above result is summarized
in the following Theorem.
Theorem 3.1. For system (1) satisfying Assumption 1, controller (10) with adaptive law (14) guarantees the boundedness of all the signals in the closed-loop system and the
globally asymptotic tracking, i.e., lim
y(t)"y (t).
R
B
4. Performance analysis
As shown in the preceding section, a key step in the
design procedure is the choice of WF a(x) and WCLF » .
C
It should be pointed out that for a given system, different
WF can be found to construct different WCLF. Therefore, the resulting controller is not unique and the control
performance also varies with the choice of WCLFs. This
brings the designer some degrees of freedom in controller
design. In the following, we show that for controller (10)
744
S.S. Ge et al./Automatica 35 (1999) 741—747
with a suitably chosen WF a(x), transient performance of
the closed-loop system can be guaranteed.
Theorem 4.1. For the closed-loop adaptive system (1), (10)
and (14), if ¼F a(x) is chosen such that b (x)4c with
?
c a positive constant, then
(i) ¸ transient bound of the filtered error
1
e (q) dq4 [c e(0)#hI 2(0)!\hI (0)],
(16)
Q
2k Q
(ii) for the systems with n52, the ¸ tracking error
bound
#f(t)#4k #f(0)# e\HR
k
# (c e (0)#hI 2(0)!\hI (0)
(17)
Q
2(kj
with computable constants k , j '0 which depend on
the design parameter j.
Proof. (i) If a(x) is chosen such that 0(b (x)4c ,
?
then
e
e
Q
pb? (xN , p#l) dp4c p dp" 2 eQ .
Q
c
»"
C
(18)
Integrating Eq. (15) over [0, t] and applying Eq. (18), we
obtain
»Q dq"»(0)!»(t)
t
t
ke(q) dq4!
Q
c
1
4 e (0)# hI 2(0)!\hI (0), ∀t50
2 Q
2
(19)
from which ¸ bound (16) can be concluded.
(ii) For the systems with order n52, Remark 2.1
shows that fQ "A f#b e with stable matrix A . It is not
Q
QQ
Q
difficult to find two constants k , j '0 which depend
on the design parameter j such that #eQR#4k e\HR
(Ioannou and Sun, 1996). The solution for f can be
written as
e R\ObQ eQ (q) dq.
t
f(t)"eQRf(0)#
Q
Therefore
#f(t)#4k #f(0)#e\HR#k
e\H R\O"eQ (q)" dq.
t
(20)
Applying the following Schwartz inequality (Ioannou
and Sun, 1996)
t
"a(q)b(q)" dq4
t
a(q) dq
t
b(q) dq
,
(21)
we have
#f(t)#4k #f(0)#e\HR
#k
t
e\HR\O dq
k
4k #f(0)#e\HR# (2j
t
t
e(q) dq
Q
e(q) dq
Q
/
.
Using Eq. (19), the inequality (17) follows. )
Remark 4.1. The ¸ bound of the error vector f in
Theorem 4.1 is obtained for high-order systems (n52).
For a first-order system, to get an explicit bound of the
tracking error, an additional condition b (x)5c is
?
needed for the choice of a(x). In this case (e "e for
Q
n"1)
e
»"
C
pb (xN , p#l ) dp5c
?
e
p dp" 2 e.
c
(22)
Noticing » (t)4»(t)4»(0), we have e (t)42»(0)/c .
C
From » (0)4c e(0)/2, the ¸ tracking bound for the
C
first-order system can be found
c
1
"e (t)"4 e(0)# hI 2(0) !\hI (0).
c c
(23)
Remark 4.2. Theorem 4.1 shows that different choices
of WF a(x) may produce different control performance.
As b (x)"[h2w (x)#b (x)]a(x) with known functions
?
@
w (x) and b (x), it is not difficult to design a WF a(x)
@
to make 0(c 4b (x)4c . For example, if b(x)"
?
exp(!x) (h #h x) with constant parameters h ,
L L
h '0, then one may take a(x)"exp(x)/(1#x) which
L
L
leads to
h #h x
L 4max[h , h ].
min[h , h ]4b (x)" ?
1#x
L
Remark 4.3. From a practical point of view, Assumption 1 holds on whole space might be a strong restriction
for many physical plants. If Assumption 1 holds only on
a compact subset )L1L, the proposed approach is still
applicable if the controller parameters are designed appropriately. The reason is that by suitably choosing the
design parameters, upper bounds of the states (derived
from Eqs. (17) and (23)) are adjustable by the designer,
and subsequently can be guaranteed within the given
compact set ) in which Assumption 1 is satisfied for all
time. The second example given in Section 5 illustrates
such an application.
S.S. Ge et al./Automatica 35 (1999) 741—747
745
5. Case study
Example 5.1. To show the controller design procedure
and validate the effectiveness of the developed scheme,
we consider a second-order system
xR "x ,
x#u
xR "
(24)
exp (!x) (h #h x)
with unknown parameters h , h '0. The objective is
to control the output y"x to follow the reference
y (t)"sin(0.5t). Plant (24) can be expressed in the
B
form of system (1) with f (x)"x, g(x)"1 and
b(x)"exp(!x) (h #h x). Comparing with Eq. (2), we
have f (x)"x, h"[h h ]2 and w (x)"[exp(!x)
@
exp(!x) x]2. In view of Remark 4.2, we choose the WF
a(x)"exp(x)/(1#x). It follows from Eqs. (8) and (9),
that
e
e
Q
Q
l
1
(p#l )
2
w(z)"
dp
dp
e a(x)
1#(p#l )
1#(p#l ) Q
l
[tan\x !tan\l e !tan\x #tan\l ]2
"
Q
e a(x)
Q
with l "yR !j(x !y ), l"!y¨ #j(x !yR ), and
B
B
B
B
h(z)"x. Then, Eqs. (10) and (14) suggest the following
controller
u"!k(1#x) exp(!x) e !hK 2w(z)!x
Q
with adaptive laws
hKQ "c l (tan\x !tan\l ),
hQK "c l(e !tan\x #tan\l ).
Q
In the simulation, the true values of the system parameters are [h h ]2"[2.0 0.5]2 and the initial condition
is [x (0) x (0)]2"[0.5 0.0]2. The parameters of the
adaptive controller are j"1.0, k"1.0, c "10.0,
c "25.0 and [hK (0) hK (0)]2"[0.0 0.0]2. The simula
tion result given in Fig. 1a indicates that the output
tracking error converges to zero asymptotically. The
responses of the estimated parameters and control input
are shown in Fig. 1b and c, respectively.
Example 5.2. In this example, we apply the proposed
approach to an inverted pendulum plant (Cannon, 1967)
described by
xR "x ,
m¸x sin x cos x
cos x
g sin x !
M#m
M#m
xR "
#
u,
4 m cos x
4 m cos x
¸ !
¸ !
3
M#m
3
M#m
y"x ,
(25)
Fig. 1. Responses of the adaptive system in Example 5.1. (a) Tracking
error y!y (b) hK (‘‘—’’) and hK (‘‘- -’’) (c) Control input u(t).
B
746
S.S. Ge et al./Automatica 35 (1999) 741—747
where x and x are the angular displacement and velo
city of the pendulum, respectively; g"9.8 m/s is the
gravity acceleration coefficient; M and m are the masses
of the cart and the pole, respectively; ¸ is the half-length
of the pole, and u is the applied force control. The true
values of the plant are M"1.0 kg, m"0.2 kg and
¸"0.5 m, initial states are [x (0) x (0)]2"[0 0]2, and
reference signal is y (t)"n/6 sin(t). Let
B
h
M#m
g sin x
w (x)" x sin x cos x ,
h" h " !m¸ ,
D
¸(M#m)
h
0
0
w (x)" cos x .
@
1
The plant (25) can be written in the form of system (1)
with f (x)"h2w (x), g(x)"cos x and b(x)"h2w (x).
D
@
Although the pendulum plant (25) does not satisfy
Assumption 1 for x31, it can be checked that
g(x)/b(x)O0 for all "x "(n/2. In order to apply the
proposed method, the design parameters should be
specified such that "x "(n/2 holds for all time as dis
cussed in Remark 4.3. Take WF a(x)"1, it follows from
Eqs. (8) and (9) that
g sin x
w(z)" l x sin x cos x #l cos x ,
l
h(z)"0.
According to Eqs. (10) and (14), the adaptive controller
can be chosen as
1
u"
[!ke !hK 2w(z)], with hKQ "!w(z)e .
Q
Q
cosx
In the simulation, controller parameters are set as j"1.0
and !"diag+0.2,, and initial condition hK (0)"0.0. To
avoid possible controller singularity when "x ""n/2,
the design parameter k is chosen as follows. Suppose
that very conservative bounds of plant parameters M,
m and ¸ are known as M41.5 kg, m40.3 kg and
¸40.75 m. It can be shown that b(x)"¸[ (M#m)!
m cos x ]41.8 and #hI (0)#46.5306. According to
Eq. (17) in Theorem 4.1, the upper bound of x can be
obtained
"x (t)"4"y "#"e (0)"
B
1
1
#
1.8e(0)# #hI (0)#, ∀t50. (26)
Q
0.2
2(k
Since
y "n/6 sin(t)
and
the
initial
states
B
[x (0) x (0)]2"[0 0]2, we know that "y "4L, e (0)"0
B
and "e (0)""n/6. It can be calculated from Eq. (26) that if
Q
the gain k'7.6, then "x "(n/2 can be guaranteed. In
the simulation test, we let k"10.0. Fig. 2a shows that
Fig. 2. Responses of the adaptive system in Example 5.2. (a) Output
y (‘‘—’’) follows y (‘‘- -’’) (b) hK (‘‘—’’), hK (‘‘2’’) and hK (‘‘- -’’)
B
(c) Control input u(t).
S.S. Ge et al./Automatica 35 (1999) 741—747
although the tracking error is large during the initial 5 s
due to the inadequate initial parameter hK (0)"0.0, the
transient bound of "x " is smaller than n/2. As the para
meters are adaptively tuned on-line, the output y(t)
tracks the reference y (t) asymptotically. The boundedB
ness of the estimated parameters and control signal are
also presented in Fig. 2b and c, respectively.
6. Conclusion
We have presented a direct adaptive controller for
a class of dynamic systems with nonlinear parameterization. The main feature of the paper is the construction of
the weighted control Lyapunov function, which can be
used to remove the nonlinear parameterization for adaptive controller design. Global stability and asymptotic
convergence of tracking error have been obtained and
the control performance of the resulting adaptive system
has been investigated.
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