- Email: [email protected]
CONVEX PARAMETRIZATIONS
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
www.elsevier.com/locate/ifac
ADAPTIVE CONTROL OF DISCRETE TIME SYSTEMS WITH CONCAVE/CONVEX PARAMETRIZATIONS C Y Qu A P Loh K F Fong A M Annaswamy
Dept of Electrical & Computer Engineering National University of Singapore Singapore 119260 Dept of Mechanical Engineering Massachussetts Institute of Technology Cambridge, MA 02139 Abstract: This paper considers the adaptive control of a class of nonlinear discrete time system with nonlinearly parametrized functions. In particular, the focus is on concave or convex parametrizations with unknown parameters. The solutions involved 2 tuning functions which are determined by a minmax optimization approach much like the continuous time counterparts found in the literature. Direct extension from continuous time case do not work very well due to the premature termination of the adaptive algorithm before zero tracking error can be achieved. In this paper, this problem is solved. The proposed algorithm can be shown to be stable and in some cases, achieve zero tracking error in steady state. Copyright © 2002 IFAC Keywords: adaptive control, discrete time systems, nonlinear systems not been very successful. This is largely due to the Lyapunov approach that is adopted in the analysis. The direct extension of (Loh et al., 1999) does not work very well because the Lyapunov approach does not allow one to construct additional signals to ensure closed loop stability. While it is possible to construct two tuning functions for the parameter adaptation as in the estimation problem in (Skantze et al., 1998), the use of such tuning functions cannot guarantee zero tracking error. Adopting the approach in (Skantze et al., 1998) however can guarantee closed loop stability, as will be seen in this paper.
1. INTRODUCTION Adaptive control of nonlinear dynamic systems is currently an area of great interest among control theorists. Much of the work thus far has been for continuous time systems (Kristic et al., 1995), (Kristic and Kokotovic, 1995) involving linear parametrizations. Very few results are available in the literature that addresses adaptive control in the presence of nonlinear parametrizations (Ortega, 1996). One such result can be found in (Loh et al., 1999) where the adaptation scheme was changed significantly from the linear case in order to accomodate the nonlinear parametrization. Specifically, two tuning functions were included, one that determines the direction of parameter updates and the other that ensures stability of the closed loop system. These two tuning functions were derived from a minimax criterion that was solved on-line when the nonlinear parametrization involves convex or concave functions.
Some recent results on discrete time systems have appeared in (Zhang et al., 1999) and (Zhang et al., 2001) for linearly parametrized systems. The main approach in these two papers involves the backstepping technique. Their convergence analysis did not make use of a Lyapunov function. In this paper, the adaptation algorithms follow closely those in (Loh et al., 1999) and (Skantze et al., 1998). However, some modifications were made to ensure
Although results exists for continuous time systems, the extension of these to discrete time systems have
343
T
that zero tracking error is achieved. This analysis was made using a Lyapunov function. The paper is organised as follows. In Section 2, the class of nonlinear system is defined along with the control objectives. Section 3 presents the algorithm for the adaptive controller, followed by Section 4 which shows the stability and convergence analysis. Simulation results are shown in Section 5 and concluding remarks are given in Section 6.
o
or
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:
m
T
(8)
C
jl
R
l
l
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otherwise
l
l
0
s
ln
2
2
c
sgn e
V
M
X
(9)
_ _
u
v
w
v
x
0
X
arg
s
2
2
c
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V
M
X
(10)
_
u
v
w v
x
C
X
(11)
C
7
7
X
7
_
2. PROBLEM STATEMENT
The class of nonlinear discrete time system we consider is as follows :
7
(13)
-
The minmax solution in (9) and (10) are given in the Appendix for ease of reference.
where and are the plant’s input and output at and are time instant , vectors of measureable states, and are the unknown parameters, and is a known convex or concave function of its arguments. The objective is to determine a bounded control input, , such that the output, , of the plant asymptotically tracks a specified bounded reference trajectory, . More precisely, we require
M
(
(1)
(12)
C C
"
$
"
(
4. STABILITY ANALYSIS
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Using the Lyapunov’s second method of stability analysis, the Lyapunov candidate was chosen as
+
(
/
M
M
M G
I
where
M
3
C
|
-
The following assumptions are made regarding the plant in (1) (A1)
M
(14) ~
7
. Defining
M
7
K
M
N
M
X
=
~
N
~
M
~
G
I
X
~
G
V >
is a bounded concave or convex funtion of
+
K
~
N
~
M
~
7
K
~
N
~
M
~
.
(A2)
~
|
K
N
|
7
5
(
~
and , it can be shown that 7
:
+
K
C
7
G
M
|
~
-
V
0
2
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is assumed to lie in an interval given by , where and are known.
=
>
K
M
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X
7
M
N
M
K
7
<
Y _
K
~
N
~
M
(15)
~
(16)
K
M
N
X
7
M
N
M
K
To ensure that be considered.
In this section, the adaptive algorithm for the nonlinear discrete time system in (1) is presented for a convex . The case of a concave is similar and will be discuss later.
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3. CONTROLLER DESIGN
Case (a)
M
for all , two cases of
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:
M
will
In this case, we require :
+
+
(17)
K
M
N
X
7
M
N
M
K
The adaptive mechanism together with the construction of the control input, can be specified as :
which implies that and satisfies
:
Y _
N
should be chosen such that
N
R
:
X
(2)
C
7
7
C
(
-
N
7 Y
_
+
Y K
M
Y
C
7
G
I
K
M
N
G
C
T
G
I
R
since , to choose
(3) :
I
K
M
R
N
. Thus, for any where
:
\
and X
, it suffices
<
T
7
G
V
K
M
N
X
G
R
(4) :
V
0
s
X
\
Y
7 2
(18)
_
V
K
R Y Z
X G
V
X
G
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:
(5)
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v
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x
Y
Z
^
Y
Y 7
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c
e
X
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Y
7
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K
Y
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M
a
(6)
K
Y
M
_
v
x
Y
Y
Since is a free variable that can be further optimized, can finally be chosen as X
N
2
c
e
f
:
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i
(7)
\
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0
s
N
(23)
Y
+ \
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2 Y
2
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e
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Y
K
M
_
u
v
w v
x
Y
Y
Y
7
Y K
With this choice of
(19) _
M
N
,
then becomes
|
Y
0
s
` `
|
7
N
K
M
X
X
arg
2 2
c
X e
v
w v
_
K _
u
(20)
V
x
=
`
`
M
7
7
7
K
X
7
>
According to the minmax solution in Lemma A.1 in the Appendix, for convex , and . Hence indeed in (19). With this choice of and in (19) and (20) respectively,
X
:
Y
K
+
=
` `
V
7
M
7
_
N
N
X
\
Y K
:
Y
(24)
7
>
V
|
V
K
M
N
X
7
K
N ~
~
M ~
V
where , and . By this choice of , and using the property of convex functions,
_
V
7
K
~
Case (b)
M
o
N
~
M
~
:
+
In this case, we require :
7
:
+
Y
Once again, after the reset and this ensures that the whole algorithm carries on in a stable manner. In other instants when , will again be reset but in the next instant, -adaptation will resume. Thus, in general, we have that except in some instances during reset. |
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` `
`
`
`
7
K
N
~
K
M
N
X
M
7
N
M
K
:
+
Y _
Following the same analysis as case (a) above, we require
M
~
7
~
C
\
o
:
|
X
N
Y o
_
+
`
K
`
Y
M
7
K
N
~
Y
N
can thus be chosen as
0
M
~
~
s
It is important to note that -reset will only happen when , in the case of convex , as shown in case (b). In addition, this happens only when . In other words as long as , there exists a such that . This implies that in the worst case, will be bounded by
2
V
X
v
\
x
_
`
`
Y
:
Y
K
M
M
+
R
Y
_
`
`
With further optimisation, \
can be chosen as
M
`
7 Y
Y
2
c
sgn e
`
V
Y
M
M
_
X
\
R
:
:
`
M
2
R
s
2
`
K
|
\
M
N
`
0
`
c
o e
£
+
_
u
v
w
v
_ V
x
v
x
Y
Y 7
(21)
_
`
`
Y K
M
|
Hence, we conclude that
o
for all £
R
. :
Y
in (21) is positive, then it follows that If according to (7) and this guarantees that \
N
\
Suppose that the time instants can be separated into three sets, where contains all time instants where -resets have occurred and , contains all time instants where and contains all ’normal’ instances where . Then, summing all , including , we have
§
¨
|
.
:
¤
¥
¦
¤
¥
Y
|
|
in (21), then by the choice However, if in (7). At the same time, according to (8), is reset to . This reset does not apply to but since , no adaptation of occurs. In other words, at the instant , when
\
:
N
©
:
N
o
o
:
~
¥
¤
|
|
:
:
|
\
¥
~
ª
C
7
C
:
¤
C
R
~
:
5
¯
¯
|
|
|
7 5
7 ±
K
~
C
and
C
(22)
C
+
°
N
~
M
~
±
v
²
³
¯
¯
|
in (16) is no longer valid for further analysis at this point. Due to the reset in (22), the modified will be as follows :
7
ª
© ~
~
+
|
v
v
²
´
²
µ
If -reset occurs only a finite number of times, then , thus implying that
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o
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~
v
²
µ
M
M
|
~
G
7
~
G
7
~ K
V
~ N
~ M
V
¯
~ K
~ N
~ M
o
£
+
M
7
~ G
7
~
V
G
7
~ K
~ N
~ M
+
V
From the analysis, and after each reset. In fact, the algorithm can never end with because if , reset will occur and if is not zero at that instant, adaptation will carry on. It follows therefore that K
|
|
may not be negative. However, if will be bounded because of assumption A.2.
:
, it
C
\
R
:
N
:
R
:
/
0
Y
N
N
Due to the reset of at time instant , at the next instant, can be chosen as again as in (21). But due to the reset, and hence N
2
M
:
+
:
_
345
3
5
M
:
Discussion A stable algorithm is also possible without -reset. This is the algorithm in (Skantze et al., 1998) used in an estimation problem. In this case, will simply be set to zero whenever as in (7). Once this happens, all adaptation stops and the whole algorithm may end prematurely before . In order to prevent this from happening, some mechanism had to be developed to keep the adaptation alive without losing stability. The reset was one way to restore the value of to a non-zero value so that adaptation can carry on until goes to zero.
−3
x 10
N
o
N
0
:
M
t
∆V
\
−5
:
−10
40
60
M
100
|
Fig. 1.
1
+
ρt
0
+
without -reset.
+
a
−1
t
80
time
for This paper only addresses the case of convex ease of presentation. The proof for concave functions of is the same. When is concave, the reset should occur at instead of as in the convex case. When changes convexity or concavity, then combinations of the convex and concave algorithm should work accordingly.
20
0
+
−2 0
20
60
40
80
100
time
Fig. 2. \
and N
without -reset.
5. SIMULATIONS 1.15
º
»
1.1
1.05
(
θ estimate
In this section, the performance of the proposed algorithm is illustrated via a simulation example applied to the following system :
1 0 ¼
-
20
40
60
80
100
60
80
100
(
(
time -
C
7
(
7
C
-
¾
¿
¾
Fig. 3.
V
À C C
À
½
M
(
7
+
Á
:
:
+
0.01
+
tilde y
t
0
−0.01
Á
-
The numerical values of the parameters in the system are chosen as
without -reset.
C
Ã
+
Á
:
:
+
Á
+
Á
(
Y
Y
−0.02 0
C
¨
±
±
:
:
C
+
+
(
Y
Y
Y
Y
20
40
Y
time
G
I
Z
:
Ä
~
Y
Å
G
V
:
~
Y
Y
Fig. 4.
Simulation results are shown for both with and without -reset. The problems in each case are illustrated clearly in these simulations. N
2
c
e
:
\
C
|
:
N
the reset, again. Adaptation of then resumes after the reset because following the reset. This is illustrated in Figure 6. The reset helps to ensure that remains above zero so that the adaptation can carry on until . adaptation and are shown in Figures 7 and 8 respectively.
o
:
N
C
C
:
without -reset.
|
Case 1 : without -reset In this case, . , and stops updating Hence, whenever and ensuring stability of the closed loop algorithm. This can be seen in Figures 1 – 4. It can be observed from Figure 2 that was reduced to zero before . Following this, stopped updating (see Figure 3) and so does which is not shown. Hence the algorithm was not able to further reduce the tracking error and the error remained at approximately -0.008 as shown in Figure 4. This is the main problem of the algorithm when no reset was used. \
M
N
M
\
R
:
M
:
C
M
:
C
Case 2 : with -reset In this case, was reset each time . It can be seen from Figure 5 that when this happens, but in the next instant after C
\
o
:
|
R
:
346
6. CONCLUSION In summary, we have thus proposed an adaptive control algorithm that works well for a system with convex parametrizations. The algorithm includes a mechanism for resetting the nonlinear parameter estimates whenever some tuning function goes to zero. This ensured that adaptation could carry on until tracking error approaches zero. Such a closed loop system is
proach. IEEE Trans Automatic Control pp. 1634– 1652. Ortega, R. (1996). Some remarks on adaptive neurofuzzy systems. Int. Journal of Adaptive Control and Signal Processing 10, 79–83. Skantze, F. P., A. P. Loh and A. M. Annaswamy (1998). Adaptive estimation of nonlinear discrete systems. Proceedings of the American Control Conference 1, 594–598. Zhang, Y., C. Wen and Y. C. Soh (1999). Robust adaptive control of uncertain discrete time systems. Automatica 35, 321–329. Zhang, Y., C. Wen and Y. C. Soh (2001). Robust adaptive control of nonlinear discrete time systems by backstepping without overparametrization. Automatica 37, 551–558.
−4
∆ Vt
1
x 10
0
−1
20 time
10
0
40
30
|
Fig. 5.
with -reset.
2
ρt
1
0 a
−1
t
−2
Fig. 6. \
Appendix
20 time
10
0
and N
40
30
Lemma A.1 : Let
with -reset.
C
X
Ë
C
7
0
7
X
7
s
_
θ estimates
1.1
2
2
c
sgn e
V
0 _
u
reset X
v
M
X
_
v
arg
s
w
x
2
2
c
sgn e
V
M
X
_
u
v
w v
x
1.05
where
and
C
<
Ë
sgn
M
. Then
1 ^
0
20 time
10
30
40
is concave is convex
C C
Ë
7 7
X
7
Ë
: _
Ë
Fig. 7.
with -reset.
C
7
is concave
m
Ë
X
−5
1
x 10
7
jl
is convex
Ë
V
À
Î
µ
tilde yt
0.5 ln
where
0
−1 20
0
60
40
80
100
time
M
and
Proof : See (Loh et al., 1999).
−0.5
Fig. 8.
with -reset.
thus able to ensure zero error tracking if the resets do not occur repeatedly. It is argued that in general, the resets will stop after some time because would be reduced to zero and all adaptation will stop.
M
7. REFERENCES Kristic, M. and P. V. Kokotovic (1995). Adaptive control of nonlinear systems : a tutorial. Adaptive Control, Filtering and Signal Processing pp. 165–198. Kristic, M., I. Kanellakopoulous and P. V. Kokotovic (1995). Nonlinear Adaptive Control Design. Wiley, New York. Loh, A. P., A. M. Annaswamy and F. P. Skantze (1999). Adaptation in the presence of a general nonlinear parametrization : An error model ap-
347
. Ï
www.elsevier.com/locate/ifac
ADAPTIVE CONTROL OF DISCRETE TIME SYSTEMS WITH CONCAVE/CONVEX PARAMETRIZATIONS C Y Qu A P Loh K F Fong A M Annaswamy
Dept of Electrical & Computer Engineering National University of Singapore Singapore 119260 Dept of Mechanical Engineering Massachussetts Institute of Technology Cambridge, MA 02139 Abstract: This paper considers the adaptive control of a class of nonlinear discrete time system with nonlinearly parametrized functions. In particular, the focus is on concave or convex parametrizations with unknown parameters. The solutions involved 2 tuning functions which are determined by a minmax optimization approach much like the continuous time counterparts found in the literature. Direct extension from continuous time case do not work very well due to the premature termination of the adaptive algorithm before zero tracking error can be achieved. In this paper, this problem is solved. The proposed algorithm can be shown to be stable and in some cases, achieve zero tracking error in steady state. Copyright © 2002 IFAC Keywords: adaptive control, discrete time systems, nonlinear systems not been very successful. This is largely due to the Lyapunov approach that is adopted in the analysis. The direct extension of (Loh et al., 1999) does not work very well because the Lyapunov approach does not allow one to construct additional signals to ensure closed loop stability. While it is possible to construct two tuning functions for the parameter adaptation as in the estimation problem in (Skantze et al., 1998), the use of such tuning functions cannot guarantee zero tracking error. Adopting the approach in (Skantze et al., 1998) however can guarantee closed loop stability, as will be seen in this paper.
1. INTRODUCTION Adaptive control of nonlinear dynamic systems is currently an area of great interest among control theorists. Much of the work thus far has been for continuous time systems (Kristic et al., 1995), (Kristic and Kokotovic, 1995) involving linear parametrizations. Very few results are available in the literature that addresses adaptive control in the presence of nonlinear parametrizations (Ortega, 1996). One such result can be found in (Loh et al., 1999) where the adaptation scheme was changed significantly from the linear case in order to accomodate the nonlinear parametrization. Specifically, two tuning functions were included, one that determines the direction of parameter updates and the other that ensures stability of the closed loop system. These two tuning functions were derived from a minimax criterion that was solved on-line when the nonlinear parametrization involves convex or concave functions.
Some recent results on discrete time systems have appeared in (Zhang et al., 1999) and (Zhang et al., 2001) for linearly parametrized systems. The main approach in these two papers involves the backstepping technique. Their convergence analysis did not make use of a Lyapunov function. In this paper, the adaptation algorithms follow closely those in (Loh et al., 1999) and (Skantze et al., 1998). However, some modifications were made to ensure
Although results exists for continuous time systems, the extension of these to discrete time systems have
343
T
that zero tracking error is achieved. This analysis was made using a Lyapunov function. The paper is organised as follows. In Section 2, the class of nonlinear system is defined along with the control objectives. Section 3 presents the algorithm for the adaptive controller, followed by Section 4 which shows the stability and convergence analysis. Simulation results are shown in Section 5 and concluding remarks are given in Section 6.
o
or
N
:
m
T
(8)
C
jl
R
l
l
T
otherwise
l
l
0
s
ln
2
2
c
sgn e
V
M
X
(9)
_ _
u
v
w
v
x
0
X
arg
s
2
2
c
sgn e
V
M
X
(10)
_
u
v
w v
x
C
X
(11)
C
7
7
X
7
_
2. PROBLEM STATEMENT
The class of nonlinear discrete time system we consider is as follows :
7
(13)
-
The minmax solution in (9) and (10) are given in the Appendix for ease of reference.
where and are the plant’s input and output at and are time instant , vectors of measureable states, and are the unknown parameters, and is a known convex or concave function of its arguments. The objective is to determine a bounded control input, , such that the output, , of the plant asymptotically tracks a specified bounded reference trajectory, . More precisely, we require
M
(
(1)
(12)
C C
"
$
"
(
4. STABILITY ANALYSIS
"
$
Using the Lyapunov’s second method of stability analysis, the Lyapunov candidate was chosen as
+
(
/
M
M
M G
I
where
M
3
C
|
-
The following assumptions are made regarding the plant in (1) (A1)
M
(14) ~
7
. Defining
M
7
K
M
N
M
X
=
~
N
~
M
~
G
I
X
~
G
V >
is a bounded concave or convex funtion of
+
K
~
N
~
M
~
7
K
~
N
~
M
~
.
(A2)
~
|
K
N
|
7
5
(
~
and , it can be shown that 7
:
+
K
C
7
G
M
|
~
-
V
0
2
|
is assumed to lie in an interval given by , where and are known.
=
>
K
M
N
X
7
M
N
M
K
7
<
Y _
K
~
N
~
M
(15)
~
(16)
K
M
N
X
7
M
N
M
K
To ensure that be considered.
In this section, the adaptive algorithm for the nonlinear discrete time system in (1) is presented for a convex . The case of a concave is similar and will be discuss later.
Y _
3. CONTROLLER DESIGN
Case (a)
M
for all , two cases of
|
:
M
will
In this case, we require :
+
+
(17)
K
M
N
X
7
M
N
M
K
The adaptive mechanism together with the construction of the control input, can be specified as :
which implies that and satisfies
:
Y _
N
should be chosen such that
N
R
:
X
(2)
C
7
7
C
(
-
N
7 Y
_
+
Y K
M
Y
C
7
G
I
K
M
N
G
C
T
G
I
R
since , to choose
(3) :
I
K
M
R
N
. Thus, for any where
:
\
and X
, it suffices
<
T
7
G
V
K
M
N
X
G
R
(4) :
V
0
s
X
\
Y
7 2
(18)
_
V
K
R Y Z
X G
V
X
G
I
:
(5)
K
v
Y
M
x
Y
Z
^
Y
Y 7
2
c
e
X
+
V
\
Y
7
_
`
K
Y
`
M
a
(6)
K
Y
M
_
v
x
Y
Y
Since is a free variable that can be further optimized, can finally be chosen as X
N
2
c
e
f
:
\
i
(7)
\
344
0
s
N
(23)
Y
+ \
7 Y
2 Y
2
c
e
X
V
K
Y
K
M
_
u
v
w v
x
Y
Y
Y
7
Y K
With this choice of
(19) _
M
N
,
then becomes
|
Y
0
s
` `
|
7
N
K
M
X
X
arg
2 2
c
X e
v
w v
_
K _
u
(20)
V
x
=
`
`
M
7
7
7
K
X
7
>
According to the minmax solution in Lemma A.1 in the Appendix, for convex , and . Hence indeed in (19). With this choice of and in (19) and (20) respectively,
X
:
Y
K
+
=
` `
V
7
M
7
_
N
N
X
\
Y K
:
Y
(24)
7
>
V
|
V
K
M
N
X
7
K
N ~
~
M ~
V
where , and . By this choice of , and using the property of convex functions,
_
V
7
K
~
Case (b)
M
o
N
~
M
~
:
+
In this case, we require :
7
:
+
Y
Once again, after the reset and this ensures that the whole algorithm carries on in a stable manner. In other instants when , will again be reset but in the next instant, -adaptation will resume. Thus, in general, we have that except in some instances during reset. |
`
` `
`
`
`
7
K
N
~
K
M
N
X
M
7
N
M
K
:
+
Y _
Following the same analysis as case (a) above, we require
M
~
7
~
C
\
o
:
|
X
N
Y o
_
+
`
K
`
Y
M
7
K
N
~
Y
N
can thus be chosen as
0
M
~
~
s
It is important to note that -reset will only happen when , in the case of convex , as shown in case (b). In addition, this happens only when . In other words as long as , there exists a such that . This implies that in the worst case, will be bounded by
2
V
X
v
\
x
_
`
`
Y
:
Y
K
M
M
+
R
Y
_
`
`
With further optimisation, \
can be chosen as
M
`
7 Y
Y
2
c
sgn e
`
V
Y
M
M
_
X
\
R
:
:
`
M
2
R
s
2
`
K
|
\
M
N
`
0
`
c
o e
£
+
_
u
v
w
v
_ V
x
v
x
Y
Y 7
(21)
_
`
`
Y K
M
|
Hence, we conclude that
o
for all £
R
. :
Y
in (21) is positive, then it follows that If according to (7) and this guarantees that \
N
\
Suppose that the time instants can be separated into three sets, where contains all time instants where -resets have occurred and , contains all time instants where and contains all ’normal’ instances where . Then, summing all , including , we have
§
¨
|
.
:
¤
¥
¦
¤
¥
Y
|
|
in (21), then by the choice However, if in (7). At the same time, according to (8), is reset to . This reset does not apply to but since , no adaptation of occurs. In other words, at the instant , when
\
:
N
©
:
N
o
o
:
~
¥
¤
|
|
:
:
|
\
¥
~
ª
C
7
C
:
¤
C
R
~
:
5
¯
¯
|
|
|
7 5
7 ±
K
~
C
and
C
(22)
C
+
°
N
~
M
~
±
v
²
³
¯
¯
|
in (16) is no longer valid for further analysis at this point. Due to the reset in (22), the modified will be as follows :
7
ª
© ~
~
+
|
v
v
²
´
²
µ
If -reset occurs only a finite number of times, then , thus implying that
¶
©
o
£
~
v
²
µ
M
M
|
~
G
7
~
G
7
~ K
V
~ N
~ M
V
¯
~ K
~ N
~ M
o
£
+
M
7
~ G
7
~
V
G
7
~ K
~ N
~ M
+
V
From the analysis, and after each reset. In fact, the algorithm can never end with because if , reset will occur and if is not zero at that instant, adaptation will carry on. It follows therefore that K
|
|
may not be negative. However, if will be bounded because of assumption A.2.
:
, it
C
\
R
:
N
:
R
:
/
0
Y
N
N
Due to the reset of at time instant , at the next instant, can be chosen as again as in (21). But due to the reset, and hence N
2
M
:
+
:
_
345
3
5
M
:
Discussion A stable algorithm is also possible without -reset. This is the algorithm in (Skantze et al., 1998) used in an estimation problem. In this case, will simply be set to zero whenever as in (7). Once this happens, all adaptation stops and the whole algorithm may end prematurely before . In order to prevent this from happening, some mechanism had to be developed to keep the adaptation alive without losing stability. The reset was one way to restore the value of to a non-zero value so that adaptation can carry on until goes to zero.
−3
x 10
N
o
N
0
:
M
t
∆V
\
−5
:
−10
40
60
M
100
|
Fig. 1.
1
+
ρt
0
+
without -reset.
+
a
−1
t
80
time
for This paper only addresses the case of convex ease of presentation. The proof for concave functions of is the same. When is concave, the reset should occur at instead of as in the convex case. When changes convexity or concavity, then combinations of the convex and concave algorithm should work accordingly.
20
0
+
−2 0
20
60
40
80
100
time
Fig. 2. \
and N
without -reset.
5. SIMULATIONS 1.15
º
»
1.1
1.05
(
θ estimate
In this section, the performance of the proposed algorithm is illustrated via a simulation example applied to the following system :
1 0 ¼
-
20
40
60
80
100
60
80
100
(
(
time -
C
7
(
7
C
-
¾
¿
¾
Fig. 3.
V
À C C
À
½
M
(
7
+
Á
:
:
+
0.01
+
tilde y
t
0
−0.01
Á
-
The numerical values of the parameters in the system are chosen as
without -reset.
C
Ã
+
Á
:
:
+
Á
+
Á
(
Y
Y
−0.02 0
C
¨
±
±
:
:
C
+
+
(
Y
Y
Y
Y
20
40
Y
time
G
I
Z
:
Ä
~
Y
Å
G
V
:
~
Y
Y
Fig. 4.
Simulation results are shown for both with and without -reset. The problems in each case are illustrated clearly in these simulations. N
2
c
e
:
\
C
|
:
N
the reset, again. Adaptation of then resumes after the reset because following the reset. This is illustrated in Figure 6. The reset helps to ensure that remains above zero so that the adaptation can carry on until . adaptation and are shown in Figures 7 and 8 respectively.
o
:
N
C
C
:
without -reset.
|
Case 1 : without -reset In this case, . , and stops updating Hence, whenever and ensuring stability of the closed loop algorithm. This can be seen in Figures 1 – 4. It can be observed from Figure 2 that was reduced to zero before . Following this, stopped updating (see Figure 3) and so does which is not shown. Hence the algorithm was not able to further reduce the tracking error and the error remained at approximately -0.008 as shown in Figure 4. This is the main problem of the algorithm when no reset was used. \
M
N
M
\
R
:
M
:
C
M
:
C
Case 2 : with -reset In this case, was reset each time . It can be seen from Figure 5 that when this happens, but in the next instant after C
\
o
:
|
R
:
346
6. CONCLUSION In summary, we have thus proposed an adaptive control algorithm that works well for a system with convex parametrizations. The algorithm includes a mechanism for resetting the nonlinear parameter estimates whenever some tuning function goes to zero. This ensured that adaptation could carry on until tracking error approaches zero. Such a closed loop system is
proach. IEEE Trans Automatic Control pp. 1634– 1652. Ortega, R. (1996). Some remarks on adaptive neurofuzzy systems. Int. Journal of Adaptive Control and Signal Processing 10, 79–83. Skantze, F. P., A. P. Loh and A. M. Annaswamy (1998). Adaptive estimation of nonlinear discrete systems. Proceedings of the American Control Conference 1, 594–598. Zhang, Y., C. Wen and Y. C. Soh (1999). Robust adaptive control of uncertain discrete time systems. Automatica 35, 321–329. Zhang, Y., C. Wen and Y. C. Soh (2001). Robust adaptive control of nonlinear discrete time systems by backstepping without overparametrization. Automatica 37, 551–558.
−4
∆ Vt
1
x 10
0
−1
20 time
10
0
40
30
|
Fig. 5.
with -reset.
2
ρt
1
0 a
−1
t
−2
Fig. 6. \
Appendix
20 time
10
0
and N
40
30
Lemma A.1 : Let
with -reset.
C
X
Ë
C
7
0
7
X
7
s
_
θ estimates
1.1
2
2
c
sgn e
V
0 _
u
reset X
v
M
X
_
v
arg
s
w
x
2
2
c
sgn e
V
M
X
_
u
v
w v
x
1.05
where
and
C
<
Ë
sgn
M
. Then
1 ^
0
20 time
10
30
40
is concave is convex
C C
Ë
7 7
X
7
Ë
: _
Ë
Fig. 7.
with -reset.
C
7
is concave
m
Ë
X
−5
1
x 10
7
jl
is convex
Ë
V
À
Î
µ
tilde yt
0.5 ln
where
0
−1 20
0
60
40
80
100
time
M
and
Proof : See (Loh et al., 1999).
−0.5
Fig. 8.
with -reset.
thus able to ensure zero error tracking if the resets do not occur repeatedly. It is argued that in general, the resets will stop after some time because would be reduced to zero and all adaptation will stop.
M
7. REFERENCES Kristic, M. and P. V. Kokotovic (1995). Adaptive control of nonlinear systems : a tutorial. Adaptive Control, Filtering and Signal Processing pp. 165–198. Kristic, M., I. Kanellakopoulous and P. V. Kokotovic (1995). Nonlinear Adaptive Control Design. Wiley, New York. Loh, A. P., A. M. Annaswamy and F. P. Skantze (1999). Adaptation in the presence of a general nonlinear parametrization : An error model ap-
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