Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
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. Keywords: adaptive control, discrete time systems, nonlinear systems 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 online when the nonlinear parametrization involves convex or concave functions. Although results exists for continuous time systems, the extension of these to discrete time systems have
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. 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
m UC Lkjll l
X
where and are the plant’s input and output at time instant , !#"%$ and &'"%( are 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 /10 2 4365
879;: + (.
The following assumptions are made regarding the plant in (1) (A1)
,+
(A2) = is assumed to lie in an interval given by < ?> , where and are known.
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. The adaptive mechanism together with the construction of the control input, can be specified as : C E
D C E F7 @A7B
(.
C D
C F7HGJILK?M NO PQG8 I
T U@
T U F7HGJVWK?M NOXJ
K
X GPVXP
Z
\O@ Y
Y
^ K]
Y
N H2dc?eLfW:gh\ i
Y
7)` _
G V
E GPI[P
RA:
(3)
M where 
 7
C 7 
E Uh
(12) (13)
M ~ G V ]
E
K ~ N~ M ~
M
C
XJ UM
 LS7 K?M NOy
M E
K ~ O N ~ M ~ = P E GPIgP K ~ N ~ M ~ 7 [
(14)
7 and . Defining , it can be shown that
XBh7M _ M ~ E K[ ~ NO ~ E
K?M NO
To ensure that be considered. Case (a) M

:
K]?]M NO
N M
:
K L7
K],
Y
(15)
XBh7M _
X ~ GJV >
K ~ E N ~ M ~
K M N L
NOW]M Y
(16)
for all , two cases of ]M will
In this case, we require XBh7M
_
NO?]M
K,
Y
:
(17)
which implies that NO should be chosen such that NR : and satisfies NO Y
Y
7
K Y
XBh _
M
+
R : since K , M A . Thus, for any X and N r \ to choose where 0ts Y
2 K V ] v x
(5)
(7)
E {
(11)
7z
 q} M G I M
\
(6)
(10)
F7zX UC F79
(4) Z RS:
sgn M _ XBh
2dc?e V v]x
Using the Lyapunov’s second method of stability analysis, the Lyapunov candidate was chosen as
Y
]M `ba
(.
(9)
4. STABILITY ANALYSIS
(2) ;GJIRS:
sgn ]M , _ XBh
The minmax solution in (9) and (10) are given in the Appendix for ease of reference.
3. CONTROLLER DESIGN
(8)
E ? C
M 9
is a bounded concave or convex funtion of
.
2dc?e V v]x 0ts
C E y7
C P
The class of nonlinear discrete time system we consider is as follows :
2 arg
[email protected] v w
XB
or Nqr:
otherwise
p2 uLvw _
2. PROBLEM STATEMENT
(1)
T FR W
T U ll 0ts ln
_
T WFop
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.
Y
Y
K* Y
7
Y
7_
Y d 2 c?e ]M V v]x
<
, it suffices
XBO M
(18)
_
XB
+
Since X is a free variable that can be further optimized, \ can finally be chosen as
0ts \OL
Y
Y X
Y
Y
7Y 2 K % M L Y u vw 7
K]% Y 01s 2 u vw @
arg
_
2dc?e V v]x
XBh _
(19)
M D
2dc?e V v]x
 J
(20)

K M N
X h7 _
Case (b) ]M oS: 7 K ` M ` NO
Y
Y
NO
Y
Y
\
Y
Y
K Y
K* Y
_
can thus be chosen as
X B ` M ` +
01s
2
01s Y
Y
Y
2 ` M Y ` L u vw 7
` _ M `
Y
K
Y
7
K]% Y
2dc?e V v]x
(21)
and
7
UC ; +

(22)
M ~ G V 7 K ~ N ~ E M ~ E E E 7 M ~ E G V 7 K ~ N ~ M ~ + 79 ~ G V
M ~ G V
7


may not be negative. However, if will be bounded because of assumption A.2. C
XBO _
7
` M J ` =
7zXJ 7z>
7@ Y
>
(24)

¢ g Y
:
, it
Due to the reset of Ur at time instant , at the next N Y instant, can be chosen as \ again as in (21). : But _ due to the reset, and hence
or£ _
7@ O
: +
+
 Hence, we conclude that or£
for all RA: .
Suppose that the time instants can be separated into three sets, ¤8¥¦ p§ Y h¨ where ¤8¥ contains all  time instants where resets have occurred and  © ~ R: , ¤¥ ~ contains all time instants where 7«ª ~ o¬: and ¤ ¥ contains all ’normal’ instances  :  where . Then, summing all , including ® , we have 5 °±

 5 7
 ±

` ]M J ` =
then becomes
J 7 K ~ N ~ M ~ Once again, after the reset and this ensures that the whole algorithm carries on in a C stable manner. In other instants when \ o : , will again be reset but in the next instant, adaptation  will resume. Thus, in general, we have that 7 K ~ N ~ M ~ except in some instances during reset.
¯
in (16) is no longer valid for further analysis at  this point. Due to the reset in (22), the modified will be as follows :
2dce V v]x
: However, if \O in (21), then NO: by the choice C in (7). At the same time, according to (8), U is reset to . This reset does not apply to C but since NOq;: , no adaptation of C occurs. In other words, at the instant when \ oS: , C
K]J
sgn M , _ XB
N \O If \ in (21) is positive, then it follows that  : according to (7) and this guarantees that .
C q
 J
(23)
It is important to note that reset will only happen : when ]M , in the case of convex + , as shown in case (b). In addition, this happens only when _ R ` M ` ` ]M ` R . In other words as long as _ , there exists  : a NOq;\FRS: such that . This implies that in ` ` the worst case, ]M will be bounded by
V v]x XB ` M _ `
With further optimisation, \ can be chosen as \O@
Y
: +
Following the same analysis as case (a) above, we require NFo
K J
¢
K]
K J
+
K J
where , h and . By this choice of , and using the property of convex functions,
: +
` ]M ` 7zNO ` M `
XBh
K]J
S7
In this case, we require
_
Y
V¡DV V¡DV
K ~ N ~ M ~
7 K ~ N ~ M ~
Y
7 N J K J ` M J `
S7
According to the minmax solution in Lemma A.1 in the Appendix, for convex ,+ , _ ;: and X EV . Hence indeed N \ : in (19). With this choice of N and X in (19) and (20) respectively,
With this choice of NJ ,
XB _
NOJ {
¯ v
?²4³
7 K ~ N ~ M ~ ¯ v
?²¢´
7«ª ~
¯ v
?² µ
© ~ +
If reset occurs only a finite number of times, then ¶ © ~ or£ v , thus implying that
?² µ ¯
K ~ E N ~ M ~ or£ +
From the analysis, K ®R: and N E ®R: after each reset. In fact, the algorithm can never end with N;: because if N·¸: , reset will occur and if ]M is not zero at that instant, adaptation will carry on. It follows /t0 therefore that 2 43®5
M r: +
This paper only addresses the case of convex ,+ for 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.
−3
x 10
t
0
∆V
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, N will simply be set to zero whenever \FoA: as in (7). Once this happens, all adaptation stops and the whole algorithm may end prematurely before ]M %¹: . 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 NO to a nonzero value so that adaptation can carry on until ]M goes to zero.
−5
−10
0
20
40
60
80
100
time
Fig. 1.

without reset.
1 ρ
t
0 −1 −2
at 0
20
40
60
80
100
time
Fig. 2. \O and NO without reset. 5. SIMULATIONS In this section, the performance of the proposed algorithm is illustrated via a simulation example applied to the following system :
θ estimate
1.15 1.1 1.05
]@ p¢ (.
@
(.
(
1
]º»,¼
@H (. 87H C E y7 C D¾?¿ ¾ À V À h½ C ¢ C ]M @H
(.
0
20
40
60
80
100
60
80
100
time
C
Fig. 3. W without reset.
879
0.01
The numerical values of the parameters in the system are chosen as
0
tilde yt
−0.01
{
Y
+ÂÁ
Z
: +ÂÁ : + Ã
· ¨[h Y
Y
G I
Y
( : Ä ]
~UÅ[~
Y
®
UC . ± G V
Y
+ÂÁ ( Y C ±. : + Y
:
Y
: + Á : +ÂÁ
: +
Y
Simulation results are shown for both with and without reset. The problems in each case are illustrated clearly in these simulations. Case 1 : without reset In this case, Nqr2dce :[h\O, . : C Hence, whenever \ , and C stops updating  Æ: and ensuring stability of the closed loop algorithm. This can be seen in Figures 1 – 4. It can be observed from Figure 2 that N was reduced to zero C before ]M ÇÈ: . Following this, U stopped updating (see Figure 3) and so does C 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. C
Case 2 : with reset In this case, U was reset each time \ o: . It can be seen from Figure 5 that when  RÉ: this happens, but in the next instant after
−0.02
0
20
40 time
Fig. 4. M without reset.

C
J o': the reset, again. Adaptation of U then resumes after the reset because N J Ê\ P Rk: following the reset. This is illustrated in Figure 6. The reset helps to ensure that N remains above zero so that the adaptation can carry on until M : . adaptation and M are shown in Figures 7 and 8 respectively.
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
−4
∆ Vt
1
0
−1
Fig. 5.
x 10
0
10

20 time
30
40
with reset.
2
ρ
t
1 0 a
−1 −2
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.
t
Appendix 0
10
20 time
30
40
Lemma A.1 : Let
Fig. 6. \O and NO with reset.
2 uLvw _
1.1 θ estimates
C E 7 0ts
XBzË _
XPL
reset
2Ìce V 0tv] s x
2 arg uL v w
1.05
C
where U E <
sgn M _ XBO 2dc?e V v]x
?
7zX C 7z
sgn ]M _ XBh
and ËÍ
sgn ]M , . Then
1 0
10
20 time
30
40
_
C
Fig. 7. U with reset. XJL
^
Ë C F7 : m jl
−5
1
x 10
ln
tilde yt
0.5
where
7
·79 8V ÀÎ µ
7zXJ UC E F7z Ë Ë
Ë
is concave
Ë
is convex
¢ O
and
0
Proof : See (Loh et al., 1999).
−0.5 −1
0
20
40
60
80
100
time
Fig. 8. M 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 M would be reduced to zero and all adaptation will stop.
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
is concave is convex
¢
.
Ï