Decentralized robust adaptive control of nonlinear systems with ...
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Decentralized Robust Adaptive Control of Nonlinear Systems With Unmodeled Dynamics Yusheng Liu and Xing-Yuan Li Abstract—The authors present a decentralized robust adaptive output feedback control scheme for a class of large-scale nonlinear systems of the output feedback canonical form with unmodeled dynamics. A modified dynamic signal is introduced for each subsystem to dominate the unmodeled dynamics and an adaptive nonlinear damping is used to counter the effects of the interconnections. It is shown that under certain assumptions, the proposed decentralized adaptive control scheme guarantees that all the signals in the closed-loop system are bounded in the presence of unmodeled dynamics, high-order interconnections and bounded disturbances. Furthermore, by choosing the design constants appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances, and unmodeled dynamics in the system. An illustration example demonstrates the effectiveness of the proposed scheme. Index Terms—Adaptive, decentralized control, large-scale systems, nonlinear, robust.
I. INTRODUCTION Much progress has been made in the field of decentralized adaptive control, see e.g., [1]–[8], [18], [19], and the references therein. Particularly, a decentralized adaptive output control scheme was presented in [8] for a class of large-scale nonlinear systems that are transformable via a global diffeomorphism into the output feedback canonical form. The scheme guarantees global uniform boundedness of the tracking error and all the states of the closed-loop system in the presence of parametric and dynamic uncertainties in the interconnections and bounded disturbances. However, the scheme cannot apply to the systems with unmodeled dynamics. The work in [18] presented a decentralized adaptive output feedback control scheme for large-scale systems with nonlinear interconnections. The scheme of [18] has several advantages: 1) it achieves asymptotic tracking; 2) the considered large-scale systems may possess an unknown, nonzero equilibrium. But, the scheme cannot apply to the systems with unmodeled dynamics and disturbances. On the other hand, the robust adaptive control of nonlinear systems has emerged as an active research area recently, e.g., [10]–[13]. Especially, the problem of robust adaptive control of nonlinear systems with unmodeled dynamics was studied in [13]. Based on the concept of input-to-state practical stability, a dynamic signal was introduced in [13] to dominate the unmodeled dynamics. It has been shown that such a signal is a useful tool for handling unmodeled dynamics. Using a combined backstepping and small-gain approach, paper [20] presented an adaptive output feedback control scheme for nonlinear systems with unmodeled dynamics. As an extension of the centralized case in [20], a decentralized robust adaptive output feedback regulation scheme was presented for a class of large-scale nonlinear systems in [19]. It should be pointed out that the class of large-scale nonlinear systems considered in [19] is broader than the class of nonlinear systems with polynomial bounds as in [7], [8], [16] and [15], in the sense that the interconnections are subject to general nonlinear bounds. Although, [17] studies a different problem—decentralized H1 almost disturbance decoupling for a class of large-scale nonlinear systems, it Manuscript received May 11, 2001; revised November 26, 2001. Recommended by Associate Editor M. M. Polycarpou. This work was supported by the National Key Basic Research Special Fund of China under Grant G1998020312. Y. Liu is with the Department of Automation, Sichuan University, Chengdu, Sichuan 610065, P. R. China (e-mail: [email protected]). X.-Y. Li is with Department of Electrical Engineering, Sichuan University, Chengdu, Sichuan 610065, P. R. China (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)04751-7.
is also assumed that the uncertain interconnections are bounded by general nonlinear functions. Theoretically, to consider a broader class of nonlinear systems is important, but some other issues such as controller design and practical problems must be considered. With the general nonlinear bounds, the design of the controllers depends on a qualitative approach (as in [19] and [20]). For large-scale nonlinear systems with many decentralized controllers, it is difficult to choose so many design parameters and functions by qualitative approach. Furthermore, with the general nonlinear bounds, the design of the ith decentralized controller is related to that of the rest decentralized controllers (see [19, (25) and (10)]). This implies that redesigns of controllers are needed when subsystems are appended to original system or taken offline (e.g., during faults in a power system), which is neither economical nor feasible for the control of large-scale systems. Besides, many practical nonlinear interconnected systems do have some uncertainties and interconnections subject to polynomial bounds, e.g., robot manipulators with the presence of Coriolis and centripetal terms have uncertainties bounded by second-order polynomials [16] and the electric power deviations between subsystems are bounded by first order polynomials [21]. Therefore, for the practical purpose and the above reasons, we still consider the case where uncertainties and interconnections are subject to polynomial bounds in this work. As in [8], with our approach, no controller redesign is needed if subsystems are added online or taken offline as long as the order of the interconnections of the appended system is less or equal to that of the original system. This work presents a decentralized robust adaptive control scheme for large-scale nonlinear systems of the output feedback canonical form with unmodeled dynamics. First, in each subsystem, a modified dynamic signal is introduced to dominate the unmodeled dynamics and a nonlinear adaptive damping is used to counter the effects of the interconnections. Then, we employ the systematic design procedure to obtain the decentralized robust adaptive output feedback controllers. It is shown that under certain assumptions, the proposed decentralized adaptive control scheme guarantees that all the signals in the closed-loop system are bounded in the presence of unmodeled dynamics, high order interconnections and bounded disturbances. Furthermore, the tracking error can be made arbitrarily small by choosing the design constants appropriately. II. PROBLEM STATEMENT Consider a large-scale nonlinear system of the output feedback canonical form with unmodeled dynamics given by the following equations:
_ =q ( ; y ) z_ 1 =z 2 + 1 (y1 ; . . . ; yN ) + i1 (y1 ; . . . ; yN ) !(t) + 1i1 (i ; yi )
_
i
i
i
i
i
i
i
.. .
+ i; 01 (y1 ; . . . ; yN ) + i; 01 (y1 ; . . . ; yN ) !(t) + 1i; 01 (i ; yi ) z_ i; =zi; +1 + i; (y1 ; . . . ; yN ) + i; (y1 ; . . . ; yN ) !(t) + bi;k 0 i (yi ) ui + 1i; (i ; yi )
zi;
01 =zi;
.. .
_ = i;k (y1; . . . ; yN ) + i;k (y1 ; . . . ; yN ) !(t) + bi0 i (yi ) ui + 1i;k (i ; yi ) yi =zi1 1 i N
zi;k
0018-9286/02$17.00 © 2002 IEEE
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...
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where zi zi1 ; ; zi;k T , yi 2 R and ui 2 R are the state, the output and the control input of the ith subsystem, respectively; ! t is a bounded unmeasurable disturbance; ij y1 ; ; yN and ij y1 ; ; yN are the dynamic and parameter uncertainties related to the interconnections between subsystems; i is the unmeasurable unmodeled dynamics and ij i ; yi represents the unmeasurable uncertainties related to the unmodeled dynamics. We assume that ij , ij and ij are unknown, but satisfy
() ( ...
( ...
)
1 (
where
)
Ai =
)
i =
1
k ij (y ; . . . ; yN )k
p
N
1
kij (y ; . . . ; yN )k
k=1 l=1 p N
1
k1ij (i ; yi )k
k=1 l=1 p
k &ijl kyl kk
1i =
k #ijl kyl kk
ijk ki kk + 'ijk kyi kk
k=1
(2)
k , #k , where &ijl ijk and 'ijk are unknown constants, p = ijl maxfpij ; 1 i N; 1 j ki g is known. To highlight the main idea of this work, it is assumed that the parameters bij , 1 i N , 0 j ki 0 i , are also known. As indicated in [8], the case of
unknown bij can be treated identically. The objective here is to design a decentralized robust adaptive output controller for each subsystem such that the output yi t tracks a given reference signal yi;r t and all the signals of the closed-loop system are bounded in the presence of unmodeled dynamics, high order interconnections and bounded disturbances. We need the following assumptions. Assumption 1: The zero dynamics of system (1) is exponentially stable, i.e., the polynomial Pi s bi;k 0 sk 0 1 1 1 bi1 s bi0 is strict Hurwitz. Assumption 2: i yi 6 ; 8t . Assumption 3: The reference signal yi;r t is bounded with ( ) bounded derivatives up to the i th order and yi;r t is piecewise continuous. Assumption 4: The unmodeled dynamics is exponentially input-tostate practically stable (exp-ISpS) [13]; i.e., the system i qi i ; yi has an exp-ISpS Lyapunov function Vi i which satisfies
()
()
( )= ( )=0 0
+ +
()
+
()
_= (
( )
)
i1 (ki k) Vi (i ) i2 (ki k) @Vi (i ) q ( ; y ) 0 c V ( ) + (ky k) + d i0 i i i i i0 @i i i i
(3) (4)
where i ; i are known functions of class K1 and ci > 0, di 0 are known constants. Without loss of generality, we assume i (1) has the following form i (s) = s i (s ) where i is a nonnegative smooth function. Otherwise, as indicated in [13], it suffices to replace
i in (4) by kyi k i (kyi k ) + "i with "i > 0 being a sufficiently 1
2
0
2
2
0
2
0
0
2
0
0
0
small-real number.
III. THE DESIGN OF DECENTRALIZED ROBUST ADAPTIVE OUTPUT FEEDBACK CONTROLLERS
1
.. .
0ki;k 0 1 1 1 0 i1
i =
.. .
i;k
1i .. .
1
.. .
ki;k
i1 .. .
i;k
bi = [ 0
1i;k
ki1
ki =
I
1 1 1 0 bi;k 0 1 1 1 bi ]T 0
and ki is chosen such that Ai is a strict Hurwitz matrix. Thus, given a Qi > , there exists a Pi > satisfying
0
0
ATi Pi + Pi Ai = 0Qi :
z^_ i = Ai z^i + ki yi + bi i (yi ) ui : Let ei
(7)
= zi 0 z^i . Then
e^_ i = Ai ei + i (y1 ; . . . ; yN ) +i (y1; . . . ; yN ) !(t) + 1i (i ; yi ) :
(8)
Define a dynamic signal
r_ i = 0ci0 ri + zi21 i0 zi21
+ di ri (t ) = ri > 0 0
0
0
(9)
(0 )
where ci0 2 ; ci0 The properties of the dynamic signal are given by the following lemma. Lemma 1 [13]: If the system i qi i ; yi is exp-ISpS, then for any constant ci0 2 ; ci0 , any initial instant t0 , any initial condition i0 i t0 and ri0 > , for any function i0 such that
i0 yi i kyi k , there exist a finite Ti0 Ti0 ci0 ; ri0 ; i0 and a nonnegative function Di t0 ; t defined for all t t0 such that Di t0 ; t , 8t t0 Ti0 and Vi i t ri t Di t0 ; t for all t t0 where the solutions are defined. In this work, we introduce the following modified dynamic signal in the design of decentralized robust adaptive feedback controllers:
_= ( ) (0 ) 0 = ( ) 0 = ( ) ( ) ( ) ( )=0 + ( ( )) ( )+ ( )
r_i =
0ci ri + zi i zi + di
0;
0
2 1
0
2 1
ri (t0 ) =ri0 > 0
= ( ) ( )
+
0
if mri if mri
0
>0
0 (10)
+
_ 0
where mri 0ci0 ri zi21 i0 zi21 di0 . It can be seen that ri and ri t ri t , 8t t0 if ri0 ri0 . Using Lemma 1, we have
=
Vi (i (t)) ri (t) + Di (t0 ; t) ;
+
(11)
where Di t0 ; t , 8t t0 Ti0 . Next, we employ the systematic design procedure of [9], [14], and [8] to obtain the decentralized robust adaptive feedback controllers. To reduce notational complexity, as in [8], we assume that all subsystems have a uniform relative degree, i.e., i , i N . However, it can be treated similarly without any difficulties when the subsystems have different relative degrees.
= 1
(5)
(6)
To estimate the states of the system, we use the following observer for the ith subsystem:
=( )=0
First, we rewrite the ith subsystem (1) as follows:
_i =qi (i ; yi ) z_i =Ai zi + ki yi + i (y1 ; . . . ; yN ) + i (y1 ; . . . ; yN ) !(t) + 1i (i ; yi ) + bi i (yi ) ui
0ki
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Since z^i2 is not the actual control, we define
Step 0: Define p
i1 = zi1 0 yi ; e = 1 + ;r
22 011 (2r ) k
i
01=2
2k
i
i
e
2 = z^ 2 0 M 1 1 ; ^ ; r i
(12)
i
k=1
i;r
i
N
p
+1 1 ( ; y ) + 1 + i
i
i
i
e 2 :
i
i
(13)
i
i
p
2
2k
i
0i11 (2ri ) 2k
01=2
1
0 1+
i
i
i
N
i
i
i
1=2
22 011 (2r ) 2
e 2 :
k
k
i
i
N
p
1
i
(18)
i
i
01
k
k
i
T i
01 2 k 0i11 (2ri ) 2k01 : @ i1 (2ri ) 2r_i ei : @ri k=1 p
2k
(14)
ik
i
ik
i1
i;r
^_ = 0
p
22 21 k
i
i
i
k
0 0 ^ ^ (t ) > 0 i
i
k lji
# !max k lji
2
j =1 k
kP k ( ) 2
i
j =1
ijk
2
2
2
i;m+1 i;m
0 [ (Q ) 0 4] ke k + i
p
i1
i1
i1k
ik
i k
ik
i k
y ; z^ 1 ; . . . ; z^ ; ^ ; r ; y ; . . . ; y ( i
i
i;m
i
i
i;r
m) i;r
:
(22)
22 d + v + '
2
i
2
2
k j i
0M
i
min
ijk
# 1 !max ; 1 = ( 1 )2 ; '1 = (' 1 )2 :
= z^
> 0, > 0 are design constants.
i=1
2
Differentiating (17) gives (21), shown at the bottom of the next page. Step m (1 m 0 2): Assume that in step m 0 1, we designed a virtual control Mi;m and defined i;m+1
N
(19)
j =1
k=1
V_ 0
N
ik
(16)
0
i
2
i
kP k (' ) ; = max ( ; 1 k p ) i
v1 =
i;r
i
k
j =1
(15) where i1 > 0 is a design constant and ^i is an adaptive gain used to counter the effects of the interconnections between subsystems. We present the following adaptive law for ^i : i
k
&1
ik
M 1 1 ; ^ ; r + y_ i
0 3
j =1
k j i
' =
k=1 i
i
i
j =1
i
22 21 01 + y_ k
N
ik
k
i
i
i
2
l
d1 =
22 011 (2r ) 2 k
k
i
i
ik
k
k=1 p
0 ^
p
kP k
&
i
i
1+
i
!max = sup (j!(t)j) ; =
k
0 21
N
v =
2
p
i
2
l
l=1
i
k
k
kP k
l=1
z^ 2 = 0 1 1 0 (N + 1)p 1 0 21 1 2 011 (2r ) =1 i
N
d =
We propose the following virtual decentralized control law for (13): i
i
where i3 is the desired value of ^i . Differentiating V0 along the solutions of (14), (16). and (18) yields (20), as shown at the bottom of the page, where
i
i
i
e P e + 21 + 001 ^
i=1
N
22 011 (2r ) 2
N
k=1
where 0i
(17)
i;r
N
V0 =
k=1
1
0 y_ :
Choose the composite Lyapunov function candidate as follows:
1 [ (y ; . . . ; y ) + (y ; . . . ; y ) !(t) + 1 ( ; y )] i
i
i
e_ =A e + 1 + i
i
k=1
= z^ 2 + e 2 , we have i
i
p
+ 1+
1=2
k
k
i
i
k=1
Since zi2
i
N
22 011 (2r ) 2
i
i
_ 1 = 2 + M 1 1 ; ^ ; r + 1 (y1 ; . . . ; y ) + 1 1 ( ; y ) + 1 (y1 ; . . . ; y ) !(t)
_ 1 = z^ 2 0 y_ + 1 (y1 ; . . . ; y ) + 1 (y1 ; . . . ; y ) !(t) i
i
Thus
1 where 0 is the inverse function of 1i1 and is again a function of class i1 K1 . Then, the dynamics of the tracking error is i
i
k
ik
ik
ik
k=1
k k + ky k
+ d1 + v1 + '1 ik
ik
i1
ik
2k
i;r
2k
+
p
+ 21 ik
i
k=1
+
p
( + 1 ) 22 011 (2D (t0 ; t)) ik
k
ik
2k
i
i
k=1
+ 2 1 2 0 2 1 21 i
0 2 3
p
i
i
22 21 k
i
k=1
i
k
i
0 ^ 0 3 + 3 2
i
i
i
i
i
2
(20)
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^
Since zi;m+2 is not the actual control, we define
Thus, (23), shown at the bottom of the next page, holds true, where the definitions of i;m+1 and Hi;m+1 are similar to those of i2 and Hi2 given by (21), which are obtained by differentiating Mi;m and zi;m+1 with respect to t. We choose the virtual control (24), shown at the bottom of the page.
= z^
^
i;m+2
i;m+2
0M
i;m+1
1 y ; z^ ; . . . ; z^ i1
i
i;m+1
; ^ ; r ; y ; . . . ; y ( i
i
i;r
m+1) i;r
:
(25)
_ 2 =^z 3 + k 2 (y 0 z^ 1 ) 0 @M 1 2 + M 1 1 ; ^ ; r @ 1 @M @M 1 ^ 0 y 0 ^ _ 0 @r 1 r_ @ @M 1 0 @ 1 1 (y1 ; . . . ; y ) + 1 (y1 ; . . . ; y ) !(t) i
i
i
i
i
i
i
i
i
i
i
i
i
i;r
i
i
i
i
i
i
i
N
i
N
i
1=2
p
2 0 (2r ) e z^ + y ; z^ ; z^ ; ^ ; r ; y ; y_ ; y + H ; ^ ; r (y ; . . . ; y ) + (y ; . . . ; y ) !(t)
+ 1 ( ; y ) + 1 + i1
i
i
i3
i2
i2
i1
i
i1
i
_
i;m+1
i
i
i1
i
i2
i
i;r
p
i;r
i1
N
1
N
1=2
011 (2r ) 2
2
k
2k
k=1
i;r
i
i
e 2 :
(21)
i
=^z +
y ; z^ ; . . . ; z^ ; ^ ; r ; y ; . . . ; y +H y ; z^ ; . . . ; z^ ; ^ ; r ; y ; . . . ; y (y ; y ; . . . ; y ) + (y ; y ; . . . ; y ) !(t) i;m+2
i;m+1
i1
1
i1
i
i1
i
2
i1
i
=0
i1
N
p
i;m+1
1H
2
i;m+1
i;m+1
i;m+1
i
+ 21
i
+ (p + 1) H + 21 1+ 2 i;m+1
2
i;m+1 p
2k
i;m+1
k=1
1H
2
i;m+1
M
i;m+1
y ; z^ 1 ; . . . ; z^ i
i
i
m+1) i;r
i;r
011 (2r )
2
(23)
i
+
i;m
2 k
i
i
; ^ ; r ; y ; . . . ; y( i
i
i
i;r
i;m+1
i
m+1) i;r
; ^ ; r ; y ; . . . ; y( i
i
i;r
m+1) i;r
011 (2r ) 2
k
i;m+1
i
i
i;m+1
i
e 2
k
i
y ; z^ 1 ; . . . ; z^
y ; z^ 1 ; . . . ; z^ i
N
k
k=1
i;m+1
(m+1) i;r
(m+1) i;r
i;r
i
i
p
i
i;r
; ^ ; r ; y ; . . . ; y(
i;m+1
y ; z^ 1 ; . . . ; z^
i
011 (2r ) 2
2k
k=1
i
i
2
2
i
i
i
1
y ; z^ 1 ; . . . ; z^
i;m+1
+
i;m+1
i;m+1
+ 1 ( ; y ) + 1 +
i;m+2
i
1
i
i;m+1
z^
2k
1 i1
k=1
i2
+ 1 ( ; y ) + 1 + i1
2k
; ^ ; r ; y ; . . . ; y( i
i
m+1) i;r
i;r
; ^ ; r ; y ; . . . ; y( i
i
i;r
m+1) i;r
:
(24)
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Choose the composite Lyapunov function candidate as follows:
Vm = Vm01 +
N i=1
2i;m+1 :
We choose the composite Lyapunov function candidate as follows
V01 = V02 +
(26)
Then, (27), shown at the bottom of the page, holds. Step 0 : Assume that in step 0 , we designed a virtual control Mi;01 for subsystem ei ; i1 ; ; i;01 ; i and defined
1
(
2
...
( )
1
1
ui = 0
(28)
bi;k 0 i (yi )
1 i;0 + z^i; + i; (1) + i; i; 1
2
2
1 i (y ; . . . ; yN ) + 1i (i ; yi ) 1
+ 21 i; Hi; (1) 2
+ i (y ; . . . ; yN ) !(t) 1
1
+ 1+
p
k=1
2k 2
0i11 (2ri ) 2k
V_ m
N
ei2 :
2
min
i=1
+
p
p k=1
p
2 1
+1
2
p
2
k=1
2
0 2 i3
+
p
k=1
2 k i k 0 i ^i 0 i3 2
2 1
2
:
(32)
1
01 j =1
2k
:
(27)
ij ij2
2
2 k [dik + vik + 'ik + (d ik + v ik + ' ik )] k i k k + kyi;r k k 2
k=1 p
1
0i11 (2Di (t0 ; t))
1
0 [ (Qi ) 0 ( + 3)] kei k 0 2 p
0i11 (2ri ) k
2k
ik + (m + 1)i21 + i i32
2 k [ik + (m + 1) ik ] 1
min
i=1
k=1
0i11 (2ri ) 2k
i;m+2
1
2
k=1
2
2 k [dik + vik + 'ik + (m + 1)(d ik + v ik + ' ik )] 1
N
j =1
2
k=1
k=1
2k
ij ij2
2
1 k i k k + kyi;r k k +
+
m+1
2 k i k 0 i ^i 0 i3 + 2 i;m 2
p
p
Theorem 3.1: Under Assumptions 1)–4), the decentralized adaptive control (32) guarantees that all the signals in the closed-loop system
(29)
0 [ (Qi ) 0 (m + 4)] kei k 0 2 0 2 i3
V_ 01
+1
+ (p + 1) i; Hi; (1) + 12 i; Hi; (1) 1 +
() yi ; z^i1 ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r
1
(1) =
We propose the following decentralized robust adaptive control for the ith subsystem:
_ i; =bi;k 0 i (yi ) ui + z^i;+1 () + i; yi ; z^i1 ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r
1
(30)
We have (31), shown at the bottom of the page, where
Thus
+ Hi;
i=1
2 i; :
(yi ; z^i ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r ):
^)
i; = z^i; (01) 0Mi;01 yi ; z^i1 ; . . . ; z^i;01 ; ^i ; ri ; yi;r ; . . . ; yi;r :
N
1
p
1
1
1
2
2
+ ik + 2 k (ik + ik ) 0i (2Di (t ; t)) k + ( 0 1)i k k + i i3 + 2 i; f i;0 + bi;k 0 i (yi ) ui + z^i; + i; (1)g p + 2(p + 1) i;Hi; (1) + i; Hi; (1) 1 + 2 k 0i (2ri ) k =1
2
1 1
1
2
0
2 1
=1
2
1
2
2
+1
2
2
2
k=1
+ 2i i; Hi; (1)
p
1
k=1
2k
0i11 (2ri ) k
1 1
2
(31)
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0; 8t t0 , we can see from (10) that whenever r_i > 0, ci0 [zi21 i0 (zi21 ) + di0 ]. Since zi21 i0 (zi21 ) is bounded, we have ri < 1= it follows that ri is bounded. The boundedness of ei follows from that of ei and ri . From (15), we can see that Mi1 ( i1 ; ^i ; ri ) is bounded since i1 ; ^i ; ri are bounded. Since i2 is bounded, the boundedness of z^i2 is established from (17). Thus, by using (22) iteratively, the boundedness of z^i;m+1 ; 1 m 0 1, is followed (m) from the boundedness of yi ; z^i1 ; . . . ; z^i;m ; ^i ; ri ; yi;r ; . . . ; yi;r . Consequently, zi1 ; . . . ; zi; are bounded since ei is bounded. In addition, since yi is bounded and, according to Assumption 4), the unmodeled dynamics _i = qi (i ; yi ) is exp-ISpS, i is bounded. Next, we prove that zi;+1 ; . . . ; zi;k are bounded. Let
r_i
consisting of (1), (7), (10), and (16) are bounded in the presence of unmodeled dynamics, high-order interconnections and bounded disturbances. Furthermore, by choosing the design constants i , ij (j = 0; 1; . . . ; ) and 0i appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances and unmodeled dynamics in the system. Proof: Choose the desired value i3 and Qi satisfying
i3 1 [dik + vik + 'ik 2 + (d1ik + v1ik + '1ik )] min (Qi ) 2i0 + ( + 3)
(33) (34)
where i0 > 0 is a design constant. From (31)–(34), we get (35), shown at the bottom of the page, where e = [e1 ; . . . ; eN ]T 2 Rn , n = Ni=1 ki ; ^ = [ ^1 ; . . . ; ^N ]T ,
wij = zij 0 bi;k 0j zi; + 1 j ki : bi;k 0
= [ 11 ; . . . ; 1; ; . . . ; N 1 . . . ; N; ]T : Since Di (t0 ; t) = 0; 8t t0 + Ti0 (see Lemma 1) and i011 is a function of class K1 , we have p
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
k=1
=0;
0
0
Then, we have
w_ ij =wi;j +1 0 bi;k 0j wi;+1 bi;k 0 + zi; bi;k 0j 01 0 bi;k 02j bi;k 001 bi;k 0 bi;k 0 + ij (y1 ; . . . ; yN ) 0 bi;k 0j i; (y1 ; . . . ; yN ) bi;k 0 b + ij (y1 ; . . . ; yN ) 0 i;k 0j i; (y1 ; . . . ; yN ) !(t) bi;k 0 b i;k 0j + 1ij (i ; yi ) 0 1 ( ; y ) bi;k 0 i; i i + 1 j ki 0 1; (42) b b i0 i0 bi;k 001 w_ i;k = 0 w 0z 2 bi;k 0 i;+1 i; bi;k 0 b i0 + i;k (y1 ; . . . ; yN ) 0 (y ; . . . ; yN ) bi;k 0 i; 1 + i;k (y1 ; . . . ; yN ) 0 bi0 i; (y1 ; . . . ; yN ) !(t) bi;k 0 b i0 + 1i;k (i ; yi ) 0 1 ( ; y ) : (43) bi;k 0 i; i i
2k
8t t + Ti :
(36)
Denote (37) and (38), shown at the bottom of the page. Notice that 1 since yi;r and 0 i1 (2Di (t0 ; t)) are bounded, S is bounded. Thus
V_ 01 e; ; ^
0V0 e; ; ^ + S:
(39)
1
^ decreases monotonically until Therefore, V01 ( e; ; ) reaches the compact set Rs =
e; ; ^
^ (e; ; )
2 Rn 2 RN 2 RN :
V01 e; ; ^
0 S : 1
(40)
This means that e; ; ^ are bounded, i.e., ei ; i1 ; . . . ; i; ; ^i ; i = 1; . . . ; N , are bounded. Thus, zi1 is bounded. Although
V_ 01 e; ; ^
N
02i kei k 0 2 0
i=1
+ +
p k=1 p k=1
2
j =1
ij ij2 0 i ^i 0 i3
2
22k [dik + vik + 'ik + (d1ik + v1ik + '1ik )] kyi;r k2k ik +
p k=1
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
2k
+ i21 + i i32
(35)
01 (Pi ) ; 2i1 ; . . . ; 2i ; 0i i ; = 1min min 2i0 min iN
S=
N
p
i=1
k=1
+
(41)
(37)
22k [dik + vik + 'ik + (d1ik + v1ik + '1ik )] kyi;r k2k
p k=1
ik +
p k=1
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
2k
+ i21 + i i32 :
(38)
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Since ij (y1 ; . . . ; yN ), ij (y1 ; . . . ; yN ), 1ij (i ; yi ), zi; and !(t) are bounded and, according to Assumption 1), the zero dynamics of system (1) is exponentially stable, it follows from (42) and (43) that wi;+1 ; . . . ; wi;k are bounded. Thus, from (41), zi;+1 ; . . . ; zi;k are bounded. Hence, z^i;+1 ; . . . ; z^i;k are bounded since ei is bounded. The boundedness of ui is established by the fact that z^i;+1 is bounded and bi;k 0 i (yi ) is bounded away from zero. Therefore, all the signals in the closed-loop system consisting of (1), (7), (10), and (16) are bounded. Furthermore, it can be seen from (37)–(40) that reducing i and increasing ij (j = 0; 1; . . . ; ) and 0i will reduce the residual error bound 01 S . This implies that by choosing the design constants i , ij (j = 0; 1; . . . ; ) and 0i appropriately, the tracking error can be made arbitrarily small. This completes the proof of Theorem 3.1. From the above design procedure and Theorem 3.1, we can see easily that the proposed decentralized adaptive control scheme requires adaptation of only one scalar parameter for each subsystem and no controller redesign is needed if subsystems are added online or taken offline as long as the order of the interconnections of the appended system is less or equal to that of the original system.
0:625; i (kzi1 k) = 2:5zi41 and i0 (kzi1 k) = 2:5zi21 . Taking ci0 = 0:6 2 (0; ci0 ), we define the modified dynamic signal as follows: r_i =
0 ri (0) =ri0 > 0
z^_ i =
dynamics,
11 (y1 ; y2 ) =y12 + y1 y2 ; 11 (y1 ; y2 ) = y1 + y1 y2 111 (1 ; y1 ) =212 ; 12 (y1 ; y2 ) = y22 + y1 12 (y1 ; y2 ) =y2 + y1 y2 ; 112 (1 ; y1 ) = 312 21 (y1 ; y2 ) =y22 cos(t) + y1 ; 21 (y1 ; y2 ) = y2 + y1 y2 121 (2 ; y2 ) =222 cos(t); 22 (y1 ; y2 ) = y12 + y2 22 (y1 ; y2 ) =y12 + y1 y2 ; 122 (2 ; y2 ) = 322 sin(t): The objective is to design a decentralized robust adaptive output controller for each subsystem such that y1 tracks y1;r = sin(2t) and y2 tracks y2;r = cos(2t) and all the signals in the closed-loop system are bounded. In the design of the controllers, we assume that ij (y1 ; y2 ), ij (y1 ; y2 ), 1ij (i ; yi )(i; j = 1; 2) are unknown, but the upper bound of their orders is known, i.e., p = 2. The relative degrees of the tow subsystems are 1 = 2 = = 2. First, we show that the unmodeled dynamics fulfils the Assumption 4). Let Vi (i ) = i2 . Then V_ i (i ) = 02i2 + 2i yi2 + i . Using [13, Lemma 3.2], we have 2i yi2 + i (1=4"1 )(2i )2 + "1 yi4 + (1=4"2 ) + "2 i2 . Taking "1 = 2:5; "2 = 0:4, we get 2i yi2 + i 0:8i2 + 2:5 kyi k4 + 0:625. Thus, Vi (i ) 01:2i2 + 2:5 kyi k4 + 0:625, i.e., the unmodeled dynamics is exp-ISpS with ci0 = 1:2; di0 =
mri 0 mri < 0
03 1 z^i + 3 yi + 0 ui : 02 0 2 1
Applying the decentralized robust adaptive output control scheme presented herein, we have
i1 =zi1 0 yi;r Mi1 = 0 i1 i1 0 ^i 4 i1 + 16 i21 p 0 21 i1 10ri + 10ri 2 0 6 i1 0 21 i1 1 + 10ri + 100ri2 ^_ i =0i 4 i21 + 16 i41 0 i ^i ; ^i (0) > 0 i2 =^zi2 0 Mi1 0 y_i;r
i2 =2 (yi 0 z^i1 ) 0 @Mi1 ( i2 + Mi1 ) @ i1 @M i1 ^ i1 0 yi;r 0 ^ _ i 0 @M r_ @ri i @ i Hi2 = 0 @Mi1 @ i1
Consider an interconnected nonlinear system consisting of two subsystems
where i ; i = 1; 2 is the unmeasurable and unmodeled !(t) = sin2 (t) is the unmeasurable disturbance and
if if
where mri = 00:6ri + 2:5zi41 + 0:625. Take i1 (ki k) = 0:8i2 Vi (i ), then 0i11 (2ri ) = 2ri =0:8. Define the observer for the ith subsystem as follows:
IV. AN ILLUSTRATION EXAMPLE
_1 = 0 1 + y12 + 0:5 z_11 =z12 + 11 (y1 ; y2 ) + 11 (y1 ; y2 ) !(t) + 111 (1 ; y1 ) z_12 = 12 (y1 ; y2 ) + 12 (y1 ; y2 ) !(t) + 112 (1 ; y1 ) + u1 y1 =z11 _2 = 0 2 + y22 + 0:5 z_21 =z22 + 21 (y1 ; y2 ) + 21 (y1 ; y2 ) !(t) + 121 (2 ; y2 ) z_22 = 22 (y1 ; y2 ) + 22 (y1 ; y2 ) !(t) + 122 (2 ; y2 ) + u2 y2 =z21 (44)
00:6ri + 2:5zi41 + 0:625
where
@Mi1 = 0 0 6 0 ^ (4 + 48 ) 0 1 p10r + 10r 2 i1 i i1 i i @ i1 2 0 21 1 + 10ri + 100ri2 @Mi1 = 0 4 + 16 3 i1 i1 @ ^i @Mi1 = 0 1 10 + 200r + 30p10r i i @ri 2 i1 0 21 i1 (10 + 200ri ): The decentralized adaptive control for the ith subsystem is ui = 0 i1 + i2 i2 + i2 + 3 i2 Hi22 + 1 i2 Hi22 (1 + 10ri + 100ri2) 2 p + 1 i2 Hi22 ( 10ri + 10ri )2 ; i = 1; 2: 2 With the following choice of the initial conditions and design constants:
1 (0) =0; z11 (0) = 1; z12 (0) = 0; z^11 (0) = 0 z^12 (0) =0; r1 (0) = 0:1; ^1 (0) = 1 2 (0) =0; z21 (0) = 2; z22 (0) = 0 z^21 (0) =0; z^22 (0) = 0; r2 (0) = 0:1 ^2 (0) =1; 01 = 1; 1 = 0:1 11 =1; 12 = 1; 02 = 1; 2 = 0:1 21 =1; 22 = 1
(45)
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(a)
(a)
(b)
(b) Fig. 1. Simulation results with the control scheme of this work. (a) Tracking error . (b) Tracking error .
we performed the simulation using MATLAB and obtained the results shown in Fig. 1. It can be seen from Fig. 1 that the decentralized controllers are robust to the unmodeled dynamics, bounded disturbances and high order interconnections with good tracking performance. With the same initial conditions as in (45), we chose the design constants as follows:
01 =100; 1 = 0:001; 11 = 100; 12 = 100; 02 = 100 2 =0:001; 21 = 100; 22 = 100
(46)
and obtained the simulation results shown in Fig. 2. The tracking errors 11 ; 21 in Fig. 2(a) and (b) are much smaller than those in Fig. 1(a) and (b). This demonstrates that by choosing the design constants i , ij and 0i appropriately, the tracking errors can be made arbitrarily small. Finally, to compare the decentralized robust adaptive output control scheme presented in this work with that in [8], we applied the decentralized adaptive output control scheme presented in [8] to system (44) with the following initial conditions and design constants (using the
Fig. 2. Simulation results with the control scheme of this work by choosing the design constants appropriately. (a) Tracking error . (b) Tracking error .
same notations as in [8]):
1 (0) =0; z11 (0) = 1; z12 (0) = 0; z^11 (0) = 0; z^12 (0) = 0 ^1 (0) =1; 2 (0) = 0; z21 (0) = 2; z22 (0) = 0; z^21 (0) = 0 z^22 (0) =0; ^2 (0) = 1; `1 = 1; 01 = 1; 1 = 0:1; 11 = 1 12 =1; `2 = 1; 02 = 1; 2 = 0:1; 21 = 1; 22 = 1: (47) The simulation results are shown in Fig. 3, from which we can see that the tracking errors are unbounded and the decentralized adaptive output control scheme presented in [8] is not robust to unmodeled dynamics. V. CONCLUSION A new decentralized robust adaptive control scheme is presented for a class of large-scale nonlinear systems of the output feedback canonical form. The scheme can be used in the systems with unmodeled dynamics, high order interconnections and bounded disturbances. Under certain assumptions, it is shown that the scheme guarantees that all the signals in the closed-loop system are bounded. By choosing the design constants appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances and unmodeled
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
(a)
(b) Fig. 3. Simulation results with the control scheme in [8]. (a) Tracking error . (b) Tracking error .
dynamics in the system. As an extension of the work in [8] to the case of unmodeled dynamics, the proposed decentralized adaptive control scheme of this work retains all the advantages of the scheme in [8]. The effectiveness of the proposed scheme is demonstrated by simulation results. It should be pointed out that the unmodeled dynamics described in this work do not depend on the outputs of other subsystems. For the more general case where the unmodeled dynamics depends on the outputs of other subsystems, how to generate decentralized dynamic signals to dominate the unmodeled dynamics is a subject for further research.
REFERENCES [1] P. A. Iaonnou, “Decentralized adaptive control of interconnected systems,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 291–298, Apr. 1986. ˘ [2] D. T. Gavel and D. D. Siljak, “Decentralized adaptive control: Structural conditions for stability,” IEEE Trans. Automat. Contr., vol. 34, pp. 413–426, Apr. 1989.
[3] L. Shi and S. K. Singh, “Decentralized adaptive controller design for large scale systems with higher order interconnections,” IEEE Trans. Automat. Contr., vol. 37, pp. 1106–1118, Aug. 1992. [4] C. Wen, “Decentralized adaptive regulation,” IEEE Trans. Automat. Contr., vol. 39, pp. 2163–2166, Oct. 1994. , “Indirect robust totally decentralized adaptive control of contin[5] uous time interconnected systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1122–1126, June 1995. [6] J. T. Spooner and K. M. Passino, “Adaptive control of a class of decentralized nonlinear systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 280–284, Feb. 1996. [7] S. Jain and F. Khorrami, “Decentralized adaptive control of a class of large scale interconnected nonlinear systems,” IEEE Trans. Automat. Contr., vol. 42, pp. 136–154, Feb. 1997. [8] , “Decentralized adaptive output feedback design for large scale nonlinear systems,” IEEE Trans. Automat. Contr., vol. 42, pp. 729–735, May 1997. [9] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, Nov. 1991. [10] D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopoulos, “Adaptive regulation of nonlinear systems with unmodeled dynamics,” IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989. [11] R. Marino and P. Tomei, “Robust adaptive state-feedback tracking for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 84–89, Jan. 1998. [12] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, pp. 423–427, 1995. [13] Z. P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,” Automatica, vol. 34, pp. 825–840, 1998. [14] R. Marino and P. Tomei, “Global adaptive output feedback control of nonlinear systems—Part II: Nonlinear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 33–48, Jan. 1993. [15] S. Jain and F. Khorrami, “Decentralized adaptive output feedback control of large scale interconnected nonlinear systems,” in Proc. 1995 American Control Conf., Seattle, WA, pp. 1600–1604. [16] L. Shi and S. K. Singh, “Decentralized control design for interconnected uncertain systems: Extensions to higher-order uncertainties,” Int. J. Control, vol. 57, pp. 1453–1468, 1993. [17] Y. Guo, Z. P. Jiang, and D. J. Hill, “Decentralized robust disturbances attenuation for a class of large-scale nonlinear systems,” Syst. Control Lett., vol. 37, pp. 71–85, 1999. [18] Z. P. Jiang, “Decentralized and adaptive nonlinear tracking of large-scale systems via output feedback,” IEEE Trans. Automat. Contr., vol. 45, pp. 2122–2128, Nov. 2000. , “New results in decentralized adaptive control with output-feed[19] back,” in Proc. 38th Conf. Decision Control, Phoenix, AZ, 1999, pp. 4772–4777. [20] , “A combined backstepping and small-gain approach to adaptive output feedback control,” Automatica, vol. 35, pp. 1131–1139, 1999. [21] S. Jain and F. Khorrami, “Robust decentralized control of power systems utilizing only swing angle measurements,” Int. J. Control, vol. 66, pp. 581–601, 1997.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Decentralized Robust Adaptive Control of Nonlinear Systems With Unmodeled Dynamics Yusheng Liu and Xing-Yuan Li Abstract—The authors present a decentralized robust adaptive output feedback control scheme for a class of large-scale nonlinear systems of the output feedback canonical form with unmodeled dynamics. A modified dynamic signal is introduced for each subsystem to dominate the unmodeled dynamics and an adaptive nonlinear damping is used to counter the effects of the interconnections. It is shown that under certain assumptions, the proposed decentralized adaptive control scheme guarantees that all the signals in the closed-loop system are bounded in the presence of unmodeled dynamics, high-order interconnections and bounded disturbances. Furthermore, by choosing the design constants appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances, and unmodeled dynamics in the system. An illustration example demonstrates the effectiveness of the proposed scheme. Index Terms—Adaptive, decentralized control, large-scale systems, nonlinear, robust.
I. INTRODUCTION Much progress has been made in the field of decentralized adaptive control, see e.g., [1]–[8], [18], [19], and the references therein. Particularly, a decentralized adaptive output control scheme was presented in [8] for a class of large-scale nonlinear systems that are transformable via a global diffeomorphism into the output feedback canonical form. The scheme guarantees global uniform boundedness of the tracking error and all the states of the closed-loop system in the presence of parametric and dynamic uncertainties in the interconnections and bounded disturbances. However, the scheme cannot apply to the systems with unmodeled dynamics. The work in [18] presented a decentralized adaptive output feedback control scheme for large-scale systems with nonlinear interconnections. The scheme of [18] has several advantages: 1) it achieves asymptotic tracking; 2) the considered large-scale systems may possess an unknown, nonzero equilibrium. But, the scheme cannot apply to the systems with unmodeled dynamics and disturbances. On the other hand, the robust adaptive control of nonlinear systems has emerged as an active research area recently, e.g., [10]–[13]. Especially, the problem of robust adaptive control of nonlinear systems with unmodeled dynamics was studied in [13]. Based on the concept of input-to-state practical stability, a dynamic signal was introduced in [13] to dominate the unmodeled dynamics. It has been shown that such a signal is a useful tool for handling unmodeled dynamics. Using a combined backstepping and small-gain approach, paper [20] presented an adaptive output feedback control scheme for nonlinear systems with unmodeled dynamics. As an extension of the centralized case in [20], a decentralized robust adaptive output feedback regulation scheme was presented for a class of large-scale nonlinear systems in [19]. It should be pointed out that the class of large-scale nonlinear systems considered in [19] is broader than the class of nonlinear systems with polynomial bounds as in [7], [8], [16] and [15], in the sense that the interconnections are subject to general nonlinear bounds. Although, [17] studies a different problem—decentralized H1 almost disturbance decoupling for a class of large-scale nonlinear systems, it Manuscript received May 11, 2001; revised November 26, 2001. Recommended by Associate Editor M. M. Polycarpou. This work was supported by the National Key Basic Research Special Fund of China under Grant G1998020312. Y. Liu is with the Department of Automation, Sichuan University, Chengdu, Sichuan 610065, P. R. China (e-mail: [email protected]). X.-Y. Li is with Department of Electrical Engineering, Sichuan University, Chengdu, Sichuan 610065, P. R. China (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)04751-7.
is also assumed that the uncertain interconnections are bounded by general nonlinear functions. Theoretically, to consider a broader class of nonlinear systems is important, but some other issues such as controller design and practical problems must be considered. With the general nonlinear bounds, the design of the controllers depends on a qualitative approach (as in [19] and [20]). For large-scale nonlinear systems with many decentralized controllers, it is difficult to choose so many design parameters and functions by qualitative approach. Furthermore, with the general nonlinear bounds, the design of the ith decentralized controller is related to that of the rest decentralized controllers (see [19, (25) and (10)]). This implies that redesigns of controllers are needed when subsystems are appended to original system or taken offline (e.g., during faults in a power system), which is neither economical nor feasible for the control of large-scale systems. Besides, many practical nonlinear interconnected systems do have some uncertainties and interconnections subject to polynomial bounds, e.g., robot manipulators with the presence of Coriolis and centripetal terms have uncertainties bounded by second-order polynomials [16] and the electric power deviations between subsystems are bounded by first order polynomials [21]. Therefore, for the practical purpose and the above reasons, we still consider the case where uncertainties and interconnections are subject to polynomial bounds in this work. As in [8], with our approach, no controller redesign is needed if subsystems are added online or taken offline as long as the order of the interconnections of the appended system is less or equal to that of the original system. This work presents a decentralized robust adaptive control scheme for large-scale nonlinear systems of the output feedback canonical form with unmodeled dynamics. First, in each subsystem, a modified dynamic signal is introduced to dominate the unmodeled dynamics and a nonlinear adaptive damping is used to counter the effects of the interconnections. Then, we employ the systematic design procedure to obtain the decentralized robust adaptive output feedback controllers. It is shown that under certain assumptions, the proposed decentralized adaptive control scheme guarantees that all the signals in the closed-loop system are bounded in the presence of unmodeled dynamics, high order interconnections and bounded disturbances. Furthermore, the tracking error can be made arbitrarily small by choosing the design constants appropriately. II. PROBLEM STATEMENT Consider a large-scale nonlinear system of the output feedback canonical form with unmodeled dynamics given by the following equations:
_ =q ( ; y ) z_ 1 =z 2 + 1 (y1 ; . . . ; yN ) + i1 (y1 ; . . . ; yN ) !(t) + 1i1 (i ; yi )
_
i
i
i
i
i
i
i
.. .
+ i; 01 (y1 ; . . . ; yN ) + i; 01 (y1 ; . . . ; yN ) !(t) + 1i; 01 (i ; yi ) z_ i; =zi; +1 + i; (y1 ; . . . ; yN ) + i; (y1 ; . . . ; yN ) !(t) + bi;k 0 i (yi ) ui + 1i; (i ; yi )
zi;
01 =zi;
.. .
_ = i;k (y1; . . . ; yN ) + i;k (y1 ; . . . ; yN ) !(t) + bi0 i (yi ) ui + 1i;k (i ; yi ) yi =zi1 1 i N
zi;k
0018-9286/02$17.00 © 2002 IEEE
(1)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
=[
...
849
]
where zi zi1 ; ; zi;k T , yi 2 R and ui 2 R are the state, the output and the control input of the ith subsystem, respectively; ! t is a bounded unmeasurable disturbance; ij y1 ; ; yN and ij y1 ; ; yN are the dynamic and parameter uncertainties related to the interconnections between subsystems; i is the unmeasurable unmodeled dynamics and ij i ; yi represents the unmeasurable uncertainties related to the unmodeled dynamics. We assume that ij , ij and ij are unknown, but satisfy
() ( ...
( ...
)
1 (
where
)
Ai =
)
i =
1
k ij (y ; . . . ; yN )k
p
N
1
kij (y ; . . . ; yN )k
k=1 l=1 p N
1
k1ij (i ; yi )k
k=1 l=1 p
k &ijl kyl kk
1i =
k #ijl kyl kk
ijk ki kk + 'ijk kyi kk
k=1
(2)
k , #k , where &ijl ijk and 'ijk are unknown constants, p = ijl maxfpij ; 1 i N; 1 j ki g is known. To highlight the main idea of this work, it is assumed that the parameters bij , 1 i N , 0 j ki 0 i , are also known. As indicated in [8], the case of
unknown bij can be treated identically. The objective here is to design a decentralized robust adaptive output controller for each subsystem such that the output yi t tracks a given reference signal yi;r t and all the signals of the closed-loop system are bounded in the presence of unmodeled dynamics, high order interconnections and bounded disturbances. We need the following assumptions. Assumption 1: The zero dynamics of system (1) is exponentially stable, i.e., the polynomial Pi s bi;k 0 sk 0 1 1 1 bi1 s bi0 is strict Hurwitz. Assumption 2: i yi 6 ; 8t . Assumption 3: The reference signal yi;r t is bounded with ( ) bounded derivatives up to the i th order and yi;r t is piecewise continuous. Assumption 4: The unmodeled dynamics is exponentially input-tostate practically stable (exp-ISpS) [13]; i.e., the system i qi i ; yi has an exp-ISpS Lyapunov function Vi i which satisfies
()
()
( )= ( )=0 0
+ +
()
+
()
_= (
( )
)
i1 (ki k) Vi (i ) i2 (ki k) @Vi (i ) q ( ; y ) 0 c V ( ) + (ky k) + d i0 i i i i i0 @i i i i
(3) (4)
where i ; i are known functions of class K1 and ci > 0, di 0 are known constants. Without loss of generality, we assume i (1) has the following form i (s) = s i (s ) where i is a nonnegative smooth function. Otherwise, as indicated in [13], it suffices to replace
i in (4) by kyi k i (kyi k ) + "i with "i > 0 being a sufficiently 1
2
0
2
2
0
2
0
0
2
0
0
0
small-real number.
III. THE DESIGN OF DECENTRALIZED ROBUST ADAPTIVE OUTPUT FEEDBACK CONTROLLERS
1
.. .
0ki;k 0 1 1 1 0 i1
i =
.. .
i;k
1i .. .
1
.. .
ki;k
i1 .. .
i;k
bi = [ 0
1i;k
ki1
ki =
I
1 1 1 0 bi;k 0 1 1 1 bi ]T 0
and ki is chosen such that Ai is a strict Hurwitz matrix. Thus, given a Qi > , there exists a Pi > satisfying
0
0
ATi Pi + Pi Ai = 0Qi :
z^_ i = Ai z^i + ki yi + bi i (yi ) ui : Let ei
(7)
= zi 0 z^i . Then
e^_ i = Ai ei + i (y1 ; . . . ; yN ) +i (y1; . . . ; yN ) !(t) + 1i (i ; yi ) :
(8)
Define a dynamic signal
r_ i = 0ci0 ri + zi21 i0 zi21
+ di ri (t ) = ri > 0 0
0
0
(9)
(0 )
where ci0 2 ; ci0 The properties of the dynamic signal are given by the following lemma. Lemma 1 [13]: If the system i qi i ; yi is exp-ISpS, then for any constant ci0 2 ; ci0 , any initial instant t0 , any initial condition i0 i t0 and ri0 > , for any function i0 such that
i0 yi i kyi k , there exist a finite Ti0 Ti0 ci0 ; ri0 ; i0 and a nonnegative function Di t0 ; t defined for all t t0 such that Di t0 ; t , 8t t0 Ti0 and Vi i t ri t Di t0 ; t for all t t0 where the solutions are defined. In this work, we introduce the following modified dynamic signal in the design of decentralized robust adaptive feedback controllers:
_= ( ) (0 ) 0 = ( ) 0 = ( ) ( ) ( ) ( )=0 + ( ( )) ( )+ ( )
r_i =
0ci ri + zi i zi + di
0;
0
2 1
0
2 1
ri (t0 ) =ri0 > 0
= ( ) ( )
+
0
if mri if mri
0
>0
0 (10)
+
_ 0
where mri 0ci0 ri zi21 i0 zi21 di0 . It can be seen that ri and ri t ri t , 8t t0 if ri0 ri0 . Using Lemma 1, we have
=
Vi (i (t)) ri (t) + Di (t0 ; t) ;
+
(11)
where Di t0 ; t , 8t t0 Ti0 . Next, we employ the systematic design procedure of [9], [14], and [8] to obtain the decentralized robust adaptive feedback controllers. To reduce notational complexity, as in [8], we assume that all subsystems have a uniform relative degree, i.e., i , i N . However, it can be treated similarly without any difficulties when the subsystems have different relative degrees.
= 1
(5)
(6)
To estimate the states of the system, we use the following observer for the ith subsystem:
=( )=0
First, we rewrite the ith subsystem (1) as follows:
_i =qi (i ; yi ) z_i =Ai zi + ki yi + i (y1 ; . . . ; yN ) + i (y1 ; . . . ; yN ) !(t) + 1i (i ; yi ) + bi i (yi ) ui
0ki
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Since z^i2 is not the actual control, we define
Step 0: Define p
i1 = zi1 0 yi ; e = 1 + ;r
22 011 (2r ) k
i
01=2
2k
i
i
e
2 = z^ 2 0 M 1 1 ; ^ ; r i
(12)
i
k=1
i;r
i
N
p
+1 1 ( ; y ) + 1 + i
i
i
i
e 2 :
i
i
(13)
i
i
p
2
2k
i
0i11 (2ri ) 2k
01=2
1
0 1+
i
i
i
N
i
i
i
1=2
22 011 (2r ) 2
e 2 :
k
k
i
i
N
p
1
i
(18)
i
i
01
k
k
i
T i
01 2 k 0i11 (2ri ) 2k01 : @ i1 (2ri ) 2r_i ei : @ri k=1 p
2k
(14)
ik
i
ik
i1
i;r
^_ = 0
p
22 21 k
i
i
i
k
0 0 ^ ^ (t ) > 0 i
i
k lji
# !max k lji
2
j =1 k
kP k ( ) 2
i
j =1
ijk
2
2
2
i;m+1 i;m
0 [ (Q ) 0 4] ke k + i
p
i1
i1
i1k
ik
i k
ik
i k
y ; z^ 1 ; . . . ; z^ ; ^ ; r ; y ; . . . ; y ( i
i
i;m
i
i
i;r
m) i;r
:
(22)
22 d + v + '
2
i
2
2
k j i
0M
i
min
ijk
# 1 !max ; 1 = ( 1 )2 ; '1 = (' 1 )2 :
= z^
> 0, > 0 are design constants.
i=1
2
Differentiating (17) gives (21), shown at the bottom of the next page. Step m (1 m 0 2): Assume that in step m 0 1, we designed a virtual control Mi;m and defined i;m+1
N
(19)
j =1
k=1
V_ 0
N
ik
(16)
0
i
2
i
kP k (' ) ; = max ( ; 1 k p ) i
v1 =
i;r
i
k
j =1
(15) where i1 > 0 is a design constant and ^i is an adaptive gain used to counter the effects of the interconnections between subsystems. We present the following adaptive law for ^i : i
k
&1
ik
M 1 1 ; ^ ; r + y_ i
0 3
j =1
k j i
' =
k=1 i
i
i
j =1
i
22 21 01 + y_ k
N
ik
k
i
i
i
2
l
d1 =
22 011 (2r ) 2 k
k
i
i
ik
k
k=1 p
0 ^
p
kP k
&
i
i
1+
i
!max = sup (j!(t)j) ; =
k
0 21
N
v =
2
p
i
2
l
l=1
i
k
k
kP k
l=1
z^ 2 = 0 1 1 0 (N + 1)p 1 0 21 1 2 011 (2r ) =1 i
N
d =
We propose the following virtual decentralized control law for (13): i
i
where i3 is the desired value of ^i . Differentiating V0 along the solutions of (14), (16). and (18) yields (20), as shown at the bottom of the page, where
i
i
i
e P e + 21 + 001 ^
i=1
N
22 011 (2r ) 2
N
k=1
where 0i
(17)
i;r
N
V0 =
k=1
1
0 y_ :
Choose the composite Lyapunov function candidate as follows:
1 [ (y ; . . . ; y ) + (y ; . . . ; y ) !(t) + 1 ( ; y )] i
i
i
e_ =A e + 1 + i
i
k=1
= z^ 2 + e 2 , we have i
i
p
+ 1+
1=2
k
k
i
i
k=1
Since zi2
i
N
22 011 (2r ) 2
i
i
_ 1 = 2 + M 1 1 ; ^ ; r + 1 (y1 ; . . . ; y ) + 1 1 ( ; y ) + 1 (y1 ; . . . ; y ) !(t)
_ 1 = z^ 2 0 y_ + 1 (y1 ; . . . ; y ) + 1 (y1 ; . . . ; y ) !(t) i
i
Thus
1 where 0 is the inverse function of 1i1 and is again a function of class i1 K1 . Then, the dynamics of the tracking error is i
i
k
ik
ik
ik
k=1
k k + ky k
+ d1 + v1 + '1 ik
ik
i1
ik
2k
i;r
2k
+
p
+ 21 ik
i
k=1
+
p
( + 1 ) 22 011 (2D (t0 ; t)) ik
k
ik
2k
i
i
k=1
+ 2 1 2 0 2 1 21 i
0 2 3
p
i
i
22 21 k
i
k=1
i
k
i
0 ^ 0 3 + 3 2
i
i
i
i
i
2
(20)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
851
^
Since zi;m+2 is not the actual control, we define
Thus, (23), shown at the bottom of the next page, holds true, where the definitions of i;m+1 and Hi;m+1 are similar to those of i2 and Hi2 given by (21), which are obtained by differentiating Mi;m and zi;m+1 with respect to t. We choose the virtual control (24), shown at the bottom of the page.
= z^
^
i;m+2
i;m+2
0M
i;m+1
1 y ; z^ ; . . . ; z^ i1
i
i;m+1
; ^ ; r ; y ; . . . ; y ( i
i
i;r
m+1) i;r
:
(25)
_ 2 =^z 3 + k 2 (y 0 z^ 1 ) 0 @M 1 2 + M 1 1 ; ^ ; r @ 1 @M @M 1 ^ 0 y 0 ^ _ 0 @r 1 r_ @ @M 1 0 @ 1 1 (y1 ; . . . ; y ) + 1 (y1 ; . . . ; y ) !(t) i
i
i
i
i
i
i
i
i
i
i
i
i
i;r
i
i
i
i
i
i
i
N
i
N
i
1=2
p
2 0 (2r ) e z^ + y ; z^ ; z^ ; ^ ; r ; y ; y_ ; y + H ; ^ ; r (y ; . . . ; y ) + (y ; . . . ; y ) !(t)
+ 1 ( ; y ) + 1 + i1
i
i
i3
i2
i2
i1
i
i1
i
_
i;m+1
i
i
i1
i
i2
i
i;r
p
i;r
i1
N
1
N
1=2
011 (2r ) 2
2
k
2k
k=1
i;r
i
i
e 2 :
(21)
i
=^z +
y ; z^ ; . . . ; z^ ; ^ ; r ; y ; . . . ; y +H y ; z^ ; . . . ; z^ ; ^ ; r ; y ; . . . ; y (y ; y ; . . . ; y ) + (y ; y ; . . . ; y ) !(t) i;m+2
i;m+1
i1
1
i1
i
i1
i
2
i1
i
=0
i1
N
p
i;m+1
1H
2
i;m+1
i;m+1
i;m+1
i
+ 21
i
+ (p + 1) H + 21 1+ 2 i;m+1
2
i;m+1 p
2k
i;m+1
k=1
1H
2
i;m+1
M
i;m+1
y ; z^ 1 ; . . . ; z^ i
i
i
m+1) i;r
i;r
011 (2r )
2
(23)
i
+
i;m
2 k
i
i
; ^ ; r ; y ; . . . ; y( i
i
i
i;r
i;m+1
i
m+1) i;r
; ^ ; r ; y ; . . . ; y( i
i
i;r
m+1) i;r
011 (2r ) 2
k
i;m+1
i
i
i;m+1
i
e 2
k
i
y ; z^ 1 ; . . . ; z^
y ; z^ 1 ; . . . ; z^ i
N
k
k=1
i;m+1
(m+1) i;r
(m+1) i;r
i;r
i
i
p
i
i;r
; ^ ; r ; y ; . . . ; y(
i;m+1
y ; z^ 1 ; . . . ; z^
i
011 (2r ) 2
2k
k=1
i
i
2
2
i
i
i
1
y ; z^ 1 ; . . . ; z^
i;m+1
+
i;m+1
i;m+1
+ 1 ( ; y ) + 1 +
i;m+2
i
1
i
i;m+1
z^
2k
1 i1
k=1
i2
+ 1 ( ; y ) + 1 + i1
2k
; ^ ; r ; y ; . . . ; y( i
i
m+1) i;r
i;r
; ^ ; r ; y ; . . . ; y( i
i
i;r
m+1) i;r
:
(24)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Choose the composite Lyapunov function candidate as follows:
Vm = Vm01 +
N i=1
2i;m+1 :
We choose the composite Lyapunov function candidate as follows
V01 = V02 +
(26)
Then, (27), shown at the bottom of the page, holds. Step 0 : Assume that in step 0 , we designed a virtual control Mi;01 for subsystem ei ; i1 ; ; i;01 ; i and defined
1
(
2
...
( )
1
1
ui = 0
(28)
bi;k 0 i (yi )
1 i;0 + z^i; + i; (1) + i; i; 1
2
2
1 i (y ; . . . ; yN ) + 1i (i ; yi ) 1
+ 21 i; Hi; (1) 2
+ i (y ; . . . ; yN ) !(t) 1
1
+ 1+
p
k=1
2k 2
0i11 (2ri ) 2k
V_ m
N
ei2 :
2
min
i=1
+
p
p k=1
p
2 1
+1
2
p
2
k=1
2
0 2 i3
+
p
k=1
2 k i k 0 i ^i 0 i3 2
2 1
2
:
(32)
1
01 j =1
2k
:
(27)
ij ij2
2
2 k [dik + vik + 'ik + (d ik + v ik + ' ik )] k i k k + kyi;r k k 2
k=1 p
1
0i11 (2Di (t0 ; t))
1
0 [ (Qi ) 0 ( + 3)] kei k 0 2 p
0i11 (2ri ) k
2k
ik + (m + 1)i21 + i i32
2 k [ik + (m + 1) ik ] 1
min
i=1
k=1
0i11 (2ri ) 2k
i;m+2
1
2
k=1
2
2 k [dik + vik + 'ik + (m + 1)(d ik + v ik + ' ik )] 1
N
j =1
2
k=1
k=1
2k
ij ij2
2
1 k i k k + kyi;r k k +
+
m+1
2 k i k 0 i ^i 0 i3 + 2 i;m 2
p
p
Theorem 3.1: Under Assumptions 1)–4), the decentralized adaptive control (32) guarantees that all the signals in the closed-loop system
(29)
0 [ (Qi ) 0 (m + 4)] kei k 0 2 0 2 i3
V_ 01
+1
+ (p + 1) i; Hi; (1) + 12 i; Hi; (1) 1 +
() yi ; z^i1 ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r
1
(1) =
We propose the following decentralized robust adaptive control for the ith subsystem:
_ i; =bi;k 0 i (yi ) ui + z^i;+1 () + i; yi ; z^i1 ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r
1
(30)
We have (31), shown at the bottom of the page, where
Thus
+ Hi;
i=1
2 i; :
(yi ; z^i ; . . . ; z^i; ; ^i ; ri ; yi;r ; . . . ; yi;r ):
^)
i; = z^i; (01) 0Mi;01 yi ; z^i1 ; . . . ; z^i;01 ; ^i ; ri ; yi;r ; . . . ; yi;r :
N
1
p
1
1
1
2
2
+ ik + 2 k (ik + ik ) 0i (2Di (t ; t)) k + ( 0 1)i k k + i i3 + 2 i; f i;0 + bi;k 0 i (yi ) ui + z^i; + i; (1)g p + 2(p + 1) i;Hi; (1) + i; Hi; (1) 1 + 2 k 0i (2ri ) k =1
2
1 1
1
2
0
2 1
=1
2
1
2
2
+1
2
2
2
k=1
+ 2i i; Hi; (1)
p
1
k=1
2k
0i11 (2ri ) k
1 1
2
(31)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
853
0; 8t t0 , we can see from (10) that whenever r_i > 0, ci0 [zi21 i0 (zi21 ) + di0 ]. Since zi21 i0 (zi21 ) is bounded, we have ri < 1= it follows that ri is bounded. The boundedness of ei follows from that of ei and ri . From (15), we can see that Mi1 ( i1 ; ^i ; ri ) is bounded since i1 ; ^i ; ri are bounded. Since i2 is bounded, the boundedness of z^i2 is established from (17). Thus, by using (22) iteratively, the boundedness of z^i;m+1 ; 1 m 0 1, is followed (m) from the boundedness of yi ; z^i1 ; . . . ; z^i;m ; ^i ; ri ; yi;r ; . . . ; yi;r . Consequently, zi1 ; . . . ; zi; are bounded since ei is bounded. In addition, since yi is bounded and, according to Assumption 4), the unmodeled dynamics _i = qi (i ; yi ) is exp-ISpS, i is bounded. Next, we prove that zi;+1 ; . . . ; zi;k are bounded. Let
r_i
consisting of (1), (7), (10), and (16) are bounded in the presence of unmodeled dynamics, high-order interconnections and bounded disturbances. Furthermore, by choosing the design constants i , ij (j = 0; 1; . . . ; ) and 0i appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances and unmodeled dynamics in the system. Proof: Choose the desired value i3 and Qi satisfying
i3 1 [dik + vik + 'ik 2 + (d1ik + v1ik + '1ik )] min (Qi ) 2i0 + ( + 3)
(33) (34)
where i0 > 0 is a design constant. From (31)–(34), we get (35), shown at the bottom of the page, where e = [e1 ; . . . ; eN ]T 2 Rn , n = Ni=1 ki ; ^ = [ ^1 ; . . . ; ^N ]T ,
wij = zij 0 bi;k 0j zi; + 1 j ki : bi;k 0
= [ 11 ; . . . ; 1; ; . . . ; N 1 . . . ; N; ]T : Since Di (t0 ; t) = 0; 8t t0 + Ti0 (see Lemma 1) and i011 is a function of class K1 , we have p
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
k=1
=0;
0
0
Then, we have
w_ ij =wi;j +1 0 bi;k 0j wi;+1 bi;k 0 + zi; bi;k 0j 01 0 bi;k 02j bi;k 001 bi;k 0 bi;k 0 + ij (y1 ; . . . ; yN ) 0 bi;k 0j i; (y1 ; . . . ; yN ) bi;k 0 b + ij (y1 ; . . . ; yN ) 0 i;k 0j i; (y1 ; . . . ; yN ) !(t) bi;k 0 b i;k 0j + 1ij (i ; yi ) 0 1 ( ; y ) bi;k 0 i; i i + 1 j ki 0 1; (42) b b i0 i0 bi;k 001 w_ i;k = 0 w 0z 2 bi;k 0 i;+1 i; bi;k 0 b i0 + i;k (y1 ; . . . ; yN ) 0 (y ; . . . ; yN ) bi;k 0 i; 1 + i;k (y1 ; . . . ; yN ) 0 bi0 i; (y1 ; . . . ; yN ) !(t) bi;k 0 b i0 + 1i;k (i ; yi ) 0 1 ( ; y ) : (43) bi;k 0 i; i i
2k
8t t + Ti :
(36)
Denote (37) and (38), shown at the bottom of the page. Notice that 1 since yi;r and 0 i1 (2Di (t0 ; t)) are bounded, S is bounded. Thus
V_ 01 e; ; ^
0V0 e; ; ^ + S:
(39)
1
^ decreases monotonically until Therefore, V01 ( e; ; ) reaches the compact set Rs =
e; ; ^
^ (e; ; )
2 Rn 2 RN 2 RN :
V01 e; ; ^
0 S : 1
(40)
This means that e; ; ^ are bounded, i.e., ei ; i1 ; . . . ; i; ; ^i ; i = 1; . . . ; N , are bounded. Thus, zi1 is bounded. Although
V_ 01 e; ; ^
N
02i kei k 0 2 0
i=1
+ +
p k=1 p k=1
2
j =1
ij ij2 0 i ^i 0 i3
2
22k [dik + vik + 'ik + (d1ik + v1ik + '1ik )] kyi;r k2k ik +
p k=1
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
2k
+ i21 + i i32
(35)
01 (Pi ) ; 2i1 ; . . . ; 2i ; 0i i ; = 1min min 2i0 min iN
S=
N
p
i=1
k=1
+
(41)
(37)
22k [dik + vik + 'ik + (d1ik + v1ik + '1ik )] kyi;r k2k
p k=1
ik +
p k=1
22k (ik + 1ik ) 0i11 (2Di (t0 ; t))
2k
+ i21 + i i32 :
(38)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Since ij (y1 ; . . . ; yN ), ij (y1 ; . . . ; yN ), 1ij (i ; yi ), zi; and !(t) are bounded and, according to Assumption 1), the zero dynamics of system (1) is exponentially stable, it follows from (42) and (43) that wi;+1 ; . . . ; wi;k are bounded. Thus, from (41), zi;+1 ; . . . ; zi;k are bounded. Hence, z^i;+1 ; . . . ; z^i;k are bounded since ei is bounded. The boundedness of ui is established by the fact that z^i;+1 is bounded and bi;k 0 i (yi ) is bounded away from zero. Therefore, all the signals in the closed-loop system consisting of (1), (7), (10), and (16) are bounded. Furthermore, it can be seen from (37)–(40) that reducing i and increasing ij (j = 0; 1; . . . ; ) and 0i will reduce the residual error bound 01 S . This implies that by choosing the design constants i , ij (j = 0; 1; . . . ; ) and 0i appropriately, the tracking error can be made arbitrarily small. This completes the proof of Theorem 3.1. From the above design procedure and Theorem 3.1, we can see easily that the proposed decentralized adaptive control scheme requires adaptation of only one scalar parameter for each subsystem and no controller redesign is needed if subsystems are added online or taken offline as long as the order of the interconnections of the appended system is less or equal to that of the original system.
0:625; i (kzi1 k) = 2:5zi41 and i0 (kzi1 k) = 2:5zi21 . Taking ci0 = 0:6 2 (0; ci0 ), we define the modified dynamic signal as follows: r_i =
0 ri (0) =ri0 > 0
z^_ i =
dynamics,
11 (y1 ; y2 ) =y12 + y1 y2 ; 11 (y1 ; y2 ) = y1 + y1 y2 111 (1 ; y1 ) =212 ; 12 (y1 ; y2 ) = y22 + y1 12 (y1 ; y2 ) =y2 + y1 y2 ; 112 (1 ; y1 ) = 312 21 (y1 ; y2 ) =y22 cos(t) + y1 ; 21 (y1 ; y2 ) = y2 + y1 y2 121 (2 ; y2 ) =222 cos(t); 22 (y1 ; y2 ) = y12 + y2 22 (y1 ; y2 ) =y12 + y1 y2 ; 122 (2 ; y2 ) = 322 sin(t): The objective is to design a decentralized robust adaptive output controller for each subsystem such that y1 tracks y1;r = sin(2t) and y2 tracks y2;r = cos(2t) and all the signals in the closed-loop system are bounded. In the design of the controllers, we assume that ij (y1 ; y2 ), ij (y1 ; y2 ), 1ij (i ; yi )(i; j = 1; 2) are unknown, but the upper bound of their orders is known, i.e., p = 2. The relative degrees of the tow subsystems are 1 = 2 = = 2. First, we show that the unmodeled dynamics fulfils the Assumption 4). Let Vi (i ) = i2 . Then V_ i (i ) = 02i2 + 2i yi2 + i . Using [13, Lemma 3.2], we have 2i yi2 + i (1=4"1 )(2i )2 + "1 yi4 + (1=4"2 ) + "2 i2 . Taking "1 = 2:5; "2 = 0:4, we get 2i yi2 + i 0:8i2 + 2:5 kyi k4 + 0:625. Thus, Vi (i ) 01:2i2 + 2:5 kyi k4 + 0:625, i.e., the unmodeled dynamics is exp-ISpS with ci0 = 1:2; di0 =
mri 0 mri < 0
03 1 z^i + 3 yi + 0 ui : 02 0 2 1
Applying the decentralized robust adaptive output control scheme presented herein, we have
i1 =zi1 0 yi;r Mi1 = 0 i1 i1 0 ^i 4 i1 + 16 i21 p 0 21 i1 10ri + 10ri 2 0 6 i1 0 21 i1 1 + 10ri + 100ri2 ^_ i =0i 4 i21 + 16 i41 0 i ^i ; ^i (0) > 0 i2 =^zi2 0 Mi1 0 y_i;r
i2 =2 (yi 0 z^i1 ) 0 @Mi1 ( i2 + Mi1 ) @ i1 @M i1 ^ i1 0 yi;r 0 ^ _ i 0 @M r_ @ri i @ i Hi2 = 0 @Mi1 @ i1
Consider an interconnected nonlinear system consisting of two subsystems
where i ; i = 1; 2 is the unmeasurable and unmodeled !(t) = sin2 (t) is the unmeasurable disturbance and
if if
where mri = 00:6ri + 2:5zi41 + 0:625. Take i1 (ki k) = 0:8i2 Vi (i ), then 0i11 (2ri ) = 2ri =0:8. Define the observer for the ith subsystem as follows:
IV. AN ILLUSTRATION EXAMPLE
_1 = 0 1 + y12 + 0:5 z_11 =z12 + 11 (y1 ; y2 ) + 11 (y1 ; y2 ) !(t) + 111 (1 ; y1 ) z_12 = 12 (y1 ; y2 ) + 12 (y1 ; y2 ) !(t) + 112 (1 ; y1 ) + u1 y1 =z11 _2 = 0 2 + y22 + 0:5 z_21 =z22 + 21 (y1 ; y2 ) + 21 (y1 ; y2 ) !(t) + 121 (2 ; y2 ) z_22 = 22 (y1 ; y2 ) + 22 (y1 ; y2 ) !(t) + 122 (2 ; y2 ) + u2 y2 =z21 (44)
00:6ri + 2:5zi41 + 0:625
where
@Mi1 = 0 0 6 0 ^ (4 + 48 ) 0 1 p10r + 10r 2 i1 i i1 i i @ i1 2 0 21 1 + 10ri + 100ri2 @Mi1 = 0 4 + 16 3 i1 i1 @ ^i @Mi1 = 0 1 10 + 200r + 30p10r i i @ri 2 i1 0 21 i1 (10 + 200ri ): The decentralized adaptive control for the ith subsystem is ui = 0 i1 + i2 i2 + i2 + 3 i2 Hi22 + 1 i2 Hi22 (1 + 10ri + 100ri2) 2 p + 1 i2 Hi22 ( 10ri + 10ri )2 ; i = 1; 2: 2 With the following choice of the initial conditions and design constants:
1 (0) =0; z11 (0) = 1; z12 (0) = 0; z^11 (0) = 0 z^12 (0) =0; r1 (0) = 0:1; ^1 (0) = 1 2 (0) =0; z21 (0) = 2; z22 (0) = 0 z^21 (0) =0; z^22 (0) = 0; r2 (0) = 0:1 ^2 (0) =1; 01 = 1; 1 = 0:1 11 =1; 12 = 1; 02 = 1; 2 = 0:1 21 =1; 22 = 1
(45)
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(a)
(a)
(b)
(b) Fig. 1. Simulation results with the control scheme of this work. (a) Tracking error . (b) Tracking error .
we performed the simulation using MATLAB and obtained the results shown in Fig. 1. It can be seen from Fig. 1 that the decentralized controllers are robust to the unmodeled dynamics, bounded disturbances and high order interconnections with good tracking performance. With the same initial conditions as in (45), we chose the design constants as follows:
01 =100; 1 = 0:001; 11 = 100; 12 = 100; 02 = 100 2 =0:001; 21 = 100; 22 = 100
(46)
and obtained the simulation results shown in Fig. 2. The tracking errors 11 ; 21 in Fig. 2(a) and (b) are much smaller than those in Fig. 1(a) and (b). This demonstrates that by choosing the design constants i , ij and 0i appropriately, the tracking errors can be made arbitrarily small. Finally, to compare the decentralized robust adaptive output control scheme presented in this work with that in [8], we applied the decentralized adaptive output control scheme presented in [8] to system (44) with the following initial conditions and design constants (using the
Fig. 2. Simulation results with the control scheme of this work by choosing the design constants appropriately. (a) Tracking error . (b) Tracking error .
same notations as in [8]):
1 (0) =0; z11 (0) = 1; z12 (0) = 0; z^11 (0) = 0; z^12 (0) = 0 ^1 (0) =1; 2 (0) = 0; z21 (0) = 2; z22 (0) = 0; z^21 (0) = 0 z^22 (0) =0; ^2 (0) = 1; `1 = 1; 01 = 1; 1 = 0:1; 11 = 1 12 =1; `2 = 1; 02 = 1; 2 = 0:1; 21 = 1; 22 = 1: (47) The simulation results are shown in Fig. 3, from which we can see that the tracking errors are unbounded and the decentralized adaptive output control scheme presented in [8] is not robust to unmodeled dynamics. V. CONCLUSION A new decentralized robust adaptive control scheme is presented for a class of large-scale nonlinear systems of the output feedback canonical form. The scheme can be used in the systems with unmodeled dynamics, high order interconnections and bounded disturbances. Under certain assumptions, it is shown that the scheme guarantees that all the signals in the closed-loop system are bounded. By choosing the design constants appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances and unmodeled
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(b) Fig. 3. Simulation results with the control scheme in [8]. (a) Tracking error . (b) Tracking error .
dynamics in the system. As an extension of the work in [8] to the case of unmodeled dynamics, the proposed decentralized adaptive control scheme of this work retains all the advantages of the scheme in [8]. The effectiveness of the proposed scheme is demonstrated by simulation results. It should be pointed out that the unmodeled dynamics described in this work do not depend on the outputs of other subsystems. For the more general case where the unmodeled dynamics depends on the outputs of other subsystems, how to generate decentralized dynamic signals to dominate the unmodeled dynamics is a subject for further research.
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