- Email: [email protected]
Output feedback stabilization of nonlinear systems with delays in the input
Applied Mathematics and Computation 167 (2005) 1026–1040
www.elsevier.com/locate/amc
Output feedback stabilization of nonlinear systems with delays in the input Xianfu Zhang a
a,b,*
, Zhaolin Cheng
b
Department of Mathematics and Physics, Shandong Institute of Architecture and Engineering, Jinan 250014, PR China b School of Mathematics and System Science, Shandong University, Jinan 250100, PR China
Abstract It is proposed a novel and simple design scheme of output feedback controller for a class of nonlinear systems with delays in the input. By constructing appropriate Lyapunov–Krasovskii functionals (LKF) and solving linear matrix inequality (LMI), the delay-dependent controller making the closed-loop system globally asymptotically stable (GAS) is explicitly constructed. A simulation example is given to show the effectiveness of the proposed design procedure. 2004 Elsevier Inc. All rights reserved. Keywords: Feedforward nonlinear systems; Time-delay systems; Output feedback; Lyapunov– Krasovskii functionals; Linear matrix inequality
* Corresponding author. Address: Department of Mathematics and Physics, Shandong Institute of Architecture and Engineering, Jinan 250014, PR China. E-mail address: [email protected] (X. Zhang).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.08.001
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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1. Introduction In real control systems, input delays are often encountered because of transmission of the measurement information. The existence of these delays may be the source of instability or serious deterioration in the performance of the closed-loop system. Hence, the control design problem of input delayed systems has attracted considerable attention (see [2,4,5,13]). In [4], backstepping is applied to design an H1 controller for a class of lower-triangular linear systems with delay in the input. In [5], the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved. On the other hand, feedforward systems are nonlinear systems described by equations having a specific triangular structure. The problem of the asymptotic stabilization by state feedback of these triangular equations in the absence of delay has been studied by many researchers (see [6,7,15]) during the last decade mainly for two reasons: first, the problem of the global asymptotic stabilization of these systems is challenging from a theoretical point of view. Note in particular that they are in general not feedback linearizable and cannot be stabilized by applying the backstepping method. Second, a number of physical devices, such as the system ball and beam with a friction term (see [14]), the TORA system (see [14]), are described, after a preliminary change of feedback, by equations with the feedforward structure. In a recent paper [13], the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved. Up to now, few paper has considered the problem of output feedback controller design for nonlinear feedforward systems with delays in the input. Motivated by [8] and [9], in this paper, by constructing the appropriate Lyapunov– Krasovskii functionals (LKF), we investigate the output feedback controller design for a class of nonlinear systems with delays in the input. Indeed, the nonlinear systems considered in this paper is more general than feedforward systems (upper-triangular form). The method used here is very deferent from that used in [13], and our method is simpler because the most work of our design procedure can be completed by MATLAB toolbox. Our stabilizing controller is delay-dependent, and hence the proposed controller is less conservative.
2. Preliminaries We consider the nonlinear system of dimension n P 2 described by the equations
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
x_ 1 ðtÞ ¼ x2 ðtÞ þ /1 ðt; xðtÞ; uðt d 1 ÞÞ; x_ 2 ðtÞ ¼ x3 ðtÞ þ /2 ðt; xðtÞ; uðt d 1 ÞÞ; .. . x_ n2 ðtÞ ¼ xn1 ðtÞ þ /n2 ðt; xðtÞ; uðt d 1 ÞÞ;
ð1Þ
x_ n1 ðtÞ ¼ xn ðtÞ þ /n1 ðt; xðtÞ; uðt d 1 ÞÞ; x_ n ðtÞ ¼ uðt d 2 Þ; y ¼ x1 ðtÞ; where x = [x1, . . ., xn] T 2 Rn is the state, u 2 R is the input, y 2 R, is the output, constants d1, d2 P 0 are the delays. The argument of the functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote xi(t) by xi. The mappings /i : R Rn R ! R; i ¼ 1; 2; . . . ; n 1 are continuous and satisfy the following growth condition: Assumption 1. For k = 1, 2, . . ., n 1, l = 1, 2, . P . ., Nk, j = 1, 2, . . ., n + 1, there exist constants ck,l P 0 and aj,k,l P 0 satisfying nþ1 j¼1 aj;k;l ¼ 1, such that j/k ðt; xðtÞ; uðt d 1 ÞÞj 6 with ck,l = 0 for integers.
Pnþ1 i¼1
Nk X
ck;l jx1 ja1;k;l jxn jan;k;l juðt d 1 Þjanþ1;k;l ;
ð2Þ
l¼1
i ai;k;l < k þ 2, where Nk(k = 1, 2, . . ., n 1) are positive
Remark 1. When some ai,k,l = 0, it can be viewed that there is no xi in the term ck,ljx1ja1,k,l jxnjan,k,lju(t d1)jan+1,k,l. When k = n 1, we will get a1,n1,l = = an,n1,l = 0, an+1,n1,l = 1, and consequently we have ! N n1 X j/n1 ðt; xðtÞ; uðt d 1 ÞÞj 6 cn1;l juðt d 1 Þj: l¼1
Compared to condition (2) with the following linear growth condition: j/i ðt; xðtÞ; uðt d 1 ÞÞj 6 cðjxiþ2 j þ þ jxn jÞ þ c1 juðt d 1 Þj i ¼ 1; 2; . . . ; n 2;
ð2Þ
j/n1 ðt; xðtÞ; uðt d 1 ÞÞj 6 c1 juðt d 1 Þj; where c P 0 and c1 P 0 are constants, it is easy to see that the system (1) satisfying condition (2*) which is indeed a feedforward nonlinear system obviously satisfies the condition (2). Now we give an example which is not a feedforward system, but satisfies the condition (2),
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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1 2 2 1 1 1 x_ 1 ¼ x2 þ ðx1 Þ3 ðuðt 1ÞÞ3 ðx3 Þ3 ðuðt 1ÞÞ3 ; 2 3 1 uðt 1Þ; x_ 2 ¼ x3 4 þ 4et x_ 3 ¼ uðt 0:1Þ; y ¼ x1 :
Up to now, to my best knowledge, there is no efficient method to design the output feedback controller for this system. Compared with the system considered in [4,5], the system (1) is more general. Ref. [6] has considered the asymptotic stabilization for the system (1) with no delay in input. In this paper, for the vector n = (n1, n2, . . ., nn) (or matrix B = [bij]m·n), we denote vector (jn1j, jn2j, . . ., jnnj) (or matrix [jbijj]m·n) by jnj (or jBj). The matrix B = [bij]m·n is said to be a nonnegative matrix if bij P 0, for 1 6 i 6 m, 1 6 j 6 n. The property about the nonnegative matrix can be found in [11]. we let kÆk denote the Euclidean norm for vector, or the induced Euclidean norm for matrix. For any real matrix A, AT denotes the transpose, In is the n-dimensional identity matrix. The following lemmas are useful in the proof of our main result. Lemma 1. For any a(t) 2 Rn and symmetrical matrix R P 0, the following inequality holds: Z t Z t Z t aT ðsÞdsR aðsÞds 6 d aT ðsÞRaðsÞds: td
td
td
1
1
Proof. Let R2 aðsÞ ¼: nðsÞ ¼ ½n1 ðsÞ; n2 ðsÞ; . . . ; nn ðsÞT , where R2 P 0 satisfying 1 1 R2 R2 ¼ R, then Z t Z t Z t Z t 1 1 aT ðsÞdsR aðsÞds ¼ aT ðsÞR2 ds R2 aðsÞds td td td 2 td Z t 2 Z t n1 ðsÞds þ n1 ðsÞds þ ¼ td td 2 Z t nn ðsÞds þ Z t td Z t 2 6d n1 ðsÞds þ d n22 ðsÞds þ td td Z t n2n ðsÞds þd td Z t ¼d nT ðsÞnðsÞds td Z t aT ðsÞRaðsÞds: ¼d td
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
LemmaP2 (see [16]). For any yi P 0 (i = 1, 2, . . ., k) and ri > 0 (i = 1, 2, . . ., k) satk isfying i¼1 ri ¼ 1, the following inequality holds: y r11 y r22 . . . y rkk 6 r1 y 1 þ r2 y 2 þ þ rk y k :
3. Output feedback controller In practice, when some state variables are unavailable, the output variables are needed to control the system. Under the Assumption 1, we will design the linear output dynamic compensator z_ ðtÞ ¼ MzðtÞ þ NyðtÞ;
M 2 Rnn ; N 2 Rn ;
ð3Þ
F 2 R1n ;
uðtÞ ¼ FzðtÞ;
such that the closed-loop systems (1) and (3) is GAS at the equilibrium (x, z) = (0, 0). Theorem 1. Under the Assumption 1, for any given delays d1, d2 P 0, there exists a linear output feedback controller of the form (3), which is dependent on the delay d2 P 0, such that the closed-loop systems (1) and (3) have globally asymptotically stability (GAS). Proof. Let ~xn ¼ xn þ
Rt
uðsÞds, then the system (1) can be covered into Z t uðsÞds; uðt d 1 Þ ; x_ 1 ðtÞ ¼ x2 ðtÞ þ /1 t; x1 ; . . . ; xn1 ; ~xn td 2
Z x_ 2 ðtÞ ¼ x3 ðtÞ þ /2 t; x1 ; . . . ; xn1 ; ~xn
td 2 t
uðsÞds; uðt d 1 Þ ;
td 2
.. . Z x_ n2 ðtÞ ¼ xn1 ðtÞ þ /n2 t; x1 ; . . . ; xn1 ; ~xn x_ n1 ðtÞ ¼ ~xn ðtÞ
Z
Z
t
uðsÞds; uðt d 1 Þ ; td 2
t
uðsÞds þ /n1 t; x1 ; . . . ; xn1 ; ~xn
td 2
t
uðsÞds; uðt d 1 Þ ;
td 2
~x_ n ðtÞ ¼ uðtÞ; y ¼ x1 ðtÞ: ð1Þ
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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We begin with by designing the following linear observer of the system (1*): a1 z_ 1 ¼ z2 þ ðy z1 Þ; L a2 z_ 2 ¼ z3 þ 2 ðy z1 Þ; L .. ð4Þ . an1 z_ n1 ¼ zn þ n1 ðy z1 Þ; L an z_ n ¼ uðtÞ þ n ðy z1 Þ: L where L > 1 will be determined later, aj > 0 (j = 1, 2, . . ., n) are coefficients of the Hurwitz polynomial pðsÞ ¼ sn þ a1 sn1 þ þ an1 s þ an : Now we define ei ¼ Li1 ðxi zi Þ; xi ¼ Li1 xi ; i ¼ 1; 2; . . . ; n 1; en ¼ Ln1 ð~xn zn Þ;
xn ¼ Ln1~xn ;
zi ¼ Li1 zi ; i ¼ 1; 2; . . . ; n: From (1*) and (4), a simple calculation gives 1 e_ ¼ Ae þ U; L
ð5Þ
1 1 z_ ¼ J 0z þ Ln1 GuðtÞ þ De; L L where T
e ¼ ½e1 ; e2 ; . . . ; en 0 a1 1 B a2 0 B B . .. A ¼ B .. . B @an1 0 an 0 0
a1 B a2 B B . D ¼ B .. B @an1 an
; 0 1 .. . 0 0
0 0 .. .
0 0 .. .
0 0
0 0
ð6Þ T
T
z ¼ ½z1 ; z2 ; . . . ; zn ; G ¼ ½0; 0; . . . ; 0; 1 ; 1 0 1 ... 0 0 1 0 ... 0 C ... 0C B0 0 1 ... 0C B. . . . C .. C .. . . ... C; .. .. .. ; J0 ¼ B . .C B C C @0 0 0 ... 1A ... 1A 0 0 0 ... 0 ... 0 1
... 0 ... 0C C .. C .. ; . .C C A ... 0 ... 0
0
/1 L/2 .. .
1
C B C B C B C B U¼B C: n3 C B L / C B n2 R n2 @ L ð/ t uðsÞdsÞ A n1 td 2 0
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
Let bj > 0 (j = 1, 2, . . ., n) are coefficients of the Hurwitz polynomial qðsÞ ¼ sn þ bn sn1 þ þ b2 s þ b1 : Next we will choose an appreciate L > 1, such that the closed-loop system (5), (6) and 1 ðb1z1 þ b2z2 þ þ bnzn Þ; ð7Þ Ln is asymptotically stable at e = 0 and z ¼ 0. From the definitions of zi ði ¼ 1; 2; . . . ; nÞ and (7), we can get 1 u ¼ n b1 z1 þ b2 Lz2 þ þ bn Ln1 zn : ð8Þ L u¼
From (5)–(7), we have 1 e_ ¼ Ae þ U; L
ð9Þ
1 1 z_ ¼ Bz þ De; L L where
0 B B B B¼B B B @
ð10Þ
0 0 .. . 0
1 0 .. . 0
0 1 .. . 0
b1
b2
b3
... ... .. . ...
0 0 .. . 1
1 C C C C: C C A
. . . bn
Since A is a stable matrix, there exists a positive definite matrix P such that PA þ AT P ¼ I: Let V1 = eTPe, we have T 1 1 1 2 Ae þ U P e þ eT P Ae þ U ¼ kek þ 2eT P U: V_ 1 jð9Þ ¼ L L L
ð11Þ
From Assumption 1, the definitions of ei and zi , L > 1 and Lemma 2, we can get Nk X k1 L /k 6 Lk1 ðck;l jx1 ja1;k;l jx2 ja2;k;l jxn jan;k;l juðt d 1 Þjanþ1;k;l Þ l¼1
xn1 an1;k;l 6L ck;l jx1 j n2 L L l¼1 an;k;l Z t xn anþ1;k;l n1 uðsÞds juðt d 1 Þj L td 2 k1
Nk X
x a2;k;l 2 a1;k;l
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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Nk x a2;k;l X xn1 an1;k;l 2 a1;k;l 6L ck;l jx1 j n2 L L l¼1 an;k;l Z t xn 1 n1 þ n ðb1 ; b2 ; . . . ; bn ÞzðsÞds L td 2 L anþ1;k;l 1 n ðb1 ; b2 ; . . . ; bn Þzðt d 1 Þ L P nþ1 Nk k iai;k;l X a a i¼1 6L ck;l jx1 j 1;k;l jxn1 j n1;k;l k1
Z jxn j þ
l¼1
t
an;k;l anþ1;k;l ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds jðb1 ; b2 ; . . . ; bn Þzðt d 1 Þj
td 2
1 6 2 tk;1 jx1 j þ tk;2 jx2 j þ þ tk;n1 jxn1 j þ tk;n jxn j L Z t þ tk;n ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds þ tk;nþ1 jðb1 ; b2 ; . . . ; bn Þzðt d 1 Þj td 2
k ¼ 1; 2; . . . ; n 2; Z n2 L / n1
! Z t N n1 X n2 uðsÞds 6 L cn1;l juðt d 1 Þj uðsÞds td 2 td 2 l¼1 1 6 2 tn1;nþ1 ðb1 ; b2 ; . . . ; bn Þjzðt d 1 Þj L Z t þ ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds t
td 2
where tk,j (k = 1, 2, . . ., n 1, j = 1, 2, . . ., n + 1) are nonnegative real numbers dependent on ck,l, aj,k,l and Nk. So we have Z t 2 T 2e P U 6 2 jej jP j T 0 jxj þ T 1 jzðsÞjds þ T 2 jzðt d 1 Þj L td 2 Z t 2 T 2 T 2 T 6 2 jej jP jT 0 jxj þ 2 jej jP jT 1 jzðsÞjds þ 2 jej jP jT 2 jzðt d 1 Þj L L L td 2 T
6
2 T d2 T jej jP jT 0 jxj þ 2 jejT jP jT 1 R1 1 T 1 jP jjej 2 L L Z t Z t 1 1 T T þ jzðsÞj dsR1 jzðsÞjds þ 2 jejT jP jT 2 R1 2 T 2 jP jjej 2 d 2 L td 2 L td 2 þ
1 jzðt d 1 ÞjT R2 jzðt d 1 Þj; L2
ð12Þ
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
where 0
t1;1
B t2;1 B B B .. B T0 ¼ B . Bt B n2;1 B @ 0
t1;2
...
t2;2 .. .
... .. .
t2;n .. .
...
0
0
t1;n t2;n .. .
1
C C C C C C; . . . tn2;n C C C ... 0 A
tn2;2 0
0
t1;n
0 1
0
C B C B C B C B C B T1 ¼ B Cðb1 ; . . . ; bn Þ; C Bt B n2;n C C B @ 1 A
t1;nþ1 t2;nþ1 .. .
1
C B C B C B C B C B T2 ¼ B Cðb1 ; . . . ; bn Þ; C Bt B n2;nþ1 C C B @ tn1;nþ1 A
0
0
R1 > 0 and R2 > 0 are two matrices to be determined later. Obviously, (12) also holds with d2 = 0. From Lemma 1, (11) and (12), we can get 1 2 T d2 T 2 T V_ 1 jð9Þ 6 kek þ 2 jej jP jT 0 jxj þ 2 jej jP jT 1 R1 1 T 1 jP jjej L L L Z t 1 1 T T T þ 2 jzðsÞj R1 jzðsÞjds þ 2 jej jP jT 2 R1 2 T 2 jP jjej L td 2 L 1 T þ 2 jzðt d 1 Þj R2 jzðt d 1 Þj: L Choose LKF 1 V~ 1 ¼ eT P e þ 2 L
Z
0
d 2
Z
t
1 jzðsÞj R1 jzðsÞjds dh þ 2 L hþt T
Z
t
jzðsÞjT R2 jzðsÞjds;
td 1
we have 1 2 T d2 T d2 T _ 2 T Ve 1 jð9Þ 6 kek þ 2 jej jP jT 0 jxj þ 2 jej jP jT 1 R1 zj R1 jzj 1 T 1 jP jjej þ 2 j L L L L 1 1 T T þ 2 jejT jP jT 2 R1 zj R2 jzj: 2 T 2 jP jjej þ 2 j L L Because B is a stable matrix, there exists a Q > 0 such that QB þ BT Q ¼ I:
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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Choose V 2 ¼ zT Qz, we have T 1 1 1 1 Bz þ De Qz þ zT Q Bz þ De V_ 2 jð10Þ ¼ L L L L 1 2 T 2 6 kzk þ jzj jQDj jej: L L Choose V ¼ Ve 1 þ Lk V 2 , where k > 0 is a constant to be determined later, then we have 1 2 T 2 T 2 V_ jð9Þð10Þ 6 kek þ 2 jej jP jT 0 jej þ 2 jej jP jT 0 jzj L L L d2 d2 T T þ 2 jejT jP jT 1 R1 zj R1 jzj 1 T 1 jP jjej þ 2 j L L 1 1 T T þ 2 jejT jP jT 2 R1 zj R2 jzj 2 T 2 jP jjej þ 2 j L L k 2k T 2 2 kzk þ 2 jzj jQDj jej L L jej 1 6 2 ðjejT ; jzjT ÞX ; jzj L where T
X¼
W jP jT 0 þ kjQDj T T 0 jP j þ kjQDj kI þ d 2 R1 þ R2
! ;
T 1 T W ¼ LI þ jP jT 0 þ T T0 jP j þ d 2 jP jT 1 R1 1 T 1 jP j þ jP jT 2 R2 T 2 jP j:
Choose L > 1 and k > 0 satisfying X < 0, we can get V_ jð9Þð10Þ < 0 which indicate that (9) and (10) is asymptotically stable at e = 0 and z ¼ 0 (see [12]). The trivial solution of the closed-loop system (1*), (4) and R t (8) is also asymptotically stable. From the state transformation, ~xn ¼ xn þ td 2 uðsÞds, it is easy to see that the closed-loop system (1), (4) and (8) is also asymptotically stable at x = 0 and z = 0 (see [17]). Up to now, we can conclude that (4) and (8) is the linear output dynamic compensator of the system (1). Since the inequality X < 0 could not be solved by MATLAB, we give the following consideration. From Schur complements, we know that finding L > 1 (as small as possible) satisfying X < 0 is equivalent to solving the following optimization problem with linear matrix inequality (LMI) constraints: min L;
R1 ;R2 ;k
s:t: L > 1 and
W11
W12
WT12
W22
< 0;
ð13Þ
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
where W11 ¼ LI n þ jP jT 0 þ T T0 jP j; T
W12 ¼ ½jP jT 0 þ kjQDj ;
pffiffiffiffiffi d 2 jP jT 1 ; jP jT 2 ;
W22 ¼ diag½kI n þ d 2 R1 þ R2 ; R1 ; R2 : LMI (13) can be easily solved by LMI toolbox in MATLAB. Our LMI (13) always has a solution, and our design method is always efficient for the system (1) satisfying the condition (2), however, the LMIs in [1,3,10] maybe have no solution, and when the LMIs have no solution, the design method proposed there will not work. Hence our design method is better. The proof of the Theorem is completed. h Remark 2. For any given delay di > 0 (i = 1,2), we can always find a constant L satisfying (13). The purpose of finding a small L > 1 satisfying (13) is that we would like get a bigger control gain. There may be many sets aj, bj (j = 1,2, . . ., n) satisfying our design requirements, but we always choose those such that L > 1 is as small as possible. Consequently the gain of the controller is as big as possible and the time of stabilizing system will be as short as possible. Remark 3. The control gain we get is dependent on the delay d2. From a theoretical point of view, for any delay d2 > 0, we can always find a stabilizing controller. But when the delay d2 > 0 is bigger, the control gain of system will be much less. The time of stabilizing system will be very long, and hence this will lower efficacy of the controller in practical. So, when we design a stabilizing controller, we should decrease the delay as far as possible. Remark 4. Our delay controller has good robust performance, that is, a stabilizing controller designed for the system (1) with delay d2 > 0, will be also efficient for the system (1) with delay d~ satisfying d~ < d 2 .
4. Example Example 1. Consider the following nonlinear system: 1 2 2 1 1 1 x_ 1 ¼ x2 þ ðx1 Þ3 ðuðt 1ÞÞ3 ðx3 Þ3 ðuðt 1ÞÞ3 ; 2 3 1 x_ 2 ¼ x3 uðt 1Þ; 4 þ 4et x_ 3 ¼ uðt 0:1Þ;
y ¼ x1 :
ð14Þ
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
1037
For the system (14) satisfying Assumption 1, it is easy to see that t1;1 ¼ 16 ; t1;2 ¼ 0; t1;3 ¼ 29 ; t1;4 ¼ 49 ; t2;4 ¼ 14. We can construct the output feedback controller for the system (14) by following procedures. Choose a1 ¼ 1:0; a2 ¼ 0:8; a3 ¼ 0:10; b1 ¼ 0:28; b2 ¼ 1:2; b3 ¼ 0:65: Solving the Lyapunov equations PA þ AT P ¼ I we can get 0
1:1071
B B P ¼ B 0:5000 @ 2:0714 0
2:8258
B B Q ¼ B 2:4267 @ 1:7857
and
QB þ BT Q ¼ I;
0:5000 2:0714 0:5000 2:4267 5:2253 2:4389
2:0714
1
C C 0:5000 C > 0; A 19:7143
1:7857
1
C C 2:4389 C > 0: A 4:5214
By solving (13), we can get L = 69.5197. So the outputfeedback controller of the system (14) is uðtÞ ¼
1 b1 z1 þ b2 Lz2 þ b3 L2 z3 ; 3 L
where z1, z2 and z3 are the states of the following system a1 ðy z1 Þ; L a2 z_ 2 ¼ z3 þ 2 ðy z1 Þ; L a3 1 z_ 3 ¼ 3 b1 z1 þ b2 Lz2 þ b3 L2 z3 þ 3 ðy z1 Þ: L L
z_ 1 ¼ z2 þ
ð15Þ
Figs. 1–3 show the state response of the closed-loop system (14) and (15) with the initial condition ½x1 ðtÞ; x2 ðtÞ; x3 ðtÞ ¼ ½30; 000; 60; 1; ½z1 ðtÞ; z2 ðtÞ; z3 ðtÞ ¼ ½2000; 200; 3; for t 2 ½1; 0:
1038
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040 4
4.5
x 10
x1 z1 4
3.5
3
2.5
2
1.5
1
0.5
0
−0.5 0
500
1000
1500
2000
2500
3000
3500
time [s]
Fig. 1. Trajectories of x1 and z1. 100 x2 z2 50
0
−50
−100
−150
−200 0
500
1000
1500
2000
2500
time [s]
Fig. 2. Trajectories of x2 and z2.
3000
3500
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
1039
2 x3 z3 1
0
−1
−2
−3
−4 0
500
1000
1500
2000
2500
3000
3500
time [s]
Fig. 3. Trajectories of x3 and z3.
Remark 5. From this example, it is easy to find that the time of stabilizing the system (14) is much longer, this is mainly because the system (14) has higher order and nonlinear terms. Indeed, the gains of stabilizing controller for the strict feedforward system are, in general, lower (see [13,15]), and the time of stabilizing such system will be longer. On the other hand, the gains of stabilizing controller for the strict feedback (lower-triangular form) system are, in general, very high (see [8,9,18]).
5. Conclusion In this paper, we have studied the problem of global stabilization by output feedback for a class of nonlinear system with delays in input. The method used here may be feasible for the more general nonlinear delay systems.
References [1] Xi. Li, C.E. de Souza, Criteria for robust stabilization of uncertain linear systems with state delay, Automatica 33 (9) (1997) 1657–1662.
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[2] X.-F. Jiang, S.-m. Fei, C.-B. Feng, Delay-dependent feedback control for linear time-delay system with input delay, Acta Automatica Sinica 27 (1) (2001) 108–114. [3] Xi. Li, C.E. de Souza, Delay-dependent robust stability and systems: a linear matrix inequality approach, IEEE Transaction Automatic Control 42 (8) (1997) 144–148. [4] Y. Guo, W. Zhou, Peter L. Lee, H1 control for a class of structured time-delay systems, Systems Control Letters 45 (2002) 35–47. [5] F. Mazenc, S. Mondie, S. Niculescu, Global asymptotic stabilization for chains of integrators with a delay in the input, IEEE Transaction Automatic Control 48 (1) (2003) 57–63. [6] X. Ye, Universal stabilization of feedforward nonlinear systems, Automatica 39 (2003) 141– 147. [7] C. Qian, W. Lin, Using small feedback to stabilize a wider class of feedforward systems, in: Proceedings of the IFAC World Congress, vol. E, Beijing, China, 1999, pp. 309–314. [8] C. Qian, W. Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Transaction Automatic Control 47 (11) (2002) 1710–1715. [9] L. Praly, Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate, IEEE Transaction Automatic Control 48 (6) (2003) 1103– 1108. [10] S. Xu, P. Van Dooren, Robust H1 filtering for a class of nonlinear systems with state delay and parameter uncertainty, International Journal of Control 75 (10) (2002) 766–774. [11] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [12] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, NewYork, 1993. [13] F. Mazenc, S. Mondie, R. Francisco, Global asymptotic stabilization of feedforward systems with delay in the input, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003, pp. 4020-4025. [14] R. Sepulchre, M. Jankovic´, P.V. Kokotovic, Constructive Nonlinear Control, SpringerVerlag, London, 1997. [15] X. Sepulchre, M. Jankovic, P.V. Kokotovic, Integrator forwarding: a new recursive nonlinear robust design, Automatica 33 (5) (1997) 979–984. [16] E. beckenbach, R. Bellman, An Introduction to Inequalities, Yale University, 1961. [17] D. Yue, Robust stabilization of uncertain systems with unknown input delay, Automatica 40 (2004) 331–336. [18] X. Zhang, Q. Liu, Z. Cheng, Output feedback control of a class of time-delay nonlinear systems, in: Proceedings of the 5th World Congress on Intelligent Control and Automation, 15–19 June, 2004, Hangzhou, PR China, pp. 897–901.
www.elsevier.com/locate/amc
Output feedback stabilization of nonlinear systems with delays in the input Xianfu Zhang a
a,b,*
, Zhaolin Cheng
b
Department of Mathematics and Physics, Shandong Institute of Architecture and Engineering, Jinan 250014, PR China b School of Mathematics and System Science, Shandong University, Jinan 250100, PR China
Abstract It is proposed a novel and simple design scheme of output feedback controller for a class of nonlinear systems with delays in the input. By constructing appropriate Lyapunov–Krasovskii functionals (LKF) and solving linear matrix inequality (LMI), the delay-dependent controller making the closed-loop system globally asymptotically stable (GAS) is explicitly constructed. A simulation example is given to show the effectiveness of the proposed design procedure. 2004 Elsevier Inc. All rights reserved. Keywords: Feedforward nonlinear systems; Time-delay systems; Output feedback; Lyapunov– Krasovskii functionals; Linear matrix inequality
* Corresponding author. Address: Department of Mathematics and Physics, Shandong Institute of Architecture and Engineering, Jinan 250014, PR China. E-mail address: [email protected] (X. Zhang).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.08.001
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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1. Introduction In real control systems, input delays are often encountered because of transmission of the measurement information. The existence of these delays may be the source of instability or serious deterioration in the performance of the closed-loop system. Hence, the control design problem of input delayed systems has attracted considerable attention (see [2,4,5,13]). In [4], backstepping is applied to design an H1 controller for a class of lower-triangular linear systems with delay in the input. In [5], the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved. On the other hand, feedforward systems are nonlinear systems described by equations having a specific triangular structure. The problem of the asymptotic stabilization by state feedback of these triangular equations in the absence of delay has been studied by many researchers (see [6,7,15]) during the last decade mainly for two reasons: first, the problem of the global asymptotic stabilization of these systems is challenging from a theoretical point of view. Note in particular that they are in general not feedback linearizable and cannot be stabilized by applying the backstepping method. Second, a number of physical devices, such as the system ball and beam with a friction term (see [14]), the TORA system (see [14]), are described, after a preliminary change of feedback, by equations with the feedforward structure. In a recent paper [13], the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved. Up to now, few paper has considered the problem of output feedback controller design for nonlinear feedforward systems with delays in the input. Motivated by [8] and [9], in this paper, by constructing the appropriate Lyapunov– Krasovskii functionals (LKF), we investigate the output feedback controller design for a class of nonlinear systems with delays in the input. Indeed, the nonlinear systems considered in this paper is more general than feedforward systems (upper-triangular form). The method used here is very deferent from that used in [13], and our method is simpler because the most work of our design procedure can be completed by MATLAB toolbox. Our stabilizing controller is delay-dependent, and hence the proposed controller is less conservative.
2. Preliminaries We consider the nonlinear system of dimension n P 2 described by the equations
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
x_ 1 ðtÞ ¼ x2 ðtÞ þ /1 ðt; xðtÞ; uðt d 1 ÞÞ; x_ 2 ðtÞ ¼ x3 ðtÞ þ /2 ðt; xðtÞ; uðt d 1 ÞÞ; .. . x_ n2 ðtÞ ¼ xn1 ðtÞ þ /n2 ðt; xðtÞ; uðt d 1 ÞÞ;
ð1Þ
x_ n1 ðtÞ ¼ xn ðtÞ þ /n1 ðt; xðtÞ; uðt d 1 ÞÞ; x_ n ðtÞ ¼ uðt d 2 Þ; y ¼ x1 ðtÞ; where x = [x1, . . ., xn] T 2 Rn is the state, u 2 R is the input, y 2 R, is the output, constants d1, d2 P 0 are the delays. The argument of the functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote xi(t) by xi. The mappings /i : R Rn R ! R; i ¼ 1; 2; . . . ; n 1 are continuous and satisfy the following growth condition: Assumption 1. For k = 1, 2, . . ., n 1, l = 1, 2, . P . ., Nk, j = 1, 2, . . ., n + 1, there exist constants ck,l P 0 and aj,k,l P 0 satisfying nþ1 j¼1 aj;k;l ¼ 1, such that j/k ðt; xðtÞ; uðt d 1 ÞÞj 6 with ck,l = 0 for integers.
Pnþ1 i¼1
Nk X
ck;l jx1 ja1;k;l jxn jan;k;l juðt d 1 Þjanþ1;k;l ;
ð2Þ
l¼1
i ai;k;l < k þ 2, where Nk(k = 1, 2, . . ., n 1) are positive
Remark 1. When some ai,k,l = 0, it can be viewed that there is no xi in the term ck,ljx1ja1,k,l jxnjan,k,lju(t d1)jan+1,k,l. When k = n 1, we will get a1,n1,l = = an,n1,l = 0, an+1,n1,l = 1, and consequently we have ! N n1 X j/n1 ðt; xðtÞ; uðt d 1 ÞÞj 6 cn1;l juðt d 1 Þj: l¼1
Compared to condition (2) with the following linear growth condition: j/i ðt; xðtÞ; uðt d 1 ÞÞj 6 cðjxiþ2 j þ þ jxn jÞ þ c1 juðt d 1 Þj i ¼ 1; 2; . . . ; n 2;
ð2Þ
j/n1 ðt; xðtÞ; uðt d 1 ÞÞj 6 c1 juðt d 1 Þj; where c P 0 and c1 P 0 are constants, it is easy to see that the system (1) satisfying condition (2*) which is indeed a feedforward nonlinear system obviously satisfies the condition (2). Now we give an example which is not a feedforward system, but satisfies the condition (2),
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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1 2 2 1 1 1 x_ 1 ¼ x2 þ ðx1 Þ3 ðuðt 1ÞÞ3 ðx3 Þ3 ðuðt 1ÞÞ3 ; 2 3 1 uðt 1Þ; x_ 2 ¼ x3 4 þ 4et x_ 3 ¼ uðt 0:1Þ; y ¼ x1 :
Up to now, to my best knowledge, there is no efficient method to design the output feedback controller for this system. Compared with the system considered in [4,5], the system (1) is more general. Ref. [6] has considered the asymptotic stabilization for the system (1) with no delay in input. In this paper, for the vector n = (n1, n2, . . ., nn) (or matrix B = [bij]m·n), we denote vector (jn1j, jn2j, . . ., jnnj) (or matrix [jbijj]m·n) by jnj (or jBj). The matrix B = [bij]m·n is said to be a nonnegative matrix if bij P 0, for 1 6 i 6 m, 1 6 j 6 n. The property about the nonnegative matrix can be found in [11]. we let kÆk denote the Euclidean norm for vector, or the induced Euclidean norm for matrix. For any real matrix A, AT denotes the transpose, In is the n-dimensional identity matrix. The following lemmas are useful in the proof of our main result. Lemma 1. For any a(t) 2 Rn and symmetrical matrix R P 0, the following inequality holds: Z t Z t Z t aT ðsÞdsR aðsÞds 6 d aT ðsÞRaðsÞds: td
td
td
1
1
Proof. Let R2 aðsÞ ¼: nðsÞ ¼ ½n1 ðsÞ; n2 ðsÞ; . . . ; nn ðsÞT , where R2 P 0 satisfying 1 1 R2 R2 ¼ R, then Z t Z t Z t Z t 1 1 aT ðsÞdsR aðsÞds ¼ aT ðsÞR2 ds R2 aðsÞds td td td 2 td Z t 2 Z t n1 ðsÞds þ n1 ðsÞds þ ¼ td td 2 Z t nn ðsÞds þ Z t td Z t 2 6d n1 ðsÞds þ d n22 ðsÞds þ td td Z t n2n ðsÞds þd td Z t ¼d nT ðsÞnðsÞds td Z t aT ðsÞRaðsÞds: ¼d td
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
LemmaP2 (see [16]). For any yi P 0 (i = 1, 2, . . ., k) and ri > 0 (i = 1, 2, . . ., k) satk isfying i¼1 ri ¼ 1, the following inequality holds: y r11 y r22 . . . y rkk 6 r1 y 1 þ r2 y 2 þ þ rk y k :
3. Output feedback controller In practice, when some state variables are unavailable, the output variables are needed to control the system. Under the Assumption 1, we will design the linear output dynamic compensator z_ ðtÞ ¼ MzðtÞ þ NyðtÞ;
M 2 Rnn ; N 2 Rn ;
ð3Þ
F 2 R1n ;
uðtÞ ¼ FzðtÞ;
such that the closed-loop systems (1) and (3) is GAS at the equilibrium (x, z) = (0, 0). Theorem 1. Under the Assumption 1, for any given delays d1, d2 P 0, there exists a linear output feedback controller of the form (3), which is dependent on the delay d2 P 0, such that the closed-loop systems (1) and (3) have globally asymptotically stability (GAS). Proof. Let ~xn ¼ xn þ
Rt
uðsÞds, then the system (1) can be covered into Z t uðsÞds; uðt d 1 Þ ; x_ 1 ðtÞ ¼ x2 ðtÞ þ /1 t; x1 ; . . . ; xn1 ; ~xn td 2
Z x_ 2 ðtÞ ¼ x3 ðtÞ þ /2 t; x1 ; . . . ; xn1 ; ~xn
td 2 t
uðsÞds; uðt d 1 Þ ;
td 2
.. . Z x_ n2 ðtÞ ¼ xn1 ðtÞ þ /n2 t; x1 ; . . . ; xn1 ; ~xn x_ n1 ðtÞ ¼ ~xn ðtÞ
Z
Z
t
uðsÞds; uðt d 1 Þ ; td 2
t
uðsÞds þ /n1 t; x1 ; . . . ; xn1 ; ~xn
td 2
t
uðsÞds; uðt d 1 Þ ;
td 2
~x_ n ðtÞ ¼ uðtÞ; y ¼ x1 ðtÞ: ð1Þ
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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We begin with by designing the following linear observer of the system (1*): a1 z_ 1 ¼ z2 þ ðy z1 Þ; L a2 z_ 2 ¼ z3 þ 2 ðy z1 Þ; L .. ð4Þ . an1 z_ n1 ¼ zn þ n1 ðy z1 Þ; L an z_ n ¼ uðtÞ þ n ðy z1 Þ: L where L > 1 will be determined later, aj > 0 (j = 1, 2, . . ., n) are coefficients of the Hurwitz polynomial pðsÞ ¼ sn þ a1 sn1 þ þ an1 s þ an : Now we define ei ¼ Li1 ðxi zi Þ; xi ¼ Li1 xi ; i ¼ 1; 2; . . . ; n 1; en ¼ Ln1 ð~xn zn Þ;
xn ¼ Ln1~xn ;
zi ¼ Li1 zi ; i ¼ 1; 2; . . . ; n: From (1*) and (4), a simple calculation gives 1 e_ ¼ Ae þ U; L
ð5Þ
1 1 z_ ¼ J 0z þ Ln1 GuðtÞ þ De; L L where T
e ¼ ½e1 ; e2 ; . . . ; en 0 a1 1 B a2 0 B B . .. A ¼ B .. . B @an1 0 an 0 0
a1 B a2 B B . D ¼ B .. B @an1 an
; 0 1 .. . 0 0
0 0 .. .
0 0 .. .
0 0
0 0
ð6Þ T
T
z ¼ ½z1 ; z2 ; . . . ; zn ; G ¼ ½0; 0; . . . ; 0; 1 ; 1 0 1 ... 0 0 1 0 ... 0 C ... 0C B0 0 1 ... 0C B. . . . C .. C .. . . ... C; .. .. .. ; J0 ¼ B . .C B C C @0 0 0 ... 1A ... 1A 0 0 0 ... 0 ... 0 1
... 0 ... 0C C .. C .. ; . .C C A ... 0 ... 0
0
/1 L/2 .. .
1
C B C B C B C B U¼B C: n3 C B L / C B n2 R n2 @ L ð/ t uðsÞdsÞ A n1 td 2 0
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
Let bj > 0 (j = 1, 2, . . ., n) are coefficients of the Hurwitz polynomial qðsÞ ¼ sn þ bn sn1 þ þ b2 s þ b1 : Next we will choose an appreciate L > 1, such that the closed-loop system (5), (6) and 1 ðb1z1 þ b2z2 þ þ bnzn Þ; ð7Þ Ln is asymptotically stable at e = 0 and z ¼ 0. From the definitions of zi ði ¼ 1; 2; . . . ; nÞ and (7), we can get 1 u ¼ n b1 z1 þ b2 Lz2 þ þ bn Ln1 zn : ð8Þ L u¼
From (5)–(7), we have 1 e_ ¼ Ae þ U; L
ð9Þ
1 1 z_ ¼ Bz þ De; L L where
0 B B B B¼B B B @
ð10Þ
0 0 .. . 0
1 0 .. . 0
0 1 .. . 0
b1
b2
b3
... ... .. . ...
0 0 .. . 1
1 C C C C: C C A
. . . bn
Since A is a stable matrix, there exists a positive definite matrix P such that PA þ AT P ¼ I: Let V1 = eTPe, we have T 1 1 1 2 Ae þ U P e þ eT P Ae þ U ¼ kek þ 2eT P U: V_ 1 jð9Þ ¼ L L L
ð11Þ
From Assumption 1, the definitions of ei and zi , L > 1 and Lemma 2, we can get Nk X k1 L /k 6 Lk1 ðck;l jx1 ja1;k;l jx2 ja2;k;l jxn jan;k;l juðt d 1 Þjanþ1;k;l Þ l¼1
xn1 an1;k;l 6L ck;l jx1 j n2 L L l¼1 an;k;l Z t xn anþ1;k;l n1 uðsÞds juðt d 1 Þj L td 2 k1
Nk X
x a2;k;l 2 a1;k;l
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
1033
Nk x a2;k;l X xn1 an1;k;l 2 a1;k;l 6L ck;l jx1 j n2 L L l¼1 an;k;l Z t xn 1 n1 þ n ðb1 ; b2 ; . . . ; bn ÞzðsÞds L td 2 L anþ1;k;l 1 n ðb1 ; b2 ; . . . ; bn Þzðt d 1 Þ L P nþ1 Nk k iai;k;l X a a i¼1 6L ck;l jx1 j 1;k;l jxn1 j n1;k;l k1
Z jxn j þ
l¼1
t
an;k;l anþ1;k;l ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds jðb1 ; b2 ; . . . ; bn Þzðt d 1 Þj
td 2
1 6 2 tk;1 jx1 j þ tk;2 jx2 j þ þ tk;n1 jxn1 j þ tk;n jxn j L Z t þ tk;n ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds þ tk;nþ1 jðb1 ; b2 ; . . . ; bn Þzðt d 1 Þj td 2
k ¼ 1; 2; . . . ; n 2; Z n2 L / n1
! Z t N n1 X n2 uðsÞds 6 L cn1;l juðt d 1 Þj uðsÞds td 2 td 2 l¼1 1 6 2 tn1;nþ1 ðb1 ; b2 ; . . . ; bn Þjzðt d 1 Þj L Z t þ ðb1 ; b2 ; . . . ; bn ÞjzðsÞjds t
td 2
where tk,j (k = 1, 2, . . ., n 1, j = 1, 2, . . ., n + 1) are nonnegative real numbers dependent on ck,l, aj,k,l and Nk. So we have Z t 2 T 2e P U 6 2 jej jP j T 0 jxj þ T 1 jzðsÞjds þ T 2 jzðt d 1 Þj L td 2 Z t 2 T 2 T 2 T 6 2 jej jP jT 0 jxj þ 2 jej jP jT 1 jzðsÞjds þ 2 jej jP jT 2 jzðt d 1 Þj L L L td 2 T
6
2 T d2 T jej jP jT 0 jxj þ 2 jejT jP jT 1 R1 1 T 1 jP jjej 2 L L Z t Z t 1 1 T T þ jzðsÞj dsR1 jzðsÞjds þ 2 jejT jP jT 2 R1 2 T 2 jP jjej 2 d 2 L td 2 L td 2 þ
1 jzðt d 1 ÞjT R2 jzðt d 1 Þj; L2
ð12Þ
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
where 0
t1;1
B t2;1 B B B .. B T0 ¼ B . Bt B n2;1 B @ 0
t1;2
...
t2;2 .. .
... .. .
t2;n .. .
...
0
0
t1;n t2;n .. .
1
C C C C C C; . . . tn2;n C C C ... 0 A
tn2;2 0
0
t1;n
0 1
0
C B C B C B C B C B T1 ¼ B Cðb1 ; . . . ; bn Þ; C Bt B n2;n C C B @ 1 A
t1;nþ1 t2;nþ1 .. .
1
C B C B C B C B C B T2 ¼ B Cðb1 ; . . . ; bn Þ; C Bt B n2;nþ1 C C B @ tn1;nþ1 A
0
0
R1 > 0 and R2 > 0 are two matrices to be determined later. Obviously, (12) also holds with d2 = 0. From Lemma 1, (11) and (12), we can get 1 2 T d2 T 2 T V_ 1 jð9Þ 6 kek þ 2 jej jP jT 0 jxj þ 2 jej jP jT 1 R1 1 T 1 jP jjej L L L Z t 1 1 T T T þ 2 jzðsÞj R1 jzðsÞjds þ 2 jej jP jT 2 R1 2 T 2 jP jjej L td 2 L 1 T þ 2 jzðt d 1 Þj R2 jzðt d 1 Þj: L Choose LKF 1 V~ 1 ¼ eT P e þ 2 L
Z
0
d 2
Z
t
1 jzðsÞj R1 jzðsÞjds dh þ 2 L hþt T
Z
t
jzðsÞjT R2 jzðsÞjds;
td 1
we have 1 2 T d2 T d2 T _ 2 T Ve 1 jð9Þ 6 kek þ 2 jej jP jT 0 jxj þ 2 jej jP jT 1 R1 zj R1 jzj 1 T 1 jP jjej þ 2 j L L L L 1 1 T T þ 2 jejT jP jT 2 R1 zj R2 jzj: 2 T 2 jP jjej þ 2 j L L Because B is a stable matrix, there exists a Q > 0 such that QB þ BT Q ¼ I:
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
1035
Choose V 2 ¼ zT Qz, we have T 1 1 1 1 Bz þ De Qz þ zT Q Bz þ De V_ 2 jð10Þ ¼ L L L L 1 2 T 2 6 kzk þ jzj jQDj jej: L L Choose V ¼ Ve 1 þ Lk V 2 , where k > 0 is a constant to be determined later, then we have 1 2 T 2 T 2 V_ jð9Þð10Þ 6 kek þ 2 jej jP jT 0 jej þ 2 jej jP jT 0 jzj L L L d2 d2 T T þ 2 jejT jP jT 1 R1 zj R1 jzj 1 T 1 jP jjej þ 2 j L L 1 1 T T þ 2 jejT jP jT 2 R1 zj R2 jzj 2 T 2 jP jjej þ 2 j L L k 2k T 2 2 kzk þ 2 jzj jQDj jej L L jej 1 6 2 ðjejT ; jzjT ÞX ; jzj L where T
X¼
W jP jT 0 þ kjQDj T T 0 jP j þ kjQDj kI þ d 2 R1 þ R2
! ;
T 1 T W ¼ LI þ jP jT 0 þ T T0 jP j þ d 2 jP jT 1 R1 1 T 1 jP j þ jP jT 2 R2 T 2 jP j:
Choose L > 1 and k > 0 satisfying X < 0, we can get V_ jð9Þð10Þ < 0 which indicate that (9) and (10) is asymptotically stable at e = 0 and z ¼ 0 (see [12]). The trivial solution of the closed-loop system (1*), (4) and R t (8) is also asymptotically stable. From the state transformation, ~xn ¼ xn þ td 2 uðsÞds, it is easy to see that the closed-loop system (1), (4) and (8) is also asymptotically stable at x = 0 and z = 0 (see [17]). Up to now, we can conclude that (4) and (8) is the linear output dynamic compensator of the system (1). Since the inequality X < 0 could not be solved by MATLAB, we give the following consideration. From Schur complements, we know that finding L > 1 (as small as possible) satisfying X < 0 is equivalent to solving the following optimization problem with linear matrix inequality (LMI) constraints: min L;
R1 ;R2 ;k
s:t: L > 1 and
W11
W12
WT12
W22
< 0;
ð13Þ
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
where W11 ¼ LI n þ jP jT 0 þ T T0 jP j; T
W12 ¼ ½jP jT 0 þ kjQDj ;
pffiffiffiffiffi d 2 jP jT 1 ; jP jT 2 ;
W22 ¼ diag½kI n þ d 2 R1 þ R2 ; R1 ; R2 : LMI (13) can be easily solved by LMI toolbox in MATLAB. Our LMI (13) always has a solution, and our design method is always efficient for the system (1) satisfying the condition (2), however, the LMIs in [1,3,10] maybe have no solution, and when the LMIs have no solution, the design method proposed there will not work. Hence our design method is better. The proof of the Theorem is completed. h Remark 2. For any given delay di > 0 (i = 1,2), we can always find a constant L satisfying (13). The purpose of finding a small L > 1 satisfying (13) is that we would like get a bigger control gain. There may be many sets aj, bj (j = 1,2, . . ., n) satisfying our design requirements, but we always choose those such that L > 1 is as small as possible. Consequently the gain of the controller is as big as possible and the time of stabilizing system will be as short as possible. Remark 3. The control gain we get is dependent on the delay d2. From a theoretical point of view, for any delay d2 > 0, we can always find a stabilizing controller. But when the delay d2 > 0 is bigger, the control gain of system will be much less. The time of stabilizing system will be very long, and hence this will lower efficacy of the controller in practical. So, when we design a stabilizing controller, we should decrease the delay as far as possible. Remark 4. Our delay controller has good robust performance, that is, a stabilizing controller designed for the system (1) with delay d2 > 0, will be also efficient for the system (1) with delay d~ satisfying d~ < d 2 .
4. Example Example 1. Consider the following nonlinear system: 1 2 2 1 1 1 x_ 1 ¼ x2 þ ðx1 Þ3 ðuðt 1ÞÞ3 ðx3 Þ3 ðuðt 1ÞÞ3 ; 2 3 1 x_ 2 ¼ x3 uðt 1Þ; 4 þ 4et x_ 3 ¼ uðt 0:1Þ;
y ¼ x1 :
ð14Þ
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
1037
For the system (14) satisfying Assumption 1, it is easy to see that t1;1 ¼ 16 ; t1;2 ¼ 0; t1;3 ¼ 29 ; t1;4 ¼ 49 ; t2;4 ¼ 14. We can construct the output feedback controller for the system (14) by following procedures. Choose a1 ¼ 1:0; a2 ¼ 0:8; a3 ¼ 0:10; b1 ¼ 0:28; b2 ¼ 1:2; b3 ¼ 0:65: Solving the Lyapunov equations PA þ AT P ¼ I we can get 0
1:1071
B B P ¼ B 0:5000 @ 2:0714 0
2:8258
B B Q ¼ B 2:4267 @ 1:7857
and
QB þ BT Q ¼ I;
0:5000 2:0714 0:5000 2:4267 5:2253 2:4389
2:0714
1
C C 0:5000 C > 0; A 19:7143
1:7857
1
C C 2:4389 C > 0: A 4:5214
By solving (13), we can get L = 69.5197. So the outputfeedback controller of the system (14) is uðtÞ ¼
1 b1 z1 þ b2 Lz2 þ b3 L2 z3 ; 3 L
where z1, z2 and z3 are the states of the following system a1 ðy z1 Þ; L a2 z_ 2 ¼ z3 þ 2 ðy z1 Þ; L a3 1 z_ 3 ¼ 3 b1 z1 þ b2 Lz2 þ b3 L2 z3 þ 3 ðy z1 Þ: L L
z_ 1 ¼ z2 þ
ð15Þ
Figs. 1–3 show the state response of the closed-loop system (14) and (15) with the initial condition ½x1 ðtÞ; x2 ðtÞ; x3 ðtÞ ¼ ½30; 000; 60; 1; ½z1 ðtÞ; z2 ðtÞ; z3 ðtÞ ¼ ½2000; 200; 3; for t 2 ½1; 0:
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X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040 4
4.5
x 10
x1 z1 4
3.5
3
2.5
2
1.5
1
0.5
0
−0.5 0
500
1000
1500
2000
2500
3000
3500
time [s]
Fig. 1. Trajectories of x1 and z1. 100 x2 z2 50
0
−50
−100
−150
−200 0
500
1000
1500
2000
2500
time [s]
Fig. 2. Trajectories of x2 and z2.
3000
3500
X. Zhang, Z. Cheng / Appl. Math. Comput. 167 (2005) 1026–1040
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2 x3 z3 1
0
−1
−2
−3
−4 0
500
1000
1500
2000
2500
3000
3500
time [s]
Fig. 3. Trajectories of x3 and z3.
Remark 5. From this example, it is easy to find that the time of stabilizing the system (14) is much longer, this is mainly because the system (14) has higher order and nonlinear terms. Indeed, the gains of stabilizing controller for the strict feedforward system are, in general, lower (see [13,15]), and the time of stabilizing such system will be longer. On the other hand, the gains of stabilizing controller for the strict feedback (lower-triangular form) system are, in general, very high (see [8,9,18]).
5. Conclusion In this paper, we have studied the problem of global stabilization by output feedback for a class of nonlinear system with delays in input. The method used here may be feasible for the more general nonlinear delay systems.
References [1] Xi. Li, C.E. de Souza, Criteria for robust stabilization of uncertain linear systems with state delay, Automatica 33 (9) (1997) 1657–1662.
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