- Email: [email protected]
Global output-feedback stabilization for a class of switched uncertain nonlinear systems
Applied Mathematics and Computation 256 (2015) 551–564
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Global output-feedback stabilization for a class of switched uncertain nonlinear systems Erpeng Li, Lijun Long ⇑, Jun Zhao State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, PR China
a r t i c l e
i n f o
Keywords: Uncertain nonlinear systems Output feedback Common Lyapunov function (CLF) Switched systems
a b s t r a c t This paper is mainly concerned with the problem of global output feedback stabilization for a class of switched nonlinear systems with uncertain control coefficients. First of all, it is shown that the unknown control coefficients are lumped together via a common coordinate transformation, thus the original switched nonlinear system is transformed into a new switched system for which control design is feasible. Second, by constructing a common Lyapunov function relying on the designed state observer, the common output feedback controller independent of switching signals is designed based on the backstepping method so as to guarantee the global asymptotic stability of the resulting closed-loop switched system under arbitrary switchings. Two examples are included for verifying the effectiveness of the method proposed. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Global output-feedback stabilization of non-switched nonlinear systems with linearly unmeasured states dependent growth is an important topic in control theory and has been extensively investigated over the last decade [5,9– 11,14,20,23,24,30,33,43]. However, for switched nonlinear systems with linearly unmeasured states dependent growth, the global output-feedback stabilization problem has been limitedly investigated in the past few years [12,13,15,17,19,28,31]. It is well-known that the output feedback control is one of the most important problems of nonlinear systems since only the plant output can be measured in many cases [1,2,21,27,26,40,41]. Generally speaking, the separation principle does not work for nonlinear systems which makes the design of output feedback control more complicated and difficult [12]. Furthermore, a more challenging issue for output feedback in the field of nonlinear control is global stabilization [10,20]. The nonlinear systems in output feedback form are a kind of significant nonlinear systems. The feedback controllers of such a kind of nonlinear systems can be constructed systematically by the backstepping approach and its variations [9,11,14,18,24,43]. On the other hand, switched systems have been attracting a great deal of attention in the past decades due to the high performance requirements of control systems for handling nonlinearities, uncertainties and operating condition variations and also fault-induced dynamic changes [16,29,32,34–36,39,42]. A switched system is a kind of hybrid systems that comprises a collection of subsystems together with a switching rule that specifies the switching among these subsystems [28]. Stability is the first requirement for a system to work normally; thus, stability of switched systems is also the first and important task in researches on switched systems [8]. Due to the interaction between continuous dynamics and discrete ⇑ Corresponding author. E-mail addresses: [email protected] (L. Long), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.amc.2015.01.039 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
552
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
dynamics, switched systems may have a very complicated behavior. Consequently, the study of stability for switched systems is more difficult than the study for continuous systems or discrete systems, and so becomes a challenging issue. Moreover, a common Lyapunov function (CLF) for all subsystems guarantees stability under arbitrary switchings [13]. Naturally, how to construct a CLF to study stability is of great importance. Recently, numerous methods have been proposed to construct such a CLF. For example, global stabilization for switched nonlinear systems in triangular structure by state feedback under arbitrary switchings is achieved via backstepping or forwarding which provides a CLF [15,19,31]. Besides, in [4], an adaptive control scheme via state feedback for switched nonlinear systems in strict-feedback form was proposed for switching with a certain dwell-time. It is worth pointing out that the results [10,15,19,20,31] are based on the assumption that full state measurements are available. However, in many systems, this assumption does not hold and an output-feedback control scheme such as observer has to be used. However, to the best of our knowledge, outputfeedback stabilization of switched nonlinear systems with unknown control coefficients has only been studied in [17], due to the lack of effective tools and the complexity arising the impact of the system structure on switching. When we consider full state measurements are unavailable, the key problem is how to achieve the global stabilization via design of a proper state observer and output feedback controllers of subsystems which have to be based on state estimates ? In this paper, we consider the problem of global stabilization via output feedback for a class of switched nonlinear systems with unknown control coefficients by exploiting the CLF method and backstepping. It is assumed that the system studied admits a certain structure, which is much more general than strict-feedback form in switched nonlinear systems. The remarkableness of the paper, compared with the closely related works, can be shown from the following two aspects: Firstly, we investigate the global stabilization for a class of switched nonlinear systems with unknown control coefficients by output feedback. When considering the case where full state may not be measurable and thus state feedback is not implemented, we designed an appropriate state observer. Secondly, we also constructed a common coordinate transformation at each step of the backstepping. Based on the observer designed and the common coordinate transformation, we constructed output feedback controllers for subsystems via state estimates. Moreover, it is worth pointing out that, since these results [15,19,31] are based on the assumption that full state measurements are available. Their methods cannot be used in the system studied in this paper. The remainder of the paper is organized as follows. Section 2 formulates the system model and the control objective. Section 3 provides the globally stabilizing control design scheme via output feedback and summarizes the main results. In Section 4, a numerical example is given to illustrate the effectiveness of the theoretical results. The paper ends with some concluding remarks. 2. Problem formulation Consider a class of switched nonlinear systems with unknown control coefficients as follows:
8 > < g_ i ¼ g i giþ1 þ wirðtÞ ðt; g; urðtÞ Þ; i ¼ 1; . . . ; n 1; g_ n ¼ g n urðtÞ þ wnrðtÞ ðt; g; urðtÞ Þ; > : y ¼ g1 ;
ð1Þ
where g ¼ ½g1 ; . . . ; gn T 2 Rn is the system state vector with the initial value g0 ¼ gðt0 Þ; u 2 R and y 2 R are the control input and system output, respectively; g i – 0; i ¼ 1; . . . ; n are unknown constant called unknown control coefficients and wirðtÞ : ½t0 ; þ1 Rn R; i ¼ 1; . . . ; n:rðtÞ : ½0; þ1Þ ! M ¼ f1; 2; . . . ; mg is the switching signals which is assumed to be a piecewise continuous (from the right) function of time; In what follows, suppose only the system output is measurable and available for feedback, and for notational simplicity, let t 0 ¼ 0. More specifically, the goal of the paper is to design output-feedback controllers for subsystems and state observer which globally stabilizes the system (1) under arbitrary switchings. To this end, the following two assumptions are imposed on the system (1). Assumption 1. There exist some unknown positive constants h0k ; k 2 M such that
jwi k ðt; g; uk Þj 6 h0k ðjg1 j þ jg2 j þ . . . þ jgi jÞ;
i ¼ 1; . . . ; n; 8k 2 M:
ð2Þ
Assumption 2. The signs of g i ; i ¼ 1; . . . ; n are known, and there exist known positive constants g and g satisfying
g 6 g i 6 g;
i ¼ 1; . . . ; n:
Remark 1. From Assumptions 1 and 2, it is easy to see that system (1) has uncertain control coefficients and unmeasured states dependent nonlinearities. Assumption 1 is a linear growth condition. It gives some information about the nonlinearities. Assumption 2 is a standard assumption in the study of non-switched nonlinear systems. Assumption 2 means that the known signs of g i play an important role in controller design. Otherwise, we cannot decide the direction along which the control designs, and the closed-loop system may be unstable.
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
553
3. Main results We introduce the following scaling transformation for the system (1):
xi ¼
gn g; g in i
uik ¼
gn w ; g in ik
i ¼ 1; . . . ; n;
ð3Þ
where
g in ¼
n Y
gj ;
i ¼ 1; . . . ; n;
cij ¼
j¼i
dij ¼
i Y
j Y
ck ;
1 6 i 6 j 6 n;
k¼i
dk ; 1 6 i 6 j 6 n;
c2ij ¼
j Y
c2k ;
1 6 i 6 j 6 n:
k¼i
k¼i
Under this transformation, the system (1) becomes
8 > < x_ i ¼ xiþ1 þ /irðtÞ ðt; x; urðtÞ Þ; i ¼ 1; . . . ; n 1; x_ n ¼ g n urðtÞ þ /nrðtÞ ðt; x; urðtÞ Þ; > : y ¼ gx1 ;
ð4Þ
where g ¼ gg1n n and h ¼ maxfh1k g; k 2 M. Since h1k is an unknown constant, h is also an unknown constant. By Assumption 1 and 2, it is easy to find a known positive constant g M P 1 such that
8 > < 1 6 jg j 6 g M ; j/ik j 6 h1k ðjx1 j þ jx2 j þ . . . þ jxi jÞ > : 6 hðjx1 j þ jx2 j þ . . . þ jxi jÞ; i ¼ 1; 2; . . . ; n:
ð5Þ
Remark 2. It is necessary to stress that the stabilization of system (1) is implied by that of transformed system (4) which has known control coefficients and is more convenient to design an observer. 3.1. Observer design Suppose that ai > 0ði ¼ 1; . . . ; nÞ are coefficients of the Hurwitz polynomial sn þ a1 sn1 þ þ an1 s þ an , and then in terms of Lemma 1 in [25], we can choose a ¼ ða1 ; a2 ; . . . ; an ÞT such that
AT P þ PA 6 I;
DP þ PD P 0;
ð6Þ
where
3 a1 1 0 6 . .. .. 7 6 .. . . 7 7 2 Rnn ; A¼6 7 6 4 an1 1 0 5 2
D ¼ diagf1; 2; . . . ; ng;
an 1 0 P is a symmetric and positive matrix, I denotes the n-dimensional identity matrix. Motivated by [5,7,17], the dynamic high-gain observer, and the observer-based output-feedback controller are designed as
8_ ^x ¼ ^x_ iþ1 Li ai ^x1 ; i ¼ 1; . . . ; n 1; > > < i ^x_ n ¼ g n urðtÞ Ln an ^x1 ; > > 2 ^x2 : L_ ¼ yL2 þ L12 þ n2n ; Lð0Þ ¼ 1;
ð7Þ
uk ¼ g n Lnþ1 an :
ð8Þ
and
nn is given by
8 n ¼ y; > > > 1 L > < n ¼ ^xi a ; i ¼ 2; . . . ; n; i1 i Li > a1 ¼ sgnðgÞc1 n1 ; > > > : ai ¼ c1 ni ; i ¼ 2; . . . ; n:
ð9Þ
554
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
and denote
ei ¼ xi Li ^xi ; zi ¼ ^xLii ; i ¼ 1; . . . ; n; ^xi ¼ ½^x1 ; . . . ; ^xn T ;
T
n ¼ ½^n1 ; . . . ; ^nn ;
e ¼ ½e1 ; . . . ; en T ; z ¼ ½z1 ; . . . ; zn T : Actually, in the paper, ci ði ¼ 1; . . . ; nÞ are required to satisfy the following inequalities:
8 c1 P 1 þ 2kPak2 þ g 2M ; > > > > > 3a2 > > c2 P 5 þ 32 g 2M d1 þ 22 þ 4g 2M c21 þ g M c1 þ 32 g 2M c41 ; > > > !2 > > i1 > X > > i i2 3 > P 1 þ 2 þ d þ 2 a þ a c c > i i j j 2 i1 < j¼2
> > > > > > > > > > > > > > > > > :
ð10Þ
i2 X þci1 ð1 þ 32 c3i1 Þ þ 3 2i3j ð1 þ c2j Þc2jði1Þ j¼2
2i4 g 2M c21ði1Þ
þ19
þ 2i2 g 2M c41 c22ði1Þ ;
i ¼ 3; . . . ; n;
where a ¼ ½a1 ; . . . ; an T and
8 d1 ¼ 1 þ 14 c21 ; > > > 8 > > > > > > < > > < > di ¼ 1 þ max 14 c2i ; > > > > > > > > > > : :
4dj c2j
9 > > > > = c2ji !! ; j ¼ 1; . . . ; i 1 ; i1 2 X > c > ðjþ1Þk > > 1þ dðjþ1Þði1Þ dðjþ1Þk ;
i ¼ 2; . . . ; n 1:
ð11Þ
k¼jþ1
3.2. Controller design By construction, we firstly construct V l ðÞ; l ¼ 1; . . . ; n , as
8 V ¼ eT P e; > < 0 V 1 ¼ V 0 þ 12 n21 þ 2L13 ^x21 ; > : V l ¼ dl1 V l1 þ 12 n2l ; l ¼ 2; . . . ; n;
ð12Þ
where P and dl satisfy (6) and (11), respectively. By view of (4), (7), (8), (9) and the definitions of ei ði ¼ 1; . . . ; nÞ and zi ði ¼ 1; . . . ; nÞ, we obtain the following switched system
8 > e_ ¼ LAe þ UrðtÞ ðt; x; urðtÞ ; LÞ þ ax1 LL_ ^x21 ; > < _ z_ ¼ LAz en Lcn nn LL Dz; > > :_ L ¼ n2 þ 12 ^x2 þ n2 ; Lð0Þ ¼ 1; 1
L
1
ð13Þ
n
where
UrðtÞ ¼
1 1 1 / ; / ; . . . ; /nrðtÞ L 1rðtÞ L 2rðtÞ L
T ;
en ¼ ½0; . . . ; 0; 1T 2 Rn : Observe that by construction, L_ P 0; Lð0Þ ¼ 1 and hence L P 1. As a result, the vector field of the closed-loop system (13) satisfies a local Lipschitz condition in a neighborhood of the initial condition ðeð0Þ; zð0Þ; Lð0ÞÞ 2 Rn Rn Rþ , and the corresponding solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) exists and is unique on ½0; T f Þ for some T f 2 ð0; þ1. Without loss of generality, we assume from now on that ½0; T f Þ is the maximally extended interval of the solution of (13). This fact will be used in the rest of the proof. Noting that L_ P 0 and L P 1, and in view of (6), the time derivative of V 0 satisfies
V_ 0 6 Lkek2 þ 2eT PUrðtÞ þ 2eT Pax1 : Observing that
ð14Þ
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
555
n X pffiffiffi UrðtÞ 6 UrðtÞ 6 nhjn1 j þ n nhkek þ nh jzi j; 1 i¼2
we have
2eT PUrðtÞ 6
n pffiffiffi 1 2 1X n1 þ 2n2 h2 kPk2 kek2 þ ð4n3 h2 kPk2 þ 2n nhkPkÞkek2 þ z2 ; 2 4 i¼2 i
ð15Þ
moreover, one has
L 1 2eT Pax1 ¼ 2eT P n1 6 Lkek2 þ 2kPak2 Ln21 ; g 2
ð16Þ
Therefore, according to (3), (15) and (16), one gets
n 1 1 1X þ 2kPak2 L n21 þ z2 ; V_ 0 6 L H kek2 þ 2 2 4 i¼2 i
ð17Þ
pffiffiffi where H ¼ 4n3 h2 kP k2 þ 2n nhkP k þ 2n2 h2 kP k2 . When l ¼ 1 , by (13) and (17), the time derivative of V 1 defined in (12) satisfies n
1X V_ 1 6 12 L H kek2 þ 12 þ 2kPak2 L n21 þ z2 4 i¼2 i
3L_ a1 1 _ þn1 Lg e2 þ Lgz2 þ gL /1k LL n1 4 ^x21 2 ^x21 þ ^x1 z2 : L 2L L
ð18Þ
By (5) and completing the square, we have
n1 Lg e2 6
L 2 L e þ Lg2 n21 6 kek2 þ Lg2M n21 ; 4 2 4
g n / 6 hn21 ; L 1 1k 1 a 1 2 ^x1 z2 6 12 ^x21 þ z2 : L 2a 1 2L
Substituting this inequality above into (18) yields
V_ 1 6 14 L H kek2 þ ððc1 1ÞL þ 12 þ hÞn21 n X _ a1 2 1 2 1 ^ LL n21 2L z2i þ Lgn1 z2 ; 2 x1 þ 2a z2 þ 4 1 i¼2
where c1 has been defined in (10). Moreover, by (9) and noting jg j P 1, we obtain
z22 2c21 n21 þ 2n22 ;
gn1 z2 6 gn1 n2 c1 n21 ;
and hence,
_ a1 2 ^ V_ 1 6 14 L H kek2 þ ðL h1 Þn21 LL n21 2L 2 x1 n
X þ 14 z2i þ 12 þ a11 n22 þ Lgn1 n2 ;
ð19Þ
i¼3
where h1 ¼ h þ 12 þ
1 2
þ a11 c21 . From (9) and (13), one can get
_ n_ 2 ¼ Lz3 a2 ^x1 2 LL n2 þ sgnðgÞc1
_ Lg e2 þ Lgz2 þ gL /1k þ LL n1 :
ð20Þ
Then, it follows from (12), (19) and (20) that
_ a1 2 ^ V_ 2 6 d1 14 L H kek2 ðL h1 Þn21 LL n21 2L 2 x1 ! n
X z2i þ 12 þ a11 n22 þ Lgn1 n2 þ n2 ðLz3 a2 ^x1 þ 14 i¼3 _ 2 LL n2
_
þ sgnðgÞc1 ðLg e2 þ Lgz2 þ gL /1k þ LL n1 ÞÞ:
ð21Þ
556
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
According to (3), (5), (9) and the definitions of e1 and z2 and d1 > 1, we have
8 Lgd1 n1 n2 6 d61 Ln21 þ 32 g 2M d1 Ln2 ; > > > > >
> > d1 > Lkek2 þ 4Ln22 ; > a2 ^x1 n2 ¼ La2 n2 ng1 e1 n22 6 d61 Ln21 þ 32 a22 Ln2 þ 16 > > > > > > d1 > < sgnðgÞ c1 Lg e2 n2 6 16 Lkek2 þ 4g 2M c21 Ln22 ; > 3g 2 > > sgnðgÞ c1 Lgz2 n2 ¼ sgnðgÞ c1 Lgn2 ðn2 sgnðgÞ c1 n1 Þ 6 ðg M c1 þ 2M c41 ÞLn22 þ d61 Ln21 ; > > > > > > > > sgnðgÞ c1 gL /1k n2 6 h1k c1 jn1 j jn2 j 6 d21 h21k c21 n21 þ d21 n22 6 d21 hc21 n21 þ d21 n22 ; > > > > > : c2 _ _ _ sgnðgÞ c1 LL n1 n2 6 41 LL n21 þ LL n22 :
ð22Þ
Besides, by (9) and the definition of z3 , we have
d d1 2 d1 2 d z 6 a þ n23 ¼ 1 c22 n22 þ 1 n23 : 4 3 2 2 2 2
ð23Þ
Besides, by (10), (22) and (23), (21) becomes
n L L d1 d1 2 c2 L 2 L 2 a1 d1 2 d1 X d1 V_ 2 6 d1 H kek2 d1 h2 n21 L c22 d1 n2 d1 1 n1 n2 2 ^x1 þ z2 þ n2 þ Ln2 n3 ; 8 2 2 a1 4 L L 4 i¼4 i 2 3 2L where h2 ¼ h1 þ 12 c21 h2 . When l ¼ 3; . . . ; n, we will prove the following conclusion by induction that the virtual controller (8) can guarantee that the time derivative of the augmented function V l defined in (12) satisfies
L L d1 d1 2 n H kek2 d1ðl1Þ l1 hl n21 d2ðl1Þ l2 c22 d1 2 a1 2 2 2 2 ! l l1 l1 c 2 X X X c2 c2 L_ L_ L a1 ðiþ1Þj diðl1Þ li i þ 1 d1ði1Þ n2i dðiþ1Þðl1Þ ðdi i 1 þ Þ n2i n2l d1ðl1Þ 2 ^x21 L L 2 4 d ðiþ1Þj 2L 2 i¼3 i¼1 j¼iþ1
V_ l 6 d1ðl1Þ
þ
L
lþ1
n d1ðl1Þ X d1ðl1Þ 2 z2 þ nlþ1 þ Lnl nlþ1 ; 4 i¼lþ2 i 2
ð24Þ
2
where hl ¼ hl1 þ h2 c21ðl1Þ . In fact, it is assumed that (24) holds when l 1 replace l. Differentiating V l yields that V_ l ¼ dl1 V_ l1 þ nl n_ l where
! ! L_ L_ L_ n_ l ¼ Lzlþ1 al ^x1 l nl þ cl1 Lzl al1 ^x1 þ nl1 þ cðk2Þðl1Þ Lzl1 al2 ^x1 þ nl2 þ L L L ! g L_ þ sgn ðgÞ c1ðl1Þ Lg e2 þ Lgz2 þ /1l þ n1 : L L Similar to Step 2, according to (3), (5), (9), the definition of ei ði ¼ 1; . . . ; nÞ; zi ði ¼ 1; . . . ; nÞ and the fact that di > 1ði ¼ 1; . . . ; l 1Þ , by completing the square, we obtain
! !2 l1 l1 X X d1ðl1Þ 2 d1ðl1Þ dl1 2 3 l2 Lnl1 þ dl1 Ln2l ; al ai ci ^x1 nl 6 Ln þ 2 a þ a c Ln2l þ lþ2 Lkek2 þ 2l Ln2l ; i i l 1 2 6 2k 2 i¼2 i¼2 3 3 1 2 2 cl1 Lzl nl 6 cl1 1 þ cl1 Lnl þ dl1 Lnl1 ; 2 6 l2 l2
l2 X X X diðl1Þ 2 d2ðl1Þ 2 1 l3i 2 ciðl1Þ Lziþ1 nl 3 2 ciðl1Þ ð1 þ c2i Þ Ln2l þ dl1 Ln2l1 þ Lni þ Ln2 ; li 6 3 2l3 i¼2 i¼2 i¼3 2 l1 l1 X L_ L_ L_ L_ 1X ciðl1Þ ni nl 6 c2iðl1Þ n2i þ ðl 1Þ n2l ; sgnðgÞ c1ðl1Þ n1 nl þ L L L L 4 i¼2 i¼1 d1ðk1Þ sgnðgÞ c1ðl1Þ Lg e2 nl 6 lþ2 Lkek2 þ 2l g 2M c21ðl1Þ Ln2l ; 2 d1ðl1Þ 2 d2ðl1Þ 2 3 l2 2 2 Ln1 þ Ln2 ; sgnðgÞ c1ðl1Þ Lgz2 nl 2 g M c1ðl1Þ þ c21 Ln2l þ 4 2l 6 2l3 d1ðl1Þ 2 d1 2 2 d1ðl1Þ 2 g d1 nl 6 h c1ðl1Þ n21 þ nl ; sgnðgÞ c1ðl1Þ /1l nl 6 h1l c1ðl1Þ jn1 j jnl j 6 h21k c21ðl1Þ n21 þ L 2 2 2 2
dl1 Lnl1 nl 6
557
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
1 1 1 d1ðl1Þ z2lþ1 6 d1ðl1Þ c2l n2l þ d1ðl1Þ n2lþ1 : 4 2 2
ð25Þ
It is also easy to obtain
L_ L_ n_ n ¼ z_ n þ cn1 n_ n1 ¼ Lcn nn an ^x1 n nn þ ncn1 nn1 ; L L
L_ L_ nn nn ¼ Lcn n2n an ^x1 nn n n2n þ ncn1 nn1 nn : L L
It is not difficult to obtain the following
L L d1 2 d1 2 2 2 n n H e d h d c d k k 1ðn1Þ n 2ðn1Þ 1 1 2 a1 2 2nþ1 2n1 2n2 2 ! ! 2 n n1 n1 c 2 X X X c c2 L_ 2 L_ 2 L a1 ðiþ1Þj diðn1Þ ni i þ 1 d1ði1Þ n2i dðiþ1Þðn1Þ di i 1 þ ni nn d1ðn1Þ z21 : L L 2 4 d 2 2 i¼3 i¼1 j¼iþ1 ðiþ1Þj
V_ n 6 d1ðn1Þ
L
ð26Þ n
n
In view of (26), it is not difficult to prove that, starting from any initial condition ðeð0Þ; zð0ÞÞ 2 R R and Lð0Þ ¼ 1 , the closed-looped system (13) has the properties: (i) the corresponding solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) existing on ½0; þ1Þ is unique and bounded; (ii) limt!þ1 ðzðtÞ; eðtÞÞ ¼ 0; limt!þ1 LðtÞ ¼ L 2 Rþ . Recall that the solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) exists and is unique on the maximally extended interval ½0; T f Þ. Thus conclusions (i) and (ii) follow immediately if one can prove that eðtÞ; zðtÞ and LðtÞ are bounded on ½0; T f Þ. This can be done by a contradiction argument. Suppose limt!T f sup ðzðtÞ; eðtÞ; LðtÞÞT ¼ þ1 , We first claim that LðtÞ cannot escape at t ¼ T f . To 2
^ x prove this claim, suppose that limt!T f LðtÞ ¼ þ1 , since L_ ¼ n21 þ L21 þ n2n P 0 , LðtÞ is a monotone non-decreasing function. From the Lemma 1 below in this paper we know that there exists a finite time t 2 ½0; T f Þ, such that V_ 6 cðkek2 þ knk2 þ z2 Þ. 1
As a consequence,
þ1 ¼ LðT f Þ Lðt Þ ¼
Z
Tf
_ LðtÞdt ¼
t
Z
Tf t
ðn21 ðtÞ þ n2n ðtÞ þ z21 ðtÞÞdt 6
Vðt Þ < þ1: c
This contradiction shows that L is bounded on ½0; T f Þ. Next, we claim that z is well defined and bounded on the interval ½0; T f Þ. To see why, consider the Lyapunov function V z ¼ zT Pz for the z-dynamic system of (14). Clearly, a direct computation gives
1 1 L_ _ V_ z ¼ LzT ðAT P þ PAÞz 2Lcn zT Pen nn zT ðDP þ PDÞz 6 Lkzk2 þ 2Lc2n kP k2 n2 6 kzk2 þ 2c2n kPk2 LL: 2 2 L This, in turn, leads to 8t 2 ½0; T f Þ, kmin ðPÞkzðtÞk2 zð0ÞT Pzð0Þ 6 zt ðtÞPzð0Þ 6 12 that 2 kzðtÞk 6
R X
ð27Þ
kzðtÞk2 dt þ c2n kP k2 L2 Therefore, it follows
1 ðzT ð0ÞPzð0Þ þ c2n kPk2 L2 Þ; 8t 2 ½0; T f Þ; kmin ðPÞ
and
Z X
2 kzðtÞk dt 2 zT ð0ÞPzð0Þ þ c2n kPk2 L2 < þ1;
where L denotes the supremum of L on X .i.e., L ¼ supt2X LðtÞ since L is bounded on ½0; T f Þ , the inequalities above imply boundedness of z. Finally, we prove that e is bounded on ½0; T f Þ . To this end, we introduce the following change of coordinates
ei ¼
xi ^xi ðL Þ
i
;
i ¼ 1; . . . ; n;
where L is a constant satisfying
L P maxfL; H þ 4g:
ð28Þ
Then, the error dynamic (13) is transformed into
e_ ¼ L Ae þ L ae1 LC1 ae1 þ C2 ax1 þ Uk ; where
e ¼ ½e1 . . . en T ;
ð29Þ
558
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
(
C1 ¼ diag 1;
C2 ¼ diag
n1 ) L L ; ; . . . ; L L
n L L ; ;...; L L
"
UrðtÞ ¼
U1rðtÞ U2rðtÞ ;
L
2
ðL Þ
...
UnrðtÞ
#T :
n
ðL Þ
Along the solutions of (29), differentiating the function T V e ¼ eT Pe on X yields V_ e 6 L kek2 þ 2e1 L aT Pe 2e1 LaT C1 Pe þ 2x1 aT C2 Pe þ 2Uk Pe. By completing the square, we get
2 2 2e1 L aT Pe 6 ðL Þ aT P e21 þ kek2 ; 2 2e1 LaT C1 Pe 6 L2 aT C1 P e21 þ kek2 ; 2 2 2x1 aT C2 Pe 6 aT C2 P x21 þ kek2 6 aT C2 P L2 n21 þ kek2 ; T 2UrðtÞ Pe 6 Hkek2 þ
n 1X 1 z2 þ n 2 ; 4 i¼2 i 2 1
which, together with (28), means that on X,
1 V_ e 6 ðL H 3Þkek2 þ 4 6 kek2 þ
n
X 2 2 2 1 n21 þ ðL2 aT P þ L2 aT C1 P e21 z2i þ aT C2 P L2 þ 2 i¼2
n 1X 1 þ H1 L_ þ ðsupt2X kzðtÞkÞ2 ; z2 þ H1 n21 þ H1 z21 6 4 i¼2 i kmaxðPÞ
ð30Þ
with
H1 ¼
2 2 2 1 þ aT C2 P L2 þ 2ððL2 aT P þ L2 aT C1 P ; 2
from (30), we easily have
keðtÞk2 6
1 ðsupt2X kzðtÞkÞ2 kmax ðPÞ þ eT ð0ÞPeð0Þ þ H1 L; kmin ðPÞ
8t 2 X;
and
Z
kbar eðtÞk2 6 eT ð0ÞPeð0Þ þ
Z
X
kzðtÞk2 þ H1 L < þ1:
X
This, together with the definitions of ei ði ¼ 1; . . . ; nÞ and ei ði ¼ 1; . . . ; nÞ and the boundedness of L, implies that e is bounded on R 2 X keðtÞk < þ1. On the other hand, using the boundedness of ðL; e; zÞ on ½0; þ1Þ, it is straightforward to deduce that e_ and z_ are bounded R þ1 R þ1 lat Lemma [6], we have limt!þ1 zðtÞ ¼ 0 on the ½0; þ1Þ. Noticing that 0 keðtÞk2 < þ1 and 0 kzðtÞk2 dt < þ1, by the Barba
X and
and limt!þ1 eðtÞ ¼ 0. 3.3. Stabilization theorem From (25), we can see that to realize global stabilization of the closed-loop switched system by output feedback, the controllers designed should ensure the negative definition of the time derivative of V n . This can be guaranteed by choosing the appropriate di P 1; i ¼ 1; . . . ; n satisfying (11) and c; h such that n1 c 2 X c2 ðiþ1Þj di i 1 þ 4 d j¼iþ1 ðiþ1Þj
c6 and
a1 ; 2
c6
1 2nþ1
;
! > 0;
i ¼ 1; . . . ; n 1;
ð31Þ
ð32Þ
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
h P H; h P hn ; h P d1 c2 þ d1 þ d1 ; 2 2 a1
2
c 2 c3 i h P 2 þ 1 d12 ; þ 1 d1ði1Þ ; i ¼ 3; . . . ; n; 2
559
ð33Þ
meanwhile,
diðn1Þ ¼ di diþ1 . . . dn1 P 1;
i ¼ 1; 2; . . . ; n 1;
ð34Þ
since it is easily obtained di P 1; i ¼ 1; 2; . . . ; n from (11). For these constants di P 1; i ¼ 1; . . . ; n; c; h, we have to verify the following Lemma. Lemma 1. There exist constants di P 1; i ¼ 1; . . . ; n; c; h satisfying inequality V_ 6 cðkek2 þ knk2 Þ cz21 . Proof. We prove the Lemma by construction. First of all, we choose
c ¼ min
a1 1 ; nþ1 : 2 2
ð35Þ
Next, we choose
d1 d1 c 2 h ¼ max H; hn ; c22 þ d1 þ ; i þ 1 d1ði1Þ ; i ¼ 3; . . . ; n : 2 a1 2
ð36Þ
Then we can obtain from (26)
V_ n 6 ðc hÞkek2 ðc hÞn21 ðc hÞn22
n
X ðc hÞn2i cz21 ¼ ðc hÞ kek2 þ knn k2 cz21 : i¼3
We have known that 0; T f is the maximal interval X on which a unique closed-loop system solution exists, where 0 < T f 6 þ1 (see Theorem 2.1page 17, [3]). Since L is a continuous and monotone nondecreasing function, there exists a finite time t 2 0; T f ; 8t 2 ½t ; T f Þ , then from (26), it follows that on ½t ; tf Þ we have such that LðtÞ P hþc c _V 6 cðkek2 þ knk2 Þ cz2 . 1
It is easy to verify that any constants di P 1; c; h ði ¼ 1; . . . ; nÞ to be de designed by (35), (36) satisfy (32), (33). By Lemma 1, if we choose di P 1; c; hði ¼ 1; . . . ; nÞ satisfy (35), (36), the right-hand side of (26) becomes negative definite, which indicates that V n ðxÞ is qualified as a CLF for the system (4). Therefore, the closed-loop switched system (4), (7), (8) and (9) which implies that, by the coordinate transformation (3), the closed-loop switched system (1), (7), (8) and (9) is globally asymptotically stable under arbitrary switchings. So far, the entire design procedure has been completed. Global stabilization of the closed-loop switched system by output feedback is summarized in the following result.
Theorem 1. Under Assumptions 1 and 2, there exist output feedback controllers (8) with positive constants di P 1; c; hði ¼ 1; . . . ; nÞ satisfying (10), (11), (32) and (33), such that the closed-loop switched system (1), (7), (8) and (9), for any initial condition, is globally asymptotically stable under arbitrary switchings. Remark 3. In this paper, a key point is to construct the common coordinate transformation in order to effectively avoid different coordinate transformations for different subsystems when applying backstepping. This is realized by design of an appropriate state observer of the switched system studied. For switched nonlinear systems in triangular form, we investigate the global stabilization problem of the system (4) based on the assumption that only the system output is measurable, while [15,19,31] present some effective methods of the global stabilization problem based on the assumption that full-state measurements are available. However, these methods in [15,19] cannot be used in the system studied in this paper. Meanwhile, in order to reduce the conservativeness caused by adoption of a common observer for all subsystems, we need to design an appropriate switched observer for the system studied. In this paper, a switched observer (7) is constructed for the switched system (1). Based on the switched observer designed (7), we construct the output-feedback controller of individual subsystem by using backstepping. Remark 4. When we study switched systems, we need to handle each subsystem by applying some non-switched or switched system theory in the existing literature, which is commonly used approach (e.g., see [13,15]). Thus, in the proof procedure of Theorem 1, for each subsystem, we adopt a similar method of the non-switch system literature [33]. However, the method of [33] cannot give any results on the topic in this paper even if [33] presents an effective method to achieve the stability of non-switched systems. For example, a common coordinate transformation for all subsystems of the switched system under study has been constructed to avoid different coordinate transformations for different subsystems when applying the backstepping technique.
560
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
4. Example In this section, we present two illustrative examples to demonstrate the applicability and effectiveness of the proposed approach. 4.1. Example 1
8 > < g_ 1 g_ 2 > : y
¼ g 1 g2 þ DhrðtÞ ðg1 ; g2 Þ; ¼ g 2 urðtÞ þ Dg rðtÞ ðg1 ; g2 Þ;
ð37Þ
¼ g1 : g g g2
g 1 g 2 g1 where Mh1 ðg1 ; g2 Þ ¼ g2 ðð1d ; Mh2 ðg1 ; g2 Þ ¼ ð1þ2 g12 Þg22 ; Mg 1 ðg1 ; g2 Þ ¼ g Þ2 þg2 Þ 1 2
2d2
lnð1þg2
g2
2
2
Þg 1 g 2
g2 ; Mg 2 ðg1 ; g2 Þ ¼ g2 g2gsin ; d1 ; d2 are unknown 2
constants. Under the same kind of transformation (3) for the above system, this system becomes
8 _ > < x1 x_ 2 > : y
¼ x2 þ Mf rðtÞ ðx1 ; x2 Þ; ¼ g 2 urðtÞ þ Mg rðtÞ ðx1 ; x2 Þ;
ð38Þ
¼ gx1 ;
As a matter of fact, the following estimations
g 2 ð1 þ d2 Þjg j g 1 g 2 g1 1 1 ; 6 2 2 2 2 g ðð1 d1 g Þ þ g Þ g 2 2 lnð1 þ g2d2 Þg g g 2 ð2d 1Þjg j 2 1 2 2 2 ; 6 g2 g2 g g g2 gjg j 2 1 2 6 21 ; ð1 þ g22 Þg 2 g
g g sing gjg j 2 2 2 6 22 ; g2 g
can be easily obtained. Hence, Assumption 1 holds with d ¼
g 2 maxf2d2 1;1þd21 g , g2
where d is an unknown constant. Suppose that
control coefficients g 1 and g 2 satisfy 0:9 6 g i 1:5; i ¼ 1; 2 and choose a1 ¼ 10; a2 ¼ 1; d1 ¼ 1; d2 ¼ 2, then g ¼ 0:9 and g ¼ 1:5 , g M can be chosen as g M ¼ 2:7778 and there exists a positive definite matrix
P¼
0:1000
0:5000
0:5000
5:1000
such that the matrix inequality (6) holds, moreover, in view of (10) and (11), by simple calculation, we choose
g 1 ¼ 1;
g 2 ¼ 1;
c1 ¼ 9:237;
c2 ¼ 87183;
d1 ¼ 22:3305:
Thus, by Theorem 1 and the design procedure provided in Section 3, we design the observer as:
¼ ^x_ 2 La1 ^x1 ; ¼ g 2 urðtÞ L2 a2 ^x1 ; ¼
y2 L2
^x2 1
þ L2 þ
n22 ; Lð0Þ
ð39Þ ¼ 1;
2
states
8_ ^x > > < 1 ^x_ 2 > > :_ L
1 0 −1 0
10
time(sec)
Fig. 1. State responses: x1 (
20 ) and ^ x1 ð Þ.
30
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
states
10 0 −10 −20 0
10
time(sec)
Fig. 2. State responses: x2 (
20
30
) and ^ x2 ð Þ.
3
L
2.5 2 1.5 1 0
10
time(sec)
20
30
Fig. 3. The trajectory of the high gain L of Example 1.
Fig. 4. Schematic diagram of the process.
states
2 1 0 −1 0
10
20 time(sec)
Fig. 5. State responses: x1 (
) and ^ x1 ð Þ.
30
561
562
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
and the controller is
u ¼ g 2 L3 c2 n2 ;
ð40Þ
n2 is given by
^x2
n2 ¼
L
2
þ sgnðgÞc1 n1 ¼
^x2
y þ sgnðgÞc1 : L L
ð41Þ
2
T x1 ð0Þ; ^ x2 ð0Þ; Lð0ÞÞ ¼ ð1; 5; 1:5; 9:24; 1Þ and The results are depicted in Figs. 1–3 with the initial conditions ðx1 ð0Þ; x2 ð0Þ; ^ the switching time between switching the system on and off: 0.001s, which shows the asymptotic stability of the closed-loop system (37), (39) and (40). Thus, the simulation results illustrate the theory presented.
4.2. Example 2. Next, we apply the proposed method to a practical example, namely a continuous stirred tank reactor(CSTR) with two modes feed stream(see Fig. 4), which is molded as a switched nonlinear system [22,37,38]: E C_ A ¼ qVr ðC Afr C A Þ a0 eRT C A ; E T_ ¼ qVr ðT f r TÞ a1 eRT C A þ VUA ðT c TÞ: q Cq
The physical meaning of the parameters of this system can be easily found in [37,38]. Under a coordinate transformation and smooth feedback [37,38], the new system is
8 > < g_ 1 g_ 2 > : y
¼ g 1 g2 þ hrðtÞ ðg1 Þ; ¼ g 2 urðtÞ ;
ð42Þ
¼ g1 ;
where h1 ¼ 0:5gg12g2 x1 ; h2 ¼ 2gg22x1 ; g ¼ gg1 g2 2 . Furthermore, we make the same coordinate transformation (3) about the system (41), which does not affect the stabilization objective. We obtain
8 _ > < x1 x_ 2 > : y
¼ x2 þ f rðtÞ ðx1 Þ; ¼ g 2 urðtÞ ;
ð43Þ
¼ gx1 ;
where f 1 ¼ 0:5x1 ; f 2 ¼ 2x1 ; g ¼ g1gg2 2 . Then the last Eqs. (43) have the same form and dimension as Eqs. (37), therefore, we may as well design the same controller and observer as Example 1. It is easy to prove that the Eqs. (41) satisfy the first assumption above in this paper. For the second assumption, we still suppose that g ¼ 0:9 and g ¼ 1:5 thus the second assumption also holds. We also choose the same parameters such as a1 ¼ 10; a2 ¼ 1 , then, there exists the same positive definite matrix
P¼
0:1000
0:5000
0:5000
5:1000
such that the matrix inequality (6) holds, moreover, in view of (10) and (11), by simple calculation, we choose
c1 ¼ 9:5;
c2 ¼ 97364;
d1 ¼ 23:5625: T
x1 ð0Þ; ^ x2 ð0Þ; Lð0ÞÞ ¼ ð1:5; 6:5; 2; 10; 1Þ; The results are depicted in Figs. 5–7 with the initial conditions ðx1 ð0Þ; x2 ð0Þ; ^ g 1 ¼ 1; g 2 ¼ 1 and the switching time between switching the system on and off: 0.001s, which shows the asymptotic stability of the closed-loop system (39), (40) and (42). Thus, the simulation results illustrate the theory presented.
10
states
0 −10 −20 −30 0
10
time(sec)
Fig. 6. State responses: x2 (
20 ) and ^ x2 ð Þ.
30
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
563
4
L
3 2 1 0
10
time(sec)
20
30
Fig. 7. The trajectory of the high gain L of Example 2.
Remark 5. The continuous stirred tank reactor has been studied from the point of view of switching control in [37,38]. These results above are based on the assumption that full state measurements are available. However, this assumption above does not hold in many practical systems. In this paper, we consider full state measurements are unavailable for the continuous stirred tank reactor, and only the system output is available for measurement. Clearly, these methods in [37,38] cannot be used in the continuous stirred tank reactor in this paper. But, based on the approach proposed in this paper, the outputfeedback control problem of the continuous stirred tank reactor is solvable.
5. Conclusion As a further investigation based on [17,33], the global asymptotic stabilization by output feedback has been investigated for a class of uncertain nonlinear switched systems with unmeasured states dependent growth. Based on backstepping and CLF, this paper has developed a constructive design approach for designing the output-feedback controllers of individual subsystems, which guarantees global asymptotic stability of the corresponding closed-loop switched system under arbitrary switchings. Finally, It was demonstrated, by means of a numerical example and its simulation, that the proposed adaptive output feedback controller can be easily designed and implemented. Our future work will be directed at extending the control design method to more general nonlinear switched systems, such as the switched systems with unknown control coefficients, and at obtaining the global (practical) tracking result for system. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grants 61304058 and 61233002, andIAPI Fundamental Research Funds under Grant 2013ZCX03–02, and Fundamental Research Funds for the Central Universities under Grant N130404026, and China Postdoctoral Science Foundation under Grant 2013M540231. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Zhiyong Chen, Jie Huang, Global robust servomechanism of a class of nonlinear systems using output feedback, Asian J. Control 9 (3) (2007) 292–305. Farzad Esfandiari, Hassan K Khalil, Output feedback stabilization of fully linearizable systems, Int. J. Control 56 (5) (1992) 1007–1037. J.K. Hale, second ed., Ordinary Differential Equations, Huntington, Krieger, 1980. Thanh-Trung Han, Shuzhi Sam Ge, Tong Heng Lee, Adaptive neural control for a class of switched nonlinear systems, Syst. Control Lett. 58 (2009) 109– 118. Lei Hao, Lin Wei, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica 42 (10) (2006) 1783– 1789. H.K. Khalil, 3rd ed., Nonlinear Systems, Prentice Hall, Englewood Cliff, 2002. Zhong-Ping Jiang, I. Mareels, D.J. Hill, Jie Huang, A unifying framework for global regulation via nonlinear output feedback: from iss to iiss, IEEE Trans. Autom. Control 49 (4) (2004) 549–562. Ying Jin, Jun Fu, Youmin Zhang, Yuanwei Jing, Reliable control of a class of switched cascade nonlinear systems with its application to flight control, Nonlinear Anal. Hybrid Syst. 11 (2014) 11–21. Prashanth Krishnamurthy, Farshad Khorrami, Ramu Sharat Chandra, Global high-gain-based observer and backstepping controller for generalized output-feedback canonical form, IEEE Trans. Autom. Control 48 (12) (2003) 2277–2283. Miroslav Krstic, Petar V. KokotoviC, Adaptive nonlinear output-feedback schemes with marino-tomei controller, IEEE Trans. Autom. Control 41 (2) (1996) 274–280. Miroslav Krstic, Petar V. Kokotovic, Ioannis Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc, 1995. Xiao-Ling Liang, Ming-Zhe Hou, Guang-Ren Duan, Output feedback stabilization of switched stochastic nonlinear systems under arbitrary switchings, Int. J. Autom. Comput. 10 (6) (2013) 571–577. Daniel Liberzon, A. Stephen Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. 19 (5) (1999) 59–70. Chang-Liang Liu, Shao-Cheng Tong, Yong-Ming Li, Yuan-Qing Xia, Adaptive fuzzy backstepping output feedback control of nonlinear time-delay systems with unknown high-frequency gain sign, Int. J. Autom. Comput. 8 (1) (2011) 14–22. Lijun Long, Jun Zhao, Global stabilization for a class of switched nonlinear feedforward systems, Syst. Control Lett. 60 (9) (2011) 734–738. Lijun Long, Jun Zhao, H1 control of switched nonlinear systems in p -normal form using multiple lyapunov functions, IEEE Trans. Autom. Control 57 (5) (May 2012) 1285–1291. Lijun Long, Jun Zhao, Output-feedback stabilisation for a class of switched nonlinear systems with unknown control coefficients, Int. J. Control 86 (3) (2013) 386–395.
564
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
[18] Lijun Long, Jun Zhao, A small-gain theorem for switched interconnected nonlinear systems and its applications, IEEE Trans. Autom. Control 59 (4) (April 2014) 1082–1088. [19] Ruicheng Ma, Jun Zhao, Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings, Automatica 46 (11) (2010) 1819–1823. [20] Riccardo Marino, Patrizio Tomei, An adaptive output feedback control for a class of nonlinear systems with time-varying parameters, IEEE Trans. Autom. Control 44 (11) (1999) 2190–2194. [21] F. Mazenc, L. Praly, W.P. Dayawansa, Global stabilization by output feedback: examples and counterexamples, Syst. Control Lett. 23 (2) (1994) 119– 125. [22] P. Mhaskar, N.H. El-Farra, P.D. Christofides, Predictive control of switched nonlinear systems with scheduled mode transitions, IEEE Trans. Autom. Control 50 (11) (Nov 2005) 1670–1680. [23] Ying-Ying Min, Yun-Gang Liu. Barbalat lemma and its application in analysis of system stability. Journal of Shandong University (engineering science), 1:012, 2007. [24] Kokotovic´ Petar, Arcak Murat, Constructive nonlinear control: a historical perspective, Automatica 37 (5) (2001) 637–662. [25] Laurent Praly, Zhong-Ping Jiang, Linear output feedback with dynamic high gain for nonlinear systems, Syst. Control Lett. 53 (2) (2004) 107–116. [26] Chunjiang Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, in: Proceedings of the American Control Conference, 2005, IEEE, 2005, pp. 4708–4715. [27] Chunjiang Qian, Wei Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Trans. Autom. Control 47 (10) (2002) 1710–1715. [28] Min Wang, Jun Zhao, Georgi M. Dimirovski, Dynamic output feedback robust H1 control of uncertain switched nonlinear systems, Int. J. Control Autom. Syst. 9 (1) (2011) 1–8. [29] Rui Wang, Gang Jin, Jun Zhao, Robust fault-tolerant control for a class of switched nonlinear systems in lower triangular form, Asian J. Control 9 (1) (2007) 68–72. [30] Chunling Wei, Qiangde Wang, Global output feedback asymptotic stabilization for a class of nonlinear systems, in: 29th Chinese Control Conference (CCC), IEEE, 2010, pp. 369–374. [31] Wu. Jenq-Lang, Stabilizing controllers design for switched nonlinear systems in strict-feedback form, Automatica 45 (4) (2009) 1092–1096. [32] Guangming Xie, Long Wang, Controllability and stabilizability of switched linear-systems, Syst. Control Lett. 48 (2) (2003) 135–155. [33] Xuehua Yan, Yungang Liu, New results on global output-feedback stabilization for nonlinear systems with unknown growth rate, J. Control Theory Appl. 11 (3) (2013) 401–408. [34] Guang-Hong Yang, Wei-Wei Che, Non-fragile H1 filter design for linear continuous-time systems, Automatica 44 (11) (2008) 2849–2856. [35] H. Yang, B. Jiang, M. Staroswiecki, Observer-based fault-tolerant control for a class of switched nonlinear systems, IET Control Theory Appl. 1 (5) (2007) 1523–1532. [36] Hao Yang, Bin Jiang, Vincent Cocquempot, Huaguang Zhang, Stabilization of switched nonlinear systems with all unstable modes: application to multiagent systems, IEEE Trans. Autom. Control 56 (9) (2011) 2230–2235. [37] M.B. Yazdi, M.R. Jahed-Motlagh, Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization, Chem. Eng. J. 155 (3) (2009) 838–843. [38] M.B. Yazdi, M.R. Jahed-Motlagh, S.A. Attia, J. Raisch. Switched feedback equivalence of a class of planar switched nonlinear systems. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. in: Proceedings of the 48th IEEE Conference on, pages 4264– 4269. IEEE, Dec 2009. [39] Evren Yurtseven, W.P.M.H. Heemels, M. Kanat Camlibel, Disturbance decoupling of switched linear systems, Syst. Control Lett. 61 (1) (2012) 69–78. [40] J. Zhai, S. Fei, Global practical tracking control for a class of uncertain non-linear systems, IET Control Theory Appl. 5 (11) (2011) 1343–1351. [41] Junyong Zhai, Chunjiang Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlinear Control 22 (14) (2012) 1543–1561. [42] Jun Zhao, David J. Hill, Dissipativity theory for switched systems, IEEE Trans. Autom. Control 53 (4) (2008) 941–953. [43] Jing Zhou, Changyun Wen, Ying Zhang, Adaptive output control of nonlinear systems with uncertain dead-zone nonlinearity, IEEE Trans. Autom. Control 51 (3) (2006) 504–511.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Global output-feedback stabilization for a class of switched uncertain nonlinear systems Erpeng Li, Lijun Long ⇑, Jun Zhao State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, PR China
a r t i c l e
i n f o
Keywords: Uncertain nonlinear systems Output feedback Common Lyapunov function (CLF) Switched systems
a b s t r a c t This paper is mainly concerned with the problem of global output feedback stabilization for a class of switched nonlinear systems with uncertain control coefficients. First of all, it is shown that the unknown control coefficients are lumped together via a common coordinate transformation, thus the original switched nonlinear system is transformed into a new switched system for which control design is feasible. Second, by constructing a common Lyapunov function relying on the designed state observer, the common output feedback controller independent of switching signals is designed based on the backstepping method so as to guarantee the global asymptotic stability of the resulting closed-loop switched system under arbitrary switchings. Two examples are included for verifying the effectiveness of the method proposed. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Global output-feedback stabilization of non-switched nonlinear systems with linearly unmeasured states dependent growth is an important topic in control theory and has been extensively investigated over the last decade [5,9– 11,14,20,23,24,30,33,43]. However, for switched nonlinear systems with linearly unmeasured states dependent growth, the global output-feedback stabilization problem has been limitedly investigated in the past few years [12,13,15,17,19,28,31]. It is well-known that the output feedback control is one of the most important problems of nonlinear systems since only the plant output can be measured in many cases [1,2,21,27,26,40,41]. Generally speaking, the separation principle does not work for nonlinear systems which makes the design of output feedback control more complicated and difficult [12]. Furthermore, a more challenging issue for output feedback in the field of nonlinear control is global stabilization [10,20]. The nonlinear systems in output feedback form are a kind of significant nonlinear systems. The feedback controllers of such a kind of nonlinear systems can be constructed systematically by the backstepping approach and its variations [9,11,14,18,24,43]. On the other hand, switched systems have been attracting a great deal of attention in the past decades due to the high performance requirements of control systems for handling nonlinearities, uncertainties and operating condition variations and also fault-induced dynamic changes [16,29,32,34–36,39,42]. A switched system is a kind of hybrid systems that comprises a collection of subsystems together with a switching rule that specifies the switching among these subsystems [28]. Stability is the first requirement for a system to work normally; thus, stability of switched systems is also the first and important task in researches on switched systems [8]. Due to the interaction between continuous dynamics and discrete ⇑ Corresponding author. E-mail addresses: [email protected] (L. Long), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.amc.2015.01.039 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
552
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
dynamics, switched systems may have a very complicated behavior. Consequently, the study of stability for switched systems is more difficult than the study for continuous systems or discrete systems, and so becomes a challenging issue. Moreover, a common Lyapunov function (CLF) for all subsystems guarantees stability under arbitrary switchings [13]. Naturally, how to construct a CLF to study stability is of great importance. Recently, numerous methods have been proposed to construct such a CLF. For example, global stabilization for switched nonlinear systems in triangular structure by state feedback under arbitrary switchings is achieved via backstepping or forwarding which provides a CLF [15,19,31]. Besides, in [4], an adaptive control scheme via state feedback for switched nonlinear systems in strict-feedback form was proposed for switching with a certain dwell-time. It is worth pointing out that the results [10,15,19,20,31] are based on the assumption that full state measurements are available. However, in many systems, this assumption does not hold and an output-feedback control scheme such as observer has to be used. However, to the best of our knowledge, outputfeedback stabilization of switched nonlinear systems with unknown control coefficients has only been studied in [17], due to the lack of effective tools and the complexity arising the impact of the system structure on switching. When we consider full state measurements are unavailable, the key problem is how to achieve the global stabilization via design of a proper state observer and output feedback controllers of subsystems which have to be based on state estimates ? In this paper, we consider the problem of global stabilization via output feedback for a class of switched nonlinear systems with unknown control coefficients by exploiting the CLF method and backstepping. It is assumed that the system studied admits a certain structure, which is much more general than strict-feedback form in switched nonlinear systems. The remarkableness of the paper, compared with the closely related works, can be shown from the following two aspects: Firstly, we investigate the global stabilization for a class of switched nonlinear systems with unknown control coefficients by output feedback. When considering the case where full state may not be measurable and thus state feedback is not implemented, we designed an appropriate state observer. Secondly, we also constructed a common coordinate transformation at each step of the backstepping. Based on the observer designed and the common coordinate transformation, we constructed output feedback controllers for subsystems via state estimates. Moreover, it is worth pointing out that, since these results [15,19,31] are based on the assumption that full state measurements are available. Their methods cannot be used in the system studied in this paper. The remainder of the paper is organized as follows. Section 2 formulates the system model and the control objective. Section 3 provides the globally stabilizing control design scheme via output feedback and summarizes the main results. In Section 4, a numerical example is given to illustrate the effectiveness of the theoretical results. The paper ends with some concluding remarks. 2. Problem formulation Consider a class of switched nonlinear systems with unknown control coefficients as follows:
8 > < g_ i ¼ g i giþ1 þ wirðtÞ ðt; g; urðtÞ Þ; i ¼ 1; . . . ; n 1; g_ n ¼ g n urðtÞ þ wnrðtÞ ðt; g; urðtÞ Þ; > : y ¼ g1 ;
ð1Þ
where g ¼ ½g1 ; . . . ; gn T 2 Rn is the system state vector with the initial value g0 ¼ gðt0 Þ; u 2 R and y 2 R are the control input and system output, respectively; g i – 0; i ¼ 1; . . . ; n are unknown constant called unknown control coefficients and wirðtÞ : ½t0 ; þ1 Rn R; i ¼ 1; . . . ; n:rðtÞ : ½0; þ1Þ ! M ¼ f1; 2; . . . ; mg is the switching signals which is assumed to be a piecewise continuous (from the right) function of time; In what follows, suppose only the system output is measurable and available for feedback, and for notational simplicity, let t 0 ¼ 0. More specifically, the goal of the paper is to design output-feedback controllers for subsystems and state observer which globally stabilizes the system (1) under arbitrary switchings. To this end, the following two assumptions are imposed on the system (1). Assumption 1. There exist some unknown positive constants h0k ; k 2 M such that
jwi k ðt; g; uk Þj 6 h0k ðjg1 j þ jg2 j þ . . . þ jgi jÞ;
i ¼ 1; . . . ; n; 8k 2 M:
ð2Þ
Assumption 2. The signs of g i ; i ¼ 1; . . . ; n are known, and there exist known positive constants g and g satisfying
g 6 g i 6 g;
i ¼ 1; . . . ; n:
Remark 1. From Assumptions 1 and 2, it is easy to see that system (1) has uncertain control coefficients and unmeasured states dependent nonlinearities. Assumption 1 is a linear growth condition. It gives some information about the nonlinearities. Assumption 2 is a standard assumption in the study of non-switched nonlinear systems. Assumption 2 means that the known signs of g i play an important role in controller design. Otherwise, we cannot decide the direction along which the control designs, and the closed-loop system may be unstable.
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
553
3. Main results We introduce the following scaling transformation for the system (1):
xi ¼
gn g; g in i
uik ¼
gn w ; g in ik
i ¼ 1; . . . ; n;
ð3Þ
where
g in ¼
n Y
gj ;
i ¼ 1; . . . ; n;
cij ¼
j¼i
dij ¼
i Y
j Y
ck ;
1 6 i 6 j 6 n;
k¼i
dk ; 1 6 i 6 j 6 n;
c2ij ¼
j Y
c2k ;
1 6 i 6 j 6 n:
k¼i
k¼i
Under this transformation, the system (1) becomes
8 > < x_ i ¼ xiþ1 þ /irðtÞ ðt; x; urðtÞ Þ; i ¼ 1; . . . ; n 1; x_ n ¼ g n urðtÞ þ /nrðtÞ ðt; x; urðtÞ Þ; > : y ¼ gx1 ;
ð4Þ
where g ¼ gg1n n and h ¼ maxfh1k g; k 2 M. Since h1k is an unknown constant, h is also an unknown constant. By Assumption 1 and 2, it is easy to find a known positive constant g M P 1 such that
8 > < 1 6 jg j 6 g M ; j/ik j 6 h1k ðjx1 j þ jx2 j þ . . . þ jxi jÞ > : 6 hðjx1 j þ jx2 j þ . . . þ jxi jÞ; i ¼ 1; 2; . . . ; n:
ð5Þ
Remark 2. It is necessary to stress that the stabilization of system (1) is implied by that of transformed system (4) which has known control coefficients and is more convenient to design an observer. 3.1. Observer design Suppose that ai > 0ði ¼ 1; . . . ; nÞ are coefficients of the Hurwitz polynomial sn þ a1 sn1 þ þ an1 s þ an , and then in terms of Lemma 1 in [25], we can choose a ¼ ða1 ; a2 ; . . . ; an ÞT such that
AT P þ PA 6 I;
DP þ PD P 0;
ð6Þ
where
3 a1 1 0 6 . .. .. 7 6 .. . . 7 7 2 Rnn ; A¼6 7 6 4 an1 1 0 5 2
D ¼ diagf1; 2; . . . ; ng;
an 1 0 P is a symmetric and positive matrix, I denotes the n-dimensional identity matrix. Motivated by [5,7,17], the dynamic high-gain observer, and the observer-based output-feedback controller are designed as
8_ ^x ¼ ^x_ iþ1 Li ai ^x1 ; i ¼ 1; . . . ; n 1; > > < i ^x_ n ¼ g n urðtÞ Ln an ^x1 ; > > 2 ^x2 : L_ ¼ yL2 þ L12 þ n2n ; Lð0Þ ¼ 1;
ð7Þ
uk ¼ g n Lnþ1 an :
ð8Þ
and
nn is given by
8 n ¼ y; > > > 1 L > < n ¼ ^xi a ; i ¼ 2; . . . ; n; i1 i Li > a1 ¼ sgnðgÞc1 n1 ; > > > : ai ¼ c1 ni ; i ¼ 2; . . . ; n:
ð9Þ
554
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
and denote
ei ¼ xi Li ^xi ; zi ¼ ^xLii ; i ¼ 1; . . . ; n; ^xi ¼ ½^x1 ; . . . ; ^xn T ;
T
n ¼ ½^n1 ; . . . ; ^nn ;
e ¼ ½e1 ; . . . ; en T ; z ¼ ½z1 ; . . . ; zn T : Actually, in the paper, ci ði ¼ 1; . . . ; nÞ are required to satisfy the following inequalities:
8 c1 P 1 þ 2kPak2 þ g 2M ; > > > > > 3a2 > > c2 P 5 þ 32 g 2M d1 þ 22 þ 4g 2M c21 þ g M c1 þ 32 g 2M c41 ; > > > !2 > > i1 > X > > i i2 3 > P 1 þ 2 þ d þ 2 a þ a c c > i i j j 2 i1 < j¼2
> > > > > > > > > > > > > > > > > :
ð10Þ
i2 X þci1 ð1 þ 32 c3i1 Þ þ 3 2i3j ð1 þ c2j Þc2jði1Þ j¼2
2i4 g 2M c21ði1Þ
þ19
þ 2i2 g 2M c41 c22ði1Þ ;
i ¼ 3; . . . ; n;
where a ¼ ½a1 ; . . . ; an T and
8 d1 ¼ 1 þ 14 c21 ; > > > 8 > > > > > > < > > < > di ¼ 1 þ max 14 c2i ; > > > > > > > > > > : :
4dj c2j
9 > > > > = c2ji !! ; j ¼ 1; . . . ; i 1 ; i1 2 X > c > ðjþ1Þk > > 1þ dðjþ1Þði1Þ dðjþ1Þk ;
i ¼ 2; . . . ; n 1:
ð11Þ
k¼jþ1
3.2. Controller design By construction, we firstly construct V l ðÞ; l ¼ 1; . . . ; n , as
8 V ¼ eT P e; > < 0 V 1 ¼ V 0 þ 12 n21 þ 2L13 ^x21 ; > : V l ¼ dl1 V l1 þ 12 n2l ; l ¼ 2; . . . ; n;
ð12Þ
where P and dl satisfy (6) and (11), respectively. By view of (4), (7), (8), (9) and the definitions of ei ði ¼ 1; . . . ; nÞ and zi ði ¼ 1; . . . ; nÞ, we obtain the following switched system
8 > e_ ¼ LAe þ UrðtÞ ðt; x; urðtÞ ; LÞ þ ax1 LL_ ^x21 ; > < _ z_ ¼ LAz en Lcn nn LL Dz; > > :_ L ¼ n2 þ 12 ^x2 þ n2 ; Lð0Þ ¼ 1; 1
L
1
ð13Þ
n
where
UrðtÞ ¼
1 1 1 / ; / ; . . . ; /nrðtÞ L 1rðtÞ L 2rðtÞ L
T ;
en ¼ ½0; . . . ; 0; 1T 2 Rn : Observe that by construction, L_ P 0; Lð0Þ ¼ 1 and hence L P 1. As a result, the vector field of the closed-loop system (13) satisfies a local Lipschitz condition in a neighborhood of the initial condition ðeð0Þ; zð0Þ; Lð0ÞÞ 2 Rn Rn Rþ , and the corresponding solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) exists and is unique on ½0; T f Þ for some T f 2 ð0; þ1. Without loss of generality, we assume from now on that ½0; T f Þ is the maximally extended interval of the solution of (13). This fact will be used in the rest of the proof. Noting that L_ P 0 and L P 1, and in view of (6), the time derivative of V 0 satisfies
V_ 0 6 Lkek2 þ 2eT PUrðtÞ þ 2eT Pax1 : Observing that
ð14Þ
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
555
n X pffiffiffi UrðtÞ 6 UrðtÞ 6 nhjn1 j þ n nhkek þ nh jzi j; 1 i¼2
we have
2eT PUrðtÞ 6
n pffiffiffi 1 2 1X n1 þ 2n2 h2 kPk2 kek2 þ ð4n3 h2 kPk2 þ 2n nhkPkÞkek2 þ z2 ; 2 4 i¼2 i
ð15Þ
moreover, one has
L 1 2eT Pax1 ¼ 2eT P n1 6 Lkek2 þ 2kPak2 Ln21 ; g 2
ð16Þ
Therefore, according to (3), (15) and (16), one gets
n 1 1 1X þ 2kPak2 L n21 þ z2 ; V_ 0 6 L H kek2 þ 2 2 4 i¼2 i
ð17Þ
pffiffiffi where H ¼ 4n3 h2 kP k2 þ 2n nhkP k þ 2n2 h2 kP k2 . When l ¼ 1 , by (13) and (17), the time derivative of V 1 defined in (12) satisfies n
1X V_ 1 6 12 L H kek2 þ 12 þ 2kPak2 L n21 þ z2 4 i¼2 i
3L_ a1 1 _ þn1 Lg e2 þ Lgz2 þ gL /1k LL n1 4 ^x21 2 ^x21 þ ^x1 z2 : L 2L L
ð18Þ
By (5) and completing the square, we have
n1 Lg e2 6
L 2 L e þ Lg2 n21 6 kek2 þ Lg2M n21 ; 4 2 4
g n / 6 hn21 ; L 1 1k 1 a 1 2 ^x1 z2 6 12 ^x21 þ z2 : L 2a 1 2L
Substituting this inequality above into (18) yields
V_ 1 6 14 L H kek2 þ ððc1 1ÞL þ 12 þ hÞn21 n X _ a1 2 1 2 1 ^ LL n21 2L z2i þ Lgn1 z2 ; 2 x1 þ 2a z2 þ 4 1 i¼2
where c1 has been defined in (10). Moreover, by (9) and noting jg j P 1, we obtain
z22 2c21 n21 þ 2n22 ;
gn1 z2 6 gn1 n2 c1 n21 ;
and hence,
_ a1 2 ^ V_ 1 6 14 L H kek2 þ ðL h1 Þn21 LL n21 2L 2 x1 n
X þ 14 z2i þ 12 þ a11 n22 þ Lgn1 n2 ;
ð19Þ
i¼3
where h1 ¼ h þ 12 þ
1 2
þ a11 c21 . From (9) and (13), one can get
_ n_ 2 ¼ Lz3 a2 ^x1 2 LL n2 þ sgnðgÞc1
_ Lg e2 þ Lgz2 þ gL /1k þ LL n1 :
ð20Þ
Then, it follows from (12), (19) and (20) that
_ a1 2 ^ V_ 2 6 d1 14 L H kek2 ðL h1 Þn21 LL n21 2L 2 x1 ! n
X z2i þ 12 þ a11 n22 þ Lgn1 n2 þ n2 ðLz3 a2 ^x1 þ 14 i¼3 _ 2 LL n2
_
þ sgnðgÞc1 ðLg e2 þ Lgz2 þ gL /1k þ LL n1 ÞÞ:
ð21Þ
556
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
According to (3), (5), (9) and the definitions of e1 and z2 and d1 > 1, we have
8 Lgd1 n1 n2 6 d61 Ln21 þ 32 g 2M d1 Ln2 ; > > > > >
> > d1 > Lkek2 þ 4Ln22 ; > a2 ^x1 n2 ¼ La2 n2 ng1 e1 n22 6 d61 Ln21 þ 32 a22 Ln2 þ 16 > > > > > > d1 > < sgnðgÞ c1 Lg e2 n2 6 16 Lkek2 þ 4g 2M c21 Ln22 ; > 3g 2 > > sgnðgÞ c1 Lgz2 n2 ¼ sgnðgÞ c1 Lgn2 ðn2 sgnðgÞ c1 n1 Þ 6 ðg M c1 þ 2M c41 ÞLn22 þ d61 Ln21 ; > > > > > > > > sgnðgÞ c1 gL /1k n2 6 h1k c1 jn1 j jn2 j 6 d21 h21k c21 n21 þ d21 n22 6 d21 hc21 n21 þ d21 n22 ; > > > > > : c2 _ _ _ sgnðgÞ c1 LL n1 n2 6 41 LL n21 þ LL n22 :
ð22Þ
Besides, by (9) and the definition of z3 , we have
d d1 2 d1 2 d z 6 a þ n23 ¼ 1 c22 n22 þ 1 n23 : 4 3 2 2 2 2
ð23Þ
Besides, by (10), (22) and (23), (21) becomes
n L L d1 d1 2 c2 L 2 L 2 a1 d1 2 d1 X d1 V_ 2 6 d1 H kek2 d1 h2 n21 L c22 d1 n2 d1 1 n1 n2 2 ^x1 þ z2 þ n2 þ Ln2 n3 ; 8 2 2 a1 4 L L 4 i¼4 i 2 3 2L where h2 ¼ h1 þ 12 c21 h2 . When l ¼ 3; . . . ; n, we will prove the following conclusion by induction that the virtual controller (8) can guarantee that the time derivative of the augmented function V l defined in (12) satisfies
L L d1 d1 2 n H kek2 d1ðl1Þ l1 hl n21 d2ðl1Þ l2 c22 d1 2 a1 2 2 2 2 ! l l1 l1 c 2 X X X c2 c2 L_ L_ L a1 ðiþ1Þj diðl1Þ li i þ 1 d1ði1Þ n2i dðiþ1Þðl1Þ ðdi i 1 þ Þ n2i n2l d1ðl1Þ 2 ^x21 L L 2 4 d ðiþ1Þj 2L 2 i¼3 i¼1 j¼iþ1
V_ l 6 d1ðl1Þ
þ
L
lþ1
n d1ðl1Þ X d1ðl1Þ 2 z2 þ nlþ1 þ Lnl nlþ1 ; 4 i¼lþ2 i 2
ð24Þ
2
where hl ¼ hl1 þ h2 c21ðl1Þ . In fact, it is assumed that (24) holds when l 1 replace l. Differentiating V l yields that V_ l ¼ dl1 V_ l1 þ nl n_ l where
! ! L_ L_ L_ n_ l ¼ Lzlþ1 al ^x1 l nl þ cl1 Lzl al1 ^x1 þ nl1 þ cðk2Þðl1Þ Lzl1 al2 ^x1 þ nl2 þ L L L ! g L_ þ sgn ðgÞ c1ðl1Þ Lg e2 þ Lgz2 þ /1l þ n1 : L L Similar to Step 2, according to (3), (5), (9), the definition of ei ði ¼ 1; . . . ; nÞ; zi ði ¼ 1; . . . ; nÞ and the fact that di > 1ði ¼ 1; . . . ; l 1Þ , by completing the square, we obtain
! !2 l1 l1 X X d1ðl1Þ 2 d1ðl1Þ dl1 2 3 l2 Lnl1 þ dl1 Ln2l ; al ai ci ^x1 nl 6 Ln þ 2 a þ a c Ln2l þ lþ2 Lkek2 þ 2l Ln2l ; i i l 1 2 6 2k 2 i¼2 i¼2 3 3 1 2 2 cl1 Lzl nl 6 cl1 1 þ cl1 Lnl þ dl1 Lnl1 ; 2 6 l2 l2
l2 X X X diðl1Þ 2 d2ðl1Þ 2 1 l3i 2 ciðl1Þ Lziþ1 nl 3 2 ciðl1Þ ð1 þ c2i Þ Ln2l þ dl1 Ln2l1 þ Lni þ Ln2 ; li 6 3 2l3 i¼2 i¼2 i¼3 2 l1 l1 X L_ L_ L_ L_ 1X ciðl1Þ ni nl 6 c2iðl1Þ n2i þ ðl 1Þ n2l ; sgnðgÞ c1ðl1Þ n1 nl þ L L L L 4 i¼2 i¼1 d1ðk1Þ sgnðgÞ c1ðl1Þ Lg e2 nl 6 lþ2 Lkek2 þ 2l g 2M c21ðl1Þ Ln2l ; 2 d1ðl1Þ 2 d2ðl1Þ 2 3 l2 2 2 Ln1 þ Ln2 ; sgnðgÞ c1ðl1Þ Lgz2 nl 2 g M c1ðl1Þ þ c21 Ln2l þ 4 2l 6 2l3 d1ðl1Þ 2 d1 2 2 d1ðl1Þ 2 g d1 nl 6 h c1ðl1Þ n21 þ nl ; sgnðgÞ c1ðl1Þ /1l nl 6 h1l c1ðl1Þ jn1 j jnl j 6 h21k c21ðl1Þ n21 þ L 2 2 2 2
dl1 Lnl1 nl 6
557
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
1 1 1 d1ðl1Þ z2lþ1 6 d1ðl1Þ c2l n2l þ d1ðl1Þ n2lþ1 : 4 2 2
ð25Þ
It is also easy to obtain
L_ L_ n_ n ¼ z_ n þ cn1 n_ n1 ¼ Lcn nn an ^x1 n nn þ ncn1 nn1 ; L L
L_ L_ nn nn ¼ Lcn n2n an ^x1 nn n n2n þ ncn1 nn1 nn : L L
It is not difficult to obtain the following
L L d1 2 d1 2 2 2 n n H e d h d c d k k 1ðn1Þ n 2ðn1Þ 1 1 2 a1 2 2nþ1 2n1 2n2 2 ! ! 2 n n1 n1 c 2 X X X c c2 L_ 2 L_ 2 L a1 ðiþ1Þj diðn1Þ ni i þ 1 d1ði1Þ n2i dðiþ1Þðn1Þ di i 1 þ ni nn d1ðn1Þ z21 : L L 2 4 d 2 2 i¼3 i¼1 j¼iþ1 ðiþ1Þj
V_ n 6 d1ðn1Þ
L
ð26Þ n
n
In view of (26), it is not difficult to prove that, starting from any initial condition ðeð0Þ; zð0ÞÞ 2 R R and Lð0Þ ¼ 1 , the closed-looped system (13) has the properties: (i) the corresponding solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) existing on ½0; þ1Þ is unique and bounded; (ii) limt!þ1 ðzðtÞ; eðtÞÞ ¼ 0; limt!þ1 LðtÞ ¼ L 2 Rþ . Recall that the solution ðeðtÞ; zðtÞ; LðtÞÞ of (13) exists and is unique on the maximally extended interval ½0; T f Þ. Thus conclusions (i) and (ii) follow immediately if one can prove that eðtÞ; zðtÞ and LðtÞ are bounded on ½0; T f Þ. This can be done by a contradiction argument. Suppose limt!T f sup ðzðtÞ; eðtÞ; LðtÞÞT ¼ þ1 , We first claim that LðtÞ cannot escape at t ¼ T f . To 2
^ x prove this claim, suppose that limt!T f LðtÞ ¼ þ1 , since L_ ¼ n21 þ L21 þ n2n P 0 , LðtÞ is a monotone non-decreasing function. From the Lemma 1 below in this paper we know that there exists a finite time t 2 ½0; T f Þ, such that V_ 6 cðkek2 þ knk2 þ z2 Þ. 1
As a consequence,
þ1 ¼ LðT f Þ Lðt Þ ¼
Z
Tf
_ LðtÞdt ¼
t
Z
Tf t
ðn21 ðtÞ þ n2n ðtÞ þ z21 ðtÞÞdt 6
Vðt Þ < þ1: c
This contradiction shows that L is bounded on ½0; T f Þ. Next, we claim that z is well defined and bounded on the interval ½0; T f Þ. To see why, consider the Lyapunov function V z ¼ zT Pz for the z-dynamic system of (14). Clearly, a direct computation gives
1 1 L_ _ V_ z ¼ LzT ðAT P þ PAÞz 2Lcn zT Pen nn zT ðDP þ PDÞz 6 Lkzk2 þ 2Lc2n kP k2 n2 6 kzk2 þ 2c2n kPk2 LL: 2 2 L This, in turn, leads to 8t 2 ½0; T f Þ, kmin ðPÞkzðtÞk2 zð0ÞT Pzð0Þ 6 zt ðtÞPzð0Þ 6 12 that 2 kzðtÞk 6
R X
ð27Þ
kzðtÞk2 dt þ c2n kP k2 L2 Therefore, it follows
1 ðzT ð0ÞPzð0Þ þ c2n kPk2 L2 Þ; 8t 2 ½0; T f Þ; kmin ðPÞ
and
Z X
2 kzðtÞk dt 2 zT ð0ÞPzð0Þ þ c2n kPk2 L2 < þ1;
where L denotes the supremum of L on X .i.e., L ¼ supt2X LðtÞ since L is bounded on ½0; T f Þ , the inequalities above imply boundedness of z. Finally, we prove that e is bounded on ½0; T f Þ . To this end, we introduce the following change of coordinates
ei ¼
xi ^xi ðL Þ
i
;
i ¼ 1; . . . ; n;
where L is a constant satisfying
L P maxfL; H þ 4g:
ð28Þ
Then, the error dynamic (13) is transformed into
e_ ¼ L Ae þ L ae1 LC1 ae1 þ C2 ax1 þ Uk ; where
e ¼ ½e1 . . . en T ;
ð29Þ
558
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
(
C1 ¼ diag 1;
C2 ¼ diag
n1 ) L L ; ; . . . ; L L
n L L ; ;...; L L
"
UrðtÞ ¼
U1rðtÞ U2rðtÞ ;
L
2
ðL Þ
...
UnrðtÞ
#T :
n
ðL Þ
Along the solutions of (29), differentiating the function T V e ¼ eT Pe on X yields V_ e 6 L kek2 þ 2e1 L aT Pe 2e1 LaT C1 Pe þ 2x1 aT C2 Pe þ 2Uk Pe. By completing the square, we get
2 2 2e1 L aT Pe 6 ðL Þ aT P e21 þ kek2 ; 2 2e1 LaT C1 Pe 6 L2 aT C1 P e21 þ kek2 ; 2 2 2x1 aT C2 Pe 6 aT C2 P x21 þ kek2 6 aT C2 P L2 n21 þ kek2 ; T 2UrðtÞ Pe 6 Hkek2 þ
n 1X 1 z2 þ n 2 ; 4 i¼2 i 2 1
which, together with (28), means that on X,
1 V_ e 6 ðL H 3Þkek2 þ 4 6 kek2 þ
n
X 2 2 2 1 n21 þ ðL2 aT P þ L2 aT C1 P e21 z2i þ aT C2 P L2 þ 2 i¼2
n 1X 1 þ H1 L_ þ ðsupt2X kzðtÞkÞ2 ; z2 þ H1 n21 þ H1 z21 6 4 i¼2 i kmaxðPÞ
ð30Þ
with
H1 ¼
2 2 2 1 þ aT C2 P L2 þ 2ððL2 aT P þ L2 aT C1 P ; 2
from (30), we easily have
keðtÞk2 6
1 ðsupt2X kzðtÞkÞ2 kmax ðPÞ þ eT ð0ÞPeð0Þ þ H1 L; kmin ðPÞ
8t 2 X;
and
Z
kbar eðtÞk2 6 eT ð0ÞPeð0Þ þ
Z
X
kzðtÞk2 þ H1 L < þ1:
X
This, together with the definitions of ei ði ¼ 1; . . . ; nÞ and ei ði ¼ 1; . . . ; nÞ and the boundedness of L, implies that e is bounded on R 2 X keðtÞk < þ1. On the other hand, using the boundedness of ðL; e; zÞ on ½0; þ1Þ, it is straightforward to deduce that e_ and z_ are bounded R þ1 R þ1 lat Lemma [6], we have limt!þ1 zðtÞ ¼ 0 on the ½0; þ1Þ. Noticing that 0 keðtÞk2 < þ1 and 0 kzðtÞk2 dt < þ1, by the Barba
X and
and limt!þ1 eðtÞ ¼ 0. 3.3. Stabilization theorem From (25), we can see that to realize global stabilization of the closed-loop switched system by output feedback, the controllers designed should ensure the negative definition of the time derivative of V n . This can be guaranteed by choosing the appropriate di P 1; i ¼ 1; . . . ; n satisfying (11) and c; h such that n1 c 2 X c2 ðiþ1Þj di i 1 þ 4 d j¼iþ1 ðiþ1Þj
c6 and
a1 ; 2
c6
1 2nþ1
;
! > 0;
i ¼ 1; . . . ; n 1;
ð31Þ
ð32Þ
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
h P H; h P hn ; h P d1 c2 þ d1 þ d1 ; 2 2 a1
2
c 2 c3 i h P 2 þ 1 d12 ; þ 1 d1ði1Þ ; i ¼ 3; . . . ; n; 2
559
ð33Þ
meanwhile,
diðn1Þ ¼ di diþ1 . . . dn1 P 1;
i ¼ 1; 2; . . . ; n 1;
ð34Þ
since it is easily obtained di P 1; i ¼ 1; 2; . . . ; n from (11). For these constants di P 1; i ¼ 1; . . . ; n; c; h, we have to verify the following Lemma. Lemma 1. There exist constants di P 1; i ¼ 1; . . . ; n; c; h satisfying inequality V_ 6 cðkek2 þ knk2 Þ cz21 . Proof. We prove the Lemma by construction. First of all, we choose
c ¼ min
a1 1 ; nþ1 : 2 2
ð35Þ
Next, we choose
d1 d1 c 2 h ¼ max H; hn ; c22 þ d1 þ ; i þ 1 d1ði1Þ ; i ¼ 3; . . . ; n : 2 a1 2
ð36Þ
Then we can obtain from (26)
V_ n 6 ðc hÞkek2 ðc hÞn21 ðc hÞn22
n
X ðc hÞn2i cz21 ¼ ðc hÞ kek2 þ knn k2 cz21 : i¼3
We have known that 0; T f is the maximal interval X on which a unique closed-loop system solution exists, where 0 < T f 6 þ1 (see Theorem 2.1page 17, [3]). Since L is a continuous and monotone nondecreasing function, there exists a finite time t 2 0; T f ; 8t 2 ½t ; T f Þ , then from (26), it follows that on ½t ; tf Þ we have such that LðtÞ P hþc c _V 6 cðkek2 þ knk2 Þ cz2 . 1
It is easy to verify that any constants di P 1; c; h ði ¼ 1; . . . ; nÞ to be de designed by (35), (36) satisfy (32), (33). By Lemma 1, if we choose di P 1; c; hði ¼ 1; . . . ; nÞ satisfy (35), (36), the right-hand side of (26) becomes negative definite, which indicates that V n ðxÞ is qualified as a CLF for the system (4). Therefore, the closed-loop switched system (4), (7), (8) and (9) which implies that, by the coordinate transformation (3), the closed-loop switched system (1), (7), (8) and (9) is globally asymptotically stable under arbitrary switchings. So far, the entire design procedure has been completed. Global stabilization of the closed-loop switched system by output feedback is summarized in the following result.
Theorem 1. Under Assumptions 1 and 2, there exist output feedback controllers (8) with positive constants di P 1; c; hði ¼ 1; . . . ; nÞ satisfying (10), (11), (32) and (33), such that the closed-loop switched system (1), (7), (8) and (9), for any initial condition, is globally asymptotically stable under arbitrary switchings. Remark 3. In this paper, a key point is to construct the common coordinate transformation in order to effectively avoid different coordinate transformations for different subsystems when applying backstepping. This is realized by design of an appropriate state observer of the switched system studied. For switched nonlinear systems in triangular form, we investigate the global stabilization problem of the system (4) based on the assumption that only the system output is measurable, while [15,19,31] present some effective methods of the global stabilization problem based on the assumption that full-state measurements are available. However, these methods in [15,19] cannot be used in the system studied in this paper. Meanwhile, in order to reduce the conservativeness caused by adoption of a common observer for all subsystems, we need to design an appropriate switched observer for the system studied. In this paper, a switched observer (7) is constructed for the switched system (1). Based on the switched observer designed (7), we construct the output-feedback controller of individual subsystem by using backstepping. Remark 4. When we study switched systems, we need to handle each subsystem by applying some non-switched or switched system theory in the existing literature, which is commonly used approach (e.g., see [13,15]). Thus, in the proof procedure of Theorem 1, for each subsystem, we adopt a similar method of the non-switch system literature [33]. However, the method of [33] cannot give any results on the topic in this paper even if [33] presents an effective method to achieve the stability of non-switched systems. For example, a common coordinate transformation for all subsystems of the switched system under study has been constructed to avoid different coordinate transformations for different subsystems when applying the backstepping technique.
560
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
4. Example In this section, we present two illustrative examples to demonstrate the applicability and effectiveness of the proposed approach. 4.1. Example 1
8 > < g_ 1 g_ 2 > : y
¼ g 1 g2 þ DhrðtÞ ðg1 ; g2 Þ; ¼ g 2 urðtÞ þ Dg rðtÞ ðg1 ; g2 Þ;
ð37Þ
¼ g1 : g g g2
g 1 g 2 g1 where Mh1 ðg1 ; g2 Þ ¼ g2 ðð1d ; Mh2 ðg1 ; g2 Þ ¼ ð1þ2 g12 Þg22 ; Mg 1 ðg1 ; g2 Þ ¼ g Þ2 þg2 Þ 1 2
2d2
lnð1þg2
g2
2
2
Þg 1 g 2
g2 ; Mg 2 ðg1 ; g2 Þ ¼ g2 g2gsin ; d1 ; d2 are unknown 2
constants. Under the same kind of transformation (3) for the above system, this system becomes
8 _ > < x1 x_ 2 > : y
¼ x2 þ Mf rðtÞ ðx1 ; x2 Þ; ¼ g 2 urðtÞ þ Mg rðtÞ ðx1 ; x2 Þ;
ð38Þ
¼ gx1 ;
As a matter of fact, the following estimations
g 2 ð1 þ d2 Þjg j g 1 g 2 g1 1 1 ; 6 2 2 2 2 g ðð1 d1 g Þ þ g Þ g 2 2 lnð1 þ g2d2 Þg g g 2 ð2d 1Þjg j 2 1 2 2 2 ; 6 g2 g2 g g g2 gjg j 2 1 2 6 21 ; ð1 þ g22 Þg 2 g
g g sing gjg j 2 2 2 6 22 ; g2 g
can be easily obtained. Hence, Assumption 1 holds with d ¼
g 2 maxf2d2 1;1þd21 g , g2
where d is an unknown constant. Suppose that
control coefficients g 1 and g 2 satisfy 0:9 6 g i 1:5; i ¼ 1; 2 and choose a1 ¼ 10; a2 ¼ 1; d1 ¼ 1; d2 ¼ 2, then g ¼ 0:9 and g ¼ 1:5 , g M can be chosen as g M ¼ 2:7778 and there exists a positive definite matrix
P¼
0:1000
0:5000
0:5000
5:1000
such that the matrix inequality (6) holds, moreover, in view of (10) and (11), by simple calculation, we choose
g 1 ¼ 1;
g 2 ¼ 1;
c1 ¼ 9:237;
c2 ¼ 87183;
d1 ¼ 22:3305:
Thus, by Theorem 1 and the design procedure provided in Section 3, we design the observer as:
¼ ^x_ 2 La1 ^x1 ; ¼ g 2 urðtÞ L2 a2 ^x1 ; ¼
y2 L2
^x2 1
þ L2 þ
n22 ; Lð0Þ
ð39Þ ¼ 1;
2
states
8_ ^x > > < 1 ^x_ 2 > > :_ L
1 0 −1 0
10
time(sec)
Fig. 1. State responses: x1 (
20 ) and ^ x1 ð Þ.
30
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
states
10 0 −10 −20 0
10
time(sec)
Fig. 2. State responses: x2 (
20
30
) and ^ x2 ð Þ.
3
L
2.5 2 1.5 1 0
10
time(sec)
20
30
Fig. 3. The trajectory of the high gain L of Example 1.
Fig. 4. Schematic diagram of the process.
states
2 1 0 −1 0
10
20 time(sec)
Fig. 5. State responses: x1 (
) and ^ x1 ð Þ.
30
561
562
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
and the controller is
u ¼ g 2 L3 c2 n2 ;
ð40Þ
n2 is given by
^x2
n2 ¼
L
2
þ sgnðgÞc1 n1 ¼
^x2
y þ sgnðgÞc1 : L L
ð41Þ
2
T x1 ð0Þ; ^ x2 ð0Þ; Lð0ÞÞ ¼ ð1; 5; 1:5; 9:24; 1Þ and The results are depicted in Figs. 1–3 with the initial conditions ðx1 ð0Þ; x2 ð0Þ; ^ the switching time between switching the system on and off: 0.001s, which shows the asymptotic stability of the closed-loop system (37), (39) and (40). Thus, the simulation results illustrate the theory presented.
4.2. Example 2. Next, we apply the proposed method to a practical example, namely a continuous stirred tank reactor(CSTR) with two modes feed stream(see Fig. 4), which is molded as a switched nonlinear system [22,37,38]: E C_ A ¼ qVr ðC Afr C A Þ a0 eRT C A ; E T_ ¼ qVr ðT f r TÞ a1 eRT C A þ VUA ðT c TÞ: q Cq
The physical meaning of the parameters of this system can be easily found in [37,38]. Under a coordinate transformation and smooth feedback [37,38], the new system is
8 > < g_ 1 g_ 2 > : y
¼ g 1 g2 þ hrðtÞ ðg1 Þ; ¼ g 2 urðtÞ ;
ð42Þ
¼ g1 ;
where h1 ¼ 0:5gg12g2 x1 ; h2 ¼ 2gg22x1 ; g ¼ gg1 g2 2 . Furthermore, we make the same coordinate transformation (3) about the system (41), which does not affect the stabilization objective. We obtain
8 _ > < x1 x_ 2 > : y
¼ x2 þ f rðtÞ ðx1 Þ; ¼ g 2 urðtÞ ;
ð43Þ
¼ gx1 ;
where f 1 ¼ 0:5x1 ; f 2 ¼ 2x1 ; g ¼ g1gg2 2 . Then the last Eqs. (43) have the same form and dimension as Eqs. (37), therefore, we may as well design the same controller and observer as Example 1. It is easy to prove that the Eqs. (41) satisfy the first assumption above in this paper. For the second assumption, we still suppose that g ¼ 0:9 and g ¼ 1:5 thus the second assumption also holds. We also choose the same parameters such as a1 ¼ 10; a2 ¼ 1 , then, there exists the same positive definite matrix
P¼
0:1000
0:5000
0:5000
5:1000
such that the matrix inequality (6) holds, moreover, in view of (10) and (11), by simple calculation, we choose
c1 ¼ 9:5;
c2 ¼ 97364;
d1 ¼ 23:5625: T
x1 ð0Þ; ^ x2 ð0Þ; Lð0ÞÞ ¼ ð1:5; 6:5; 2; 10; 1Þ; The results are depicted in Figs. 5–7 with the initial conditions ðx1 ð0Þ; x2 ð0Þ; ^ g 1 ¼ 1; g 2 ¼ 1 and the switching time between switching the system on and off: 0.001s, which shows the asymptotic stability of the closed-loop system (39), (40) and (42). Thus, the simulation results illustrate the theory presented.
10
states
0 −10 −20 −30 0
10
time(sec)
Fig. 6. State responses: x2 (
20 ) and ^ x2 ð Þ.
30
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
563
4
L
3 2 1 0
10
time(sec)
20
30
Fig. 7. The trajectory of the high gain L of Example 2.
Remark 5. The continuous stirred tank reactor has been studied from the point of view of switching control in [37,38]. These results above are based on the assumption that full state measurements are available. However, this assumption above does not hold in many practical systems. In this paper, we consider full state measurements are unavailable for the continuous stirred tank reactor, and only the system output is available for measurement. Clearly, these methods in [37,38] cannot be used in the continuous stirred tank reactor in this paper. But, based on the approach proposed in this paper, the outputfeedback control problem of the continuous stirred tank reactor is solvable.
5. Conclusion As a further investigation based on [17,33], the global asymptotic stabilization by output feedback has been investigated for a class of uncertain nonlinear switched systems with unmeasured states dependent growth. Based on backstepping and CLF, this paper has developed a constructive design approach for designing the output-feedback controllers of individual subsystems, which guarantees global asymptotic stability of the corresponding closed-loop switched system under arbitrary switchings. Finally, It was demonstrated, by means of a numerical example and its simulation, that the proposed adaptive output feedback controller can be easily designed and implemented. Our future work will be directed at extending the control design method to more general nonlinear switched systems, such as the switched systems with unknown control coefficients, and at obtaining the global (practical) tracking result for system. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grants 61304058 and 61233002, andIAPI Fundamental Research Funds under Grant 2013ZCX03–02, and Fundamental Research Funds for the Central Universities under Grant N130404026, and China Postdoctoral Science Foundation under Grant 2013M540231. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Zhiyong Chen, Jie Huang, Global robust servomechanism of a class of nonlinear systems using output feedback, Asian J. Control 9 (3) (2007) 292–305. Farzad Esfandiari, Hassan K Khalil, Output feedback stabilization of fully linearizable systems, Int. J. Control 56 (5) (1992) 1007–1037. J.K. Hale, second ed., Ordinary Differential Equations, Huntington, Krieger, 1980. Thanh-Trung Han, Shuzhi Sam Ge, Tong Heng Lee, Adaptive neural control for a class of switched nonlinear systems, Syst. Control Lett. 58 (2009) 109– 118. Lei Hao, Lin Wei, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica 42 (10) (2006) 1783– 1789. H.K. Khalil, 3rd ed., Nonlinear Systems, Prentice Hall, Englewood Cliff, 2002. Zhong-Ping Jiang, I. Mareels, D.J. Hill, Jie Huang, A unifying framework for global regulation via nonlinear output feedback: from iss to iiss, IEEE Trans. Autom. Control 49 (4) (2004) 549–562. Ying Jin, Jun Fu, Youmin Zhang, Yuanwei Jing, Reliable control of a class of switched cascade nonlinear systems with its application to flight control, Nonlinear Anal. Hybrid Syst. 11 (2014) 11–21. Prashanth Krishnamurthy, Farshad Khorrami, Ramu Sharat Chandra, Global high-gain-based observer and backstepping controller for generalized output-feedback canonical form, IEEE Trans. Autom. Control 48 (12) (2003) 2277–2283. Miroslav Krstic, Petar V. KokotoviC, Adaptive nonlinear output-feedback schemes with marino-tomei controller, IEEE Trans. Autom. Control 41 (2) (1996) 274–280. Miroslav Krstic, Petar V. Kokotovic, Ioannis Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc, 1995. Xiao-Ling Liang, Ming-Zhe Hou, Guang-Ren Duan, Output feedback stabilization of switched stochastic nonlinear systems under arbitrary switchings, Int. J. Autom. Comput. 10 (6) (2013) 571–577. Daniel Liberzon, A. Stephen Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. 19 (5) (1999) 59–70. Chang-Liang Liu, Shao-Cheng Tong, Yong-Ming Li, Yuan-Qing Xia, Adaptive fuzzy backstepping output feedback control of nonlinear time-delay systems with unknown high-frequency gain sign, Int. J. Autom. Comput. 8 (1) (2011) 14–22. Lijun Long, Jun Zhao, Global stabilization for a class of switched nonlinear feedforward systems, Syst. Control Lett. 60 (9) (2011) 734–738. Lijun Long, Jun Zhao, H1 control of switched nonlinear systems in p -normal form using multiple lyapunov functions, IEEE Trans. Autom. Control 57 (5) (May 2012) 1285–1291. Lijun Long, Jun Zhao, Output-feedback stabilisation for a class of switched nonlinear systems with unknown control coefficients, Int. J. Control 86 (3) (2013) 386–395.
564
E. Li et al. / Applied Mathematics and Computation 256 (2015) 551–564
[18] Lijun Long, Jun Zhao, A small-gain theorem for switched interconnected nonlinear systems and its applications, IEEE Trans. Autom. Control 59 (4) (April 2014) 1082–1088. [19] Ruicheng Ma, Jun Zhao, Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings, Automatica 46 (11) (2010) 1819–1823. [20] Riccardo Marino, Patrizio Tomei, An adaptive output feedback control for a class of nonlinear systems with time-varying parameters, IEEE Trans. Autom. Control 44 (11) (1999) 2190–2194. [21] F. Mazenc, L. Praly, W.P. Dayawansa, Global stabilization by output feedback: examples and counterexamples, Syst. Control Lett. 23 (2) (1994) 119– 125. [22] P. Mhaskar, N.H. El-Farra, P.D. Christofides, Predictive control of switched nonlinear systems with scheduled mode transitions, IEEE Trans. Autom. Control 50 (11) (Nov 2005) 1670–1680. [23] Ying-Ying Min, Yun-Gang Liu. Barbalat lemma and its application in analysis of system stability. Journal of Shandong University (engineering science), 1:012, 2007. [24] Kokotovic´ Petar, Arcak Murat, Constructive nonlinear control: a historical perspective, Automatica 37 (5) (2001) 637–662. [25] Laurent Praly, Zhong-Ping Jiang, Linear output feedback with dynamic high gain for nonlinear systems, Syst. Control Lett. 53 (2) (2004) 107–116. [26] Chunjiang Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, in: Proceedings of the American Control Conference, 2005, IEEE, 2005, pp. 4708–4715. [27] Chunjiang Qian, Wei Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Trans. Autom. Control 47 (10) (2002) 1710–1715. [28] Min Wang, Jun Zhao, Georgi M. Dimirovski, Dynamic output feedback robust H1 control of uncertain switched nonlinear systems, Int. J. Control Autom. Syst. 9 (1) (2011) 1–8. [29] Rui Wang, Gang Jin, Jun Zhao, Robust fault-tolerant control for a class of switched nonlinear systems in lower triangular form, Asian J. Control 9 (1) (2007) 68–72. [30] Chunling Wei, Qiangde Wang, Global output feedback asymptotic stabilization for a class of nonlinear systems, in: 29th Chinese Control Conference (CCC), IEEE, 2010, pp. 369–374. [31] Wu. Jenq-Lang, Stabilizing controllers design for switched nonlinear systems in strict-feedback form, Automatica 45 (4) (2009) 1092–1096. [32] Guangming Xie, Long Wang, Controllability and stabilizability of switched linear-systems, Syst. Control Lett. 48 (2) (2003) 135–155. [33] Xuehua Yan, Yungang Liu, New results on global output-feedback stabilization for nonlinear systems with unknown growth rate, J. Control Theory Appl. 11 (3) (2013) 401–408. [34] Guang-Hong Yang, Wei-Wei Che, Non-fragile H1 filter design for linear continuous-time systems, Automatica 44 (11) (2008) 2849–2856. [35] H. Yang, B. Jiang, M. Staroswiecki, Observer-based fault-tolerant control for a class of switched nonlinear systems, IET Control Theory Appl. 1 (5) (2007) 1523–1532. [36] Hao Yang, Bin Jiang, Vincent Cocquempot, Huaguang Zhang, Stabilization of switched nonlinear systems with all unstable modes: application to multiagent systems, IEEE Trans. Autom. Control 56 (9) (2011) 2230–2235. [37] M.B. Yazdi, M.R. Jahed-Motlagh, Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization, Chem. Eng. J. 155 (3) (2009) 838–843. [38] M.B. Yazdi, M.R. Jahed-Motlagh, S.A. Attia, J. Raisch. Switched feedback equivalence of a class of planar switched nonlinear systems. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. in: Proceedings of the 48th IEEE Conference on, pages 4264– 4269. IEEE, Dec 2009. [39] Evren Yurtseven, W.P.M.H. Heemels, M. Kanat Camlibel, Disturbance decoupling of switched linear systems, Syst. Control Lett. 61 (1) (2012) 69–78. [40] J. Zhai, S. Fei, Global practical tracking control for a class of uncertain non-linear systems, IET Control Theory Appl. 5 (11) (2011) 1343–1351. [41] Junyong Zhai, Chunjiang Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlinear Control 22 (14) (2012) 1543–1561. [42] Jun Zhao, David J. Hill, Dissipativity theory for switched systems, IEEE Trans. Autom. Control 53 (4) (2008) 941–953. [43] Jing Zhou, Changyun Wen, Ying Zhang, Adaptive output control of nonlinear systems with uncertain dead-zone nonlinearity, IEEE Trans. Autom. Control 51 (3) (2006) 504–511.