- Email: [email protected]
Global state regulation by output feedback for feedforward systems with input and output dependent incremental rate
Available online at www.sciencedirect.com
Journal of the Franklin Institute 352 (2015) 2526–2538 www.elsevier.com/locate/jfranklin
Global state regulation by output feedback for feedforward systems with input and output dependent incremental rate Xianglei Jiaa, Shengyuan Xua,n, Ticao Jiaoa, Yuming Chub, Yun Zoua a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China b School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, PR China Received 19 May 2014; received in revised form 16 February 2015; accepted 16 March 2015 Available online 2 April 2015
Abstract In this paper, the problem of global state regulation by output feedback is considered for a class of uncertain feedforward nonlinear systems with free-delay or time-delay states. Compared with existing results, we construct a novel observer-based controller with gain exponent to achieve the global state regulation via single output feedback under less conservative assumptions. Further, by using the Lyapunov– Krasovskii theorem, we show that the control scheme proposed is available to a class of time-delay systems with input and output dependent incremental rate. Finally, a numerical example is given to illustrate the usefulness of our results. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The problem of global output feedback control is one of the most important and challenging problems in the field of nonlinear control, which has received a lot of attention over the past decades. Compared with linear systems, the separation principle does not hold for nonlinear systems, as shown in [1]. Thus, it is necessary to impose some extra restrictive conditions on nonlinear terms to obtain global output feedback controller. In the case when uncertain nonlinear systems dominated by a lower-triangular system with linear growth in unmeasured states, the n
Corresponding author. E-mail address: [email protected] (S. Xu).
http://dx.doi.org/10.1016/j.jfranklin.2015.03.035 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2527
Fig. 1. The trajectories of x1 and x^ 1 .
problems of global output feedback stabilization or regulation have been addressed in [2–4,6–13]. Specifically, using a feedback domination design method, the global exponential stabilizer was constructed under the linear growth condition with known growth rate in [2]. When the growth rate was a known smooth function of system output, the global state regulation was achieved by an adaptive output feedback controller in [3], where the results of [4,5] were generalized. With the help of switching logic, an adaptive output feedback controller was proposed in the case of unknown constant growth rate in [6]. Unlike the control scheme proposed by [6], universal adaptive high-gain observers were introduced to achieve global output feedback stabilization in [7,8]. For a larger class of nonlinear systems with unknown control coefficients, the global state regulation problem was investigated by output feedback in [9–11]. Furthermore, a universal adaptive output feedback controller was constructed for a class of nonlinear systems with unknown time delays and output function in [12], and an output feedback controller was proposed by introducing double dynamic gains in [13]. To deal with the case of high-order growth, several attempts have been made such as [14–20]. In particular, for a class of nonlinear systems with uncontrollable/unobservable linearization, the problem of output feedback stabilization was handled in [14–16]. In view of homogeneous system theory, a homogeneous domination approach was introduced to achieve output feedback stabilization in [17]. Subsequently, a generalized homogeneous domination approach was established in [18]. In the case of polynomial growth conditions, a recursive design algorithm was developed to achieve output feedback stabilization by constructing a reduced-order observer in [19]. By using dual observers, the global output feedback stabilization was achieved for nonlinear systems with lowerorder and high-order nonlinearities in [20]. On the other hand, the problem of global output feedback control has been investigated for a class of feedforward nonlinear systems in [21–28]. Specifically, when triangular type restriction was not satisfied, a linear output feedback control scheme was proposed to achieve global exponential stabilization in [22]. By introducing dynamic high-gain scaling technique, an adaptive output feedback control scheme was proposed for feedforward systems in [21]. With the aid of the idea of
2528
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Fig. 2. The trajectories of x2 and x^ 2 .
universal control and the dynamic high-gain scaling technique, the global stabilization was achieved by linear output feedback in [23,24]. By the homogeneous domination approach, the problem of output feedback stabilization was handled for a class of nonlinear systems with high-order growth in unmeasurable states in [25]. Further, in [26], such problem was solved for a class of nonlinear systems with uncertain output function. In the presence of time delays, the global state regulation was achieved via linear output feedback in [27,28], where the growth rate depended on unknown constant or nonlinearities of system input. Notably, in the case of lower triangular, [8] has generalized the result of [7]. Similarly, in the feedforward case, the results of [28,23] can be also generalized under suitable assumptions. In this paper, we will consider this question. The main contributions of this paper can be summarized as: First, we construct a high-gain observer with adjustable high-gain exponential, which can be regarded as an extension of [28]. By adopting a new coordinate change, we show that the output feedback controller of this paper is capable of applying to a class of uncertain nonlinear systems, where the growth rate depends not only on system input but also on system output. Second, with the aid of the Lyapunov–Krasovskii theorem, we solve the problem of global state regulation for a class of nonlinear time-delay system by the controller proposed. Notations: In this paper, the argument of the functions will be omitted whenever no confusion can arise from the context. For any real vector or matrix A, AT denotes its transpose; J J denotes the Euclidean norm for the matrix and vector. 2. Preliminaries and key technical lemmas In this section, we briefly discuss what are feedforward nonlinear systems. As shown in [21], the systems of the form x_ i ¼ xiþ1 þ f i ðt; y; xiþ2 ; …; xn ; uÞ; x_ n ¼ u
i ¼ 1; …; n 1
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2529
Fig. 3. The trajectories of x3 and x^ 3 .
y ¼ x1 ; are referred to as feedforward nonlinear systems. Moreover, the following systems are also referred to as feedforward nonlinear systems (see Remark 1 in [28]): x_ i ¼ xiþ1 þ f i ðd; x; uÞ;
i ¼ 1; …; n 1
x_ n ¼ u y ¼ x1 ; with jf i ðÞj r cðuÞðjxiþ2 j þ ⋯ þ jxn j þ jujÞ. In what follows, we introduce three key lemmas. Lemma 2.1 (Qian and Lin [29]). For any positive integers m; n and any real-valued function αðx; yÞ40, then jxjm jyjn r
m n αðx; yÞjxjmþn þ αðx; yÞ m=n ðx; yÞjyjmþn : mþn mþn
Lemma 2.2 (Qian and Lin [29]). For xA R, y A R, p Z 1 is a constant, the following inequalities hold: jx þ yjp r 2p 1 jxp þ yp j: Lemma 2.3 (Praly and Jiang [30]). If matrices A and B are Hurwitz, D ¼ diagðν þ ðn 1Þμ; …; νÞ, then there exist P ¼ PT 40, Q ¼ QT 40 and α40 satisfying ( AT P þ PA r I; DP þ PD Z αI BT Q þ QB r I;
DQ þ QD Z αI;
2530
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Fig. 4. The trajectory of r.
where μ40, ν40, ai 40, 0 1 a1 B ⋮ ⋮ B A¼B @ an 1 0
ki 40 ði ¼ 1; …; nÞ, are suitable design parameters, and 1 0 1 ⋯ 0 0 1 ⋯ 0 B ⋮ ⋱ ⋮C ⋮ ⋱ ⋮ C C B C C A Rnn ; B ¼ B C A Rnn : 0 ⋯ 1 A ⋯ 1A @ 0 k1 k2 ⋯ kn 0 ⋯ 0
an 3. Main result
In this section, an observer-based controller will be constructed to solve the problem of global state regulation by output feedback. Consider the following uncertain nonlinear system: x_ i ¼ xiþ1 þ f i ðd; x; uÞ; x_ n ¼ u y ¼ x1 ;
i ¼ 1; …; n 1 ð1Þ
where x ¼ ðx1 ; …; xn Þ A R is the system state, u A R and y A R are the system input and measured output, respectively, and d : R-Rs is a continuous bounded mapping that represents a family of time-varying parameters or disturbances. The functions f i ; i ¼ 1; …; n 1; are C0 functions with f i ðd; 0Þ ¼ 0, which need not to be precisely known. To solve the problem of global state regulation by output feedback for system (1), we introduce an assumption as follows: T
n
Assumption 3.1. There exist a known constant 0opo1=ðn 1Þ and a known positive continuous function cðÞ such that ! nþ1 X jf i ðd; x; uÞj r cðuÞð1 þ jyjp Þ jxj j þ juj ; xnþ1 ¼ 0; i ¼ 1; …; n 1: j ¼ iþ2
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2531
Remark 3.1. As stated in Remark 1 of [28], system (1) satisfying Assumption 3.1 is referred to as feedforward nonlinear system. In the case when the growth rate was an unknown constant, instead of cðuÞð1 þ jyjp Þ, the problem of global state regulation was solved by output feedback in [23,27]. Moreover, when the growth rate was c(u), such problem was addressed in [28]. Theorem 3.1. Under Assumption 3.1, the global state regulation of system (1) can be achieved by the dynamic output feedback controller described by x^_ i ¼ x^ iþ1 þ ai r iμ ðy x^ 1 Þ; i ¼ 1; …; n 1 x^_ n ¼ u þ an r nμ ðy x^ 1 Þ u ¼ r nμ k 1 x^ 1 þ r μ k 2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n ; ð2Þ with the gain r updated by 8 μ > < r_ ¼ 1 max ϖ ðu; yÞ r ; 0 ; αr 2μ 1 2 > : rð0Þ ¼ r 0 Z 1;
ð3Þ
where ai 40, ki 40, i¼ 1,…,n, μ40 and α40 are determined by Lemma 2.3, x^ ¼ ð^x 1 ; …; x^ n ÞT is the observer state vector with the initial value x^ ð0Þ ¼ x^ 0 . Remark 3.2. Compared with the controller in [28], an extra design parameter μ is introduced in Eqs. (2) and (3). This parameter plays a key role in constructing suitable positive definite matrices P, Q and D by Lemma 2.3, to avoid too large control gain. Proof. As shown in [28], the state r(t) of Eq. (3) has the following properties: ðiÞ : r_ Z 0;
rμ Z ϖ ðÞ: 2 Define ei ¼ xi x^ i , then the error dynamics is given by ai e_ i ¼ eiþ1 iμ ðy x^ 1 Þ þ f i ðÞ; i ¼ 1; …; n 1 r an e_ n ¼ nμ ðy x^ 1 Þ: r Further, consider the following change of coordinates ði ¼ 1; …; nÞ: ðiiÞ : αr 2μ 1 r_ þ
εi ¼
ei νþðn iÞμ r
and
zi ¼
x^ i νþðn iÞμ r
;
ð4Þ
ð5Þ
ð6Þ
by which the closed-loop system (2) and (5) can be represented in the following compact form: r_ ε_ ¼ r μ Aε Dε þ F r r_ z_ ¼ r μ Bz Dz þ r μ aε1 ; ð7Þ r where ε ¼ ðε1 ; …; εn ÞT , z ¼ ðz1 ; …; zn ÞT , a ¼ ða1 ; …; an ÞT and F ¼ f 1 ðÞ=r νþðn 1Þμ ; …; f n 1 ðÞ= r νþμ ; 0ÞT . A, B and D are given by Lemma 2.3. Next, we will prove that all states of system (3) and (7) are bounded on ½0; t f Þ for some t f A ð0; þ1, and system (7), for any gain function r(t) generated by Eq. (3), is asymptotically stable at ε ¼ 0 and z ¼ 0. Without loss of generality, we assume that ½0; t f Þ is the maximally extended interval of the solution of Eqs. (3) and (7).
2532
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Choose the Lyapunov function V : Rn Rn -Rþ Vðε; zÞ ¼ σV 1 ðεÞ þ V 2 ðzÞ;
ð8Þ
where V 1 ðεÞ ¼ εT Pε; V 2 ðzÞ ¼ zT Qz; P and Q are given by Lemma 2.3, and σ ¼ JM J 2 is to be determined by Eq. (10). With the aid of Lemma 2.3 and the fact that r_ Z140, on the interval ½0; t f Þ, the derivative of V along the solution of Eqs. (3) and (7) is given by r_ σ V_ r μ ‖ε‖2 σα ‖ε‖2 þ 2σεT PF r r 1 2 r_ 2 μ ‖z‖ α ‖z‖2 þ μ ε1 zT Qa: r r r In what follows, we estimate the inequality above. Firstly, based on Lemma 2.1, we have 2 1 1 μ ε1 zT Qar μ ‖z‖2 þ μ ‖M‖2 ‖ε‖2 ; r 2r 2r where M ¼ Qða1 ; a2 ; …; an ÞT ð2; 0; …; 0Þ. In addition, from the definitions of εi and zi, one gets f ðÞ j^x iþ2 j þ jeiþ2 j juj i p ð Þ þ ⋯ þ c ð u Þ 1 þ jyj νþðn r iÞμ r r νþðn iÞμ r νþðn iÞμ 1 r 2μ cðuÞð1 þ jyjp Þ n J εJ þ k~ Jz J ; r where k~ ¼ nmaxi ¼ 1;…;n fk i þ 1g is a known constant. By using Eq. (11) and Lemma 2.1, one has j2σεT PFj r 2σ J PJ J ε J JF J σ r 2μ h1 ðu; yÞ‖ε‖2 þ h2 ðu; yÞJ ε J JzJ ; r where h1 ðu; yÞ ¼ 2nðn 1Þ JP JcðuÞð1 þ jyjp Þ and h2 ðu; yÞ ¼ 2ðn 1Þk~ JP JcðuÞð1 þ jyjp Þ. Based on Lemma 2.2, substituting Eqs. (10) and (12) into Eq. (9) leads to
1 1 h2 ðu; yÞ r_ 2 2 2 _ V r σ μ ‖ε‖ α ‖ε‖ þ 2μ h1 ðu; yÞ þ ‖ε‖ 2r r 2ϵ r 1 1 ϵ r_ μ ‖z‖2 α ‖z‖2 þ 2μ σh2 ðu; yÞ‖z‖2 2r r 2 r σ rμ 1 rμ þ αr 2μ 1 r_ ζ 1 ðu; yÞ ‖ε‖2 2μ þ αr 2μ 1 r_ ζ 2 ðu; yÞ ‖z‖2 ; r 2μ r r 2 2
ð9Þ
ð10Þ
ð11Þ
ð12Þ
ð13Þ
where ϵ is any positive constant. ζ 1 ðu; yÞ ¼ h1 ðu; yÞ þ h2 ðu; yÞ=2ϵ and ζ 2 ðu; yÞ ¼ ðϵ=2Þσh2 ðu; yÞ. In view of Eq. (4), choose a positive continuous function ϖðu; yÞ as follows: ϖðu; yÞ ¼ maxfζ 1 ðu; yÞ; ζ 2 ðu; yÞg þ κ; where κ is any positive constant. Integrating Eqs. (13) and (14), we have σκ κ V_ r 2μ ‖ε‖2 2μ ‖z‖2 : r r
ð14Þ
ð15Þ
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2533
From Eq. (15), it is concluded that V is bounded on the maximal interval of existence of solution ½0; t f Þ. By the definition of V, we can get that the states ε and z are bounded, which implies the following property: Property 3.1. r(t) is bounded on the interval ½0; t f Þ. In fact, for 8 t A ½0; t f Þ y νþðn 1Þμ r jz1 j þ jε1 j r ω1 ¼ constant; r from which we can deduce that jyj r ω1 r νþðn 1Þμ :
ð16Þ
In addition, by Eq. (6) and the boundedness of z and r ν μ r 1 (since ν r μ), we have u ¼ r nμ ðk 1 x^ 1 þ r μ k2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n Þ ¼ r ν μ ðk1 z1 þ k 2 z2 þ ⋯ þ k n zn Þ r ω2 ¼ constant;
ð17Þ
which implies that c(u) is bounded. By using Eq. (16) and the boundedness of c(u) yields ζ 1 ðu; yÞ r ω3 r pνþpðn 1Þμ þ ω4 ζ 2 ðu; yÞ r ω5 r pνþpðn 1Þμ þ ω6
ð18Þ
where ω3, ω4, ω5 and ω6 are suitable unknown constants. By means of Eqs. (14) and (18), one has ϖ ðu; yÞr ω7 r pνþpðn 1Þμ þ ω8 rμ 1 ϖ ðu; yÞ r r pνþpðn 1Þμ r μ pðn 1Þμ pν 2ω7 þ ω8 ; 2 2 where ω7 and ω8 are unknown constants. Since po1=ðn 1Þ, we can always find a pair of constants ðν; μÞ such that μ pðn 1Þμ pν40:
ð19Þ
ð20Þ
With this in mind, suppose r is unbounded, together with r Z 1 is C , then limt-tf ¼ þ 1, and there exists a finite time t 0 A ð0; t f Þ, when t 0 r t rt f , such that rμ 1 ϖ ðu; yÞ r r pνþpðn 1Þμ r μ pðn 1Þμ pν 2ω7 þ ω8 o0; 2 2 1
i.e., r_ ¼ 0, which leads to a contradiction. Hence, r(t) is bounded on ½0; t f Þ. Up to now, it has been proved that all signals of the closed system (3) and (7) are bounded. Thus, t f ¼ þ 1 and solutions exist for all time. In view of Eq. (15), we can conclude that the closed-loop system (7), for any control gain function r(t) generated by Eq. (3), is asymptotically stable at ε ¼ 0 and z¼ 0. Further, by Eq. (6) and the boundedness of r, one gets that the closedloop system (1) and (2), for any control gain function r generated by Eq. (3), is asymptotically stable at x ¼ x^ ¼ ð^x 1 ; …; x^ n ÞT ¼ 0. The proof of Theorem 3.1 has been completed. □
2534
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
4. Extending and discussion The control scheme in the previous section can be applied to time-delay nonlinear system as follows: x_ i ¼ xiþ1 þ f i ðd; x; u; xðt τi Þ; uðt τi ÞÞ; x_ n ¼ u y ¼ x1 ;
i ¼ 1; …; n 1 ð21Þ
where x ¼ ðx1 ; …; xn Þ A R is the system state, u A R and y A R are the system input and measured output, respectively, and d : R-Rs is a continuous bounded mapping that represents a family of time-varying parameters or disturbances. The functions f i ; i ¼ 1; …; n 1; are C0 functions with f i ðd; 0Þ ¼ 0, which need not to be precisely known. Constants τi Z 0, i ¼ 1; …; n 1; are unknown delays. For system (21), we introduce the following assumption, which is less conservative than the one in [28]. T
n
~ the following Assumption 4.1. For i ¼ 1; …; n 1; and any ðd; x1 ; …; xn ; u; x~ 1 ; …; x~ n ; uÞ, conditions hold: ! nþ1 X p ~ r ci ðuÞð1 þ jx1 j Þ jxj j þ juj jf i ðd; x1 ; …; xn ; u; x~ 1 ; …; x~ n ; uÞj ~ þ c~ i ðuÞð1 þ j~x 1 jp Þ
nþ1 X
!
j ¼ iþ2
~ ; j~x j j þ juj
j ¼ iþ2
where 0 r po1=ðn 1Þ is a known constant, xnþ1 ¼ x~ nþ1 ¼ 0, ci ðÞ and c~ i ðÞ are known positive continuous functions. Remark 4.1. Assumption 4.1 means that the growth rate is related with not only system input but also system output. For such feedforward nonlinear system with time delays, the output feedback controller of [28] is unavailable. Therefore, inspired by the case of lower triangular [8], we construct a new output feedback controller, where the dynamic gain is dependent on system output. Theorem 4.1. Under Assumption 4.1, the global state regulation of system (21) can be achieved by a dynamic output feedback controller described by x^_ i ¼ x^ iþ1 þ ai r iμ ðy x^ 1 Þ; i ¼ 1; …; n 1 x^_ n ¼ u þ an r nμ ðy x^ 1 Þ u ¼ r nμ k1 x^ 1 þ r μ k2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n ; with the gain r updated by 8 μ > < r_ ¼ 1 max ϖ ðu; yÞ r ; 0 ; αr 2μ 1 2 > : rð0Þ ¼ r 0 Z 1;
ð22Þ
ð23Þ
where ai 40, ki 40, i¼ 1,…,n, μ40 and α40 are determined by Lemma 2.3, x^ ¼ ð^x 1 ; …; x^ n ÞT is the observer state vector with the initial value x^ ð0Þ ¼ x^ 0 .
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2535
Proof. Theorem 4.1 can be proved in a fashion similar to that of Theorem 3.1. Thus, we give only a brief illustration in the following. Observing that by construction, we have r_ ðtÞ Z 0 and rðtÞ Z rðt τi Þ, for t A ½0; þ1Þ. Under Assumption 4.1, we give the following estimate instead of Eq. (12): σ T 2σε PF r 2μ h0;1 ðu; yÞ‖ε‖2 þ h0;2 ðu; yÞ Jε J Jz J r 1 σ nX þ 2μ g ðuðt τi Þ; yðt τi ÞÞJε J Jεðt τi ÞJ r i ¼ 1 i;1 þgi;2 ðuðt τi Þ; yðt τi ÞÞ Jε J Jzðt τi ÞJ;
ð24Þ
where h0;1 ðu; yÞ ¼ 2nðn 1ÞJ P JcðuÞð1 þ jyjp Þ, h0;2 ðu; yÞ ¼ 2ðn 1Þk~ J PJ cðuÞð1 þ jyjp Þ, gi;1 ðu; yÞ ¼ 2nðn 1ÞJP J c~ ðuðt τi ÞÞð1 þ jyðt τi Þjp Þ and gi;2 ðu; yÞ ¼ 2ðn 1Þk~ JP J c~ ðuðt τi ÞÞ ð1 þ jyðt τi Þjp Þ. Substituting Eqs. (10) and (24) into Eq. (9), we have " !# Pn 1 _ 1 1 h ðu; yÞ ðc þ c Þ r 0;2 i;1 i;2 i ¼ 1 þ V_ r σ μ α þ 2μ h0;1 ðu; yÞ þ ‖ε‖2 2r 2ϵ 2 r r þ þ
1 1 ϵ r_ ‖z‖2 α ‖z‖2 þ 2μ σh0;2 ðu; yÞ‖z‖2 μ 2r r 2 r nX 1 gi;1 ðuðt τi Þ; yðt τi ÞÞ‖εðt τi Þ‖ 2 σ i¼1 nX 1
2ci;1
r μ ðt τi Þ
gi;2 ðuðt τi Þ; yðt τi ÞÞ‖zðt τi Þ‖ 2 σ ; r μ ðt τi Þ 2ci;2 i¼1
where ϵ, ci;1 , ci;2 , i ¼ 1; 2; …; n 1, are any positive constants. Further, we choose a Lyapunov–Krasovskii function as follows: Z t nX 1 gi;1 ðuðsÞ; yðsÞÞ JεðsÞJ 2 σ W ¼V þ ds r μ ðsÞ 2ci;1 t τi i¼1 Z t nX 1 gi;2 ðuðsÞ; yðsÞÞ JzðsÞJ 2 σ þ ds; r μ ðsÞ 2ci;2 t τi i¼1 where V is defined by Eq. (8). With the aid of Eq. (25), on the interval ½0; t f Þ, the derivative of W is given by μ μ _ r σ r þ αr 2μ 1 r_ η1 ðu; yÞ ‖ε‖2 1 r þ αr 2μ 1 r_ η2 ðu; yÞ J zJ 2 W r 2μ 2 r 2μ 2 with
! 1 g2i;1 ðu; yÞ h0;2 ðu; yÞ 1 nX þ η1 ðu; yÞ ¼ h0;1 ðu; yÞ þ ci;1 þ ci;2 þ ; 2ϵ 2i¼1 ci;1 1 ϵ nX σ 2 g ðu; yÞ: η2 ðu; yÞ ¼ σh0;2 ðu; yÞ þ 2 i ¼ 1 2ci;2 i;2
ð25Þ
ð26Þ
ð27Þ
2536
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
In view of Eq. (4), choose a positive continuous function ϖðu; yÞ as follows: ϖðu; yÞ ¼ maxfη1 ðu; yÞ; η2 ðu; yÞg þ κ;
ð28Þ
where κ is any positive constant. Combining Eq. (27) with Eq. (28), one gets _ r σκ ‖ε‖2 κ J zJ 2 : W r 2μ r 2μ
ð29Þ
By Eq. (29) and the Lyapunov–Krasovskii theorem, similar to the counterpart of the proof of Theorem 3.1, it can be concluded that Eqs. (22) and (23) are the output feedback controller of Eq. (21). The proof of Theorem 4.1 has been completed. □ 5. Numerical example Consider the feedforward nonlinear time-delay system: x_ 1 ¼ x2 þ θ1 y1=5 ðt 5Þuðt 5Þx3 ðt 5Þ x_ 2 ¼ x3 þ θ2 u2 x_ 3 ¼ u y ¼ x1 ;
ð30Þ
where 1 r θ1 r 1 and 1r θ2 r 1 are unknown constants. Due to the existence of y1=5 ðt 5Þ, the problem of global regulation of system (30) via output feedback cannot be solved by using the control scheme of [28]. But it is easy to check that system (30) satisfies Assumption 4.1. Thus, choosing ν ¼ μ ¼ 1; a ¼ ða1 ; a2 ; a3 Þ ¼ ð2; 2; 1Þ; k ¼ ðk1 ; k2 ; k3 Þ ¼ ð1; 2; 1Þ; α ¼ 10, the global output feedback controller can be constructed as follows: 2 x^_ 1 ¼ x^ 2 þ ðy x^ 1 Þ r 2 x^_ 2 ¼ x^ 3 þ 2 ðy x^ 1 Þ r _x^ 3 ¼ u þ 1 ðy x^ 1 Þ r3 1 u ¼ 3 x^ 1 þ 2r x^ 2 þ r 2 x^ 3 ; r with the gain r updated by 8 ( ! 2 2 2 2 2 2 1=2 > r > < r_ ¼ 1 max 8 x^ 1 þ x^ 2 þ x^ 3 1 þ y2=5 þ 90 x^ 1 þ x^ 2 þ x^ 3 2=5 1=2 1þy þ 90 ; 0 ; 10r 2 r6 r4 r2 r6 r4 r2 > > : rð0Þ ¼ 1:
ð31Þ By choosing the system initial states to be ðx1 ð0Þ; x2 ð0Þ; x3 ð0Þ; x^ 1 ð0Þ; x^ 2 ð0Þ; x^ 3 ð0ÞÞ ¼ ð2; 3; 5; 1; 1; 1Þ and θ1 ¼ θ2 ¼ 1, the simulation results are shown in Figs. 1–4.
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2537
6. Conclusion This paper has shown that it is possible to achieve global regulation of feedforward delay-free systems and feedforward time-delay systems by output feedback. Motivated by the related literature [8,28], we construct an output feedback controller which can be regarded as an extension of [28]. Ascribed to the introduction of the design parameters μ, the output feedback controller has the ability to deal with the feedforward nonlinear system under less conservative assumptions. Acknowledgments This work was supported by the National Natural Science Foundation of PR China under Grants 61174038, 61374086, 61374087, 61403178, the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT13072, a project funded by the priority academic program development of Jiangsu Higher Education Institutions.
References [1] F. Mazenc, L. Praly, W.P. Dayawansa, Global stabilization by output feedback: examples and counterexamples, Syst. Control Lett. 23 (2) (1994) 119–125. [2] C.J. Qian, W. Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Trans. Autom. Control 47 (10) (2002) 1710–1715. [3] B. Yang, W. Lin, Further results on global stabilization of uncertain nonlinear systems by output feedback, Int. J. Robust Nonlinear Control 15 (2005) 247–268. [4] L. Praly, Z.P. Jiang, On global output feedback stabilization of uncertain nonlinear systems, in: Proceedings of the 42nd IEEE CDC, Maui, Hawaii, 2003, pp. 1544–1549. [5] C.J. Qian, W. Lin, Smooth output feedback stabilization of planar systems without controllable/observable linearization, IEEE Trans. Autom. Control 47 (2002) 2068–2073. [6] H.L. Choi, J.T. Lim, Stabilization of a class of nonlinear systems by adaptive output feedback, Automatica 41 (2005) 1091–1097. [7] H. Lei, W. Lin, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica 42 (10) (2006) 1783–1789. [8] H. Lei, W. Lin, Adaptive regulation of uncertain nonlinear systems by output feedback: a universal control approach, Syst. Control Lett. 56 (2007) 529–537. [9] Y.G. Liu, Global stabilization by output feedback for a class of nonlinear systems with uncertain control coefficients and unmeasured states dependent growth, Sci. China: Inf. Sci. 51 (2008) 1508–1520. [10] F. Shang, Y.G. Liu, C.H. Zhang, Adaptive output feedback stabilization for a class of nonlinear systems with inherent nonlinearities and uncertainties, Int. J. Robust Nonlinear Control 21 (2011) 157–176. [11] Y.G. Liu, Global asymptotic regulation via time-varying output feedback for a class of uncertain nonlinear systems, SIAM J. Control Optim. 51 (2013) 4318–4342. [12] W.Q. Ai, J.Y. Zhai, Universal adaptive regulation for a class of nonlinear systems with unknown time delays and output function via output feedback, J. Frankl. Inst. 350 (2013) 3168–3187. [13] J.W. Lei, Y.D. Wang, Research on two kinds of universal output feedback adaptive controller, J. Frankl. Inst. 350 (2013) 637–650. [14] B. Yang, W. Lin, Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback, IEEE Trans. Autom. Control 49 (2004) 1069–1080. [15] B. Yang, W. Lin, Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization, IEEE Trans. Autom. Control 50 (2005) 619–630. [16] C. Qian, W. Lin, Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems, IEEE Trans. Autom. Control 51 (2006) 1457–1471.
2538
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
[17] C. Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, in: Proceedings of American Control Conference, June 2005, pp. 4708–4715. [18] J. Polendo, C. Qian, A generalized homogeneous approach for global stabilization of inherently nonlinear systems via output feedback, Int. J. Robust Nonlinear Control 17 (2007) 605–629. [19] H. Lei, W. Lin, Robust control of uncertain systems with polynomial nonlinearity by output feedback, Int. J. Robust Nonlinear Control 19 (2009) 692–723. [20] J. Li, C. Qian, M. Frye, A dual-observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities, Int. J. Robust Nonlinear Control 19 (2009) 1697–1720. [21] P. Krishnamurthy, F. Khorrami, A high-gain scaling technique for adaptive output feedback control of feedforward systems, IEEE Trans. Autom. Control 49 (2004) 2286–2292. [22] H.L. Choi, J.T. Lim, Global exponential stabilization of a class of nonlinear systems by output feedback, IEEE Trans. Autom. Control 50 (2005) 255–257. [23] X. Zhang, Y. Lin, Global adaptive stabilisation of feedforward systems by smooth output feedback, IET Control Theory Appl. 6 (2012) 2134–2141. [24] X. Zhang, Y. Lin, Global output feedback stabilisation of nonlinear systems with a unifying linear controller structure, Int. J. Control 87 (2014) 320–329. [25] C. Qian, J. Li, Global output feedback stabilization of upper-triangular nonlinear systems using a homogeneous domination approach, Int. J. Robust Nonlinear Control 16 (2006) 441–463. [26] J.Y. Zhai, C. Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlinear Control 22 (2012) 1543–1561. [27] J.Y. Zhai, W. Zha, Global adaptive output feedback control for a class of nonlinear time-delay systems, ISA Trans. 53 (2014) 2–9. [28] X.F. Zhang, L. Baron, Q. Liu, E.K. Boukas, Design of stabilizing controllers with a dynamic gain for feedforward nonlinear time-delay systems, IEEE Trans. Autom. Control 56 (2011) 692–697. [29] C.J. Qian, W. Lin, Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Syst. Control Lett. 42 (2001) 185–200. [30] L. Praly, Z.P. Jiang, Linear output feedback with dynamic high gain for nonlinear systems, Syst. Control Lett. 53 (2004) 107–116.
Journal of the Franklin Institute 352 (2015) 2526–2538 www.elsevier.com/locate/jfranklin
Global state regulation by output feedback for feedforward systems with input and output dependent incremental rate Xianglei Jiaa, Shengyuan Xua,n, Ticao Jiaoa, Yuming Chub, Yun Zoua a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China b School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, PR China Received 19 May 2014; received in revised form 16 February 2015; accepted 16 March 2015 Available online 2 April 2015
Abstract In this paper, the problem of global state regulation by output feedback is considered for a class of uncertain feedforward nonlinear systems with free-delay or time-delay states. Compared with existing results, we construct a novel observer-based controller with gain exponent to achieve the global state regulation via single output feedback under less conservative assumptions. Further, by using the Lyapunov– Krasovskii theorem, we show that the control scheme proposed is available to a class of time-delay systems with input and output dependent incremental rate. Finally, a numerical example is given to illustrate the usefulness of our results. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The problem of global output feedback control is one of the most important and challenging problems in the field of nonlinear control, which has received a lot of attention over the past decades. Compared with linear systems, the separation principle does not hold for nonlinear systems, as shown in [1]. Thus, it is necessary to impose some extra restrictive conditions on nonlinear terms to obtain global output feedback controller. In the case when uncertain nonlinear systems dominated by a lower-triangular system with linear growth in unmeasured states, the n
Corresponding author. E-mail address: [email protected] (S. Xu).
http://dx.doi.org/10.1016/j.jfranklin.2015.03.035 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2527
Fig. 1. The trajectories of x1 and x^ 1 .
problems of global output feedback stabilization or regulation have been addressed in [2–4,6–13]. Specifically, using a feedback domination design method, the global exponential stabilizer was constructed under the linear growth condition with known growth rate in [2]. When the growth rate was a known smooth function of system output, the global state regulation was achieved by an adaptive output feedback controller in [3], where the results of [4,5] were generalized. With the help of switching logic, an adaptive output feedback controller was proposed in the case of unknown constant growth rate in [6]. Unlike the control scheme proposed by [6], universal adaptive high-gain observers were introduced to achieve global output feedback stabilization in [7,8]. For a larger class of nonlinear systems with unknown control coefficients, the global state regulation problem was investigated by output feedback in [9–11]. Furthermore, a universal adaptive output feedback controller was constructed for a class of nonlinear systems with unknown time delays and output function in [12], and an output feedback controller was proposed by introducing double dynamic gains in [13]. To deal with the case of high-order growth, several attempts have been made such as [14–20]. In particular, for a class of nonlinear systems with uncontrollable/unobservable linearization, the problem of output feedback stabilization was handled in [14–16]. In view of homogeneous system theory, a homogeneous domination approach was introduced to achieve output feedback stabilization in [17]. Subsequently, a generalized homogeneous domination approach was established in [18]. In the case of polynomial growth conditions, a recursive design algorithm was developed to achieve output feedback stabilization by constructing a reduced-order observer in [19]. By using dual observers, the global output feedback stabilization was achieved for nonlinear systems with lowerorder and high-order nonlinearities in [20]. On the other hand, the problem of global output feedback control has been investigated for a class of feedforward nonlinear systems in [21–28]. Specifically, when triangular type restriction was not satisfied, a linear output feedback control scheme was proposed to achieve global exponential stabilization in [22]. By introducing dynamic high-gain scaling technique, an adaptive output feedback control scheme was proposed for feedforward systems in [21]. With the aid of the idea of
2528
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Fig. 2. The trajectories of x2 and x^ 2 .
universal control and the dynamic high-gain scaling technique, the global stabilization was achieved by linear output feedback in [23,24]. By the homogeneous domination approach, the problem of output feedback stabilization was handled for a class of nonlinear systems with high-order growth in unmeasurable states in [25]. Further, in [26], such problem was solved for a class of nonlinear systems with uncertain output function. In the presence of time delays, the global state regulation was achieved via linear output feedback in [27,28], where the growth rate depended on unknown constant or nonlinearities of system input. Notably, in the case of lower triangular, [8] has generalized the result of [7]. Similarly, in the feedforward case, the results of [28,23] can be also generalized under suitable assumptions. In this paper, we will consider this question. The main contributions of this paper can be summarized as: First, we construct a high-gain observer with adjustable high-gain exponential, which can be regarded as an extension of [28]. By adopting a new coordinate change, we show that the output feedback controller of this paper is capable of applying to a class of uncertain nonlinear systems, where the growth rate depends not only on system input but also on system output. Second, with the aid of the Lyapunov–Krasovskii theorem, we solve the problem of global state regulation for a class of nonlinear time-delay system by the controller proposed. Notations: In this paper, the argument of the functions will be omitted whenever no confusion can arise from the context. For any real vector or matrix A, AT denotes its transpose; J J denotes the Euclidean norm for the matrix and vector. 2. Preliminaries and key technical lemmas In this section, we briefly discuss what are feedforward nonlinear systems. As shown in [21], the systems of the form x_ i ¼ xiþ1 þ f i ðt; y; xiþ2 ; …; xn ; uÞ; x_ n ¼ u
i ¼ 1; …; n 1
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2529
Fig. 3. The trajectories of x3 and x^ 3 .
y ¼ x1 ; are referred to as feedforward nonlinear systems. Moreover, the following systems are also referred to as feedforward nonlinear systems (see Remark 1 in [28]): x_ i ¼ xiþ1 þ f i ðd; x; uÞ;
i ¼ 1; …; n 1
x_ n ¼ u y ¼ x1 ; with jf i ðÞj r cðuÞðjxiþ2 j þ ⋯ þ jxn j þ jujÞ. In what follows, we introduce three key lemmas. Lemma 2.1 (Qian and Lin [29]). For any positive integers m; n and any real-valued function αðx; yÞ40, then jxjm jyjn r
m n αðx; yÞjxjmþn þ αðx; yÞ m=n ðx; yÞjyjmþn : mþn mþn
Lemma 2.2 (Qian and Lin [29]). For xA R, y A R, p Z 1 is a constant, the following inequalities hold: jx þ yjp r 2p 1 jxp þ yp j: Lemma 2.3 (Praly and Jiang [30]). If matrices A and B are Hurwitz, D ¼ diagðν þ ðn 1Þμ; …; νÞ, then there exist P ¼ PT 40, Q ¼ QT 40 and α40 satisfying ( AT P þ PA r I; DP þ PD Z αI BT Q þ QB r I;
DQ þ QD Z αI;
2530
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Fig. 4. The trajectory of r.
where μ40, ν40, ai 40, 0 1 a1 B ⋮ ⋮ B A¼B @ an 1 0
ki 40 ði ¼ 1; …; nÞ, are suitable design parameters, and 1 0 1 ⋯ 0 0 1 ⋯ 0 B ⋮ ⋱ ⋮C ⋮ ⋱ ⋮ C C B C C A Rnn ; B ¼ B C A Rnn : 0 ⋯ 1 A ⋯ 1A @ 0 k1 k2 ⋯ kn 0 ⋯ 0
an 3. Main result
In this section, an observer-based controller will be constructed to solve the problem of global state regulation by output feedback. Consider the following uncertain nonlinear system: x_ i ¼ xiþ1 þ f i ðd; x; uÞ; x_ n ¼ u y ¼ x1 ;
i ¼ 1; …; n 1 ð1Þ
where x ¼ ðx1 ; …; xn Þ A R is the system state, u A R and y A R are the system input and measured output, respectively, and d : R-Rs is a continuous bounded mapping that represents a family of time-varying parameters or disturbances. The functions f i ; i ¼ 1; …; n 1; are C0 functions with f i ðd; 0Þ ¼ 0, which need not to be precisely known. To solve the problem of global state regulation by output feedback for system (1), we introduce an assumption as follows: T
n
Assumption 3.1. There exist a known constant 0opo1=ðn 1Þ and a known positive continuous function cðÞ such that ! nþ1 X jf i ðd; x; uÞj r cðuÞð1 þ jyjp Þ jxj j þ juj ; xnþ1 ¼ 0; i ¼ 1; …; n 1: j ¼ iþ2
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2531
Remark 3.1. As stated in Remark 1 of [28], system (1) satisfying Assumption 3.1 is referred to as feedforward nonlinear system. In the case when the growth rate was an unknown constant, instead of cðuÞð1 þ jyjp Þ, the problem of global state regulation was solved by output feedback in [23,27]. Moreover, when the growth rate was c(u), such problem was addressed in [28]. Theorem 3.1. Under Assumption 3.1, the global state regulation of system (1) can be achieved by the dynamic output feedback controller described by x^_ i ¼ x^ iþ1 þ ai r iμ ðy x^ 1 Þ; i ¼ 1; …; n 1 x^_ n ¼ u þ an r nμ ðy x^ 1 Þ u ¼ r nμ k 1 x^ 1 þ r μ k 2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n ; ð2Þ with the gain r updated by 8 μ > < r_ ¼ 1 max ϖ ðu; yÞ r ; 0 ; αr 2μ 1 2 > : rð0Þ ¼ r 0 Z 1;
ð3Þ
where ai 40, ki 40, i¼ 1,…,n, μ40 and α40 are determined by Lemma 2.3, x^ ¼ ð^x 1 ; …; x^ n ÞT is the observer state vector with the initial value x^ ð0Þ ¼ x^ 0 . Remark 3.2. Compared with the controller in [28], an extra design parameter μ is introduced in Eqs. (2) and (3). This parameter plays a key role in constructing suitable positive definite matrices P, Q and D by Lemma 2.3, to avoid too large control gain. Proof. As shown in [28], the state r(t) of Eq. (3) has the following properties: ðiÞ : r_ Z 0;
rμ Z ϖ ðÞ: 2 Define ei ¼ xi x^ i , then the error dynamics is given by ai e_ i ¼ eiþ1 iμ ðy x^ 1 Þ þ f i ðÞ; i ¼ 1; …; n 1 r an e_ n ¼ nμ ðy x^ 1 Þ: r Further, consider the following change of coordinates ði ¼ 1; …; nÞ: ðiiÞ : αr 2μ 1 r_ þ
εi ¼
ei νþðn iÞμ r
and
zi ¼
x^ i νþðn iÞμ r
;
ð4Þ
ð5Þ
ð6Þ
by which the closed-loop system (2) and (5) can be represented in the following compact form: r_ ε_ ¼ r μ Aε Dε þ F r r_ z_ ¼ r μ Bz Dz þ r μ aε1 ; ð7Þ r where ε ¼ ðε1 ; …; εn ÞT , z ¼ ðz1 ; …; zn ÞT , a ¼ ða1 ; …; an ÞT and F ¼ f 1 ðÞ=r νþðn 1Þμ ; …; f n 1 ðÞ= r νþμ ; 0ÞT . A, B and D are given by Lemma 2.3. Next, we will prove that all states of system (3) and (7) are bounded on ½0; t f Þ for some t f A ð0; þ1, and system (7), for any gain function r(t) generated by Eq. (3), is asymptotically stable at ε ¼ 0 and z ¼ 0. Without loss of generality, we assume that ½0; t f Þ is the maximally extended interval of the solution of Eqs. (3) and (7).
2532
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
Choose the Lyapunov function V : Rn Rn -Rþ Vðε; zÞ ¼ σV 1 ðεÞ þ V 2 ðzÞ;
ð8Þ
where V 1 ðεÞ ¼ εT Pε; V 2 ðzÞ ¼ zT Qz; P and Q are given by Lemma 2.3, and σ ¼ JM J 2 is to be determined by Eq. (10). With the aid of Lemma 2.3 and the fact that r_ Z140, on the interval ½0; t f Þ, the derivative of V along the solution of Eqs. (3) and (7) is given by r_ σ V_ r μ ‖ε‖2 σα ‖ε‖2 þ 2σεT PF r r 1 2 r_ 2 μ ‖z‖ α ‖z‖2 þ μ ε1 zT Qa: r r r In what follows, we estimate the inequality above. Firstly, based on Lemma 2.1, we have 2 1 1 μ ε1 zT Qar μ ‖z‖2 þ μ ‖M‖2 ‖ε‖2 ; r 2r 2r where M ¼ Qða1 ; a2 ; …; an ÞT ð2; 0; …; 0Þ. In addition, from the definitions of εi and zi, one gets f ðÞ j^x iþ2 j þ jeiþ2 j juj i p ð Þ þ ⋯ þ c ð u Þ 1 þ jyj νþðn r iÞμ r r νþðn iÞμ r νþðn iÞμ 1 r 2μ cðuÞð1 þ jyjp Þ n J εJ þ k~ Jz J ; r where k~ ¼ nmaxi ¼ 1;…;n fk i þ 1g is a known constant. By using Eq. (11) and Lemma 2.1, one has j2σεT PFj r 2σ J PJ J ε J JF J σ r 2μ h1 ðu; yÞ‖ε‖2 þ h2 ðu; yÞJ ε J JzJ ; r where h1 ðu; yÞ ¼ 2nðn 1Þ JP JcðuÞð1 þ jyjp Þ and h2 ðu; yÞ ¼ 2ðn 1Þk~ JP JcðuÞð1 þ jyjp Þ. Based on Lemma 2.2, substituting Eqs. (10) and (12) into Eq. (9) leads to
1 1 h2 ðu; yÞ r_ 2 2 2 _ V r σ μ ‖ε‖ α ‖ε‖ þ 2μ h1 ðu; yÞ þ ‖ε‖ 2r r 2ϵ r 1 1 ϵ r_ μ ‖z‖2 α ‖z‖2 þ 2μ σh2 ðu; yÞ‖z‖2 2r r 2 r σ rμ 1 rμ þ αr 2μ 1 r_ ζ 1 ðu; yÞ ‖ε‖2 2μ þ αr 2μ 1 r_ ζ 2 ðu; yÞ ‖z‖2 ; r 2μ r r 2 2
ð9Þ
ð10Þ
ð11Þ
ð12Þ
ð13Þ
where ϵ is any positive constant. ζ 1 ðu; yÞ ¼ h1 ðu; yÞ þ h2 ðu; yÞ=2ϵ and ζ 2 ðu; yÞ ¼ ðϵ=2Þσh2 ðu; yÞ. In view of Eq. (4), choose a positive continuous function ϖðu; yÞ as follows: ϖðu; yÞ ¼ maxfζ 1 ðu; yÞ; ζ 2 ðu; yÞg þ κ; where κ is any positive constant. Integrating Eqs. (13) and (14), we have σκ κ V_ r 2μ ‖ε‖2 2μ ‖z‖2 : r r
ð14Þ
ð15Þ
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2533
From Eq. (15), it is concluded that V is bounded on the maximal interval of existence of solution ½0; t f Þ. By the definition of V, we can get that the states ε and z are bounded, which implies the following property: Property 3.1. r(t) is bounded on the interval ½0; t f Þ. In fact, for 8 t A ½0; t f Þ y νþðn 1Þμ r jz1 j þ jε1 j r ω1 ¼ constant; r from which we can deduce that jyj r ω1 r νþðn 1Þμ :
ð16Þ
In addition, by Eq. (6) and the boundedness of z and r ν μ r 1 (since ν r μ), we have u ¼ r nμ ðk 1 x^ 1 þ r μ k2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n Þ ¼ r ν μ ðk1 z1 þ k 2 z2 þ ⋯ þ k n zn Þ r ω2 ¼ constant;
ð17Þ
which implies that c(u) is bounded. By using Eq. (16) and the boundedness of c(u) yields ζ 1 ðu; yÞ r ω3 r pνþpðn 1Þμ þ ω4 ζ 2 ðu; yÞ r ω5 r pνþpðn 1Þμ þ ω6
ð18Þ
where ω3, ω4, ω5 and ω6 are suitable unknown constants. By means of Eqs. (14) and (18), one has ϖ ðu; yÞr ω7 r pνþpðn 1Þμ þ ω8 rμ 1 ϖ ðu; yÞ r r pνþpðn 1Þμ r μ pðn 1Þμ pν 2ω7 þ ω8 ; 2 2 where ω7 and ω8 are unknown constants. Since po1=ðn 1Þ, we can always find a pair of constants ðν; μÞ such that μ pðn 1Þμ pν40:
ð19Þ
ð20Þ
With this in mind, suppose r is unbounded, together with r Z 1 is C , then limt-tf ¼ þ 1, and there exists a finite time t 0 A ð0; t f Þ, when t 0 r t rt f , such that rμ 1 ϖ ðu; yÞ r r pνþpðn 1Þμ r μ pðn 1Þμ pν 2ω7 þ ω8 o0; 2 2 1
i.e., r_ ¼ 0, which leads to a contradiction. Hence, r(t) is bounded on ½0; t f Þ. Up to now, it has been proved that all signals of the closed system (3) and (7) are bounded. Thus, t f ¼ þ 1 and solutions exist for all time. In view of Eq. (15), we can conclude that the closed-loop system (7), for any control gain function r(t) generated by Eq. (3), is asymptotically stable at ε ¼ 0 and z¼ 0. Further, by Eq. (6) and the boundedness of r, one gets that the closedloop system (1) and (2), for any control gain function r generated by Eq. (3), is asymptotically stable at x ¼ x^ ¼ ð^x 1 ; …; x^ n ÞT ¼ 0. The proof of Theorem 3.1 has been completed. □
2534
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
4. Extending and discussion The control scheme in the previous section can be applied to time-delay nonlinear system as follows: x_ i ¼ xiþ1 þ f i ðd; x; u; xðt τi Þ; uðt τi ÞÞ; x_ n ¼ u y ¼ x1 ;
i ¼ 1; …; n 1 ð21Þ
where x ¼ ðx1 ; …; xn Þ A R is the system state, u A R and y A R are the system input and measured output, respectively, and d : R-Rs is a continuous bounded mapping that represents a family of time-varying parameters or disturbances. The functions f i ; i ¼ 1; …; n 1; are C0 functions with f i ðd; 0Þ ¼ 0, which need not to be precisely known. Constants τi Z 0, i ¼ 1; …; n 1; are unknown delays. For system (21), we introduce the following assumption, which is less conservative than the one in [28]. T
n
~ the following Assumption 4.1. For i ¼ 1; …; n 1; and any ðd; x1 ; …; xn ; u; x~ 1 ; …; x~ n ; uÞ, conditions hold: ! nþ1 X p ~ r ci ðuÞð1 þ jx1 j Þ jxj j þ juj jf i ðd; x1 ; …; xn ; u; x~ 1 ; …; x~ n ; uÞj ~ þ c~ i ðuÞð1 þ j~x 1 jp Þ
nþ1 X
!
j ¼ iþ2
~ ; j~x j j þ juj
j ¼ iþ2
where 0 r po1=ðn 1Þ is a known constant, xnþ1 ¼ x~ nþ1 ¼ 0, ci ðÞ and c~ i ðÞ are known positive continuous functions. Remark 4.1. Assumption 4.1 means that the growth rate is related with not only system input but also system output. For such feedforward nonlinear system with time delays, the output feedback controller of [28] is unavailable. Therefore, inspired by the case of lower triangular [8], we construct a new output feedback controller, where the dynamic gain is dependent on system output. Theorem 4.1. Under Assumption 4.1, the global state regulation of system (21) can be achieved by a dynamic output feedback controller described by x^_ i ¼ x^ iþ1 þ ai r iμ ðy x^ 1 Þ; i ¼ 1; …; n 1 x^_ n ¼ u þ an r nμ ðy x^ 1 Þ u ¼ r nμ k1 x^ 1 þ r μ k2 x^ 2 þ ⋯ þ r ðn 1Þμ kn x^ n ; with the gain r updated by 8 μ > < r_ ¼ 1 max ϖ ðu; yÞ r ; 0 ; αr 2μ 1 2 > : rð0Þ ¼ r 0 Z 1;
ð22Þ
ð23Þ
where ai 40, ki 40, i¼ 1,…,n, μ40 and α40 are determined by Lemma 2.3, x^ ¼ ð^x 1 ; …; x^ n ÞT is the observer state vector with the initial value x^ ð0Þ ¼ x^ 0 .
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2535
Proof. Theorem 4.1 can be proved in a fashion similar to that of Theorem 3.1. Thus, we give only a brief illustration in the following. Observing that by construction, we have r_ ðtÞ Z 0 and rðtÞ Z rðt τi Þ, for t A ½0; þ1Þ. Under Assumption 4.1, we give the following estimate instead of Eq. (12): σ T 2σε PF r 2μ h0;1 ðu; yÞ‖ε‖2 þ h0;2 ðu; yÞ Jε J Jz J r 1 σ nX þ 2μ g ðuðt τi Þ; yðt τi ÞÞJε J Jεðt τi ÞJ r i ¼ 1 i;1 þgi;2 ðuðt τi Þ; yðt τi ÞÞ Jε J Jzðt τi ÞJ;
ð24Þ
where h0;1 ðu; yÞ ¼ 2nðn 1ÞJ P JcðuÞð1 þ jyjp Þ, h0;2 ðu; yÞ ¼ 2ðn 1Þk~ J PJ cðuÞð1 þ jyjp Þ, gi;1 ðu; yÞ ¼ 2nðn 1ÞJP J c~ ðuðt τi ÞÞð1 þ jyðt τi Þjp Þ and gi;2 ðu; yÞ ¼ 2ðn 1Þk~ JP J c~ ðuðt τi ÞÞ ð1 þ jyðt τi Þjp Þ. Substituting Eqs. (10) and (24) into Eq. (9), we have " !# Pn 1 _ 1 1 h ðu; yÞ ðc þ c Þ r 0;2 i;1 i;2 i ¼ 1 þ V_ r σ μ α þ 2μ h0;1 ðu; yÞ þ ‖ε‖2 2r 2ϵ 2 r r þ þ
1 1 ϵ r_ ‖z‖2 α ‖z‖2 þ 2μ σh0;2 ðu; yÞ‖z‖2 μ 2r r 2 r nX 1 gi;1 ðuðt τi Þ; yðt τi ÞÞ‖εðt τi Þ‖ 2 σ i¼1 nX 1
2ci;1
r μ ðt τi Þ
gi;2 ðuðt τi Þ; yðt τi ÞÞ‖zðt τi Þ‖ 2 σ ; r μ ðt τi Þ 2ci;2 i¼1
where ϵ, ci;1 , ci;2 , i ¼ 1; 2; …; n 1, are any positive constants. Further, we choose a Lyapunov–Krasovskii function as follows: Z t nX 1 gi;1 ðuðsÞ; yðsÞÞ JεðsÞJ 2 σ W ¼V þ ds r μ ðsÞ 2ci;1 t τi i¼1 Z t nX 1 gi;2 ðuðsÞ; yðsÞÞ JzðsÞJ 2 σ þ ds; r μ ðsÞ 2ci;2 t τi i¼1 where V is defined by Eq. (8). With the aid of Eq. (25), on the interval ½0; t f Þ, the derivative of W is given by μ μ _ r σ r þ αr 2μ 1 r_ η1 ðu; yÞ ‖ε‖2 1 r þ αr 2μ 1 r_ η2 ðu; yÞ J zJ 2 W r 2μ 2 r 2μ 2 with
! 1 g2i;1 ðu; yÞ h0;2 ðu; yÞ 1 nX þ η1 ðu; yÞ ¼ h0;1 ðu; yÞ þ ci;1 þ ci;2 þ ; 2ϵ 2i¼1 ci;1 1 ϵ nX σ 2 g ðu; yÞ: η2 ðu; yÞ ¼ σh0;2 ðu; yÞ þ 2 i ¼ 1 2ci;2 i;2
ð25Þ
ð26Þ
ð27Þ
2536
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
In view of Eq. (4), choose a positive continuous function ϖðu; yÞ as follows: ϖðu; yÞ ¼ maxfη1 ðu; yÞ; η2 ðu; yÞg þ κ;
ð28Þ
where κ is any positive constant. Combining Eq. (27) with Eq. (28), one gets _ r σκ ‖ε‖2 κ J zJ 2 : W r 2μ r 2μ
ð29Þ
By Eq. (29) and the Lyapunov–Krasovskii theorem, similar to the counterpart of the proof of Theorem 3.1, it can be concluded that Eqs. (22) and (23) are the output feedback controller of Eq. (21). The proof of Theorem 4.1 has been completed. □ 5. Numerical example Consider the feedforward nonlinear time-delay system: x_ 1 ¼ x2 þ θ1 y1=5 ðt 5Þuðt 5Þx3 ðt 5Þ x_ 2 ¼ x3 þ θ2 u2 x_ 3 ¼ u y ¼ x1 ;
ð30Þ
where 1 r θ1 r 1 and 1r θ2 r 1 are unknown constants. Due to the existence of y1=5 ðt 5Þ, the problem of global regulation of system (30) via output feedback cannot be solved by using the control scheme of [28]. But it is easy to check that system (30) satisfies Assumption 4.1. Thus, choosing ν ¼ μ ¼ 1; a ¼ ða1 ; a2 ; a3 Þ ¼ ð2; 2; 1Þ; k ¼ ðk1 ; k2 ; k3 Þ ¼ ð1; 2; 1Þ; α ¼ 10, the global output feedback controller can be constructed as follows: 2 x^_ 1 ¼ x^ 2 þ ðy x^ 1 Þ r 2 x^_ 2 ¼ x^ 3 þ 2 ðy x^ 1 Þ r _x^ 3 ¼ u þ 1 ðy x^ 1 Þ r3 1 u ¼ 3 x^ 1 þ 2r x^ 2 þ r 2 x^ 3 ; r with the gain r updated by 8 ( ! 2 2 2 2 2 2 1=2 > r > < r_ ¼ 1 max 8 x^ 1 þ x^ 2 þ x^ 3 1 þ y2=5 þ 90 x^ 1 þ x^ 2 þ x^ 3 2=5 1=2 1þy þ 90 ; 0 ; 10r 2 r6 r4 r2 r6 r4 r2 > > : rð0Þ ¼ 1:
ð31Þ By choosing the system initial states to be ðx1 ð0Þ; x2 ð0Þ; x3 ð0Þ; x^ 1 ð0Þ; x^ 2 ð0Þ; x^ 3 ð0ÞÞ ¼ ð2; 3; 5; 1; 1; 1Þ and θ1 ¼ θ2 ¼ 1, the simulation results are shown in Figs. 1–4.
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
2537
6. Conclusion This paper has shown that it is possible to achieve global regulation of feedforward delay-free systems and feedforward time-delay systems by output feedback. Motivated by the related literature [8,28], we construct an output feedback controller which can be regarded as an extension of [28]. Ascribed to the introduction of the design parameters μ, the output feedback controller has the ability to deal with the feedforward nonlinear system under less conservative assumptions. Acknowledgments This work was supported by the National Natural Science Foundation of PR China under Grants 61174038, 61374086, 61374087, 61403178, the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT13072, a project funded by the priority academic program development of Jiangsu Higher Education Institutions.
References [1] F. Mazenc, L. Praly, W.P. Dayawansa, Global stabilization by output feedback: examples and counterexamples, Syst. Control Lett. 23 (2) (1994) 119–125. [2] C.J. Qian, W. Lin, Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm, IEEE Trans. Autom. Control 47 (10) (2002) 1710–1715. [3] B. Yang, W. Lin, Further results on global stabilization of uncertain nonlinear systems by output feedback, Int. J. Robust Nonlinear Control 15 (2005) 247–268. [4] L. Praly, Z.P. Jiang, On global output feedback stabilization of uncertain nonlinear systems, in: Proceedings of the 42nd IEEE CDC, Maui, Hawaii, 2003, pp. 1544–1549. [5] C.J. Qian, W. Lin, Smooth output feedback stabilization of planar systems without controllable/observable linearization, IEEE Trans. Autom. Control 47 (2002) 2068–2073. [6] H.L. Choi, J.T. Lim, Stabilization of a class of nonlinear systems by adaptive output feedback, Automatica 41 (2005) 1091–1097. [7] H. Lei, W. Lin, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica 42 (10) (2006) 1783–1789. [8] H. Lei, W. Lin, Adaptive regulation of uncertain nonlinear systems by output feedback: a universal control approach, Syst. Control Lett. 56 (2007) 529–537. [9] Y.G. Liu, Global stabilization by output feedback for a class of nonlinear systems with uncertain control coefficients and unmeasured states dependent growth, Sci. China: Inf. Sci. 51 (2008) 1508–1520. [10] F. Shang, Y.G. Liu, C.H. Zhang, Adaptive output feedback stabilization for a class of nonlinear systems with inherent nonlinearities and uncertainties, Int. J. Robust Nonlinear Control 21 (2011) 157–176. [11] Y.G. Liu, Global asymptotic regulation via time-varying output feedback for a class of uncertain nonlinear systems, SIAM J. Control Optim. 51 (2013) 4318–4342. [12] W.Q. Ai, J.Y. Zhai, Universal adaptive regulation for a class of nonlinear systems with unknown time delays and output function via output feedback, J. Frankl. Inst. 350 (2013) 3168–3187. [13] J.W. Lei, Y.D. Wang, Research on two kinds of universal output feedback adaptive controller, J. Frankl. Inst. 350 (2013) 637–650. [14] B. Yang, W. Lin, Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback, IEEE Trans. Autom. Control 49 (2004) 1069–1080. [15] B. Yang, W. Lin, Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization, IEEE Trans. Autom. Control 50 (2005) 619–630. [16] C. Qian, W. Lin, Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems, IEEE Trans. Autom. Control 51 (2006) 1457–1471.
2538
X. Jia et al. / Journal of the Franklin Institute 352 (2015) 2526–2538
[17] C. Qian, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, in: Proceedings of American Control Conference, June 2005, pp. 4708–4715. [18] J. Polendo, C. Qian, A generalized homogeneous approach for global stabilization of inherently nonlinear systems via output feedback, Int. J. Robust Nonlinear Control 17 (2007) 605–629. [19] H. Lei, W. Lin, Robust control of uncertain systems with polynomial nonlinearity by output feedback, Int. J. Robust Nonlinear Control 19 (2009) 692–723. [20] J. Li, C. Qian, M. Frye, A dual-observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities, Int. J. Robust Nonlinear Control 19 (2009) 1697–1720. [21] P. Krishnamurthy, F. Khorrami, A high-gain scaling technique for adaptive output feedback control of feedforward systems, IEEE Trans. Autom. Control 49 (2004) 2286–2292. [22] H.L. Choi, J.T. Lim, Global exponential stabilization of a class of nonlinear systems by output feedback, IEEE Trans. Autom. Control 50 (2005) 255–257. [23] X. Zhang, Y. Lin, Global adaptive stabilisation of feedforward systems by smooth output feedback, IET Control Theory Appl. 6 (2012) 2134–2141. [24] X. Zhang, Y. Lin, Global output feedback stabilisation of nonlinear systems with a unifying linear controller structure, Int. J. Control 87 (2014) 320–329. [25] C. Qian, J. Li, Global output feedback stabilization of upper-triangular nonlinear systems using a homogeneous domination approach, Int. J. Robust Nonlinear Control 16 (2006) 441–463. [26] J.Y. Zhai, C. Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlinear Control 22 (2012) 1543–1561. [27] J.Y. Zhai, W. Zha, Global adaptive output feedback control for a class of nonlinear time-delay systems, ISA Trans. 53 (2014) 2–9. [28] X.F. Zhang, L. Baron, Q. Liu, E.K. Boukas, Design of stabilizing controllers with a dynamic gain for feedforward nonlinear time-delay systems, IEEE Trans. Autom. Control 56 (2011) 692–697. [29] C.J. Qian, W. Lin, Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Syst. Control Lett. 42 (2001) 185–200. [30] L. Praly, Z.P. Jiang, Linear output feedback with dynamic high gain for nonlinear systems, Syst. Control Lett. 53 (2004) 107–116.