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Semi-Global Stabilization for a Class of Nonlinear Time-Delay Systems by Linear Output Feedback∗
Proceedings Proceedings of of the the 12th 12th IFAC IFAC Workshop Workshop on on Time Time Delay Delay Systems Systems June 28-30, Ann Arbor, USA Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems June 28-30, 2015. 2015. Ann Arbor, MI, USA Proceedings of 12th IFAC Workshop on Available online at www.sciencedirect.com June June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USA
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Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Nonlinear Time-Delay Systems by Linear Nonlinear Time-Delay Systems by Linear Nonlinear Time-Delay Systems by Linear ⋆⋆ by Linear Nonlinear Time-Delay Systems Output Feedback Output Feedback Output Feedback Feedback ⋆⋆ Output
∗ ∗ ∗ ∗ Pan Pan Wang Wang ∗∗ ,, Lin Lin Chai Chai ∗∗ ,, Shumin Shumin Fei Fei ∗∗ ,, Guquan Guquan Liu Liu ∗∗ Pan Pan Wang Wang ∗ ,, Lin Lin Chai Chai ∗ ,, Shumin Shumin Fei Fei ∗ ,, Guquan Guquan Liu Liu ∗ ∗ ∗ Key Laboratory of Measurement and Control of Complex Systems of ∗ Key Laboratory of Measurement and Control of Complex Systems of ∗Engineering, Key of and Control of of Ministry of of Southeast Key Laboratory Laboratory of Measurement Measurement andSchool Control of Complex Complex Systems Systems of Engineering, Ministry of Education; Education; School of Automation, Automation, Southeast Engineering, Ministry of Education; School of Automation, Southeast University, Nanjing, 210096,P.R.China (e-mails: [email protected]; Engineering, Ministry of Education; School of Automation, Southeast University, Nanjing, 210096,P.R.China (e-mails: [email protected]; University, 210096,P.R.China (e-mails: [email protected]; chailin [email protected]; [email protected]; [email protected]). University, Nanjing, 210096,P.R.China (e-mails: [email protected]; chailinNanjing, [email protected]; [email protected]; [email protected]). chailin [email protected]; [email protected]; [email protected]). chailin [email protected]; [email protected]; [email protected]). Abstract: The The semi-global semi-global stabilization stabilization problem problem for for aa class class of of nonlinear nonlinear systems systems with with state state Abstract: Abstract: The semi-global stabilization problem for a class of nonlinear systems with state time-delay is addressed in this paper. Sufficient conditions are derived by using LyapunovAbstract: The semi-global stabilization problem conditions for a class are of nonlinear with state time-delay is addressed in this paper. Sufficient derived bysystems using Lyapunovtime-delay is in Sufficient conditions derived by using Krasovskii functional functional method andpaper. adjusting the scaling scaling gain to toare guarantee the systems as well well as as time-delay is addressed addressed in this this paper. Sufficient conditions are derivedthe by systems using LyapunovLyapunovKrasovskii method and adjusting the gain guarantee as Krasovskii method and adjusting the to the as the resulted resultedfunctional closed-loop systems under linear output gain feedback controller tosystems be semi-globally semi-globally Krasovskii functional method andunder adjusting the scaling scaling gain to guarantee guarantee theto systems as well well as as the closed-loop systems aa linear output feedback controller be the resulted closed-loop systems under output controller be stable. Numerical example is presented presented tolinear illustrate the feedback effectiveness of the the to obtained results in in the resulted closed-loop systems under aato linear output feedback controller to be semi-globally semi-globally stable. Numerical example is illustrate the effectiveness of obtained results stable. Numerical example is presented to illustrate the effectiveness of the obtained results in this paper. stable. Numerical example is presented to illustrate the effectiveness of the obtained results in this paper. this paper. this paper. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Semi-global, nonlinear nonlinear time-delay time-delay systems, systems, Lyapunov-Krasovskii Lyapunov-Krasovskii functional, functional, linear linear Keywords: Semi-global, Keywords: Semi-global, nonlinear time-delay systems, Lyapunov-Krasovskii functional, linear observer, output feedback. Keywords: Semi-global, nonlinear time-delay systems, Lyapunov-Krasovskii functional, linear observer, output feedback. observer, output output feedback. feedback. observer, 1. INTRODUCTION INTRODUCTION of 1. of instability instability and and poor poor performance performance of of many many industrial industrial 1. INTRODUCTION INTRODUCTION of instability and poor performance of many industrial industrial and engineering systems. Consequently, stability analysis 1. of instability and poor performance of many and engineering systems. Consequently, stability analysis and engineering systems. Consequently, stability analysis synthesis for time-delay systems have been one of of Stabilization by output feedback is one of the important and engineering systems. Consequently, stability analysis synthesis for time-delay systems have been one Stabilization by output feedback is one of the important and synthesis for time-delay systems have been of the most challenging issues and many nice results have Stabilization by output feedback is one of the important problems in in the the field of offeedback nonlinearis control. control. Unlike linear and synthesis for time-delay systems one of the most challenging issues and many have nice been resultsone have Stabilization by output one of the important problems field nonlinear Unlike linear the most challenging issues and many nice results have been proposed (see Chai (2013), Zhai et al. (2014), Ye problems in the field of nonlinear control. Unlike linear systems, separation principle usually does not hold for challenging issues and many niceal.results beenmost proposed (see Chai (2013), Zhai et (2014),have Ye problems in the fieldprinciple of nonlinear control. linear systems, separation usually does Unlike not hold for the been proposed (see(2004), Chai (2013), (2013), Zhai et (2007) al. (2014), (2014), Ye (2011), Wu Yakoubi et and systems, separation principle usually does feedback not hold holdplus for been nonlinear separation systems which which leads to to that state state feedback plus proposed (see Chai al. Ye (2011), Wu et et al. al. (2004), YakoubiZhai et al. al.et (2007) and the the systems, principle usually does not for nonlinear systems leads that (2011), Wu et al. (2004), Yakoubi et al. (2007) and the references therein). The most commonly used method for nonlinear systems which leads to that state feedback plus observability simply do not imply stabilizability by output (2011), Wu et al. (2004), Yakoubi et al. (2007) and the references therein). The most commonly used method for nonlinear systems which leads to that state feedback plus observability simply do not imply stabilizability by output references therein). The used the observability simply do not not implythe stabilizability by output solving feedback. For Forsimply the above above reason, the problem of of by stabilizareferences therein). problems The most most is commonly used method method for for solving time-delay time-delay problems iscommonly the Lyapunov-Krasovskii Lyapunov-Krasovskii observability do imply stabilizability output feedback. the reason, problem stabilizasolving time-delay problems is the Lyapunov-Krasovskii functional approach (Gu et al. (2003)). The feedback. For the above reason, the problem of stabilization by output feedback is much more difficult and chaltime-delay Lyapunov-Krasovskii functional approachproblems (Gu et is al.the (2003)). The controller controller feedback. For the above reason, problem of and stabilization by output feedback is muchthe more difficult chal- solving approach et (2003)). design for class nonlinear tion by than output feedback is much much morefeedback difficultfor and challenging than thefeedback stabilization by state state feedback for nonlinfunctional approach (Gu et of al.strict-feedback (2003)). The The controller controller design problem problem for aa(Gu class ofal. strict-feedback nonlinear tion by output is more difficult and chal- functional lenging the stabilization by nonlindesign problem for a class of strict-feedback nonlinear systems with time delays is solved in Hua et al. (2009). lenging than the stabilization by state feedback for nonlinear systems. Over the years, a large amount of papers have design problem for a class of strict-feedback nonlinear systems with time delays is solved in Hua et al. (2009). lenging than the stabilization by state feedback for nonlinear systems. Over the years, a large amount of papers have systems with time delays is solved in Hua et al. Sun et al. (2011) investigated a class of time-delay highear systems. Over the years, a large amount of papers have investigated global stabilization of nonlinear systems using with timeinvestigated delays is solved in ofHua et al. (2009). (2009). Sun et al. (2011) a class time-delay highear systems. global Over the years, a large amount of papersusing have systems investigated stabilization of nonlinear systems Sun et al. (2011) investigated a class of time-delay highorder nonlinear systems. The recursive design procedure investigated global stabilization of nonlinear systems using output feedback and obtained interesting results (see Gong al. (2011)systems. investigated a class of design time-delay highorderetnonlinear The recursive procedure investigated global of nonlinear systems using Sun output feedback andstabilization obtained interesting results (see Gong order nonlinear systems. The recursive design procedure of a continuous state-feedback controller is derived using output feedback and obtained interesting results (see Gong et al. (2003), Qian et al. (2002), Qian (2005a), Gong et al. order nonlinear systems. The recursive design procedure output feedback and interesting resultsGong (see Gong et al. (2003), Qian etobtained al. (2002), Qian (2005a), et al. of a continuous state-feedback controller is derived using of continuous state-feedback is using of aa power integrator, and et al. (2003), (2003), Qianet etal. al.(2001) (2002), Qian (2005a), Gong et al. al. the (2007) and Qian Qian etet al.al. (2001) and the(2005a), references therein). of continuous state-feedback controller is derived derived using theaa method method of adding adding powercontroller integrator, and globally globally et al. Qian (2002), Qian Gong et (2007) and and the references therein). the method of adding a power integrator, and globally asymptotic stability of the closed-loop system is proven (2007) and Qian et al. (2001) and the references therein). Moreover, it has been proved that restrictive conditions method stability of adding integrator, andisglobally asymptotic of athepower closed-loop system proven (2007) and itQian al. (2001) referencesconditions therein). the Moreover, has et been provedand thatthe restrictive asymptotic of is of construction of Moreover, itthe hassituation been proved proved that can restrictive conditions assumed in init the situation of global global can be relaxed relaxed when with asymptotic stability of the the closed-loop closed-loop system is proven proven with the the aid aidstability of the the delicate delicate constructionsystem of an an appropriappropriMoreover, has been that restrictive conditions assumed of be when with the aid of the delicate construction of an appropriate Lyapunov-Krasovskii functional. Reference Chai assumed in the situation of global can be relaxed when semi-global stabilization is considered. The works Qian with the aid of the delicate construction of an appropriet al. al. assumed in stabilization the situation isof considered. global can The be relaxed semi-global works when Qian ate Lyapunov-Krasovskii functional. Reference Chai et ate Lyapunov-Krasovskii functional. Reference Chai et (2014) dealt with the problem of global output feedback semi-global stabilization is considered. The works Qian (2005b), Zhai et al. (2013), Jia et al. (2010), Yu et al. ate Lyapunov-Krasovskii functional. Reference Chai et al. al. (2014) dealt with the problem of global output feedback semi-global stabilization is considered. The works Qian (2005b), Zhai et al. (2013), Jia et al. (2010), Yu et al. (2014) dealt problem of output for class of systems (2005b), Zhai et al. (2013), Jiaet etal. al. (2010), Yu et al. stabilization (2012), Zhai Zhai et et al. al. (2013b), Tian etet al.al. (2012) andYu Jia et et al. (2014) dealt with with the problem of global globalnonlinear output feedback feedback stabilization for aathe class of uncertain uncertain nonlinear systems (2005b), Zhai (2013), Jia (2010), (2012), et al. (2013b), Tian (2012) and Jia et stabilization for aa class classby ofthe uncertain nonlinear systems subject homogeneous domination (2012), Zhai et et al. (2013b), (2013b), Tian et et stabilization al. (2012) (2012) and andby Jia et al. al. stabilization (2012) dealt dealt with the semi-global semi-global stabilization byJia output for uncertain nonlinear systems subject to to time-delay time-delay byof the homogeneous domination (2012), Zhai al. Tian al. et (2012) with the output subject to time-delay by the homogeneous domination approach. (2012) dealt with the semi-global stabilization by output feedback controller of nonlinear systems. In Qian (2005b), subject to time-delay by the homogeneous domination approach. (2012) dealt with the semi-global stabilization by output feedback controller of nonlinear systems. In Qian (2005b), feedback controller of nonlinear nonlinear systems. In Qian Qian (2005b),aa approach. by extending extending the feedback feedback domination design techniques, approach. feedback controller of systems. In (2005b), by the domination design techniques, In this this paper, paper, we we deal deal with with the the semi-global semi-global stabilization stabilization by extending the feedback domination design techniques, systematic design scheme is developed to construct linearaa In by extending the feedback domination techniques, systematic design scheme is developeddesign to construct linear In this paper, we deal with the semi-global stabilization problem of aa class class of nonlinear time-delay systems. Firstly, In this paper, we deal with the semi-global stabilization problem of of nonlinear time-delay systems. Firstly, systematic design scheme is developed to construct linear output feedback controllers that semi-globally stabilize a systematic designcontrollers scheme is that developed to construct lineara we output feedback semi-globally stabilize problem of a class of nonlinear time-delay systems. Firstly, propose a linear output feedback controller with problem of a class of nonlinear time-delay systems. Firstly, we propose a linear output feedback controller with aa output feedback controllers that semi-globally semi-globally stabilize general feedback class of of uncertain uncertain nonlinear systems under under less rere-aa scaling output controllers that stabilize general class nonlinear systems less we propose a linear output feedback controller gain for the nonlinear time-delay system. Then propose output feedback controller with scaling gain aforlinear the nonlinear time-delay system.with Thenaa general class of uncertain uncertain nonlinear systems under less rere- we strictiveclass conditions. Reference Jia et et systems al. (2010) (2010) constructed general of nonlinear under less strictive conditions. Reference Jia al. constructed scaling gain for the nonlinear time-delay system. Then by using the Lyapunov-Krasovskii functional method, an scaling gain for the nonlinear time-delay system. Then using the Lyapunov-Krasovskii functional method, an conditions. Reference Jia and et al. al.controller (2010) constructed constructed a scaled scaled conditions. homogeneous observer and controller to semisemi- by strictive Reference Jia et (2010) astrictive homogeneous observer to by using the Lyapunov-Krasovskii functional method, an appropriate scaling gain in in the the linear linear output feedback by using the Lyapunov-Krasovskii functional method, an appropriate scaling gain output feedback a scaled homogeneous observer and controller to semiglobally stabilize a class of uncertain nonlinear systems aglobally scaled stabilize homogeneous and controller semi- controller a classobserver of uncertain nonlinear to systems appropriate scaling gain in the linear output feedback is obtained to semi-globally stabilize the closedappropriate scaling gain in the linear output feedback controller is obtained to semi-globally stabilize the closedglobally stabilize class of of an uncertain nonlinear systems controller is obtained to semi-globally stabilize the closedin aa finite finitestabilize time by by aachoosing choosing an appropriate gain. systems globally class uncertain nonlinear in time appropriate gain. loop Finally, numerical to controller is obtained stabilize is closedloop system. system. Finally, toan ansemi-globally numerical example example isthegiven given to in aa finite finite time time by by choosing choosing an an appropriate appropriate gain. gain. in loop system. Finally, an numerical example is given to validate our results presented in this paper. On the other hand, time-delay systems get many reloop system. Finally, an numerical example is given to On the other hand, time-delay systems get many re- validate our results presented in this paper. validate our results presented in this paper. On the other hand, time-delay systems get many researchers’ attentions since delays are often the main causes validate our results presented in this paper. On the other hand, time-delay systems get many researchers’ attentions since delays are often the main causes The paper paper is is organized organized as as follows. follows. Section Section 22 presents presents the the searchers’ attentions attentions since since delays delays are are often often the the main main causes causes The searchers’ ⋆ The paper is organized as follows. Section 2 presents the problem fomulation and some preliminaries. The main project was supported by the National Natural Science ⋆ This The paper is organized as follows. Section 2 presents the problem fomulation and some preliminaries. The main This project was supported by the National Natural Science ⋆ problem fomulation and some preliminaries. The main Foundation of China (61374038; 61473079; 61374060), and a project This project was supported by the National Natural Science result is proposed in Section 3. Section 4 extends the result ⋆ problem fomulation and some preliminaries. main Foundation of China 61374060), and a Science project This project was (61374038; supported 61473079; by the National Natural result is proposed in Section 3. Section 4 extendsThe the result funded Priority Academic Program Development Foundation of (61374038; and project result is in 3. extends the to time-delay systems under non-triangular funded by by the the Priority Academic61473079; Program 61374060), Development ofa Jiangsu Foundation of China China (61374038; 61473079; 61374060), andof a Jiangsu project result is proposed proposed in Section Section 3. Section Section extends the result result to nonlinear nonlinear time-delay systems under44the the non-triangular Higher Institutions (1108007002). funded by Academic Program to nonlinear time-delay systems under the non-triangular Higher Education Education Institutions (1108007002). funded by the the Priority Priority Academic Program Development Development of of Jiangsu Jiangsu to nonlinear time-delay systems under the non-triangular Higher Education Institutions (1108007002).
Higher Education Institutions (1108007002). Copyright IFAC 2015 2015 185 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 185 Copyright IFAC 2015 185 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 185Control. 10.1016/j.ifacol.2015.09.375
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growth condition. Section 5 illustrates the obtained results by a numerical example, which is followed by the conclusion in Section 6. 2. PROBLEM STATEMENT AND PRELIMINARIES Consider the following nonlinear time-delay system x˙ 1 (t) = x2 (t) + ϕ1 (x1 (t), x1 (t − τ )), x˙ 2 (t) = x3 (t) + ϕ2 (x1 (t), x2 (t), x1 (t − τ ), x2 (t − τ )), .. . x˙ n−1 (t) = xn (t) + ϕn−1 (x1 (t), . . . , xn−1 (t), (1) x1 (t − τ ), . . . , xn−1 (t − τ )), x˙ n (t) = u(t) + ϕn (x1 (t), . . . , xn (t), x1 (t − τ ), . . . , xn (t − τ )), y = x (t), 1 x(t) = φ(t), −τ ≤ t ≤ 0
where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ Rn is the system state; u(t) ∈ R is the control input; y(t) ∈ R is the system output; τ is a constant which represents time delay; x(t − τ ) = (x1 (t − τ ), x2 (t − τ ), . . . , xn (t − τ ))T ∈ Rn stands for the delayed state; φ(t) is the initial function of the system state vector; ϕi (·), i = 1, 2, . . . , n,represent nonlinear perturbations that are not guaranteed to be precisely known. In this paper, our objective is to develop a linear output feedback controller which can semi-globally stabilize the uncertain nonlinear time-delay system (1). The problem of semi-global stabilization for system (1) can be described as follows. System (1) is said to be semi-global stabilized by linear output feedback controller if for each compact subset Ωx ⊂ Rn , there exists another nonempty set Ωxˆ ⊂ Rn and a linear output feedback controller { x ˆ˙ = F (ˆ x, y, u), (2) u = u(ˆ x)
Remark 2. Assumption 1 is just a theoretical hypothesis. In practical applications, if the nonlinear function can be simplified as the linear forms, the works will become simple. As a consequence, we are considering the practical significance of this assumption and combining Assumption 1 with practical systems. 3. MAIN RESULTS In this section, we are devoted in the problem of semiglobal stabilization of the nonlinear time-delay system (1). That is to say, we show that there exists a linear output feedback controller which can stabilized the nonlinear time-delay system (1) semi-globally. A linear output feedback controller of system (1) will be obtained under Assumption 1. Theorem 1. For a given compact subset Ωx = {x| ∥x∥ ≤ r} ⊂ Rn , suppose that nonlinear time-delay system (1) satisfies Assumption 1, then for any initial condition x(t) = φ(t) ∈ Ωx , there exists an appropriate gain L ≥ 1 such that system (1) can be semi-globally stabilized by the following output feedback controller x ˆ˙ 1 (t) = x ˆ2 (t) + La1 (x1 (t) − x ˆ1 (t)), 2 ˙ 2 (t) = x ˆ (t) + L a (x (t) − x ˆ1 (t)), x ˆ 3 2 1 .. (5) . n−1 x ˆ˙ (t) = x ˆn (t) + L an−1 (x1 (t) − x ˆ1 (t)), ˙ n−1 n ˆ1 (t)), x ˆn (t) = u(t) + L an (x1 (t) − x and
u(t) = −Ln [b1 x ˆ1 (t) +
1 1 b2 x ˆ2 (t) + · · · + n−1 bn x ˆn (t)] (6) L L
where ai > 0 and bi > 0, i = 1, 2, . . . , n are the coefficients of the Hurwitz polynomials p1 (s) = sn + a1 sn−1 + · · · + an−1 s + an (7) and p2 (s) = sn + bn sn−1 + · · · + b2 s + b1 .
(8)
such that the solution of the closed-loop system is locally stable and [ ] [ ] x(0) x(t) 2n ∈ Ω ⊂ R ⇒ lim = 0. (3) x ˆ(0) ˆ(t) t→∞ x
Proof. For i = 1, 2, . . . , n, new coordinates transformation with a constant scaling gain L ≥ 1 to be determined are given as follows zi (t) = xi (t)/Li−1 , zˆi (t) = x ˆi (t)/Li−1 , v(t) = u(t)/Ln (9)
In order to solve the problem better, the following assumption is given. Assumption 1. For i = 1, 2, . . . , n, ϕi (·) satisfies
and εi (t) = zi (t) − zˆi (t), ε(t) = (ε1 (t), ε2 (t), . . . , εn (t))T . (10)
|ϕi (x1 (t), . . . , xi (t), x1 (t − τ ), . . . , xi (t − τ ))|
≤ c1 (y)(|x1 (t)| + . . . + |xi (t)|)
+c2 (y)(|x1 (t − τ )| + . . . + |xi (t − τ )|)
(4)
where c1 (y) and c2 (y) are smooth functions. Remark 1. Assumption 1 is a linear growth condition and similar assumptions can be find in Qian et al. (2002) and Chai (2013). Under this hypothesis, the nonlinear function ϕi (x1 (t), · · · , xi (t), x1 (t − τ ), · · · , xi (t − τ )) for i = 1, · · · , n should be bounded by linear terms. Assumption 1 is more general than the linear growth condition imposed in Qian et al. (2002) for the reason that Assumption 1 reduces to the nonlinear system in Qian et al. (2002) when c2 = 0. 186
Under the above definition, a simple calculation is given as ε(t) ˙ = LA1 ε(t) + Φ (11) with
−a1 .. . A1 = −a n−1 −an
1 ··· .. . . . . 0 ··· 0 ···
0 .. . ,Φ = 1 0
L
ϕ1 (·) 1 ϕ2 (·) L . .. . 1 ϕ (·) n n−1
Moreover, with the help of the coordinates change (9), the output feedback controller (6) can be rewritten as u(t) = Ln v(t), v(t) = −b1 zˆ1 (t) − · · · − bn zˆn (t). (12)
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Then zˆ˙ (t) = LB zˆ(t) + LA2 Cε(t) where
0 1 .. .. . . B= 0 0 −b1 −b2
(13)
T ··· 0 a1 1 .. .. a2 0 . . , A2 = . , C = . . .. .. ··· 1 an 0 · · · −bn
In the following, we prove that the closed-loop systems (11) and (13) are semi-global stable. According to the definition of ai and bi , it is not difficult to verify that matrices A1 and B are Hurwitz matrices. Consequently, there exist two positive definite matrices P1 = P1T > 0 and P2 = P2T > 0 such that T AT 1 P1 + P1 A1 = −I, B P2 + P2 B = −I.
(14)
Choose a Lyapunov function described as T
T
V1 (ε(t), zˆ(t)) = (d + 1)ε (t)P1 ε(t) + zˆ (t)P2 zˆ(t),
(15)
where d = 2∥P2 A2 C∥2 . The derivative of V1 (ε(t), zˆ(t)) along the trajectory of the closed-loop systems (11) and (13) is given as V˙ 1 (ε(t), zˆ(t)) = (d + 1)ε˙T (t)P1 ε(t) + (d + 1)εT (t)P1 ε(t) ˙ T T ˙ ˙ +zˆ (t)P2 zˆ(t) + zˆ (t)P2 zˆ(t) T = (d + 1)εT (t)AT 1 LP1 ε(t) + (d + 1)Φ P1 ε(t)
+(d + 1)εT (t)P1 LAT 1 ε(t)
+c2 (y)(|x1 (t − τ )| + · · · ]} +|xn (t − τ )|) [ 1 = 2(d + 1)∥ε(t)∥ · ∥P1 ∥ (1 + + · · · L 1 + n−1 )(c1 (y)|x1 (t)| + c2 (y) L 1 1 ·|x1 (t − τ )|) + (1 + + · · · + n−2 ) L L |x2 (t)| |x2 (t − τ )| + c2 (y) ) ·(c1 (y) L L 1 |xn−1 (t)| + · · · + (1 + )(c1 (y) L Ln−2 |xn−1 (t − τ )| +c2 (y) ) Ln−2 |xn (t)| |xn (t − τ )| ] +(c1 (y) n−1 + c2 (y) ) L Ln−1 { ≤ 2(d + 1)c3 (y)∥ε(t)∥ |x1 (t)|
|x2 (t)| |x2 (t − τ )| + L L |xn (t)| |xn (t − τ )| } , (17) + · · · + n−1 + L Ln−1 where c3 (y) > 0 is a smooth function. Remark 3. c1 (y) and c2 (y) are bounded for the reason that y = x1 and x1 is in the compact subset Ωx . Consequently, c3 (y) is existed and dependent on Ωx see Qian (2005b). Similar conclusion can be find in reference Zhai et al. (2013) and Jia et al. (2010). +|x1 (t − τ )| +
For any arbitrarily large number r > 1, define compact sets Ωx = {x|∥x∥ ≤ r, x ∈ Rn }, N = {(ε, zˆ)|∥ε∥ ≤ 2r, ∥ˆ z∥ ≤ 2r} and Ω = {(ε, zˆ)|V1 (ε, zˆ) ≤ α, α = max V1 (ε, zˆ)}.
+(d + 1)ε(t)T P1 Φ + zˆT (t)B T LP2 zˆ(t) ˆ(t) +εT (t)C T AT 2 LP2 z
(ε,ˆ z )∈N
+ˆ z T (t)P2 LB T zˆ(t) + zˆT (t)P2 LA2 Cε(t)
For any x(t) ∈ Ωx , we obtain (ε, zˆ) ∈ N by using the coordinates transformation given as (9) and (10). It is obvious that Ω is a nonempty compact set which is in the interior of N and (ε, zˆ) are bounded in Ω.
z (t)∥2 = −(d + 1)L∥ε(t)∥2 − L∥ˆ +2(d + 1)εT (t)P1 Φ
+2Lˆ z T (t)P2 A2 Cε(t)
187
(16)
For L ≥ 1, using Assumption 1, the following inequality holds 2(d + 1)εT (t)P1 Φ ≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ · ∥Φ∥
xi (t) From εi (t) = zi (t) − zˆi (t) = L ˆi (t), we know that i−1 − z � x (t) � � i � zi (t)| + |εi (t)|. (18) � i−1 � ≤ |ˆ L
Consequently, for any (ε, zˆ) ∈ Ω, (17) can be rewritten as
= 2(d + 1)∥ε(t)∥ · ∥P1 ∥ √ 1 1 · |ϕ1 |2 + | ϕ2 |2 + · · · + | n−1 ϕn |2 L L 1 ≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ · (|ϕ1 | + |ϕ2 | L 1 + · · · + n−1 |ϕn |) L {
2(d + 1)εT (t)P1 Φ ≤ 2(d + 1)c3 (y)∥ε(t)∥
n ∑ i=1
[|ˆ zi (t)| + |εi (t)|
+|ˆ zi (t − τ )| + |εi (t − τ )|]
z (t)∥2 + α3 ∥ε(t − τ )∥2 ≤ α1 ∥ε(t)∥2 + α2 ∥ˆ
+α4 ∥ˆ z (t − τ )∥2 , (19) where αi (i = 1, 2, 3, 4) are constants depending on Ω.
≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ c1 (y)|x1 (t)| 1[ c1 (y)(|x1 (t)| +c2 (y)|x1 (t − τ )| + L +|x2 (t)|) + c2 (y)(|x1 (t − τ )| ] 1 [ +|x2 (t − τ )|) + · · · + n−1 c1 (y) L ·(|x1 (t)| + · · · + |xn (t)|)
On the other hand, √ 1 2Lˆ z T (t)P2 A2 Cε(t) ≤ 2L( √ ∥ˆ z (t)∥)( 2∥P2 A2 C∥ · ∥ε(t)∥) 2 1 z (t)∥2 + Ld∥ε(t)∥2 . (20) ≤ L∥ˆ 2 Substituting (19) and (20) into (16), we have 187
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z (t)∥2 V˙ 1 (ε(t), zˆ(t)) ≤ −(d + 1)L∥ε(t)∥2 − L∥ˆ
+α1 ∥ε(t)∥2 + α2 ∥ˆ z (t)∥2 + α3 ∥ε(t − τ )∥2 1 z (t)∥2 + Ld∥ε(t)∥2 +α4 ∥ˆ z (t − τ )∥2 + L∥ˆ 2 L = −(L − α1 )∥ε(t)∥2 − ( − α2 )∥ˆ z (t)∥2 2 +α3 ∥ε(t − τ )∥2 + α4 ∥ˆ z (t − τ )∥2 . (21)
In Assumption 1, a lower-triangular growth condition for system (1) is necessary. Actually, the following more general condition can also be established which means the nonlinear conditions in the nonlinear time-delay system (1) are bounded under the non-triangular growth condition instead of lower-triangular growth condition. Assumption 2. For i = 1, 2, . . . , n, ϕi (·) satisfies |ϕi (x(t), x(t − τ ))| n n ∑ ∑ |xj (t)| + c2 (y) |xj (t − τ )| ≤ c1 (y)
Choose a Lyapunov-Krasovskii functional described as V2 (ε(t), zˆ(t)) = V1 (ε(t), zˆ(t)) +
∫t
t−τ
j=1
(α3 ∥ε(s)∥2
+α4 ∥ˆ z (s)∥2 )ds. (22) Combine with (21), the derivative of V2 (ε(t), zˆ(t)) satisfies L z (t)∥2 V˙ 2 (ε(t), zˆ(t)) ≤ −(L − α1 )∥ε(t)∥2 − ( − α2 )∥ˆ 2 +α3 ∥ε(t − τ )∥2 + α4 ∥ˆ z (t − τ )∥2 +α3 (∥ε(t)∥2 − ∥ε(t − τ )∥2 )
z (t)∥2 − ∥ˆ z (t − τ )∥2 ) +α4 (∥ˆ
= −(L − α1 − α3 )∥ε(t)∥2 L −( − α2 − α4 )∥ˆ z (t)∥2 . (23) 2 The right hand of (23) is negative definite under the condition that L > max{α1 + α3 , 2(α2 + α4 ), 1}, which means V˙ 2 (ε(t), zˆ(t)) ≤ −β1 ∥ε(t)∥2 − β2 ∥ˆ z (t)∥2 (24) holds for two positive constants β1 and β2 . As a result, starting from Ω, the trajectory will stay in the compact set forever. Moreover lim (ε(t), zˆ(t)) = 0. (25) t→∞
This implies that Ω is a domain of attraction. Remark 4. L is related to αi (i = 1, 2, 3, 4) which are dependent on Ωx . Thus, L is also dependent on Ωx and the property of semi-global is incarnated here. This is different from the global results in which L is just appropriate but not dependent on Ωx . From the definition, we know that for any arbitrarily large set Ωx , (x, x ˆ) ∈ Ωx × Ωx ⇒ ∥ε(t)∥ ≤ 2r, ∥ˆ z (t)∥ ≤ 2r
⇒ (ε(t), zˆ(t)) ∈ Ω. (26) Then we know that starting from any points in Ωx × Ωx , the trajectory will stay in the compact set Ω and tend to the origin. In conclusion, Theorem 1 comes into existence that means the problem of semi-global stabilization for the nonlinear time-delay system (1) can be solved by the linear output feedback controller (5) and (6). 4. EXTENSION In this section, we extend the result to nonlinear timedelay systems under the non-triangular growth condition. 188
(27)
j=1
where c1 (y) and c2 (y) are smooth functions. Theorem 2. Under Assumption 2, for a given compact subset Ωx = {x|∥x∥ ≤ r} ⊂ Rn , for any initial condition x(t) = φ(t) ∈ Ωx , there exists an appropriate gain L ≥ 1 such that system (1) can be semi-globally stabilized by the output feedback controller (5) and (6). Proof. The proof process is similar to that of Theorem 1 by using the same output feedback controller (5) and (6). (17) is hold under Assumption 2 although the nonlinear conditions are not triangular form. Consequently, the semiglobal stabilization problem can be solved by designing the output feedback controller as the form (5) and (6) with an appropriate gain L. The detailed proof is omitted here for brevity. 5. ILLUSTRATIVE EXAMPLE In this section, we give an example to show how to apply the results proposed in this paper to investigate the semi-global stabilization of a class of nonlinear time-delay system. Let us consider the following 2-dimensional nonlinear timedelay system x˙ 1 (t) = x2 (t) + x1 (t) ln(1 + x21 (t)) + 0.5 sin(x1 (t − 0.8)), (28) x˙ 2 (t) = u(t) + 0.5x1 (t) + 0.4x2 (t) x21 (t) 2 ) cos(x2 (t − 0.8)). + 0.5x1 (t)(1 − 2
It is obvious that the nonlinear time-delay system (28) satisfies Assumption 1. Next we check whether the output feedback controller { x ˆ˙ 1 (t) = x ˆ2 (t) + La1 (x1 − x ˆ1 ), (29) ˆ1 ), x ˆ˙ 2 (t) = u(t) + L2 a2 (x1 − x and u(t) = −L2 [b1 x ˆ1 (t) +
1 b2 x ˆ2 (t)] L
(30)
can stabilize the system (28) semi-globally according to Theorem 1. In this example, the compact set Ωx and N in Theorem 1 are given as Ωx = {x|∥x∥ ≤ 5} and N = {(ε, zˆ)|∥ε∥ ≤ 10, ∥ˆ z ∥ ≤ 10} correspondingly. The gains are selected as a1 = 1, a2 = 0.5, b1 = 5, b2 = 4, αi (i = 1, 2, 3, 4) and L can be obtained as α1 = α2 = α3 = α4 = 1.6 and L = 6.8 relatively through calculating, and for −0.8 ≤ t ≤ 0 the initial functions are chosen as ) ( ) ( ) ( ) ( x1 (t) x ˆ1 (t) 0.5 sin(t − 0.8) 0.6 = , = . x2 (t) x ˆ2 (t) −0.8 sin(t − 0.8) −1
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1 x1 0.5
x ˆ1
0 −0.5
0
1
2
3
4
5 Time(s)
6
7
8
9
10
5 x2 x ˆ2 0
−5
0
1
2
3
4
5 Time(s)
6
7
8
9
10
Fig. 1. State response. 50 u 0 −50 −100 −150
0
1
2
3
4
5 Time(s)
6
7
8
9
10
Fig. 2. Time history of the control signal. Obviously, the initial functions are in the given set Ωx ×Ωx . Remark 5. The region of Ωx is not given in the examples of many existing semi-global works and L is selected as an appropriate gain in not only semi-global works but also global results. However, in this example, we give the specific region of Ωx and L is calculated using Theorem 1. It is easy to obtain by MATLAB, ( ) ( ) 0.7500 −0.500 1.1500 0.100 , P2 = . P1 = −0.500 2.500 0.100 0.1500 The simulation is given in Fig.1 and Fig.2. Fig.1 is the swing curve of the state response and Fig. 2 shows the trajectory of the input control u. It is clear that the nonlinear time-delay system (28) under the linear output feedback control law u = −L2 [b1 x ˆ1 (t) + L1 b2 x ˆ2 (t)] is asymptotically stable. From the simulations, we can conclude that the result presented in Theorem 1 is very practicable in semiglobal stabilization of nonlinear time-delay system. 6. CONCLUSIONS In this paper, the semi-global stabilization problem of a class of nonlinear lower-triangular and non-triangular systems with time delay in the state and a linear output feedback controller with a scaling gain has been investigated through using a Lyapunov-Krasovskii technique. The output feedback control law, by which the semiglobal asymptotic stability of the closed-loop system is guaranteed, is memoryless and easy for implementation. Simulations show that the results obtained in this paper are very practicable in semi-globally analyzing the stability of a class of nonlinear time-delay systems. REFERENCES Q. Gong, and W. Lin. A note on global output regulation of nonlinear systems in the output feedback form. IEEE Transactions on Automatic Control, volume 48, number 6, pages 1049–1054, 2003. 189
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K. Gu, V. Kharitonov, and J. Chen. Stability of time-delay systems. Birkhauser: Boston, 2003. C.C. Hua, P.X. Liu, and X.P. Guan. Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems. IEEE Transactions on Industrial Electroics, volume 56, number 9, pages 3723– 3732, 2009. Z.Y. Sun, and Y.G. Liu. State-feedback stabilization control design for a class of time-delay high order nonlinear systems. Proceedings of the 30th Chinese control Conference, pages 476–480, July 2011. L. Chai, and C.J. Qian. Global stabilization via homogeneous output feedback for a class of uncertain nonlinear systems subject to time-delays. Transactions of the Institute of Measurement and Control, volume 36, number 4, pages 478–486, 2014.
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Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Semi-Global Stabilization for a Class of Nonlinear Time-Delay Systems by Linear Nonlinear Time-Delay Systems by Linear Nonlinear Time-Delay Systems by Linear ⋆⋆ by Linear Nonlinear Time-Delay Systems Output Feedback Output Feedback Output Feedback Feedback ⋆⋆ Output
∗ ∗ ∗ ∗ Pan Pan Wang Wang ∗∗ ,, Lin Lin Chai Chai ∗∗ ,, Shumin Shumin Fei Fei ∗∗ ,, Guquan Guquan Liu Liu ∗∗ Pan Pan Wang Wang ∗ ,, Lin Lin Chai Chai ∗ ,, Shumin Shumin Fei Fei ∗ ,, Guquan Guquan Liu Liu ∗ ∗ ∗ Key Laboratory of Measurement and Control of Complex Systems of ∗ Key Laboratory of Measurement and Control of Complex Systems of ∗Engineering, Key of and Control of of Ministry of of Southeast Key Laboratory Laboratory of Measurement Measurement andSchool Control of Complex Complex Systems Systems of Engineering, Ministry of Education; Education; School of Automation, Automation, Southeast Engineering, Ministry of Education; School of Automation, Southeast University, Nanjing, 210096,P.R.China (e-mails: [email protected]; Engineering, Ministry of Education; School of Automation, Southeast University, Nanjing, 210096,P.R.China (e-mails: [email protected]; University, 210096,P.R.China (e-mails: [email protected]; chailin [email protected]; [email protected]; [email protected]). University, Nanjing, 210096,P.R.China (e-mails: [email protected]; chailinNanjing, [email protected]; [email protected]; [email protected]). chailin [email protected]; [email protected]; [email protected]). chailin [email protected]; [email protected]; [email protected]). Abstract: The The semi-global semi-global stabilization stabilization problem problem for for aa class class of of nonlinear nonlinear systems systems with with state state Abstract: Abstract: The semi-global stabilization problem for a class of nonlinear systems with state time-delay is addressed in this paper. Sufficient conditions are derived by using LyapunovAbstract: The semi-global stabilization problem conditions for a class are of nonlinear with state time-delay is addressed in this paper. Sufficient derived bysystems using Lyapunovtime-delay is in Sufficient conditions derived by using Krasovskii functional functional method andpaper. adjusting the scaling scaling gain to toare guarantee the systems as well well as as time-delay is addressed addressed in this this paper. Sufficient conditions are derivedthe by systems using LyapunovLyapunovKrasovskii method and adjusting the gain guarantee as Krasovskii method and adjusting the to the as the resulted resultedfunctional closed-loop systems under linear output gain feedback controller tosystems be semi-globally semi-globally Krasovskii functional method andunder adjusting the scaling scaling gain to guarantee guarantee theto systems as well well as as the closed-loop systems aa linear output feedback controller be the resulted closed-loop systems under output controller be stable. Numerical example is presented presented tolinear illustrate the feedback effectiveness of the the to obtained results in in the resulted closed-loop systems under aato linear output feedback controller to be semi-globally semi-globally stable. Numerical example is illustrate the effectiveness of obtained results stable. Numerical example is presented to illustrate the effectiveness of the obtained results in this paper. stable. Numerical example is presented to illustrate the effectiveness of the obtained results in this paper. this paper. this paper. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Semi-global, nonlinear nonlinear time-delay time-delay systems, systems, Lyapunov-Krasovskii Lyapunov-Krasovskii functional, functional, linear linear Keywords: Semi-global, Keywords: Semi-global, nonlinear time-delay systems, Lyapunov-Krasovskii functional, linear observer, output feedback. Keywords: Semi-global, nonlinear time-delay systems, Lyapunov-Krasovskii functional, linear observer, output feedback. observer, output output feedback. feedback. observer, 1. INTRODUCTION INTRODUCTION of 1. of instability instability and and poor poor performance performance of of many many industrial industrial 1. INTRODUCTION INTRODUCTION of instability and poor performance of many industrial industrial and engineering systems. Consequently, stability analysis 1. of instability and poor performance of many and engineering systems. Consequently, stability analysis and engineering systems. Consequently, stability analysis synthesis for time-delay systems have been one of of Stabilization by output feedback is one of the important and engineering systems. Consequently, stability analysis synthesis for time-delay systems have been one Stabilization by output feedback is one of the important and synthesis for time-delay systems have been of the most challenging issues and many nice results have Stabilization by output feedback is one of the important problems in in the the field of offeedback nonlinearis control. control. Unlike linear and synthesis for time-delay systems one of the most challenging issues and many have nice been resultsone have Stabilization by output one of the important problems field nonlinear Unlike linear the most challenging issues and many nice results have been proposed (see Chai (2013), Zhai et al. (2014), Ye problems in the field of nonlinear control. Unlike linear systems, separation principle usually does not hold for challenging issues and many niceal.results beenmost proposed (see Chai (2013), Zhai et (2014),have Ye problems in the fieldprinciple of nonlinear control. linear systems, separation usually does Unlike not hold for the been proposed (see(2004), Chai (2013), (2013), Zhai et (2007) al. (2014), (2014), Ye (2011), Wu Yakoubi et and systems, separation principle usually does feedback not hold holdplus for been nonlinear separation systems which which leads to to that state state feedback plus proposed (see Chai al. Ye (2011), Wu et et al. al. (2004), YakoubiZhai et al. al.et (2007) and the the systems, principle usually does not for nonlinear systems leads that (2011), Wu et al. (2004), Yakoubi et al. (2007) and the references therein). The most commonly used method for nonlinear systems which leads to that state feedback plus observability simply do not imply stabilizability by output (2011), Wu et al. (2004), Yakoubi et al. (2007) and the references therein). The most commonly used method for nonlinear systems which leads to that state feedback plus observability simply do not imply stabilizability by output references therein). The used the observability simply do not not implythe stabilizability by output solving feedback. For Forsimply the above above reason, the problem of of by stabilizareferences therein). problems The most most is commonly used method method for for solving time-delay time-delay problems iscommonly the Lyapunov-Krasovskii Lyapunov-Krasovskii observability do imply stabilizability output feedback. the reason, problem stabilizasolving time-delay problems is the Lyapunov-Krasovskii functional approach (Gu et al. (2003)). The feedback. For the above reason, the problem of stabilization by output feedback is much more difficult and chaltime-delay Lyapunov-Krasovskii functional approachproblems (Gu et is al.the (2003)). The controller controller feedback. For the above reason, problem of and stabilization by output feedback is muchthe more difficult chal- solving approach et (2003)). design for class nonlinear tion by than output feedback is much much morefeedback difficultfor and challenging than thefeedback stabilization by state state feedback for nonlinfunctional approach (Gu et of al.strict-feedback (2003)). The The controller controller design problem problem for aa(Gu class ofal. strict-feedback nonlinear tion by output is more difficult and chal- functional lenging the stabilization by nonlindesign problem for a class of strict-feedback nonlinear systems with time delays is solved in Hua et al. (2009). lenging than the stabilization by state feedback for nonlinear systems. Over the years, a large amount of papers have design problem for a class of strict-feedback nonlinear systems with time delays is solved in Hua et al. (2009). lenging than the stabilization by state feedback for nonlinear systems. Over the years, a large amount of papers have systems with time delays is solved in Hua et al. Sun et al. (2011) investigated a class of time-delay highear systems. Over the years, a large amount of papers have investigated global stabilization of nonlinear systems using with timeinvestigated delays is solved in ofHua et al. (2009). (2009). Sun et al. (2011) a class time-delay highear systems. global Over the years, a large amount of papersusing have systems investigated stabilization of nonlinear systems Sun et al. (2011) investigated a class of time-delay highorder nonlinear systems. The recursive design procedure investigated global stabilization of nonlinear systems using output feedback and obtained interesting results (see Gong al. (2011)systems. investigated a class of design time-delay highorderetnonlinear The recursive procedure investigated global of nonlinear systems using Sun output feedback andstabilization obtained interesting results (see Gong order nonlinear systems. The recursive design procedure of a continuous state-feedback controller is derived using output feedback and obtained interesting results (see Gong et al. (2003), Qian et al. (2002), Qian (2005a), Gong et al. order nonlinear systems. The recursive design procedure output feedback and interesting resultsGong (see Gong et al. (2003), Qian etobtained al. (2002), Qian (2005a), et al. of a continuous state-feedback controller is derived using of continuous state-feedback is using of aa power integrator, and et al. (2003), (2003), Qianet etal. al.(2001) (2002), Qian (2005a), Gong et al. al. the (2007) and Qian Qian etet al.al. (2001) and the(2005a), references therein). of continuous state-feedback controller is derived derived using theaa method method of adding adding powercontroller integrator, and globally globally et al. Qian (2002), Qian Gong et (2007) and and the references therein). the method of adding a power integrator, and globally asymptotic stability of the closed-loop system is proven (2007) and Qian et al. (2001) and the references therein). Moreover, it has been proved that restrictive conditions method stability of adding integrator, andisglobally asymptotic of athepower closed-loop system proven (2007) and itQian al. (2001) referencesconditions therein). the Moreover, has et been provedand thatthe restrictive asymptotic of is of construction of Moreover, itthe hassituation been proved proved that can restrictive conditions assumed in init the situation of global global can be relaxed relaxed when with asymptotic stability of the the closed-loop closed-loop system is proven proven with the the aid aidstability of the the delicate delicate constructionsystem of an an appropriappropriMoreover, has been that restrictive conditions assumed of be when with the aid of the delicate construction of an appropriate Lyapunov-Krasovskii functional. Reference Chai assumed in the situation of global can be relaxed when semi-global stabilization is considered. The works Qian with the aid of the delicate construction of an appropriet al. al. assumed in stabilization the situation isof considered. global can The be relaxed semi-global works when Qian ate Lyapunov-Krasovskii functional. Reference Chai et ate Lyapunov-Krasovskii functional. Reference Chai et (2014) dealt with the problem of global output feedback semi-global stabilization is considered. The works Qian (2005b), Zhai et al. (2013), Jia et al. (2010), Yu et al. ate Lyapunov-Krasovskii functional. Reference Chai et al. al. (2014) dealt with the problem of global output feedback semi-global stabilization is considered. The works Qian (2005b), Zhai et al. (2013), Jia et al. (2010), Yu et al. (2014) dealt problem of output for class of systems (2005b), Zhai et al. (2013), Jiaet etal. al. (2010), Yu et al. stabilization (2012), Zhai Zhai et et al. al. (2013b), Tian etet al.al. (2012) andYu Jia et et al. (2014) dealt with with the problem of global globalnonlinear output feedback feedback stabilization for aathe class of uncertain uncertain nonlinear systems (2005b), Zhai (2013), Jia (2010), (2012), et al. (2013b), Tian (2012) and Jia et stabilization for aa class classby ofthe uncertain nonlinear systems subject homogeneous domination (2012), Zhai et et al. (2013b), (2013b), Tian et et stabilization al. (2012) (2012) and andby Jia et al. al. stabilization (2012) dealt dealt with the semi-global semi-global stabilization byJia output for uncertain nonlinear systems subject to to time-delay time-delay byof the homogeneous domination (2012), Zhai al. Tian al. et (2012) with the output subject to time-delay by the homogeneous domination approach. (2012) dealt with the semi-global stabilization by output feedback controller of nonlinear systems. In Qian (2005b), subject to time-delay by the homogeneous domination approach. (2012) dealt with the semi-global stabilization by output feedback controller of nonlinear systems. In Qian (2005b), feedback controller of nonlinear nonlinear systems. In Qian Qian (2005b),aa approach. by extending extending the feedback feedback domination design techniques, approach. feedback controller of systems. In (2005b), by the domination design techniques, In this this paper, paper, we we deal deal with with the the semi-global semi-global stabilization stabilization by extending the feedback domination design techniques, systematic design scheme is developed to construct linearaa In by extending the feedback domination techniques, systematic design scheme is developeddesign to construct linear In this paper, we deal with the semi-global stabilization problem of aa class class of nonlinear time-delay systems. Firstly, In this paper, we deal with the semi-global stabilization problem of of nonlinear time-delay systems. Firstly, systematic design scheme is developed to construct linear output feedback controllers that semi-globally stabilize a systematic designcontrollers scheme is that developed to construct lineara we output feedback semi-globally stabilize problem of a class of nonlinear time-delay systems. Firstly, propose a linear output feedback controller with problem of a class of nonlinear time-delay systems. Firstly, we propose a linear output feedback controller with aa output feedback controllers that semi-globally semi-globally stabilize general feedback class of of uncertain uncertain nonlinear systems under under less rere-aa scaling output controllers that stabilize general class nonlinear systems less we propose a linear output feedback controller gain for the nonlinear time-delay system. Then propose output feedback controller with scaling gain aforlinear the nonlinear time-delay system.with Thenaa general class of uncertain uncertain nonlinear systems under less rere- we strictiveclass conditions. Reference Jia et et systems al. (2010) (2010) constructed general of nonlinear under less strictive conditions. Reference Jia al. constructed scaling gain for the nonlinear time-delay system. Then by using the Lyapunov-Krasovskii functional method, an scaling gain for the nonlinear time-delay system. Then using the Lyapunov-Krasovskii functional method, an conditions. Reference Jia and et al. al.controller (2010) constructed constructed a scaled scaled conditions. homogeneous observer and controller to semisemi- by strictive Reference Jia et (2010) astrictive homogeneous observer to by using the Lyapunov-Krasovskii functional method, an appropriate scaling gain in in the the linear linear output feedback by using the Lyapunov-Krasovskii functional method, an appropriate scaling gain output feedback a scaled homogeneous observer and controller to semiglobally stabilize a class of uncertain nonlinear systems aglobally scaled stabilize homogeneous and controller semi- controller a classobserver of uncertain nonlinear to systems appropriate scaling gain in the linear output feedback is obtained to semi-globally stabilize the closedappropriate scaling gain in the linear output feedback controller is obtained to semi-globally stabilize the closedglobally stabilize class of of an uncertain nonlinear systems controller is obtained to semi-globally stabilize the closedin aa finite finitestabilize time by by aachoosing choosing an appropriate gain. systems globally class uncertain nonlinear in time appropriate gain. loop Finally, numerical to controller is obtained stabilize is closedloop system. system. Finally, toan ansemi-globally numerical example example isthegiven given to in aa finite finite time time by by choosing choosing an an appropriate appropriate gain. gain. in loop system. Finally, an numerical example is given to validate our results presented in this paper. On the other hand, time-delay systems get many reloop system. Finally, an numerical example is given to On the other hand, time-delay systems get many re- validate our results presented in this paper. validate our results presented in this paper. On the other hand, time-delay systems get many researchers’ attentions since delays are often the main causes validate our results presented in this paper. On the other hand, time-delay systems get many researchers’ attentions since delays are often the main causes The paper paper is is organized organized as as follows. follows. Section Section 22 presents presents the the searchers’ attentions attentions since since delays delays are are often often the the main main causes causes The searchers’ ⋆ The paper is organized as follows. Section 2 presents the problem fomulation and some preliminaries. The main project was supported by the National Natural Science ⋆ This The paper is organized as follows. Section 2 presents the problem fomulation and some preliminaries. The main This project was supported by the National Natural Science ⋆ problem fomulation and some preliminaries. The main Foundation of China (61374038; 61473079; 61374060), and a project This project was supported by the National Natural Science result is proposed in Section 3. Section 4 extends the result ⋆ problem fomulation and some preliminaries. main Foundation of China 61374060), and a Science project This project was (61374038; supported 61473079; by the National Natural result is proposed in Section 3. Section 4 extendsThe the result funded Priority Academic Program Development Foundation of (61374038; and project result is in 3. extends the to time-delay systems under non-triangular funded by by the the Priority Academic61473079; Program 61374060), Development ofa Jiangsu Foundation of China China (61374038; 61473079; 61374060), andof a Jiangsu project result is proposed proposed in Section Section 3. Section Section extends the result result to nonlinear nonlinear time-delay systems under44the the non-triangular Higher Institutions (1108007002). funded by Academic Program to nonlinear time-delay systems under the non-triangular Higher Education Education Institutions (1108007002). funded by the the Priority Priority Academic Program Development Development of of Jiangsu Jiangsu to nonlinear time-delay systems under the non-triangular Higher Education Institutions (1108007002).
Higher Education Institutions (1108007002). Copyright IFAC 2015 2015 185 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 185 Copyright IFAC 2015 185 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 185Control. 10.1016/j.ifacol.2015.09.375
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growth condition. Section 5 illustrates the obtained results by a numerical example, which is followed by the conclusion in Section 6. 2. PROBLEM STATEMENT AND PRELIMINARIES Consider the following nonlinear time-delay system x˙ 1 (t) = x2 (t) + ϕ1 (x1 (t), x1 (t − τ )), x˙ 2 (t) = x3 (t) + ϕ2 (x1 (t), x2 (t), x1 (t − τ ), x2 (t − τ )), .. . x˙ n−1 (t) = xn (t) + ϕn−1 (x1 (t), . . . , xn−1 (t), (1) x1 (t − τ ), . . . , xn−1 (t − τ )), x˙ n (t) = u(t) + ϕn (x1 (t), . . . , xn (t), x1 (t − τ ), . . . , xn (t − τ )), y = x (t), 1 x(t) = φ(t), −τ ≤ t ≤ 0
where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ Rn is the system state; u(t) ∈ R is the control input; y(t) ∈ R is the system output; τ is a constant which represents time delay; x(t − τ ) = (x1 (t − τ ), x2 (t − τ ), . . . , xn (t − τ ))T ∈ Rn stands for the delayed state; φ(t) is the initial function of the system state vector; ϕi (·), i = 1, 2, . . . , n,represent nonlinear perturbations that are not guaranteed to be precisely known. In this paper, our objective is to develop a linear output feedback controller which can semi-globally stabilize the uncertain nonlinear time-delay system (1). The problem of semi-global stabilization for system (1) can be described as follows. System (1) is said to be semi-global stabilized by linear output feedback controller if for each compact subset Ωx ⊂ Rn , there exists another nonempty set Ωxˆ ⊂ Rn and a linear output feedback controller { x ˆ˙ = F (ˆ x, y, u), (2) u = u(ˆ x)
Remark 2. Assumption 1 is just a theoretical hypothesis. In practical applications, if the nonlinear function can be simplified as the linear forms, the works will become simple. As a consequence, we are considering the practical significance of this assumption and combining Assumption 1 with practical systems. 3. MAIN RESULTS In this section, we are devoted in the problem of semiglobal stabilization of the nonlinear time-delay system (1). That is to say, we show that there exists a linear output feedback controller which can stabilized the nonlinear time-delay system (1) semi-globally. A linear output feedback controller of system (1) will be obtained under Assumption 1. Theorem 1. For a given compact subset Ωx = {x| ∥x∥ ≤ r} ⊂ Rn , suppose that nonlinear time-delay system (1) satisfies Assumption 1, then for any initial condition x(t) = φ(t) ∈ Ωx , there exists an appropriate gain L ≥ 1 such that system (1) can be semi-globally stabilized by the following output feedback controller x ˆ˙ 1 (t) = x ˆ2 (t) + La1 (x1 (t) − x ˆ1 (t)), 2 ˙ 2 (t) = x ˆ (t) + L a (x (t) − x ˆ1 (t)), x ˆ 3 2 1 .. (5) . n−1 x ˆ˙ (t) = x ˆn (t) + L an−1 (x1 (t) − x ˆ1 (t)), ˙ n−1 n ˆ1 (t)), x ˆn (t) = u(t) + L an (x1 (t) − x and
u(t) = −Ln [b1 x ˆ1 (t) +
1 1 b2 x ˆ2 (t) + · · · + n−1 bn x ˆn (t)] (6) L L
where ai > 0 and bi > 0, i = 1, 2, . . . , n are the coefficients of the Hurwitz polynomials p1 (s) = sn + a1 sn−1 + · · · + an−1 s + an (7) and p2 (s) = sn + bn sn−1 + · · · + b2 s + b1 .
(8)
such that the solution of the closed-loop system is locally stable and [ ] [ ] x(0) x(t) 2n ∈ Ω ⊂ R ⇒ lim = 0. (3) x ˆ(0) ˆ(t) t→∞ x
Proof. For i = 1, 2, . . . , n, new coordinates transformation with a constant scaling gain L ≥ 1 to be determined are given as follows zi (t) = xi (t)/Li−1 , zˆi (t) = x ˆi (t)/Li−1 , v(t) = u(t)/Ln (9)
In order to solve the problem better, the following assumption is given. Assumption 1. For i = 1, 2, . . . , n, ϕi (·) satisfies
and εi (t) = zi (t) − zˆi (t), ε(t) = (ε1 (t), ε2 (t), . . . , εn (t))T . (10)
|ϕi (x1 (t), . . . , xi (t), x1 (t − τ ), . . . , xi (t − τ ))|
≤ c1 (y)(|x1 (t)| + . . . + |xi (t)|)
+c2 (y)(|x1 (t − τ )| + . . . + |xi (t − τ )|)
(4)
where c1 (y) and c2 (y) are smooth functions. Remark 1. Assumption 1 is a linear growth condition and similar assumptions can be find in Qian et al. (2002) and Chai (2013). Under this hypothesis, the nonlinear function ϕi (x1 (t), · · · , xi (t), x1 (t − τ ), · · · , xi (t − τ )) for i = 1, · · · , n should be bounded by linear terms. Assumption 1 is more general than the linear growth condition imposed in Qian et al. (2002) for the reason that Assumption 1 reduces to the nonlinear system in Qian et al. (2002) when c2 = 0. 186
Under the above definition, a simple calculation is given as ε(t) ˙ = LA1 ε(t) + Φ (11) with
−a1 .. . A1 = −a n−1 −an
1 ··· .. . . . . 0 ··· 0 ···
0 .. . ,Φ = 1 0
L
ϕ1 (·) 1 ϕ2 (·) L . .. . 1 ϕ (·) n n−1
Moreover, with the help of the coordinates change (9), the output feedback controller (6) can be rewritten as u(t) = Ln v(t), v(t) = −b1 zˆ1 (t) − · · · − bn zˆn (t). (12)
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Then zˆ˙ (t) = LB zˆ(t) + LA2 Cε(t) where
0 1 .. .. . . B= 0 0 −b1 −b2
(13)
T ··· 0 a1 1 .. .. a2 0 . . , A2 = . , C = . . .. .. ··· 1 an 0 · · · −bn
In the following, we prove that the closed-loop systems (11) and (13) are semi-global stable. According to the definition of ai and bi , it is not difficult to verify that matrices A1 and B are Hurwitz matrices. Consequently, there exist two positive definite matrices P1 = P1T > 0 and P2 = P2T > 0 such that T AT 1 P1 + P1 A1 = −I, B P2 + P2 B = −I.
(14)
Choose a Lyapunov function described as T
T
V1 (ε(t), zˆ(t)) = (d + 1)ε (t)P1 ε(t) + zˆ (t)P2 zˆ(t),
(15)
where d = 2∥P2 A2 C∥2 . The derivative of V1 (ε(t), zˆ(t)) along the trajectory of the closed-loop systems (11) and (13) is given as V˙ 1 (ε(t), zˆ(t)) = (d + 1)ε˙T (t)P1 ε(t) + (d + 1)εT (t)P1 ε(t) ˙ T T ˙ ˙ +zˆ (t)P2 zˆ(t) + zˆ (t)P2 zˆ(t) T = (d + 1)εT (t)AT 1 LP1 ε(t) + (d + 1)Φ P1 ε(t)
+(d + 1)εT (t)P1 LAT 1 ε(t)
+c2 (y)(|x1 (t − τ )| + · · · ]} +|xn (t − τ )|) [ 1 = 2(d + 1)∥ε(t)∥ · ∥P1 ∥ (1 + + · · · L 1 + n−1 )(c1 (y)|x1 (t)| + c2 (y) L 1 1 ·|x1 (t − τ )|) + (1 + + · · · + n−2 ) L L |x2 (t)| |x2 (t − τ )| + c2 (y) ) ·(c1 (y) L L 1 |xn−1 (t)| + · · · + (1 + )(c1 (y) L Ln−2 |xn−1 (t − τ )| +c2 (y) ) Ln−2 |xn (t)| |xn (t − τ )| ] +(c1 (y) n−1 + c2 (y) ) L Ln−1 { ≤ 2(d + 1)c3 (y)∥ε(t)∥ |x1 (t)|
|x2 (t)| |x2 (t − τ )| + L L |xn (t)| |xn (t − τ )| } , (17) + · · · + n−1 + L Ln−1 where c3 (y) > 0 is a smooth function. Remark 3. c1 (y) and c2 (y) are bounded for the reason that y = x1 and x1 is in the compact subset Ωx . Consequently, c3 (y) is existed and dependent on Ωx see Qian (2005b). Similar conclusion can be find in reference Zhai et al. (2013) and Jia et al. (2010). +|x1 (t − τ )| +
For any arbitrarily large number r > 1, define compact sets Ωx = {x|∥x∥ ≤ r, x ∈ Rn }, N = {(ε, zˆ)|∥ε∥ ≤ 2r, ∥ˆ z∥ ≤ 2r} and Ω = {(ε, zˆ)|V1 (ε, zˆ) ≤ α, α = max V1 (ε, zˆ)}.
+(d + 1)ε(t)T P1 Φ + zˆT (t)B T LP2 zˆ(t) ˆ(t) +εT (t)C T AT 2 LP2 z
(ε,ˆ z )∈N
+ˆ z T (t)P2 LB T zˆ(t) + zˆT (t)P2 LA2 Cε(t)
For any x(t) ∈ Ωx , we obtain (ε, zˆ) ∈ N by using the coordinates transformation given as (9) and (10). It is obvious that Ω is a nonempty compact set which is in the interior of N and (ε, zˆ) are bounded in Ω.
z (t)∥2 = −(d + 1)L∥ε(t)∥2 − L∥ˆ +2(d + 1)εT (t)P1 Φ
+2Lˆ z T (t)P2 A2 Cε(t)
187
(16)
For L ≥ 1, using Assumption 1, the following inequality holds 2(d + 1)εT (t)P1 Φ ≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ · ∥Φ∥
xi (t) From εi (t) = zi (t) − zˆi (t) = L ˆi (t), we know that i−1 − z � x (t) � � i � zi (t)| + |εi (t)|. (18) � i−1 � ≤ |ˆ L
Consequently, for any (ε, zˆ) ∈ Ω, (17) can be rewritten as
= 2(d + 1)∥ε(t)∥ · ∥P1 ∥ √ 1 1 · |ϕ1 |2 + | ϕ2 |2 + · · · + | n−1 ϕn |2 L L 1 ≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ · (|ϕ1 | + |ϕ2 | L 1 + · · · + n−1 |ϕn |) L {
2(d + 1)εT (t)P1 Φ ≤ 2(d + 1)c3 (y)∥ε(t)∥
n ∑ i=1
[|ˆ zi (t)| + |εi (t)|
+|ˆ zi (t − τ )| + |εi (t − τ )|]
z (t)∥2 + α3 ∥ε(t − τ )∥2 ≤ α1 ∥ε(t)∥2 + α2 ∥ˆ
+α4 ∥ˆ z (t − τ )∥2 , (19) where αi (i = 1, 2, 3, 4) are constants depending on Ω.
≤ 2(d + 1)∥ε(t)∥ · ∥P1 ∥ c1 (y)|x1 (t)| 1[ c1 (y)(|x1 (t)| +c2 (y)|x1 (t − τ )| + L +|x2 (t)|) + c2 (y)(|x1 (t − τ )| ] 1 [ +|x2 (t − τ )|) + · · · + n−1 c1 (y) L ·(|x1 (t)| + · · · + |xn (t)|)
On the other hand, √ 1 2Lˆ z T (t)P2 A2 Cε(t) ≤ 2L( √ ∥ˆ z (t)∥)( 2∥P2 A2 C∥ · ∥ε(t)∥) 2 1 z (t)∥2 + Ld∥ε(t)∥2 . (20) ≤ L∥ˆ 2 Substituting (19) and (20) into (16), we have 187
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z (t)∥2 V˙ 1 (ε(t), zˆ(t)) ≤ −(d + 1)L∥ε(t)∥2 − L∥ˆ
+α1 ∥ε(t)∥2 + α2 ∥ˆ z (t)∥2 + α3 ∥ε(t − τ )∥2 1 z (t)∥2 + Ld∥ε(t)∥2 +α4 ∥ˆ z (t − τ )∥2 + L∥ˆ 2 L = −(L − α1 )∥ε(t)∥2 − ( − α2 )∥ˆ z (t)∥2 2 +α3 ∥ε(t − τ )∥2 + α4 ∥ˆ z (t − τ )∥2 . (21)
In Assumption 1, a lower-triangular growth condition for system (1) is necessary. Actually, the following more general condition can also be established which means the nonlinear conditions in the nonlinear time-delay system (1) are bounded under the non-triangular growth condition instead of lower-triangular growth condition. Assumption 2. For i = 1, 2, . . . , n, ϕi (·) satisfies |ϕi (x(t), x(t − τ ))| n n ∑ ∑ |xj (t)| + c2 (y) |xj (t − τ )| ≤ c1 (y)
Choose a Lyapunov-Krasovskii functional described as V2 (ε(t), zˆ(t)) = V1 (ε(t), zˆ(t)) +
∫t
t−τ
j=1
(α3 ∥ε(s)∥2
+α4 ∥ˆ z (s)∥2 )ds. (22) Combine with (21), the derivative of V2 (ε(t), zˆ(t)) satisfies L z (t)∥2 V˙ 2 (ε(t), zˆ(t)) ≤ −(L − α1 )∥ε(t)∥2 − ( − α2 )∥ˆ 2 +α3 ∥ε(t − τ )∥2 + α4 ∥ˆ z (t − τ )∥2 +α3 (∥ε(t)∥2 − ∥ε(t − τ )∥2 )
z (t)∥2 − ∥ˆ z (t − τ )∥2 ) +α4 (∥ˆ
= −(L − α1 − α3 )∥ε(t)∥2 L −( − α2 − α4 )∥ˆ z (t)∥2 . (23) 2 The right hand of (23) is negative definite under the condition that L > max{α1 + α3 , 2(α2 + α4 ), 1}, which means V˙ 2 (ε(t), zˆ(t)) ≤ −β1 ∥ε(t)∥2 − β2 ∥ˆ z (t)∥2 (24) holds for two positive constants β1 and β2 . As a result, starting from Ω, the trajectory will stay in the compact set forever. Moreover lim (ε(t), zˆ(t)) = 0. (25) t→∞
This implies that Ω is a domain of attraction. Remark 4. L is related to αi (i = 1, 2, 3, 4) which are dependent on Ωx . Thus, L is also dependent on Ωx and the property of semi-global is incarnated here. This is different from the global results in which L is just appropriate but not dependent on Ωx . From the definition, we know that for any arbitrarily large set Ωx , (x, x ˆ) ∈ Ωx × Ωx ⇒ ∥ε(t)∥ ≤ 2r, ∥ˆ z (t)∥ ≤ 2r
⇒ (ε(t), zˆ(t)) ∈ Ω. (26) Then we know that starting from any points in Ωx × Ωx , the trajectory will stay in the compact set Ω and tend to the origin. In conclusion, Theorem 1 comes into existence that means the problem of semi-global stabilization for the nonlinear time-delay system (1) can be solved by the linear output feedback controller (5) and (6). 4. EXTENSION In this section, we extend the result to nonlinear timedelay systems under the non-triangular growth condition. 188
(27)
j=1
where c1 (y) and c2 (y) are smooth functions. Theorem 2. Under Assumption 2, for a given compact subset Ωx = {x|∥x∥ ≤ r} ⊂ Rn , for any initial condition x(t) = φ(t) ∈ Ωx , there exists an appropriate gain L ≥ 1 such that system (1) can be semi-globally stabilized by the output feedback controller (5) and (6). Proof. The proof process is similar to that of Theorem 1 by using the same output feedback controller (5) and (6). (17) is hold under Assumption 2 although the nonlinear conditions are not triangular form. Consequently, the semiglobal stabilization problem can be solved by designing the output feedback controller as the form (5) and (6) with an appropriate gain L. The detailed proof is omitted here for brevity. 5. ILLUSTRATIVE EXAMPLE In this section, we give an example to show how to apply the results proposed in this paper to investigate the semi-global stabilization of a class of nonlinear time-delay system. Let us consider the following 2-dimensional nonlinear timedelay system x˙ 1 (t) = x2 (t) + x1 (t) ln(1 + x21 (t)) + 0.5 sin(x1 (t − 0.8)), (28) x˙ 2 (t) = u(t) + 0.5x1 (t) + 0.4x2 (t) x21 (t) 2 ) cos(x2 (t − 0.8)). + 0.5x1 (t)(1 − 2
It is obvious that the nonlinear time-delay system (28) satisfies Assumption 1. Next we check whether the output feedback controller { x ˆ˙ 1 (t) = x ˆ2 (t) + La1 (x1 − x ˆ1 ), (29) ˆ1 ), x ˆ˙ 2 (t) = u(t) + L2 a2 (x1 − x and u(t) = −L2 [b1 x ˆ1 (t) +
1 b2 x ˆ2 (t)] L
(30)
can stabilize the system (28) semi-globally according to Theorem 1. In this example, the compact set Ωx and N in Theorem 1 are given as Ωx = {x|∥x∥ ≤ 5} and N = {(ε, zˆ)|∥ε∥ ≤ 10, ∥ˆ z ∥ ≤ 10} correspondingly. The gains are selected as a1 = 1, a2 = 0.5, b1 = 5, b2 = 4, αi (i = 1, 2, 3, 4) and L can be obtained as α1 = α2 = α3 = α4 = 1.6 and L = 6.8 relatively through calculating, and for −0.8 ≤ t ≤ 0 the initial functions are chosen as ) ( ) ( ) ( ) ( x1 (t) x ˆ1 (t) 0.5 sin(t − 0.8) 0.6 = , = . x2 (t) x ˆ2 (t) −0.8 sin(t − 0.8) −1
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1 x1 0.5
x ˆ1
0 −0.5
0
1
2
3
4
5 Time(s)
6
7
8
9
10
5 x2 x ˆ2 0
−5
0
1
2
3
4
5 Time(s)
6
7
8
9
10
Fig. 1. State response. 50 u 0 −50 −100 −150
0
1
2
3
4
5 Time(s)
6
7
8
9
10
Fig. 2. Time history of the control signal. Obviously, the initial functions are in the given set Ωx ×Ωx . Remark 5. The region of Ωx is not given in the examples of many existing semi-global works and L is selected as an appropriate gain in not only semi-global works but also global results. However, in this example, we give the specific region of Ωx and L is calculated using Theorem 1. It is easy to obtain by MATLAB, ( ) ( ) 0.7500 −0.500 1.1500 0.100 , P2 = . P1 = −0.500 2.500 0.100 0.1500 The simulation is given in Fig.1 and Fig.2. Fig.1 is the swing curve of the state response and Fig. 2 shows the trajectory of the input control u. It is clear that the nonlinear time-delay system (28) under the linear output feedback control law u = −L2 [b1 x ˆ1 (t) + L1 b2 x ˆ2 (t)] is asymptotically stable. From the simulations, we can conclude that the result presented in Theorem 1 is very practicable in semiglobal stabilization of nonlinear time-delay system. 6. CONCLUSIONS In this paper, the semi-global stabilization problem of a class of nonlinear lower-triangular and non-triangular systems with time delay in the state and a linear output feedback controller with a scaling gain has been investigated through using a Lyapunov-Krasovskii technique. The output feedback control law, by which the semiglobal asymptotic stability of the closed-loop system is guaranteed, is memoryless and easy for implementation. Simulations show that the results obtained in this paper are very practicable in semi-globally analyzing the stability of a class of nonlinear time-delay systems. REFERENCES Q. Gong, and W. Lin. A note on global output regulation of nonlinear systems in the output feedback form. IEEE Transactions on Automatic Control, volume 48, number 6, pages 1049–1054, 2003. 189
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