Finite-time stability analysis and H∞ control for a class of nonlinear time-delay Hamiltonian systems

Automatica 49 (2013) 390–401

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Finite-time stability analysis and H∞ control for a class of nonlinear time-delay Hamiltonian systems✩ Renming Yang a,b , Yuzhen Wang a,1 a

School of Control Science and Engineering, Shandong University, Jinan 250061, PR China

b

School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan 250357, PR China

article

info

Article history: Received 18 October 2010 Received in revised form 5 March 2012 Accepted 23 September 2012 Available online 8 December 2012 Keywords: Nonlinear time-delay Hamiltonian system Finite-time stability Finite-time H∞ control Razumikhin approach Energy shaping

abstract This paper investigates the finite-time stability (FTS) analysis and finite-time H∞ control design for a class of nonlinear time-delay Hamiltonian systems, and proposes some delay-dependent results on both the FTS and finite-time control design. First, a criterion on the FTS is proposed for general time-delay nonlinear systems via the Razumikhin approach. Then, based on the criterion, some delay-independent and delay-dependent conditions on the FTS are derived for the nonlinear time-delay Hamiltonian systems by constructing a suitable Lyapunov function. Third, we use the obtained FTS results to investigate the finite-time H∞ control problem, and present a control design procedure for a class of nonlinear time-delay port-controlled Hamiltonian systems by the energy-shaping approach. Study of two illustrative examples shows that the results obtained in this paper work very well in the FTS analysis and control design of some nonlinear time-delay Hamiltonian systems. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction As well known, time delay inevitably occurs in many practical control systems, such as communication systems, neural networks, engineering systems, process control systems, etc. The delay is a source of the instability and oscillatory response, and thus the stability analysis and control design of the time-delay systems are of great importance for both theoretical and practical reasons (Gu, Kharitonov, & Chen, 2003; Zhong, 2006). In the past two decades, many nice results have been obtained for the analysis and synthesis of both linear time-delay systems (Bagchi, 1976; Boyd, Ghaoui, Feron, & Balakrishnan, 1994; Gu et al., 2003; Han, 2005; He, Wu, She, & Liu, 2004; Park, 1999; Xu & Lam, 2005; Zhang, Tsiotras, & Knospe, 2002) and nonlinear time-delay systems (Coutinho & de Souza, 2008; Fridman, Dambrine, & Yeganefar, 2008; Mazenc & Bliman, 2006; Nguang, 2000; Papachristodoulou, 2005; Teel, 1998; Wu, 1999). The stability criteria for time-delay systems

✩ This work is supported by the National Natural Science Foundation of China (G61074068, G61034007, G61174036), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the Natural Science Foundation of Shandong Province (ZR2010FM013). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hendrik Nijmeijer under the direction of Editor Andrew R. Teel. E-mail addresses: [email protected] (R. Yang), [email protected] (Y. Wang). 1 Tel.: +86 531 88392515; fax: +86 531 88392205.

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.11.034

can be divided into two cases: the delay-independent and delaydependent ones. Since the delay-independent criterion can lead to more conservatism for small-size delay systems, considerable attention has been devoted to the delay-dependent one. Apart from the study of time delay, another important research issue in practice is the so-called fast convergence. In general, an asymptotically stable controller cannot guarantee that the system under study achieves the control performance of fast convergence, while a finite-time controller possesses such a property. As indicated by Hong, Huang, and Xu (2001), the finitetime controller possesses not only fast convergence but also better robustness and disturbance attention properties. It is noted that the problems on the finite-time stability (FTS) and stabilization have drawn considerable attention, see, e.g. Bhat and Bernstein (2000), Haimo (1986), Hong and Jiang (2006), Hong et al. (2001); Hong, Xu, and Huang (2002), Moulay and Perruquetti (2006), Orlov (2005) and Wang and Feng (2008). Although much of the work has been focused on the FTS and finite-time stabilization for nonlinear systems, there are few results on the FTS for nonlinear time-delay ones except Karafyllis (2006) and Moulay, Dambrine, Yeganefar, and Perruquetti (2008). Karafyllis (2006) investigated the finite-time stabilization of a class of time-delay systems in the triangular form by using the back-stepping technique. The FTS of the nonlinear time-delay systems was studied by Moulay et al. (2008), and a delay-independent result was derived via the Lyapunov–Krasovskii (L–K) functional method. It should be pointed out that the study of FTS and finite-time H∞ control design for nonlinear time-delay systems is more difficult, because

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

nonlinear time-delay systems have more complicated dynamic behaviors than the systems without delay. Furthermore, it is very difficult to find an L–K functional satisfying the derivative condition of the FTS as shown in Moulay et al. (2008), which is the very reason why there are fewer results on the FTS for nonlinear time-delay systems. In recent years, the port-controlled Hamiltonian (PCH) system, proposed in Maschke and van der Schaft (1992) and van der Schaft and Maschke (1995), has been well studied in Escobar, vander Schaft, and Ortega (1999), Fujimoto, Sakurama, and Sugie (2003), Ortega, van der Schaft, Maschke, and Escobar (2002), Wang, Li, and Cheng (2003), and applied to many practical control problems (Cheng, Xi, Hong, & Qin, 1999; Wang et al., 2003; Xi, Cheng, Lu, & Mei, 2002). The Hamiltonian function in a PCH system is considered as the total energy and can be used as a candidate of Lyapunov function in many physical systems. Due to this and its nice structure, the energy-based approach was extended to study the FTS of the PCH systems without delay in Wang and Feng (2008) and the stability of the nonlinear timedelay Hamiltonian systems in Pasumarthy and Kao (2009), Sun and Wang (2007), respectively. In Sun and Wang (2007), the authors investigated the stability of a class of time-delay Hamiltonian systems and proposed a constant Hamiltonian realization for a class of nonlinear systems with a constant time delay. The stability of a class of nonlinear Hamiltonian systems with time-varying delay was studied in Pasumarthy and Kao (2009), and some sufficient conditions on the stability were derived via constructing corresponding L–K functionals. Although the nonlinear time-delay Hamiltonian system is a kind of important nonlinear time-delay system, there are, to the authors’ best knowledge, few results on the FTS and finite-time H∞ control design for the systems. In this paper, we investigate the FTS and finite-time H∞ control problems of a class of nonlinear time-delay Hamiltonian systems, and propose a number of delay-dependent results on the FTS and control design. First, a criterion on the FTS is derived by Razumikhin approach (R-approach, for short) for general nonlinear time-delay systems. Then, based on the criterion obtained, the FTS of the nonlinear time-delay Hamiltonian systems is investigated, and several delay-dependent sufficient conditions are derived for the systems by constructing a suitable Lyapunov function. Third, we use the obtained FTS results to investigate the H∞ control problem, and present a control design procedure for a class of nonlinear time-delay PCH systems by the energy-shaping approach (Ortega et al., 2002). Study of two illustrative examples shows that the results obtained in this paper work very well in the FTS analysis and finite-time H∞ control design of the time-delay Hamiltonian systems. The main contributions of this paper are as follows. A new criterion on the FTS is established for nonlinear time-delay systems by using the R-approach. In the criterion, the key is to find a suitable Lyapunov function satisfying the derivative condition of the FTS, which is more difficult than that of the asymptotic stability. To apply the new criterion to a class of nonlinear time-delay Hamiltonian systems, a specific form of the Lyapunov function is also presented in this paper. In comparison with the existing results such as the L–K functional method, our method has some advantages in establishing the derivative condition of the FTS for nonlinear timedelay PCH systems in practice, which is shown in Sections 3 and 4 in this paper (also see Remark 10). Another contribution of this paper is that we present some delay-dependent results on both the FTS and finite-time H∞ control design. It is noted that there are fewer delay-dependent results and also fewer results on the finitetime H∞ control design for nonlinear time-delay systems in the literature. Generally, it is more difficult to obtain delay-dependent results than delay-independent ones, and delay-dependent results have less conservatism than delay-independent ones for small-size delay systems.

391

The remainder of the paper is organized as follows. Section 2 is the problem formulation and preliminaries. In Section 3, several FTS results are proposed for the nonlinear time-delay Hamiltonian systems, and the finite-time H∞ control problem is discussed in Section 4. In Section 5, we give two illustrative examples to support our new results, which is followed by the conclusion in Section 6. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of n × m real matrices. For symmetric matrices P and Q , the notation P > Q (P > Q ) means that matrix P − Q is positive definite (positive semi-definite), and similarly, P < Q (P 6 Q ) means that matrix P − Q is negative definite (negative semi-definite). A matrix-valued function P : X → Rn×n is positive definite if P (x) is positive definite for each x ∈ X and there exists α > 0 such that P (x) > α In for all x ∈ X , where In is the n × n identity matrix. ∥x∥ denotes the Euclidean norm of x. The space of continuously differentiable function φ : [−h, 0] → Rn with finite norm ∥φ∥h = sup−h6t 60 ∥φ(t )∥ is denoted by £nh , and xt ∈ £nh is a segment of the function x(·) given by xt (θ ) = x(t + θ ), ∀θ ∈ [−h, 0] and xt ∈ Ω ⊂ Rn means that x(t + θ ) ∈ Ω , ∀θ ∈ [−h, 0]. 2. Problem formulation and preliminaries In this section, we first give the problem’s formulation, and then present some preliminaries, which will be used in the sequel. Consider the following nonlinear time-delay Hamiltonian system described as Pasumarthy and Kao (2009) and Sun and Wang (2007):



x˙ (t ) = R(x)∇x H (x) + T (x)∇x˜ H (˜x), x(τ ) = φ(τ ), ∀τ ∈ [−2h, 0],

(1)

where x(t ) ∈ Ω ⊆ Rn is the state, R(x) ∈ Rn×n satisfies R(x) + RT (x) < 0 for all x ∈ Ω , T (x) ∈ Rn×n , x˜ (t ) := x(t − h), h > 0 is a constant time delay, H (x) is the Hamiltonian function with x = 0 as its minimum point, ∇x H (x) is the gradient vector of H (x) at x, φ(τ ) is a vector-valued initial-condition function, τ = 2θ (θ ∈ [−h, 0]), and Ω is some bounded convex neighborhood of the origin, in which local stability analysis will be investigated. Similar to Moulay et al. (2008), we assume that the system (1) possesses uniqueness of the solution in forward time. Remark 1. It is easy to see that the system (1) can be regarded as a generalization of the ones in Pasumarthy and Kao (2009) and Sun and Wang (2007). The main objective of the paper is to study the FTS of the system (1) and the finite-time H∞ control design problem of the system (1) with control inputs and disturbances via the R-approach. To facilitate the analysis, we give some preliminaries first. Consider the following nonlinear time-delay system:



x˙ (t ) = f (x, x˜ ), x(θ ) = φ(θ ), ∀θ ∈ [−h, 0],

(2)

where x˜ = x(t − h) and f (x, x˜ ) is a continuous vector field satisfying f (0, 0) = 0. Definition 1 (Moulay et al., 2008). Assume that the system (2) possesses uniqueness of the solution in forward time. The system (2) is finite-time stable if (1) the system (2) is stable, and (2) there exists δ > 0 such that, for any φ ∈ Cδ , there exists 0 6 T (φ) < +∞ for which x(t , φ) = 0 for all t > T (φ), where

392

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

Cδ := {φ ∈ £nh : ∥φ∥£n < δ}.T0 (φ) = inf{T (φ) > 0 : x(t , φ) = h 0, ∀t > T (φ)} is a functional called the settling time of the system (2). Lemma 1. Consider the system (2) with uniqueness of the solution in forward time. If there exist real numbers β > 1, k > 0, a Class-K function σ and a C 1 Lyapunov function, V (x), of the system (2) such that (1) σ (∥x∥) 6 V (x), and 1

(2) V˙ 6 −kV β (x), x ∈ Ω hold along the trajectory of the system whenever V (x(t + θ )) 6 V (x(t )) for θ ∈ [−h, 0], then the system (2) is finite-time stable. If Ω = Rn and σ is a Class-K∞ function, then the origin is a globally finite-time stable equilibrium of the system (2). Furthermore, the settling time of the system (2) with respect to the initial condition φ ∈ Cδ satisfies

β

T0 (φ) 6

k(β − 1)

V

β−1 β

(φ)

for all t > 0. Proof. Since V (x) is a Lyapunov function for the system (2), applying Razumikhin Theorem (Gu et al., 2003), it is easy to know that the system (2) is asymptotically stable under the conditions of the lemma. Next, we need to prove that T0 (φ) < +∞. Based on Condition 0 t (2), one can obtain V (φ) dz1 6 −k 0 dτ , from which it follows that T0 (φ) completed.

β−1

β 6 k(β−1) V β

zβ

(φ) for all t > 0. Thus, the proof is



Remark 2. The derivative condition on the FTS is more restrictive than that of the asymptotic stability (see Condition (2), which is known as the derivative condition of the FTS). In Bhat and Bernstein (2000), it has been shown that Condition (2) is necessary for studying the FTS of continuous autonomous systems. Thus, the key in applying the lemma is to find a Lyapunov function satisfying Condition (2). Since the FTS of nonlinear time-delay systems is more difficult to study than the asymptotic stability of nonlinear time-delay systems (or nonlinear systems), we will develop a new criterion on the FTS by using the R-approach, and give a specific form of Lyapunov functions for the FTS (see Sections 3, 4 and Remark 10, below).

Theorem 1. Consider the system (1). Assume that (1) the Hamiltonian function is given as H (x) =

+

Σ2T Σ1

6ϵ

Σ1T Σ3 Σ1

+ϵ

−1

Σ2T Σ3−1 Σ2

where α > 1 is a real number; (2) there exist constant symmetric matrices L and S of appropriate dimensions such that RT (x) + R(x) 6 L,

n

p−1 p

n

 i =1

 1p |xi |

n

>



1

| xi | p >

i =1



m := λmax (L) + λmax (S ) + n



|xi |

α−1 α

< 0,

Then, the system (1) is locally finite-time stable whenever H (x(t + θ )) 6 H (x(t )) for any θ ∈ [−h, 0]. Furthermore, the settling time satisfies T0 (φ) 6

(2α − 1)2 α−1 H α (φ). −2mα(α − 1)

Proof. To prove Theorem 1, we need to show that the system satisfies all the conditions of Lemma 1. Choose H (x) as a Lyapunov function candidate, then it is easy to obtain that Condition (1) of Lemma 1 is satisfied. Next, we prove Condition (2) of Lemma 1 is satisfied. Computing the derivative of H (x) along the trajectory of the system (1), and based on Lemma 2, we have

˙ (x) = 2∇xT H (x)R(x)∇x H (x) + 2∇xT H (x)T (x)∇x˜ H (˜x) 2H 6 2∇xT H (x)R(x)∇x H (x) + ∇x˜T H (˜x)∇x˜ H (˜x)

+ ∇xT H (x)T (x)T T (x)∇x H (x) = ∇xT H (x)[R(x) + RT (x) + T (x)T T (x)]∇x H (x) 2   n 1 2α (˜x2i ) 2α−1 + 2α − 1 i =1 6 (λmax (L) + λmax (S ))∇xT H (x)∇x H (x)

2α

 +

2α − 1

2  n 1 (˜x2i ) 2α−1 i =1

2α

 + (3)

2α − 1

2α



6 (λmax (L) + λmax (S ))

.

i =1

2α − 1

2  n 1 (x2i ) 2α−1



3. The FTS of time-delay Hamiltonian systems In this section, we study the FTS of the system (1), and propose several FTS results by choosing a kind of suitable Lyapunov function. For the FTS of the system (1), based on Lemma 1, we have the following results.

i =1

2  n 1 (˜x2i ) 2α−1 .

(5)

i =1

Using Lemma 3 and noticing that H (x(t + θ )) 6 H (x(t )) (θ ∈

[−h, 0]), it is easy to obtain that

holds, where | · | denotes the absolute value sign. Proof. See Appendix.

x ∈ Ω;

where n is the state’s dimension, and λmax (L) and λmax (S ) denote the maximum eigenvalues of L and S, respectively.

 1p

n

T (x)T T (x) 6 S ,

(3) the following inequality holds

Lemma 3. For any given constant number p > 1,



(4)

i =1

Lemma 2 (Liao, Chen, & Sanchezm, 2002). For any real matrices Σ1 , Σ2 , 0 < Σ3 = Σ3T of appropriate dimensions and a scalar ϵ > 0, the following inequality holds:

Σ1T Σ2

n  α (x2i ) 2α−1 ,

n 

1

(˜x2i ) 2α−1 =

i=1

i=1

1

(x2i ) 2α−1

α x2i 2α−1

(˜ )

i =1 1 α−1 n α Hα

 α1

i =1 1

= H α (x).

(˜x) 6

6

α−1 n α

α−1 1 n α Hα

  α1 n  α (˜x2i ) 2α−1 i =1

(x),   α1 n  n 1   α α 2 2α−1 α 2 2α−1 = (xi ) > ( xi ) =

n 

n  

(6)

i =1

(7)

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

Substituting (6) and (7) into (5) and based on λmax (L) + λmax (S ) < 0, we have

˙ (x) 6 (λmax (L) + λmax (S )) 2H α−1 α

+n

2

2α − 1

2α

 =m

2α



2

2α − 1

2α



2

2α − 1

1

H α (x)

Proof. Since the FTS of the system (10) is equivalent to that of the system (1), we only need to show the FTS of the system (10) under the conditions of the theorem. Choose H (x) as a Lyapunov function candidate. Computing the derivative of H (x) along the trajectory of the system (10), similar to the proof of Theorem 1, we have

˙ (x) = 2∇xT H (x)R(x)∇x H (x) + 2∇xT H (x)˜x 2H

1

H α (x)

+ 2∇xT H (x)T (x)∇x˜ H (˜x) − 2∇xT H (x)x  t  ∇xT H (x(t )) R(x(s))∇x H (x(s)) +2 t −h  + T (x(s))∇x˜ H (˜x(s)) ds    2α 2 1 α−1 α 6 λmax (L) + λmax (S ) + n H α (x) 2α − 1

1

H α (x).

From Condition (3) and Lemma 1, it is easy to see that the theorem follows, and thus the proof is completed.  Remark 3. From the proof of Theorem 1, it is easy to see that Theorem 1 still holds if R(x) in both the system and condition (2) is replaced by R(˜x).

− 2∇xT H (x)x + 2∇xT H (x)˜x  t  ∇xT H (x(t )) R(x(s))∇x H (x(s)) +2

Remark 4. It is noted that the FTS criterion stated in Theorem 1 does not depend on the delay h, that is, it is a delay-independent stability criterion. As shown in Gu et al. (2003), such a criterion often gives very conservative stability assessment when the delay is small. In the following, we propose a delay-dependent result on the FTS of the system (1). From (1), it is easy to obtain that



t −h

 + T (x(s))∇x˜ H (˜x(s)) ds.

[−˙x(s) + R(x(s))∇x H (x(s)) + T (x(s))∇x˜ H (˜x(s))]ds = 0,

∇xT H (x)x =

2α 2α − 1

Substituting (9) into (1), the system (1) can be expressed as follows

T

x˜ x˜ = r

=

t

 R(x(s))∇x H (x(s)) + T (x(s))∇x˜ H (˜x(s)) ds.

(10)

t −h

Obviously, the FTS of the system (10) is equivalent to that of the system (1).

n  α (x2i ) 2α−1 , i=1

where α > 1 is a real number; (2) there exist a constant real number ϵ > 0 and constant symmetric matrices L, Z and S of appropriate dimensions such that 4α 2α − 1

RT (xt )R(xt ) 6 Z ,

,

RT (x) + R(x) 6 L, T (xt )T T (xt ) 6 S ,

(11)

xt ∈ Ω ;

(3) the following inequality holds m := λmax (L) + λmax (S ) + {h(λmax (Z ) + λmax (S ))

+ 2h + 1 + ϵ −1 }n

α−1 α

< 0,

where r := max{∥˜x∥ : x˜ ∈ Ω }. Then, the system (1) is locally finite-time stable whenever H (x(t + τ )) 6 H (x(t )), τ ∈ [−2h, 0]. Furthermore, the settling time satisfies

(2α − 1)2 α−1 T0 (φ) 6 H α (φ). −2mα(α − 1)

r

r

2

2α

6r

2

α    2α− 1 2 n  x˜ i

r

i=1

n

 α (˜x2i ) 2α−1 ,

r 2α−1 i=1

x˜ ∈ Ω ,

which implies that x˜ T x˜ 6

r2 2α

H (˜x) 6

r2 2α

r 2α−1

2α−2

H (x) = r 2α−1 H (x)

(15)

holds on Ω . For the terms ∇xT H (x)∇x H (x) and ∇x˜T H (˜x)∇x˜ H (˜x), similar to the proof of Theorem 1, one can obtain

(1) the Hamiltonian function is given as

2α−2

n  2  x˜ i

r 2α−1

Theorem 2. Consider the system (1). Assume that

ϵ r 2α−1 6

2

i =1

x˙ = R(x)∇x H (x) + T (x)∇x˜ H (˜x) − x + x˜



i =1

(9)

t −h

(14)

α < 1, we obtain x˜ 2i and 2α− 1

n

Noting that x˜ T x˜ =

[R(x(s))∇x H (x(s)) + T (x(s))∇x˜ H (˜x(s))]ds − x + x˜ = 0.

H (x) =

(13)

2∇xT H (x)˜x 6 ϵ −1 ∇xT H (x)∇x H (x) + ϵ x˜ T x˜ .

t

+

H (x).

Based on Lemma 2, we have

(8)

from which it follows that



(12)

For the term ∇xT H (x)x, one can obtain

t t −h



393

T xH

∇

(x)∇x H (x) 6

α−1 n α



α−1 α



∇x˜T H (˜x)∇x˜ H (˜x) 6 n

2α

2

2α − 1 2α

2

2α − 1

1

H α (x), 1

H α (x).

(16)

On the other hand, using Lemma 2 and (16), and noting that H (x(t + τ )) 6 H (x(t )) (τ ∈ [−2h, 0]), we have 2∇xT H (x(t ))R(x(s))∇x H (x(s)) 6 ∇xT H (x(t ))∇x H (x(t )) + ∇xT H (x(s))Z ∇x H (x(s)) 6 [1 + λmax (Z )]n

α−1 α



2α

2

2α − 1

1

H α (x);

(17)

2∇xT H (x(t ))T (x(s))∇x˜ (s) H (˜x(s)) 6 ∇xT H (x(t ))∇x H (x(t )) + ∇x˜T(s) H (˜x(s))S ∇x˜ (s) H (˜x(s)) 6 [1 + λmax (S )]

α−1 n α



2α 2α − 1

2

1

H α (x).

(18)

394

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

Substituting (13)–(18) into (12) and using Conditions (2) and (3), we have

˙ ( x) 6 2H



based on which and (21), we have

˙ 6 2∇xT H (x)R(x)∇x H (x) + b∇x˜T H (˜x)∇x˜ H (˜x) 2H

λmax (L) + λmax (S )   2α 2 1 α−1 −1 α + n (1 + ϵ ) H α (x) 2α − 1 2α−2 4α − H (x) + ϵ r 2α−1 H (x) 2α − 1 2  t  2α {2 + λmax (Z ) + 2α − 1 t −h α−1

+ b−1 ∇xT H (x)T (x)T T (x)∇x H (x)  t  ∇xT H (x(s)) R(x(s))∇x H (x(s)) +2 t −h  + T (x(s))∇x˜ H (˜x(s)) − x˙ (s) ds = 2∇xT H (x)R(x)∇x H (x) + b∇x˜T H (˜x)∇x˜ H (˜x) + b−1 ∇xT H (x)T (x)T T (x)∇x H (x)  t  ∇xT H (x(s)) R(x(s))∇x H (x(s)) +2 t −h  + T (x(s))∇x˜ H (˜x(s)) ds

1

+ λmax (S )}n α H α (x(t ))ds 2  2α 1 H α (x), =m 2α − 1 from which and Lemma 1 it is easy see that the theorem follows, and thus the proof is completed.  Remark 5. Note that it is very difficult to obtain the delaydependent FTS results by using R-approach. In fact, to obtain the delay-dependent result on the FTS of the system (1) via R-approach, based on Gu et al. (2003), one needs to obtain the model transformation form of the system (1). However, ∂ 2 H (x) ∂ x2

since Hess(H (x)) =

equation = ∇x H (x) − ∇x˜ H (˜x) to obtain the model transformation form of the system (1), which is the reason why we have introduced the equivalent form (10) for the system (1). The FTS criterion stated in Theorem 2 depends on the delay h, which is different from the one in Moulay et al. (2008). In the following, to facilitate the application, we present another delaydependent result.

t



∇x˜T H (˜x(s))∇x˜ H (˜x(s))ds

a t −h

6 an

6

2α



α−1 α

α−1 ahn α

i=1

where α > 1 is a real number; (2) there exist constant symmetric matrices L and S of appropriate dimensions such that T (xt )T T (xt ) 6 S ,

xt ∈ Ω ;

2α − 1

1

H α (x(t )),

from which it follows that t



∇x˜T H (˜x(s))∇x˜ H (˜x(s))ds

0 6 −a t −h

α−1 α

2α



2

2α − 1

1

H α (x(t )).

(23)

L + a− 1 S 6 0 ,

(19) α−1 n α

S ) + (b + ha)

< 0.

(20)

Then, the system (1) is locally finite-time stable whenever H (x(t + τ )) 6 H (x(t )), τ ∈ [−2h, 0]. Furthermore, the settling time satisfies

(2α − 1)2 α−1 H α (φ). −2mα(α − 1)

˙ 6 2∇xT H (x)R(x)∇x H (x) + b∇x˜T H (˜x)∇x˜ H (˜x) 2H From (1), we obtain R(x)∇x H (x) + T (x)∇x˜ H (˜x) − x˙ = 0,

+ 2∇xT H (x(s))T (x(s))∇x˜ H (˜x(s))  − a∇x˜T H (˜x(s))∇x˜ H (˜x(s)) ds 6 ∇xT H (x)[L + b−1 S ]∇x H (x) + b∇x˜T H (˜x)∇x˜ H (˜x)

+ ahn

α−1 α

2α



 +b +

(21)

2α 2α − 1

α−1 ahn α

2

2α − 1

6 λmax (L + b−1 S )

Proof. Choose H (x) as a Lyapunov function candidate. Computing the derivative of H (x) along the trajectory of the system (1), based on Lemma 2, we have

+ b−1 ∇xT H (x)T (x)T T (x)∇x H (x).

+ b−1 ∇xT H (x)T (x)T T (x)∇x H (x)  2 2α 1 α−1 + b∇x˜T H (˜x)∇x˜ H (˜x) + ahn α H α (x) 2α − 1  t  + ∇xT H (x(s))[R(x(s)) + RT (x(s))]∇x H (x(s)) t −h

(3) there exist positive constant numbers a and b such that

T0 (φ) 6

2

˙ 6 ∇xT H (x)[R(x) + RT (x)]∇x H (x) 2H

n  α (x2i ) 2α−1 ,

m := λmax (L + b

1

H α (˜x(s))ds

Substituting (23) into (22) and based on Condition (2), we have

(1) the Hamiltonian function is given as

−1

t t −h

2α



Theorem 3. Consider the system (1). Assume that

RT (xt ) + R(xt ) 6 L,

2 

2α − 1

+ ahn

H (x) =

(22)

Noting that (6) and H (x(t + τ )) 6 H (x(t )) for any τ ∈ [−2h, 0], one can obtain

is not continuous, one cannot use the

∂ 2 H (x) x˙ (s)ds t −h ∂ x 2

t

− 2H (x(t )) + 2H (x(t − h)).



1

H α (x) +



t

ξ T Υ ξ ds t −h

2α



2  n 1 (x2i ) 2α−1

2α − 1 i =1 2  n 1 (˜x2i ) 2α−1 i =1

2α 2α − 1

2

1

H α (x) +



t

ξ T Υ ξ ds, t −h

where ξ := [∇xT H (x(s)), ∇x˜T H (˜x(s))]T and R(x(s)) + RT (x(s)) Υ := T T (x(s))



T (x(s)) . −aIn



R. Yang, Y. Wang / Automatica 49 (2013) 390–401

delay, φ(τ ) is a vector-valued initial-condition function, and

According to Schur complement lemma and (19), we have R(x(s)) + R (x(s)) + a T

−1

T (x(s))T (x(s)) 6 0, T

which implies that Υ 6 0. On the other hand, using (6), (7) and Condition (3), it is easy to obtain that

˙ 6 λmax (L + b−1 S ) 2H + (b + ah)n 2α

 =m

2α − 1

α−1 α

2

2α



2



2α

1

H (x) α

2α − 1

2

2α − 1

1

H α (x)

1

H α (x).

Remark 6. It is easy to see that if a > b holds in Theorem 3, then the condition (19) can be omitted. In fact, when a > b, one can derive (19) from (20) directly. Remark 7. To obtain the delay-dependent FTS result of the system (1),  t different from Theorem 2, we have introduced the equation 2 t −h ∇xT H (x(s)){R(x(s))∇x H (x(s)) + T (x(s))∇x˜ H (˜x(s)) − x˙ (s)}ds = 0 in the proof of Theorem 3. Moreover, it is easy to see that the conditions of Theorem 3 are more easily checked than the ones in Theorem 2. Remark 8. It is noted that the Hamiltonian function given in this section can be replaced by the one in the form of n 

|xi |β ,

1 < β < 2.

H (x) =

(24)

i=1 2α < 2, it is easy to see that the In fact, since α > 1 H⇒ 1 < 2α− 1 form given in (24) is equivalent to that given in (4).

Remark 9. It should be pointed out that the FTS results proposed in this section can be reduced to the case of conventional asymptotic stability, as long as one takes α = 1 in (4) or β > 2 in (24). Remark 10. In this section, a kind of specific form of Lyapunov function is used to obtain the results on the FTS, which is different from the method in Moulay et al. (2008). This makes it more convenient to apply our results to investigate the finitetime stability and stabilization for the nonlinear time-delay PCH systems in practice (see, Sections 4 and 5, below). Furthermore, from the proofs of Theorems 1–3, it can be seen that one can apply Lemma 1 and the given Lyapunov function to easily obtain the derivative condition of the FTS, which is an advantage of the Rapproach used in this paper.

α > 1.

(26)

i =1

Without loss of generality, we assume that g1 (x) has full column rank, which implies that g1T (x)g1 (x) ̸= 0 for any x ∈ Ω . Furthermore, to study the finite-time H∞ control problem, we give the following assumption with respect to w :

4. Finite-time H∞ control design



(25)

where x(t ) ∈ Ω ⊂ Rn is the state vector, u ∈ Rm is the control input, w ∈ Rq (q < n) is the external disturbance, h is a constant



w w dt 6 1 , T

0

where µ > 0 is a real number. Remark 11. In this section, to facilitate the analysis, the Hamiltonian function of the system (25) is assumed as (26). In fact, for a general nonlinear time-delay PCH system, its Hamiltonian function can be changed into the form of (26) by using the ‘‘energy shaping’’ technique (Ortega et al., 2002). Please see the following example. Example. Consider the following nonlinear time-delay PCH system with any Hamiltonian function: x˙ = R(x)∇x H1 (x) + T (x)∇x˜ H1 (˜x) + g1 (x)u + g2 (x)w.

(27)

α 2 2α−1 i=1 xi

, Ha (x) := H (x) − H1 (x). To change Let H (x) = ( ) the Hamiltonian function into the form of (26), we can design the control law u such that n

g1 (x)u = R(x)∇x Ha (x) + T (x)∇x˜ Ha (˜x) + g1 v,

(28)

where v is a new input. Substituting (28) into the system (27), one can obtain a Hamiltonian system with its Hamiltonian function in the form of (26). Therefore, in this section, we always assume that the Hamiltonian function has the form of (26). Now, consider the system (25). Let the disturbance attenuation level γ > 0 be given, and choose z = h(x)g1T (x)∇x H (x)

(29)

as the penalty signal, where h(x) is a weighted matrix with appropriate dimension. The finite-time H∞ control problem of the system (25) is to find a control law u = α(x) such that the closed-loop system consisting of the system (25) and the control law is finite-time stable when w vanishes, and meantime, for any non-zero w ∈ Θ , the zero state response of the closed-loop system (φ(θ ) = 0, w(θ ) = 0, θ ∈ [−h, 0]) satisfies T

∥z (t )∥2 dt 6 γ 2 0

In this section, we use the results obtained in Section 3 to investigate the finite-time H∞ control problem for a class of nonlinear time-delay PCH systems, and present a new control design procedure. Consider the following nonlinear time-delay PCH system:

+∞

  Θ = w ∈ R q : µ2



x˙ = R(x)∇x H (x) + T (x)∇x˜ H (˜x) + g1 (x)u + g2 (x)w, x(τ ) = φ(τ ), ∀τ ∈ [−2h, 0],

n  α (x2i ) 2α−1 ,

Assumption 1. The disturbance w in the system (25) belongs to the set (Coutinho & de Souza, 2008)

From (20) and Lemma 1, it is easy to see that the theorem holds, and thus the proof is completed. 

H (x) =

395

T



∥w(t )∥2 dt ,

∞ > T > 0.

(30)

0

To facilitate the analysis, we consider a typical set Ω as follows (Boyd et al., 1994)

  Ω := x : αiT x 6 1, i = 1, 2, . . . , n ,

(31)

where αi , i = 1, 2, . . . , n, are known constant vectors, which define n edges of Ω . For the finite-time H∞ control problem of the system (25), we have the following result. Theorem 4. Under Assumption 1, assume that, for the given γ > 0,

396

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

(i) there exist constant symmetric matrices K , L, Z and S of appropriate dimensions such that g1 (xt )h (xt )h( ) T

xt g1T

γ2

2s −

xt ∈ Ω ;

2 µ2

−sαi

(32)

(33)

+ [ahl + b]n

α−1 α

κµ2

r



2α − 1

2

2α

< 0,

where l := maxx∈Ω {H (x)} and r := max{∥˜x∥ : x˜ ∈ Ω }. Then, a finite-time H∞ control law of the system (25) can be designed as



1

u=−

2



1

hT (x)h(x) +

I g1T (x)∇x H (x). 2 m

2γ

(34)

Proof. Substituting (34) into (25) leads to



x˙ = F (x)∇x H (x) + T (x)∇x˜ H (˜x) + g2 (x)w, z = h(x)g1T (x)∇x H (x),

1 2

g1 (x)hT (x)h(x)g1T (x) −

Choose H (x) = date, and set

n

1 2γ

By Assumption 1 and noting that w(θ ) = 0, θ ∈ [−h, 0], we obtain



g (x)g1T (x). 2 1

t

w (s)w(s)ds 6 T

(x2i ) 2α−1 as a Lyapunov function candi-

D(t , x) := 2H (x) +

t

Next, we prove that D(t , x) 6 0. Computing the derivative of H (x) along the trajectory of the closed-loop system (35) yields

˙ (x) 6 2∇xT H (x)F (x)∇x H (x) + b∇x˜T H (˜x)∇x˜ H (˜x) 2H + ϵ∇

(x)g2 (x) (x)∇x H (x).

(36)

based on which and (38) it is easy to know that t

 + 2F (x(s))]∇x H (x(s)) + 2T (x(s))∇x˜ H (˜x(s)) ds +

1

.

κµ2

(39)

0 6 H (x(t ))



t

 ∇xT H (x(s)) [κ g2 (x(s))g2T (x(s))

 + 2F (x(s))]∇x H (x(s)) + 2T (x(s))∇x˜ H (˜x(s)) ds

2∇xT H (x)F (x)∇x H (x)



= 2∇xT H (x) R(x) −

1 2γ

+

 T g (x)g1 (x) ∇x H (x) 2 1

(x)g1 (x)h (x)h(x)g1T (x)∇x H (x)   1 T T = 2∇x H (x) R(x) − g1 (x)g1 (x) ∇x H (x) − ∥z ∥2 , 2γ 2 −∇

 ∇xT H (x(s)) [κ g2 (x(s))g2T (x(s))

t −h

On one hand,

T xH

,

Therefore, from (39) and H (x(t )) > 0, we have

+ b−1 ∇xT H (x)T (x)T T (x)∇x H (x) + ϵ −1 w T w g2T

1

µ2

t −h

0

T xH

wT (s)w(s)ds 6 0

0≤

(∥z (s)∥2 − γ 2 ∥w(s)∥2 )ds.

+∞



t −h

 

(38)

(35)

α

i=1

∇xT H (x(s))˙x(s)ds

t −h

where F (x) := R(x) −



t −h t

 ∇xT H (x(s)) F (x(s))∇x H (x(s)) t −h  + T (x(s))∇x˜ H (˜x(s)) + g2 (x(s))w(s) ds  t  ∇xT H (x(s)) F (x(s))∇x H (x(s)) ≤2 t −h  + T (x(s))∇x˜ H (˜x(s)) ds  t  + κ∇xT H (x(s))g2 (x(s))g2T (x(s))∇x H (x(s)) t −h  + κ −1 w T (s)w(s) ds  t  ≤ ∇xT H (x(s)) [κ g2 (x(s))g2T (x(s)) t −h  + 2F (x(s))]∇x H (x(s)) + 2T (x(s))∇x˜ H (˜x(s)) ds  t + κ −1 wT (s)w(s)ds.

ϵ −1 6 γ 2 , 2α−2 2α−1

t

=2

(iii) there exist positive real numbers ϵ, κ, a and b such that

L + κ Z − K + a−1 S 6 0,

 =2

In

m := λmax {L + b−1 S + ϵ Z } +

F (x(s))∇x H (x(s))

t −h

0 6 2H (x(t )) − 2H (x(t − h))

i = 1, 2, . . . , n;

1



from which and Lemma 2 it follows that

 −sαiT  > 0,

t

 + T (x(s))∇x˜ H (˜x(s)) + g2 (x(s))w(s) ds,

(ii) there exist positive real numbers s, µ such that 2α−2



t −h

T (xt )T T (xt ) 6 S ,

r 2α−1 γ 2

x˙ (s)ds =

g1 (xt )g1T (xt ) 6 L,

g2 (xt )g2T (xt ) 6 Z ,



t



(xt ) 6 K ,

1

RT (xt ) + R(xt ) −

and on the other hand, based on (35), we have

1

κµ2

H (x(t )).

(40)

Next, we prove the following inequality:

T

(37)

H ( x) 6 r

2α−2 2α−1



2α − 1 2α

2

∇xT H (x)∇x H (x),

x ∈ Ω.

(41)

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

Noting that (26) and α > 1, we have n 

H (x) =

with which and Condition (i) we have α

2α

xi2α−1 = r 2α−1

i

i =1

r2

i=1 1

n  2  2α−1  x

2α

i

6 r 2α−1

=r



2α−2 2α−1

=r

r2

i=1

 T ˙ 2H (x) 6 ∇x H (x) L + b−1 S + ϵ Z

n  2  2α−1  x

2α

2α−2 2α−1

n  1 (x2i ) 2α−1

1

+

κµ2

i =1

2α − 1

2 ∇

2α

T xH

(x)∇x H (x),

+ H (x(t )) + alhn

t

 ∇xT H (x(s)) [0.5κ g2 (x(s))g2T (x(s)) 0 6 2H (x(t )) t −h  + F (x(s))]∇x H (x(s)) + T (x(s))∇x˜ H (˜x(s)) ds 1

κµ

2α − 1



2α−2

r 2α−1 2

∇xT H (x)∇x H (x)

2α

2α 2α



2α − 1

2  In ∇x H (x)

2  n 1 (˜x2i ) 2α−1 + ϵ −1 wT w i =1

t



ξ T Υ ξ ds

α−1 α

2α



2

2α − 1

1

H α (x),

where ξ := [∇xT H (x(s)), ∇x˜T H (˜x(s))]T and

Υ :=

2

2α − 1

t −h

Substituting (41) into (40), it is easy to obtain that

+

r



2α−2 2α−1

− ∥z ∥2 + b

which implies that the inequality (41) holds on Ω .



397

(42)

L + κZ − K T T (x(s))

T (x(s)) . −aIn





Based on (33) and Schur complement lemma, one can obtain Υ 6 0.

holds on Ω .

In addition, based on (6), we have

Based on (23), we have H (x(t ))



 2  1 2α−2 2α − 1 T ˙ 2 α− 1 r In 2H (x) 6 ∇x H (x) L + κµ2 2α 

t

a∇x˜T H (˜x(s))∇x˜ H (˜x(s))ds t −h

6 ahn

α−1 α

6 alhn

α−1 α

2α



2

2α − 1 2α



+ b−1 S + ϵ Z ∇x H (x) − ∥z ∥2 + ϵ −1 w T w

1

H 1+ α (x(t ))

2

2α − 1

1

H α (x(t )),

x ∈ Ω,

+ [ahl + b]n

t



a∇x˜T H (˜x(s))∇x˜ H (˜x(s))ds



t −h

+

α−1 alhn α



2α

2

2α − 1

˙ ( t , x) 6 ∇ D

T xH

1

H α (x(t ))



˙ (x) 6 2∇xT H (x) R(x) − 2H

1 2γ

6

κµ2

T xH

+

T

+ ϵ∇xT H (x)g2 (x)g2T (x)∇x H (x) + ϵ −1 wT w  t  + H (x(t )) ∇xT H (x(s)) [κ g2 (x(s))g2T (x(s)) t −h  + 2F (x(s))]∇x H (x(s)) + 2T (x(s))∇x˜ H (˜x(s)) ds  t − H (x(t )) a∇x˜T H (˜x(s))∇x˜ H (˜x(s))ds

1

κµ2

r

2α−2 2α−1

+ [ahl + b]n  6



+

1

κµ2

2α−2

r 2α−1

2α − 1 2α

2

r



2α−2 2α−1

2α − 1 2α

α−1 α

(44)

2α − 1

∇xT H (x)∇x H (x),



2 In + b

2α α−1 α

2α



2

2α − 1

−1

S + ϵZ

1

H α (x(t ))

2  ∇xT H (x)∇x H (x)

2α



2

2α − 1

1

H α (x(t ))

λmax {L + b−1 S + ϵ Z } +

1

κµ

r 2

2α−2 2α−1

+ [ahl + b]n



2α − 1

 6 m

2α 2α − 1

2  

2α α−1 α



2

1

2α

2

2α − 1 2α

2α − 1

t −h



1

H α (x).

λmax {L + b−1 S + ϵ Z }



g (x)g1T (x) ∇x H (x) 2 1

(x)T (x)T (x)∇x H (x)  2 2α α−1 1 + b∇x˜T H (˜x)∇x˜ H (˜x) + alhn α H α (x) 2α − 1

− ∥z ∥ + b ∇

(x) L +

1

× ∇x H (x) + [ahl + b]n 

Substituting (37), (42) and (43) into (36), one can obtain

−1

2α − 1

(43)

holds on Ω , where l = maxx∈Ω {H (x)}.

2

2

˙ (t , x) and noting (7) and Condition (iii), Substituting (44) into D we have

from which it follows that 0 6 −H (x(t ))

2α



α−1 α

H α (x).

2

1

H α (x(t ))

1

H α (x(t ))

(45)

398

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

Integrating (45) from 0 to T and using the zero state response condition, we obtain T



2H (x) +

(∥z (s)∥ − γ ∥w(s)∥ )ds 6 0, 2

2

2

(46)

0

from which and the fact that H (x) > 0 we have T



∥z (s)∥2 ds 6 γ 2

T



In addition, in view of xT x = that xT x = r 2

n  2  xi

r

i=1

n 

r2

= r

2α

2α−1

n

i =1

α x2i and 2α− < 1, we know 1

n  2  2α−1  xi

r

i =1 α

2α−2

(x2i ) 2α−1 = r 2α−1 H (x)

(47)

xT x 6 r

2α−2 2α−1

H (x) 6 r

γ2 2

∥w(s)∥2 ds 6

r 2α−1 γ 2

0

from which it follows that ∥x∥2 6

2µ2

where H (x) = ( ) , R(˜x) = − (1 + x˜ ) and T (x) = 0. It is easy to obtain that R(˜x) + RT (˜x) 6 −

,

2α−2 r 2α−1 γ 2 . 2µ2

4

λmax (L) = − ,

2α−2

x x−

r 2α−1 γ 2

6 0,

2µ2

s.t. 2 − 2αiT x > 0,

i = 1, 2, . . . , n.

2α−2 2α−1



γ2 r 2s −  ξ 2µ2 −sαi

−sαiT  ξ > 0,

T

i = 1, 2, . . . , n,

In

which is equivalent to Condition (32), then (48) holds, where ξ = [1, xT ]T and s > 0 is a free scalar introduced by the S-procedure. Thus, one can see from Condition (32) that the trajectory x(t ) remains in Ω for all t > 0, φ = 0 and w ∈ Θ . On the other hand, it is easy to obtain from (44) and m < 0 2α

2

1

˙ 6 that H H α (x) when w = 0, which implies that the 2α−1 closed-loop system (35) is finite-time stable. Therefore, the proof is completed.  m 2



3

3

:= L,

T (x)T T (x) = 0 := S ,

Remark 12. To obtain the delay-dependent result on the finitetime H∞ control problem in Theorem 4, similar to Theorem 3, we have introduced the inequalities (42) and (43) in the proof of Theorem 4. Besides, it is well worth pointing out that the selection of l in (43) may lead to conservatism to some extent, and the larger l may lead to the more conservatism. Remark 13. It is noted that Condition (2) of Theorem 3 as well as Condition (i) of Theorem 4 are only with respect to xt , that is, these conditions are not the mixed variable case of x and xt . Thus, to check these conditions, one only needs to test the matrix inequalities for all x ∈ Ω . In addition, it is well worth pointing out that we only need to find the upper bounds of these matrices inequalities, not to solve them. There are many methods for checking these matrix inequalities, e.g., the convex optimization method (Papachristodoulou, 2005), the convex-vertex method (Coutinho & de Souza, 2008) (only check whether the vertices of Ω satisfy these inequalities), and so on.

λmax (S ) = 0,

and m = − 43 < 0. Thus, according to Remark 3, all the (replaced) conditions of Theorem 1 are satisfied for the system. From the theorem, the system (49) is globally finite-time stable, and the settling time is given as

(48)

Applying the S-procedure (Boyd et al., 1994), if



4

2

from which it follows that

Next, we prove that x(t ) ∈ Ω for ∀t > 0, φ = 0 and all w ∈ Θ . Based on (31), we need to prove T

(50)

2 3

x2 4

2α−2

T



(49)

x˙ = R(˜x)∇x H (x) + T (x)∇x˜ H (˜x), 3

i=1

holds on Ω . Based on Assumption 1, (46) and (47), we have 2α−2 2α−1

x˙ (t ) = −|x(t )|β sgn(x(t ))(1 + x2 (t − h)),

where x(t ) ∈ R1 and β = 0.5. In what follows, we apply Theorem 1 to study the finite-time stability of the system. First, we express the system as the following Hamiltonian form:

α

6 r2

In this section, we give two illustrative examples to show how to use the results obtained in this paper to study the FTS and finitetime H∞ control design for nonlinear time-delay (Hamiltonian) systems. Example 1. Consider the following nonlinear time-delay system (Moulay et al., 2008):

∥w(s)∥2 ds. 0

0

5. Illustrative examples

T0 (φ) 6

(2α − 1)2 α−1 H α (φ) = 2|φ|1−β , −2mα(α − 1)

α=

3 2

,

(51)

which is the same as that in Moulay et al. (2008). From the example, it is easy to see that, by applying the specific form of Lyapunov function given in the paper, one can investigate the finite-time stability of the nonlinear time-delay systems which can be expressed in the Hamiltonian form. This is an advantage of the method proposed in the paper. Example 2. Consider the following forced mathematical pendulum system with a constant time-delay (Pasumarthy & Kao, 2009): y˙ = R(y)∇y H (y) + T (y)∇y˜ H (y(t − h)) + u + g w, where H (y) = 12 y22 +(1 − cos y1 )+ 2 y3 − y∗3 − y∗1 − (y3 − y∗3 ) − sin y∗1 )2 ,

 3

 R(y) =

0 −1 0

0 −1.55 0

1 0 , −1



 T (y) =

0 0 0

1 3

(52)

 ∗ 2

sin y1

0 0 1

+ 21 (y1 −

0 −1 , 0



g = [0.2 0.08 0.15]T , (y∗1 , 0, y∗3 ) is an equilibrium point of the system (52) with y∗1 = π given as in Pasumarthy and Kao (2009), and w is the external disturbance. Note that the asymptotic stability of the system (52) with u ≡ 0 and w ≡ 0 was studied in Pasumarthy and Kao (2009), and a delay-dependent stability result was obtained with 0 < h 6 0.132. For further details, please see Pasumarthy and Kao (2009). In the following, we use Theorem 4 to study the finite-time H∞ control design for the system (52). By taking x1 := y1 − y∗1 , x2 := y2 and x3 := y3 − y∗3 as a coordinate transformation, we have x˙ = R(x)∇x H1 (x) + T (x)∇x˜ H1 (˜x) + u + g w,

(53)

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

399

where H1 (x) = 0.5x22 + (1 + cos x1 ) + 1.5x23 + 0.5(x1 − x3 )2 and 0 −1 0

0 −1.55 0

 R(x) =

1 0 , −1



 T (x) =

0 0 0

0 0 1

0 −1 . 0



Note that the state variables of the mathematical pendulum system are bounded in the real application. Thus, in this example, we assume that the state is a bounded one, say, x = [x1 , x2 , x3 ]T ∈ Ω = {(x1 , x2 , x3 ) : |x1 | 6 2, |x2 | 6 2, |x3 | 6 2}. Now, we design a finite-time H∞ control law for the systems (53). First, by the ‘‘energy shaping’’ technique (Ortega et al., 2002), ¯ (x) = it is easy to see that H1 (x) can be changed to the form of H 4

4

4

x13 + x23 + x33 under the following control: u = R(x)∇x Ha (x) + T (x)∇x˜ Ha (˜x) + v,

Fig. 1. Responses of the state.

(54)

¯ (x) − H1 (x). In this case, the where v is a new input and Ha (x) := H system (53) is expressed as ¯ (x) + T (x)∇x˜ H¯ (˜x) + v + g w. x˙ = R(x)∇x H

(55)

Furthermore, let a disturbance attenuation level γ = 0.4 be given, and choose

¯ (x) = z = h(x)I3T ∇x H

4



1

1

1



x13 + x23 + x33

3

(56)

as the penalty signal, where h(x) = [1, 1, 1] is a weighting vector. Second, a straightforward computation shows that

 I3 h (xt )h( ) T

xt I3T

=

1 1 1

1 1 1

1 1 1



 −6.25 −1 =

1

R (xt ) + R(xt ) − T

γ

:= K ,

I IT 2 3 3

−1 −9.35

1

0

:= L,  T

gg =

0.04 0.016 0.03

0.016 0.0064 0.012 0 0 0

 T T (xt )T (xt ) =

0 1 0

0.03 0.012 0.0225

1 0 −8.25



0 0 1



λmax (Z ) = 0.0689, λmax (S ) = 1. Third, choosing a = , b = 1, ϵ = 6.25, µ = 0.8, κ = 9 and αi = 0.5(i = 1, 2, 3), it is easy to obtain that ϵ −1 6 γ 2 , L + κ Z − K + a−1 S 6 0, and αi |xi | 6 1(i = 1, 2, 3) for any x ∈ Ω . On the other hand, for x ∈ Ω , we have α = 2, r = 2 and l = 7.5595 such  2α−2  α−1 1 2α−1 2 2α−1 that m := λmax {L+b−1 S +ϵ Z }+ κµ +[ahl+b]n α < 2r 2α 0 for 0 < h < 1.95, which implies that Condition (iii) of Theorem 4 holds for the system. In addition, there exists s = 0.3 such that (32) holds. Thus, all the conditions of Theorem 4 are satisfied for this system when 0 < h < 1.95. From Theorem 4, a finite-time H∞ controller of the system (55) can be designed as

 29 1  2 1 2 1 − x13 − x23 − x33  6 3 3    29 13 2 1  2 31 v = − x1 − x2 − x33  .  3 6 3   2 1  2 13 29 13 3 − x1 − x2 − x3

x1 − 4x3 −

29

1

x13 −

2

1

x23 +

2

1



x33

 6 3 3  P (x, x˜ )  2 13 37 31 4 13 2 13 −x1 + 4x3 − x1 − x2 − x3 + x˜ 2 − x˜ 2 3 3 6 3



u= 

:= S ,

1 8

6



:= Z ,

λmax (L) = −5.6049,

3

Therefore, a finite-time H∞ controller of the system (53) is given as



which implies that Condition (i) of Theorem 4 holds for the system. Moreover, it follows from the above that

3

Fig. 2. Swing curves of the control u.

where P (x, x˜ ) := x1 − x3 + 1

31 x 20 2

1

− sin x1 − 2x13 −

1

207 3 x 30 2

(57)

1

− 23 x33 −

4 3 x 3 3

˜ − x˜ 1 + 4x˜ 3 .

To show the effectiveness of the control law (57), we carry out some numerical simulations for the system (53) with choices as follows. Initial condition: φ(t ) = (−0.1, 0.1, − 0.15); Time delay: h = 0.1. To test the robustness of the controller with respect to external disturbances, a disturbance of amplitude 5 is added to the system in the time duration [0.05–0.1 s]. The simulation results are shown in Figs. 1 and 2, which are the state’s responses and the control signal u used in this example, respectively. It can be observed from Figs. 1 and 2 that the state converges to the equilibrium quickly after the disturbance is removed. The simulation results show that the controller (57) is very effective in the finite-time robust stabilization of the system (53), and has strong robustness against external disturbances. 6. Conclusion In this paper, we have investigated the FTS and H∞ control design of a class of nonlinear time-delay Hamiltonian systems via Razumikhin approach, and obtained a number of new results on the FTS and robust finite-time stabilization. Different from the

400

R. Yang, Y. Wang / Automatica 49 (2013) 390–401

existing results, we have obtained some delay-dependent results on the FTS and proposed a finite-time H∞ control design procedure for the systems. Moreover, a specific form of Lyapunov functions is presented to study the FTS of the systems, which makes it more convenient to apply our new results to investigate the FTS and control design for the nonlinear time-delay PCH systems in practice. The study of two illustrative examples has shown that the results obtained in this paper work very well in the FTS analysis and robust finite-time stabilization of some nonlinear time-delay (PCH) systems. Appendix. The proof of Lemma 3 It is easy to know from Jensen’s inequality (Wang & Feng, 2008) 1

n

n

1

p that i=1 |xi | p > holds. i=1 |xi | It is noted that the first part of (3) holds true when n = 2 (Qian & Lin, 2001). Now, we show that it holds for arbitrarily given n. Obviously, the first part of (3) holds for p = 1. Next, we consider the case p > 1, which is divided into the following two steps:

Step 1: We prove that the inequality

 f

n 

 λ i xi

i =1

≥

n 

λi f (xi )

(58)

i=1

holds for ∀xi ∈ I ⊆ R1 (i = 1, 2, . . . , n) and i=1 λi = 1 (λi > 0), where f (·) ∈ R1 is a concave function on the interval I (i.e. f (λx1 + (1 − λ)x2 ) ≥ λf (x1 ) + (1 − λ)f (x2 ), λ ∈ (0, 1)). We apply the mathematical induction to prove the inequality (58). When n = 2, it is obvious that the inequality (58) holds. Assume that the inequality (58) holds when n = k. Next, we prove k+1 that the inequality (58) holds for n = k + 1. Let i=1 λi = 1 (λi >

n

0) and αi =

λi

1−λk+1

(i = 1, . . . , k), then

k

i=1

αi = 1. Thus, we

have f (λ1 x1 + λ2 x2 + · · · + λk xk + λk+1 xk+1 )

 λ 1 x1 + · · · + λ k xk = f (1 − λk+1 ) + λk+1 xk+1 1 − λk+1   λ1 λk ≥ (1 − λk+1 )f x1 + · · · + xk 1 − λk+1 1 − λk+1 + λk+1 f (xk+1 ) = (1 − λk+1 )f (α1 x1 + · · · + αk xk ) + λk+1 f (xk+1 ) ≥ (1 − λk+1 )[α1 f (x1 ) + · · · + αk f (xk )] + λk+1 f (xk+1 )  λ1 = (1 − λk+1 ) f (x1 ) + · · · 1 − λk+1  k+1  λk + f (xk ) + λk+1 f (xk+1 ) = λi f (xi ), 1 − λk+1 i=1 

which implies that the inequality (58) holds for n = k + 1. Therefore, it holds for any n. Step 2: We prove that the first part of (3) holds for p > 1. Letting 1

f (x) = x p (p > 1, x > 0) in the inequality (58), it is easy to see that f (x) is a concave function. Furthermore, letting all λi = 1 (i = 1, . . . , n), then the first part of (3) holds. Thus, the proof is n completed.  References Bagchi, A. (1976). A martingale approach to state estimation in delay-differential systems. Journal of Mathematical Analysis and Applications, 56, 195–210.

Bhat, S. P., & Bernstein, D. S. (2000). Finite time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38(3), 751–766. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. Cheng, D., Xi, Z., Hong, Y., & Qin, H. (1999). Energy-based stabilization of forced Hamiltonian systems with its application to power systems. In Proceedings of the 14th IFAC world congress, Vol. O, Beijing, PR China (pp. 297–303). Coutinho, D. F., & de Souza, C. E. (2008). Delay-dependent robust stability and L2-gain analysis of a class of nonlinear time-delay systems. Automatica, 44, 2006–2018. Escobar, G., vander Schaft, A. J., & Ortega, R. (1999). A Hamiltonian viewpoint in the modelling of switching power converters. Automatica, 35(3), 445–452. Fridman, E., Dambrine, M., & Yeganefar, N. (2008). On input-to-state stability of systems with time-delay: a matrix inequalities approach. Automatica, 44, 2364–2369. Fujimoto, K., Sakurama, K., & Sugie, T. (2003). Trajectory tracking control of portcontrolled Hamiltonian systems via generalized canonical transformations. Automatica, 39(12), 2059–2069. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Berlin: Springer. Haimo, V. T. (1986). Finite-time controllers. SIAM Journal on Control and Optimization, 24, 760–770. Han, Q. (2005). On stability of linear neutral systems with mixed time delays: a discretized Lyapunov functional approach. Automatica, 41(7), 1209–1218. He, Y., Wu, M., She, J., & Liu, G. (2004). Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Transactions on Automatic Control, 49, 828–832. Hong, Y., Huang, J., & Xu, Y. (2001). On an output feedback finite-time stabilization problem. IEEE Transactions on Automatic Control, 46(2), 305–309. Hong, Y., & Jiang, Z. (2006). Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties. IEEE Transactions on Automatic Control, 51(12), 1950–1956. Hong, Y., Xu, Y., & Huang, J. (2002). Finite-time control for robot manipulators. Systems & Control Letters, 46, 243–253. Karafyllis, I. (2006). Finite-time global stabilization by means of time-varying distributed delay feedback. SIAM Journal on Control and Optimization, 45(1), 320–342. Liao, X., Chen, G., & Sanchezm, E. N. (2002). Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Networks, 15, 855–866. Maschke, B.M., & van der Schaft, A. (1992). Port-controlled Hamiltonian systems: modelling origins and system theoretic properties. In Proceedings of the IFAC symposium on nonlinear control systems design, Bordeaux, France (pp. 282–288). Mazenc, F., & Bliman, P. (2006). Backstepping design for time-delay nonlinear systems. IEEE Transactions on Automatic Control, 51(1), 149–154. Moulay, E., Dambrine, M., Yeganefar, N., & Perruquetti, W. (2008). Finite time stability and stabilization of time-delay systems. Systems & Control Letters, 57, 561–566. Moulay, E., & Perruquetti, W. (2006). Finite time stability and stabilization of a class of continous systems. Journal of Mathematical Analysis and Applications, 323, 1430–1443. Nguang, S. K. (2000). Robust stabilization of a class of time-delay nonlinear systems. IEEE Transactions on Automatic Control, 45(4), 756–762. Orlov, Y. (2005). Finite time stability and robust control synthesis of uncertain switched systems. SIAM Journal on Control and Optimization, 43(4), 1253–1271. Ortega, R., van der Schaft, A. J., Maschke, B. M., & Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4), 585–596. Papachristodoulou, A. (2005). Robust stabilization of nonlinear time delay systems using convex optimization. In Proc, 2005, IEEE conference on decision and control (pp. 5788–5793). Park, P. (1999). A delay-dependent stability criterion for system with uncertain time-invairiant delay. IEEE Transactions on Automatic Control, 44(4), 876–877. Pasumarthy, R., & Kao, C.Y. (2009). On stability of time-delay Hamiltonian systems. In Proc, 2009 American control conf. (pp. 4909–4914). Qian, C., & Lin, W. (2001). Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. Systems & Control Letters, 42(3), 185–200. Sun, W., & Wang, Y. (2007). Stability analysis for some class of time-delay nonlinear Hamiltonian systems. Journal of Shandong University (Natural Science), 42(12), 1–9. Teel, A. R. (1998). Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Transactions on Automatic Control, 43(7), 960–964. van der Schaft, A. J., & Maschke, B. M. (1995). The Hamiltonian formulation of energy conserving physical systems with external ports. Archive für Elektronik und Übertragungstechnik, 49, 362–371. Wang, Y., & Feng, G. (2008). Finite-time stabilization of port-controlled Hamiltonian systems with applicatioin to nonlinear affine systems. In Proceedings of the 2008 American contral conference, Westin Seattle Hotel, Seattle, Washington, USA (pp. 1202–1207). Wang, Y., Li, C., & Cheng, D. (2003). Generalized Hamiltonian realization of timeinvariant nonlinear systems. Automatica, 39, 1437–1443. Wu, W. (1999). Robust linearizing controllers for nonlinear time-delay systems. IEEE Proceedings Control Theory and Applications, 146(1), 91–97.

R. Yang, Y. Wang / Automatica 49 (2013) 390–401 Xi, Z., Cheng, D., Lu, Q., & Mei, S. (2002). Nonlinear decentralized controller design for multimachine power systems using Hamiltonian function method. Automatica, 38(3), 527–534. Xu, S., & Lam, J. (2005). Improved delay-dependent stability criteria for time-delay systems. IEEE Transactions on Automatic Control, 50(3), 384–387. Zhang, X., Tsiotras, P., & Knospe, C. (2002). Stability analysis of LPV time-delayed systems. International Journal of Control, 75, 538–558. Zhong, Q. (2006). Robust control of time-delay systems. Berlin: Springer.

Renming Yang received his M.S. degree from the Shandong Normal University, China in 2006. Now, he is a Ph.D. candidate majoring in control theory and control engineering at Shandong University, China. His research interests include the stability analysis and control design for nonlinear and nonlinear time-delay systems.

401

Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. From 2001 to 2003, he worked as a Postdoctoral Fellow in Tsinghua University, Beijing, China. Since 2003, he has been a full professor with the School of Control Science and Engineering, Shandong University, China, and the director of the Institute of Automatic Control, Shandong University. From Mar. 2004 to Jun. 2004, from Feb. 2006 to May 2006 and from Nov. 2008 to Jan. 2009, he visited City University of Hong Kong as a Research Fellow. From Sept. 2004 to May 2005, he worked as a visiting Research Fellow at the National University of Singapore. His research interests include nonlinear control systems, Hamiltonian systems and robust control. Prof. Wang received the Award of Guan Zhaozhi in 2002, the Award of Huawei from the Chinese Academy of Sciences in 2001, the Award of Natural Science from Chinese Education Ministry in 2005, and the National Award of Natural Science of China in 2008. He was selected as a Taishan Scholar of Shandong Province of China in 2011. Currently, he is an associate editor of Asian Journal of Control, and IMA Journal of Mathematic Control and Information, and a Technical Committee member of IFAC (TC 2.3).