Global finite-time adaptive stabilization for nonlinear systems with multiple unknown control directions

Automatica 69 (2016) 298–307

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Global finite-time adaptive stabilization for nonlinear systems with multiple unknown control directions✩ Jian Wu a , Weisheng Chen a , Jing Li b a

School of Aerospace Science and Technology, Xidian University, Xi’an 710071, China

b

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

article

info

Article history: Received 10 July 2014 Received in revised form 11 November 2015 Accepted 20 February 2016

Keywords: Global finite-time stability Adaptive control Multiple unknown control directions Logic switching rule Uncertain nonlinear systems

abstract In this paper, the problem of global finite-time stabilization is addressed for a class of nonlinear systems with multiple unknown control directions. Unlike all the existing works on finite-time stabilization, our work allows the control directions of the considered systems to be unknown. This paper is the first to focus on the global finite-time control of such uncertain systems. To overcome the main obstacle arising from unknown control directions, we develop a novel Lyapunov-based logic switching rule, and then the desired adaptive switching controllers are designed, where the controller parameters are to be tuned online in a switching manner according to the proposed switching logic. It is shown that the obtained controllers guarantee the closed-loop systems being globally finite-time stable. A simulation example is provided to illustrate the effectiveness of the proposed control algorithms. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction During the past two decades, an important research issue in the practical control systems, so-called finite-time convergence (Bhat & Bernstein, 1998, 2000), has received a great deal of attention. It is well-known that an asymptotically stable controller generally cannot guarantee the systems achieving the control performance of fast convergence. Therefore, increasing effort has been devoted to the problems of finite-time stability and finite-time stabilization for various types of nonlinear dynamical systems, e.g., see Bhat and Bernstein (1998), Bhat and Bernstein (2000), Hong (2002), Hong, Huang, and Yu (2001), Hong, Wang, and Chen (2006), Moulay and Perruquetti (2006), Qian and Li (2005) and references therein, to name just a few. Particularly, Bhat and Bernstein (2000) have established the theory of finite-time stability for continuous autonomous systems based on Lyapunov stability theorem. Subsequently, the problems of finite-time control have been addressed for nonsmooth nonlinear systems (Li & Qian, 2006), time-delay systems (Yang & Wang, 2012), switched systems (Orlov, 2005), stochastic systems (Chen & Jiao, 2010; Khoo, Yin,

✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Changyun Wen under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (J. Wu), [email protected] (W. Chen), [email protected] (J. Li).

http://dx.doi.org/10.1016/j.automatica.2016.03.005 0005-1098/© 2016 Elsevier Ltd. All rights reserved.

Man, & Yu, 2013; Yin & Khoo, 2014; Yin, Khoo, Man, & Yu, 2011), and so on. In this paper, we focus on solving the global finite-time stabilization problem for a class of uncertain nonlinear systems with multiple unknown control directions, which are described by

 x˙ 1 = h1 (t )x2 + f1 (x1 , t )   x˙ 2 = h2 (t )x3 + f2 (x1 , x2 , t ) ..    . x˙ n = hn (t )u + fn (x1 , . . . , xn , t )

(1)

where the system state vector x = [x1 , . . . , xn ]T ∈ Rn is measurable and u ∈ R is the control input; for i = 1, . . . , n, fi : Ri × R+ → R are unknown and locally Lipschitz functions with fi (0, t ) = 0, and hi : R+ → R/{0} are unknown but bounded. For the purposes of convenience, in the later analysis, we denote x¯ i := [x1 , . . . , xi ]T and u := xn+1 . In addition, the signs of the functions hi (t ), called the control directions of the system (1), are assumed to be unknown. When hi (t ) = 1 with i = 1, . . . , n, by using the addinga-power-integrator technique originally developed in Lin (2000), the problem of global finite-time stabilization by output feedback for a class of planar systems has been presented in Qian and Li (2005) with p being an odd integer. Subsequently, employing the same design technique, Li and Qian (2006) have solved the global finite-time control problem for a class of nonsmooth nonlinear systems with 0 < p < 1 and Huang, Lin, and Yang

J. Wu et al. / Automatica 69 (2016) 298–307

(2005) have addressed the finite-time stabilization for higherdimensional uncertain nonlinear systems in a lower-triangular form with p = 1. In addition, the stabilization problems of the system (1) with stochastic disturbances have been addressed by using the backstepping design method (e.g., see Deng & Krstić, 1997, Deng & Krstić, 2000 and Deng, Krstić, & Williams, 2001). Very recently, based on the previous works on stochastic systems, the problems of finite-time stabilization for some stochastic nonlinear systems have been investigated in Khoo et al. (2013) and Yin and Khoo (2014) where hi (t ) = 1 are still required. When the control directions of the uncertain systems are unknown, the control problems become very difficult, and therefore this is an unfading research interest in the adaptive control community. With the introduction of Nussbaum-gain function, this problem has been originally solved by Nussbaum in Nussbaum (1983). Since then, plentiful results have been obtained on this topic for various uncertain systems, e.g., see Ding (1998), Ding and Ye (2002), Ge, Yang, and Lee (2008), Liu and Huang (2008), Liu and Tong (2015), Tiago, Alessandro, and Liu (2010), Yang, Ge, and Lee (2009), Yan and Liu (2010), Ye (2011b), Ye, Chen, and Li (2005), Ye and Ding (2001), Zhang and Ge (2007) and Zhang, Wen, and Soh (2000). Especially, Ding and Ye have addressed the adaptive control problems for several classes of uncertain nonlinear systems with unknown control directions in Ding (1998), Ding and Ye (2002) and Ye and Ding (2001). By using the discrete Nussbaum-type gain, the difficulty arising from unknown control directions has been overcome for uncertain nonlinear discrete-time systems in Ge et al. (2008) and Yang et al. (2009). Recently, by the approaches of Nussbaum-gain and adding-a-power-integrator, Yan and Liu (2010) have addressed the global practical tracking problem for a class of high-order uncertain nonlinear systems with unknown control directions under fairly weak conditions. Unfortunately, aforementioned control methods under Nussbaum-gain functions just guarantee that the signals of the closed-loop systems are bounded and the regulation/tracking errors asymptotically converge to zero. This is because these Nussbaum approaches cannot ensure the time derivative of overall Lyapunov function being negative definite. Due to this inherent drawback, the Nussbaum methods cannot solve the finite-time control problem for nonlinear systems with unknown control directions. Based on the above discussions, naturally, a challenging and practical problem is: How to solve the control problem of the global finite-time convergence for the system (1) with multiple unknown control directions? In this paper, we will address this issue. Note that in Ge et al. (2008), Tiago et al. (2010), Ye (1999), Ye (2003) and Ye (2011a), some adaptive switching controllers have been proposed based on the developed logic switching rules. The feature of these controllers is that they all contain a tuning parameter which is to be tuned online in a switching manner by following the logic switching rules, then the whole closed-loop control systems are proven to be asymptotically stable under such adaptive switching controllers. Mainly inspired by the works presented in Ye (1999), Ye (2003), Ye (2005), Ye (2009) and Ye (2011a), the global finitetime stabilization problem is to be solved for the system (1) by borrowing the adaptive switching control and the adding-apower-integrator technique in this paper. For details, the main contributions of this paper are highlighted as follows. The restriction that the control directions of the systems are known in the related works (e.g., see Bhat & Bernstein, 1998, Bhat & Bernstein, 2000, Chen & Jiao, 2010, Hong, 2002, Hong et al., 2001, Hong et al., 2006, Huang et al., 2005, Khoo et al., 2013, Li & Qian, 2006, Li, Qian, & Ding, 2010, Moulay & Perruquetti, 2006, Orlov, 2005, Qian & Li, 2005, Shen & Huang, 2012, Yang & Wang, 2012, Yin & Khoo, 2014 and Yin et al., 2011) has been successfully

299

removed. That is, we allow the control directions of the system (1) to be unknown. Obviously, these control schemes are no longer applicable to the case of multiple unknown control directions. As far as we know, this work is the first to address the global finitetime stabilization problem for nonlinear systems with multiple unknown control directions. To solve the problem of global finitetime stabilization for the system (1) with multiple unknown control directions, we propose a novel switching control scheme by developing a crucial Lyapunov-based logic switching rule. The obtained adaptive switching controllers contain a parameter to be tuned online in a switching manner based on the proposed logic switching rule. It is proved that the states of the system (1) converge to the equilibrium point x = 0 in finite time under the obtained controllers. The rest of this paper is organized as follows. In Section 2, we present some preliminaries. Section 3 proposes a logic switching rule, then a globally finite-time stable controller is developed for the system (1). In Section 4, the control scheme is extended to system (27) with ratio of odd integers. A simulation example is provided in Section 5, and we conclude the work of this paper in Section 6. Notation. The following notations are used throughout this paper. R denotes the set of all real numbers; R+ denotes the set of nonnegative real numbers; Rn denotes the real n-dimensional space; Rn×n denotes the set of n × n real matrices; For a given vector or matrix X , X T denotes its transpose; inf(·) denotes the greatest lower bound; A := B means that B is defined as A; |x| is the absolute value of the real number x; for P ≥ 0, ⌈P ⌉ denotes the maximum integer not larger than P; C i denotes a set of functions whose ithorder derivatives are continuous and differentiable; if the inverse function of function H (·) exists, it is denoted by H −1 (·). 2. Problem formulation and preliminaries 2.1. Problem formulation In this paper, the control objective is to design an adaptive switching state-feedback controller of the form u = v(x, k)

(2)

such that the system (1) is globally finite-time stable (GFTS), where k = 0, 1, 2, . . . , is a tuning parameter assigned by the logic switching rule proposed in Section 3.2. Therefore, in fact the design of our controller includes a two-step procedure: we first design a controller containing a tuning parameter k, and then develop a logic switching rule to tune this parameter online in a switching manner. Remark 1. It should be mentioned that the problem of global finite-time stabilization of the system (1) has already been considered in Huang et al. (2005), Li and Qian (2006) and Qian and Li (2005). However, the control schemes in Huang et al. (2005), Li and Qian (2006) and Qian and Li (2005) are proposed under hi (t ) = 1, i = 1, . . . , n, i.e., they all require a priori knowledge of the control directions of the system (1). Obviously, they are no longer applicable to the case of multiple unknown control directions. This paper attempts to solve such an open control problem by developing an adaptive switching controller for the system (1) with multiple unknown control directions. 2.2. Global finite-time stability Consider the following nonlinear switching system x˙ = f (x, uσ (t ) (x, t ), t ),

x(t0 ) = x0 ,

(3)

300

J. Wu et al. / Automatica 69 (2016) 298–307

where x ∈ Rn is the system state, and x0 is the initial state with the initial time t0 ; the switching function σ : [t0 , +∞) → B := {0, 1, 2, . . .}, also called a logic switching rule, can be designed according to the practical requirements; U := {ui (x, t ) ∈ Rn × [t0 , +∞) → Rm }+∞ i=0 is a set of candidate controllers and f : Rn × Rm × [t0 , +∞) → Rn is the continuous function satisfying f (0, 0, t ) = 0. Specifically, let σ (t ) = k, t ∈ [tk , tk+1 ), and {tk }+∞ k=0 is a sequence of switching time. For simplicity, the solution of the system (3) starting with (t0 , x0 ) is denoted by x(t ). In this paper, we assume that the system states do not jump at each switching time tk , thus the solution x(t ) is continuous. Definition 1 (Bhat & Bernstein, 2000). The equilibrium point x = 0 of the system (3) is finite-time convergent if there is an open neighborhood S ⊂ Rn of the origin and a function T : S \ {0} → (t0 , +∞), such that ∀ x0 ∈ S , the solution x(t ) of the system (3) with x0 as the initial condition is defined and unique in forward time for t ∈ [0, T (x0 , t0 )), and x(t ) ∈ S \ {0} for t ∈ [t0 , T (x0 , t0 )) and limt →T x(t ) = 0 and x(t ) = 0, ∀t > T . Then, T (x0 , t0 ) := min{t |x(t ) = 0, t ≥ t0 } > 0 is called the settling time. The equilibrium point x = 0 of the system (3) is finite-time stable if it is Lyapunov stable and finite-time convergent. Particularly, when S = Rn , the equilibrium point x = 0 of the system (3) is said to be GFTS.

When c = p∗ /q∗ ≤ 1, where p∗ > 0 and q∗ > 0 are odd integers, then

|z c − yc | ≤ 21−c |z − y|c . Lemma 4 (Huang et al., 2005). Suppose κ and λ are two positive real numbers, and φ ≥ 0, ϕ ≥ 0 and π ≥ 0 are continuous functions. Then, for any constant ς > 0,

φ κ ϕ λ π ≤ ς φ κ+λ +

  κλ λ κ κ+λ ϕ κ+λ π λ . κ + λ ς (κ + λ)

3. GFTS adaptive switching control In this section, the global finite-time control problem is addressed for the system (1). We first design a state-feedback controller with a tuning parameter k. Then, a logic switching rule is constructed to tune this parameter online. Finally, under the obtained adaptive switching controller, the global finite-time stability of the closed-loop system can be proved, and the main result is given by a theorem. 3.1. Controller design

2.3. Some important lemmas

To achieve the control objective, we introduce the following assumptions.

Lemma 1. Consider nonlinear system x˙ = f (x, t ), x(t0 ) = x0 , where x ∈ Rn is the system state, and x0 is the initial state with the initial time t0 . Let the function f (x, t ) in a domain D be discontinuous only on a set M of measure zero. In addition, if the following condition is satisfied

Assumption 1 (Huang et al., 2005). For i = 1, . . . , n, the following inequality holds:

(x − y) · (f (x, t ) − f (y, t )) ≤ ρ(|x − y| ), 2

where x, y ∈ Rn , function ρ(u) ≥ 0,as u ≥ 0, is strictly increasing, continuous and concave such that 0+ du/ρ(u) = ∞. Then, the solution of the system has uniqueness in forward time. The above lemma can be obtained by combining Theorem 178 in Situ (2005) and Theorem 1 in Filippov (1988, Page 106). The detailed proofs are omitted here due to space limitation. Lemma 2. Suppose that W (t ) is a nonnegative function defined on [t0 , +∞), and dW (t ) dt

ν

≤ −ρ[W (t )] ,

(4)

where ρ > 0 and 0 < ν < 1 are the constants. Then, we have



1

W (t ) ≤ [(W (t0 ))1−ν − ρ(1 − ν)(t − t0 )] 1−ν ,

W (t ) = 0,

t ∈ [t0 , T¯0 ),

t ∈ [T¯0 , +∞), (W (t ))1−ν

where we define T¯0 := t0 + ρ(10−ν) . The proof of Lemma 2 can be completed by integrating both sides of (4), and the detailed derivation is omitted here. Actually, it is also implied by some established results on finite-time stability (e.g., see Theorem 4.2 in Bhat & Bernstein, 2000). In the remainder of this subsection, we introduce two lemmas which are the key tools for the adding-a-power-integrator technique. Lemma 3 (Huang et al., 2005). For any real numbers zi , i = 1, . . . , n, y, z and 0 < c ≤ 1, the following inequality holds:

(|z1 | + · · · + |zn |)c ≤ |z1 |c + · · · + |zn |c .

|fi (¯xi , t )| ≤ (|x1 | + · · · + |xi |)γi (¯xi ),

(5)

where γi (¯xi ) ≥ 0 are the known C functions. 1

Assumption 2 (Nguang, 2000). For i = 1, . . . , n, the functions hi (t ) satisfy h¯ i ≥ |hi (t )| ≥ h0 , where h0 > 0 is an unknown constant and h¯ i > 0 are known functions. In the following part, based on the above assumptions, we will construct the finite-time controller with a tuning parameter k. To this end, we define a set of virtual controllers x∗1 (¯x0 , k), . . . , x∗n (¯xn−1 , k) and new variables ξ1 , . . . , ξn as follows

   x∗1 = 0,      x∗ = −Γ ξ q2 β (¯x ), 1 ,k 1 1 1 2 ..    .    ∗ qn xn = −Γn−1,k ξn− xn−1 ), 1 βn−1 (¯

1 q

∗ q1

1 q

∗ q1

ξ 1 = x 1 1 − x1 ξ2 = x2 2 − x2 .. . 1

∗

1

2

(6) 1

ξn = xnqn − xn qn

+3−2i where the parameters 1 = q1 > q2 > · · · > qi := 2n2n with +1 i = 1, . . . , n; for j = 1, . . . , n − 1, the functions βj (·) > 0 will be designed later; define Γj,k := ζj,k H (k) with H (·) ≥ 0 being strictly increasing with respect to k and the switching functions ζj,k ∈ {−1, +1} being dependent on the tuning parameter k which is to be tuned via a logic switching rule given in the following subsection. In this subsection, k is assumed to be fixed, thus ζj,k and H (k) are also fixed. Now, consider the following Lyapunov function

V (x) =

1 2

x21 +

n 

Wi (¯xi ),

(7)

i=2

where Wi (¯xi ) :=



xi

x∗ i



1

∗ q1

s q i − xi

i

2−qi

ds.

J. Wu et al. / Automatica 69 (2016) 298–307

From Proposition 1 in Huang et al. (2005), it can be seen that Wi (¯xi ), i = 2, . . . , n, have the following properties: ∂W

2 −q i

C-1: Wi (¯xi ) is C 1 ; moreover, ∂ x i = ξi i i − 1,



∂ Wi = −(2 − qi ) ∂ xj

∗ q1

∂ xi

, and for j = 1, . . . ,

Substituting (9)–(11) into (8) yields dV (x) dt

≤ h1 (t )x1 x2 + ξ1d ρ˜ 1 (¯x1 ) + +

xi



∂ xj

x∗ i



∗ q1

1

s q i − xi

1−qi

i

dt

= x1 [h1 (t )x2 + f1 (¯x1 , t )] +

2−qi

hi (t )ξi

+ξ



+

+

(9)

≤

3

dV (x) dt

ξ



+ ξ ρ˜ (¯xi ) d i i

(10)

3

3

j =1

where ρ¯ i ≥ 0 is a C 1 function.

(13)

n 

2−qi−1 ∗ xi

hi−1 (t )ξi−1

i=2

+

i −1 n  2 

3

n   ξid−1 ξjd + + ξ1d ρ˜ 1 (¯x1 )

j=1

i=2

3

n  [ρ˜ i (¯xi ) + ρ¯ i (¯xi ) + ρˆ i (¯xi )]ξid .

(14)

i =2

q

x∗i+1 (¯xi , k) := −Γi,k ξi i+1 βi (¯xi ), i = 1, . . . , n − 1, q u(x, k) := −Γn,k ξn n+1 βn (x),

(15)

where β1 (¯x1 ) = n + ρ˜ 1 (¯x1 ) > 0, βj (¯xj ) = n − j + 1 + ρ˜ j (¯xj )+ ρ¯ j (¯xj )+ ρˆ j (¯xj ) > 0, j = 2, . . . , n, are the C 1 functions. By substituting (15) into (14), the following inequality holds dV (x) dt

≤−

n  i=1

ξid −

n  [Γi,k hi (t ) − 1]ξid βi (¯xi ).

(16)

i =1

V 2n+1 (x) ≤ 2(ξ1d + · · · + ξnd ). Let µ := dV (x) dt

2n . 2n+1

1

≤ − V µ (x) − 4

(11)

(17)

It is easy to see that n  [Γi,k hi (t ) − 1]ξid βi (¯xi ) i =1

1 = − V µ (x) − (ζkT H − 1T )Ξ , 4

i −1 i i −1     2n−1 ∂ Wi   x˙ j  ≤ 2(2 − qi )|ξi | |ξj | 2n+1 Ci,j  ∂ xj j =1 j =1 j =1

 ξjd + ξid ρ¯ i (¯xi ),

(12)

i=2

2n

d j

with C functions γ¯i ≥ 0 and ρ˜ i ≥ 0. Similar to the equality (3.10) in Huang et al. (2005), using Lemma 4, we observe from C-1 and C-3 that

i−1

j =1

On the other hand, noting V (x) ≤ 2(ξ12 + · · · + ξn2 ) and using Lemma 3, we have

 2  |ξl |qi − 2n+1 γ¯i

j =1

1 

3

n   ξjd + [ρ˜ i (¯xi ) + ρ¯ i (¯xi )]ξid .

≤ hn (t )ξn2−qn u + +



1

≤

+ ξ1d ρ˜ 1 (¯x1 )

Based on inequality (14), we design the virtual controllers and the actual control law as

The results of C-2 and C-3 are very similar to Propositions 3 and 4 in Huang et al. (2005), and thus the detailed proofs are omitted here. According to C-2 and using Lemma 4, we can have the following inequalities

l =1

n i−1  2 

i =2

  ∗1       ∂ xi q i    2n−1 2n−1   2n+1 + · · · + |ξ | 2n+1 ˙ x Ci,j (¯xi ). j  ≤ |ξ1 | i  ∂x j    

i−1 1 

2−qi−1 ∗ xi

hi−1 (t )ξi−1

where ρˆ i (¯xi ) ≥ 0 is a C 1 function. By using the inequality (13), (12) becomes

C-3: For j = 1, . . . , i − 1, there are C 1 functions Ci,j (¯xi ) ≥ 0, such that

i  

n 

i=2

qi

fi (¯xi , t )| ≤ |ξi |2−qi 

(xi − x∗i ) + hn (t )ξn2−qn u

n n  d     ξi−1 2 −q   hi−1 (t )ξi−1 i−1 (xi − x∗i ) ≤ + ρˆ i (¯xi )ξid , 

|fi (¯xi , t )| ≤ (|ξ1 | + · · · + |ξi | )γ˜i (¯xi ).

2 −q i

2−qi−1

i =2

where d = 2n4n , and ρ˜ 1 (¯x1 ) = (1 + x21 )γ1 (¯x1 ) ≥ 0 is a C 1 function. +1 To further bound the last two terms in (8), we need the following results. C-2: For i = 1, . . . , n, there exist the C 1 functions γ˜i (¯xi ) ≥ 0, such that

|ξi

i=2

Under Assumption 2, by using Lemmas 3 and 4, the following inequality holds

x1 (h1 (t )x2 + f1 (¯x1 , t )) ≤ h1 (t )x1 x2 + x21 γ1 (¯x1 )

qi

j =1

hi−1 (t )ξi−1

i=2

(8)

Next, we bound each term on the right-hand side of (8). First, under Assumption 2, we have

≤ h1 (t )x1 x2 + ξ1d ρ˜ 1 (¯x1 ),

3

n   ξjd + (ρ˜ i (¯xi ) + ρ¯ i (¯xi ))ξid

i=2

xi+1

i−1  ∂ Wi  (¯xi , t ) + x˙ j . ∂ xj j =1

x i +1

i=2

i=2 2−qi fi i

2−qi

hi (t )ξi

n

Based on the above conclusions, the time derivative of V (x) can be calculated as dV (x)

n i−1  2  i =2

ds.

=

n  

n  i=2



i

301

(18)

where ζk := H (k)[ζ1,k , . . . , ζn,k ]T ∈ Rn , 1 := [1, . . . , 1]T ∈ Rn , H := diag{h1 (t ), . . . , hn (t )} ∈ Rn×n and Ξ := [ξ1d β1 (x1 ), . . . , ξnd βn (¯xn )]T ∈ Rn . In the next subsection, we shall develop a tuning mechanism to tune the parameter k online, such that the closed-loop system is GFTS.

302

J. Wu et al. / Automatica 69 (2016) 298–307

Remark 2. Compared with the finite-time controller developed in Huang et al. (2005) where a priori knowledge of the control directions is known hi (·) = 1 with i = 1, . . . , n, an extra tuning parameter k is added in our controller (15). Such an adaptively tuning parameter is used to deal with the obstacle arising from the unknown control directions of the system (1), i.e., the term −(ζkT H − 1T )Ξ in (18). The detailed analysis can be found in Sections 3.2 and 3.3. The same design idea is also employed to develop the finite-time controller for the system (1) with p ∈ (0, 1) in Section 4. Remark 3. Similar to the design methods proposed in Ye (1999), Ye (2003), Ye (2005), Ye (2009) and Ye (2011a), we also assume that the tuning parameter k is a fixed constant in this subsection. Then, according to a logic switching rule proposed in the following subsection, this parameter will be tuned online in a switching manner. 3.2. Lyapunov-based logic switching rule and candidate control laws design Although the signs of the functions hj (t ), j = 1, . . . , n are unknown, it is clear that for each hj (t ), its sign is either positive (denoted by +1) or negative (denoted by −1). For a given nth-order system, the signs of these functions hj (t ) have 2n possible cases which can be explicitly determined, without n loss of generality, denoted by ζ (0) , ζ (1) , . . . , ζ (2 −1) , where ζ (l) := T n [al1 , al2 , . . . , aln ] ∈ R with ali ∈ {+1, −1}, l = 0, 1, . . . , 2n − 1, i = 1, . . . , n. For example, for a second nonlinear system, the signs of the functions hi (t ), i = 1, 2 have 4 possible cases which can be denoted by ζ (0) = [+1, +1]T , ζ (1) = [+1, −1]T , ζ (2) = [−1, +1]T , ζ (3) = [−1, −1]T . Suppose that the actual signs of these functions hj (t ) are expressed by a vector ζ := [a1 , a2 , . . . , an ]T ∈ Rn with ai ∈ {−1, +1}, i = 1, . . . , n. ′ Obviously, there exists a l′ ∈ {0, 1, . . . , 2n − 1} such that ζ (l ) = ζ holds. Note that |hj (t )| ≥ h0 > 0 and H (k) is strictly increasing with respect to k, then we assign the switching vector ζk to be switched according to the following condition, for t ∈ [t0 , +∞) and k = 0, 1, 2, . . . ,

 H (k)ζ (0) ,    H (k)ζ (1) , ζk (t ) = . ..     n H (k)ζ (2 −1) ,

if k = τ · 2n if k = τ · 2n + 1

.. .

(19)

if k = τ · 2n + (2n − 1)

with the number of period τ := ⌈ 2kn ⌉. From (6) and (15), we can see that the actual controller depends on the switching vector ζk , thus, the set of the candidate controllers is given as follows

U = {uk (t ) = u(x, t , ζk )}+∞ k=0 .

(20)

We start with k = 0, switch to k = 1, then switch to k = 2, and so on. If and when to switch is based on a logic switching rule to tune the parameter k online, which is designed as follows

σ (t ) =

k, if V (x(t )) ≤ ω(t , tk , x(tk ), ϵ), t ≥ tk , k + 1, otherwise,



(21)

where ϵ > 0 is a preselected constant, V (x(t )) is given by (7), and 1

ω(t , tk , x(tk ), ϵ) = [(V (x(tk )) + ϵ)1−µ − 0.25(1 − µ)(t − tk )] 1−µ 4(V (x(tk ))+ϵ)1−µ for tk ≤ t ≤ tk + , otherwise, ω(t , tk , x(tk ), ϵ) = 0. 1−µ The sequence of the switching time is defined iteratively tk+1 := inf{t |t > tk , V (x(t )) > ω(t , tk , x(tk ), ϵ)}.

Remark 4. It should be mentioned that some logic switching rules have been proposed in the existing works, e.g., see Ye (1999), Ye (2003), Ye (2005), Ye (2009) and Ye (2011a). However, based on these logic switching rules, the developed adaptive switching controllers just can guarantee that the closed-loop control systems are asymptotically stable. In real applications, the convergence speed is one of the key control performance indexes. Thus, in this subsection, a novel logic switching rule is proposed. Then, it is used to design our adaptive switching controller which ensures the finite-time stability of the closed-loop system. The detailed proof of the closed-loop stability is presented in the following subsection. 3.3. Global finite-time stability analysis Theorem 1. Considering the closed-loop system consisting of the plant (1), the actual control law and the virtual control variables shown in (15) with the logic switching rule (21) and the set of the candidate controllers (20), for any initial conditions, we can show that (i) There exists a finite number k′ such that (ζkT′ H − 1T )Ξ in (18) is nonnegative for all t ≥ tk′ , and only a finite number of switchings can occur and is no more than k′ , i.e., σ (t ) ≤ k′ ; (ii) the solution x(t ) to the closed-loop system exists and is bounded on [t0 , ∞), and the solution x(t ) is unique in forward time; (iii) the transient performance satisfies V (x(t )) ≤ V (x(t0 )) + k′ ϵ for all t ≥ t0 ; (iv) the equilibrium point x = 0 is finite-time stable, and the settling time satisfies T (x0 , t0 ) ≤ t0 +

1

k′

4[V (x0 )+(k+1)ϵ] 1−µ k=0 1−µ

Proof. From the logic switching rule (21) and the definition of the switching time (22), it can be seen that ‘‘=’’ cannot hold at each switching time tk . Hence, the switching time sequence {tk }+∞ k=0 is strictly increasing. (i) We first show that there exists a finite number k′ such that Πk′ := (ζkT′ H − 1T )Ξ is nonnegative for (x0 , t ) ∈ Rn × [tk′ , +∞). Since Ξ is a nonnegative vector all the time, then we just need to find a k′ such that ζkT′ H − 1T is also a nonnegative vector. According to the analysis presented in Section 3.2, it can be seen that there ′ must exist a l′ such that ζ (l ) = ζ holds with l′ ∈ {0, 1, . . . , 2n − 1}. Note the facts that |hi (t )| ≥ h0 > 0, i = 1, . . . , n, and H (k) is strictly increasing with respect to k. From (19), taking k′ = 2n · (⌈H −1 ( h1 )⌉ + 1) + l′ , we can easily verify that ζkT′ H − 1T ≥ 0 0 holds for all t ≥ tk′ . Next, we show that indeed only a finite number of switchings can occur and is no more than k′ , i.e., σ (t ) ≤ k′ ;, i.e., σ (t ) ≤ k′ . Seeking a contradiction, suppose that switching cannot end dV (x(t )) ≤ at k′ . When σ (t ) = k′ , the inequality (18) becomes dt

− 14 V µ (x(t )). Note that according to the logic switching rule (21), V (x(tk′ )) ̸= 0. By using Lemma 2, it can be seen that for tk′ < t < tk′ + 4[V (x(tk′ ))]1−µ /(1 − µ), we have 1

V (x(t )) ≤ [(V (x(tk′ )))1−µ − 0.25(1 − µ)(t − tk′ )] 1−µ 1

< [(V (x(tk′ )) + ϵ)1−µ − 0.25(1 − µ)(t − tk′ )] 1−µ , and for t ≥ tk′ + 4[V (x(tk′ ))]1−µ /(1 − µ), V (x(t )) = 0 holds. Therefore, the switching condition in (21) can be never satisfied after tk′ , i.e., σ (t ) = k′ for t ≥ tk′ . This yields a contradiction. Thus, σ (t ) ≤ k′ holds, which implies that the Zeno phenomenon is excluded consideration here. (ii) We first show that the solution x(t ) to the system (1) exists and is bounded on [t0 , +∞). According to logic switching rule (21), we have 1

V (x(t )) ≤ [(V (x(tk )) + ϵ)1−µ − 0.25(1 − µ)(t − tk )] 1−µ (22)

.

≤ V (x(tk )) + ϵ

J. Wu et al. / Automatica 69 (2016) 298–307

on each interval [tk , tk+1 ), k = 0, 1, 2, . . . , x(t ) is uniformly bounded. Then, the closed-loop solution of the system (1) exists. The whole solution can be defined as the concatenation of solutions between successive switchings. Let its maximum interval of existence be [t0 , tf ). In addition, suppose that the maximum of σ (t ) ¯ Then, following the logic switching rule (21), for t ≥ tk¯ , we is k. have V (x(t )) = 0 or 1

V (x(t )) < [(V (x(tk¯ )) + ϵ)1−µ − 0.25(1 − µ)(t − tk¯ )] 1−µ

≤ V (x(tk¯ )) + ϵ,

(23)

which shows the boundedness of V (x(t )). Thus, x(t ) is bounded, and no finite time escape phenomenon occurs and tf = +∞. Next, we show that the solution x(t ) is unique in forward time. Substituting (15) into (1) gives rise to x˙ = F (x(t ), t ),

(24)

where h1 (t )x2 + f1 (x1 , t )  h2 (t )x3 + f2 (x1 , x2 , t ) 





F (x(t ), t ) =  

.. .

 

hn (t )u + fn (x1 , . . . , xn , t ) q

with u = −Γn,k ξn n+1 βn (x). According to the presentation in (i) above, we can see that the function F (x, t ) is discontinuous on a set W ⊂ M := {t1 , . . . , tk′ }. For a given system, k′ is a finite number, so the measure of set W is zero. In addition, as stated in Huang et al. (2005), for given systems, it can be verified that the controller u(x) satisfies |u(x, t )− u(y, t )| ≤ L0 |x − y|α , where x, y ∈ Rn , α ∈ (0, 1) and L0 is a constant. Then, by using Lemma 3 and Assumption 2, since functions fi (·, ·) are locally Lipschitz, we have

when t > tk¯ , we can have V (x(t )) = 0 or V (x(t )) < V (x(tk¯ )) + ϵ

≤ V (x0 ) + (k¯ + 1)ϵ ≤ V (x0 ) + k′ ϵ. Hence, we also conclude that V (x(t )) ≤ V (x0 ) + k′ ϵ holds for all t ≥ t0 . (iv) Based on the above arguments, we now show that the conclusion (iv) of Theorem 1 holds. Considering t ≥ tk¯ , we claim Vn (x(T (x0 , t0 ))) = 0 where T (x0 , t0 ) = tk¯ +4[V (x(tk¯ ))+ϵ]1−µ /(1− µ). Seeking a contradiction, we assume that V (x(T (x0 , t0 ))) ̸= 0, which means V (x(t )) ̸= 0 for all t ∈ [tk¯ , T (x0 , t0 )]. Then, according to (23), one has 1

V (x(t )) < [(V (x(tk¯ )) + ϵ)1−µ − 0.25(1 − µ)(t − tk¯ )] 1−µ over this interval. However, when t = T (x0 , t0 ), the right-hand side of the above inequality is zero, which yields a contradiction. Since the origin is the equivalent of the system (1), then x(t ) = 0 holds for t ≥ T (x0 , t0 ). On the other hand, it can be seen from the logic switching rule (21) that the length of interval [tk , tk+1 ) with 0 ≤ k ≤ k¯ −1 is less than 4[V (x(tk ))+ϵ]1−µ /(1−µ), and the length of [tk¯ , T (x0 , t0 )] is less than 4[V (x(tk¯ )) + ϵ]1−µ /(1 − µ). Noticing ¯ we can see that the settling V (x(tk )) ≤ V (x0 ) + kϵ with 0 ≤ k ≤ k, time satisfies T (x0 , t0 ) ≤ t0 +

k¯  4[V (x(t0 )) + (k + 1)ϵ]1−µ k=0

≤ t0 +

1−µ

k′  4[V (x(t0 )) + (k + 1)ϵ]1−µ k=0

1−µ

.

(26)

This completes the whole proof.

(x − y) · (F (x, t ) − F (y, t )) ≤ c1 (|x1 − y1 |1+α + · · · + |xn − yn |1+α ) + c2 (|x1 − y1 |2 + · · · + |xn − yn |2 ) ≤ c2 |x − y|2 + c3 (|x − y|2 )

303

4. Extension of the proposed control algorithm

1+α 2

= ρ(|x − y|2 ),

(25)

where ci , i = 1, 2, 3 are constants. That is, we can have ρ(z ) =

In this section, we show that the method proposed in the above section can be further extended to solve the finite-time stabilization problem of more general systems which are described by

1+α

c2 z + c3 z 2 for z ≥ 0. It can be verified that ρ(z) ≥ 0, as z ≥ 0, is strictly increasing, continuous and concave, and 0+ dz /ρ(z ) = ∞ holds. Based on the above observations, according to Lemma 1, we can conclude that the uniqueness in forward time of the solution x(t ) is guaranteed. (iii) We show that the conclusion (iii) of Theorem 1 holds. Following the logic switching rule (21), 1

V (x(t )) ≤ [(V (x(tk )) + ϵ)1−µ − 0.25(1 − µ)(t − tk )] 1−µ

≤ V (x(tk )) + ϵ over each interval [tk , tk+1 ). Thus, V (x(t )) ≤ V (x(tk )) + kϵ . Based on the fact k¯ ≤ k′ , only two cases are needed to be considered, namely, k¯ = k′ and k¯ < k′ . For the former, when t0 ≤ t ≤ tk′ , we have V (x(t )) ≤ V (x0 )+k′ ϵ ; when t > tk′ , we also have V (x(t )) = 0 or V (x(t )) < [(V (x(tk′ )) + ϵ)1−µ −

1−µ

4 ≤ V (x(tk′ )) + ϵ ≤ V (x0 ) + k′ ϵ.

Thus, V (x(t )) ≤ V (x0 ) + k′ ϵ holds for all t ≥ t0 . For the latter, when t0 ≤ t ≤ tk¯ , one obtains V (x(t )) ≤ V (x0 ) + k¯ ϵ ≤ V (x0 ) + k′ ϵ;

1

(t − tk′ )] 1−µ

 p x˙ 1 = h1 (t )x2 + f1 (x1 , t )   x˙ 2 = h2 (t )xp + f2 (x1 , x2 , t ) 3 ..   .  x˙ n = hn (t )u + fn (x1 , . . . , xn , t )

(27)

where p ∈ P ∗ := {ℓ ∈ R|ℓ = ϖν ≤ 1 for any positive odd integers ν and ϖ }, and all the rest of variables and functions have the same definitions as shown in system (1). Assumption 3 (Li & Qian, 2006). For i = 1, . . . , n, the following inequality holds:

|fi (¯xi , t )| ≤ a(|x1 |p + · · · + |xi |p ),

(28)

where p ∈ (0, 1) is defined as in (27) and a is a positive constant. Assumption 4 (Zhang & Ge, 2007). For i = 1, . . . , n, the functions hi (t ) satisfy h¯ i ≥ |hi (t )| ≥ h0 , where h0 > 0 is an unknown constant and h¯ i > 0 are known constants. Now based on the above assumptions, by combining the result obtained in Li and Qian (2006) with the design procedure presented in the above section, some brief derivations to design the finite-time controller are provided as follows. First, we define a set

304

J. Wu et al. / Automatica 69 (2016) 298–307

of virtual controllers x∗1 (¯x0 , k), . . . , x∗n (¯xn−1 , k) and new variables ξ1 , . . . , ξn satisfying

(ii) the transient performance satisfies V (x(t )) ≤ V (x(t0 )) + k′ ϵ for all t ≥ t0 .

 ∗ x1 = 0,   x∗2 = −Γ1,k b1 ξ1 , ..   . ∗ xn = −Γn−1,k bn−1 ξn−1 ,

Proof. Note that Ξ in (32) is also a nonnegative vector. Hence, following the same derivation as shown in the proof of Theorem 1, it is easy to show that k′ also exists and can be determined. Furthermore, the whole proof of Theorem 2 is very similar to that of Theorem 1, thus it is omitted here.

ξ1 = x1 − x∗1 ξ2 = x2 − x∗2 .. .

(29)

ξn = xn − x∗n

where for i = 1 · · · , n − 1, bi > 0 are design parameters to be 1

determined later, and Γi,k = ζi,k H p (k) where ζi,k and H (k) are defined as shown in (6). In this section, k is still assumed to be fixed, and thus Γi,k is also fixed. n 1 2 Consider Lyapunov function V (x) = i=1 2 ξi . By the change of coordinates (29) and adopting some similar derivations in Li and Qian (2006), the time derivative of V (x) can be concluded as dV (x) dt

≤ aξ11+p +

n   1 i=2

2

(|ξ1 |1+p + · · · + |ξi−1 |1+p ) + g˜i |ξi |1+p

n −1

+

 1 2

i =1

+

n −1 

|ξi |1+p + gˆi+1 |ξi+1 |1+p





∗p

hi (t )ξi xi+1 + hn (t )ξn u,

where g˜i > 0 and gˆi > 0, i = 2, . . . , n are design parameters. Based on inequality (30), we design the virtual controllers and the actual control law as ∗p

p

p p

xi+1 (¯xi , k) := −Γi,k bi ξi ,

i = 1, . . . , n − 1,

(31)

p

u(x, k) := −Γn,k bpn ξnp , 1

1

where b1 = (n + a) p , bj = (n − j + 1 + g˜j + gˆj ) p , j = 2, . . . , n. Substituting (31) into (30) yields dV (x) dt

≤ −V µ (x) − (ζkT H − 1T )Ξ , 1+p p

(32) 1+p p

where µ := 2 , Ξ := [ξ1 b1 , . . . , ξn bn ]T ∈ Rn , ζk ∈ Rn , 1 ∈ Rn and H ∈ Rn×n have the same definitions as given in (18). Next, we still employ the logic switching rule (21) with the switching time sequence (22) to tune parameter k online, where 1 +p

V (x) =

ξ and ω(t , tk , x(tk ), ϵ) = [(V (x(tk )) + ϵ)

n

1 2 i =1 2 i 2

(t − tk )] for tk ≤ t ≤ tk + ω(t , tk , x(tk ), ϵ) = 0. 1−p 2

1−p

Remark 6. From the design processes presented in Sections 3 and 4, it can be seen that there are two differences between the cases p = 1 (i.e., system (1)) and p < 1 (i.e., system (27)). The first is that the available control schemes are developed under different assumptions (see Assumptions 1–4). The other is that by means of different change of coordinates (see (6) and (29)), two kinds of switching controllers are designed.

(30)

i=1



Remark 5. Compared with the finite-time controller designed in Li and Qian (2006) where the information of the control direction is required to be known a priori, an extra tuning parameter k is added in our controller (31). As mentioned in Remark 2, this adaptively tuning parameter is used to deal with the difficultly arising from the unknown control directions of the system (27), i.e., the term −(ζkT H − 1T )Ξ in (32).

1−p 2(V (x(tk ))+ϵ) 2

1 −p

1−p 2

−

, otherwise,

Remark 7. In this paper, the considered systems (1) and (27) are allowed the control directions to be unknown. For this case, the existing schemes cannot be applied to solve the finite-time stabilization problem of the system (1), e.g., see Hong (2002), Hong et al. (2001), Hong et al. (2006), Huang et al. (2005), Li and Qian (2006), Li et al. (2010), Orlov (2005), Qian and Li (2005), Shen and Huang (2012) and Yang and Wang (2012). To overcome the main obstacle arising from the unknown control directions, a switching design idea is introduced to design a switching controller with a parameter k. By establishing a new Lyapunov-based logic switching rule to tune parameter k adaptively, an appropriate controller can be found from a set of the candidate controllers such that the closed-loop systems are finite-time stable. This is the main technique of this paper, which is different from all the existing designs in dealing with the finite-time control of nonlinear systems. 5. Simulation example In this section, consider the following second-order nonlinear system of the form



From the change of coordinates (29), it can be seen that the actual controller depends on the switching vector ζk which is defined as shown in (19), therefore, the set of the candidate controllers is given as follows

U = {uk (t ) = u(x, t , ζk )}+∞ k=0 .

(33)

Similarly, we are in a position to give the stability result of the closed-loop system by the following theorem.

x˙ 1 = h1 (t )x2 + f1 (x1 ) x˙ 2 = h2 (t )u + f2 (x)

(34)

where x = [x1 , x2 ]T , the signs of h1 (t ) and h2 (t ) are assumed to be unknown, and suppose that the system functions satisfy Assumptions 1 and 2. Following the design procedure presented in Section 3, let 5

∗5

3

ξ1 = x1 , ξ2 = x23 − x2 3 , x∗2 = −Γ1,k ξ15 β1 (x1 ), and the control law is designed as

Theorem 2. Consider the closed-loop system consisting of the plant (27), the actual control law and the virtual control variables presented in (31) with the logic switching rule (21) and the set of the candidate controllers (33). Then, for any initial conditions, we can show that (i) the equilibrium point x = 0 is finite-time stable, and the settling

k′

(35)

where for i = 1, 2, Γi,k := H (k)ζi,k with H (k) being strictly increasing with respect to k and ζi,k ∈ {1, −1} being dependent on the tuning parameter k, and βi (¯xi ) ≥ 0 will be selected below. Consider the overall Lyapunov function as

2

2[V (x )+(k+1)ϵ] 1−p

0 time satisfies T (x0 , t0 ) ≤ t0 + k=0 1−p has the same definition as given in Theorem 1;

1

u = −Γ2,k ξ25 β2 (x),

, where k′

V (x) =

1 2

ξ12 +



x2 x∗ 2



5

∗5

s 3 − x2 3

 75

ds,

(36)

J. Wu et al. / Automatica 69 (2016) 298–307

305

whose time derivative satisfies dV (x) dt

1

4

≤ − (V (x)) 5 − (ζkT H − 1T )Ξ ,

(37)

4

where ζk = H (k)[ζ1,k , ζ2,k ]T ∈ R2 , 1 = [1, 1]T ∈ R2 , H = 8

8

diag{h1 (t ), h2 (t )} ∈ R2×2 , and Ξ = [ξ15 β1 (x1 ), ξ25 β2 (x)]T ∈ R2 , β1 (x1 ) = 2 + ρ˜ 1 (x1 ), β2 (x) = 1 + ρˆ 2 (x) + ρ˜ 2 (x) + ρ¯ 2 (x) with 2

5

7

8

1

ρ˜ 1 (x1 ) = γ1 (x1 )|ξ1 | 5 , ρˆ 2 (x) = 83 ·( 15 ) 3 ·[2 5 h¯ 1 ] 3 , ρ˜ 2 (x) = 87 ·( 34 ) 7 · 8 4 5

2 5

8 7

2 5

8 7

[(|ξ1 | γ2 (x)) + (H (k)|ξ1 | β1 (x1 )γ2 (x)) ] + |ξ2 | γ2 (x), ρ¯ 2 (x) =  ∗ 53  8 2 8 8  ∂ x2  5 5 10 85 9 53 · ( · · ( ) )  ∂ x1  [(H (k)h¯ 1 β1 (x1 )) 5 + (γ1 (x1 )ξ15 ) 5 ] + 8 4 7  ∗ 53  14  ∂ x2  ¯  h1 . ∂ x1

5

As analyzed in Section 3.2, for a second-order system considered in this example, the signs of the elements of the switching vector ζk exist for 22 cases, without loss of generality, then we assume

ζ (0) := [−1, +1]T , ζ (2) := [+1, +1]T ,

Fig. 1. The system states x1 and x2 .

ζ (1) := [+1, −1]T , ζ (3) := [−1, −1]T .

Based on the above fact, the set of the candidate controllers is given as follows

U = {uk (t ) = u(x, t , ζk )}+∞ k=0 , where u(x, t , ζk ) is designed as (35) and the vector ζk is switched by

 H (k)ζ (0) ,    H (k)ζ (1) , ζk (t ) = H (k)ζ (2) ,    H (k)ζ (3) ,

if k if k if k if k

=τ =τ =τ =τ

· 22 · 22 + 1 · 22 + 2 · 22 + 3

(38)

Fig. 2. The virtual control variable x∗2 .

where τ = ⌈ 2k2 ⌉ with k = 0, 1, 2, . . . . The logic switching rule (21) is used to tune the parameter k online, where V (x) is given by (36) and ω(t , tk , x(tk ), ϵ) = 1

[(V (x(tk )) + ϵ) 5 − 1 5

1 20

(t − tk )]5 for tk ≤ t ≤ tk + 20(V (x(tk )) +

ϵ) , and otherwise ω(t , tk , x(tk ), ϵ) = 0 with ϵ > 0 a design parameter. Based on Theorem 1, it can be concluded that there exists a k′ such that (ζkT′ H − 1T )Ξ ≥ 0 holds, and the system state x converges to zero in finite time. For the simulation, a practical system—Two-Stage Chemical Reactor with Delay-Free Recycle Streams (Deng & Krstić, 1997) is considered in the following. The functions in the mass balance 1−R equations are h1 (t ) = V 2 , f1 (x1 ) = −( θ1 + k1 )x1 , h2 (t ) =

, f1 (x) =

1

1

− ( θ12 + k2 )x2 , where for i = 1, 2, xi are the compositions, Ri are the recycle flow rates, θi are the reactor F2 V2

R1 2 x V2 1

residence times, ki are the reaction constants, Vi are the reactor volumes, and F2 is the feed rate. In the simulation, let θ1 = θ2 = 1, k1 = k2 = 1, R1 = R2 = 0.55, V1 = V2 = 0.3, F2 = 0.3. It is easy to verify that they satisfy Assumptions 1 and 2, where the known functions are chosen as γ1 (x1 ) = h¯ 1 = 2.5 and γ2 (x) = x21 + 2, h¯ 2 = 2. We take H (k) = 0.37(k + 1) which is strictly increasing with respect to k. The design parameter is selected as ϵ = 0.01, and the initial condition is set to be x0 = [x1 (0), x2 (0)]T = [1, −1.5]T . Then, the settling-time function satisfies 1

T (x0 , 0) ≤

4 2  4[V (x0 ) + (k + 1)ϵ] 1− 5

k=0

1−

4 5

≈ 31.717.

The simulation results are shown in Figs. 1–6, from which it can be seen that the system states indeed converge to zero before 31.717 s (see Fig. 1). This accords with Theorem 1 and sufficiently verifies the effectiveness of the proposed control scheme.

Fig. 3. The control input u.

6. Conclusions In this paper, the problem of global finite-time stabilization was addressed for a class of uncertain nonlinear systems with multiple unknown control directions. To overcome the main obstacle arising from unknown control directions, a novel Lyapunov-based logic switching rule is developed. Then, the desired state-feedback controllers with an adaptively tuning parameter have been designed. It has been proven that the closed-loop control systems are GFTS under the obtained adaptive switching controllers. In the future, there are still some interesting problems on this topic to further study, such as how to extend the proposed

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J. Wu et al. / Automatica 69 (2016) 298–307

Excellent Talents in University (NCET-10-0665), the Fundamental Research Funds for the Central Universities (K5051370014), Natural Science Foundation of Anhui Province (1608085QF131), and University Team Program for Innovative Research Platform of Intelligent Perception and Computing of Anhui Province. References

Fig. 4. The tuning parameter k.

Fig. 5. The signs of the elements of switching vector ζk .

Fig. 6. The switching condition.

control scheme to the uncertain system (27) with an odd-type order p > 1, how to solve the problem of global finite-time control for nonlinearly parameterized systems with multiple unknown control directions, how to design the global finite-time control laws for feedforward nonlinear systems with multiple unknown control directions. Acknowledgments This work is supported by National Natural Science Foundation of China (61174213, 61203074), the Program for New Century

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307 Jian Wu received the Ph.D. degree in applied mathematics from Xidian University, Xi’an, China, in 2015. He is currently a Lecturer with the School of Computer and Information, Anqing Normal University. His current research interests include intelligent control, adaptive control and adaptive switching control.

Weisheng Chen received his B.Sc. degree from the Department of Mathematics, Qufu Normal University, Qufu, China, in 2000, his M.Sc. and Ph.D. degrees from the Department of Applied Mathematics, Xidian University, Xi’an, China, in 2004 and 2007, respectively. From 2008 to 2009, he was a visiting scholar in the School of Automation at Southeast University, Nanjing, China. From 2013–2014, he was a visiting scholar in the Department of Electrical and Computer Engineering, University of California, Riverside, USA. He is currently a professor in the School of Aerospace Science and Technology, Xidian University, Xi’an, China. His current research interests include multi-agent systems, nonlinear systems, distributed strategy, adaptive and learning control. Jing Li is an Associate Professor in Xidian University, China. From Sep. 2009 to Jul. 2010, she was a Visiting Scholar with School of Control, Shandong University, China. From Oct. 2011 to Oct. 2012, she was an Academic Visitor with Plymouth University in Robotics and with the Imperial College London in human–robot interaction respectively. Her current research interests include adaptive neural network control, stochastic control, robotics and human–robot interaction.