Systems & Control Letters 62 (2013) 930–936

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Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Controller design for nonlinear affine systems by controlLyapunov functions

Shihong Ding a,b, Wei Xing Zheng b,∗

a School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, Chinab School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith NSW 2751, Australia

a r t i c l e i n f o

Article history:Received 6 September 2012Received in revised form2 April 2013Accepted 4 July 2013Available online 15 August 2013

Keywords:Affine systemsControl Lyapunov functionFinite-time stabilizationHomogeneous property

a b s t r a c t

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicitfeedback controller is developed for a nonlinear multiple-input affine system by assuming that thereexists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditionsfor the continuity of the derived controller are developed. And then the developed control designmethodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that undersome homogeneous assumptions on control Lyapunov functions and the interconnection term, thecascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilizationfor nonlinear affine systems are also obtained.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

A system is called a homogeneous system if the vector field pos-sesses homogeneity, which is the property whereby objects suchas functions and vector fields scale in a consistent fashion with re-spect to a scaling operation on the underlying space [1]. Homo-geneous systems often have some nice and interesting properties.For instance, in the case of time-invariant homogeneous systems,the local asymptotic stability implies the global asymptotic sta-bility. Another important property is that a nonlinear system islocally asymptotically stable if the local stability of its homoge-neous approximation can be established [2]. In addition, it is alsoshown in [3] that an asymptotically stable homogeneous systemadmits a homogeneous Lyapunov function. These important prop-erties make the analysis and synthesis of homogeneous systemsmuch easier than that of non-homogeneous systems.

Recently, homogeneous stabilization has attracted much atten-tion [4–11]. The homogeneous stabilization aims to find a con-troller such that the resulting closed-loop system is homogeneousand asymptotically stable. In [6], the explicit feedback laws are

This work was supported in part by a research grant from the AustralianResearch Council, the NSF of China (No. 61203014 and No. 61203054), the NSF ofJiangsu Province (No. BK2012283), the Priority Academic Program Development ofJiangsu Higher Education Institutions and the Specialized Research Fund for theDoctoral Program of Higher Education of China (No. 20113227120002).∗ Corresponding author. Fax: +61 2 4736 0867.

E-mail addresses: dsh@mail.ujs.edu.cn (S. Ding), w.zheng@uws.edu.au(W.X. Zheng).

0167-6911/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.sysconle.2013.07.001

developed for a class of single-input systems, and sufficient condi-tions for the existence of homogeneous controllers are proposed.In [5], it is discussed how to obtain a homogeneous control systemby redesigning the feedback controller. Later, the homogeneousstabilization problem for multiple-input systems is investigatedin [7,9]. Moreover, the existence of homogeneous Lyapunov func-tions for homogeneous systems is also considered in [12,3]. It canbe clearly seen that the results in [4,5,12,6,7,9–11,3] are all aboutcontinuous-time control systems. In [13,14], the homogeneousfeedback stabilization problem is further examined for discrete-time control systems.

It can be observed that most of the aforementioned results areconcerned with how to construct special homogeneous feedbacklaws for stabilization of nonlinear systems with some typicalstructures and then to find a Lyapunov function for testing thestability of the resulting closed-loop system. Different from theseresults, the homogeneous feedback control design by using controlLyapunov functions is recently proposed in [7,9], where theexistence of a control Lyapunov function is first assumed and then astabilizing controller is constructed by using the control Lyapunovfunction.

The control Lyapunov function method introduced in [15]shows a way to consider the choice of the Lyapunov function andthe design of control simultaneously. To test the stabilizabilityof a system with a control input, the existence of a control Lya-punov function is necessary and sufficient. In fact, it was pointedout in [15] that for a class of nonlinear control systems, the sta-bilizability is equivalent to the existence of a control Lyapunovfunction. Meanwhile, the control Lyapunov function method is an

S. Ding, W.X. Zheng / Systems & Control Letters 62 (2013) 930–936 931

effective tool to analyze the asymptotic stability properties of equi-librium points or input to state stability property [16]. In addition,the control Lyapunov function has played an important role in thedesign of inverse optimal stabilizing control laws, which has at-tracted much attention in recent years.

In this paper, we will consider the controller design for non-linear affine systems by control Lyapunov functions. The main re-sults of this paper are motivated by the previous works in [7,9].First, we extend the results in [7,9] to design a controller for anonlinear multiple-input affine system under the assumption onthe existence of a control Lyapunov function. Then an introduc-tion of some additional homogeneous assumptions on the controlLyapunov function and the nonlinear affine system yields that theobtained controller is not only continuous at the origin but alsohomogeneous, thus achieving the homogeneous stabilization. Fur-thermore, the designed controller is also utilized to stabilize a classof cascaded systems by imposing some sufficient conditions on thedriven subsystem. Finally, as a by-product of the newly obtainedstabilization results, the finite-time stabilization problem for non-linear affine systems is discussed.

2. Preliminaries

In this section, we present several useful definitions and alemma,whichwill play an important role in the subsequent proofs.

Throughout the paper, denote

Rn+

= x : x = (x1, . . . , xn)T ∈ Rn and xi > 0.

A mapping

∆rεx = (εr1x1, . . . , εrnxn)T , ∀ε > 0, ∀x ∈ Rn

\ 0

is said to be a dilation on Rn\ 0, where r = (r1, . . . , rn)T and

ri > 0, i = 1, . . . , n.We first introduce the definition of homogeneity, which

originates from [17,3].

Definition 1. Consider the following affine system

x = f (x)+ g(x)u, x ∈ Rn, u ∈ Rm (1)

where x ∈ Rn is the system state, u ∈ Rm is the input, andf (x) : Rn

→ Rn and g(x) : Rn→ Rn×m are continuous functions

with fi being the ith element of f and g i being the ith row of g .System (1) is said to be homogeneous of degree d with respect to(rT , sT )T if for all ε > 0,

fi(∆rεx)+ g i(∆r

εx)∆sεu = εd+ri(fi(x)+ g i(x)u),

i = 1, . . . , n

where r ∈ Rn+, s ∈ Rm

+, and d ≥ −minr1, . . . , rn. Let V (x) :

Rn→ R be a continuous function. V (x) is said to be homogeneous

of degree σ > 0 with respect to r = (r1, . . . , rn)T if for all ε > 0,

V (∆rεx) = εσV (x), ∀x ∈ Rn

\ 0.

Then the definition of the control Lyapunov function given by[7,18] can be stated as follows.

Definition 2. A continuously differentiable, positive-definite andradially unbounded Lyapunov function V (x) : Rn

→ R is called acontrol Lyapunov function of system (1) if

infu∈Rm

Lf V (x)+ LgV (x) · u < 0, ∀x = 0,

where Lf V and LgV are Lie derivatives of V (x) along f (x) and g(x),respectively.

The following definition shows when a control Lyapunov functionsatisfies a small control property.

Definition 3 ([18]). A control Lyapunov function for system (1) issaid to satisfy the small control property if for any ε > 0, thereexists δ > 0 such that when ∥x∥ < δ and x = 0, there exists∥u∥ < ε such that Lf V (x)+ LgV (x) · u < 0.

Finally, an important result about when a system is finite-timestable is given below.

Lemma 1 ([19]). Consider system x = f (x), x ∈ Rn with f (0) = 0and f (x) : Rn

→ Rn being a continuous function. If there exist acontinuously differentiable, positive definite and radially unboundedLyapunov function V (x) and real numbers c > 0, 0 < α < 1 suchthat V (x) ≤ −cV α(x), then the origin is a globally finite-time stableequilibrium point.

3. Main results

In this section, the stabilization problems for nonlinear affinesystems are investigated. The main results include three parts.First, a control design methodology for nonlinear affine system(1) is proposed by using the control Lyapunov function suchthat system (1) can be globally stabilized. Second, the designedcontroller is applied to stabilize a class of nonlinear cascadedaffine systems, and it is shown that under the same controller, thecascaded affine system is globally stabilizable. Finally, we discussthe finite-time stabilization problem.

3.1. Stabilization of nonlinear affine system (1)

In this subsection, we will present a design methodology toconstruct a stabilizing law for system (1) under the followingassumption.

Assumption 1. There exists a control Lyapunov function V1(x) forsystem (1).

Remark 1. Usually even when the exact information of f (x) andg(x) in (1) is known, it is still not easy to construct an effectivestabilizing controller. The reason is that it is difficult to find aproper Lyapunov function to test the stability of the closed-loopsystem. However, from the theoretical point of view, it is relativelyeasy to construct a control Lyapunov function for system (1) beforecontrol design, since the control Lyapunov functionmainly focuseson the local property of the system. Meanwhile, Assumption 1is a commonly used assumption to guarantee that there existsa controller to stabilize system (1). Under Assumption 1, thecontroller can be constructed by using the control Lyapunovfunction. Similar assumptions can also be found in [7,9], etc.

Now we give the first result of the paper, which can be consideredas an extended version of Moulay’s feedback control [7].

Theorem 1. Under Assumption 1, the following controller

ui =

−k1 · βi(x)

α(x)+ (|α(x)|p + k2 · β(x)q)1/p

β(x),

for β(x) = 00, for β(x) = 0

(2)

globally stabilizes system (1), where α(x) = Lf V1(x), β(x) =m

i=1|Lg iV1(x)|1+ai , βi(x) = sign

Lg iV1(x)

|Lg iV1(x)|ai , ai > 0, i =

1, . . . ,m, and k1 ≥ 1, k2 > 0, q > 1, p ≥ 1 are real numbers.

Proof. We first prove that controller (2) is continuous on Rn\ 0,

whose proof is inspired by [7]. And then, we will prove the globalasymptotic stability of the closed-loop system (1)–(2).

932 S. Ding, W.X. Zheng / Systems & Control Letters 62 (2013) 930–936

Let

Q1 = (ν, t) ∈ R2: ν ∈ R and t > 0

∪(ν, t) ∈ R2: ν < 0 and t = 0.

Define the following equation on Q1 [7]:

ψ(ν, t) =

ν + (|ν|p + k2|t|q)1/p

t, for ν ∈ R, t > 0

0, for ν < 0, t = 0.

Then we will show that ψ(ν, t) is continuous on Q1.Noting that (|ν|p + k2|t|q)1/p ≥ |ν| for p ≥ 1, it is not difficult

to verify that

ν + (|ν|p + k2|t|q)1/p

t≥ν + |ν|

t≥ 0,

∀(ν, t) ∈ (ν, t) ∈ R2: ν ∈ R and t > 0 .

Thus, we have ψ(ν, t) ≥ 0 for all (ν, t) ∈ Q1. In addition, since1 +

k2|t|q

|ν|p ≥ 1 and 0 < 1/p ≤ 1, the following inequality alwaysholds:1 +

k2|t|q

|ν|p

1/p

≤ 1 +k2|t|q

|ν|p.

From this, we get

0 ≤ limν<0,t→0

ψ(ν, t) = limν<0,t→0

ν + |ν|1 +

k2|t|q

|ν|p

1/pt

≤ limν<0,t→0

ν + |ν| +k2|t|q

|ν|p−1

t= lim

ν<0,t→0

k2|t|q

t|ν|p−1= 0

which immediately leads to

limν<0,t→0

ψ(ν, t) = 0. (3)

So it can be concluded from (3) that ψ(ν, t) is continuous on Q1.Note from Assumption 1 that V1(x) is a control Lyapunov

function of system (1). According toDefinition 2, if LgV1(x) = 0 andx = 0, then we have Lf V1(x) < 0. It also implies that β(x) = 0 andx = 0 yields Lf V1(x) < 0. Consequently, it can be easily checkedthat for all x ∈ Rn

\ 0, (α(x), β(x)) ∈ Q1, which shows thatcontroller (2) is continuous for all x ∈ Rn

\ 0 by the fact thatψ(ν, t) is continuous on Q1.

Then we will show that the closed-loop system (1)–(2) isglobally asymptotically stable. The proof can be divided into twocases.

First of all, take the candidate Lyapunov function to be V1(x).We will show

V1(x) < 0, ∀x ∈ Rn\ 0 ∩ x : LgV1(x) = 0. (4)

Since V1(x) is a control Lyapunov function for system (1), from thedefinition of the control Lyapunov function, we deduce that (4)holds automatically.

Next we will show

V1(x) < 0, ∀x ∈ Rn\ 0 ∩ x : LgV1(x) = 0. (5)

Taking the derivative of V1(x) along the closed-loop system (1)–(2)yields

V1(x) = Lf V1(x)+ LgV1(x) · u

= α(x)− k1mi=1

Lg iV1(x)

·βi(x)α(x)+ (|α(x)|p + k2 · β(x)q)1/p

β(x)

= α(x)− k1α(x)+ (|α(x)|p + k2 · β(x)q)1/p

β(x)

×

mi=1

Lg iV1(x) · βi(x)

= α(x)− k1α(x)+ (|α(x)|p + k2 · β(x)q)1/p

β(x)β(x)

= −(k1 − 1)α(x)− k1(|α(x)|p + k2 · β(x)q)1/p. (6)

With k1 ≥ 1 in mind, it follows that (5) also holds.Finally, by combining (4) and (5) together, we have

V1(x) < 0, ∀x ∈ Rn\ 0,

thus completing the proof.

Remark 2. It can be shown that Theorem 1 encompasses theresults in [7]. If we let gi(x) be the ith column of g , i.e., g(x) =

(g1(x), . . . , gm(x)), then system (1) can be rewritten as

x = f (x)+

mi=1

gi(x)ui, x ∈ Rn, u ∈ Rm,

which is just the system considered in [7]. In addition, by takingai = 1 (i = 1, . . . ,m) and k1 = k2 = 1, controller (2) will reduceto the controller (4) proposed in [7].

According to Theorem1 in [18], if the control Lyapunov functionV1(x) proposed in Assumption 1 satisfies the small control property,then controller (2) is continuous at the origin. However, the smallcontrol property is usually not easy to verify. Therefore, we give asufficient condition in the following to guarantee the continuity ofcontroller (2).

Assumption 2. System (1) is homogeneous of degree d withrespect to (rT , sT )T and V1(x) is homogeneous of degree σ1 withrespect to r.

Then we have the following result.

Theorem 2. Under Assumptions 1–2, controller (2) with p = q > 1and ai =

siσ1+d−si

is continuous at the origin and homogeneouslystabilizes system (1).

Proof. First of all, we will prove that under Assumption 2,controller (2) with p = q and ai =

siσ1+d−si

is continuous at theorigin.

Note from Assumption 2 that V1(x) is homogeneous of degreeσ1 with respect to r = (r1, . . . , rn)T . Let Q2 = x : V1(x) = 1 andε = V 1/σ1

1 (x). Since V1(x) is a positive definite function, we haveε = V 1/σ1

1 (x) > 0 for any x = 0. Then, ∀x = 0,

V1(ε−r1x1, . . . , ε−rnxn)

= V1((V−1/σ11 (x))r1x1, . . . , (V

−1/σ11 (x))rnxn)

= (V−1/σ11 (x))σ1V1(x) = 1. (7)

Denote x1 = V−r1/σ11 (x)x1, . . . , xn = V−rn/σ1

1 (x)xn. It follows from(7) that V1(x1, . . . , xn) = 1, which implies x = (x1, . . . , xn)T ∈ Q2.This alsomeans that for all x ∈ Rn

\0, there exist ε = V1(x)1/σ1 =

0 and x ∈ Q2 such that

x = (x1, . . . , xn)T = (εr1 x1, . . . , εrn xn)T . (8)

It is assumed that system (1) is homogeneous of degree d withrespect to (r1, . . . , rn, s1, . . . , sm)T > 0. So it can be easily shownthat f (x) and g i(x) are homogeneous of degrees d and d − si with

S. Ding, W.X. Zheng / Systems & Control Letters 62 (2013) 930–936 933

respect to (r1, . . . , rn)T , respectively. Noting that p = q and ai =si

σ1+d−si, it is straightforward to verify by (2) that

ui(∆rεx) = εsiui(x). (9)

Based on this, according to (8), for any x ∈ Rn\ 0, there exist

x ∈ Q2 and ε > 0 such that

ui(x) = ui(∆rεx) = εsiui(x).

Thus, we have

x = (εr1 x1, . . . , εrn xn) → 0H⇒ ε → 0H⇒ ui(x) = ui(∆

rεx) = εsiui(x) → 0,

which implies that ui(x) is continuous at the origin.Moreover, combining the homogeneous assumptions of system

(1) and V1(x), it is clear that the closed-loop system (1)–(2) ishomogeneous of degree d. By Theorem 1, controller (2) stabilizessystem (1), and thus homogeneously stabilizes system (1).

Remark 3. It should be pointed out that system (1) is alsoconsidered in [9] under the assumption that it is homogeneous.Assume that system (1) is homogeneous of degree d with respectto (rT , sT )T and V1(x) is homogeneous of degree σ1 with respect tor . It was proved in Theorem 2 of [9] that when ai =

siσ1+d−si

(i =

1, . . . ,m), k1 =12 , and p = q = 1, controller (2) is continuous

on Rn and globally stabilizes system (1). This implies that evenq = 1, by appropriately choosing ai, ki and q, controller (2) maystill continuously stabilize system (1).

Remark 4. As a matter of fact, the homogeneity assumptions ofsystem (1) plus the conditions of p = q and ai =

siσ1+d−si

implythat the homogeneous Lyapunov function V1(x) satisfies the smallcontrol property. According to Proposition 2 in [9], to prove thesmall control property, it is only required to prove that for anyε > 0, there is a δ > 0 such that

∥x∥ ≤ δ and LgV1(x) = 0

H⇒ P(x) =Lf V1(x)

max1≤i≤m

|Lg iV1(x)|< ε.

Actually, it is easy to show that

P(x) = P(∆rε x) = min

1≤i≤m

Lf V1(x)εsi

|Lg iV1(x)|

, x ∈ Q2.

Thus,

x = (εr1 x1, . . . , εrn xn) → 0 ⇒ ε → 0 ⇒ P(x) → 0.

It implies that V1(x) satisfies the small control property.

Example 1. Let us consider the following nonlinear affine system

x1 = −x32 + x2u1, x2 = x31 − x32 − x1u1 + x22u2. (10)

By choosing V1(x) =14x

41+

14x

42, we can verify that V1(x) is homoge-

neous of degree σ1 = 4with respect to r = (1, 1)T . In addition, wecan also verify that system (10) is homogeneous of degree d = 2with respect to (rT , sT )T = (1, 1, 2, 1)T . Let f (x) = [−x32, x

31 − x32]

T

and g(x) =

x2 0

−x1 x22

, and take a1 =

12 , a2 =

15 . According to

Theorems 1–2, the following continuous controller globally homo-geneously stabilizes system (10):

ui = −k1 · βi(x)α(x)+ (|α(x)|p + k2 · β(x)p)1/p

β(x), i = 1, 2 (11)

with p > 1, k1 ≥ 1, k2 > 0, α(x) = −x62, β1(x) = sign(x31x2 −

x1x32)|x31x2 − x1x32|

1/2, β2(x) = x2, β(x) = |x31x2 − x1x32|3/2

+ x62.

3.2. Stabilization of cascaded systems

Systems with cascaded dynamical structures appear in manycontrol applications. Cascaded structures often reflect configura-tions of major system components, especially when each of thesecomponents constitutes a dynamical subsystem [20]. Usually, thestability of individual subsystems is not sufficient to identifythe stability property of an entire cascaded system. As describedin [20], it is also necessary to characterize the nature of the inter-connection of the subsystems. In recent years, global stabilizationof nonlinear cascaded systems has received extensive attentionsince cascaded design can be utilized to reduce the complexity ofcontroller design and stability analysis (see, e.g., [21–24]).

In this subsection, the control designmethodology proposed forsystem (1) will be extended to stabilize the following nonlinearaffine cascaded system

z = f0(z)+ h(z, x) (12a)x = f (x)+ g(x)u (12b)

where (zT , xT )T ∈ Rk+n and u ∈ Rm are the system state andcontrol input, respectively, h(z, x) : Rk+n

→ Rk is a continuousfunction, and the definitions of f (x) and g(x) are the same as thosefor system (1). We will design a controller to globally stabilizesystem (12) under Assumption 1 and the following two additionalconditions.

Assumption 3. For system

z = f0(z) (13)

there exist two positive-definite, radially unbounded functionsV2(z) and ω2(z) such that V2(z)|(13) ≤ −ω2(z), with V2(z) beinghomogeneous of degree σ2 > 0 and ω2(z) being homogeneous ofdegree σ ′

2 > 0 with respect to l = (l1, . . . , lk)T , respectively.

Assumption 4. Let (zT , xT )T ∈ Rk+n. For h(z, x) = (h1(z, x), . . . ,hk(z, x))T , there exist continuous functions Mi(z, x) ≥ 0, i =

1, . . . , k with Mi(z, 0) = 0 and constant τ satisfying −

minl1, . . . , lk ≤ τ < σ ′

2 − σ2 such that for any z, x,

|hi(εl1z1, . . . , εlkzk, x)| ≤ Mi(z, x)εli+τ . (14)

In order to stabilize cascaded system (12), except for Assumption 1,two additional assumptions, i.e., Assumptions 3–4, are alsorequired. It can be seen that Assumption 3 guarantees that the zerodynamics of the driven subsystem is globally asymptotically stable,while Assumption 4 means that the interconnection term h(z, x)satisfies a homogeneous growth condition which ensures that theinterconnection term cannot grow too fast. Moreover, it should benoted that comparedwith the common assumptions for the drivensubsystem, the matching conditions, which are usually not easy toverify, are not required here. Instead, we only need to verify thestability of subsystem (13) and the homogeneous growth condition(14) in this paper, thus the sufficient conditions given here are notdifficult to verify to some extent.

Now, by utilizing Theorem 1, we are ready to give the followingresult.

Theorem 3. Under Assumptions 1, 3 and 4, cascaded system (12) canbe globally asymptotically stabilized by controller (2).

Proof. From Theorem 1, it is clear that cascaded system (12b) canbe globally stabilized by controller (2). To prove the global asymp-totic stability of the closed-loop cascaded system (12) and (2),according to [20], we only need to show the global boundedness.

934 S. Ding, W.X. Zheng / Systems & Control Letters 62 (2013) 930–936

Define Ω1 = z : V2(z) = 1. Note from Assumption 3 thatV2(z) is homogeneous of degree σ2. Then letting ε = V 1/σ2

2 (z),∀z = 0 leads to

V2(ε−l1z1, . . . , ε−lkzk)

= V2((V−1/σ22 (z))l1z1, . . . , (V

−1/σ22 (z))lkzk)

= (V−1/σ22 (z))σ2V2(z) = 1. (15)

Denote z1 = V−l1/σ22 (z)z1, . . . , zk = V−lk/σ2

2 (z)zk. It follows readilyfrom (15) that V2(z1, . . . , zk) = 1, which implies z = (z1, . . . , zk)T

∈ Ω1. Consequently, for any z ∈ Rk\ 0, there exist ε = V 1/σ2

2 (z)> 0 and z = (z1, . . . , zk)T ∈ Ω1 such that

z = (εl1 z1, . . . , εlk zk)T . (16)

In addition, note that

V2(z)|(12a) = −ω2(z)+ hT (z, x)∂V2(z)∂z

. (17)

By Eq. (4) of [3], we obtain that if V2(z) is a homogeneous Lyapunovfunction of degree σ2 with respect to (l1, . . . , lk)T , then

∂V2(z)∂zi

is ho-mogeneous of degree σ2 − li with the same weight. With this inmind, by letting φ(z) =

∂V2(z)∂z , φi(z) =

∂V2(z)∂zi

, i = 1, . . . , k, ac-cording to Assumption 3, we can show thathT (z, x)

∂V2(z)∂z

=hT (∆l

ε z, x)φ(∆lε z)

=

ki=1

hi(∆lε z, x)φi(∆

lε z)

≤

ki=1

Mi(z, x)εli+τ εσ2−liφi(z)

= εσ2+τk

i=1

Mi(z, x)∂V2(z)∂ zi

(18)

with z ∈ Ω1.

Recall that V2(z) andω2(z) are positive definite Lyapunov func-tions. Thus, there exist K∞ functions ψ1(∥z∥) and ψ2(∥z∥) suchthat V2(z) ≤ ψ1(∥z∥) and ω2(z) ≥ ψ2(∥z∥). This implies that forany z ∈ Ω1, we get

1 = V2(z) ≤ ψ1(∥z∥) H⇒ ∥z∥ ≥ ψ−11 (1).

This further means

ω2(z) ≥ ψ2(∥z∥) ≥ ψ2(ψ−11 (1)). (19)

Combining (16) and (18) together, by the definition of φ(·) andφi(·), we obtain from (17) that

V1(z)|(12a) = −ω2(z)+ hT (z, x)∂V2(z)∂z

= −ω2(∆lε z)+ hT (∆l

ε z, x)φ(∆lε z)

≤ −εσ′2ω2(z)+

ki=1

εli+τMi(z, x) · εσ2−liφi(z)

= −εσ′2ω2(z)+ εσ2+τ

ki=1

Mi(z, x)∂V2(z)∂ zi

. (20)

Note from Theorem 1 that under controller (2), the driving subsys-tem (12b) of cascaded system (12) is globally asymptotically stable.

This means that xwill converge to zero as the time goes to infinity.Then there exists a positive real number γ1 > 0 such that

ki=1

Mi(z, x)∂V2(z)∂ zi

≤ γ1

due to the fact that z is also bounded. This, together with (19) and(20), yields

V1(z)|(12a) ≤ −εσ′2ψ2(ψ

−11 (1))+ γ1ε

σ2+τ (21)

with −minl1, . . . , lk ≤ τ < σ ′

2 − σ2 being guaranteed by As-sumption 4.

Let

ε0 = max

1,

γ1

ψ2(ψ−11 (1))

1/(σ ′2−σ2−τ)

Ω2 = z : z = (εl1 z1, . . . , εlk zk), ε ≥ ε0.

From (21), it is clear that

V1(z)|(12a) ≤ 0, ∀z ∈ Ω2,

which implies that the states of cascaded system (12) are bounded.This completes the proof.

Remark 5. It should be noted that the stabilization of cascadedsystem by using the control Lyapunov function and homogeneousproperty was already reported in [25]. However, in comparison ofthis paper with [25], there are three main differences. The firstone is that the constructions of control laws by using controlLyapunov functions are different. The second one is that thecascaded system in this paper is not required to be homogeneousalthough the subsystems (13) and (12b) are homogeneous; incontrast, the homogeneous assumption for the cascaded systemin [25] is a precondition. The third one is that this paper considersthe global stabilization of general cascaded systems, while [25]only considers the stabilization of second-order and third-ordercascaded systems (see Theorems 3 and 4 in [25]).

Remark 6. For cascaded systems, there exists a natural phe-nomenon, which is called peaking phenomenon [26]. According to(20), when τ approaches σ ′

2−σ2, the growth rate ofω2(z) is similarto that of term hT (z, x) ∂V2(z)

∂z . Under this case, the intensity of peak-ing phenomenon is enhanced. On the contrary, when τ approaches−minl1, . . . , lk, the growth rate of ω2(z) is faster than that ofterm hT (z, x) ∂V2(z)

∂z . Then, the intensity of peaking phenomenon isreduced. This will also be verified by Fig. 1 in Example 2 in the se-quel.

In Theorem 3, the parameter τ plays a very important role inthe stability analysis. In the following, we will illustrate that anyrelaxation on τ will lead to the case that the closed-loop system(12) and (2) is no longer globally stable.

Example 2. Consider the following system

z1 = −z31 + zc1 · x2 (22a)

x1 = −x32 + x2u1, x2 = x31 − x32 − x1u1 + x22u2 (22b)

where c > 0. Take V1(x) =12 (x

21 + x22). From Example 1, we

obtain that the driving subsystem (22b) can be globally stabilizedby controller (11). Thismeans that to stabilize the cascaded system(22), we only need to verify Assumptions 3–4.

Taking V2(z1) =12 z

21 , then the derivative of V2(z1) along sys-

tem z1 = −z31 is given by V2(z1) = −ω2(z1) with ω2(z1) = z41 .Thus, V2(z1) and ω2(z1) are homogeneous of degrees σ2 = 2 and

S. Ding, W.X. Zheng / Systems & Control Letters 62 (2013) 930–936 935

Fig. 1. Response curves of z1 under controller (11) with k1 = 1, k2 = 5, p = q = 2and initial states be (5, 1, 5).

σ ′

2 = 4 with respect to l1 = 1, respectively, which implies thatAssumption 3 holds. Furthermore, for all ε > 0, we have

|h(εl1z1, x2)| = εl1c |zc1 · x2|.

Since 0 ≤ τ < σ ′

2 − σ2 = 2, according to Assumption 4, we needto have

|h(εl1z1, x2)| = εl1c |zc1 · x2| ≤ εl1+τ |zc1x2|.

Thismeans that onlywhen the parameter c satisfies l1c < l1+σ ′

2−

σ2 = 3l1, i.e., 0 ≤ c < 3, Assumption 4 can be satisfied. Actually, ifc = 3, then for some initial states, the driven subsystem (22a) willdiverge to infinity in finite time. The similar analysis can also befound in Example 4.29 of [20]. Under controller (11), the simula-tion result for the state z1 under different c is shown in Fig. 1. FromFig. 1, it can be clearly seen that when c approaches 3, the peakingphenomenon is strengthened.

3.3. Finite-time stabilization

Finite-time control systems usually offer good convergenceand disturbance rejection properties [19,27], and thus finite-timecontrol has drawn a considerable interest in recent years (see,e.g., [28,27,29–35]). There are mainly three types of methods forfinite-time control. The first type is the finite-time homogeneousmethod, which tells us that if a stable system possesses a negativedegree of homogeneity, then the system is finite-time stable [1].The second type is the Lyapunov method [19], which states thatfor an autonomous system, if a positive-definite and radiallyunbounded Lyapunov function V (x), positive constants c and α ∈

(0, 1) can be found such that V (x) + cV α(x) ≤ 0, then theautonomous system is finite-time stable. The last type of finite-time control is the terminal sliding mode method [36,37]. Thismethod introduces a nonlinear sliding manifold, which yields thatthe states slide to the origin along the manifold in a finite timewhen they reach the manifold.

In this subsection, we will show some results on finite-timestabilization. According to Bhat’s homogeneous theory [1], fora homogeneous system, if it is asymptotically stable and has anegative degree of homogeneity, then it is also finite-time stable.For system (1), according to Theorem2, if it has a negative degree ofhomogeneity, then we have the following finite-time stabilizationresult.

Corollary 1. Under Assumptions 1–2, if d < 0, then by takingp = q and ai =

siσ1+d−si

, controller (2) globally finite-time stabilizessystem (1).

Proof. From Theorem 1, we know that the closed-loop system(1)–(2) is globally asymptotically stable. Moreover, by Assump-tion 2 and d < 0, we can verify that ui(∆

rεx) = εsiui(x). This im-

plies that the closed-loop system (1)–(2) is homogeneous of degreed < 0. Hence, the closed-loop system (1)–(2) is globally finite-timestable.

Similarly, for Theorem 3, we also have the corresponding finite-time control result.

Corollary 2. Under Assumptions 1–4, if σ2 > σ ′

2 > 0 and d < 0,then by taking p = q and ai =

siσ1+d−si

(i = 1, . . . , n), undercontroller (2), cascaded system (12) can be globally finite-time sta-bilized.

Proof. From Corollary 1, it can be concluded that under the givenconditions, the driving subsystem (12b) of cascaded system (12)is globally finite-time stable. This implies that there exists a finitetime instant t∗ such that x = 0 for t ≥ t∗. Thus, we alsoobtain h(z, x) = 0 for t ≥ t∗. Note from Theorem 3 that underAssumptions 1, 3 and 4, the states of cascaded system (12) arealways bounded. Nowwe only need to prove the global finite-timestability of subsystem (13).

According to Assumption 3, we get

V2(z)|(13) ≤ −ω2(z) (23)

withV2(z) andω2(z)beinghomogeneous of degreesσ2 andσ ′

2 withrespect to l = (l1, . . . , lk)T , respectively. Furthermore, by a simple

calculation, we can also verify that Vσ ′2/σ2

2 (z) is homogeneous of

degree σ ′

2 with respect to l. Since ω2(z) and Vσ ′2/σ2

2 (z) have thesame degree of homogeneity with respect to the same weight,according to the homogeneous theory (Lemma 2.3 of [11]), thereexists a constant γ > 0 such that

ω2(z) ≥ γ Vσ ′2/σ2

2 (z).

Then by (23), we have

V2(z)|(13) ≤ −γ Vσ ′2/σ2

2 (z). (24)

Since 0 < σ ′

2 < σ2, we have 0 < σ ′

2/σ2 < 1. Finally, according toLemma 1, subsystem (13) is globally finite-time stable.

4. Conclusion

This paper has studied the global stabilization problems for aclass of nonlinear affine systems by using the control Lyapunovfunction. Our first result has extended the main theorem proposedin [7], thus being a more general result. Our second result hasaddressed the control design for a class of nonlinear cascadedaffine systems by using partial-state feedbacks. By imposing someconditions on the driven subsystem, we have shown that thestabilization of the driving subsystem will guarantee the globalstabilization of the whole cascaded affine system. The finite-timestabilization proposed in our third result has been obtained byimposing some homogeneous properties on the considered systemand utilizing the control Lyapunov function. During the controldesign, the control Lyapunov function has played a significantrole and can be considered as a key ingredient of the controldesign. Further research will be focused on how to systematicallyconstruct such a kind of control Lyapunov functions.

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