Control for a class of non-linear singularly perturbed systems subject to actuator saturation

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Published in IET Control Theory and ApplicationsReceived on 7th October 2012Revised on 12th April 2013Accepted on 28th April 2013doi: 10.1049/iet-cta.2013.0084

ISSN 1751-8644

Brief Paper

Control for a class of non-linear singularly perturbedsystems subject to actuator saturationLinna Zhou1, ChunyuYang1, Weidong Zhang2

1School of Information and Electrical Engineering, China University of Mining andTechnology, Xuzhou 221116,People’s Republic of China2Department of Automation, Shanghai JiaoTong University, Shanghai 200240, People’s Republic of ChinaE-mail: [email protected]

Abstract: This study considers the control problem for a class of non-linear singularly perturbed systems (SPSs) subjectto actuator saturation. A sufficient condition for the existence of state-feedback controllers to achieve a prescribed stabilitybound is proposed and the corresponding basin of attraction is estimated. Then a convex optimisation problem is formulated,by which an optimal controller can be obtained to achieve a prescribed stability bound and simultaneously maximise theestimate of the basin of attraction of the SPSs for any allowable singular perturbation parameter. Furthermore, a stabilitycondition is established, which improves the existing stability bound analysis methods for a class of non-linear SPSs. Finally,examples are given to show the advantages and effectiveness of the obtained results.

1 Introduction

Multiple time-scale systems are often encountered in engi-neering fields, such as electronic circuits, motor controlsystems, magnetic-ball suspension systems and so on. Theyare usually modelled as singularly perturbed systems (SPSs)with a small singular perturbation parameter ε determiningthe degree of separation between the slow and fast modesof the systems [1]. The stability bound problem, which isknown as the problem of determining an allowable upperbound such that the system is stable for all ε lower thanthe upper bound, is a fundamental problem for SPSs andhas attracted much attention in the past decades. A numberof approaches to estimating the stability bound were pro-posed for various kinds of SPSs in [2–8] and some controllerdesign methods to enlarge the stability bound of SPSs wereproposed in [9–12]. However, very few works have consid-ered the problems of stability bound analysis and synthesisfor non-linear SPSs subject to actuator saturation.

Saturation non-linearity is widely present in practical con-trol systems and thus intensive research efforts have beendevoted to control systems subject to actuator saturation(see, e.g. [13–16]). One of the significant topics for controlsystems subject to actuator saturation is estimating the basinof attraction of the closed-loop systems, which has beendiscussed in great depth (see, e.g. [14, 15, 17]). Recently,some researchers have paid their attention to the problemsof analysis and synthesis of SPSs subject to actuator satura-tion. By the so-called reduction technique, Liu [18] proposeda controller design method for SPSs subject to actuatorsaturation under the assumption that the fast dynamics is

IET Control Theory Appl., 2013, Vol. 7, Iss. 10, pp. 1415–1421doi: 10.1049/iet-cta.2013.0084

stable and Garcia and Tarbouriech [19] formulated a convexoptimisation problem to estimate the basin of attraction ofSPSs subject to actuator saturation. Xin et al. [20, 21] intro-duced the so-called reduced-order adjoint systems, by whichsome methods to estimate the basin of attraction of SPSswere proposed. An alternative approach that is independentof system decomposition was proposed in [22]. However,the above-mentioned results were limited to linear SPSs anddid not consider the stability bound problem.

This paper will consider the control problem for a classof non-linear SPSs subject to actuator saturation. The objec-tive is to propose a state-feedback controller design methodto achieve a given stability bound and simultaneously max-imise the estimate of the basin of attraction of the SPSs.First, a sufficient condition for the existence of state-feedback controllers is proposed, such that for any ε lessthan the pre-defined stability bound, the closed-loop sys-tem is asymptotically stable. Then, a convex optimisationproblem is formulated, by which an optimal controller canbe obtained to achieve the prescribed stability bound andsimultaneously maximise the estimate of the basin of attrac-tion of the SPSs for any allowable singular perturbationparameter. Furthermore, the proposed result is specialisedto deal with stability problem of SPSs with non-linearities,which is shown to be less conservative than the existingmethods in the sense that a tighter stability bound can beobtained. Finally, some examples are given to show theeffectiveness of the obtained results. The main contributionsof the paper are as follows: (i) it addresses a more generalclass of systems that are subject to non-linearities; (ii) thestability bound and the estimate of the basin of attraction are

1415© The Institution of Engineering and Technology 2013

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considered simultaneously; and (iii) the design procedure isreduced to solving a convex optimisation problem.

The rest of this paper is organised as follows. Section 2presents system description and problem statement. InSection 3, the main results are proposed. Examples are givenin Section 4 to illustrate various features of the proposedmethods and show their advantages over the existing results.Finally, conclusions are given in Section 5.Notation: The superscript T stands for matrix transposi-tion and the notation M−T denotes the transpose of theinverse matrix of M. For vectors v, w ∈ Rp, v � w meansthat the inequalities between the vectors are componentwise.� denotes the block induced by symmetry. For a matrix M,M(i) denotes ith row of M.

2 Problem formulation

Consider the following system

E(ε)x(t) = Ax(t) + Bgg(x(t)) + Busat(u(t)) (1)

where x = [ x1x2

] ∈ Rn is the state, x1 ∈ Rn1 , x2 ∈ Rn2 , E(ε) =[In1 00 εIn2

]∈ Rn×n, A ∈ Rn×n, Bg ∈ Rn×p and Bu ∈ Rn×q are

constant matrices. Moreover, sat(·) is a componentwisesaturation map Rq �→ Rq defined as

sat(ui(t)) = sign(ui(t))min{ρi, |ui(t)|} (2)

where ρi > 0, i = 1, 2, . . . , q denote the symmetric ampli-tude bound relative to the ith control input.

For the convenience of exposition, we define the decen-tralised dead-zone non-linearity ψ(·) as

ψ(u(t)) = sat(u(t)) − u(t) (3)

The non-linear term g(·) in (1) is assumed to satisfy thefollowing condition, which is more general than the well-known globally Lipschitz condition

gT(x)g(x) ≤ xTHTHx, ∀x ∈ Rn (4)

where H ∈ Rp×n is a constant matrix.

Remark 1: It is worth mentioning that condition (4) is dif-ferent from the cone-bounded sector condition [23]. Thereexists overlapping and complementarity between them.Furthermore, the former covers the well-known matcheduncertainty [24] and Lipschitz non-linearity [25] as specialcases, but the latter does not. Therefore the non-linearity sat-isfying (4) has been widely discussed in the literature [26].However, the problems of analysis and design for system(1) have not been investigated.

In this paper, we assume that a superior limit ε0 for εis available, and that the following state-feedback controllerlaw can be designed and implemented for any value of 0 <ε ≤ ε0 to stabilise system (1)

u = K(ε)x (5)

Then, we have the closed-loop system

E(ε)x(t) = (A + BuK(ε))x(t) + Bgg(x(t)) + Buψ(u(t))(6)

It is known that in the presence of actuator saturation, globalstabilisation cannot be achieved if the open-loop system


is unstable. Thus, it is necessary to estimate the basin ofattraction. To this end, the problem under consideration isformulated as follows:

Problem 1: Given a scalar ε0 > 0, determine feedback gainmatrix K(ε) and a region �(ε) ⊆ Rn, as large as possible,such that for any initial condition x0 ∈ �(ε), the closed-loop system (6) is asymptotically stable for any ε ∈ (0, ε0]and any g(x) satisfying (4).

Remark 2: As shown in [12, 24, 27], one of the importantissues to design ε-dependent controllers is guaranteeing theobtained controller to be well-defined for any allowable ε,which will be discussed later.

The following lemmas will be used in the sequel.

Lemma 1 [12]: For a positive scalar ε0 and symmetricmatrices S1, S2 and S3 with appropriate dimensions, if

S1 ≥ 0 (7)

S1 + ε0S2 > 0 (8)

andS1 + ε0S2 + ε2

0S3 > 0 (9)

hold, then

S1 + εS2 + ε2S3 > 0, ∀ε ∈ (0, ε0] (10)

Lemma 2 [12]: If there exist matrices Zi (i = 1, 2, . . . , 5)with Zi = ZT

i (i = 1, 2, 3, 4) satisfying

Z1 > 0 (11)[Z1 + ε0Z3 ε0ZT

5

ε0Z5 ε0Z2

]> 0 (12)

and [Z1 + ε0Z3 ε0ZT

5

ε0Z5 ε0Z2 + ε20Z4

]> 0 (13)

then

E(ε)Z(ε) = ZT(ε)E(ε) > 0, ∀ε ∈ (0, ε0] (14)

where Z(ε) =[

Z1+εZ3 εZT5

Z5 Z2+εZ4

].

Lemma 3 [17]: For any diagonal positive-definite matrix� ∈ Rq×q, the non-linearity ψ(v) = sat(v) − v satisfies thefollowing inequality

ψT(v)�(ψ(v) + w) ≤ 0, ∀v, w ∈ S(v0) (15)

where S(v0) = {v, w ∈ Rq| − v0 � v − w � v0} and v0 ∈ Rq

is given.

Remark 3: In [12], fuzzy control of non-linear SPSs wasinvestigated by Lemmas 1 and 2, but the proposed resultscannot deal with actuator saturation. Lemma 3 was usedin [23] to study the control problem for a class of non-linear systems subject to actuator saturations. However,applying the proposed method in [23] to SPSs may leadto ill-conditioned numerical problems. Using Lemmas 1, 2and 3, and giving full consideration to the singular pertur-bation structure, this paper will propose an controller designmethod to achieve a prescribed stability bound and simulta-neously maximise the estimate of the basin of attraction ofthe SPSs.


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3 Main results

The main results are proposed in this section. A state-feedback controller design method is proposed and a con-vex optimisation problem is formulated to obtain the bestestimate of the basin of attraction of the closed-loop systems.

3.1 Controller design

Theorem 1: Given a scalar ε0 > 0, if there exist a diago-nal positive-definite matrix S ∈ Rq×q, matrices M1 ∈ Rq×n1 ,M2 ∈ Rq×n2 , Y ∈ Rq×n, Z1 ∈ Rn1×n1 , Z2 ∈ Rn2×n2 , Z3 ∈ Rn1×n1 ,Z4 ∈ Rn2×n2 , Z5 ∈ Rn2×n1 with Zi = ZT

i (i = 1, 2, 3, 4), suchthat linear matrix inequalities (LMIs) (11), (12), (13) and⎡

⎢⎢⎢⎣� � � �

BTg −I � �

SBTu − ME(0) − Y 0 −2S �

HU1 0 0 −I

⎤⎥⎥⎥⎦ < 0 (16)

⎡⎢⎢⎢⎣

� � �

BTg −I � �

SBTu − ME(ε0) − Y 0 −2S �

H(U1 + ε0U2) 0 0 −I

⎤⎥⎥⎥⎦ < 0 (17)

[Z1 �

M1(i) ρ2i

]≥ 0, i = 1, 2, . . . , q (18)

⎡⎣Z1 + ε0Z3 � �

ε0Z5 ε0Z2 �

M1(i) ε0M2(i) ρ2i

⎤⎦ ≥ 0, i = 1, 2, . . . , q (19)

⎡⎣Z1 + ε0Z3 � �

ε0Z5 ε0Z2 + ε20Z4 �

M1(i) ε0M2(i) ρ2i

⎤⎦ ≥ 0, i = 1, 2, . . . , q

(20)

hold, where U1 = [Z1 0Z5 Z2

], U2 =

[Z3 ZT

50 Z4

], � = UT

1 AT +AU1 + YTBT

u + BuY, = (U1 + ε0U2)TAT + A(U1 + ε0U2) +

YTBTu + BuY and M = [

M1 M2

].

Then the controller (5) with K(ε) = YZ−1(ε), Z(ε) =U1 + εU2 renders system (6) asymptotically stable for anyε ∈ (0, ε0]. Moreover, the ellipsoid

�(ε) = {x|xTZ−T (ε)E(ε)x ≤ 1}is an estimate of the basin of attraction of the closed-loopsystem.

Proof: Let v = u and w = ME(ε)Z−1(ε)x + u = (ME(ε)Z−1(ε) + K(ε))x. Then from Lemma 3, the non-linearityψ(u) satisfies

ψT(u)�(ψ(u) + (ME(ε)Z−1(ε) + K(ε))x) ≤ 0 (21)

for ∀x ∈ S(ρ, ε), where � is an arbitrary diagonal positive-definite matrix, ρ = [ρ1 ρ2 · · · ρq]T, and S(ρ, ε) = {x| −ρ � ME(ε)Z−1(ε)x � ρ}.

From Lemma 1, LMIs (18), (19) and (20) imply that[ZT(ε)E(ε) �

M(i)E(ε) ρ2i

]≥ 0, i = 1, 2, . . . , q, ∀ε ∈ (0, ε0]

(22)


which is equivalent to[E−1(ε)ZT(ε) �

M(i) ρ2i

]≥ 0, i = 1, 2, . . . , q, ∀ε ∈ (0, ε0]

(23)

Pre- and post multiplying (23) by

diag([E−1(ε)ZT(ε)]−1, I)

and its transpose, respectively, we have[E(ε)Z−1(ε) �

M(i)E(ε)Z−1(ε) ρ2i

]≥ 0, i = 1, 2, . . . , q

which implies

E(ε)Z−1(ε) ≥ Z−T (ε)E(ε)MT(i)ρ

−2i M(i)E(ε)Z−1(ε)

Then for any x ∈ �(ε), it holds that

xTZ−T (ε)E(ε)MT(i)ρ

−2i M(i)E(ε)Z−1(ε)x ≤ 1

which implies that �(ε) ⊆ S(ρ, ε).LMIs (16) and (17) imply⎡

⎢⎢⎢⎣ � � �

BTg −I � �

SBTu − ME(ε) − Y 0 −2S �

H(U1 + εU2) 0 0 −I

⎤⎥⎥⎥⎦ < 0 (24)

for ∀ε ∈ (0, ε0], where = (U1 + εU2)TAT + A(U1 +

εU2) + YTBTu + BuY, that is⎡

⎢⎢⎢⎣ � � �

BTg −I � �

SBTu − ME(ε) − Y 0 −2S �

HZ(ε) 0 0 −I

⎤⎥⎥⎥⎦ < 0 (25)

for ∀ε ∈ (0, ε0].Pre- and post-multiplying (25) by

diag(Z−T (ε), I, S−1, I)

and its transpose, respectively, we have⎡⎢⎢⎢⎣

�1 � � �

BTg Z−1(ε) −I � �

�2 0 −2S−1 �

H 0 0 −I

⎤⎥⎥⎥⎦ < 0, ∀ε ∈ (0, ε0] (26)

where �1 = ATZ−1(ε) + Z−T (ε)YTBTu Z−1(ε) + Z−T (ε)A +

Z−T (ε)BuYZ−1(ε) and �2 = BTu Z−1(ε) − S−1ME(ε)Z−1(ε) −

S−1YZ−1(ε).Letting K = YZ−1(ε), � = S−1, P(ε) = Z−1(ε), we have

� �

⎡⎢⎣

�1 � �

BTg P(ε) −I �

BTu P(ε) − �ME(ε)P(ε) − �K 0 −2�

⎤⎥⎦ < 0

(27)

for ∀ε ∈ (0, ε0], where �1 = (A + BuK)TP(ε) + PT(ε)(A +BuK) + HTH.


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By Lemma 2, LMIs (11), (12) and (13) guarantee that(14) holds, which implies

E(ε)P(ε) = PT(ε)E(ε) > 0, ∀ε ∈ (0, ε0] (28)

Define an ε-dependent Lyapunov function

V (x) = xTE(ε)P(ε)x (29)

Computing the derivative of V (x) along the trajectories ofsystem (6) and taking into account (21) and (4), we have

V |(6)

= (E(ε)x)TP(ε)x + xTPT(ε)E(ε)x

= xT((A + BuK)TP(ε) + PT(ε)(A + BuK))x

+ 2gTBTg P(ε)x + 2ψTBT

u P(ε)x

≤ xT((A + BuK)TP(ε) + PT(ε)(A + BuK))x

+ 2gTBTg P(ε)x + 2ψTBT

u P(ε)x + xTHTHx

− gTg − 2ψT(u)�(ψ(u) + (ME(ε)P(ε) + K)x)

=[

xgψ

]T

�

[xgψ

]< 0, ∀x ∈ �(ε), x = 0 (30)

Therefore the closed-loop system is asymptotically stable forany x0 ∈ �(ε) and any ε ∈ (0, ε0]. �

Remark 4: In Theorem 1, LMIs (11), (12) and (13) are usedto guarantee the positiveness of Lyapunov function (28), andLMIs (16) and (17) ensure the negativeness of the derivativeof the Lyapunov function (28) along the trajectories of sys-tem (6). And LMIs (18), (19) and (20) relate the ellipsoid�(ε) with the basin of attraction of the closed-loop system.The feasibility of these LMIs can be effectively checked bythe LMI Toolbox in MATLAB [28].

Remark 5: LMIs (11) and (12) indicate that Z1 > 0 andZ2 > 0. As a result, the matrix U1 = [

Z1 0Z5 Z2

]is non-singular.

In addition, the proof of Theorem 1 has shown that Z(ε) =U1 + εU2 is non-singular for all ε ∈ (0, ε0]. Then K(ε) =Y(U1 + εU2)

−1 is well defined for all ε ∈ (0, ε0]. Sincelimε→0+ K(ε) = YU−1

1 , controller (5) can be reduced to anε-independent one if ε is sufficiently small.

Remark 6: There are basically two methods handling satu-ration non-linearity, namely, the sector bound approach [29]and the convex Hull approach [30]. The former needs lesscomputational cost but is more conservative than the latter[31]. In this paper, the sector bound approach is used toderive the main results. In a similar way, the convex Hullapproach can be applied to obtain the parallel results.

Corollary 1: Given a scalar ε0 > 0, if there exist matri-ces Z1 ∈ Rn1×n1 , Z2 ∈ Rn2×n2 , Z3 ∈ Rn1×n1 , Z4 ∈ Rn2×n2 ,Z5 ∈ Rn2×n1 with Zi = Z

T

i (i = 1, 2, 3, 4), such that

Z1 > 0 (31)[Z1 + ε0Z3 ε0Z

T

5

ε0Z5 ε0Z2

]> 0 (32)


[Z1 + ε0Z3 ε0Z

T

5

ε0Z5 ε0Z2 + ε20Z4

]> 0 (33)

⎡⎢⎣

UT

1 AT + AU1 � �

BTg −I �

HU1 0 −I

⎤⎥⎦ < 0 (34)

and

⎡⎢⎣

(U1 + ε0U2)TAT + A(U1 + ε0U2) � �

BTg −I �

H(U1 + ε0U2) 0 −I

⎤⎥⎦ < 0 (35)

hold, where

U1 =[

Z1 0

Z5 Z2

], U2 =

[Z3 Z

T

5

0 Z4

]

Then system (1) with u = 0 is asymptotically stable for ∀ε ∈(0, ε0].

Remark 7: Stability bound problem of system (1) with u = 0was considered in [26, 32]. The Lyapunov function uponwhich Corollary 1 is based is more general than those usedin [26, 32], which provides an opportunity to reduce theconservatism of the proposed method. Thus, it is expectedthat Corollary 1 can lead to a tighter stability bound than theexisting methods in [26, 32], as will be shown by an examplein the next section. The best estimate of the stability boundε0 can be obtained by solving the following optimisationproblem

maxZ1,Z2,Z3,Z4,Z5

ε0

s.t. LMIs (31), (32), (33), (34) and (35) (36)

which can be effectively solved by a one-dimensional searchalgorithm with the aid of LMI Toolbox in MATLAB [28].

3.2 Optimisation of the estimate of the basin ofattraction

There are mainly two approaches to measuring the size ofthe estimate of the basin of attraction. One takes the shapeof a prescribed set into consideration [15, 31] and the othermeasures the size of the estimate of the basin by its volume[33]. Here, we choose the first method and the second onecan be obtained in a similar way.

Given a shape set �0 ∈ Rn and a scaling factor β, where�0 = Co{vr ∈ Rn, r = 1, 2, . . . , nr}. One can determine thebest estimate of the basin of attraction by maximising thescaling factor β, such that β�0 ⊆ �(ε), which can beformulated into the following optimisation problem

maxS,M1,M2,Y,Z1,Z2,Z3,Z4,Z5

β

s.t. (11), (12), (13), (16), (17), (18), (19), (20)

and S > 0 and β�0 ⊆ �(ε) (37)

where S is a diagonal matrix.


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Constraint β�0 ⊆ �(ε) can be converted to

β2vTr Z−T (ε)E(ε)vr ≤ 1

which is equivalent to

[E(ε)Z(ε) E(ε)vr

vTr E(ε) μ

]≥ 0 (38)

where μ = 1/β2.From Lemma 1, inequality (38) can be guaranteed by

[Z1 �

vT1r μ

]≥ 0, r = 1, 2, . . . , nr (39)

⎡⎣Z1 + ε0Z3 � �

ε0Z5 ε0Z2 �

vT1r ε0vT

2r μ

⎤⎦ ≥ 0, r = 1, 2, . . . , nr (40)

and

⎡⎣Z1 + ε0Z3 � �

ε0Z5 ε0Z2 + ε20Z4 �

vT1r ε0vT

2r μ

⎤⎦ ≥ 0, r = 1, 2, . . . , nr

(41)where vr = [vT

1r vT2r]T.

Then the optimisation problem (37) can be reformulatedto the following convex optimisation problem

minS,M1,M2,Y,Z1,Z2,Z3,Z4,Z5

μ

s.t. (11), (12), (13), (16), (17), (18), (19), (20), (39)

(40), (41) and S > 0 (42)

where S is a diagonal matrix.

Remark 8: Analysis and synthesis problems for variouskinds of SPSs subject to actuator saturation have been con-sidered in [18–22]. Compared with the existing methods, theadvantages of the proposed method are as follows: (i) it canbe applied to a more general class of systems which are sub-ject to non-linearities; (ii) it considers the stability bound andthe estimate of the basin of attraction simultaneously; and(iii) it is reduced to solving a convex optimisation problem.

4 Examples

In this section, examples are given to illustrate various fea-tures of the proposed methods and show their advantagesover the existing results.

Example 1: This example will illustrate that Corollary 1 canlead to a tighter stability bound for SPSs than the methodsof [26, 32]. Consider the example proposed in [26, 32]

x1 = x1 − x2 + |x1|x2

1 + 4x22

εx2 = 2x1 − x2 + x1|x2|1 + 4x2

1

(43)


System (43) can be transformed in the form of (1) with

E(ε) =[

1 00 ε

], A =

[1 −12 −1

], Bg =

[1 00 1

]

H =[

0.25 00 0.25

], Bu = 0, g(x) =

⎡⎢⎢⎣

|x1|x2

1 + 4x22

x1|x2|1 + 4x2

1

⎤⎥⎥⎦

It is easy to show that g(x) satisfies inequality (4).Solving the optimisation problem (36), we obtain the best

estimate of the stability bound ε0 = 0.4528 and the corre-sponding solutions to LMIs (31), (32), (33), (34) and (35)are as follows

Z1 = 2.3993, Z2 = 5.6605, Z3 = 2.5826,

Z4 = 3.2612, Z5 = 5.3670

Thus by Corollary 1, the system is asymptotically stable forany ε ∈ (0, 0.4528]. It can be seen that the obtained stabilitybound is much larger than 9.5 × 10−3 and 0.3395 computedby the methods of [26, 32], respectively.

Example 2: This example will demonstrate how the pro-posed method is applied to an inverted pendulum systemcontrolled by a DC motor via a gear train. The model, whichwas first established in [34], is described by⎧⎪⎪⎪⎨

⎪⎪⎪⎩x1(t) = x2(t)

x2(t) = g

lsin x1(t) + NKm

ml2x3(t)

Lax3(t) = −KbNx2(t) − Rax3(t) + u(t)

(44)

where x1(t) = θp(t) denotes the the angle (rad) of the pen-dulum from the vertical upward, x2(t) = θp(t), x3(t) = Ia(t)denotes the current of the motor, u(t) is the control inputvoltage, Km is the motor torque constant, Kb is the back emfconstant, N is the gear ratio, and La is the inductance, whichis usually a small positive constant.

The parameters for the plant are as follows: g =9.8 m/s2, N = 50, l = 1 m, m = 1 kg, Km = 0.1 Nm/A, Kb =0.1Vs/rad, Ra = 1 � and La = 0.05 H and the input voltageis required to satisfy |u| ≤ 5. Note that the inductance La

represents the small parameter in the system. Substitutingthe parameters into (44), we have⎧⎨

⎩x1(t) = x2(t)x2(t) = 9.8 sin x1(t) + 5x3(t)εx3(t) = −5x2(t) − x3(t) + u

(45)

where ε = La.System (45) can be transformed into the form of (1) with

E(ε) =[

1 0 00 1 00 0 ε

], A =

[0 1 00 0 50 −5 −1

]

Bg = [0 9.8 0

]T, Bu = [

0 0 1]T

H = [1 0 0

], ρ = 5, g(x) = sin(x1)

For system (45), xe = [0 0 0]T corresponds to the uprightrest position of the inverted pendulum. The control goal is


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to swing the pendulum towards the upright rest position andbalance there.

Let �0 with �0 = Co{vr ∈ Rn, r = 1, 2, 3, 4}, v1 = [−1 00]T, v2 = [1 0 0]T, v3 = [0 − 2 0]T, v4 = [0 2 0]T.

Solving the convex optimisation problem (42) with ρ = 5,ε0 = 0.5, we have

Z1 =[

1.8163 −9.3929−9.3929 174.3434

], Z2 = 143.3272

Z3 =[

5.1583 −0.9167−0.9167 0.8754

], Z4 = −0.1412

Z5 = [−25.4234 −24.7311]Y = [−6.05983.0415 − 1323.4596]

M1 = [5.8537 − 2.5507], M2 = 0.19569, S = 1324.3439

μ = 0.7698, β = 1.1397

Taking into account ε = 0.05, we have the controller gainmatrix

K = [−192.8878 −11.9894 −11.0477]and the estimate of the basin of attraction of the closed-loopsystem � = {x ∈ R3|xTPx ≤ 1}, where

P =[

0.7615 0.0422 0.00710.0422 0.0081 0.00040.0071 0.0004 0.0004

]

The ellipsoid � and the trajectory starting from x0 =[−0.5 4 20]T are shown in Fig. 1 and the control inputis shown in Fig. 2. It can be seen from Fig. 1 that the tra-jectory starting from x0 = [−0.5 4 20]T ∈ � remains in �and converges to the equilibrium point of system (45), thatis, xe = [0 0 0]T.

By simulation, we can obtain the real basin of attractionof the closed-loop system. In Fig. 3, mark × denotes theregion excluded by the basin of attraction. It can be seenthat the ellipsoid � is a conservative estimate of the basinof attraction. It is an interesting and significant topic to findless conservative methods.

It should be remarked that a prerequisite for applying theexisting methods in [18–22] to system (45) is linearising the

Fig. 1 Estimation of the basin of attraction and the convergingtrajectory starting from x0 = [−0.5 4 20]T


0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

time (sec)

sat(

u)

Fig. 2 Control input

Fig. 3 Illustration of the conservatism of the estimate of the basinof attraction

non-linear system around its equilibrium. In this case, thecontroller only work well around its equilibrium. The newlydeveloped method in this paper overcomes this problem andpresents an estimate of the basin of attraction of the closed-loop system.

5 Conclusion

In this paper, we have considered the stabilisation problemfor a class of non-linear SPSs subject to actuator satura-tion. We first proposed a state-feedback controller designmethod and constructed an ε-dependent estimate of the basinof attraction, by which a convex optimisation algorithm wasformulated to maximise the estimate of the basin of attrac-tion of the closed-loop system. The obtained result improvesthe existing methods in the sense that it achieves a prede-fined stability bound and gives consideration to maximisingthe estimate of the basin of attraction by solving a con-vex optimisation problem. Furthermore, a stability conditionthat improves the existing stability bound analysis methodsfor SPSs was established. Finally, the presented examplesdemonstrated the utility of the proposed methods and thecontributions of the results.


www.ietdl.org

6 Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (grant numbers 60904009, 60904079,61025016 and 61020106003), the Fundamental ResearchFunds for the Central Universities (grant no. 2013QNA50)and the China Postdoctoral Science Foundation fundedproject (grant number 2013M530278).

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Control for a class of non-linear singularly perturbed systems subject to actuator saturation