Robust Stability Analysis of Asymptotic Second-order Sliding

Mode Control System Using Lyapunov Function

Yaodong Pan, Guangjun Liu, and Krishna Dev KumarDepartment of Aerospace Engineering, Ryerson University350 Victoria Street, Toronto, Ontario M5B 2K3, Canada

{yaodong.pan, gjliu, kdkumar}@ryerson.ca

Abstract—This paper investigates the robust stability of anasymptotic second-order sliding mode (2nd-SM) control system,where a first-order sliding mode (1st-SM) control law is imple-mented to realize an asymptotic 2nd-SM control for a lineartime-invariant continuous-time system with a relative degree oftwo. It is found in the paper that a 2nd-SM can be reachedlocally and asymptotically by a 1st-SM control law if the sum ofthe system poles is less than the sum of the system zeros. Theasymptotic convergence to the 2nd-SM and the robust stabilityof the asymptotic 2nd-SM control system are for the first timeproved with Lyapunov functions, in the presence of matchedexternal disturbances and parameter uncertainties. Finally, theeffectiveness of the asymptotic 2nd-SM control algorithm isverified through numerical simulations.

Index Terms—Variable Structure Control, Second-order SlidingMode Control, First-order Sliding Mode Control, AsymptoticConvergence, Lyapunov Function

I. INTRODUCTION

It is well known that a variable structure (VS) control system

with a sliding mode (SM) is robust to external disturbances

and parameter uncertainties if the SM is reached in finite time

and the reduced-order system on the SM is stable [1][2]. In

case a switching function designed for the SM control has

a relative degree of two with respect to the control input, a

second-order SM (2nd-SM) controller should be designed to

ensure the finite time occurrence of a 2nd-SM [3][4]. Various

2nd-SM control algorithms such as the twisting 2nd-SM control

[4] and the sub-optimal 2nd-SM control [3] have been proposed

and implemented [5][6][7][4][3][8].

To ensure the finite time convergence to a 2nd-SM with the

system output as the switching function, both the output and

its derivative are needed in the control implementation [3][4].

For most of practical control systems, however, the derivative

of the output is unmeasurable or unavailable for the control

implementation although it may be estimated by an observer.

It has been proposed to realize an asymptotic 2nd-SM control

without using the derivative by a 1st-SM control law (1st-

SMCL), i.e., a relay control law [9][10][11]. Some features of

the asymptotic 2nd-SM control systems with an ideal relay or a

hysteresis relay have been researched in [12][13][14][15][16].

Related research results can also be found in the SM control

with fast dynamic actuators [17] or inertial sensors [18].

In the present paper, the robust stability of an asymptotic

2nd-SM control system is investigated using Lyapunov func-

tions for a linear time-invariant continuous-time system with

matched external disturbances and parameter uncertainties. The

system output that is considered as the switching function is

assumed to have a relative degree of two with respect to the

control input. As a necessary condition for a 2nd-SM to be

reached locally and asymptotically with a 1st-SMCL, it is for

the first time revealed that the sum of system poles must be

less than the sum of system zeros. In addition, as the finite time

convergence to a 2nd-SM can not be realized by a 1st-SMCL,

the stability of an asymptotic 2nd-SM control system can not

be guaranteed by the SM control theory even if the 2nd-SM

where the reduced-order system is asymptotically stable, can

be reached asymptotically. In the present paper, the asymptotic

convergence to the 2nd-SM and the robust stability of the

asymptotic 2nd-SM control system are proved using identified

Lyapunov functions.

This paper is organized as follows: Section 2 describes the

problem to be studied in this paper; Section 3 analyzes the

stability of the asymptotic 2nd-SM control system; Section 4

gives simulation results; and Section 5 concludes the paper.

II. PROBLEM DESCRIPTION

This paper considers a single input single output (SISO)

linear time-invariant system with a relative degree of two,

described by{x(t) = (A+ΔA)x(t)+Bu(t)+D(t)y(t) = Cx(t) , (1)

where x(t)∈Rn, y(t)∈R, and u(t)∈R are the state, output, and

input variables, respectively; the system matrices A, B, and Cand the parameter uncertainty ΔA are with suitable dimensions;

and D(t)∈ Rn is the external disturbance; (A, B) and (C, A) are

controllable and observable pairs, respectively.

It is assumed that the uncertainty ΔA and the disturbance

D(t) satisfy the matching condition [1][2], i.e., there exists a

function d(x, t) such that

ΔAx(t)+D(t) = Bd(x, t).

Assumption 1: d(x, t) and its derivative d(x, t) with respect

to time t are bounded as

|d(x, t)| ≤ d1, |d(x, t)| ≤ d2, ∀t ∈ [0,+∞),∀x(t) ∈ X

with two known positive constants d1 and d2, where X ⊂ Rn is

a bounded compact subset for the state x(t).

313978-1-4244-5704-5/10/$26.00 ©2010 IEEE

Proceedings of the 2010 IEEEInternational Conference on Information and Automation

June 20 - 23, Harbin, China

Without loss of generality, it is assumed that the system (A,

B, C) in (1) has been transformed to its controllable canonical

form. Thus A, B and C for the system (1) with a relative degree

of two can be described as [19]

A =

[O(n−1)×1 In−1

−a0 −a1 −a2 · · · −an−1

],

B =

[O1×(n−1)

1

], C =

[c0 c1 · · · cn−2 0

],

where ai (i = 0,1, · · · ,n− 1) and c j ( j = 0,1, · · · ,n− 2) are

constants and cn−2 �= 0. In this paper, Ii and O j×k are the i× i(i = 2,3, · · · ) identity matrix and the j× k ( j = 1,2, · · · ; k =1,2, · · · ) zero matrix, respectively.

In this case, the transfer function G(s) of the system (A, B,

C) in (1) can be represented as

G(s) = C(sIn−A)−1B

=cn−2sn−2 + cn−3sn−3 + · · ·+ c1s+ c0

sn +an−1sn−1 + · · ·+a1s+a0

= Ksn−2 + cn−3sn−3 + · · ·+ c1s+ c0

sn +an−1sn−1 + · · ·+a1s+a0(2)

= K(s− z1)(s− z2) · · ·(s− zn−2)

(s− p1)(s− p2) · · ·(s− pn), (3)

where pi (i = 1,2, · · · ,n) and z j ( j = 1,2, · · · ,n−2) are poles

and zeros of the system (1), respectively, and the open loop

gain K and coefficients ci (i = 0,1, · · · ,n− 3) are determined

by

K = cn−2, ci = ci/cn−2 = ci/K.

The denominator and numerator polynomials of G(s) are

coprime for the controllable and observable system (1). Without

loss of generality, it is assumed that the open loop gain K is

positive, i.e., K = cn−2 > 0.

For relative degree one systems, a 1st-SMCL has been

designed as [1][2]

u(t) =−ksgn(y(t)), (4)

where k is a positive constant and the signum function sgn(·)is defined as

sgn(y) ={ −1, y < 0

1, y≥ 0.

A 1st-SM can be reached in finite time, locally for a relative

degree one system with the 1st-SMCL (4) [1][2].

In this paper, the system (1) considered has a relative degree

of two. The control objective is to stabilize the relative degree

two system (1), locally and asymptotically with the 1st-SMCL

(4) such that

1) a 2nd-SM is reached locally and asymptotically;

2) the reduced-order system on the 2nd-SM is asymptoti-

cally stable; and

3) the system (1) converges to the origin locally and asymp-

totically.

III. ASYMPTOTIC 2ND-SM CONTROL

For the relative degree two system (1), the first and the

second derivatives of the output y(t) with respect to time tare determined by

y(t) = Cx(t) =CAx(t)+CB(u(t)+d(x, t)) =CAx(t)

y(t) = CAx(t) =CAAx(t)+CAB(u(t)+d(x, t)),

where CB = 0 and CAB = cn−2 = K > 0. Thus the 2nd-SM

manifold S2 where y(t) = y(t) = 0 holds, is defined as

S2 = {x(t) : Cx(t) =CAx(t)0}.Define a new state vector w(t) ∈ Rn as

w(t) =[

y(t) y(t) z(t)]= T x(t), (5)

where z(t) ∈ Rn−2 defined as

z(t) = K[

x1(t) x2(t) · · · xn−2(t)]T

is an internal state vector, and the state transformation matrix

T ∈ Rn×n is given by

T =

⎡⎣ C

CAKIn−2 O(n−2)×2

⎤⎦ . (6)

Then the system (1) can be transformed as{w(t) = Aw(t)+ B(u(t)+d(x, t))y(t) = Cw(t)

, (7)

with A = TAT−1, B = T B, and C =CT−1.

Rewrite the transformation matrix T as

T = K[

T11 T12

In−2 O(n−2)×2

],

where T11 ∈ R2×(n−2) and T12 ∈ R2×2 are defined as

T11 =

[c0 c1 · · · cn−3

0 c0 · · · cn−4

], T12 =

[1 0

cn−3 1

].

Then the inverse matrix of T is given by

T−1 =1

K

[O(n−2)×2 In−2

T−112 −T−1

12 T11

],

where T−112 and −T−1

12 T11 are determined by the following two

matrices, respectively:[1 0

−cn−3 1

],

[ −c0 −c1 · · · −cn−3

cn−3c0 cn−3c1− c0 · · · cn−3cn−3− cn−4

].

Using the above descriptions of T and T−1, A, B, and C are

obtained as

A =

⎡⎣ 0 1 O1×(n−2)

−α −λ CzBz O(n−2)×1 Az

⎤⎦

B =

⎡⎣ 0

KO(n−2)×1

⎤⎦ , C =

[1 O1×(n−1)

],

314

where Az ∈ R(n−2)×(n−2), Bz ∈ R(n−2)×1, Cz ∈ R1×(n−2), α , and

λ are respectively defined as

Az =

[O(n−3)×1 In−3

−c0 −c1 −c2 · · · −cn−3

](8)

Bz =

[O(n−3)×1

1

]

Cz =[

b0 b1 · · · bn−3

]α = an−2 + c2

n−3− cn−4− cn−3an−1

λ = an−1− cn−3. (9)

Here bi (i = 0,1, · · · ,n−3) are given by

bi = −ai + ci−2− ci(−an−2 + cn−4)

+(cn−3ci− ci−1)(−an−1 + cn−3)

with c−1 = c−2 = 0.Therefore the system (1) can be represented by two subsys-

tems as

y(t) = −λ y(t)−αy(t)+Czz(t)+K(u(t)+d(x, t)) (10)

z(t) = Azz(t)+Bzy(t). (11)

A. Stability of Reduced-order SystemConsider the system (7) with the state vector w(t) defined

in (5), which is transformed from the system (1) by the state

transformation (6). On the 2nd-SM S2, y(t) = y(t) = 0 holds.

Thus the stability of the reduced-order system on the 2nd-SM

S2 is totally determined by the dynamics (11) of the internal

state z(t). Eigenvalues of the system matrix Az defined by (8)

are zeros z j ( j = 1,2, · · · ,n−2) of the system (1), which have

negative real parts when the system (1) is minimum phase.

Therefore the reduced-order system on the 2nd-SM S2 is

asymptotically stable, i.e., Az is a stable matrix if and only

if the system (1) is minimum phase.This condition for the stability of a 2nd-SM is similar to the

stability condition for a 1st-SM and can be extended to higher-

order SM cases. Thus the second-order and higher-order SM

control algorithms with the output as the switching function can

be implemented if and only if the system is minimum phase.

B. Asymptotic Convergence to 2nd-SMLemma 1: Consider the relative degree two system (1). It

is a necessary condition for the 2nd-SM S2 to be reached

asymptotically with the 1st-SMCL (4) that the sum of system

poles is less than the sum of system zeros, i.e.,

n

∑i=1

pi <n−2

∑j=1

z j. (12)

Proof: It has been proven in [9][10] that one of necessary

conditions for the system to converge asymptotically to the

2nd-SM S2 is that the coefficient λ in (10) is positive.It follows from descriptions of G(s) in (2) and (3) that sums

of poles and zeros are equal to −an−1 and −cn−3, respectively,

i.e.,

n

∑i=1

pi =−an−1,n−2

∑j=1

z j =−cn−3.

According to the equation (9), λ is determined by

λ = an−1− cn−3 =−n

∑i=1

pi +n−2

∑j=1

z j,

which is positive if and only if the inequality (12) holds, i.e.,

the sum of system poles is less than the sum of system zeros.

Therefore it is a necessary condition for the 2nd-SM S2 to be

reached asymptotically with the 1st-SMCL (4) that the sum of

system poles is less than the sum of system zeros.Remark 1: Lemma 1 indicates that the asymptotic 2nd-SM

control system does not require the open-loop system (1) to

be stable but the inequality (12) must be satisfied for the

asymptotic convergence to the 2nd-SM. In fact, even for a

stable system (1), the inequality (12) may not hold. On the

other hand, the inequality (12) may hold even if the system (1)

is unstable.Theorem 1: Consider the relative degree two system (1),

which is transformed to the system (7) by the state trans-

formation (6). The 2nd-SM S2 can be reached locally and

asymptotically with the 1st-SMCL (4) if

1) the sum of system poles is less than the sum of system

zeros of (1);

2) the system state vector w(t) =[

y(t) y(t) zT (t)]T

is

considered in a bounded compact subset Ω of Rn, i.e.,

w(t) ∈Ω⊂ Rn,∀t ∈ [0,+∞);3) Assumption 1 holds for w(t) ∈Ω⊂ Rn; and

4) the following inequality holds with a sufficiently large

gain k:

k > max{k1(w),k2(w)},∀w(t) ∈Ω,∀t ∈ [0,+∞) (13)

where k1(w) and k2(w) are respectively determined by

k1(w) =|αy(t)−Czz(t)|+Kd1

K

k2(w) =σ + |Hw(t)|+2λKd1 +Kd2

2λK.

Here σ is a positive constant and H ∈ R1×n is defined as

H =[

2λα−CzBz α −Cz(Az +2λ In−2)].

Proof: Rewrite the dynamical equation (10) with the 1st-

SMCL (4) as

y(t) =−λ y(t)−η(t)sgn(y), (14)

where the constant λ is positive as proved in Lemma 1, and

η(t) ∈ R is defined as

η(t) = (αy(t)−Czz(t)−K(u(t)+d(x, t)))sgn(y)

= (αy(t)−Czz(t)−Kd(x, t))sgn(y)+Kk. (15)

It follows from the inequality (13) that

k > max{k1(w),k2(w)} ≥ k1(w).

Thus η(t) is positive because of

η(t) ≥ −(|αy(t)−Czz(t)|+Kd1)+Kk

> −(|αy(t)−Czz(t)|+Kd1)+Kk1(w)

> 0.

315

Choose two positive constants ε and ν satisfying⎧⎪⎪⎨⎪⎪⎩

ε ≤ λ(

σ2η2(t) −ν

)ε < λ 2

η(t)ν < σ

2η2(t)

,∀w(t) ∈Ω,∀t ∈ [0,+∞). (16)

Then a Lyapunov function candidate is defined as

Vy(t) =1

2

[y(t) y(t)

][ 1η(t)

ελ

ελ ε

][y(t)y(t)

]+ |y(t)|

=y2(t)2η(t)

+ε2

y2(t)+ελ

y(t)y(t)+ |y(t)|, (17)

which is continuously differentiable when y(t) �= 0. As ε and1

η(t) are positive and εη(t) − ε2

λ 2 is also positive according to the

second inequality in (16), the 2×2 symmetric matrix

[ 1η(t)

ελ

ελ ε

]

is positive definite. Therefore the Lyapunov function candidate

Vy(t) is positive definite for all w(t)∈Ω and for all t ∈ [0,+∞).The derivative of Vy(t) for y(t) �= 0 along the trajectories of

(14) is

Vy(t) =y(t)y(t)

η(t)− y2(t)

2η2(t)η(t)+ εy(t)y(t)

+ελ(y(t)y(t)+ y2(t))+ sgn(y)y(t)

= − y2(t)2η2(t)

(2λη(t)+ η(t))

− ελ

η(t)|y(t)|+ ελ

y2(t). (18)

After some mathematical manipulations, it follows from the

inequality (13), i.e., k > max{k1(w),k2(w)} ≥ k2(w) that

2λη(t)+ η(t) = 2λKk

+(Hw(t)−2λKd(x, t)−Kd(x, t))sgn(y)

> 2λKk−|Hw(t)|−2λKd1−Kd2

≥ σ .

Substituting the above inequality for (18) and using the first

inequality in (16) yield

Vy(t) < − σ2η2(t)

y2(t)− ελ

η(t)|y(t)|+ ελ

y2(t)

= −(

σ2η2(t)

− ελ

)y2(t)− ε

λη(t)|y(t)|

≤ −ν y2(t)− ελ

η(t)|y(t)|≤ 0, (Vy(t) = 0 ⇔ y(t) = y(t) = 0)

which means that Vy(t) keeps decreasing until both y(t) and

y(t) converge to zero. Therefore it is proved that the 2nd-SM

S2 is reached locally and asymptotically with the 1st-SMCL

(4) for w(t) ∈Ω.

C. Robust Stability

In case the system (1) is minimum phase, the reduced-order

system (11) on the 2nd-SM is asymptotically stable. Thus a

symmetric positive definite matrix Pz ∈R(n−2)×(n−2) exists such

that

PzAz +ATz Pz =−In−2.

Define a Lyapunov function candidate

Vz(t) = zT (t)Pzz(t). (19)

Then its derivative along the trajectories of (11) is given by

Vz(t) = zT (t)(PzAz +ATz Pz)z(t)+2zT (t)PzBzy(t)

= −zT (t)z(t)+2zT (t)PzBzy(t)

= −||z(t)||2 +2zT (t)PzBzy(t)

≤ −||z(t)||2 +2||z(t)||× ||PzBz||× |y(t)|= −||z(t)||(||z(t)||−2||PzBz||× |y(t)|) .

Therefore the Lyapunov function candidate Vz(t) keeps decreas-

ing as long as ||z(t)||> 2||PzBz||× |y(t)|, which means that the

internal state z(t) converges into a vicinity determined by

||z(t)|| ≤ 2||PzBz||× |y(t)|. (20)

Thus z(t) is bounded for any bounded y(t). Furthermore, the

asymptotic stability of the system (1) with the 1st-SMCL (4)

is shown in the following theorem.

Theorem 2: Consider the relative degree two system (1),

which is transformed to the system (7) by the state transfor-

mation (6). The system (1) with the 1st-SMCL (4) is locally

and asymptotically stable if

1) all conditions given in Theorem 1 are satisfied;

2) the system (1) is minimum phase;

3) the following inequality holds with a sufficiently large

gain k:

k ≥max{k3(w),k2(w)},∀w(t) ∈Ω,∀t ∈ [0,+∞) (21)

where Ω and k2(w) are defined in Theorem 1, and k3(w)is defined as

k3(w) =|αy(t)−Czz(t)|+Kd1 +2|BT

z Pzz(t)|+μK

.

Here μ is a positive constant.

Proof: Define a Lyapunov function candidate as

V (t) =Vy(t)+ελ

Vz(t), (22)

where the positive constant ε satisfies the inequalities in (16),

λ determined by (9) is the positive coefficient in the dynamical

equation (14), and Vy(t) and Vz(t) are defined by (17) and (19),

respectively. Substituting (17) and (19) for (22) yields

V (t) =1

2

[y(t) y(t)

][ 1η(t)

ελ

ελ ε

][y(t)y(t)

]+ |y(t)|

+ελ

zT (t)Pzz(t).

316

Thus V (t) is positive definite and is continuous differentiable

for y(t) �= 0 if the system (1) is minimum phase and all

conditions given in Theorem 1 are satisfied.

It follows from the definition of η(t) in (15) and the

inequality (21) given in this theorem that

η(t) = (αy(t)−Czz(t)−Kd(x, t))sgn(y)+Kk

≥ −|αy(t)−Czz(t)|−Kd1 +Kk

≥ −|αy(t)−Czz(t)|−Kd1 +Kk3(w)

= 2|BTz Pzz(t)|+μ, ∀w(t) ∈Ω,∀t ∈ [0,+∞).

With this relation, it can be confirmed that the derivative of V (t)for y(t) �= 0 along the trajectories of (10) and (11) is negative

as shown below.

V (t) = Vy(t)+ελ

Vz(t)

≤ −ν y2(t)− ελ

η(t)|y(t)|

− ελ

zT (t)z(t)+ελ

2zT (t)PzBzy(t)

= −ν y2(t)− ελ

zT (t)z(t)

− ελ|y(t)|(η(t)−2zT (t)PzBzsgn(y))

≤ −ν y2(t)− ελ

zT (t)z(t)− εμλ|y(t)|

≤ 0, (V (t) = 0 ⇔ w(t) = 0).

Therefore the system (1) with the 1st-SMCL (4) is locally and

asymptotically stable for w(t) ∈Ω.

Remark 2: It is clear that k3(w) is larger than k1(w). Thus

the following holds:

max{k3(w),k2(w)} ≥max{k1(w),k2(w)}.According to the gain conditions (13) and (21) given in

Theorem 1 and Theorem 2, respectively, the gain determined in

Theorem 1 is larger than or equal to the one given in Theorem

2. Therefore the gain k which ensures the robust stability of

the system (1) with the 1st-SMCL (4) should be larger than

or equal to the one guaranteeing the asymptotic convergence

to the 2nd-SM S2. In other words, with a gain k chosen to

ensure the asymptotic convergence to the 2nd-SM, the robust

stability of the system (1) with the 1st-SMCL (4) may not be

guaranteed.

IV. SIMULATION RESULTS

In the simulation, a 3rd-order system⎧⎪⎪⎨⎪⎪⎩

x(t) =

⎡⎣ 0 1 0

0 0 1

7 8 −9

⎤⎦x(t)+

⎡⎣ 0

0

1

⎤⎦(u(t)+d(x, t))

y(t) =[

2 1 0]

x(t)

is considered as an example, which has a relative degree of

two. The disturbance d(x, t) is assumed to be determined by

d(x, t) =[

0.1 −0.2 0.4]

x(t)+2sin(10t).

The poles pi (i = 1,2,3) and the zero z1 are:

{p1, p2, p3}= {−9.7471,−0.5526,1.2997}, z1 =−2.

Thus this unstable system can be stabilized by the asymptotic

2nd-SM control algorithm as it is a minimum phase system

and the pole-zero condition (12) is satisfied with

3

∑i=1

pi =−9 <1

∑j=1

z j =−2.

The dynamics of the output y(t) and the internal state z(t) ∈R can be respectively described in the form of (10) and (11)

with λ =−∑3i=1 pi +∑1

j=1 z j = 7 > 0 as

y(t) = −7y(t)+22y(t)−37z(t)+u(t)+d(x, t)

z(t) = −2z(t)+ y(t),

The simulation results with the 1st-SMCL (4) are shown

in Figure 1, with the control gain k chosen as k = 50 and the

initial state x(0) as[

1 1 1]T

. It is clear that the 2nd-SM is

reached asymptotically, the state variable x(t) converges to the

origin asymptotically, and u(t) is a typical 1st-SM control input

switched between ±k. Thus the system used for the simulation

with the 1st-SMCL (4) is asymptotically stable.

0 1 2 3 4 5−6

−4

−2

0

2

4

6State Variables x(t)

t [second]

x(t)

x1(t)x2(t)x3(t)

0 1 2 3 4 5−50

0

50Control Input u(t)

t [second]

u(t)

u(t)

0 1 2 3 4 5−10

−5

0

5

Switching Function y(t) and its Derivative

t [second]

y(t),

dy(t)

/dt

y(t)dy(t)/dt

−1 0 1 2 3−10

−5

0

5

Phase Plane of y(t) and dy(t)/dt

dy(t)

/dt

y(t)

Fig. 1. Simulation Result with Asymptotic 2nd-SM Control

For comparison, the twisting 2nd-SM control [4]

u(t) =−r1sgn(y(t))− r2sgn(y(t)) (23)

and the sub-optimal 2nd-SM control [3]

u(t) =−r3sgn(y(t)− y∗(t)/2)+ r4sgn(y∗(t)), (24)

are also simulated to control the system (23) with the same

initial condition and with the same disturbance d(x, t). Here

y∗(t) is the value of y(t) detected at the closest time in the

past when y(t) was 0. The initial value of y∗(t) is 0. In the

simulation, the control parameters are chosen as

r1 = r3 = k = 50, r2 = r4 = 10 (25)

to ensure the finite time convergence to the 2nd-SM.

317

The simulation results with the twisting 2nd-SM control and

the sub-optimal 2nd-SM control are shown in Figures 2 and 3,

respectively. Obviously, both of the 2nd-SM control algorithms

using the derivative of the output y(t) ensure the finite time

convergence. There is, however, no remarkable improvement

for the convergence of the system state x(t) to the origin as

it is mainly determined by the dynamics of the reduced-order

system on the 2nd-SM, i.e., the system open-loop zeros.

0 1 2 3 4 5−5

0

5State Variables x(t)

t [second]

x(t)

x1(t)x2(t)x3(t)

0 1 2 3 4 5−60

−40

−20

0

20

40

60Control Input u(t)

t [second]

u(t)

u(t)

0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4Switching Function y(t) and its Derivative

t [second]

y(t),

dy(t)

/dt

y(t)dy(t)/dt

−1 0 1 2 3−10

−8

−6

−4

−2

0

2

4Phase Plane of y(t) and dy(t)/dt

dy(t)

/dt

y(t)

Fig. 2. Simulation Result with Twisting Control

V. CONCLUSION

In this paper, we have studied the asymptotic 2nd-SM control

of linear time-invariant continuous-time systems with matched

external disturbances and parameter uncertainties, and found

that the reduced-order system on the 2nd-SM is asymptotically

stable if the considered system with a relative degree of two is

minimum phase. It has been proved in the paper that the 2nd-

SM can be reached locally and asymptotically as long as the

sum of system poles is less than the sum of system zeros. Using

the two identified Lyapunov functions, it has proven that the

asymptotic 2nd-SM control system converges to the 2nd-SM

and to the origin, locally and asymptotically. The simulation

results have confirmed the asymptotic convergence to the 2nd-

SM and the robust stability of the asymptotic 2nd-SM control

system.

ACKNOWLEDGMENT

This work is supported in part by the Canada Research Chair

program.

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