A Comprehensive Analysis of Chattering in Second Order Sliding Mode Control Systems

A Comprehensive Analysis of Chattering in
Second Order Sliding Mode Control Systems
Igor Boiko
1
, Leonid Fridman
2
, Alessandro Pisano
3
, and Elio Usai
3
1
University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, Canada

2
Department of Control, Engineering Faculty, National Autonomous University of
Mexico (UNAM)

3
Department of Electrical and Electronic Engineering (DIEE), University of
Cagliari, Piazza D’Armi, 09123 Cagliari (Italy)
{pisano,eusai}@diee.unica.it
1 Introduction
Chattering is the most problematic issue in sliding mode control applications
[30], [31], [14], [33], [36]. Among the well known approaches based on smooth
approximations of the discontinuities [11],[29] and asymptotic state observers
[10, 33], the use of the high order sliding mode control approach can attenuate
the chattering phenomenon significantly [15],[21], [2],[4], [5],[3],[22],[27].
There are different approaches to chattering analysis which take into account
different causes of it: the presence of fast actuators and sensors [32], [17], [18],
[19], [34], [6], time delays and/or hysteresis [35], [32], [33], quantization effects
(see for example [23]).

The purpose of the present chapter is to present a systematic approach to the
chattering analysis in control systems with second-order sliding mode controllers
(2-SMC) caused by the presence of fast actuators. We shall follow both the time-
domain approach, based on the state-space representation, and the frequency-
domain approach.
The estimation of the oscillation magnitude in conventional (i.e., first-order)
SMC systems with fast actuators and sensors was developed in [32], [18], [19]
via the combined use of the singularly-perturbed relay control systems theory
and Lyapunov techniques. However, Lyapunov theory is not readily applicable to
analyze 2-SMC systems, for which new decomposition techniques are demanded.
The Poincar´e maps were successfully used in [24], [13] to study the periodic
oscillations in relay control systems. In [17] a decomposition of Poincar´emaps
was proposed to analyze chattering in systems with first order sliding modes,
which led to Pontryagin-Rodygin [25] like averaging theorems providing sufficient
conditions for the existence and stability of fast periodic motions.
The describing function (DF) technique [1] offers finding approximate values
of the frequency and the amplitude of periodic motions in systems with linear
G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 23–49, 2008.
springerlink.com
c
 Springer-Verlag Berlin Heidelberg 2008
24 I. Boiko et al.
plants driven by the sliding mode controllers [37],[26]. The Tsypkin locus [35]
provides an exact solution of the periodic problem, including finding exact values
of the amplitude and the frequency of the steady-state oscillation. The above-
mentioned frequency-domain methods were developed to analyze relay feedback
systems and cannot be used directly for the analysis of 2-SMC systems. In [7],[8]
the DF method was adapted to analysis of the Twisting and the Super-twisting
2-SMC algorithms [21]. In [9], a DF based method of parameter adjustment of
the generalized sub-optimal 2-SMC algorithm [3],[5] was proposed to ensure the

desired frequency and amplitude of the periodic chattering trajectories.
In the present Chapter, a systematic approach to analysis of chattering in
2-SMC systems is developed. The presence of parasitic dynamics is considered
to be the main cause of chattering, and the corresponding effects are analyzed
by means of a few techniques. The treatment is developed by considering the
”Generalized Sub-optimal” (G-SO) algorithm [3],[5]. All the main results can be
easily generalized to the Twisting algorithm [21] with minor modifications in the
proofs.
For a class of nonlinear uncertain systems with nonlinear fast actuators:
1. It is proved that the approximability domain [38] of the 2-SMC G-SO
algorithms depends on the actuator time constant μ as O(μ
2
)andO(μ)forthe
sliding 2-SMC variable and its derivative respectively.
2. Sufficient conditions of the existence of asymptotically orbitally stable pe-
riodic solution are obtained in terms of Poincar´emaps.
For linear, possibly linearized, dynamics driven by 2-SMC G-SO algorithms,
frequency-domain methods of analysis of the periodic solutions are developed,
and, in particular:
3. The describing function method is adapted to perform an approximate
analysis of the periodic motions.
4. The Tsypkin’s method is modified for the analysis of the systems driven
by 2-SMC G-SO algorithms. This modification allows for finding exact values of
the parameters of periodic motions, without requiring for the actuator dynamics
to be fast.
The chapter is organized as follows: in Section 2, a class of nonlinear systems
with nonlinear fast actuators is introduced. In Section 3 we show that the 2-SMC
G-SO algorithm with suitably chosen parameters steers the system trajectories
in finite time towards an invariant vicinity of the second-order sliding set. We
also estimate the amplitude of chattering oscillations as a function of the actua-

tor time constant. In Section 4 sufficient conditions of the existence and stability
of fast periodic motions in a vicinity of the second-order sliding set are derived
via the Poincar´e map approach. In Section 5, frequency-domain approaches to
chattering analysis are developed. The describing function method is adapted in
subsection 5.1 to carry out analysis of periodic motions in systems with linear
plants. In Subsection 5.2, the Tsypkin’s method is modified to obtain the pa-
rameters of the periodic motion exactly. Examples illustrating the application
of the proposed methodologies are spread over the chapter. The proofs of the
Theorems are given in the Appendix.
A Comprehensive Analysis of Chattering 25
2 2-SMC Systems with Dynamic Actuators
We shall consider a nonlinear single-input system:
˙x = a(x, z
1
)(1)
with the state vector x =[x
1
,x
2
, ,x
n
] ∈ X ⊂ R
n
and the scalar “virtual”
control input z
1
∈ Z
1
⊂ R. The plant input z
1

is modifiable via the dynamic
fast actuator
μ ˙z = h(z, u), (2)
where z =[z
1
,z
2
, ,z
m
] ∈ Z ⊂ R
m
is the actuator state vector, u ∈ U ⊂ R is
the actuator input and μ ∈ R
+
is a small positive parameter. Let a : X ×Z
1
→
R
n
and h : Z × U → R
m
be vector-fields satisfying proper restrictions on their
growth and smoothness that will be specified later.
Let the control task for system (1)-(2) be the finite-time vanishing of the
scalar output variable
s
1
(x):X → S
1
⊂ R (3)

which defines the sliding manifold s
1
(x) = 0 assigning desired dynamic properties
(e.g. stability) to the constrained sliding-mode dynamics. Define
s
2
(x, z
1
)=
∂s
1
(x)
∂x
a(x, z
1
):X ×Z
1
→ S
2
(4)
and assume that the following conditions hold ∀(x, z
1
) ∈ X ×Z
1
:
∂
∂z
1
s
2

(x, z
1
)=0 (5)
∂
∂z
1

∂
∂x
[s
2
(x, z
1
)a(x, z
1
)]

=0 (6)
Laborious but straightforward computations show that conditions (5)-(6) hold
if and only if
rank J(x, z
1
)=2 ∀(x, z
1
) ∈ X ×Z
1
(7)
with the matrix J defined as follows
J(x, z
1

)=

∂s
1
∂x
1
∂s
1
∂x
2

∂s
1
∂x
n
0
∂s
2
∂x
1
∂s
2
∂x
2

∂s
2
∂x
n
∂s

2
∂z
1

(8)
By virtue of the Inverse Function Theorem, one can explicitly define a vector
w ∈ W ⊂ R
n−2
and a diffeomorfic state coordinate change x = Φ(s, w):S ×
W → X ,withs =[s
1
,s
2
] ∈ S, which is one to one at any point where condition
(7) holds [12].
Assume that vector w can be selected in such a way that its dynamics does
not depend on the plant input variable z
1
, i.e., let the transformed system (1),(2)
dynamics in the (w, s) coordinates be defined as follows
26 I. Boiko et al.
˙w = g(w, s), (9)
˙s
1
= s
2
, ˙s
2
= f (w, s, z
1

), (10)
μ ˙z = h(z,u), (11)
where g : W × S → R
n−2
, f : W × S ×Z
1
→ R, h : Z × U → R
m
are smooth
functions of their arguments such that f ∈ C
2
[
¯
W ×
¯
S ×
¯
Z
1
],g∈ C
2
[
¯
W ×
¯
S]and
h ∈ C
2
[
¯

Z ×
¯
U], where upper bar means the closure of domain. This means that
the “sliding variable” s
1
has a well–defined relative degree r = 2 with respect to
the plant input variable z
1
over the whole domain of analysis.
We consider here the case when the actuator output z
1
has the full relative
degree m, equal to the order of the actuator dynamics, with respect to the
discontinuous control u.
Remark 3. The special form (9) for the internal dynamics can be always
achieved if the original dynamics (1) has affine dependence on z
1
[20]. We are
considering in this paper the subclass of non-affine systems (1) for which such a
special choice of vector w can be found.
3 Generalized Suboptimal Algorithm: Convergence
Conditions
Consider system (9)-(11) driven by the “Generalized Suboptimal” 2-SMC algo-
rithm [5]
u = −Usign(s
1
− βs
1Mi
) (12)
where U and β are the constant controller parameters and s

1Mi
is the latest
“singular point” of s
1
, i.e., the value of s
1
at the most recent time instant t
Mi
(i =1, 2, ) such that ˙s
1
(t
Mi
) = 0. Our analysis is semi-global in the sense
that the initial conditions w(0), s(0), z(0)areassumedtobelongtotheknown,
arbitrarily large, compact domains W
0
, S
0
,andZ
0
, respectively. The solutions
of the system (9)-(12) are understood in the Filippov sense [16].
Remark 4. Since the relative degree between the sliding output s
1
and the
discontinuous control u is m + 2, only sliding modes of order m + 2, occurring
onto the following sliding set
1
[21], can take place.
s

1
= 0 (13)
˙s
1
= s
2
= 0 (14)
¨s
1
(w, 0,z
1
)=f(w, 0,z
1
) = 0 (15)
(16)
s
(m)
1
(w, 0,z) = 0 (17)
s
(m+1)
1
(w, 0,z) = 0 (18)
1
The successive total time derivatives of s
1
must be evaluated along the trajectories
of system (9)-(11) in the usual way.
A Comprehensive Analysis of Chattering 27
The internal dynamics in the (m + 2)-th order sliding-mode is described by

equation
2
˙w = g(w, 0). (19)
Suppose that for all w ∈ W and z ∈ Z there exists a unique isolated value of
u = u
0
(z,w) as a solution of equation
s
(m+2)
1
(w, 0,z,u) = 0 (20)
which maintains the system trajectories onto the (m+2)-th order sliding domain
(13)-(18). Note that the actuator input u appears explicitly as an argument of
(20), but not of (13)-(18), according to the fact that the relative degree of s
1
with respect to u is m +2.
Then, the system equilibrium point can be computed as the unique solution
w
0
, z
0
, u
0
(w
0
,z
0
) of the system of equations (15)-(20). The knowledge of the
equilibrium point will be used later in the Chapter to define a local linearization
for the system (9)-(11).

Assumption 1. The internal dynamics (9) and the actuator dynamics (11)
meet the following input-to-state stability properties for some positive constants
ξ
1
,ξ
2
[28]
w(t)≤w(0) + ξ
1
sup
0≤τ ≤t
s(τ) (21)
z(t)≤z(0)+ ξ
2
sup
0≤τ≤t
|u(τ)| (22)
Assumption 2. There exist positive constants H
0
, H
1
, H
2
G
m
, G
M
such that
function f is bounded as follows:
z

1
≤ 0: −
˜
H(s, w)+G
M
z
1
≤ f(w, s
1
,s
2
,z
1
) ≤
˜
H(s, w)+G
m
z
1
z
1
> 0: −
˜
H(s, w)+G
m
z
1
≤ f(w, s
1
,s

2
,z
1
) ≤
˜
H(s, w)+G
M
z
1
(23)
˜
H(s, w)=H
0
+ H
1
s + H
2
w (24)
Assumption 3. Consider the actuator dynamics (11) with the constant input
u(t)=
U, t ≥ t
0
.Then,∀ε ∈ (0, 1) there is γ ∈ [γ
m
,γ
M
] and N(ε) > 0 such
that
|z
1

− γU|≤εγU ∀t ≥ t
0
+ N(ε)μ (25)
Assumption 1 prescribes a linear growth of w(t) and z(t) w.r.t.s(t) and
|u|, respectively. Assumption 2 guarantees that the virtual plant control input
z
1
, with large enough magnitude, can set the sign of f (see Fig. 1). The knowl-
edge of constants ξ
1
,ξ
2
, H
0
, , G
M
is mainly a technical requirement. With
sufficiently large U,andβ ∈ [0.5, 1) sufficiently close to 1, stability can be in-
sured regardless of ξ
1
, , G
M
. Assumption 3 requires a “non-integrating” stable
2
In this case the (m+2) th order sliding dynamics do not depend on the control, i.e.
they do not depend on the definition of solutions in (m+2)-th order sliding mode.
28 I. Boiko et al.
actuator dynamics whose settling time in the step response is O(μ). Note that γ
and N are considered uncertain. Assumption 3 is always satisfied in the special
case of a linear asymptotically stable actuator dynamics.

Assume, only temporarily, that there exists a certain constant H such that
|
˜
H(s, w)|≤H, then conditions (23)-(24) reduce to the following:
z
1
≤ 0 ⇒−H + G
M
z
1
≤ f(w, s
1
,s
2
,z
1
) ≤ H + G
m
z
1
z
1
> 0 ⇒−H + G
m
z
1
≤ f (w, s
1
,s
2

,z
1
) ≤ H + G
M
z
1
(26)
which can be represented graphically as in Fig. 1. The dashed lines limit the
“admissible region” for the uncertain function f .

H
-H
H + G
M
z
1
−H+G
M
z
1
f
z
1
H + G
m
z
1
−H+G
m
z

1
Fig. 1. Bounding curves for the function f
Consider the following tuning rules:
U =
ηH
(1 −ε)γ
m
G
m
,β∈

1 −
q(η − 1)
η +1+Δ
, 1

, (27)
Δ =
γ
M
G
M
(1 + ε)
γ
m
G
m
(1 −ε)
,η>1,ε∈ (0, 1),q∈ (0, 1) (28)
In the next Theorem it is proven that with sufficiently large H and suf-

ficiently small μ, control (12), (27)-(28) actually guarantees that condition
|
˜
H(·)|≤H holds, and furthermore, it is also shown that a positively invari-
ant μ-neighborhood of the second order sliding set s
1
= s
2
= 0, defined as
O
μ
≡

(s
1
,s
2
):|s
1
|≤ρ
1
μ
2
, |s
2
|≤ρ
2
μ

(29)

attracts in finite time the system trajectories.
Theorem 1. Consider system (9)-(11), satisfying Assumptions 1-3 and driven
by the Generalized Sub-Optimal controller (12), (27)-(28). Then, if H is suffi-
ciently large and μ is sufficiently small, the closed loop system trajectories enter
in finite time the invariant domain O
μ
defined by (29), where ρ
1
and ρ
2
are
proper constants independent of μ.
A Comprehensive Analysis of Chattering 29
Proof. See the appendix.
Simulation example. Consider system
˙w = −sin(w)+s
1
+ s
2
,
˙s
1
= s
2
, ˙s
2
=
s
2
1+s

2
2
+ s
1
+ w +[2+cos(z
1
+ s
2
)]z
1
,
μ ˙z
1
= z
2
,μ˙z
2
= −z
1
− z
2
+ u.
(30)
The initial conditions and the controller parameters are s
1
(0) = 20, s
2
(0) = 5,
w(0) = 5, z
1

(0) = z
2
(0) = 0, U = 80, β =0.8. Figure 2 shows the time evolution
of s
1
and s
2
when μ =0.001. The amplitude of chattering was evaluated as
the maximum of |s
1
| and |s
2
| in the steady state, yielding |s
1
|≤7E-4 and
|s
2
|≤0.4. We performed a second test using μ =0.01. The accuracy of s
1
changed to 7E − 2, and the accuracy of s
2
changed to 4, in perfect accordance
with (29). Simulations show high-frequency periodic motions for s
1
, s
2
and w.
Below those “chattering” trajectories are investigated in further detail.
0 2 4 6
−30

−20
−10
0
10
20
30
Time [sec]
s
1
(t)
s
2
(t)
Fig. 2. The steady-state evolution of s
1
and s
2
4Poincar´e Map Analysis
We are going to derive conditions for the existence of stable periodic motions,
in the vicinity O
μ
of the second order sliding set, in terms of the properties of
some associated Poincar´e maps. We will also give a constructive procedure to
compute the parameters of such periodic chattering motions. Introducing the
new variables y
1
= μ
−2
s
1

, y
2
= μ
−1
s
2
, rewrite system (9)-(11) in the form
˙w = g(w, μ
2
y
1
,μy
2
), (31)
μ ˙y
1
= y
2
,μ˙y
2
= f(w, μ
2
y
1
,μy
2
,z
1
), (32)
μ ˙z = h(z,u(μ

2
y
1
)) (33)
Note that the Generalized Sub-Optimal algorithm (12) is endowed by the
homogeneity property u(μ
2
y
1
)=u(y
1
). Consider the Original System in the
Fast Time (OSFT)
30 I. Boiko et al.
dw/dτ = μg(w, μ
2
y
1
,μy
2
), (34)
dy
1
/dτ = y
2
,dy
2
/dτ = f(w, μ
2
y

1
,μy
2
,z
1
), (35)
dz/dτ = h(z,u(y
1
)) (36)
and the Fast Subsystem (FS) with the “frozen” slow dynamics (w ∈
W is con-
sidered here as a fixed parameter):
d¯y
1
/dτ =¯y
2
,d¯y
2
/dτ = f (w, 0, 0, ¯z
1
), (37)
d¯z/dτ = h(¯z,u(¯y
1
)) (38)
Consider the solution of system (37)-(38) with initial condition
¯y
+
1
(0,w)=¯y
0

1
, ¯y
+
2
(0,w)=0
¯z
+
(0,w)=¯z
0
,f(w, ¯y
0
1
, 0, ¯z
0
1
) < 0
(39)
such that f (w, ¯y
0
1
, 0, ¯z
0
1
) < 0 for all w ∈ W. Suppose that for all w ∈ W there
exists the smallest positive root of equation τ = T (w)forwhich¯y
+
2
(T (w),w)=0,
f(w, ¯y
+

1
(T (w),w), 0, ¯z
+
1
(T (w),w)) < 0. Now we can define for all w ∈ W the
Poincar´emap:
(¯y
0
1
, ¯z
0
) → Ξ(w, ¯y
0
1
, ¯z
0
)=

¯y
1
(T (w),w)
¯z(T (w),w)

(40)
of the of the domain f(w, y
1
, 0,z
1
) < 0onthesurfacey
2

= 0 into itself, generated
by system FS (37)-(38) (the details of this mapping are described in Appendix B).
4.1 Sufficient Conditions for the Periodic Solution Existence
Let us suppose that the FS (37)-(38) has a nondegenerated isolated periodic
solution and the following conditions hold:
Condition 1. ∀w ∈
W the FS (37)-(38) has an isolated T
0
(w)-periodic solution
(¯y
10
(τ,w), ¯y
20
(τ,w), ¯z
0
(τ,w)). (41)
Condition 2. ∀w ∈
W the Poincar´emapΞ(w, y
0
1
, z
0
) has an isolated fixed point
(¯y
∗
1
(w),z
∗
(w)) corresponding to the periodic solution (41).
Condition 3. ∀w ∈

W the eigenvalues λ
i
(w)(i =1, ,m+1)ofthematrix
∂Ξ
∂(y
1
,z)
(w, ¯y
∗
1
(w), ¯z
∗
(w)) (42)
are such that |λ
i
(w)| =1.
Condition 4. The averaged systems
dw
dt
= p(w)=
1
T
0
(w)

T
0
(w)
0
g(w, ¯y

10
(τ,w), ¯y
20
(τ,w), ¯z
0
(τ,w))dτ
(43)
has an isolated, nondegenerated, equilibrium point w
0
such that
A Comprehensive Analysis of Chattering 31
p(w
0
)=0,det




dp
dw
(w
0
)




=0. (44)
The following Theorem is demonstrated:
Theorem 2. Under conditions 1 −4, system (31)-(33) has an isolated periodic

solution near the cycle
(w
0
, ¯y
10
(t/μ, w
0
), ¯y
20
(t/μ, w
0
), ¯z
0
(t/μ, w
0
)) (45)
with period μ(T
0
(w
0
)+O(μ)).
Proof. See the Appendix.
4.2 Sufficient Conditions of Stability of Periodic Solution
Let ν
j
(w
0
)(j =1, , n − 2) be the eigenvalues of the matrix
dp
dw

(w
0
). Suppose
that the periodic solution of FS (37)-(38) is exponentially orbitally stable and
the equilibrium point of the averaged equations is exponentially stable, i.e. :
Condition 5. |λ
i
(w)| < 1, (i =1, ,m+1).
Condition 6. ν
j
(w
0
) are real negative, i.e., ν
j
(w
0
) < 0, ∀j =1, , n − 2.
Theorem 3. Under conditions 1 − 6, the periodic solution (45) of the system
(31)-(33) is orbitally asymptotically stable.
Proof. See the Appendix.
4.3 Example of Poincar´e Map Analysis
Consider the following linear dynamics
˙w = −w + y
1
˙y
1
= y
2
, ˙y
2

= z
μ ˙z = −z + u, u = −sign

y
1
−
1
2
y
1M

(46)
We have shown that analysis of periodic solutions can be performed by referring
to the decomposition into fast and slow subsystem dynamics. The FS dynamics
dy
1
/dτ = y
2
,dy
2
/dτ = z, dz/dτ = −z + u (47)
generates the following Poincar´emapΞ
+
(y
1
,z)=(Ξ
+
1
(y
1

,z),Ξ
+
2
(y
1
,z)) of the
domain z<0onthesurfacey
2
= 0 into the domain z>0ofthesamesurface
(see the Appendix for the detailed derivation):
32 I. Boiko et al.
Ξ
+
1
(y
1
,z)=
1
2
y
1
+

(z + 1)(1 − e
−T
+
sw
) − T
+
sw


T
+
p
+
+

(z +1)e
−T
+
sw
− 2

(T
+
p
+ e
−T
+
p
− 1) +
1
2
T
+
2
p
Ξ
+
2

(y
1
,z)=1+((z +1)e
−T
+
sw
− 2)e
−T
+
p
(48)
where T
+
sw
= T
+
sw
(y
1
,z)andT
+
p
= T
+
p
(y
1
,z) are the smallest positive roots of
the following equations:
1

2
y
1
+(z +1)(e
−T
+
sw
+ T
+
sw
− 1) −
1
2
T
+
sw
2
= 0 (49)
(z + 1)(1 −e
−T
+
sw
)+

(z +1)e
−T
+
sw
− 2


(1 −e
−T
+
p
)+T
+
p
− T
+
sw
= 0 (50)
Taking into account the symmetry of dynamics (47) with respect to origin
(0, 0, 0) we can skip the computation of the map Ξ
−
(y
1
,z) and rewrite condition
for the periodicity of system (47) trajectory in the form:
Ξ
+
1
(y
0
1p
,z
0
p
)=−y
0
1p

,Ξ
+
2
(y
0
1p
,z
0
p
)=−z
0
p
(51)
The fixed points are
y
0
1p
≈ 3.95,z
0
p
≈−0.96, (52)
and the switching times are
T
+
sw
≈ 2.01, T
+
p
≈ 3.93, T
+

0
= T
+
sw
+ T
+
p
≈ 5.94. (53)
The Frechet derivatives entering the Jacobian matrix are given by
J =
∂Ξ
∂(y
1
,z)
=

∂Ξ
1
∂y
1
|
(y
0
1p
,z
0
p
)
∂Ξ
1

∂z
|
(y
0
1p
,z
0
p
)
∂Ξ
2
∂y
1
|
(y
0
1p
,z
0
p
)
∂Ξ
2
∂z
|
(y
0
1p
,z
0

p
)

=

−0.4894 1.5273
0.0095 −0.0147

(54)
The eigenvalues of matrix J are eig(J)=[−0.5182, 0.0141], i.e. they are both
lying within the unit circle of the complex plane, which implies that the periodic
solution of the fast subsystem (47) is orbitally asymptotically stable.
The averaged equations for the internal dynamics has the form ˙w = −w.Now
from Theorems 1-3 it follows that: i) system (46) has an orbitally asymptotically
stable periodic solution lying in the O
μ
boundary layer (29) of the second-order
sliding set y
1
= y
2
= 0. ii) in the steady state the internal dynamics w variable
features a O(μ) deviation from the equilibrium point w = 0 of the averaged
solution. It is also expected from Theorem 2 that the period of oscillation is
O(μ). The period and amplitude of the periodic solutions of (46) can be easily
inferred from (52) and (53) via proper μ-dependent scaling.
The above results have been checked by means of computer simulations. The
initial conditions are w(0) = y
1
(0) = y

2
(0) = z(0) = 1. The value μ =0.1was
used in the first test. It is expected, on the basis of the previous considerations,
that y
1
exhibits a steady oscillation with amplitude μ
2
y
0
1p
≈ 0.0395 and period
2μT
+
p
≈ 1.18s. The plots in Fig. 3 highlight the convergence to the periodic
solution starting from initial conditions outside from the attracting O
μ
domain.
A Comprehensive Analysis of Chattering 33
0 2 4 6 8 10
−0.5
0
0.5
1
1.5
2
Time [sec]
The y
1
time evolution

−0.5 0 0.5 1 1.5 2
−2
−1
0
1
2
Time [sec]
Trajectory in the y
1
− y
2
plane
Fig. 3. Transient trajectories with μ =0.1. Left: y
1
time evolution. Right: y
1
-y
2
trajectory.
5 Frequency Domain Analysis
The Poincar´e map based analysis provides an exact but complicated approach.
Therefore, the use and adaptation of frequency methods for the chattering analy-
sis in control systems with the fast actuators driven by 2-SMC G-SO algorithms
seems expedient. However, this approach applies to linear dynamics. For this
reason in this section we assume that the plant plus actuator dynamics are
either linear or linearized in the conventional sense in a small vicinity of the
domain (29).
Subsection 5.1 discusses the problem of local linearization, and Subsection 5.1
states formally the analysis problem. Subsection 5.3 is devoted to the describing
function approach to the analysis of the G-SO algorithm in the closed loop. This

approach is approximate and requires the linearized system (actuator and plant)
being a low-pass filter. This assumption is equivalent to the hypothesis of the
actuator being fast. Thus, subsection 5.4 presents the modified Tsypkin locus
[35] method, which does not require the filtering hypothesis, and, furthermore,
provides exact values of the frequency and the amplitude of the periodic mo-
tion as a solution of an algebraic equation and the use of an explicit formula,
respectively.
5.1 Local Linearization of System (9)-(11)
We have shown that controller (12) can provide for the appearance of a stable
sliding mode of order m + 2, and that the system (9)-(12), with full state vector
ξ =[w
T
,s
T
,z
T
], has a fixed equilibrium point ξ
0
=(w
0
, 0,z
0
)(seeRemark4).
The constant equilibrium value for the actuator input is u = u
0
(w
0
,z
0
), then it

is reasonable to linearize the system (9)-(12) in the small neighborhood of the
point ξ
0
by considering the constant control value u
0
in the terms depending on
it. Simple computations yield the following linearized dynamics
˙
ξ = Aξ + Bu,
s
1
= Cξ
(55)
34 I. Boiko et al.
A =
⎡
⎢
⎢
⎣
∂g
∂s
1
∂g
∂s
2
∂g
∂w
0
0100
∂f

∂s
1
∂f
∂s
2
∂f
∂w
∂f
∂z
000
∂h
∂z
⎤
⎥
⎥
⎦
, B =
⎡
⎢
⎢
⎢
⎣
0
.
.
.
0
∂h
∂u
⎤

⎥
⎥
⎥
⎦
, C =[0, ,0, 1, 0, ,0] (56)
where the nonzero element of C is represented by its (n−1)-th entry according to
the state-output relationship s
1
= Cξ. The characteristic matrix A and control
gain vector B contain some partial derivatives of functions f , g and h which
must be evaluated in the considered equilibrium point ξ
0
and equilibrium control
value u
0
.
Because of the system trajectories will converge to an O(μ) vicinity of the
equilibrium, the accuracy of the linear approximation depends on the μ param-
eter, the smaller μ the higher the accuracy. The transfer function and harmonic
response of the linearized system (55)-(56), which is asymptotically stable by
construction, can be computed straightforwardly.
5.2 Problem Statement
Consider the transfer function W (s) of a linear stable dynamics and the asso-
ciated harmonic response W (jω). The closed-loop system with the generalized
sub-optimal algorithm is presented in the next Fig. 4-left.
W(j
ω
)
G-SO
Algorithm

s
1
u
s
1
−
u
U
−
U
p
M
s
1

β
p
M
s
1

β
−

Fig. 4. Left: Closed-loop system with the G-SO algorithm. Right: the control charac-
teristic in steady state.
The term s
1Mi
appearing in the switching function of control (12) changes
step-wise at the time instants t

Mi
(i=1,2,. ) at which ˙s
1
(t
Mi
)=0.Duringthe
periodic motion s
1Mi
is an alternating (ringing) series of positive and negative
values, i.e. s
p
1M
, −s
p
1M
, s
p
1M
, −s
p
1M
(here the label “p” stands for periodic). The
control sign change would occur at the time when the plant output is equal
to ±βs
p
1M
. Therefore, in the periodic motion, the control function (12) can be
represented by the hysteretic relay nonlinearity in Fig. 4-right. This representa-
tion opens the way for the use of the frequency-domain methods developed for
analysis of relay feedback systems [1, 35]. The main difference from the conven-

tional application of the existing methods is that the hysteresis value is unknown
a-priori.
A Comprehensive Analysis of Chattering 35
5.3 Describing-Function (DF) Analysis
DF analysis is a simple approach which can provide in most cases a sufficiently
accurate estimate of the frequency and the amplitude of a possible periodic
motion. The main difference between the considered case and a conventional
relay system is that the hysteresis value βs
p
1M
is actually unknown.Tosolvethis
problem we can consider that during a periodic motion the extreme values of the
output coincide, in magnitude, with its amplitude. Therefore, s
p
1M
is actually the
unknown amplitude of the periodic motion. The DF of the relay with a negative
hysteresis is given as follows [1]:
q(A
y
)=
4c
πA
y

1 −
b
2
A
2

y
+ j
4bc
πA
2
y
(57)
where b = βy
p
M
is a half of the hysteresis, c = U is the relay amplitude and
A
y
= y
p
M
is the amplitude of the harmonic input to the relay. Then we can exploit
the given relationships between the hysteresis parameters and the oscillation
parameters in order to obtain the following expression for the DF of the G-SO
algorithm:
q(A
y
)=
4U
πA
y


1 − β
2

+ jβ

(58)
A periodic solution can be found from the harmonic balance equation
W (jω)=−
1
q(A
y
)
[1], where the negative reciprocal of the DF (58) is as follows
−
1
q
= −
πA
y
4U


1 −β
2
− jβ

(59)
As usual, the periodic solutions correspond to the points of intersection of the
W (jω)and−1/q(A
y
) loci, the latter being a straight line backing out of the
origin with a slope that depends only on parameter β, as depicted in Fig. 5.
Therefore, a periodic motion may occur if at some frequency ω =

ω the phase
characteristic of the actuator-plant transfer function W is equal to −180
0
−
arcsin(β). If such a requirement is fulfilled, so that intersection between the two

Re
Im
W(j
ω
)
arcsin β
ω
A
O
U
A
y
()
y
Aq
1
−
Fig. 5. DF-analysis
36 I. Boiko et al.
plots occurs, then the frequency and the amplitude of the periodic solution can
be derived from the ”cross-over” frequency
ω and from the magnitude of vector
OA in Fig. 5, respectively. An intersection point will certainly exists if the overall
relative degree of the combined actuator-plant degree is three or higher.

5.4 Exact Frequency-Domain Analysis Via Modified Tsypkin Locus
The DF analysis given above provides a simple and systematic, but approximate,
evaluation of the magnitude and frequency of the periodic motions in linear
systems driven by the G-SO algorithm (12). An exact solution, which does
not require the actuator to be fast, can be obtained via application of the
Tsypkin’s method [35].
The Tsypkin locus approach involves computing the following complex func-
tion Λ(ω), called the Tsypkin locus:
Λ(ω)=
4c
π

∞
k=1
Re{W [(2k −1)ω]} + j
4c
π

∞
k=1
1
2k−1
·Im{W [(2k − 1)ω]}
(60)
where c is the magnitude of the relay output. The frequency ω of the periodic
solution can be found by solving the equation Im Λ(
ω)=−b,where2b is the
hysteresis value. Unfortunately, the hysteresis b is a function of the unknown
amplitude of the oscillations, and the explicit formula for the “true” amplitude
of the oscillations does not exist.

The problem of the exact frequency-domain analysis can be conveniently
solved by the technique presented below. Introduce the complex function Φ(ω)
as follows:
Φ(ω)=−

[A
y
(ω)]
2
− y
2

π
ω
,ω

+ jy

π
ω
,ω

(61)
where y

π
ω
,ω

is the value of the system output at the time instant when the

relay switches from −V
M
to V
M
(π/ω is half a period in the periodic motion
and t = 0 is assumed, without loss of generality, to be the time of the relay
switch from V
M
to −V
M
)andA
y
(ω) is the amplitude of the plant output in the
assumed periodic motion of frequency ω:
A
y
=max
t∈[0,T ]
|y(t, ω)| (62)
y(t, ω) can be computed by means of its Fourier series:
y(t, ω)=
4c
π

∞
k=1
1
k
sin(
1

2
πk)sin[kωt + ϕ(kω)]L(ωk)=
=
4c
π

∞
k=1
(−1)
k+1
2k−1
sin[(2k −1)ωt + ϕ((2k −1)ω)] ·L((2k − 1)ω)
(63)
where ϕ(kω)=argW (jkω), L(kω)=|W (jkω)| are the phase and magnitude of
W (jω) at the frequency kω, respectively.
The frequency-dependent variable A
y
(ω) can be computed by using (62) and
(63), and y(
π
ω
,ω) as the imaginary part of (60) or via using the Fourier series
(63). As a result, function Φ(ω) has the same imaginary part as the Tsypkin
A Comprehensive Analysis of Chattering 37
locus, and the magnitude of function Φ(ω) at the intersection point represents
the amplitude of the periodic solution.
Having computed the function Φ(ω), we can carry out the graphical analysis
of possible periodic motions the same way as it was done above via the DF
technique, simply replacing the Nyquist plot of W (jω) with the function Φ(ω)
given by (61). Let us call the function Φ(ω) given by formula (61) the modified

Tsypkin locus.
6 Examples of Frequency-Domain Analysis
6.1 Linear Case
Consider W (s) being the cascade connection of the second-order linear plant
W
p
(s) and the first-order dynamic actuator W
a
(s)
W
p
(s)=
1
s
2
+ s +1
,W
a1
(s)=
1
0.01s +1
(64)
The loop is closed via the G-SO algorithm (12) having the switching antici-
pation parameter β =0.2 and control magnitude U = 1. The approximate and
theoretically exact parameters of the periodic solution (obtained by means of
the DF and modified Tsypkin locus techniques, respectively) were computed,
and the “true” values were also found by computer simulation (see Table I).
Table 1. Periodic motion analysis of Section VI.A
Frequency Amplitude
[rad sec

−1
]
DF 24.9 0.0020
Modified Tsypkin Locus 23.26 0.0024
Simulation 22.25 0.0025
Fig. 6 provides the results of the computer simulations. The higher accuracy
of the modified Tsypkin analysis, with respect to the DF analysis, is apparent
from the inspection of the Table I, and it is justified by the theoretical analysis
presented above. The mismatch of the simulation values with respect to those
values computed via the Modified Tsypkin Locus (which are theoretically exact)
is caused by factors of numerical approximation such as truncation of the series
(63), round-offs, discrete-time integration of the simulation example.
6.2 Linearization-Based Analysis
Consider the simplified model of the rotating arm driven by a torque motor
through an elastic friction link
1
2
ML
2
¨q
1
+ B
1
˙q
1
+
1
2
MgLsin(q
1

)=K(q
2
− q
1
)+B(˙q
2
− ˙q
1
)
J ¨q
2
+ B
2
˙q
2
+ K(q
2
− q
1
)+B(˙q
2
− ˙q
1
)=τ
(65)
38 I. Boiko et al.
0 2 4 6 8 10
−0.5
0
0.5

1
1.5
Time [sec]
The y
1
time evolution
8 8.5 9 9.5 10
−3
−2
−1
0
1
2
3
x 10
−3
Time [sec]
The y
1
time evolution
(zoom on steady state)
Fig. 6. Section 6.A. The periodic solution.
where q
1
and q
2
represent the arm and motor coordinates, respectively, M and
L are the mass and length of the arm, B
1
is the arm friction term, K is the

joint stiffness coefficient, B is the link viscous friction coefficient, J and B
2
are
the motor inertia and viscous friction coefficient measured at the link-side of
the gears, τ is the electromagnetic torque exerted by the motor. The electrical
dynamics of the torque servo drive is accounted for by adding a first-order filter
between the “command” torque τ
∗
(reference input to the torque-controlled
servo drive) and the actual torque profile τ, according to the following equation
τ =
1
1+μs
τ
∗
μ =0.01 (66)
Let only the link coordinates be available for measurement and define the
sliding variable as s
1
=˙q
2
+c(q
2
−q
∗
1
), with c>0andq
∗
1
being a set-point value.

The dynamics (65) restricted onto the manifold s
1
= 0 is now briefly discussed.
If s
1
tends to zero then q
2
→ q
∗
1
and ˙q
2
→ 0 exponentially. Considering these
two conditions into the first equation (65) yields an exponentially stable error
variable q
1
− q
∗
1
, as it can be proven by standard Lyapunov analysis.
In order to obtain a continuous torque profile, we add an integrator at the
input side [5]. The time-derivative of the command torque is set via the G-
SO controller with β =0.8andU = 20. The following parameters were used:
M =1kg, L =1m, J =0.01kgm
2
, B
1
=0.1Nms, B
2
=1Nms, K = 100Nm,

B =1Nms, q
∗
1
= π/3, c = 3. The linearized dynamics can be expressed in terms
of the deviation variables
δq =[δq
1
,δq
2
,δ˙q
1
,δ˙q
2
]andδs
1
= s
1
as follows:
˙
δq = Aδq + Bτ
δs
1
= Cδq
(67)
A =
⎡
⎢
⎢
⎣
0010

0001
−

g
L
cos(q
∗
1
)+
2K
ML
2

2K
ML
2
−
2(B
1
+B)
ML
2
2B
ML
2
K
J
−
K
J

B
J
−
B+B
2
J
⎤
⎥
⎥
⎦
, B =
⎡
⎢
⎢
⎣
0
0
0
1
J
⎤
⎥
⎥
⎦
, C
T
=
⎡
⎢
⎢

⎣
0
c
0
1
⎤
⎥
⎥
⎦
(68)
A Comprehensive Analysis of Chattering 39
The linearized plant can therefore be presented as in Fig. 4-left with
W (jω)=
C(jωI − A)
−1
B
jω(jωμ +1)
(69)
As shown in the upper plot of Fig. 7, DF analysis yields the following value
of the chattering frequency
ˆ
ω = 512rad/sec. The magnitude of W in the in-
tersection point is M = |W(j512)|≈9.4e − 5, and the oscillation amplitude
is
ˆ
A
y
=4MU/π ≈ 0.0024. Analysis via the modified Tsypkin locus (see Fig.7,
lower plot) gives the following frequency and amplitude:
ˆ

ω = 442.8rad/sec,
ˆ
A
y
=0.0028.
Simulations of the nonlinear model (65)-(66) with μ =0.01 provide the steady-
mode time evolution of s
1
depicted in Fig. 8. The observed amplitude and fre-
quency of the oscillation are: A
y
≈ 0.0029 and ω ≈ 436rads
−1
.
Nyquist Diagram
Real Axis
Imaginary Axis
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
x 10
-4
0
1
2
x 10
-4
System: W
Real: -5.61e-005
Imag: 7.53e-005
Frequency (rad/sec): 512
-1/N(A

y
)
W(j
ω
)
-3 -2.5 -2 -1.5 -1 -0.5 0
x 10
-3
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-3
Real Axis
Imaginary Axis
Modified Tsypkin Analysis of the elastic arm example
Φ
(
ω
)
ω=400
ω=500
- 1/N(A
y
)

Real: - 0.0017
Imag: 0.0022
Chattering frequency: 442. 85 rad/sec
Chattering Amplitude: 0.0028
Fig. 7. The elastic arm example. Upper plot: DF analysis. Lower plot: Modified Tsyp-
kin analysis.
40 I. Boiko et al.
9.55 9.6 9.65 9.7 9.75
−5
0
5
x 10
−3
Time [sec]
The sliding variable s
1
in the
steady state
Fig. 8. The s
1
time evolution in the steady state
The test was also repeated with a smaller value of μ =0.001. The contrac-
tion of the boundary layer O
μ
was expected in accordance with (29), which is
confirmed as the reduction of the amplitude of chattering presented in Table
II. The frequency of chattering increases, which agrees with the fact that the
control system bandwidth becomes larger.
Table 2. Periodic motion analysis of Section VI.B with μ =0.001
Frequency Amplitude

[rad sec
−1
]
Modified Tsypkin Locus 1.68E+3 4.99E-4
Simulation 1.61E+3 5E-4
7 Conclusions
A systematic analysis of chattering in control systems with fast actuators driven
by the second-order sliding mode “generalized sub-optimal” controller is pro-
posed. Analysis is carried out in the state-space and frequency domains for linear
and nonlinear plants. It is shown that the system motions always converge to
the O
μ
domain (29) of approximation with respect to the fast actuator’s time
constant μ, and, under some conditions, a stable attracting limit cycle fully
contained in this domain exists.
For nonlinear plants and nonlinear actuators:
• The amplitude of chattering in a small vicinity of the sliding surface is esti-
mated.
• Sufficient conditions of the existence and orbital asymptotic stability of the
fast periodic motions are obtained in terms of the properties of corresponding
Poincar´emaps.
A Comprehensive Analysis of Chattering 41
For linear plants and linear actuators:
• A methodology of approximate analysis of the amplitude and the frequency
of chattering via application of the DF method is proposed.
• A methodology of exact analysis of the amplitude and the frequency of chat-
tering via application of the modified Tsypkin’s method is given.
Acknowledgment
This work was supported in part by Mexican CONACyT (Consejo Nacional de
Ciencia y Tecnologia), grants no. 43807-Y no. J110.418/2006, and Programa de

Apoyo a Proyectos de Investigacion e Innovacion Tecnolgica (PAPIIT) UNAM,
grant no. 107006-2, and by Italian MURST Project “Novel control systems in
high–speed railways”.
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A Comprehensive Analysis of Chattering 43
Appendices
Proof of Theorem 1
Step A. There exists a sequence of time instants t
M

j
, j ≥ 1 such that
s
2
(t
M
j
)=0.
Let t
0
= 0 be the initial time instant and assume, without loss of general-
ity that s
1
(0) > 0. By considering Assumption 3 together with the well-known
properties of the Suboptimal Algorithm in the absence of actuator dynamics,
it readily follows that there is t
M
1
≥ t
0
such that s
2
(t
M
1
)=0.Itiseasy
to show that, because of the actuator dynamics, if ˙s
2
(t
M

1
)s
1
(t
M
1
) > 0then
˙s
2
(t
M
2
)s
1
(t
M
2
) < 0. It can be therefore assumed that at t = t
M1
condition
˙s
2
(t
M
1
)s
1
(t
M
1

) < 0 holds, which implies that at t ≥ t
+
M
1
the system trajectory
enters one of the even quarters s
1
s
2
< 0. Let us assume, without loss of gen-
erality, that s
1
(t
M
1
) > 0, and refer to the plot in figure 9. All possible actual
trajectories of the uncertain system are confined between the limiting curves
a, , d in Fig. 9 compted by a worst-case analysis considering the limit values
of z
1
to which correspond limit values for ˙s
2
according to (26).
There is t
c
1
>t
M
1
such that s

1
(t
c
1
)=βs
1M
1
, then control u is
u(t)=

−Ut
M
1
≤ t<t
c
1
Ut
c
1
≤ t<t
M
2
(70)
By Assumption 3, and (26)-(28), there exists t
M
2
>t
M
1
such that s

2
(t
M
2
)=0.
By iteration the claim is proven.
Step B. There are ρ
∗
1
> 0andq ∈ (0, 1)such that if |s
1M
j
| >ρ
∗
1
μ
2
then
|s
1M
j+h
|≤max{β,q}|s
1M
j
|, 0 <q<1,h∈{1, 2} (71)
0
0
0
0
b

a
limit trajectories
actual state trajectory
s
1
(t)
c
d
1
1M
s
β
1
1M
s
2
1M
s
s
2
(t)
P
C1
P
M1
2
1M
s
22C
s

12C
s
2C
P
12C
s
22C
s
2
1M
s
Fig. 9. Actual and limit trajectories in the s
1
-s
2
plane
44 I. Boiko et al.
The property is proven for j = 1. Refer to the plot in figure 9. We aim at
evaluating the points s
1M
2
and s
1M
2
. By algebraic computations it yields that
singular point P
M2
≡ (s
1M
2

, 0), achieved at t = t
M2
when s
2
(t
M2
)=0,issuch
that
s
1M2
≤ s
1M2
≤ s
1M2
(72)
s
1M2
= βs
1M1
−
(1−β)(H+G
M
γ
M
(1+ε)U)
G
m
γ
m
(1−ε)U−H

s
1M1
− ϕ
1
μ
√
s
1M1
− ϕ
2
μ
2
s
1M2
= βs
1M1
−
(1−β)(G
m
γ
m
(1−ε)U−H)
H+G
M
γ
M
(1+ε)U
s
1M1
(73)

with ϕ
1
and ϕ
2
being positive constants. The contraction condition (71), with
j = h =1,isthenequivalenttos
1M2
≥−qs
1M1
, which can be rewritten as
Ω
1
s
1M1
+ ϕ
1
μ
√
s
1M1
+ ϕ
2
μ
2
≤ (β + q)s
1M1
Ω
1
=
(1−β)[H+G

M
γ
M
(1+ε)U
M
]
G
m
(1−ε)U−H
(74)
To solve equation (74) let us introduce the new variable ρ
1
=
s
1M1
μ
2
and rewrite
(74) as
Ω
1
ρ
1
+ ϕ
4
√
ρ
1
+ ϕ
3

≤ (β + q)ρ
1
(75)
If condition Ω
1
<β+ q holds, then the slope of the right hand side of (75)
is less than β + q and a nonempty solution interval of (75) exists. Manipulating
condition Ω
1
<β+ q one obtains directly the second of (27). The resulting
solution interval of (75) is ρ
1
≥ ρ
1
where ρ
1
is the unique positive root of
equation Ω
1
ρ
0
+ ϕ
4
√
ρ
0
+ ϕ
3
=(β + q) ρ
0

. Then, by considering (72)-(73),
there is ρ
∗
1
>ρ
1
such that, as long as
|s
1M1
|≥ρ
∗
1
μ
2
(76)
then |s
M3
| is contractive with respect to |s
M1
| according to (71). Convergence
takes place in finite time since there is k>0 such that t
M,i+1
−t
M,i
≤ k

|s
Mi
|.
Step C. There exist a positive constant H overestimating |

˜
H(s, w)| .
Let W
0
≥w(0), S
10
≥|s
1
(0)|, S
20
≥|s
2
(0)|, and define s
1
=sup
t≥0
|s
1
|,
s
2
=sup
t≥0
|s
2
|. By combining (24) and (21) it can be written
|
˜
H
(s, w)|≤F

0
+ F
1
(
s
1
+
s
2
)
(77)
where F
0
= H
0
+H
2
W
0
and F
1
= H
1
+H
2
ξ
1
. Assume temporarily that a constant
H overestimating |
˜

H(s, w)| for any t ≥ 0 exists. It can be found constants
α
0
, ,α
5
such that
s
1
≤ S
10
+ α
0
Hμ
2
+ α
1
μ +
α
2
H
, (78)
s
2
≤ S
20
+ α
2
NμH +

2(1 − β)α

3
H
√
s
1
+ α
4
NμH (79)
A Comprehensive Analysis of Chattering 45
Consider the inequality
H ≥ F
0
+ F
1
s
1
(H)+F
1
s
2
(H) (80)
We shall prove that (80) admits the semi-infinite solution interval H ∈ (H
∗
, ∞).
By simple manipulations one can rewrite (80) as
H ≥ λ
0
+ F
1
μ(α

0
Hμ + α
1
)+
F
1
α
2
G
+ λ
1
Hμ+ λ
2

α
2
+ S
10
H + α
0
H
2
μ
2
+ α
1
Hμ
(81)
where λ
0

, ,λ
2
are proper constants.
If the slope of the right-hand side of (81), viewed as a function of the variable
H, is less than one for sufficiently large H then the inequality (81) admits a
semi-infinite solution interval of the type H ∈ [H
∗
, ∞]. Considering the higher-
order terms in H one express such condition as F
1
α
0
μ
2
+[λ
1
+ λ
2
√
α
0
]μ<1,
yielding, in turns, μ ≤ μ
∗
,whereμ
∗
is the unique positive solution of equation
F
1
α

0
μ
∗
2
+[λ
1
+ λ
2
√
α
0
]μ
∗
=1.
Step D. There is ρ
1
> 0, ρ
2
> 0 such that the domain O
μ
(29) is invariant.
It has been demonstrated that
B
1
(ρ
∗
1
)={(s
1
,s

2
):|s
1
|≤ρ
∗
1
μ
2
,s
2
=0} (82)
is attracting. Consider the worst-case evolution starting from one of the neigh-
bors (says the right one) of the attracting domain B
1
. The analysis performed
in step C can be applied by setting S
10
= ρ
∗
1
μ
2
and S
20
= 0 in (78) and (79). By
evaluating the corresponding values of
s
1
and s
2

and considering the contraction
condition (71), it can be concluded that there are such ρ
1
,ρ
2
> 0 such that the
set (29) is invariant, and this concludes the proof.
Poincar´e Map Derivation: General Procedure
Let us derive the Poincar´e maps of the domain y
1
> 0,f(w, y
1
, 0,z
1
) < 0onthe
surface y
2
= 0 into the domain y
1
< 0,f(w, y
1
, 0,z
1
) > 0onthesamesurface
y
2
=0,generating by systems (34)-(36) and (37)-(38).
Let y
0
1

> 0, and denote as w
+
(τ,μ), y
+
1
(τ,μ), y
+
2
(τ,μ), z
+
(τ,μ)and¯y
+
1
(τ,w),
¯y
+
2
(τ,w), ¯z
+
(τ,w) the solution of systems (34)-(36) and (37)-(38) with the initial
conditions w
+
(0,μ)=w
0
, y
+
1
(0,μ)=y
0
1

, y
+
2
(0,μ)=0,z
+
(0,μ)=z
0
(w
0
∈ W)
and ¯y
+
1
(0,w)=¯y
0
1
,¯y
+
2
(0,w)=0,¯z
+
(0,w)=¯z
0
such that f(w, ¯y
0
1
, 0, ¯z
0
) < 0for
all w ∈

W.
Let T
+
sw
be the smallest positive root of the equation ¯y
+
1
(T
+
sw
,w)=β¯y
0
1
and such that
d¯y
+
1
dt
(T
+
sw
,w)=¯y
+
2
(T
+
sw
,w) < 0. From the implicit function the-
orem there exists a switching time-instant T
+

μsw
(μ)=T
+
sw
+ O(μ) such that
y
+
1
(T
+
sw
(μ),μ)=βy
0
1
, y
+
2
(T
+
sw
(μ),μ) < 0.
Denote as w
+
p
(τ,μ), y
+
1p
(τ,μ), y
+
2p

(τ,μ), z
+
p
(τ,μ), and ¯y
+
1p
(τ,w), ¯y
+
2p
(τ,w),
¯z
+
p
(τ,w) the solutions of systems (35)-(36) and (37)-(38) with the initial
46 I. Boiko et al.
conditions w
+
p
(T
+
μsw
,μ)=w
+
(T
+
μsw
,μ), y
+
1p
(T

+
μsw
,μ)=y
+
1
(T
+
μsw
,μ), y
+
2p
(T
+
μsw
,μ)=
y
+
2
(T
+
μsw
,μ), z
+
p
(T
+
μsw
,μ)=z
+
(T

+
μsw
,μ)and¯y
+
1p
(T
+
μsw
,w)=¯y
+
1
(T
+
μsw
,w),
¯y
+
2p
(T
+
μsw
,w)=¯y
+
2
(T
+
μsw
,w), ¯z
+
p

(T
+
μsw
,w)=¯z
+
(T
+
μsw
,w).
Let T
+
p
(w) be the smallest positive root of the equation ¯y
+
2p
(T
+
p
(w),w)=0
and such that
d¯y
+
1
dt
(T
+
p
,w)=¯y
+
2

(T
+
p
,w) < 0. From the implicit function theorem
one can conclude that there exists a switching time-instant T
+
μp
= T
+
p
(w)+O(μ)
such that y
+
2p
(T
+
μp
,μ)=0,y
+
1p
(T
+
μp
,μ) < 0.
So, we have designed the Poincar´emaps
Ξ
+
(w, ¯y
0
1

, ¯z
0
)=

¯y
+
1p
(T
+
p
(w),w)
¯z
+
p
(T
+
p
(w),w)

(83)
and
Ψ
+
(w
0
,y
0
1
,z
0

)=
⎡
⎣
w
+
p
(T
+
μp
,μ)
y
+
1p
(T
+
μp
,μ)
z
+
p
(T
+
μp
,μ)
⎤
⎦
=

w
0

+ O(μ)
Ξ
+
(w
0
,y
0
1
,z
0
)+O(μ)

(84)
Similarly, we can compute the Poincar´emaps
Ξ
−
(w, ¯y
+
1p
(T
+
p
(w),w), ¯z
+
p
(T
+
p
(w),w)) =


¯y
−
1p
(T
−
p
(w),w)
¯z
−
p
(T
−
p
(w),w)

(85)
Ψ
−
(w
0
,y
0
1
,z
0
; μ)=
⎡
⎣
w
−

p
(T
−
μp
,μ)
y
−
1p
(T
−
μp
,μ)
z
−
p
(T
−
μp
,μ)
⎤
⎦
=
⎡
⎣
w
0
+ O(μ)
¯y
−
1p

(T
−
p
(w
0
),w
0
)+O(μ)
¯z
−
p
(T
−
p
(w
0
),w
0
)+O(μ)
⎤
⎦
. (86)
of the points (¯y
+
1p
(T
+
p
(w),w), ¯z
+

p
(T
+
p
(w),w)) and (w
+
p
(T
+
μp
,μ),y
+
1p
(T
+
μp
,μ),
z
+
p
(T
+
μp
,μ)) of the domain y
1
< 0,f(w, y
1
, 0,z
1
) > 0onthesurfacey

2
=0
into the same surface y
2
= 0. Their detailed derivation is skipped for brevity.
() ()()()
++++++
swswsw
TzTyTy ,,
21
() ()()
++++
pp
TzTy ,0,
1
()()()()
−−−−−−
swswsw
TzTyTy ,,
21
()
00
1
,0, zy
y
1
y
2
z
() ()()

−−−−
pp
TzTy ,0,
1

Fig. 10. The overall Poincar´e map
A Comprehensive Analysis of Chattering 47
Combining (83)-(86) will provide for the overall Poincar´e map (see fig. 10), which
allows checking the conditions for the existence and stability of the periodic limit
cycles presented in the Section 4.
Proof of Theorem 2
From the implicit function theorem it follows that there exist some neighborhood
N of the point (w
0
, ¯y
∗
1
(w
0
), ¯z
∗
(w
0
)) and μ
0
> 0 such that Ψ ∈C[N] , ∀μ ∈
[0,μ
0
]. Moreover, we can rewrite Ψ(w, y
1

,z,μ)intheform
Ψ(w, y
1
,z,μ)=

w + μQ(w, y
1
,z,μ)
R(w, y
1
,z,μ)

(87)
where Q(w, y
1
,z,μ)andR(w, y
1
,z,μ) are sufficiently smooth functions such that
Q(w, ¯y
∗
1
(w), ¯z
∗
(w), 0) = 0, (88)
R(w, ¯y
∗
1
(w), ¯z
∗
(w), 0) = Ξ(w, ¯y

∗
1
(w), ¯z
∗
(w)) = (¯y
∗
1
(w), ¯z
∗
(w)). (89)
Let us rewrite the map Ψ = Ψ (w, ξ, η, μ) in terms of the ’error’ variables (ξ, η)=
(y
1
− ¯y
∗
1
(w),z − ¯z
∗
(w)), yielding
Ψ =

Ψ
1
(w, ξ, η, μ)
Ψ
2
(w, ξ, η, μ)

=


w + μQ(w, ξ +¯y
∗
1
(w),η+ z
∗
(w),μ)
R(w, ξ +¯y
∗
1
(w),η+ z
∗
(w),μ) −(¯y
∗
1
(w),z
∗
(w))

(90)
Clearly one has that Ψ(w, 0, 0, 0) = (w, 0, 0). Existence of the periodic solution
follows from the existence of the fixed point (w
∗
(μ), ξ
∗
(μ), η
∗
(μ)) of the Poincar´e
map (90), which we are going to prove.
Existence conditions for the fixed point are written in the form:
G(w

∗
,ξ
∗
,η
∗
,μ)=

G
1
(w
∗
,ξ
∗
,η
∗
,μ)
G
2
(w
∗
,ξ
∗
,η
∗
,μ)

=

1
μ

[w
∗
− Ψ
1
(w
∗
,ξ
∗
,η
∗
,μ)]
(ξ
∗
η
∗
)
T
− Ψ
2
(w
∗
,ξ
∗
,η
∗
,μ)

= 0 (91)
Taking into account that, if μ =0,thenw
∗

(0) = w
0
, ξ
∗
(0) = 0, η
∗
(0) = 0 and
G
1
(w
0
, 0, 0, 0) = −T
0
(w
0
)p(w
0
) = 0, it turns out that if μ = 0 conditions (91)
are fulfilled. Moreover, taking into account that G
2
(w, 0, 0, 0) = 0, ∀w ∈ W we
can conclude that
∂G
2
∂w
(w
0
, 0, 0, 0) ≡ 0.
Let us evaluate the Jacobian matrix of function G with respect to variables
w, (ξ,η)atμ = 0. It yields the following matrix

∂G
∂(w, (ξ, η))




(w
0
,0,0,0)
=

−T
0
(w
0
)
dp
dw
(w
0
)
∂G
2
∂(ξ,η)
(w
0
, 0, 0, 0)
0 I
m
−

∂Ξ
∂(y
1
,z)
(w
0
, ¯y
∗
1
(w
0
), 0, ¯z
∗
(w
0
))

(92)
which turns out to be not degenerated. This means the map G, admits an isolated
fixed point (w
∗
(μ),ξ
∗
(μ),η
∗
(μ)) corresponding to the periodic solution of system
(34)-(36) and (31)-(33), and w
∗
(μ)=w
0

+O(μ), y
∗
1
(μ)=¯y
∗
(w
0
)+O(μ), z
∗
(μ)=
¯z
∗
(w
0
)+O(μ).