Design and frequency analysis of continuous finite-time-convergent ...
Design and frequency analysis of continuous finite-time-convergent differentiator Xinhua Wang and Hai Lin Department of Electrical & Computer Engineering National University of Singapore, 4 Engineering Drive 3, Singapore 117576 [email protected]
Abstract: In this paper, a continuous finite-time-convergent differentiator is presented based on a strong Lyapunov function. The continuous differentiator can reduce chattering phenomenon sufficiently than normal sliding mode differentiator, and the outputs of signal tracking and derivative estimation are all smooth. Frequency analysis is applied to compare the continuous differentiator with sliding mode differentiator. The beauties of the continuous finite-time-convergent differentiator include its simplicity, restraining noises sufficiently, and avoiding the chattering phenomenon. Keywords: continuous finite-time-convergent, differentiator, frequency analysis. convergence compared with other typical differentiators; no
1. Introduction
chattering
phenomenon.
However,
peaking
Differentiation of signals is a well-known problem
phenomenon exist in the output of derivative estimation.
[1-8], which has attracted much attention in recent years.
Moreover, the ability of restraining noise was not
Obtaining the velocities of tracked targets is crucial for
considered.
several kinds of systems with correct and timely
In [21], we designed a hybrid differentiator with high
performances, such as the missile-interception systems
speed convergence, and it succeeds in applications to
[9] and underwater vehicle systems [10], in which
velocity estimations for low-speed regions only based
disturbances must be restrained. The simpleness and
on position measurements [22] and to quad-rotor aircraft
robustness of differentiators should be taken into
[23], in which only the convergence of signal tracking
consideration.
was described, but the convergence of derivative
The popular high-gain differentiators [4, 5, 6] provide for an exact derivative when their gains tend to infinity.
estimation was not given, and the regulation of parameters was complex.
Unfortunately, their sensitivity to small high-frequency
In this paper, a continuous finite-time-convergent
noise also infinitely grows. With any finite gain values
differentiator is presented based on a strong Lyapunov
such a differentiator has also a finite bandwidth. Thus,
function. For this differentiator we study its stability and
being not exact, it is, at the same time, insensitive with
finite time convergence characteristics by means of
respect to high-frequency noise. Such insensitivity may
Lyapunov functions. The continuous differentiator can
be considered both as advantage or disadvantage
reduce chattering phenomenon sufficiently than sliding
depending on the circumstances. Moreover, high gain
mode differentiator. The advantage of the use of
results in peaking phenomenon.
Lyapunov functions is to easily obtain the parameters of
In [7, 8], a differentiator via second-order (or
differentiator.
high-order) sliding modes algorithm was proposed. The
Frequency analysis is applied to compare the
information one needs to know on the signal is an upper
continuous finite-time-convergent differentiator with
bound for Lipschitz constant of the derivative of the
sliding mode differentiator. For nonlinear differentiators,
signal.
is
an extended version of the frequency response method,
introduced and there exists no chattering phenomenon in
describing function method [19, 24], can be used to
signal
approximately analyze and predict nonlinear behaviors
Although tracking,
second-order derivative
sliding
estimation
mode still
exist
chattering phenomenon.
of nonlinear differentiators. Even though it is only an
In [11], we presented a finite-time-convergent
approximation method, the desirable properties it
differentiator based on finite-time stability [12-14] and
inherits from the frequency response method, and the
singular perturbation technique [15-20]. The merits of
shortage of other, systematic tools for nonlinear
this
differentiator analysis, make it an indispensable
differentiator
exist
in:
rapidly
finite-time
1
component of the bag of tools of practicing control engineers. By describing function method, we can conclude that the continuous finite-time-convergent differentiator has the better ability of restraining high-frequency noises than sliding mode differentiator. This paper is organized in the following format. In section 2, preliminaries are introduced. In section 3,
where
x1 ," xn are suitable coordinates on R n and
r = ( r1 ," , rn ) with the dilation coefficients r1 " , rn positive real numbers. A
vector
f ( x ) = ( f1 ( x ) ," f n ( x ) )
field
Τ
is
problem statement of sliding mode differentiator is
homogeneous of degree k ∈ R with respect to the
given. In section 4, continuous finite-time-convergent
family of dilations δ ρr if
differentiator is presented, and its robustness analysis is
(
given in section 5. In section 6, frequency analysis of continuous differentiator is given.
)
f i ρ r1 x1 ," , ρ rn xn = ρ ri + k fi ( x )
In section 7, the
i = 1," , n, ρ > 0
simulations are given, and our conclusions are made in
(4)
System (1) is called homogeneous if its vector field f is
section 8.
homogeneous. The following lemma was presented in some
2. Preliminaries First of all, the concepts related to finite-time control
references like [12, 25, 26].
are given (See [13]).
Lemma1: Suppose that system (1) is homogeneous of
Definition 1: Consider a time-invariant system in the
degree k < 0 with respect to the family of dilations δ ρr ,
form of x = f ( x ) , f ( 0 ) = 0, x ∈ R n
where
f : Uˆ 0 → R n
is
continuous
on
an
(1)
f ( x ) is continuous and x = 0 is its asymptotically
open
stable equilibrium. Then equilibrium of system (1) is
neighborhood Uˆ 0 of the origin. The equilibrium x = 0
globally finite-time stable.
of the system is (locally) finite-time stable if (i) it is
continuous function V: D R such that the following
asymptotically stable, in Uˆ , an open neighborhood of
conditions hold:
the origin, with Uˆ ⊆ Uˆ 0 ; (ii) it is finite-time convergent
(1) V is positive definite;
in
Uˆ
, that is, for any initial condition x0 ∈Uˆ \ {0} , there
Lemma 2 [14, Theorem 4.2]. Suppose there exists a
(2) There exist real number c>0 and θ (0,1) and an open neighborhood
θ V ( x ) + c (V ( x ) ) ≤ 0 , x ∈ν \ {0}
is a settling time T > 0 such that every solution x ( t , x0 ) of system (1) is defined with x ( t , x0 ) ∈Uˆ \ {0}
for t ∈ [ 0, T ] and satisfies
lim x ( t , x0 ) = 0
(2)
t →T
origin x = 0 is globally finite-time stable.
and the settling-time function T is 1−θ 1 T ( x) ≤ V ( x )) ( c (1 − θ )
that assigns to every real ρ > 0 a diffeomorphism
δ ρ ( x1 ," , xn ) = ( ρ x1 ," , ρ xn ) r1
rn
(6)
and T is continuous. If in addition D = R n , V is proper,
V takes negative values on
R n \ {0} , then the
origin is a globally finite-time-stable equilibrium of (1). Assumption 1. For (1), there exist ρ ∈ ( 0,1] and a
Definition 2: A family of dilations δ ρr is a mapping
r
(5)
Then the origin is a finite-time-stable equilibrium of (1),
and and x ( t , x0 ) = 0 , if t ≥ T . Moreover, if Uˆ = R n , the
ν ⊂ D of the origin such that
nonnegative constant f
a such that
( z1 ) − f ( z1 )
≤ a
zi − zi
ρ
(7)
(3) 2
Let the Lyapunov function be
where z, z ∈ℜn .
1 V = k2 e1 + e22 2
Remark 1. There are a number of nonlinear functions actually satisfying Assumption 1. For example, one such function
xρ
is
ρ
x ρ − x ρ ≤ 21− ρ x − x ,
since
ρ ∈ ( 0,1] . Moreover, there are smooth functions also satisfying this property. In fact, it is easy to verify that
sin x − sin x ≤ 2 x − x
ρ
for any
Therefore,
(
)
0.5 V = k2 sgn ( e1 ) e2 − k1 e1 sgn ( e1 ) + e2 ( − k2 sgn ( e1 ) ) (15)
= −k1k2 e1
0.5
L2 . For differentiator (8), there exists a is the desired signal, and it is satisfied with time
ts > 0 such that x1 = v ( t ) , x2 = v ( t )
for
v ( t ) − v0 ( t ) ≤ σ . Therefore, for some positive constants
(9)
µ1 and µ2 the following inequalities are established [8]:
t ≥ ts . In fact, we let
1
e1 = x1 − v ( t ) , e2 = x2 − v ( t )
e1 = x1 − v0 ( t ) ≤ μ1σ , e2 = x2 − v0 ( t ) ≤ μ 2σ 2 3.3 Chattering phenomenon
Because switch function exists in the second
The tracking error system is
e1 = e2 − k1 e1
0.5
sgn ( e1 )
e2 = −k2 sgn ( e1 ) − v( t )
differential equation of differentiator (8), although the (11)
0.5
sgn ( e1 )
e2 ∈ − [ k2 − L2 , k2 + L2 ] sgn ( e1 )
output x1 is smooth, the output x2 is continuous but non-smooth, it is called as chattering phenomenon. In
We can get the following differential inclusion:
e1 = e2 − k1 e1
order to explain the problem, we give an example in the following. (12)
Example 1: For system 1
x1 = x2 − k1 x1 2 sgn ( x1 )
or
e1 = e2 − k1 e1
0.5
sgn ( e1 )
e2 = − k2 sgn ( e1 )
(19)
(10)
x2 = − k2 sgn ( x1 ) , k2 ∈ [ k2 − L2 , k2 + L2 ]
(13)
(20)
let k1 = 6, k2 = 9 , then we have the solutions x1 and x2 in
Fig. 1. 3
x1 = x2 − λ2
Ω ( x1 − v ( t ) ) A0.5
(26)
4 x2 = −λ1 ( x − v (t )) πA 1 The nature frequency of system (26) is
ωn =
2 k2
(27)
π A0.5
and we have 2ςωn =
k1Ω A0.5
(28)
Therefore, the damping coefficient is
Fig. 1 x1 and x2 of system (20)
From Fig. 1, we find that, for x2, chattering
ς=
phenomenon happens near the equilibrium. Moreover, if noises exist in signal, this chattering phenomenon will magnify noises near the equilibrium. In some velocity feedback systems, this chattering in x2 can make
k1Ω π 4 k2
(29)
With the error amplitude decreasing, the nature frequency
ωn
increases.
Moreover,
chattering
motors trembling. Therefore, chattering phenomenon must be removed sufficiently in the output x2 of a
phenomenon happen rear the equilibrium for the
differentiator.
chattering phenomenon will magnify noises.
3.4 Frequency analysis of sliding mode differentiator
We can also analyse sliding differentiator (8) from Let e1 = x1 − v ( t ) = A sin ωt , we have
π∫
π
A sin ωτ
0
= A0.5
2
π
∫
π
0
sin ωτ
1.5
restrain sufficiently high-frequency noises, we should
4. Continuous finite-time-convergent differentiator
sgn ( A sin ωτ ) sin ωτ d ωt
0.5
In order to remove chattering phenomenon and to design a continuous differentiator.
frequency characteristics.
2
discontinuous differentiator. If noises exist in signal, this
4.1 Continuous finite-time-convergent system
(21)
In order to design continuous differentiator, firstly, we give a finite-time stability Theorem as follow.
d ωt
Theorem 1: For continuous system
Then we can get
2
π
∫
π
0
sin ωτ dωt =
2
π
( − cos ωτ ) |0 = π
4
π
z1 = z2 − k1 z1
(22)
and 2
π
π
∫ ( sin ωτ ) 0
In (21), let Ω =
2
2
d ωt =
π∫
π
0
2
π
∫
π
0
sin ωτ
1 − cos 2ωτ d ωt = 1 2
1.5
1< Ω
0 , ts > 0 and α ∈ ( 0,1)
Therefore, the describing function of nonlinear function 0.5
2
α
dωt , we have
4
α +1
(25)
α +1 2k2 1 1⎛ α +1 ⎞ z1 + z22 + ⎜ k1 z1 2 sgn ( z1 ) − z2 ⎟ 2 2⎝ α +1 ⎠
2
(32)
and we can get
and the linearization system of differentiator (8) is
4
α +1 1 ⎞ α +1 ⎛ 2k V = ⎜ 2 + k12 ⎟ z1 + z22 − k1 z2 z1 2 sgn ( z1 ) ⎝ α +1 2 ⎠ ⎡ 4k2 ⎤ + k 2 − k1 ⎥ ⎡ α +1 ⎤1 = ⎢ z1 2 sgn ( z1 ) z2 ⎥ ⎢ α + 1 1 ⎥ ⎣ ⎦ 2 ⎢ −k 2 ⎦ ⎣ 1
Therefore, we have α −1
α −1
(33)
z1
2
α +1 2
2
min
⎤ sgn ( z1 ) z2 ⎥ ⎦
⎡ 4k 2 1 ⎢ 2 + k1 P = α +1 2⎢ ⎣ −k1
Τ
(34)
⎤ −k1 ⎥ ⎥ 2 ⎦
(35)
λmin { P}
(36)
From (35), we know that matrix positive-definite, and
P is symmetrical and
≤ V ≤ λmax { P} ς
2 2
3α +1
V 2(α +1)
(45)
3α + 1 3α + 1 = 2 (α + 1) 3α + 1 + (1 − α )
(46)
3α + 1 0 and 0 < α < 1 such that 5
the outputs x1 and x2 of differentiator (48) are
k12 (α + 1) α e2 e1 sgn ( e1 ) 2 ⎛ k12 (α + 1) ⎞ 3α +1 k1 (α + 1) 2 α −1 + k1 ⎜ k2 − e2 e1 2 ⎟ e1 2 − 2 2 ⎝ ⎠ k 2 (α + 1) α + 1 e2 e1 sgn ( e1 ) 2 α +1 ⎛ ⎞ + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ v( t ) − e2 v( t ) ⎝ ⎠
V = −2k1k2 e1
smooth, and
ς
2
⎛ lL2 ⎞ ≤ ⎜⎜ ⎟⎟ ⎝ λmin {Q} ⎠
α +1 2α
(49)
is satisfied. Moreover, when α = 0 , output smooth and ς = 0 for
x1 is
x2 is continuous but non-smooth, and
α −1 2
Τ
Let
⎡
ς = ⎢ e1 ⎣
2 k1 ⎡ 2k2 + k1 (α + 1) −k1 (α + 1) ⎤ , ⎢ ⎥ α +1 ⎦ 2 ⎣ −k1 (α + 1)
(51) Q=
(52)
k1 (α + 1)
lL2 < 1 can be obtained in differentiator (48) and λmin {Q}
becomes sufficiently small in a finite time. This
α +1 2
l = [ k1
−2] 2 = k12 + 4
ς
= e1
2 2
α −1
continuous differentiator (48) is continuous and its
Therefore, we can get
is
selected
to
design
the
Proof: The error system of differentiator (48) is
e1 = e2 − k1 e1
α +1 2
sgn ( e1 )
2
V ≤ −λmin {Q} ς = −λmin {Q} ς
differentiator, regulation of parameters is easier.
(53)
e2 = − k2 e1 sgn ( e1 ) − v( t )
(58)
(59)
α +1
+ e22
(60)
From (60) and 0 < α < 1 , we have
Comparing with discontinuous differentiator (8),
function
(57)
From (57), we have
e1
outputs are all smooth. Moreover, because a strong
Τ
and
will be given in the following proof.
Lyapunov
⎤ sgn ( e1 ) e2 ⎥ ⎦
2 k1 ⎡ 2k2 + k1 (α + 1) − k1 (α + 1) ⎤ ⎢ ⎥ α +1 ⎦ 2 ⎣ − k1 (α + 1)
0 < α < 1 is selected sufficiently small, therefore, 2
(56)
(50)
2 2 k1 > 0 , k2 > 2 ( k1 + 4 ) L2 2
ς
(55)
⎡ α2+1 ⎤ sgn ( z1 ) ⎥ v t + [ k1 −2] ⎢ e1 () ⎢⎣ ⎥⎦ e2
l = [ k1 −2] 2 , Q=
⎡ α2+1 ⎤k sgn ( e1 ) e2 ⎥ 1 ⎢⎣ e1 ⎦2
α +1 ⎡ 2k + k 2 (α + 1) −k1 (α + 1) ⎤ ⎡ e 2 sgn ( e ) ⎤ 1 1 ⎥ ⎢ ×⎢ 2 1 ⎥ α +1 ⎦ ⎢ ⎥⎦ e ⎣ − k1 (α + 1) ⎣ 2
⎡ α +1 ⎤ ς = ⎢ e1 2 sgn ( e1 ) e2 ⎥ , e1 = x1 − v ( t ) , ⎣ ⎦ e2 = x2 − v ( t ) ,
+
Therefore, we can get V = − e1
t ≥ ts . Where v ( t ) ≤ L2 , and
3α +1 2
≥ ς
α −1 α +1
α −1 α +1
(61)
2
2
ς 2 + lL2 ς
3α +1 α +1 2
− lL2 ς
⎛ = − ⎜ λmin {Q} ς ⎝
2
2α α +1 2
(62)
⎞ − lL2 ⎟ ς ⎠
α
Moreover, we have
Select a Lyapunov function as V=
α +1 2k 2 1 1⎛ α +1 ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ 2 2⎝ α +1 ⎠
α +1
2
The time derivative of Lyapunov function (54) along the solutions of system (53) is
ς
(54)
We can select
2
⎛ lL2 ⎞ 2α ≤ ⎜⎜ ⎟⎟ ⎝ λmin {Q} ⎠
(63)
k1 and k2 such that
6
λmin {Q} > lL2
(64)
in signal v ( t ) , i.e., v ( t ) = v0 ( t ) + δ ( t ) , where v0 ( t )
In fact, s− sI − Q =
k1 ( 2k2 + k12 (α + 1) ) k21 k1 (α + 1) 2 =0 k1 k k1 (α + 1) s − 1 (α + 1) 2 2
(65) is the desired second-order derivable signal, δ ( t ) is a
i.e.,
(
)
k k2 s − 1 2k2 + ( k12 + 1) (α + 1) s + 1 2k2 (α + 1) = 0 2 4 2
bounded noise and satisfied with δ ( t ) ≤ σ . Then, the (66) following inequality is established in finite time
The minimum eigenvalue of (66) is λmin {Q} =
−
(
2 k1 ⎛⎜ 2k2 + ( k1 + 1) (α + 1) 2⎜ 2 ⎝
( 2k + ( k + 1) (α + 1) ) 2 1
2
2
2
Theorem 3: For differentiator (48), if there exist a noise
)
ς
(67)
⎞ − 8k2 (α + 1) ⎟ ⎟ ⎟ ⎠
⎡
ς = ⎢ e1 ⎣
( 2k + ( k
2 1
2
−
+ 1) (α + 1) 2
l1 = [ k1
)
)
2
⎞ − 8k2 (α + 1) ⎟ ⎟ ⎟ ⎠
α +1 2
(72)
Τ
⎤ sgn ( e1 ) e2 ⎥ , e1 = x1 − v0 ( t ) , ⎦
⎡ k α + 1) ⎤ 1−2α α2+1 , −2 ] 2 , Ψ1 (σ ) = k1 ⎢ k2 + 1 ( l2 ⎥ 2 σ 2 ⎣ ⎦
Ψ 2 (σ ) = k2 [1 + l2 ] 21−α σ α
(68)
Q=
> L2 k12 + 4
2 k1 ⎡ 2k2 + k1 (α + 1) − k1 (α + 1) ⎤ ⎢ ⎥ α +1 ⎦ 2 ⎣ − k1 (α + 1)
(73)
Proof: Let
Therefore, we get k2 >
Ψ 1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
e2 = x2 − v0 ( t ) , v0 ( t ) ≤ L2
have
(
≤
where
Because λmin {Q} > lL2 and k1 > 0 are required, we
2 k1 ⎛⎜ 2k2 + ( k1 + 1) (α + 1) 2⎜ 2 ⎝
2
2 ( k12 + 4 ) L22
e1 = x1 − v0 ( t ) , e2 = x2 − v0 ( t )
(69)
k (α + 1) 2 1
(74)
The error system is
and k1 > 0
(70)
e1 = e2 − k1 e1 − δ
Therefore, when 0 < α < 1 is sufficiently small,
α + 1 is sufficiently large, finally, the tracking and 2α estimation errors are sufficiently small in a finite time. When α = 0 , Levant differentiator (8) is obtained, and the Lyapunov function can be designed as V=
output
1 2k 2 1 1⎛ ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ α +1 2 2⎝ ⎠
α +1 2
sgn ( e1 − δ )
e2 = − k2 e1 − δ sgn ( e1 − δ ) − v0 ( t ) Let
Δ1 = − e1 − δ
α +1 2
sgn ( e1 − δ ) + e1
(71)
proof.
2
sgn ( e1 )
(76)
α
Therefore, we have
x1 is smooth and x2 is continuous but
non-smooth, and ς = 0 for t ≥ ts . This concludes the
α +1
Δ 2 = − e1 − δ sgn ( e1 − δ ) + e1 sgn ( e1 ) α
2
(75)
α
Δ1 ≤ 2
1−α 2
δ
Δ 2 ≤ 21−α δ
α +1 2
α
≤2
1−α 2
σ
α +1 2
(77)
≤ 21−α σ α
The Lyapunov function is selected as
5. Robustness analysis of continuous differentiator 7
V=
α +1 2k 2 1 1⎛ α +1 ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ α +1 2 2⎝ ⎠
2
(78)
V ≤ − e1
Let α +1
⎡
ς = ⎢ e1 ⎣
2
Therefore, we have
+ e1
Τ
⎡ 4k 2 ⎤ 2 ⎤ sgn ( e1 ) e2 ⎥ , P = 1 ⎢ α + 1 + k1 − k1 ⎥ ⎦ ⎥ 2⎢ ⎣
− k1
(79)
2 ⎦
α −1 2
λmin {Q} ς 2 + l1L2 ς
α −1 2
2
Ψ1 (σ ) ς
Suppose there exist a positive constant
we can get
c1 ς V = ς Τ Pς
+ Ψ 2 (σ ) ς
2
(80)
and we know that matrix P is symmetrical and
2
(87)
2
2
c1 such that
> Ψ 1 (σ )
(88)
Therefore, we have V ≤ − e1
α −1 2
⎡⎣λmin {Q} − c1 ⎤⎦ ς
+ ⎡⎣l1L2 + Ψ 2 (σ ) ⎤⎦ ς
2 2
2
(89)
From (82) and 0 ⎜⎜ 1 2 ⎟⎟ ⎝ λmin {Q} − c1 ⎠
(94)
the differential inequality (93) is finite time convergent.
where 2 k ⎡ 2k + k (α + 1) −k1 (α + 1) ⎤ , l = [ k 1 Q= 1⎢ 2 1 ⎥ 1 α +1 ⎦ 2 ⎣ −k1 (α + 1)
l2 = [ k1 −1] 2
−2] 2 ,
Because α is sufficiently small,
α +1 2α
is sufficiently
α +1
(84)
2α small, we want ⎛⎜ l1 L2 + Ψ 2 (σ ) ⎞⎟ ⎜ λ {Q} − c ⎟ 1 ⎠ ⎝ min
to be sufficiently
small, therefore, it is required that 0 < l1 L2 + Ψ 2 (σ ) < 1 . λmin {Q} − c1
Let k (α + 1) ⎤ 1−2α α2+1 ⎡ l2 ⎥ 2 σ Ψ1 (σ ) = k1 ⎢ k2 + 1 2 ⎣ ⎦
(85)
Ψ 2 (σ ) = k2 [1 + l2 ] 21−α σ α
(86)
Then, we have c1 < λmin {Q} − l1 L2 − Ψ 2 (σ )
(95)
Therefore, from (88), we know that if 8
ς
2
>
Ψ 1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
(96)
Therefore, the damping coefficient of (103) is ς=
the differential inequality (93) is finite time convergent,
k1Ω1 2 k2 Ω 2
From (104), when the magnitude 1−α 2
(106) A
of tracking error
1
and the error system (75) is finite-time stable. Therefore,
is relatively small, A
we can get
with sliding mode differentiator, the nature frequency ωn can be kept small. Moreover, the proposed ς
2
≤
Ψ1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
(97)
> A 2 > A , therefore, comparing
differentiator is continuous, the outputs are all smooth. Therefore,
the
chattering
phenomenon
This concludes the proof.
high-frequency noises can be restrained sufficiently.
6. Frequency analysis of continuous differentiator
7. Simulations
For continuous differentiator (48), let x1 − v ( t ) = A sin ωt , we have 2
π∫
π
A sin ωτ
0
=A
α +1 2
2
π
∫
π
0
α +1 2
sgn ( A sin ωτ ) sin ωτ d ωτ
sin ωτ
α +3
α +3
π Denote Ω1 = 2 sin ωτ ∫ 0 π
2
1 < Ω1
Abstract: In this paper, a continuous finite-time-convergent differentiator is presented based on a strong Lyapunov function. The continuous differentiator can reduce chattering phenomenon sufficiently than normal sliding mode differentiator, and the outputs of signal tracking and derivative estimation are all smooth. Frequency analysis is applied to compare the continuous differentiator with sliding mode differentiator. The beauties of the continuous finite-time-convergent differentiator include its simplicity, restraining noises sufficiently, and avoiding the chattering phenomenon. Keywords: continuous finite-time-convergent, differentiator, frequency analysis. convergence compared with other typical differentiators; no
1. Introduction
chattering
phenomenon.
However,
peaking
Differentiation of signals is a well-known problem
phenomenon exist in the output of derivative estimation.
[1-8], which has attracted much attention in recent years.
Moreover, the ability of restraining noise was not
Obtaining the velocities of tracked targets is crucial for
considered.
several kinds of systems with correct and timely
In [21], we designed a hybrid differentiator with high
performances, such as the missile-interception systems
speed convergence, and it succeeds in applications to
[9] and underwater vehicle systems [10], in which
velocity estimations for low-speed regions only based
disturbances must be restrained. The simpleness and
on position measurements [22] and to quad-rotor aircraft
robustness of differentiators should be taken into
[23], in which only the convergence of signal tracking
consideration.
was described, but the convergence of derivative
The popular high-gain differentiators [4, 5, 6] provide for an exact derivative when their gains tend to infinity.
estimation was not given, and the regulation of parameters was complex.
Unfortunately, their sensitivity to small high-frequency
In this paper, a continuous finite-time-convergent
noise also infinitely grows. With any finite gain values
differentiator is presented based on a strong Lyapunov
such a differentiator has also a finite bandwidth. Thus,
function. For this differentiator we study its stability and
being not exact, it is, at the same time, insensitive with
finite time convergence characteristics by means of
respect to high-frequency noise. Such insensitivity may
Lyapunov functions. The continuous differentiator can
be considered both as advantage or disadvantage
reduce chattering phenomenon sufficiently than sliding
depending on the circumstances. Moreover, high gain
mode differentiator. The advantage of the use of
results in peaking phenomenon.
Lyapunov functions is to easily obtain the parameters of
In [7, 8], a differentiator via second-order (or
differentiator.
high-order) sliding modes algorithm was proposed. The
Frequency analysis is applied to compare the
information one needs to know on the signal is an upper
continuous finite-time-convergent differentiator with
bound for Lipschitz constant of the derivative of the
sliding mode differentiator. For nonlinear differentiators,
signal.
is
an extended version of the frequency response method,
introduced and there exists no chattering phenomenon in
describing function method [19, 24], can be used to
signal
approximately analyze and predict nonlinear behaviors
Although tracking,
second-order derivative
sliding
estimation
mode still
exist
chattering phenomenon.
of nonlinear differentiators. Even though it is only an
In [11], we presented a finite-time-convergent
approximation method, the desirable properties it
differentiator based on finite-time stability [12-14] and
inherits from the frequency response method, and the
singular perturbation technique [15-20]. The merits of
shortage of other, systematic tools for nonlinear
this
differentiator analysis, make it an indispensable
differentiator
exist
in:
rapidly
finite-time
1
component of the bag of tools of practicing control engineers. By describing function method, we can conclude that the continuous finite-time-convergent differentiator has the better ability of restraining high-frequency noises than sliding mode differentiator. This paper is organized in the following format. In section 2, preliminaries are introduced. In section 3,
where
x1 ," xn are suitable coordinates on R n and
r = ( r1 ," , rn ) with the dilation coefficients r1 " , rn positive real numbers. A
vector
f ( x ) = ( f1 ( x ) ," f n ( x ) )
field
Τ
is
problem statement of sliding mode differentiator is
homogeneous of degree k ∈ R with respect to the
given. In section 4, continuous finite-time-convergent
family of dilations δ ρr if
differentiator is presented, and its robustness analysis is
(
given in section 5. In section 6, frequency analysis of continuous differentiator is given.
)
f i ρ r1 x1 ," , ρ rn xn = ρ ri + k fi ( x )
In section 7, the
i = 1," , n, ρ > 0
simulations are given, and our conclusions are made in
(4)
System (1) is called homogeneous if its vector field f is
section 8.
homogeneous. The following lemma was presented in some
2. Preliminaries First of all, the concepts related to finite-time control
references like [12, 25, 26].
are given (See [13]).
Lemma1: Suppose that system (1) is homogeneous of
Definition 1: Consider a time-invariant system in the
degree k < 0 with respect to the family of dilations δ ρr ,
form of x = f ( x ) , f ( 0 ) = 0, x ∈ R n
where
f : Uˆ 0 → R n
is
continuous
on
an
(1)
f ( x ) is continuous and x = 0 is its asymptotically
open
stable equilibrium. Then equilibrium of system (1) is
neighborhood Uˆ 0 of the origin. The equilibrium x = 0
globally finite-time stable.
of the system is (locally) finite-time stable if (i) it is
continuous function V: D R such that the following
asymptotically stable, in Uˆ , an open neighborhood of
conditions hold:
the origin, with Uˆ ⊆ Uˆ 0 ; (ii) it is finite-time convergent
(1) V is positive definite;
in
Uˆ
, that is, for any initial condition x0 ∈Uˆ \ {0} , there
Lemma 2 [14, Theorem 4.2]. Suppose there exists a
(2) There exist real number c>0 and θ (0,1) and an open neighborhood
θ V ( x ) + c (V ( x ) ) ≤ 0 , x ∈ν \ {0}
is a settling time T > 0 such that every solution x ( t , x0 ) of system (1) is defined with x ( t , x0 ) ∈Uˆ \ {0}
for t ∈ [ 0, T ] and satisfies
lim x ( t , x0 ) = 0
(2)
t →T
origin x = 0 is globally finite-time stable.
and the settling-time function T is 1−θ 1 T ( x) ≤ V ( x )) ( c (1 − θ )
that assigns to every real ρ > 0 a diffeomorphism
δ ρ ( x1 ," , xn ) = ( ρ x1 ," , ρ xn ) r1
rn
(6)
and T is continuous. If in addition D = R n , V is proper,
V takes negative values on
R n \ {0} , then the
origin is a globally finite-time-stable equilibrium of (1). Assumption 1. For (1), there exist ρ ∈ ( 0,1] and a
Definition 2: A family of dilations δ ρr is a mapping
r
(5)
Then the origin is a finite-time-stable equilibrium of (1),
and and x ( t , x0 ) = 0 , if t ≥ T . Moreover, if Uˆ = R n , the
ν ⊂ D of the origin such that
nonnegative constant f
a such that
( z1 ) − f ( z1 )
≤ a
zi − zi
ρ
(7)
(3) 2
Let the Lyapunov function be
where z, z ∈ℜn .
1 V = k2 e1 + e22 2
Remark 1. There are a number of nonlinear functions actually satisfying Assumption 1. For example, one such function
xρ
is
ρ
x ρ − x ρ ≤ 21− ρ x − x ,
since
ρ ∈ ( 0,1] . Moreover, there are smooth functions also satisfying this property. In fact, it is easy to verify that
sin x − sin x ≤ 2 x − x
ρ
for any
Therefore,
(
)
0.5 V = k2 sgn ( e1 ) e2 − k1 e1 sgn ( e1 ) + e2 ( − k2 sgn ( e1 ) ) (15)
= −k1k2 e1
0.5
L2 . For differentiator (8), there exists a is the desired signal, and it is satisfied with time
ts > 0 such that x1 = v ( t ) , x2 = v ( t )
for
v ( t ) − v0 ( t ) ≤ σ . Therefore, for some positive constants
(9)
µ1 and µ2 the following inequalities are established [8]:
t ≥ ts . In fact, we let
1
e1 = x1 − v ( t ) , e2 = x2 − v ( t )
e1 = x1 − v0 ( t ) ≤ μ1σ , e2 = x2 − v0 ( t ) ≤ μ 2σ 2 3.3 Chattering phenomenon
Because switch function exists in the second
The tracking error system is
e1 = e2 − k1 e1
0.5
sgn ( e1 )
e2 = −k2 sgn ( e1 ) − v( t )
differential equation of differentiator (8), although the (11)
0.5
sgn ( e1 )
e2 ∈ − [ k2 − L2 , k2 + L2 ] sgn ( e1 )
output x1 is smooth, the output x2 is continuous but non-smooth, it is called as chattering phenomenon. In
We can get the following differential inclusion:
e1 = e2 − k1 e1
order to explain the problem, we give an example in the following. (12)
Example 1: For system 1
x1 = x2 − k1 x1 2 sgn ( x1 )
or
e1 = e2 − k1 e1
0.5
sgn ( e1 )
e2 = − k2 sgn ( e1 )
(19)
(10)
x2 = − k2 sgn ( x1 ) , k2 ∈ [ k2 − L2 , k2 + L2 ]
(13)
(20)
let k1 = 6, k2 = 9 , then we have the solutions x1 and x2 in
Fig. 1. 3
x1 = x2 − λ2
Ω ( x1 − v ( t ) ) A0.5
(26)
4 x2 = −λ1 ( x − v (t )) πA 1 The nature frequency of system (26) is
ωn =
2 k2
(27)
π A0.5
and we have 2ςωn =
k1Ω A0.5
(28)
Therefore, the damping coefficient is
Fig. 1 x1 and x2 of system (20)
From Fig. 1, we find that, for x2, chattering
ς=
phenomenon happens near the equilibrium. Moreover, if noises exist in signal, this chattering phenomenon will magnify noises near the equilibrium. In some velocity feedback systems, this chattering in x2 can make
k1Ω π 4 k2
(29)
With the error amplitude decreasing, the nature frequency
ωn
increases.
Moreover,
chattering
motors trembling. Therefore, chattering phenomenon must be removed sufficiently in the output x2 of a
phenomenon happen rear the equilibrium for the
differentiator.
chattering phenomenon will magnify noises.
3.4 Frequency analysis of sliding mode differentiator
We can also analyse sliding differentiator (8) from Let e1 = x1 − v ( t ) = A sin ωt , we have
π∫
π
A sin ωτ
0
= A0.5
2
π
∫
π
0
sin ωτ
1.5
restrain sufficiently high-frequency noises, we should
4. Continuous finite-time-convergent differentiator
sgn ( A sin ωτ ) sin ωτ d ωt
0.5
In order to remove chattering phenomenon and to design a continuous differentiator.
frequency characteristics.
2
discontinuous differentiator. If noises exist in signal, this
4.1 Continuous finite-time-convergent system
(21)
In order to design continuous differentiator, firstly, we give a finite-time stability Theorem as follow.
d ωt
Theorem 1: For continuous system
Then we can get
2
π
∫
π
0
sin ωτ dωt =
2
π
( − cos ωτ ) |0 = π
4
π
z1 = z2 − k1 z1
(22)
and 2
π
π
∫ ( sin ωτ ) 0
In (21), let Ω =
2
2
d ωt =
π∫
π
0
2
π
∫
π
0
sin ωτ
1 − cos 2ωτ d ωt = 1 2
1.5
1< Ω
0 , ts > 0 and α ∈ ( 0,1)
Therefore, the describing function of nonlinear function 0.5
2
α
dωt , we have
4
α +1
(25)
α +1 2k2 1 1⎛ α +1 ⎞ z1 + z22 + ⎜ k1 z1 2 sgn ( z1 ) − z2 ⎟ 2 2⎝ α +1 ⎠
2
(32)
and we can get
and the linearization system of differentiator (8) is
4
α +1 1 ⎞ α +1 ⎛ 2k V = ⎜ 2 + k12 ⎟ z1 + z22 − k1 z2 z1 2 sgn ( z1 ) ⎝ α +1 2 ⎠ ⎡ 4k2 ⎤ + k 2 − k1 ⎥ ⎡ α +1 ⎤1 = ⎢ z1 2 sgn ( z1 ) z2 ⎥ ⎢ α + 1 1 ⎥ ⎣ ⎦ 2 ⎢ −k 2 ⎦ ⎣ 1
Therefore, we have α −1
α −1
(33)
z1
2
α +1 2
2
min
⎤ sgn ( z1 ) z2 ⎥ ⎦
⎡ 4k 2 1 ⎢ 2 + k1 P = α +1 2⎢ ⎣ −k1
Τ
(34)
⎤ −k1 ⎥ ⎥ 2 ⎦
(35)
λmin { P}
(36)
From (35), we know that matrix positive-definite, and
P is symmetrical and
≤ V ≤ λmax { P} ς
2 2
3α +1
V 2(α +1)
(45)
3α + 1 3α + 1 = 2 (α + 1) 3α + 1 + (1 − α )
(46)
3α + 1 0 and 0 < α < 1 such that 5
the outputs x1 and x2 of differentiator (48) are
k12 (α + 1) α e2 e1 sgn ( e1 ) 2 ⎛ k12 (α + 1) ⎞ 3α +1 k1 (α + 1) 2 α −1 + k1 ⎜ k2 − e2 e1 2 ⎟ e1 2 − 2 2 ⎝ ⎠ k 2 (α + 1) α + 1 e2 e1 sgn ( e1 ) 2 α +1 ⎛ ⎞ + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ v( t ) − e2 v( t ) ⎝ ⎠
V = −2k1k2 e1
smooth, and
ς
2
⎛ lL2 ⎞ ≤ ⎜⎜ ⎟⎟ ⎝ λmin {Q} ⎠
α +1 2α
(49)
is satisfied. Moreover, when α = 0 , output smooth and ς = 0 for
x1 is
x2 is continuous but non-smooth, and
α −1 2
Τ
Let
⎡
ς = ⎢ e1 ⎣
2 k1 ⎡ 2k2 + k1 (α + 1) −k1 (α + 1) ⎤ , ⎢ ⎥ α +1 ⎦ 2 ⎣ −k1 (α + 1)
(51) Q=
(52)
k1 (α + 1)
lL2 < 1 can be obtained in differentiator (48) and λmin {Q}
becomes sufficiently small in a finite time. This
α +1 2
l = [ k1
−2] 2 = k12 + 4
ς
= e1
2 2
α −1
continuous differentiator (48) is continuous and its
Therefore, we can get
is
selected
to
design
the
Proof: The error system of differentiator (48) is
e1 = e2 − k1 e1
α +1 2
sgn ( e1 )
2
V ≤ −λmin {Q} ς = −λmin {Q} ς
differentiator, regulation of parameters is easier.
(53)
e2 = − k2 e1 sgn ( e1 ) − v( t )
(58)
(59)
α +1
+ e22
(60)
From (60) and 0 < α < 1 , we have
Comparing with discontinuous differentiator (8),
function
(57)
From (57), we have
e1
outputs are all smooth. Moreover, because a strong
Τ
and
will be given in the following proof.
Lyapunov
⎤ sgn ( e1 ) e2 ⎥ ⎦
2 k1 ⎡ 2k2 + k1 (α + 1) − k1 (α + 1) ⎤ ⎢ ⎥ α +1 ⎦ 2 ⎣ − k1 (α + 1)
0 < α < 1 is selected sufficiently small, therefore, 2
(56)
(50)
2 2 k1 > 0 , k2 > 2 ( k1 + 4 ) L2 2
ς
(55)
⎡ α2+1 ⎤ sgn ( z1 ) ⎥ v t + [ k1 −2] ⎢ e1 () ⎢⎣ ⎥⎦ e2
l = [ k1 −2] 2 , Q=
⎡ α2+1 ⎤k sgn ( e1 ) e2 ⎥ 1 ⎢⎣ e1 ⎦2
α +1 ⎡ 2k + k 2 (α + 1) −k1 (α + 1) ⎤ ⎡ e 2 sgn ( e ) ⎤ 1 1 ⎥ ⎢ ×⎢ 2 1 ⎥ α +1 ⎦ ⎢ ⎥⎦ e ⎣ − k1 (α + 1) ⎣ 2
⎡ α +1 ⎤ ς = ⎢ e1 2 sgn ( e1 ) e2 ⎥ , e1 = x1 − v ( t ) , ⎣ ⎦ e2 = x2 − v ( t ) ,
+
Therefore, we can get V = − e1
t ≥ ts . Where v ( t ) ≤ L2 , and
3α +1 2
≥ ς
α −1 α +1
α −1 α +1
(61)
2
2
ς 2 + lL2 ς
3α +1 α +1 2
− lL2 ς
⎛ = − ⎜ λmin {Q} ς ⎝
2
2α α +1 2
(62)
⎞ − lL2 ⎟ ς ⎠
α
Moreover, we have
Select a Lyapunov function as V=
α +1 2k 2 1 1⎛ α +1 ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ 2 2⎝ α +1 ⎠
α +1
2
The time derivative of Lyapunov function (54) along the solutions of system (53) is
ς
(54)
We can select
2
⎛ lL2 ⎞ 2α ≤ ⎜⎜ ⎟⎟ ⎝ λmin {Q} ⎠
(63)
k1 and k2 such that
6
λmin {Q} > lL2
(64)
in signal v ( t ) , i.e., v ( t ) = v0 ( t ) + δ ( t ) , where v0 ( t )
In fact, s− sI − Q =
k1 ( 2k2 + k12 (α + 1) ) k21 k1 (α + 1) 2 =0 k1 k k1 (α + 1) s − 1 (α + 1) 2 2
(65) is the desired second-order derivable signal, δ ( t ) is a
i.e.,
(
)
k k2 s − 1 2k2 + ( k12 + 1) (α + 1) s + 1 2k2 (α + 1) = 0 2 4 2
bounded noise and satisfied with δ ( t ) ≤ σ . Then, the (66) following inequality is established in finite time
The minimum eigenvalue of (66) is λmin {Q} =
−
(
2 k1 ⎛⎜ 2k2 + ( k1 + 1) (α + 1) 2⎜ 2 ⎝
( 2k + ( k + 1) (α + 1) ) 2 1
2
2
2
Theorem 3: For differentiator (48), if there exist a noise
)
ς
(67)
⎞ − 8k2 (α + 1) ⎟ ⎟ ⎟ ⎠
⎡
ς = ⎢ e1 ⎣
( 2k + ( k
2 1
2
−
+ 1) (α + 1) 2
l1 = [ k1
)
)
2
⎞ − 8k2 (α + 1) ⎟ ⎟ ⎟ ⎠
α +1 2
(72)
Τ
⎤ sgn ( e1 ) e2 ⎥ , e1 = x1 − v0 ( t ) , ⎦
⎡ k α + 1) ⎤ 1−2α α2+1 , −2 ] 2 , Ψ1 (σ ) = k1 ⎢ k2 + 1 ( l2 ⎥ 2 σ 2 ⎣ ⎦
Ψ 2 (σ ) = k2 [1 + l2 ] 21−α σ α
(68)
Q=
> L2 k12 + 4
2 k1 ⎡ 2k2 + k1 (α + 1) − k1 (α + 1) ⎤ ⎢ ⎥ α +1 ⎦ 2 ⎣ − k1 (α + 1)
(73)
Proof: Let
Therefore, we get k2 >
Ψ 1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
e2 = x2 − v0 ( t ) , v0 ( t ) ≤ L2
have
(
≤
where
Because λmin {Q} > lL2 and k1 > 0 are required, we
2 k1 ⎛⎜ 2k2 + ( k1 + 1) (α + 1) 2⎜ 2 ⎝
2
2 ( k12 + 4 ) L22
e1 = x1 − v0 ( t ) , e2 = x2 − v0 ( t )
(69)
k (α + 1) 2 1
(74)
The error system is
and k1 > 0
(70)
e1 = e2 − k1 e1 − δ
Therefore, when 0 < α < 1 is sufficiently small,
α + 1 is sufficiently large, finally, the tracking and 2α estimation errors are sufficiently small in a finite time. When α = 0 , Levant differentiator (8) is obtained, and the Lyapunov function can be designed as V=
output
1 2k 2 1 1⎛ ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ α +1 2 2⎝ ⎠
α +1 2
sgn ( e1 − δ )
e2 = − k2 e1 − δ sgn ( e1 − δ ) − v0 ( t ) Let
Δ1 = − e1 − δ
α +1 2
sgn ( e1 − δ ) + e1
(71)
proof.
2
sgn ( e1 )
(76)
α
Therefore, we have
x1 is smooth and x2 is continuous but
non-smooth, and ς = 0 for t ≥ ts . This concludes the
α +1
Δ 2 = − e1 − δ sgn ( e1 − δ ) + e1 sgn ( e1 ) α
2
(75)
α
Δ1 ≤ 2
1−α 2
δ
Δ 2 ≤ 21−α δ
α +1 2
α
≤2
1−α 2
σ
α +1 2
(77)
≤ 21−α σ α
The Lyapunov function is selected as
5. Robustness analysis of continuous differentiator 7
V=
α +1 2k 2 1 1⎛ α +1 ⎞ e1 + e22 + ⎜ k1 e1 2 sgn ( e1 ) − e2 ⎟ α +1 2 2⎝ ⎠
2
(78)
V ≤ − e1
Let α +1
⎡
ς = ⎢ e1 ⎣
2
Therefore, we have
+ e1
Τ
⎡ 4k 2 ⎤ 2 ⎤ sgn ( e1 ) e2 ⎥ , P = 1 ⎢ α + 1 + k1 − k1 ⎥ ⎦ ⎥ 2⎢ ⎣
− k1
(79)
2 ⎦
α −1 2
λmin {Q} ς 2 + l1L2 ς
α −1 2
2
Ψ1 (σ ) ς
Suppose there exist a positive constant
we can get
c1 ς V = ς Τ Pς
+ Ψ 2 (σ ) ς
2
(80)
and we know that matrix P is symmetrical and
2
(87)
2
2
c1 such that
> Ψ 1 (σ )
(88)
Therefore, we have V ≤ − e1
α −1 2
⎡⎣λmin {Q} − c1 ⎤⎦ ς
+ ⎡⎣l1L2 + Ψ 2 (σ ) ⎤⎦ ς
2 2
2
(89)
From (82) and 0 ⎜⎜ 1 2 ⎟⎟ ⎝ λmin {Q} − c1 ⎠
(94)
the differential inequality (93) is finite time convergent.
where 2 k ⎡ 2k + k (α + 1) −k1 (α + 1) ⎤ , l = [ k 1 Q= 1⎢ 2 1 ⎥ 1 α +1 ⎦ 2 ⎣ −k1 (α + 1)
l2 = [ k1 −1] 2
−2] 2 ,
Because α is sufficiently small,
α +1 2α
is sufficiently
α +1
(84)
2α small, we want ⎛⎜ l1 L2 + Ψ 2 (σ ) ⎞⎟ ⎜ λ {Q} − c ⎟ 1 ⎠ ⎝ min
to be sufficiently
small, therefore, it is required that 0 < l1 L2 + Ψ 2 (σ ) < 1 . λmin {Q} − c1
Let k (α + 1) ⎤ 1−2α α2+1 ⎡ l2 ⎥ 2 σ Ψ1 (σ ) = k1 ⎢ k2 + 1 2 ⎣ ⎦
(85)
Ψ 2 (σ ) = k2 [1 + l2 ] 21−α σ α
(86)
Then, we have c1 < λmin {Q} − l1 L2 − Ψ 2 (σ )
(95)
Therefore, from (88), we know that if 8
ς
2
>
Ψ 1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
(96)
Therefore, the damping coefficient of (103) is ς=
the differential inequality (93) is finite time convergent,
k1Ω1 2 k2 Ω 2
From (104), when the magnitude 1−α 2
(106) A
of tracking error
1
and the error system (75) is finite-time stable. Therefore,
is relatively small, A
we can get
with sliding mode differentiator, the nature frequency ωn can be kept small. Moreover, the proposed ς
2
≤
Ψ1 (σ ) λmin {Q} − l1 L2 − Ψ 2 (σ )
(97)
> A 2 > A , therefore, comparing
differentiator is continuous, the outputs are all smooth. Therefore,
the
chattering
phenomenon
This concludes the proof.
high-frequency noises can be restrained sufficiently.
6. Frequency analysis of continuous differentiator
7. Simulations
For continuous differentiator (48), let x1 − v ( t ) = A sin ωt , we have 2
π∫
π
A sin ωτ
0
=A
α +1 2
2
π
∫
π
0
α +1 2
sgn ( A sin ωτ ) sin ωτ d ωτ
sin ωτ
α +3
α +3
π Denote Ω1 = 2 sin ωτ ∫ 0 π
2
1 < Ω1
