Delay-Dependent Stability Analysis and Design of Input-delayed
Systems by Smooth Sliding Mode Control: Lagrange Theorem
Approach
ELBROUS M. JAFAROV
Aeronautical and Astronautical Engineering Department
Istanbul Technical University
34469, Maslak, TURKEY
Abstract: - A new smooth sliding mode control design methodology based on Lagrange mean value theorem is
proposed for stabilization of single input delayed systems. The Lagrange mean value theorem as a basic theorem of
calculus is used for the design of linear sliding mode time-delay controller for the first time. This controller satisfies the
sliding condition using a Zhou and Fisher type continuous control law eliminating the chattering effect. The
constructive delay-dependent asymptotically stable sliding conditions are obtained by using the augmented Lyapunov-
Krasovskii functionals and formulated in terms of simple (4x4)-matrix inequality with scalar elements. Developed
design approach can be extended to robust stabilization of sliding system with unknown but bounded input delay. The
maximum upper bounds of delay size can be found by using simple optimization algorithms. Helicopter hover control
is considered as a design example for illustrating the usefulness of smooth sliding mode approach. Unstable helicopter
dynamics are successfully stabilized by using linear sliding mode time-delay controller. For example, settling time is
about 20 sec. Therefore, simulation results confirmed the effectiveness of proposed design methodology. Apparently,
the proposed method has a great potential in design of time-delayed controllers.
Key-Words: - Input-delayed systems, Lagrange mean value theorem, sliding mode control, robust stabilization,
Lyapunov-Krasovskii functional method.
Received: June 15, 2021. Revised: April 17, 2022. Accepted: May 18, 2022. Published: June 16, 2022.
1 Introduction
Time-delay effect is frequently encountered in oil-
chemical systems, metallurgy and machine-tool process
control, nuclear reactors, bio-technical systems missile-
guidance and aircraft control systems, aerospace remote
control and communication control systems, etc. The
presence of delay effect complicates the analysis and
design of control systems. Moreover, time delay effects
in state vector, especially in control input degrades the
control performances and make the closed–loop
stabilization problem challenging. A common design
method of input-delayed systems is well known Smith
predictor control to cancel the effect of time-delay.
Smith predictor is a popular and very effective long
delay compensator for stable processes. The main
advantage of the Smith predictor control method is that,
the time-delay is eliminated from the characteristic
equation of the closed-loop system. Classical Smith
predictor was suggested by Smith [1], [2]. Modified
Smith predictor scheme’s have been advanced by
Marshall [3], Aleviskas and Seborg [4], Watanabe and
Ito [5], [6], Al-Sunni and Al-Neymer [7], Majhi and
Atherton [8].
The other important control design method of input-
delayed systems is the reduction method that was
suggested by Kwon and Pearson [9].
Recently several new variable structure control design
methods for stabilization of various classes of systems
without time-delay are developed, for example by Wang,
Lee and Juang [10], Lee and Xu [11], Cao and Xu [12],
[13], Choi [14], Edwards, Spurgeon and Hebden [15],
Sabanovic, Fridman and Spurgeon [16], Jafarov [17]-
[19], Yeh, Chien and Fu [20], Singh, Steinberg and Page
[21], Koshkouei and Zinober [22]. But, there is no a
large number of papers concerning the problem of
stabilization of time-delay systems by variable structure
control, for example see Shyu and Yan [23], Yan [24],
Luo, De La Sen and Rodellar [25], Gouaisbaut,
Dambrine and Richard [26], Richard [27], Perruquetti
and Barbot [28], Jafarov [29], [30], Li and De Carlo
[31], Gouaisbaut, Blango and Richard [32], Koshkouei
and Zinober [33] etc. In analysis and design of time-
delay systems by sliding mode control the Lyapunov-
Krasovskii functional method is commonly used. Recent
advances in time-delay systems are presented by Richard
[27], Fridman and Shaked [34], Jafarov [35], Niculescu
and Gu [36], Niculescu [37], Mahmoud [38], Gu,
Kharitonov and Chen [39], Boukas and Liu [40]. Some
sufficient delay-dependent stability conditions for linear
delay perturbed systems are derived using exact
Lyapunov-Krasovskii functionals by Kharitonov and
Niculescu [41]. Several new LMI delay-dependent
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Elbrous M. Jafarov
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robust stability results for linear time-delay systems with
unknown time-invariant delays by using Padé
approximation are presented by Zhang, Knospe and
Tsiotras [42]. Both delay-independent and delay-
dependent robust stability LMI’s from conditions for
linear time-delay systems with unknown delays by using
appropriately selected Lyapunov-Krasovskii functionals
are systematically investigated by Zhang, Knospe and
Tsiotras in another paper [43]. Stability of the internet
network rate control with diverse delays based on
Nyquist criterion is considered by Tian and Yang [44].
Improved delay-dependent stability conditions for time-
delay systems in terms of strict LMI’s avoiding cross
terms are developed by Xu and Lam [45]. A new state
transformation is introduced to exhibit the delay-
dependent stability condition for time-delay systems by
Mahmoud and Ismail [46].
Variable structure control is often used to handle the
worst-case control environment: parametric
perturbations, external disturbances with knowledge of
only the upper bounds etc. Sometimes we may come up
with more appropriate control approaches such as
incorporating VSC with linear control, time-delay
control etc. It is well known that classical sliding mode
control uses a discontinuous control action to drive the
state to the origin along the reaching and sliding paths
and is insensitive to parametric uncertainties and
external disturbances. However, the control chattering
due to the discontinuity in control law sometimes is
undesirable. The continuous sliding mode control
approach satisfies the sliding conditions using a
continuous control law without requiring discontinuous
switching in the controller. Therefore, it retains the
advantages of sliding control but without the chattering
phenomena. Such approach is used by Zhou and Fisher
[47], Shtessel and Buffingtonn [48] etc. Continuous
sliding mode control concept is discussed in details and
its comparison analysis with the conventional
discontinuous sliding mode control by Zhou and Fisher
[47].
Note that VSC cannot be directly applied to the control
of input-delayed system. Feng, Mian and Weibing [49],
Hu, Basker and Crisalle [50] have been successfully
used the reduction method combined with variable
structure control for stabilization of certain and uncertain
multivariable input-delayed systems with known delays.
In this paper, a new sliding mode control design
methodology for the single input delayed systems with
known or unknown but bounded delays is developed.
This design method is based on the Lagrange mean value
theorem, which is used for the first time for the
stabilization of input-delayed systems. Proposed linear
sliding mode time-delay controller also satisfies the
sliding condition, but in contrast to classical variable
structure control, uses Zhou and Fisher type of
continuous control law without requiring discontinuous
switching in the controller. Therefore, undesired control
chattering in this case is avoided.
The constructive delay-dependent asymptotical stability
and robustly stable sliding conditions are obtained by
using the Lyapunov-Krasovskii functional method and
formulated in terms of some matrix inequalities. Hence,
it is possible also to compute the maximum upper bound
of the allowable time-delay
h
using efficient convex
optimization algorithms. Helicopter hover control is
considered as a design example for illustrating the
performances of smooth sliding mode approach.
Unstable helicopter dynamics is successfully stabilized
by using linear sliding mode time-delay controller. For
example, settling time is about 20 sec. Therefore,
simulation results confirmed the effectiveness of the
proposed design methodology.
2 A New Design Method
Let us consider the following single input-delayed
system
( ) ( ) ( )x t Ax t bu t h
(1)
where
()xt
is the measurable n-state vector, u(t) is the
scalar control input,
A
is a constant real (
nn
)-matrix,
b is the constant n-vector,
0h
is a time-delay
0h const
or unknown but bounded delay
0hh
and initial condition
( ) ( )u t t
for
0ht
, where
()t
is a known scalar function.
This design method is based on the Lagrange mean value
theorem.
Remember that Lagrange mean value theorem [53], [54]
is stated as follows
( ) ( ) ( ),
f b f a f a b
ba
(2)
where
()fx
is a continuous at every point of the closed
interval [a, b] and differentiable at every point of its
interior (a, b) or in terms of delayed control input
( ) ( ) ( )u t h u t hu
(3)
where
is a point in
t h t
.
After introducing the
parameter, the constructive
delay-dependent asymptotical stability and robustly
stable sliding conditions can be derived by using the
augmented Lyapunov-Krasovskii functionals.
Now, after preparing the necessary background we can
present a new continuous sliding mode control design
methodology for input-delayed systems with known or
unknown but bounded delays.
Select a Zhou and Fisher type of continuous sliding
mode controller as
( ) ( )u t ks t
(4)
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where k is a constant gain scalar to be designed. Assume
that linear sliding mode is defined in n-dimensional state
space by the following linear function:
( ) ( )
T
s t c x t
(5)
where c is a design n-vector to be selected. This linear
control law must satisfy the sliding condition.
Using the Lagrange mean value theorem (3) let us
represent input-delayed system (1) as follows
( ) ( ) [ ( ) ( )]x t Ax t b u t hu
( ) ( ) ( )Ax t bks t bhu
( ) ( ) ( )Ax t bks t kbhs
( ) ( ) ( )
T
Ax t bks t kbhc x
(6)
( ) ( ) [ ( ) ( )]
T
Ax t bks t kbhc Ax bks h
2
( ) ( ) ( ) ( )]
TT
Ax t bks t kbhc Ax k hbc bs h
From (6) it is obvious that full delay term h already
appears in transformed system. Now, our goal is to
organize an asymptotically stable linear sliding mode on
defined hyper plane
( ) 0st
(5). Stable sliding mode
conditions are formulated in the following theorem. But,
we need to make the following assumption.
Assumption 1: Time-delay parameter
is a time-
dependent function and norm-bounded such that
0 1 ( ) 1t
(7)
where
is a scalar.
Note that time-delay Assumption 1 is conventional and
is commonly used by many authors, for example, by
Ikeda and Ashida [55], Su and Chu [56], Su, Ji and Chu
[57], Wu, He, She and Liu [58], Kim [59] etc.
Theorem 1: Suppose that Assumption 1 holds. Then the
transformed time-delay system (6) driven by continuous
sliding mode controller (4), (5), is delay-dependent
asymptotically stable relative to the manifold
( ) 0st
(5), if there are design parameters
, c, , , k
and
such that the following sliding conditions are satisfied:
22
22
11
b ( ) 0
22
1
(1 ) 0 0
2
1( ) 0 - (1- ) 0
2
0
T T T
T
T
kc khc b k h c b
khc b
H
k h c b
0 0 -
0H
(8)
or
T
kcb
(9)
1
(10)
TT cAc
(11)
where
is any left or right eigenvalue of matrix A;
,
and
are some positive adjustable scalars.
Note that, design of the manifold
( ) 0st
(5) does not
imply assigning the eigenvalue
of the matrix; it
appears only in proof of the theorem and may take an
arbitrary value as pointed by Ackermann and Utkin [52].
Proof: Choose an augmented Lyapunov-Krasovskii
functionals as
22
1
( ( ), ( ), ( ), ( )) ( ) ( )
2h
V s t s s h s t h s t s d
22
( ) ( )
tt
th
s d s d
(12)
where
,
and
are some positive adjustable scalars.
The time derivative of (12) along the state trajectory of
(6) can be calculated as follows:
2 2 2
2 2 2
( ) ( ) ( ) ( ( ) ( )) ( )
( ) ( ) ( ) ( )
V s t s t t s s h s t
t s s t s t h
2 2 2 2
2 2 2
( )[ ( ) ( ) ( )
( ) ( )] ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
T T T T
T
s t c Ax t kc bs t khc bc Ax
k h c b s h s t s t h
t s t s h t s
(13)
Since
22
( ) ( ) ( )t s s
(14)
22
( ) ( ) (1 ) ( )t s h s h
(15)
and (11) hold, then (13) reduces to
22
2 2 2
2 2 2
22
2
2 2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) (1 ) ( )
(1 ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( (1 ) ) (
TT
T
TT
T
V s t kc bs t khc b s s t
k h c b s h s t s t
s h s s h
s s t h
kc b s t khc b s s t
k h c b s t s h s
22
)
(1 ) ( ) ( )s h s t h
22
22
() 11
b ( ) 0
22
() 1
(1 ) 0 0
2
1
() ( ) 0 - (1-
2
()
T
T T T
T
T
st kc khc b k h c b
skhc b
sh k h c b
s t h
) 0
0 0 0 -
2
min
( )
( )
( ) ( ) ( ) ( ) 0
()
()
T
st
s
y t Hy t H y t
sh
s t h
(16)
where
( ) ( ) ( ) ( ) ( ) T
y t s t s s h s t h
Note that matrix
H
has its own quadratic structure
1
T
H MH M
where
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- 0 0 0
0 )-(1- 0 0
0 0 )1( 0
0 0 0
1
bkc
H
T
1 0 0 0
0 1 0 0
0 0 1 0
0
)1(
)(0.5
)1(
5.0
1
22
bchkbckh
M
TT
Since M is a nonsingular and
10H
because its leading
principle elements are always negative then
0H
.
Therefore, condition (16) means that manifold
( ) 0st
is
reached in finite time and the reaching time can be
evaluated approximately as follows:
min
(0)
()
s
y
tH
(17)
Thus, the time-delay system (6) with known delay is
delay-dependent asymptotically stable relative to the
manifold
( ) 0st
(5).
If we consider a case where the delay term is unknown
but bounded
0hh
then we can solve the following
convex optimization problem:
OP: maximize h
Subject to conditions (8) (18)
with
0
,
0
,
0
This ends the proof of Theorem 2.
Let us consider a simple analytical example to illustrate
our design approach
Example 1: Consider the first order input-delayed
system
( ) ( ) ( )x t ax t bu t h
(19)
where a and b are some constant scalars.
Define a continuous sliding mode controller as follows.
( ) ( )u t ks t
(20)
( ) ( )s t cx t
(21)
where k and c are the design scalars.
Substituting (3) with (20) and (21) into (19) we have
( ) ( ) ( )x t ax t bu t h
( ) ( ) ( )ax t bu t bhu
)()()(
sbhktbkstax
(22)
)()()(
xbhkctbkstax
)()()()( 22 hcshkbabhkcxtbkstax
)()()()( 22 hcshkbabhkstbkstax
Then the time-derivative of (12) along (22) is given by
22
2 2 2 2
( ) ( ) ( ) ( ) ( )
+ ( ) ( ) ( ) ( ) ( )
V s t s t t s s h
s t t s s t s t h
2 2 2 2 2
( )[ ( ) ( ) ( )
( )] ( ) ( ) ( ) ( )
s t acx t bkcs t abchks
b c hk s h t s t s h
2 2 2 2
( ) ( ) ( ) ( ) ( )s t t s s t s t h
2
2 2 2 2
22
( ) ( ) ( )
( ) ( ) (1 ) ( )
(1 ) ( ) ( )
a bck s t abchks t s
b c hk s t s h s
s h s t h
(23)
2 2 2
2 2 2
() 11
- 0
22
() 1
- (1 ) 0 0
2
1
()
0 - (1- )
2
()
T
st bck a abchk b c hk
sabchk
sh b c hk
s t h
0
0 0 0 -
2
min
( )
( )
( ) ( ) ( ) ( ) 0
()
()
T
st
s
y t H y t H y t
sh
s t h
(24)
or the following Sylvester’s conditions hold:
1
| | 0H bck a
or
bck a
with
0
(25)
2
2
| | (1 )
1
( ) 0
4
H bck a
abchk
(26)
3
| | 0H
and
4
| | 0H
respectively.
Then, time-delay system (22) with known h is delay-
dependent asymptotically stable relative to the
( ) 0st
.
If we consider a case where h is unknown but bounded
0hh
then the maximum upper bound can be
calculated as follows.
From min
2
| | 0H
(26) we compute
22
1
(1 ) ( ) 0
2
Habc abch k
k
(27)
Hence,
2
2 (1 )
()
bc
hk abc
(28)
with
(1 ) 0
,
0bc
.
Thus, time-delay system (22) with unknown but
bounded delay term is robustly asymptotically stable
relative to the
( ) 0st
with upper bound
h
(28).
3 Design example: Helicopter hover
control
The linearized longitudinal motion of helicopter near
hover (Fig.1) can be modeled by the normalized linear
third order system [60] with introduced pilot time-delay
h [61] as follows:
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0.4 0 0.01 6.3
1 0 0 0 ( )
1.4 9.8 0.02 9.8
qq
th
uu
(29)
where
q
is the pitch rate,
is the pitch angle of fuselage,
u
is the horizontal velocity (standard aircraft notation),
is the rotor tilt angle (control variable),
h
is the pilot’s effective time- delay, for example,
0.43h
s.
Continuous sliding mode controller is formed as (4):
( ) ( )u t ks t
(30)
where
k
is a scalar to be designed by (8), (9) and sliding
function is defined as (5):
1 2 3
()s t c q c c u
(31)
where
1 2 3
,,c c c
are design parameters to be determined.
Design procedure can be fulfilled with MATLAB
programming (which is given in Appendix 1) by the
following steps:
-0.6565
( ) 0.1183 + 0.3678i
0.1183 - 0.3678i
eig A
A is unstable with one pear conjugate complex-roots.
Calculate matrix (8)
-0.0313 0.0177 -0.0034 0
0.0177 -0.1640 0 0
-0.0034 0 -0.1820 0
0 0 0 -0.3000
H
-0.3000
-0.1821
() -0.1663
-0.0289
eig H
H is a negative definite matrix.
c1 = -0.0389
c2 = 0.0592
c3 = -0.9975
k = 0.0125
h = 0.4300
eta = 0.0900
alpha = 0.2000
beta =0.0200
gamma = 0.3000
hmax = 1.714
cTb = -10.0204
Thus all design parameters are calculated. Maximum
upper bound of time delay, hmax = 1.714, is found from
condition (8). A block diagram of continuous sliding
mode controller for helicopter input-delayed system (1),
(4), (5) or (29), (30), (31) is shown in Fig. 2. This system
is simulated by using MATLAB-Simulink. Continuous
sliding mode controller is performed by linear Simulink
blocks
()st
and
()ut
. Note that, these are not variable
structure blocks, but linear blocks satisfying the sliding
condition (16). Helicopter control performances are
shown in Fig. 3, from which can be seen that unstable
helicopter dynamics is successfully stabilized by using
linear sliding mode controller. For example, settling time
is about 20 sec. Reaching time is also 20 sec. Therefore,
simulation results confirmed the usefulness of the
developed design methodology.
4 Conclusion
A new continuous sliding mode control design
methodology based on Lagrange mean value theorem is
proposed for stabilization of single input delayed
systems. The Lagrange mean value theorem as a basic
theorem of calculus is used for the design of linear
sliding mode time-delay controller for the first time. This
controller satisfies the sliding condition using a Zhou
and Fisher type continuous control law eliminating the
chattering effect. The constructive delay-dependent
asymptotically stable sliding conditions are obtained by
using the augmented Lyapunov-Krasovskii functionals
and formulated in terms of simple (
44
)-matrix
inequality with scalar elements. Developed design
approach are extended to robust stabilization of sliding
system with unknown but bounded input delay. The
maximum upper bounds of delay size are found by using
simple optimization algorithms. Helicopter hover control
is considered as design example for illustrating the
performances of smooth sliding mode approach.
Unstable helicopter dynamics are successfully stabilized
by using linear sliding mode time-delay controller. For
example, settling time is about 20 sec. Therefore,
simulation results confirmed the effectiveness of the
proposed design methodology. Apparently, the proposed
method has a great potential in design of time-delayed
controllers.
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Elbrous M. Jafarov
E-ISSN: 2224-2856
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Volume 17, 2022
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Appendix:
clear; clc;
A = [-0.4 0 -0.01; 1 0 0;
-1.4 9.8 -0.02];
[V,D] = eig(A);
D = diag(D)
% selection according to case a):
lamda_L = D(1)
c1 = V(1,1)
c2 = V(2,1)
c3 = V(3,1)
h = 0.43
eta = 0.09
alpha = 0.2
beta = 0.2
gamma = 0.3
c_T = [c1 c2 c3];
b = [6.3; 0; 9.8];
k = 0.8*(lamda_L+beta+gamma)/(c_T*b)
h_max = 1.714 % delay
cTb = c_T*b
h11 = lamda_L-k*c_T*b+beta+gamma
h22 = alpha*eta-(1-eta)*beta
h33 =-alpha*(1-eta)
h44 = -gamma
H1 = [ lamda_L-k*c_T*b+beta+gamma;
0.5*k*h*c_T*b*lamda_L;
-0.5*k^2*h*(c_T*b)^2; 0];
H2 = [0.5*k*h*c_T*b*lamda_L;
alpha*eta-(1-eta)*beta; 0; 0];
H3 = [-0.5*k^2*h*(c_T*b)^2; 0;
-alpha*(1-eta); 0];
H4 = [0; 0; 0; -gamma];
H = [H1 H2 H3 H4];
Fig.1 Helicopter.
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Fig.2 Block diagram of linear sliding mode controller for input-delayed system
a) State time responses b) Linear sliding mode control function
c) Sliding function
Fig.3 Smooth sliding mode control
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