16th IEEE International Conference on Control Applications
Part of IEEE Multi-conference on Systems and Control
Singapore, 1-3 October 2007
MoB02.1
Design of a Robust Sliding Mode Controller using Multirate Output
Feedback
A. Inoue, M. Deng, K. Matsuda, and B. Bandyopadhyay∗
Department of Systems Engineering
Okayama University
3-1-1 Tsushima-Naka, Okayama 700-8530, Japan
{inoue, deng}@suri.sys.okayama-u.ac.jp
* Indian Institute of Technology Bombay
Mumbai, India
Abstract— This paper considers a robust sliding mode control
scheme using multirate output feedback. Using different plant
output rates including the former plant output information,
sampling error at multiple sampling interval is considered
to discrete-time output feedback control design. As a result,
control performance can be improved. Meanwhile, the multirate
output feedback can also guarantee closed-loop system stability.
I. INTRODUCTION
In real control applications, different control signal rates
and sensor output rates are often used. This kind of system
is called multirate system. Nowadays, the research on the
multirate system control design has become a challenging
work. So far, many interesting results have been proposed
for systems [1,2, and references therein]. In this paper, a
robust sliding mode control scheme using multirate output
feedback is explored. The main reason is described as
follows. The feature of a sliding mode control system is
that the controller is switched between distinctive control
structures and the system trajectory is forced to reach the
sliding surface and to slide along it. Once the states of the
controlled system enter the sliding mode, the dynamics of
the system are determined by the selection of sliding surface
and are independent of uncertainties and disturbances. As a
result, the sliding mode control is a robust control method
and it has found broad applications. Considering the sliding
mode control with multirate feedback, some studies and
research were undertaken [3, -, 6]. In this paper, although
the continuous-time sliding mode controller is robust to
disturbance, the discrete-time controller is sensitive to output
measurement error, because a switching function is included
in the controller, non-linear is strongly affected by the error at
the switching point. As a continuous research, sampling error
at multiple sampling interval is considered to discrete-time
output feedback control design by considering the former
plant information. That is, state of the sliding mode controller
is realized by the combining use of multirate output feedback
and past plant outputs, then the control gains include additional design parameters, namely, the gains have redesigned
parameters. Selecting the value of the redesigned parameters,
we can design gains robustly to observation noise. As a
1-4244-0443-6/07/$20.00 ©2007 IEEE.
result, control performance is improved. Meanwhile, the
multirate output feedback can also guarantee closed-loop
system stability.
This paper is organized as follows. Section II states
the problem setup. Section III discusses the design of the
proposed sliding mode controller. In Section IV, a numerical
example is given to show the proposed method. Conclusion
is drawn in Section V.
II. PROBLEM STATEMENT
Consider the following single input single output continuous time LTI system described by state equation
ẋ(t) = Ax(t) + Bu(t)
(1)
y(t) = Cx(t)
(2)
where x(t) is n-dimensional state, u(t) is input and y(t) is
output. Let the above system sampled at a sampling time τ ,
we have
x((k + 1)τ ) = Φτ x(kτ ) + Γτ u(kτ )
y(kτ ) = Cx(kτ )
(3)
(4)
It is assumed that the pair (Φτ , Γτ ) is controllable and the
pair (Φτ ,C) is observable. In this paper, the objective is to
design control u by using multirate output feedback.
III. D ESIGN OF MULTIRATE OUTPUT FEEDBACK
CONTROL SYSTEM
In this section, an algorithm for computing sliding mode
controller and switching surface are considered by using multirate output feedback, namely, using plant output samples.
Using sampling time ∆, ∆ = τ /N, N is an interger.
x((k + 1)∆) = Φx(k∆) + Γu(k∆)
y(k∆) = Cx(k∆)
200
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on June 5, 2009 at 06:18 from IEEE Xplore. Restrictions apply.
(5)
(6)
MoB02.1
Based on the same manner in [2],
x(kτ + ∆) = Φx(kτ ) + Γu(kτ )
(7)
2
(8)
x(kτ + 2∆) = Φ x(kτ ) + (Φ + I)Γu(kτ )
..
.
x((k + 1)τ ) = ΦN x(kτ ) +
(9)
We add past output y((k − 1)τ − ∆) to yk ,


y((k − 1)τ − ∆)
ȳk =  y((k − 1)τ ) 
y((k − 1)τ + ∆)
and consider the following relationships,
N−1
∑ Φi Γu(kτ )
y((k − 1)τ + ∆) = y(kτ − ∆) = Cx((k − 1)τ + ∆)
y(kτ ) = Cx(kτ )
y((kτ + ∆) = Cx(kτ + ∆)
(10)
i=0
= Φτ x(kτ ) + Γτ u(kτ )
(11)
Defining
(18)
(19)
(20)
(21)
and


x(kτ ) = Φx((k − 1)τ + ∆) + Γu((k − 1)τ )
y((k − 1)τ )
y((k − 1)τ + ∆)


yk = 

..


.
y(kτ − ∆)
−1
x((k + 1)τ ) = Φτ x(kτ ) + Γτ u(kτ )
yk+1 = C0 x(kτ ) + D0 u(kτ )
(13)
(14)
where

C
CΦ
..
.


0
CΓ
..
.
x((k − 1)τ + ∆) = Φ x(kτ ) − Φ Γu((k − 1)τ )
x(kτ + ∆) = Φx(kτ ) + Γu(kτ )
(12)
We obtain the multirate output sampled system as follows.









C0 = 

 , D0 = 




N−2 i
N−1
CΦ
C ∑i=0 Φ Γ
(15)
Since the original continuous-time system (1) and (2) is
observable, discrete-time system (3) and (4) is also ovservable, and C0 is rank n and n × n matrix. For observable
systems,from (13) and (14), the following expression is
given.
x(kτ ) = C0−1 yk+1 −C0−1 D0 u(kτ )
(16)
(22)
(23)
(24)
Then, we have




y((k − 1)τ + ∆)
x((k − 1)τ + ∆)
 =C

y(kτ )
x(kτ )
ȳk+1 = 
y(kτ + ∆)
x(kτ + ∆)

 −1
Φ x(kτ ) − Φ−1 Γu((k − 1)τ )

x(kτ )
=C
Φx(kτ ) + Γu(kτ )
 −1 
 −1

¸
CΦ
CΦ Γ 0 ·
u((k − 1)τ )




C
0
0
=
x(kτ ) +
u(kτ )
CΦ
0
CΓ
(25)
Further, defining
 −1
 −1 

CΦ Γ 0
CΦ
0
C̄0 ,  C  , Γ̄ ,  0
CΦ
0
CΓ
(26)
and using C̄0? such that C̄0?C̄0 = I2×2 , expression of x((k + 1)τ
can be derived as
x((k + 1)τ ) = Φτ x(kτ ) + Γτ u(kτ )
¸¶
µ
·
u((k − 1)τ )
?
?
= Φτ C̄0 ȳk+1 − C̄0 Γ̄
u(kτ )
·
¸
£
¤ u((k − 1)τ )
+ 0 Γτ
u(kτ )
·
¸
u((k − 1)τ )
= Φτ C̄0? ȳk+1 + (Γ̄τ − Φτ C̄0? Γ̄)
u(kτ )
(27)
and
x((k + 1)τ ) = Φτ x(kτ ) + Γτ u(kτ )
= Φτ (C0−1 yk+1 −C0−1 D0 u(kτ )) + Γτ u(kτ )
= Φτ C0−1 yk+1 + (Γτ − Φt auC0−1 D0 )u(kτ ) (17)
Let k + 1 be k, then
x(kτ ) = Ly yk + Lu u((k − 1)τ )
Ly = Φτ C0−1
−1
namely,
Lu = Γτ − Φτ C0−1 D0
That is, state variable x(kτ ) is calculated by using multioutput yk and input u((k − 1)τ ). Coefficients Ly and Lu are
determined by the above equation and have no redesign
parameters. To include a design parameter, we add past
output y((k − 1)τ − ∆) to yk . To be simple in explanation,
we consider the case of n = 2 and N = 2. Then
·
¸
y((k − 1)τ )
yk =
y((k − 1)τ + ∆)
¸
·
u((k − 2)τ )
x(kτ ) = L̄y ȳk + L̄u
u((k − 1)τ )
(28)
Then the control input Fx x(kτ ) can be realized as
·
¸
u((k − 2)τ )
u(kτ ) = Fx x(kτ ) = Fx (L̄y ȳk + L̄u
)
u((k − 1)τ )
Now it will be shown how feedback gain Fx L̄y can be
expressed in terms of two free parameters using the following
procedure, and using this flexibility how the gain matrix can
be robustly designed against round off error.
201
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on June 5, 2009 at 06:18 from IEEE Xplore. Restrictions apply.
MoB02.1
Using Cayley-Hamilton Theorem, we obtain that
2
Φ = φ1 I + φ2 Φ
IV. D ESIGN PROCEDURE EXPLANATION BY NUMERICAL
where characteristic equation of Φ is
λ 2 − φ2 λ − φ 1 = 0
·
(30)
¸
C
is observable and C0 is 2 × 2 matrix, the
CΦ
· −1
¸
C0
0
−1
?
inverse of C0 exists. Further, defining C̄00 , Φ
,
0
we obtain that


· −1
¸ C
C
0 
?
CΦ  Φ−1
C̄00
C̄0 = Φ 0
0
CΦ2
µ
·
¸¶
0 0
= Φ I2×2 +
Φ−1 = I2×2
(31)
0 0
¸
·
? , Φ −φ1 −φ2 1 ,
Also, defining C̄01
−φ1 −φ2 1


·
¸
C
−
φ
φ
−
1
1
2
?

 Φ−1 = 0 (32)
CΦ
C̄01
C̄0 = Φ
−φ1 −φ2 1
φ1C + φ2CΦ
¸
·
0
? + λ1
C̄? ,
As a result, we can define that C̄0? = C̄00
0 λ2 01
which satisfies C̄0?C̄0 = I2×2 and includes λ1 and λ2 as extra
design parameter, and the expression of state variables (28)
includes the extra design parameters in L̄y = Φτ C̄0? , L̄u =
Γ̄τ −Φτ C̄0? Γ̄. Then the first term of the control can be realized
as
Since C0 =
u(k) = F̄y ȳk , F̄y = (F̄y1 , F̄y2 , F̄y3 )
(33)
where in selecting the values of the parameters, gains Fyi (i =
1, 2, 3) can be with same sign and different values.
In the following, a reaching law approach can be designed
for the quasi-sliding mode control of discrete-time plant (3),
(4) by using the method in [2]. That is, design sliding surface
as
s(k) = cT x(k)
In this section, one example is considered to show the
design procedure of the proposed design method.
Consider the following plant
· ¸
· ¸
d x1
x
(39)
= A 1 + Bu
x2
dt x2
y = Cx
(40)
where
·
¸
−1.0077 2.01533
A=
−2.0153 1.00767
£
¤
B = −1.00267 1.00767
£
¤
C= 1 0
Using τ = 0.006, the discrete-time system is given as
·
¸ ·
¸·
¸ · ¸
x1 (k + 1)
0 1 x1 (k)
0
=
+
u(k)
(41)
x2 (k + 1)
−1 1 x2 (k)
1
·
¸
£
¤ x (k)
y(k) = 1 0 1
(42)
x2 (k)
Now, design sliding surface as
s(k + 1) − s(k) = −qτ s(k) − ετ sgn(s(k))
(35)
1 − qτ > 0, ε > 0
(36)
where
Then, the quasi-sliding mode control law is given as
u(k) = −(cT Γτ )−1 ((cT Φτ − cT + qτ cT )x(k)
(37)
Considering (28), the multirate output feedback-based quasisliding mode control law can be represented by
·
¸
u(k − 2)
u(k) =F̄y yk + F̄u
u(k − 1)
¸
·
u(k − 2)
T
−1
T
T
) (38)
− (c Γτ ) ετ sgn(c L̄y yk + c L̄u
u(k − 1)
The feedback gain F̄y includes extra parameters λ1 and λ2 .
£
¤
s(k) = cT x(k), cT = −0.738 1
(43)
s(k + 1) − s(k) = −qτ s(k) − ετ sgn(s(k))
(44)
From (35), we obtain the reaching law as
Selecting q = 1 and ε = 0.1, by using the proposed design
method shown in Section 3, we have that
¸
·
y((k − 1)τ )
(45)
yk =
y((k − 1)τ + ∆)
·
¸T
−1.7730
yk + 0.99953u(k − 1)
1.729
µ·
¸
¶
−1.262
− 0.0006sgn
yk + 1.0702u(k + 1)
0.454
u(k) =
(34)
and the reaching law satisfies the following condition
+ ετ sgn(cT x(k)))
EXAMPLE
(29)
(46)
In this case, coefficients of yk are Fy1 = −1.730 and Fy2 =
1.729 ane sre almost Fy1 = Fy2 . Then, if sampling period
is small, then y((k − 1)τ ) ; y((k − 1)τ + ∆) in yk , and the
feedback input Fy yk = Fy1 y((k + 1)τ ) + Fy2 ((k + 1)τ + ∆) is
strongly affected by numerical round off error, and Fy yk ; 0.
This effect is caused by the fact of Fy1 = −Fy2 . If we add
y((k − 1)τ − ∆) to yk


·
¸
y((k − 1)τ − ∆)
y((k − 1)τ )
yk =
→ ȳk =  y((k − 1)τ )  (47)
y((k − 1)τ + ∆)
y((k − 1)τ + ∆)
Then the feedback term becomes
F̄y yk = F̄y1 y((k − 1)τ − ∆) + F̄y2 y((k − 1)τ + ∆)
+ F̄y3 y((k − 1)τ + ∆)
(48)
and F̄y1 , F̄y2 and F̄y3 include extra parameters λ1 and λ2 .
Selecting the values λ1 and λ2 as λ1 = −0.5, λ2 = 1.0. And
we can avoid the case of Fyi = Fy j (i 6= j) and in this case
202
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on June 5, 2009 at 06:18 from IEEE Xplore. Restrictions apply.
MoB02.1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
0
1
2
3
4
5
6
7
8
Time[s]
Fig. 1.
Simulation result
the effect of the numerical round off error does not appear.
Then F̄y1 = −0.730, F̄y2 = −0.4656 and F̄y3 = 0.9991.
In figure 1, three lines are drawn. The dotted line shows
the output without numerical errors, using control input (46)
which is the same to the output using control (48). The
dashed line is the output by (46) and solid line is by (48)
both rounded at the third decimal digit. The case by (48) is
almost the same to the case without round-off error, where
as the case by (46) is strongly affected by round-off error.
V. CONCLUSION
In this paper, based on multirate output feedback, a sliding
mode control scheme is considered to discrete-time system.
Also, the proposed controller ensures robustness to numerical
round off error. Numerical example is given to show the
effectiveness of the proposed method.
R EFERENCES
[1] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems,
Berlin, Germany: Springer-Verlag, 1995.
[2] B. Bandyopadhyay and S. Janardhanan, Discrete-Time Sliding Mode
Control:A Multirate Output Feedback Approach, Berlin, Germany:
Springer-Verlag, 2006.
[3] V. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electromechanical Systems, Taylor & Francis, 1999.
[4] A. Inoue and M. Deng, A Design of a Partial Sliding Mode Controller
using Duality to Linear Functional Observer, in Proc. of the 2006 IEEE
International Symposium on Intelligent Control, pp. 488-491, 2006.
[5] K. Nonami and H. Tian, Sliding Mode Control - Design Theory
on Nonlinear Robust Control -, Corona Publishing Co., 1994(in
Japanese).
[6] C.-Y. Su, Q. Wang, X. Chen, S. Rakheja, Adaptive Variable Structure
Control of a class of Nonlinear Systems with Unknown Prandtlshlinskii Hysteresis, IEEE Transactions on Automatic Control, vol.
50, no. 12, pp. 2069-2074, 2005.
203
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on June 5, 2009 at 06:18 from IEEE Xplore. Restrictions apply.