Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.052
COMBINING STATE ESTIMATOR AND DISTURBANCE OBSERVER IN
DISCRETE-TIME SLIDING MODE CONTROLLER DESIGN
Jeang-Lin Chang
ABSTRACT
In response to a multiple input/multiple output discrete-time linear system with mismatched disturbances, an algorithm capable of performing estimated system states and unknown disturbances is proposed first, and then followed with the design of the controller. Attributed to the fact that both system
states and disturbances can be estimated simultaneously with our proposed
method, the estimation error is constrained at less than O(T ) as the disturbance between the two sampling points is insignificant. In addition, the estimated system states and disturbances are then to be used in the controller when
implementing our algorithm in a non-minimum phase system (with respect
to the relation between the output and the disturbance). The tracking error is
constrained in a small bounded region and the system stability is guaranteed.
Finally, a numerical example is presented to demonstrate the applicability of
the proposed control scheme.
Key Words: Discrete-time, sliding mode, state estimator, disturbance
observer.
I. INTRODUCTION
Sliding-mode control has been widely studied and
is well established with respect to continuous-time
systems [1]. Recently, the use of personal computer
(PC)-based controllers is becoming popular, and many
investigators have paid more attention to the design of
the discrete-time sliding mode controller. However, due
to the characteristics of a finite sampling rate, some of
the properties that are available to be applied to the
continuous-time sliding mode control could be considered inappropriate for a discrete-time system [2]. More
importantly, a sliding mode controller developed mainly
for continuous-time systems may be unstable when implemented with direct digital applications. Hence, many
Manuscript received February 22, 2005; revised October 4,
2007; accepted September 24, 2007.
The author is with Department of Electrical Engineering,
Oriental Institute of Technology, Pan-Chiao, Taipei County 220,
Taiwan.
This research was supported by the National Science Council, Taiwan, under Contact NSC 95-2213-E-161-002.
q
researchers [2–14] have focused their efforts on investigation into the design of discrete-time sliding mode
control (DSMC) or discrete-time variable structure
control.
Milosavljevic [3] was one of the first researchers
among others to formally propose that the sampling
process in discrete-time systems can limit the existence
of the sliding mode. Later on, Sapturk et al. [4] suggested that a reaching condition should be widely used
in current DSMC systems. By comparing hyperplane
design techniques with other investigators, a sliding
surface design based on the Lyapunov method was
proposed in designing for discrete-time systems by
Spurgeon [5]. Koshkouei and Zinober [6] clarified the
concept of DSMC by presenting several new sufficient
conditions to illustrate the necessity of the existence
of using the discrete-time sliding mode. Elmali and
Olgac [7] used the time delay approach to designing
perturbation, estimating that the upper bound of perturbation is not required. Such similar methods can be
found in other papers [8, 9]. In these papers [2–11],
however, the controller design is based on the assumption that all of the system states are available, whereas
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
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Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
in most physical systems this premise is not always
the case. With emphasis given to a robust output tracking scheme, Edwards and Spurgeon [15] proposed an
observer-based sliding mode controller for continuoustime multiple input/multiple output (MIMO) systems.
Also, a very detailed discussion for the design of output feedback sliding mode controller was published
more recently [1]. Aside from the above-mentioned reports, the pole assignment method was used to design
the sliding surface by Chang [16], and a reduced-order
observer design using output feedback sliding mode
control was also proposed. In light of designing direct
torsion control of flexible shaft, Korondi et al. [12]
proposed a scheme implementing an observer-based
DSMC. To explicitly account for computational delay
in any digital implementation, Misawa [13] presented
a prediction observer-based DSMC. In view of the
advantages, in another paper, Tang and Misawa [14]
used output feedback and made use of a single sliding
surface in the controller design for MIMO systems.
As noted before, the disturbance observer (DO),
which has been formulated in frequency domain, is
known to be very effective in compensating disturbances [17] and it has become very popular for robust
motion control [18, 19]. Conventionally, based on the
transfer function approach, the DO is designed using
the inverse dynamics of a plant and the so-called Qfilter is to be used to determine the performance of robustness and disturbance rejection. The main problems
with this approach are that it is difficult for DO to be
designed into MIMO systems and the system must be
in minimum phase (with respect to the relation between
the output and the disturbance). Of necessity, Mita
et al. [20] proposed a technique of two-delay output
control to overcome the non-minimum phase problem.
Encouraged by their findings, Chang et al. [21] used
a two-delay approach and they extended the unknown
input observer design to estimate the disturbances in
MIMO systems.
In this paper, a design technique using DSMC
by employing output information only for a class of
MIMO systems with mismatched disturbances is proposed. First, we introduce the integral term of the estimation output error in the observer design to propose
certain degrees of freedom. Using this technique and
assuming that the disturbances do not vary too much in
between two consecutive sampling instances, we shall
demonstrate that the proposed algorithm in digital implementations can not only reduce the state estimation
errors, but also restrict the disturbance estimation errors to be smaller than a size of O(T ). Hence, the
conventionally-assumed upper bound restriction on the
disturbances is relaxed to the restriction of the rate of
q
the disturbances, which is considerably slower than the
sampling rate. In a like manner, using the estimation
states and the estimation disturbances, the control law
is then designed and the closed-loop stability is analyzed. Together, some significant features are discussed
including the selection of the sliding surface, the performance of estimation disturbances and the convergence
rate to sliding mode.
This paper is organized as follows. The system
description and the problem formulation are given in
Section II. Section III presents the state estimator and
the disturbance observer. In Section IV, the sliding
surface design that can effectively reduce the influence
of mismatched disturbance during sliding motion is
demonstrated. In Section V, we develop the controller
algorithm using the estimation states and disturbances
to stabilize the system with system stability analysis being demonstrated. Section VI gives a numerical
example to exhibit the effectiveness of the proposed
controller. Finally, Section VII presents the concluding
remarks.
II. PROBLEM FORMULATION
Consider a continuous-time MIMO linear system
with mismatched disturbances described by
ẋ(t) = Hx(t) + Du(t) + Gd(t)
y(t) = Cx(t)
(1)
where x ∈ Rn are states of the system, u ∈ Rm are control
inputs, d ∈ Rl are mismatched disturbances, and y ∈ R p
are the measurable outputs of the system. For system
(1), the system matrices have appropriate dimensions
and the number of inputs is smaller than the number of
outputs. Moreover, the matrix G is of full rank. Suppose that the sampling interval is T and a zero-orderhold is adopted for the above-mentioned continuoustime model. Denoting x(k) = x(kT ), y(k) = y(kT ),
u(k) = u(kT ), and d(k) = d(kT ) where k ≥ 0 is an
integer, the discrete-time model can be then given by
x(k + 1) = Ax(k) + Bu(k) + f(k)
y(k) = Cx(k)
(2)
where these matrices A and B can be determined as
A = exp(HT ) = In + HT +
T
exp(H)Dd
B=
H2 T 2
+ ···
2
(3)
0
= DT +
1
HDT 2 + · · · = O(T )
2!
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
J.-L. Chang: Combining State Estimator and Disturbance Observer in Discrete-time Sliding Mode Controller Design
Moreover, if d(t) is bounded and smooth [22], then the
disturbance term f(k) in (2) can be written as
T
f(k) =
(4)
T
where E = 0 exp(H)Gd ∈ O(T ) and the magnitudes of f(k) and r(k) are on the order of O(T ) and
O(T 2 ), respectively. The magnitude of g is said to be
g = O(T n ) if
lim
T →0
g
= 0
Tn
and
lim
g
T →0 T n−1
=0
(5)
where n is an integer and we denote O(T 0 ) = O(1).
Although there exist unknown disturbances for system
(2), the objective of this work is to design an output
feedback DSMC such that the conditions x(k) → 0 and
y(k) → 0 as k → ∞ can be guaranteed. To illustrate this
point, the following assumptions are made throughout
this paper.
Assumption 1. System (2) is controllable and observable. Moreover, rank(CB) = m.
Assumption 2 ([8, 22]). The sampling interval T is
sufficiently small such that the disturbances do not vary
too much between two consecutive sampling instances.
From what we have assumed above, consequently, the
following relations are to be effectively approximated
as
O(T n ) + O(T n+1 ) ≈ O(T n )
O(T n )O(1) ≈ O(T n )
∀n ∈ ℵ
∀n ∈ ℵ
where ℵ is a set consisting of all integers.
Assumption 3. These matrices A, G and C can satisfy
A − In G
=n +l
rank
C
0
The assumption follows that the number of disturbances
is smaller than the number of outputs.
III. STATE ESTIMATOR AND
DISTURBANCE OBSERVER
Let x̂(k) and ŷ(k) be the estimations of x(k)
and y(k), respectively. Design state and disturbance
q
x̂(k + 1) = Ax̂(k) + Bu(k)
q(k + 1) = q(k) + L2 (y(k) − ŷ(k))
0
= Ed(k) + r(k)
observers as
+ L1 (y(k) − ŷ(k)) + Gq(k)
eH Gd((k + 1)T − )d
517
(6)
ŷ(k) = Cx̂(k)
where q ∈ Rl , L1 ∈ Rn× p and L2 ∈ Rl× p are matrices
designed by the latter. Define the errors of the estimation
states and the estimation outputs as x(k) = x(k) − x̂(k)
and y(k) = y(k) − ŷ(k) = C
x(k), respectively. From (2)
and (6) we have
x(k + 1) = (A − L1 C)
x(k) + f(k) − Gq(k)
(7)
Since the matrix G is full rank, we define e(k) = G+ f(k)
− q(k) ∈ Rl where G+ = (GT G)−1 GT ∈ Rl×n and e
represents the estimation error of unknown disturbances. Equation (7) can be then rewritten as
x(k + 1) = (A − L1 C)
x(k)
+ Ge(k) + (In − GG+ )f(k)
(8)
Moreover, the dynamic equation of e is given by
e(k + 1) = G+ f(k + 1) − q(k + 1)
= G+ f(k + 1)−q(k) − L2 (y(k)−Cx̂(k))
= G+ (f(k + 1) − f(k))
x(k)
+ e(k) − L2 C
(9)
Introducing an augmented state vector zT (k) =
[
xT (k) eT (k)] and augmenting (8) and (9) obtains
x(k + 1)
z(k + 1) =
e(k + 1)
A − L1 C G x(k)
=
e(k)
−L2 C
Il
+
(In − GG )f(k)
+
(10)
G+ (f(k + 1) − f(k))
= (M − LN)z(k)
(In − GG+ )f(k)
+
G+ (f(k + 1) − f(k))
x(k)
y(k) = [C 0]
= Nz(k)
e(k)
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Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
where M =
A G
0 Il
∈ R(n+l)×(n+l) , L = [L1T L2T ]T ∈
R(n+l)× p and N = [C 0] ∈ R p×(n+l) . From (10), we
know that, if (M, N) is observable, the matrix M − LN
can be stabilized.
Lemma 1. If the pair (A, C) is observable, then the
following statements are equivalent:
A − In G
(1) rank
=n +l
C
0
(2) The pair (M, N) is observable.
Proof. From linear system, the pair (A, C) is observable in terms of which it is equivalent to
In − A
= n ∀ ∈ C
rank
C
Hence, statement 2 holds if and only if
rank
In+l − M
Lemma 2 ([22]). If the derivative of disturbance d(t)
is bounded and it remains smooth in the interval
(k + 1)T ≤ t < (k + 2)T , then it follows that
(1) (In − GG+ )f(k) = O(T 2 )
(2) G+ (f(k + 1) − f(k)) = O(T 2 ).
From Lemma 2, if the disturbances do not vary too
much between two consecutive sampling instances, then
equation (10) becomes
z(k + 1) = (M − LN)z(k) + b
where the value b is in the order of O(T 2 ) and this
value decreases as the sampling period is increased.
If d(t) is a constant disturbance, then it follows that
d(k + 1) = d(k) and the proposed observer design can
yield the exact estimation. Since the matrix M − LN is
stable, from (11) we have
z(k)
≤ M−LN
k z(0)
+
⎜⎢
=rank ⎝⎣
In − A
0
−G
0
lim z(k)
≤
k→∞
There are to cases to be discussed: (i) = 1 and (ii)
= 1 in the following.
(i) When = 1, the above-mentioned equation holds
because the pair (A, C) is observable.
(ii) When = 1, it follows that
rank
In+l − M
N
⎛⎡
⎜⎢
= rank ⎝⎣
In − A −G
= rank
A − In
C
⎤⎞
⎥⎟
0 ⎦⎠
0
C
(12)
From (12) as k → ∞, the norm of z(k) is bounded by
∀ ∈ C.
M−LN
i
≤ M − LN
k z(0)
1 − M − LN
k
+
1 − M − LN
⎤⎞
⎥⎟
( − 1)Il ⎦⎠
C
=n +l
k−1
i=0
N
⎛⎡
(11)
ỹ(k) = Nz(k)
0
G
Theorem 1. Consider the dynamic system (1) and its
corresponding discrete-time model can be described by
(2). If the state and disturbance observers are designed
as (6), then it follows from (10) and Lemma 2 that
x̃(k + 1)
x̃(k)
= (M − LN)
+ O(T 2 ). (14)
e(k + 1)
e(k)
Since a matrix L from Lemma 1 should be found such
that M − LN is stable, the following statement can be
obtained:
(2)
Hence, the two statements (1) and (2) are equivalent.
This completes the proof of Lemma 1.
q
(13)
which indicates that the estimation errors are bounded
and, if M − LN
is small, the width of this range also
becomes small. The following theorem summarizes the
results.
(1)
0
1 − M − LN
lim x̃(k)
≤ O(T )
k→∞
lim e(k)
≤ O(T )
k→∞
where • denotes the Euclidian norm of •.
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519
J.-L. Chang: Combining State Estimator and Disturbance Observer in Discrete-time Sliding Mode Controller Design
where the matrix F1 CB ∈ Rm×m is assumed to be nonsingular. The equivalent control ueq (k) can be then obtained as
Proof. Using Tustin’s approximation [22], we have
1 − M − LN
= 1 − max
=1−
−2T p
2+ Tp
=
= O(T )
2− Tp 2− Tp
where max is the maximum eigenvalue of M − LN and
p is the corresponding eigenvalue in the continuoustime case to be assumed as O(1). Hence,
ueq (k) = −(F1 CB)−1 [(F1 CA + F2 )x(k)
+F1 Cf(k) − F2 x̃(k)]
Substituting (17) into (2) yields the system in sliding
mode as
lim z(k)
≤ O(T −1 )O(T 2 ) = O(T ).
k→∞
Since limk→∞ z(k)
= limk→∞
O(T ), we have
(1)
(2)
x(k + 1) = (A − B(F1 CB)−1 (F1 CA + F2 ))x(k)
x̃(k)
2 +
e(k)
2 ≤
+B(F1 CB)−1 F2 x̃(k)
+(I − B(F1 CB)−1 F1 C)f(k)
lim x̃(k)
≤ O(T )
k→∞
= (Ā − BK)x(k) + (I − B(F1 CB)−1
lim e(k)
≤ O(T ).
×F1 C)f(k) + B(F1 CB)−1 F2 x̃(k) (18)
k→∞
The proof of this theorem is complete.
(17)
Remark 1. As can be seen from the above-mentioned
analysis, when unknown disturbances are smooth, the
observer method proposed can be used to accurately
perform the operation with anticipated estimation. This
assumption, an indication of insignificant variation of
disturbances between two sampling points, points out
the fact that the bandwidth of disturbances is far smaller
than the control bandwidth.
IV. SLIDING SURFACE DESIGN
Since the system states cannot be measured directly, the estimation states x̂(k) need to be used in the
sliding surface. The sliding surface is designed as
where Ā = A − B(F1 CB)−1 F1 CA and K = (F1 CB)−1
F2 ∈ Rm×n . Combining with the error dynamics of the
observer, we can obtain the overall system dynamics in
sliding mode as
⎡
x(k + 1)
⎤
⎡
(Ā − BK)
B(F1 CB)−1 F2
0
⎥ ⎢
⎢
0
A − L1 C
⎣ x̃(k + 1) ⎦ = ⎢
⎣
e(k + 1)
0
−L2 C
⎤
⎤ ⎡
⎡
x(k)
(I − B(F1 CB)−1 F1 C)f(k)
⎥
⎥ ⎢
⎢
+
⎥.
(I
−
GG
)f(k)
⎣ x̃(k) ⎦ + ⎢
n
⎣
⎦
e(k)
G+ (f(k + 1) − f(k))
⎤
⎥
G⎥
⎦
Il
(19)
From (4), we have
s(k) = F1 y(k) + F2 x̂(k − 1)
= F1 y(k) + F2 (x(k − 1) − x̃(k − 1))
where F1 ∈ Rm× p and F2 ∈ Rm×n are to be designed
in the latter part of this context. Suppose system
(2) is successfully constrained in the sliding surface
s(k) = 0. According to the concept of equivalent control [1], we have s(k + 1)|u = ueq = 0, where ueq represents the equivalent control. From (15) it follows
that
q
(16)
(20)
Let p(k) = (I − B(F1 CB)−1 F1 C)Ed(k) = O(T ). Since
r(k) = O(T 2 ), (I − GG+ )f(k) = O(T 2 ) and G+ (f(k +
1) − f(k)) = O(T 2 ), it follows from Assumption 2 that
equation (19) can be written as
⎡
x(k + 1)
⎤
⎡
⎥ ⎢
⎢
⎣ x̃(k + 1) ⎦ = ⎢
⎣
s(k + 1) = F1 C(Ax(k) + Bu(k) + f(k))
+F2 x(k) − F2 x̃(k)
(I − B(F1 CB)−1 F1 C)f(k)
= (I − B(F1 CB)−1 F1 C)Ed(k)
+(I − B(F1 CB)−1 F1 C)r(k).
(15)
e(k + 1)
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
(Ā − BK)
B(F1 CB)−1 F2
0
A − L1 C
0
−L2 C
0
⎤
⎥
G⎥
⎦
Il
520
Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
⎡
x(k)
⎤
⎡
p(k)
⎤
Although the estimation states are used in the sliding surface, from (21), the separating principle in linear system is valid and the system’s stability in sliding
mode can be guaranteed.
⎥
⎥ ⎢
⎢
× ⎣ x̃(k) ⎦ + ⎣ 0 ⎦ + O(T 2 )
0
e(k)
⎡
⎢
≈⎢
⎣
(Ā − BK) B(F1 CB)−1 F2
A − L1 C
0
−L2 C
0
⎤
⎡
0
⎤
⎥
G⎥
⎦
Il
⎤
⎡
p(k)
x(k)
⎥
⎥ ⎢
⎢
× ⎣ x̃(k) ⎦ + ⎣ 0 ⎦ .
(21)
0
e(k)
Lemma 3. If the pair (A, B) is controllable, then it follows that the pair (Ā, B) is controllable.
Proof. From linear system, the pair (A, B) is controllable if and only if
rank([In − A B]) = n
∀ ∈ C.
V. CONTROLLER DESIGN
After obtaining a suitable sliding surface (15), integrating the estimation disturbances q(k) with the control law is to be designed in this section. From the above
analysis written in Section III, the disturbance observer
yields better estimation when the change in d(t) is considerably slow. Hence, the estimation tem q(k) can be
fed forward to the controller to counteract the effect
of f(k). The main advantage of this method is that the
conventionally assumed upper bound restriction on the
disturbance is relaxed. The restriction is now focused
on the changing rate of the disturbance, which is slower
than the sampling period. Let the control algorithm be
designed as
u(k) = −(F1 CB)−1 [(F1 CA + F2 )x̂(k)
−Xs(k) − v(k) + F1 CGq(k)]
Since
[I − Ā B] = [I − A + B(F1 CB)−1 F1 CA B]
= [I − A + B(F1 CB)−1 F1 CA B]
I
0
,
×
−(F1 CB)−1 F1 CA I
= [I − A B]
it follows from matrix theory that the pair (Ā, B)
can be shown to be controllable when system (2) is
controllable.
From (21), system behavior in sliding mode is
still affected by the mismatched disturbance term
(I − B(F1 CB)−1 F1 C)Ed(k). The following is a series
of procedures listed for deriving the coefficients of
sliding surface F1 and F2 .
I. Choose F1 to minimize the value of (I −
B(F1 CB)−1 F1 C)E
, where the linear programming method or the genetic algorithm [23] is to
be used to find it. Moreover, the chosen matrix F1
must satisfy the condition that F1 CB is invertible.
II. Assign the desired n eigenvalues of system
Ā − BK and then calculate K by using feedback
design method. The matrix F2 can be obtained
using F2 = (F1 CB)K.
q
(22)
where X = diag(q1 , . . . , qm ) is chosen such that
X
<1 and the control inputs v(k) are designed in
the latter part of this paper. Substituting (22) into (16)
obtains the dynamics of s(k) as
s(k + 1) = Xs(k) + v(k) + F1 CAx̃(k)
+F1 Cf(k) − F1 CGq(k)
= Xs(k) + v(k) + F1 CAx̃(k)
+F1 CGe(k) + F1 C(In − GG+ )f(k)
= Xs(k) + v(k) + h(k)
(23)
+F1 C(In − GG+ )f(k)
where h(k) = F1 CAx̃(k) + F1 CGe(k). Let = [F1 CA
F1 CG] and obtain = O(1). From Theorem 1, it
follows that
b
1 − M − LG
= O(T )O(1)≈O(T )
lim h(k)
≤
k→∞
as k→∞.
(24)
Hence, from Assumption 2, we obtain that the dynamic
equation of s(k) can be approximated as
s(k + 1) = Xs(k) + v(k) + h(k)
(25)
The behavior of s(k) will satisfy the following theorem.
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
J.-L. Chang: Combining State Estimator and Disturbance Observer in Discrete-time Sliding Mode Controller Design
Theorem 2. Consider system (2) and choose the
and
sliding surface (15). Define = 1−
M−LG
+ε
= X
(1−
M−LG
)
>0 where ε>0 is a chosen constant
and design the control inputs u(k) as
u(k) = −(F1 CB)−1 [(F1 CA + F2 )x̂(k)
−Xs(k) − v(k) + F1 CGq(k)]
(26)
where
v(k) = −Xsat(s(k), )
⎧
s(k)
⎪
⎨
for s(k)
>
sat(s(k), ) = s(k)
⎪
⎩ s(k)
for s(k)
≤ (27)
When the system satisfies s(k)
>, its trajectory overlaps into the region s(k)
≤ over a finite step, where
s(k)
≤ is called an approaching region. In addition,
the system is finally restricted in the region s(k)
≤ as k → ∞, which is named as a sliding region.
Proof. When s(k)
>, from (25) and (27) it follows
that
s(k + 1) = Xs(k) − =
Xs(k)
+ h(k)
s(k)
Xs(k)
(
s(k)
− ) + h(k).
s(k)
Taking the norm of two sides of the above equation can
result in
s(k + 1)
≤ X
(
s(k)
− ) + h(k)
+ ε
≤ X
s(k)
−
1 − M − LG
+
h(k)
.
Since ε>0, from (24), there exists a non-negative integer
+ε
k1 such that h(k)
≤ 1−
M−LG
for k ≥ k1 , and then
the above-mentioned equation becomes
s(k + 1)
≤ X
s(k)
for k ≥ k1
In the worst-case scenario, it is assumed that the condition s(k)
> holds when k ≥ k1 . From 0<
X
<1,
a non-negative integer k2 can be found such that
X
k2 < s(k1 )
. It follows that s(k)
≤ when k ≥
k1 + k2 . Hence, the system trajectory will reach to the
region s(k)
≤ in a finite number of steps. When the
q
521
system enters in the approaching region, the control
law (27) is switched to v(k) = −Xs(k) and then the
dynamics of s(k) becomes
s(k + 1) = h(k)
As k → ∞, it turns out to be obvious that s(k)
≤ .
This means that the system is finally restricted in the
region s(k)
≤ , which is described as the sliding
region. Hence, the system trajectory first moves into the
approaching region within finite steps and finally stays
in the sliding region.
In the paper reported by Eun and Cho’s [7], the
gain of d(k + 1) − d(k)
is treated as the sliding layer
and to be used to design as a desired switching controller. Unfortunately, if the value of perturbation is not
exactly known, the system may encounter a problem
of entering into the sliding layer. However, how to estimate the upper bound of d(k + 1) − d(k)
becomes
difficult work. In this paper a parameter ε is introduced
in the control law (26) to design a first layer and the
gain of is the second layer. The modified control algorithm switches in the first layer and it will slide in
the second layer. Hence, the above-mentioned problem
can be solved.
Finally, we shall analyze the system stability when
the system stays in the sliding region. From (26) and
(27), the control inputs in the sliding region can be described by
u(k) = −(F1 CB)−1 [(F1 CA + F2 )x̂(k)
+F1 CGq(k)]
(28)
Substitute (28) into (2) to obtain the closed-loop system
as
x(k + 1) = (Ā − BK)x(k)
+B(F1 CB)−1 (F1 CA + F2 )x̃(k)
+f(k) − B(F1 CB)−1 F1 CGq(k)
= (Ā − BK)x(k) + R1 x̃(k) + R2 Ge(k)
+R2 (I − GG+ )f(k)
(29)
+(I − R2 )f(k)
where R1 = B(F1 CB)−1 (F1 CA + F2 ) and R2 =
B(F1 CB)−1 F1 C. Combine both (29) and the estimation error dynamics (10) to obtain a formula of the
overall system as
⎤
x(k + 1)
⎣ x̃(k + 1) ⎦
e(k + 1)
⎡
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
522
Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
⎤
⎤⎡
x(k)
(Ā − BK)
R1
R2 G
=⎣
0
A − L1 C
G ⎦ ⎣ x̃(k) ⎦
e(k)
0
−L2 C
Il
⎡
⎤
(I − R2 )f(k)
⎣
⎦
0
+
0
⎡
⎤
R2 (In − GG+ )f(k)
+ ⎣ (In − GG+ )f(k) ⎦ .
(30)
G+ (f(k + 1) − f(k))
⎡
Using Lemma 3 and Assumption 2, the above equation
can be rewritten as
⎤
x(k + 1)
⎣ x̃(k + 1) ⎦
e(k + 1)
⎡
⎤
⎤⎡
x(k)
(Ā − BK)
R1
R2 G
≈⎣
0
A − L1 C
G ⎦ ⎣ x̃(k) ⎦
e(k)
0
−L2 C
Il
⎡
⎤
(I − R2 )f(k)
⎦
0
+⎣
(31)
0
⎡
From (31), the design of an overall system may be
accomplished by designing Ā − BK and M − LN
separately. Therefore, one can conclude that the robust
stability is guaranteed under the proposed control algorithm. The following section provides a numerical
example to demonstrate the proposed control law.
VI. SIMULATION RESULTS
Consider system (1) with the following matrices:
⎤
6
⎣ 8 ⎦,
10
⎤
−2
1
1
H = ⎣ 10 −10 −3 ⎦ , D =
−6
2
−2
⎤
⎡
5 1
1 0
⎦
⎣
G= 0 7 , C =
−1 1
8 2
0.2 sin(1.5t)
d(t) =
0.3 cos(0.5t)
⎡
⎡
−1
,
1
This system is sampled by T = 0.02s and then its
discrete-time model will lead to the following system
q
matrices as
⎡
0.9615
⎢
A = ⎣ 0.1808
0.0181
0.0187
⎥
0.8195 −0.0514 ⎦ ,
−0.1116 0.0344
⎡
0.1211
⎤
⎥
⎢
B = ⎣ 0.1509 ⎦ ,
⎤
0.9586
⎡
0.0996 0.0213
⎤
⎥
⎢
E = ⎣ 0.0050 0.1277 ⎦ ,
0.1510 0.0406
0.1919
The system has the transmission zeros at 1.2562 with
respect to the relation between the output and the disturbance; hence, it is non-minimum phase. By selecting
poles at {0.05, 0.1, 0.2, 0.3 ± 0.1 j} from the observer
design, we can obtain the system matrices L1 and L2
from (9) as
⎡
30.4071 8.1393
⎢
L1 = ⎣ 2.2344
⎤
⎥
1.7444 ⎦
and
28.1583 7.9356
L2 =
0.8543
0.2408
−0.3037 0.0033
With the help of MATLAB (The MathWorks Inc,
Natick, MA, USA), F1 = [20 0.1] was selected to
minimize the norm of E − B(F1 CB)−1 F1 CE and
F2 = [55.6207 0.6134 31.7365] was designed to stabilize the matrix A−B(F1 CB)−1 F1 CA−B(F1 CB)−1 F2 ,
where its eigenvalues are placed in {0.1, 0.7, 0.8}.
Concurrently, the controller law is designed as
u(k) = −(F1 CB)−1 [(F1 CA + F2 )x̂(k) − s(k)
−sat(s(k), ) + F1 CGq(k)]
where = 0.8 and = 0.2. Under initial states
x(0) = [0 0.1 0 ]T and x̂(0) = [0 0 0]T , the estimation error of x̃(k) is shown in Figure 1. And the error of
estimation disturbances is shown in Figure 2. Clearly,
the system states and the unknown disturbances are well
estimated after k > 10. Hence, the proposed observer
is capable of estimating the states under disturbance
consideration. Figures 3 and 4 show the trajectories
of system states x(k) and system outputs y(k), respectively. The response of |s(k)| can reach and enter in
the first layer |s(k)|<0.2 and then it is constrained in
the second layer. From Figure 5, it can be seen that the
response of |s(k)| is constrained as depicted. Based on
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
J.-L. Chang: Combining State Estimator and Disturbance Observer in Discrete-time Sliding Mode Controller Design
Fig. 1. The estimation error of x.
Fig. 4. The trajectories of y.
Fig. 2. The response of e.
Fig. 5. The response of |s(k)|.
523
VII. CONCLUSIONS
Fig. 3. The trajectories of x.
these figures, although the system is in the presence of
disturbances, the performance of system outputs y(k) is
excellent under the proposed algorithm.
q
An algorithm of utilizing output feedback method
in performing sliding mode controller design is proposed in regard to a MIMO discrete-time system with
mismatched disturbances. At first, an observer design
method estimating both the system states and disturbances simultaneously is presented. Although the perfect estimation is not achievable, when the disturbance
in between the two consecutive sampling points is not
changed significantly, our proposed algorithm is able to
render the estimation error to be restricted within O(T ).
Moreover, our proposed estimator method should be implemented in some non-minimum phase systems. Due
to the fact that the estimated system states and disturbances are to have been implemented in the controller,
the closed-loop stability is guaranteed and the tracking
error is to be constrained in a small region. Although, the
underlying system has an unstable transmission zero, in
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society
524
Asian Journal of Control, Vol. 10, No. 5, pp. 515 524, September 2008
conclusion, our control law is feasible arising from the
simulation results.
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Jeang-Lin Chang received the
B.S. and M.S. degrees in Control
Engineering, the Ph.D. degree in
Electrical and Control Engineering
from National Chiao Tung University, Taiwan, in 1992, 1994,
1999, respectively. He was with the
Mechanical Research Laboratory,
Industrial Technology Research Institute, Taiwan, during 1997–1999.
In 1999, he joined the Department
of Electrical Engineering, Oriental Institute of Technology, as an
Assistant Professor. He is now an associated professor.
His research interests include sliding mode control,
motion control, and signal processing.
2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society