IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
3
On the Design of Sliding-Mode
Static-Output-Feedback Controllers for
Systems With State Delay
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X. R. Han, Emilia Fridman, Sarah K. Spurgeon, and Chris Edwards
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Abstract—This paper considers the development of slidingmode-based output-feedback controllers for uncertain systems
which are subject to time-varying state delays. A novel method
is proposed for design of the switching surface. This method
is based on the descriptor approach and leads to a solution in
terms of linear matrix inequalities (LMIs). When compared to
existing methods (even for systems without delays), the proposed
method is efficient and less conservative than other results, giving
a feasible solution when the Kimura–Davison conditions are not
satisfied. No additional constraints are imposed on the dimensions
or structure of the reduced order triple associated with design of
the switching surface. The magnitude of the linear gain used to
construct the controller is also verified as an appropriate solution
to the reachability problem using LMIs. A stability analysis for
the full-order time-delay system with discontinuous right-hand
side is formulated. This paper facilitates the constructive design
of sliding-mode static-output-feedback controllers for a rather
general class of time-delay systems. A numerical example from the
literature illustrates the efficiency of the proposed method.
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Index Terms—Linear matrix inequalities (LMIs), sliding-mode
25 control (SMC), static output feedback (SOF), time delay.
I. I NTRODUCTION
26
S
LIDING-MODE control (SMC) [1] is known for its complete robustness to so-called matched uncertainties (which
29 can include time delays that satisfy matching conditions) and
30 disturbances [2]–[4]. The control technique has been applied in
31 many industrial areas [5]–[7]. Many early theoretical develop32 ments in SMC assume that all the system states are accessible.
33 In the case where only a subset of states are measurable, which
34 is relevant to a range of practical applications, either output
35 feedback control or the observer-based method are required.
36 Some work has considered implementation of SMC schemes
37 using observers [8]–[10]. In [11], a sliding-mode observer has
38 been shown to give a significant increase in performance in esti39 mation of the unknown variables of a boost converter compared
27
28
to a traditional current-mode control strategy. A further inter- 40
esting strand considers the fast output sampling method [12], 41
[13]. Recently, in [14], a fast sampling method is employed 42
for a discrete systems in the presence of time-varying delays 43
where a sliding-mode controller is designed using linear matrix 44
inequalities (LMIs) combined with a delta-operator approach. 45
However, all these methods require additional computation. 46
The most straightforward approach is to consider the study of 47
SMC via static output feedback (SOF).
48
One problem of interest in the development of SMC via SOF 49
is the design of the switching surface, which is effectively a 50
reduced order SOF problem for a particular subsystem. Two 51
different methods were proposed to design the sliding surface 52
using eigenvalue assignment and eigenvector techniques in [15] 53
and [16]. A canonical form was provided in [18] via which 54
the SOFSMC design problem is routinely converted to an SOF 55
stabilization problem. As stated in [20], all previous-reported 56
methods for the existence problem are, in fact, equivalent to 57
a particular SOF problem. The solution to the general SOF 58
problem, even for linear time-invariant systems, is still open. 59
LMI methods have been considered within the context of 60
sliding-mode controller design. For example, [21] and [22] 61
presented LMIs methods to design static sliding-mode output- 62
feedback controllers and [23] presented a necessary and suffi- 63
cient condition to solve the existence problem in terms of LMIs 64
for linear uncertain systems.
65
It is important to note that all the work described above 66
does not consider an existence problem involving delay and 67
many practical problems include such effects [24], [25]. In [26], 68
the problem of the development of sliding-mode controllers 69
for operation in the presence of single or multiple, constant 70
or time-varying state delays has been solved. This uses the 71
usual regular form method of solution and the uncertainty is 72
assumed to be matched, where matched describes that uncer- 73
tainty class which is implicit in the range of the input channels 74
and will be rejected by an appropriately designed SMC strategy, 75
although it is important to note that full state availability is 76
assumed. This problem has also been considered in [27] where 77
a class of uncertain time delay systems with multiple fixed 78
delays in the system states is considered. This paper considers 79
unmatched and time-varying parameter uncertainties, together 80
with matched and bounded external disturbances, but again, full 81
state information is assumed to be available to the controller. In 82
[28], Lyapunov functionals were for the first time introduced 83
for the analysis of time-varying delay. In [29], a descriptor 84
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Manuscript received October 8, 2008; revised May 13, 2009. This work was
supported by the Engineering and Physical Sciences Research Council, U.K.,
under Grant EP/E 02076311.
X. R. Han and S. K. Spurgeon are with the Department of Electronics,
University of Kent, Canterbury, CT2 7NZ, U.K. (e-mail: xh25@kent.ac.uk;
S.K.Spurgeon@kent.ac.uk).
E. Fridman is with the School of Electrical Engineering, Tel Aviv University,
Tel Aviv 69978, Israel (e-mail: emilia@eng.tau.ac.il).
C. Edwards is with the Department of Engineering, University of Leicester,
Leicester, LE1 7RH U.K. (e-mail: ce14@leicester.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2009.2023635
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0278-0046/$25.00 © 2009 IEEE
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
approach to stability and control of linear systems with time86 varying delays, which is based on the Lyapunov–Krasovskii
87 techniques, was combined with results on the SMC of such
88 systems. The systems under consideration were subjected to
89 norm-bounded uncertainties and uncertain bounded delays and
90 the solution given in terms of LMIs. Reference [30] develops
91 a SMC synthesis for a class of uncertain time-delay systems
92 with nonlinear disturbances and unknown delay values whose
93 unperturbed dynamics is linear. The synthesis was based on a
94 new delay-dependent stability criterion. The controller is found
95 to be robust against sufficiently small delay variations and
96 external disturbances.
97
It is important to emphasize that much of the aforementioned
98 literature on SMC of time-delay systems assumes full-state
99 feedback. Reference [31] considered SMC of systems with
100 time-varying delay. This paper considered a solution via LMIs
101 for the existence problem. The current paper extends this contri102 bution to consider the solution of the existence and reachability
103 problems. Specifically, the selection of parameters for stability
104 of the full-order closed-loop system are obtained via LMIs. In
105 this paper, a compensator-based design problem is considered
106 using the proposed SOF approach. Example from the literature
107 illustrates the efficiency of the method. In Section II, the
108 problem formulation is described. The existence problem is
109 formulated in Section III. In Sections IV and V, stability of
110 the full-order closed-loop system is derived via LMIs, and the
111 reachability problem is presented. Compensator-based design
112 is demonstrated in Section VI.
113
114
II. P ROBLEM F ORMULATION
Consider an uncertain time-delay system
ż(t) = Az(t) + Ad z (t − τ (t)) + B (u(t) + ξ(t, z, u))
y(t) = Cz(t)
(1)
where z ∈ Rn , u ∈ Rm , and y ∈ Rp with m ≤ p ≤ n. The
time-varying delay τ (t) is supposed to be bounded 0 ≤ τ (t) ≤
117 h, and it may be either slowly varying (i.e., differentiable
118 delay with τ̇ (t) ≤ d < 1) or fast varying (piecewise continuous
119 delay). Assume that the nominal linear system (A, Ad , B, C)
120 is known and that the input and output matrices B and C are
121 both of full rank. The unknown function ξ: R+ × Rn × Rm →
122 Rn , which represents the system nonlinearities plus any model
123 uncertainties, is assumed to satisfy the matching condition and
115
116
ξ(t, z, u) < k1 u + a(t, y)
(2)
for some known function a : R+ × Rp → R+ and positive
125 constant k1 < 1. It can be shown that if rank(CB) = m, there
126 exists a coordinate system in which the system (A, Ad , B, C)
127 has the structure
A11 A12
Ad11 Ad12
A=
Ad =
A21 A22
Ad21 Ad22
0
B=
C = [0 T ]
(3)
B2
124
where B2 ∈ Rm×m is nonsingular and T ∈ Rp×p is orthogo- 128
nal. The system can be represented as
129
ż1 (t) = A11 z1 (t) + Ad11 z1 (t − τ (t))
+ A12 z2 (t) + Ad12 z2 (t − τ (t))
2
ż2 (t) =
(A2i zi (t) + Ad2i zi (t − τ (t)))
i=1
+ B2 (u(t) + ξ(t, z, u))
y(t) = Cz(t).
(4)
Consider the following switching function:
130 AQ1
S = {z(t) ∈ Rn : F Cz(t) = 0}
(5)
for some selected matrix F ∈ Rm×p where by design 131
det(F CB) = 0. Let
132
[F1
F2 ] = F T
(6)
where F1 ∈ Rp−m and F2 ∈ Rm . As a result
IE
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F C = [F1 C1
F2 ]
133
(7)
where
C1 = 0(p−m)×(n−p)
I(p−m) .
134
(8)
Therefore, F CB = F2 B2 and the square matrix F2 is 135
nonsingular. By assumption, the uncertainty is matched, and 136
therefore the sliding motion is independent of the uncertainty 137
represented by ξ(·). In addition, because the canonical form in 138
(3), where it is necessary that the pair (A11 , A12 ) is controllable 139
and (A11 , C1 ) is observable, can be viewed as a special case 140
of the regular form normally used in sliding-mode controller 141
design, the switching function can also be expressed as
142
s(t) = z2 (t) + KC1 z1 (t)
(9)
143
where K ∈ Rm×(p−m) and is defined as K = F2−1 F1 .
Then, a simple SMC law, depending on the output informa- 144
tion F y(t) can be defined by
145
u(t) = −γF y(t) − v(t)
where
v(t) =
F y(t)
ρ(t, y) F
y(t) ,
0,
if F y(t) = 0
otherwise
(10)
146
(11)
where ρ(t, y) is some positive scalar function of the outputs
147
ρ(t, y) = (k1 γ F y(t) + α(t, y) + γ2 ) /(1 − k1 )
where γ and γ2 are positive design scalars [18]. The closed- 148
loops system (4) and (10) can be described by the following 149
equations:
150
ż1 (t) = (A11 − A12 KC1 )z1 (t)
+ (Ad11 − Ad12 KC1 )z1 (t − τ (t))
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
ṡ(t) = (A21 − γB2 KC1 )z1 (t)
3
t−τ
(t)
1
ż1T (s)Rż1 (s)ds ≥
h
+ (Ad21 − γB2 KC1 )z1 (t − τ (t))
+ (A22 − γB2 )z2 (t) + (Ad22 − γB2 )z2 (t − τ (t))
t−h
t−τ
(t)
t−τ
(t)
ż1T (s)dsR
t−h
ż1 (s)ds.
t−h
(18)
+ B2 (ξ(t, z, u) − v(t))
y(t) = Cz(t).
(12)
Then,
162
V̇ (t) ≤ 2z1T (t)P ż1T (t) + h2 ż1T (t)Rż1 (t)
− (z1 (t) − z1 (t − τ (t)))T R (z1 (t) − z1 (t − τ (t)))
151
III. E XISTENCE P ROBLEM
152
153
154
On the sliding manifold s(t) = 0, it is well known [19] that
the reduced-order sliding motion is governed by a free motion
with system matrix
× R (z1 (t − τ (t)) − z1 (t − h))
ż1 (t) = (A11 − A12 KC1 )z1 (t)
− (1 − d)z1T (t − τ (t)) Sz1 (t − τ (t)) .
+(Ad11 − Ad12 KC1 )z1 (t − τ (t)) . (13)
155
Consider a Lyapunov–Krasovskii functional
t
z1T (s)Ez1 (s)ds
+
t−τ (t)
0 t
ż1T (s)Rż1 (s)dsdθ
+h
Using the descriptor method as in [33] and the free-weighting 163
matrices technique from [34], the right-hand side of the 164
expression
165
0 ≡ 2 z1T (t)P2T + ż1T (t)P3T
+(Ad11 − Ad12 KC1 )z1 (t − τ (t))]
z1T (s)Sz1 (s)ds
+
(14)
V̇ (t) ≤ η T (t)Θη(t) ≤ 0
where the symmetric matrices P > 0 and E, S, R ≥ 0.
The condition V̇ (t) < 0 guarantees asymptotic stability of
the reduced order system as in [32]. Differentiating V (t)
159 along (13)
156
V̇ (t) = 2z1T (t)P ż1 (t) + h2 ż1T (t)Rż1 (t)
t
ż1T (s)Rż1 (s)ds + z1T (t)(E + S)z1 (t)
−h
t−h
z1T (t
−
− h)Ez1 (t − h)
− (1 − τ̇ (t)) z1T (t − τ (t)) Sz1 (t − τ (t)) .
160
Further using the identity
t
−h
(15)
θ12 = P − P2T + '(A11 − A12 KC1 )T P2
θ33 = − (E + R)
ż1T (s)Rż1 (s)ds
(16)
t
t
ż1T (s)dsR
t−τ (t)
θ34 = R
θ44 = − 2R − (1 − d)S.
and applying Jensen’s inequality
t−τ (t)
170
θ24 = 'P2T (Ad11 − Ad12 KC1 )
t−τ (t)
1
≥
h
(22)
θ22 = − 'P2 − 'P2T + h2 R
ż1T (s)Rż1 (s)ds
−h
ż1T (s)Rż1 (s)ds
θ14
θ24 
<0
θ34
θ44
+ (A11 − A12 KC1 )T P2 + E + S − R
t−h
t
0
0
θ33
∗
θ11 = P2T (A11 − A12 KC1 )
t
161
θ12
θ22
∗
∗
169

θ14 = P2T (Ad11 − Ad12 KC1 ) + R
= −h
t−h
if the matrix inequality

θ11
 ∗
Θ=
∗
∗
(21)
is feasible, where
t−τ
(t)
ż1T (s)Rż1 (s)ds
(20)
with matrix parameters P2 , P3 = 'P2 ∈ Rn−m is 166
added into the right-hand side of (19). Setting η(t) = 167
col{z1 (t), z˙1 (t), z1 (t − h), z1 (t − τ (t))}, it follows that
168
−h t+θ
157
158
(19)
× [−ż1 (t) + (A11 − A12 KC1 )z1 (t)
t−h
t
+ z1T (t)(E + S)z1 (t) − z1T (t − h)Ez1 (t − h)
IE
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V (t)
= z1T (t)P z1 (t)
− (z1 (t − τ (t)) − z1 (t − h))T
ż1 (s)ds
t−τ (t)
(17)
(23)
Multiplying matrix Θ from the right and the left by 171
diag{P2−1 , P2−1 , P2−1 , P2−1 } and its transpose, respectively, and 172
denoting
173
Q2 = P2−1
P = QT
2 P Q2
= QT
E
2 EQ2
= QT RQ2
R
2
S = QT
2 SQ2
4
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
< 0, where
it follows Θ < 0 ⇔ Θ


θ11 θ12 0 θ14


=  ∗ θ22 0 θ24  < 0
Θ
 ∗
∗ θ33 θ34 
∗
∗
∗ θ44
(24)
θ11 = (A11 − A12 KC1 )Q2
T
+ QT
2 (A11 − A12 KC1 ) + E + S − R
T
θ12 = P − Q2 + 'QT
2 (A11 − A12 KC1 )
θ14 = (Ad11 − Ad12 KC1 )Q2 + R
2
θ22 = − 'Q2 − 'QT
2 +h R
θ24 = '(Ad11 − Ad12 KC1 )Q2
−R
θ33 = − E
θ34 = R
− (1 − d)S.
θ44 = − 2R
176
177
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Define the variable Q2 in the following form:
Q11
Q12
Q2 =
Q22 M δQ22
(26)
where Q22 is a (p − m) × (p − m) matrix, and M ∈
R(p−m)×(n−p) and δ ∈ R are a priori selected tuning parameters. It follows that
KC1 Q2 = [KQ22 M δKQ22 ].
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Defining
Y = KQ22
180
it follows that
KC1 Q2 = [Y M δY ].
181
(27)
To construct K, substitute (27) into (25) to yield
θ11 = A11 Q2 − A12 [Y
207
208
209
then the matrix P̄ satisfies the structural constraint
211
Ā0 = Ā − γ B̄F C̄
= Ā − γ B̄[0 F2 ]
θ14 = Ad11 Q2 − Ad12 [Y M δY ] + R
2
θ22 = − 'Q2 − 'QT
2 +h R
210
(31)
(32)
for sufficiently large γ [18]. In the new coordinate system, the 214
uncertain system (1) can be written as
215
θ24 = 'Ad11 Q2 − 'Ad12 [Y M δY ]
ż(t) = Āz(t) + A¯d z(t − τ ) + B̄ (u(t) + ξ(t, z, u)) .
θ34 = R
− (1 − d)S
θ44 = − 2R
where Ā11 = A11 − A12 KC1 and Ād11 = Ad11 − Ad12 KC1
and F2 is a design parameter. Let P̄ be a symmetric positive
definite matrix partitioned conformably with the matrices in
(29) so that
P̄1 0
P̄ =
(30)
0 P̄2
if the design matrix F2 = B2T P̄2 . The matrix P̄ can be shown 212
to be a Lyapunov matrix for
213
T
T T
θ12 = P − Q2 + 'QT
2 A11 − '[Y M δY ] A12
(33)
The closed-loop system will have the form
(28)
with the tuning parameters δ, ', and M. The following Lemma
may now be stated.
184
Lemma 1: Given a priori selected tuning parameters ', δ,
185 and M ∈ R(p−m)×(n−p) , then (24) is an LMI in the decision
> 0, E
≥ 0, S ≥ 0, R
≥ 0 and matrices Q22 ∈
186 variables P
182
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203
204
It can be shown [18] that there exist a coordinate system in 205
which the system triple (Ā, Ād , B̄, F C̄) has the property that 206
Ā11 Ā12
Ād11 Ād12
Ād =
Ā =
Ā21 Ā22
Ād21 Ād22
0
B̄ =
(29)
F C̄ = [0 F2 ]
B2
P̄ B̄ = C̄ T F T
T
δY ] + QT
2 A11
− [Y M δY ]T AT
12 + E + S − R
−R
θ33 = − E
IV. S TABILITY OF THE F ULL -O RDER
C LOSED -L OOP S YSTEM
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(25)
R(p−m)×(p−m) , Q11 ∈ R(n−p)×(n−p) , Q12 ∈ R(n−p)×(p−m) , 187
Y ∈ Rm×(p−m) . If a solution to (24) exists, which may be read- 188
ily obtained from available LMI tools, then the reduced order 189
system (13) is asymptotically stable for all differentiable delays 190
0 ≤ τ (t) ≤ h, τ̇ (t) ≤ d < 1. Moreover, (13) is asymptotically 191
stable for all piecewise-continuous delays 0 ≤ τ (t) ≤ h, if the 192
LMI (24) is feasible with S = 0.
193
Remark: The proposed method is suitable for SOF sliding- 194
mode controller design where Kimura–Davison conditions, 195
written as n ≤ m + p − 1, are not satisfied as in [20]–[22]. No 196
constraints are imposed on the dimensions of the reduced-order 197
triple A11 , A12 , C1 . This represents a constructive and efficient 198
approach to output-feedback-based design for a relatively broad 199
class of systems, which is less conservative than existing results 200
[20]–[22]; an example to illustrate the advantages of the method 201
for systems without time-delay is presented in [17].
202
ż(t) = Ā0 z(t) + Ād z(t − τ ) + B̄ (ξ(t, yt ) − vy (t)) .
216
(34)
For large enough γ > 0, these conditions are delay independent 217
with respect to the delay in z2 .} However, for derivation of 218
this condition using Lyapunov–Krasovskii techniques, it is 219
necessary to consider the case where τ̇ ≤ d < 1. A stability 220
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
condition for the full-order closed-loop system can be derived
222 using the following Lyapunov–Krasovskii functional:
221
t
T
z T (s)Ēz(s)ds
V (t) = z (t)P̄ z(t) +
t−h
t
0 t
T
+
z (s)S̄z(s)ds + h
ż T (s)R̄ż(s)dsdθ
(35)
−h t+θ
t−τ (t)
R̄1 0
223 where the matrix Ē, S̄ ≥ 0 and R̄ =
≥ 0 as it is
0 0
224 desired to determine a stability condition for the time delay
225 system which is delay independent with respect to delay in
226 z2 (t). Differentiating V (t) along the closed-loop trajectories
V̇ (t) ≤ 2z (t)P̄ ż (t) + h ż (t)R̄ż(t)
T
T
Inequality (41) is an LMI in the decision variables P̄1 > 0, Ē ≥ 235
0, S̄ ≥ 0 and R̄1 ≥ 0. Equation (37) is valid if (41) is satisfied 236
and given
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2z T P̄ B̄ (ξ(t, z, u) − v(t))
= 2y T F T (ξ(t, z, u) − v(t))
× R̄ (z (t − τ (t)) − z(t − h))
< −2 F y(t) (ρ(t, y) − k1 u(t) − α(t, y)) .
(36)
Substitute the right-hand side of (34) into (36). Setting ς(t) =
col{z(t), z(t − h), z(t − τ (t))}, it follows that
is satisfied if ς T (t)Φh ς(t) + h2 ż T (t)R̄ż(t) < 0
2z T P̄ B̄(ξ(t, z, u) − v(t)) < 0, where


φ11 − R̄
0
P̄ A¯d + R̄

Φh = 
∗
−(Ē + R̄)
R̄
∗
∗
−2R̄ − (1 − d)S̄
φ11 = ĀT
0 P̄ + P̄ Ā0 + S̄ + Ē.
and
and so by rearranging
239
≥ k1 (v(t) + γ F y(t)) + α(t, y) + γ2
≥ k1 u(t) + α(t, y) + γ2 .
(43)
240
(38)
V̇ (t) < −2γ2 F y(t) < 0 if z(t) = 0
241
242
243
(39)
and therefore the system is asymptotically stable.
Lemma 2: Given large enough γ, let there exist n × n matrices P̄1 > 0, Ē ≥ 0, S̄ ≥ 0, R̄1 ≥ 0 from the LMI solver
such that LMI (41) holds. Given that the design parameters
k1 , α(t, y), γ2 , and P̄2 have been selected so that (44) holds,
the closed-loop system (33) is asymptotically stable for all
differentiable delays 0 ≤ τ (t) ≤ h, τ̇ (t) ≤ d ≤ 1.
V. F INITE -T IME R EACHABILITY TO THE
S LIDING M ANIFOLD
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249
h2 ż T (t)R̄ż(t)
T
= h2 z T (t)Ā0 + z T (t − τ )A¯d
+ ξ(t, z, u) − v(t)T B̄ T R̄
238
ρ(t, y) = (k1 γ F y(t) + α(t, y) + γ2 ) /(1 − k1 )
From (37), if (41) is valid, then from (42) and (43),
Setting 1(t) = col{z(t), z(t − h), z(t − τ ), ξ(t, z, u) − v(t)}
However, by definition
ρ(t, y) = k1 ρ(t, y) + k1 γ F y(t) + α(t, y) + γ2
+ 2z T P̄ B̄ (ξ(t, z, u) − v(t)) < 0 (37)
with
(42)
IE
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oo
f
= −2ρ(t, y) F y(t) + 2 F y(t) ξ(t, z, u)
V̇ (t) ≤ ς(t)T Φh ς(t) + h2 ż T (t)R̄ż(t)
232
−R̄1
− (z (t − τ (t)) − z(t − h))T
− (1 − d)z T (t − τ (t)) S̄z (t − τ (t)) .
231
∗∗∗
≤ −2y T F T v(t) + 2 F y(t) ξ(t, z, u)
+ z T (t)(Ē + S̄)z(t) − z T (t − h)Ēz(t − h)
229
230
Using the Schur complement, ξ T (t)Φh ξ(t)+h2 ż T (t)R̄ż(t) < 0 233
holds if
234


I(n−m)
R̄1
hĀT
0
0




0(n,n−m)  Φh

(41)

 < 0.
I
T


(n−m)
R̄1 
hA¯d

0
2 T
− (z(t) − z (t − τ (t)))T R̄ (z(t) − (t − τ (t)))
227
228
5
× Ā0 z + A¯d z(t − τ ) + B̄ (ξ(t, z, u) − v(t))
 T 
Ā0  0n  I(n−m) 2
T
= 1 (t)  ¯ T 
h R̄1
Ad
0
T
B̄
 T T
T Ā0
I
 0n 
× (n−qm)
(40)
 ¯ T  1(t).
Ad
0
T
B̄
(44)
244
245
246
247
Corollary: An ideal sliding motion takes place on the sur- 250
face S if
251
−1 L
−1 L
B Ā0 z(t) + B Ād z(t − τ ) < γ2 − η
(45)
2
2
L
where the matrices ĀL
0 and Ād represent the last m rows of 252
Ā0 and Ād , respectively, and η is a small scalar satisfying 253
0 < η < γ2 .
254
Proof:
255
ṡ(t) = F C̄ Ā0 z̄(t) + F C̄ Ād z(t−τ )+F2 B2 (ξ(t, z, u)−v(t)) .
(46)
6
256
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
Let Vc : Rm → R be defined by
T
Vc (s) = sT (t) F2−1 P̄2 F2−1 s(t).
257
Then, using the fact that F2T = P̄2 B2 it follows that
T
T
F2−1
258
(47)
F2−1
P̄2 F2−1 F C̄ Ā0 = B2−1 ĀL
0
P̄2 F2−1 F C̄ Ād = B2−1 ĀL
d.
(48)
Then, it can be verified that
T
−1 L
V̇c = 2sT (t)B2−1 ĀL
0 z(t) + 2s (t)B2 Ād z(t − τ )
+ 2sT (t) (ξ(t, z, u) − v(t))
≤ 2 s(t) B2−1 ĀL
0 z(t) + 2 s(t)
Fig. 1.
Output against time.
< − 2η s(t)
IE
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× B2−1 ĀL
d z(t − τ ) − 2γ2 s(t)
(49)
if z(t) and z(t − τ ) ∈ Ω. It follows that there exists a t0 such
that z(t) and z(t − τ ) ∈ Ω for all t > t0 . Consequently, (49)
261 holds for all t > t0 . A sliding motion will thus be attained in
262 finite time.
263
Example 1: The following model of a liquid monopropel264 lant rocket motor has been considered in [35]. It is assumed
265 that the variable κ = 0.8 in this case, where Ad (1, 1) = −κ
266 and A(1, 1) = κ − 1. The outputs have been chosen to be the
267 second and fourth states so that in (1)
259
260

−0.2
 0
A=
−1
0
 
0
1
B= 
0
0
0 0
0 0
0 −1
1 1
C=

0
−1 

1
0

−0.8
 0
Ad = 
0
0
0 1
0 0
0 0
0 0

0
0

0
0
0 1 0 0
.
0 0 0 1
(50)
Clearly, the Kimura–Davison conditions are not met. Here,
the rate at which the delay varies with time has been examined
with d = 0, a slow varying delay. The gain from the LMI tool
271 solver with δ = 50, ' = 0.5, h = 0.45s, P̄2 = 1, γ = 2.9, and
272 M = [5 0.2], yields F = [−1 − 2.0754]. The poles of the
273 sliding-mode dynamics are {−2.67, −0.4, −0.2}. A simu274 lation was performed with the initial state values [1 1 1 1].
275 As shown, the LMI solver gave a feasible result for stability
276 for h ≤ 0.45s, but in simulation, the closed-loop system only
277 became unstable for h ≥ 1s (Fig. 1); this is due to the conserv278 ativeness of the method. The LMI solver gave feasible closed279 loop stability results for controller gain γ ≥ 2.9 while a choice
280 of γ ≥ 1 in simulation was able to stabilize the system with the
281 compensation of longer settling time. The switching function
282 for h = 0.45s, γ = 2.9 is shown in Fig. 2.
268
269
270
Fig. 2.
Switching function.
VI. C OMPENSATOR -BASED E XISTENCE P ROBLEM
283
For certain system triples (A11 , A12 , C1 ), LMI (24) is known 284
to be infeasible. In this case, consider a dynamic compensator 285
similar to that of El-Khazali and DeCarlo [36]
286
żc (t) = Hzc (t) + Dy(t)
(51)
where the matrices H ∈ Rq×q and D ∈ Rq×p are to be deter- 287
mined. Define a new hyperplane in the augmented state space, 288
formed from the plant and compensator state spaces, as
289
Sc = (z(t), zc (t)) ∈ Rn+q : Fc zc (t) + F Cz(t) = 0
(52)
where Fc ∈ Rm×q and F ∈ Rm×p . Define D1 ∈ Rq×(p−m) 290
and D2 ∈ Rq×m as
291
[D1
D2 ] = DT
(53)
then the compensator can be written as
żc (t) = Hzc (t) + D1 C1 z1 (t) + D2 z2 (t)
292
(54)
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
293
294
7
where C1 is defined in (8). The sliding motion, obtained by
eliminating the coordinates z2 (t), can be written as
ż1 (t) = (A11 − A12 KC1 )z1 (t) − A12 Kc zc (t)
+ (Ad11 − Ad12 KC1 )z1 (t − τ ) − Ad12 Kc zc (t − τ )
żc (t) = (D1 − D2 K)C1 z1 (t) + (H − D2 Kc )zc (t)
(55)
where K = F2−1 F1 and Kc = F2−1 Fc , then similar to [37], the
296 design problem becomes one of selecting a compensator, re297 presented by the matrices D1 , D2 , and H, and a hyperplane,
298 represented by the matrices K and Kc , so that the system
ż1 (t)
z1 (t)
A11 − A12 KC1
−A12 Kc
=
(D1 − D2 K)C1 H − D2 Kc
żc (t)
zc (t)
295
Ac
Ad11 − Ad12 KC1
+
0
−Ad12 Kc
0
z1 (t − τ )
zc (t − τ )
It follows that
(56)
is stable. To obtain the compensator gains this problem can be
shown to be a new output-feedback problem with
A11 0
A12
K Kc
C1 0
0
Ac =
−
D1 H
0
0
D2 −Iq
0 Iq
Acd =
Aq
Bq
Kq
Aqd
Bqd
Cq
Ad11 0
Ad12 0
K Kc
C1 0
−
.
D1 H
0
0
0
0
0 Iq
Kq
Cq
(57)
The existence problem represented by system (56), where Ac
and Acd are partitioned as in (57) and D2 is a tuning parameter,
303 can be solved as for the noncompensated case (13). Similarly
304 to (27),
Q11
Q12
Kq Cq Q2 = Kq 0(p−m+q)×(n−p) Ip−m+q
Q22 M δQ22
301
302
= [ Kq Q22 M
= [Y M
δY ]
δKq Q22 ]
(58)
where Y = Kq Q22 , M ∈ R(p−m+q)×(n−p) is a tuning matrix.
306
Example 2: Consider the delay system




0 25 −1
0.1 0.1
0
A= 1
0
0  Ad =  0 0.3 −0.1 
−5 0
1
0 0.2
0
 
0
0 1 0
B = 0 C =
(59)
0 0 1
1
305
307
from which
A11 =
0
1
25
0
308
λ(A11 − A12 KC1 ) = ± (25 + K 2 )
and so (24) is infeasible. Now, consider designing a first-order 309
compensator. Choosing D2 = 1, it follows that
310




0 25 0
0.1 0.1 0
Aq =  1 0 0  Aqd =  0 0.3 0 
0 0 0
0
0 0




−1 0
−0.1 0
Bq =  0
0  Bqd =  0
0
1 −1
0
0
0 1 0
Cq =
.
0 0 1
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Acd
299
300
Fig. 3. Compensator-based controller design h = 2.5s.
A12 =
−1
0
Choosing δ = 50, ' = 5, M = [10 4] , and d = 0 (slowly 311
varying delay) with the maximum allowable delay h = 0.25s, 312
the LMITOOL solver returns
313
−25.38 −0.37
Kq =
.
−25.32 −5.03
The augmented system with compensator given by
H DC
0 0
Aa =
Ada =
0
A
0 Ad
0
I
0
Ba =
Ca = q
B
0 C
314
is asymptotically stablized by the controller
315
[Fc
F ] = [−0.369
− 25.38
Taking the controller in the form of (10) and (11) where γ = 10, 316
ρ = 10. Simulation results with the switching gain and initial 317
conditions [0.5, 0, 0], as shown in Figs. 3 and 4.
318
VII. C ONCLUSION
C1 = [1
0].
1].
319
A descriptor Lyapunov–Krasovskii functional method has 320
been introduced for SOF switching function design for systems 321
8
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
Fig. 4. Compensator-based controller design h = 2.5s.
with state time-varying delays. The delay is assumed bounded
with a known upper bound, either slowly or fast varying. In
addition, a novel stability analysis of the full-order closed325 loop discontinuous time-delay system has been performed via
326 the Krasovskii method, which is delay independent in z2 (t)
327 (and thus the delay is restricted to be slowly varying) and
328 delay dependent in z1 (t), i.e., in the state of the reduced-order
329 system. The proposed SOF design approach also applies to
330 compensator-based design. Examples show the effectiveness of
331 the method. For future work, the results can be extended to the
332 interval delay case, where the lower bound on the delay is taken
333 into account. The Razumikin approach can be employed for the
334 stability analysis of the full-order closed-loop system with fast
335 varying delay.
AQ2
AQ3
AQ4
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324
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X. R. Han received the degree in electrical engineering from Fushun Mining College, Fushun, China, in
2002, and the M.Sc. degree in electrical and electronic engineering from the University of Leicester,
Leicester, U.K., in 2006. He is currently working
toward the Ph.D. degree at the University of Kent,
Canterbury, U.K.
His research interests include sliding-mode control and time-delay systems.
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Emilia Fridman received the M.Sc. degree in
mathematics from Kuibyshev State University,
Kuibyshev, U.S.S.R., in 1981, and the Ph.D. degree in mathematics from Voroneg State University,
Voroneg, U.S.S.R., in 1986.
From 1986 to 1992, she was an Assistant and then
an Associate Professor in the Department of Mathematics, Kuibyshev Institute of Railway Engineers,
U.S.S.R. Since 1993, she has been with Tel Aviv
University, Tel Aviv, Israel, where she is currently
Professor of electrical engineering—systems. Her
research interests include time-delay systems, distributed parameter systems,
H∞ control, singular perturbations, nonlinear control, and asymptotic methods. She has published about 80 articles in international scientific journals.
She serves as an Associate Editor for Automatica and the IMA Journal of
Mathematical Control and Information.
9
Sarah K. Spurgeon received the B.Sc. and D.Phil.
degrees from the University of York, York, U.K., in
1985 and 1988, respectively.
She is currently a Professor of control engineering and the Director of the Electronics Department,
University of Kent, Canterbury, U.K. Her research
interests are in the area of robust nonlinear control
and estimation, particularly via sliding-mode techniques in which area she has published in excess of
200 refereed papers.
Dr. Spurgeon is a Fellow of the Institution of
Engineering and Technology, a Fellow of the Institute for Mathematics and
Applications, a Fellow of the Institute of Measurement and Control, and was
elected a Fellow of the Royal Academy of Engineering in 2008. She received
the IEEE Millennium Medal in 2000.
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Chris Edwards was born in Swansea, U.K. He
received the B.Sc. degree in mathematics from the
University of Warwick, Coventry, U.K., in 1987,
and the Ph.D. degree from Leicester University,
Leicester, U.K., in 1995.
From 1987 to 1991, he was a Research Officer
for British Steel Technical, Port Talbot, U.K., where
he was involved with mathematical modeling of
rolling and finishing processes. Since 1991, he has
been with Leicester University, where he was first a
Ph.D. student supported by a British Gas Research
Scholarship, appointed as a Lecturer with the Control Systems Research Group
in 1996, and promoted to Senior Lecturer in 2004 and Reader in 2007. He is a
coauthor of over 200 refereed papers and two books on sliding-mode control.
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HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
AUTHOR QUERIES
AUTHOR PLEASE ANSWER ALL QUERIES
AQ1 = “Assume that” was deleted. Please check if correct.
AQ2 = Please provide publication update in Ref. [9].
AQ3 = Please provide month of publication in Ref. [10].
AQ4 = Please provide publication update in Ref. [11].
AQ5 = Please provide publication update in Ref. [14].
AQ6 = Please provide issue number and month of publication in Ref. [16].
AQ7 = Please provide page range in Ref. [17].
AQ8 = Please provide page range in Ref. [31].
AQ9 = Please specify the degree received by Mr. Han from Fushun Mining College, Fushun, China, in 2002.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
3
On the Design of Sliding-Mode
Static-Output-Feedback Controllers for
Systems With State Delay
4
X. R. Han, Emilia Fridman, Sarah K. Spurgeon, and Chris Edwards
1
2
Abstract—This paper considers the development of slidingmode-based output-feedback controllers for uncertain systems
which are subject to time-varying state delays. A novel method
is proposed for design of the switching surface. This method
is based on the descriptor approach and leads to a solution in
terms of linear matrix inequalities (LMIs). When compared to
existing methods (even for systems without delays), the proposed
method is efficient and less conservative than other results, giving
a feasible solution when the Kimura–Davison conditions are not
satisfied. No additional constraints are imposed on the dimensions
or structure of the reduced order triple associated with design of
the switching surface. The magnitude of the linear gain used to
construct the controller is also verified as an appropriate solution
to the reachability problem using LMIs. A stability analysis for
the full-order time-delay system with discontinuous right-hand
side is formulated. This paper facilitates the constructive design
of sliding-mode static-output-feedback controllers for a rather
general class of time-delay systems. A numerical example from the
literature illustrates the efficiency of the proposed method.
24
Index Terms—Linear matrix inequalities (LMIs), sliding-mode
25 control (SMC), static output feedback (SOF), time delay.
I. I NTRODUCTION
26
S
LIDING-MODE control (SMC) [1] is known for its complete robustness to so-called matched uncertainties (which
29 can include time delays that satisfy matching conditions) and
30 disturbances [2]–[4]. The control technique has been applied in
31 many industrial areas [5]–[7]. Many early theoretical develop32 ments in SMC assume that all the system states are accessible.
33 In the case where only a subset of states are measurable, which
34 is relevant to a range of practical applications, either output
35 feedback control or the observer-based method are required.
36 Some work has considered implementation of SMC schemes
37 using observers [8]–[10]. In [11], a sliding-mode observer has
38 been shown to give a significant increase in performance in esti39 mation of the unknown variables of a boost converter compared
27
28
to a traditional current-mode control strategy. A further inter- 40
esting strand considers the fast output sampling method [12], 41
[13]. Recently, in [14], a fast sampling method is employed 42
for a discrete systems in the presence of time-varying delays 43
where a sliding-mode controller is designed using linear matrix 44
inequalities (LMIs) combined with a delta-operator approach. 45
However, all these methods require additional computation. 46
The most straightforward approach is to consider the study of 47
SMC via static output feedback (SOF).
48
One problem of interest in the development of SMC via SOF 49
is the design of the switching surface, which is effectively a 50
reduced order SOF problem for a particular subsystem. Two 51
different methods were proposed to design the sliding surface 52
using eigenvalue assignment and eigenvector techniques in [15] 53
and [16]. A canonical form was provided in [18] via which 54
the SOFSMC design problem is routinely converted to an SOF 55
stabilization problem. As stated in [20], all previous-reported 56
methods for the existence problem are, in fact, equivalent to 57
a particular SOF problem. The solution to the general SOF 58
problem, even for linear time-invariant systems, is still open. 59
LMI methods have been considered within the context of 60
sliding-mode controller design. For example, [21] and [22] 61
presented LMIs methods to design static sliding-mode output- 62
feedback controllers and [23] presented a necessary and suffi- 63
cient condition to solve the existence problem in terms of LMIs 64
for linear uncertain systems.
65
It is important to note that all the work described above 66
does not consider an existence problem involving delay and 67
many practical problems include such effects [24], [25]. In [26], 68
the problem of the development of sliding-mode controllers 69
for operation in the presence of single or multiple, constant 70
or time-varying state delays has been solved. This uses the 71
usual regular form method of solution and the uncertainty is 72
assumed to be matched, where matched describes that uncer- 73
tainty class which is implicit in the range of the input channels 74
and will be rejected by an appropriately designed SMC strategy, 75
although it is important to note that full state availability is 76
assumed. This problem has also been considered in [27] where 77
a class of uncertain time delay systems with multiple fixed 78
delays in the system states is considered. This paper considers 79
unmatched and time-varying parameter uncertainties, together 80
with matched and bounded external disturbances, but again, full 81
state information is assumed to be available to the controller. In 82
[28], Lyapunov functionals were for the first time introduced 83
for the analysis of time-varying delay. In [29], a descriptor 84
IE
E
Pr E
oo
f
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Manuscript received October 8, 2008; revised May 13, 2009. This work was
supported by the Engineering and Physical Sciences Research Council, U.K.,
under Grant EP/E 02076311.
X. R. Han and S. K. Spurgeon are with the Department of Electronics,
University of Kent, Canterbury, CT2 7NZ, U.K. (e-mail: xh25@kent.ac.uk;
S.K.Spurgeon@kent.ac.uk).
E. Fridman is with the School of Electrical Engineering, Tel Aviv University,
Tel Aviv 69978, Israel (e-mail: emilia@eng.tau.ac.il).
C. Edwards is with the Department of Engineering, University of Leicester,
Leicester, LE1 7RH U.K. (e-mail: ce14@leicester.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2009.2023635
1
0278-0046/$25.00 © 2009 IEEE
2
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
approach to stability and control of linear systems with time86 varying delays, which is based on the Lyapunov–Krasovskii
87 techniques, was combined with results on the SMC of such
88 systems. The systems under consideration were subjected to
89 norm-bounded uncertainties and uncertain bounded delays and
90 the solution given in terms of LMIs. Reference [30] develops
91 a SMC synthesis for a class of uncertain time-delay systems
92 with nonlinear disturbances and unknown delay values whose
93 unperturbed dynamics is linear. The synthesis was based on a
94 new delay-dependent stability criterion. The controller is found
95 to be robust against sufficiently small delay variations and
96 external disturbances.
97
It is important to emphasize that much of the aforementioned
98 literature on SMC of time-delay systems assumes full-state
99 feedback. Reference [31] considered SMC of systems with
100 time-varying delay. This paper considered a solution via LMIs
101 for the existence problem. The current paper extends this contri102 bution to consider the solution of the existence and reachability
103 problems. Specifically, the selection of parameters for stability
104 of the full-order closed-loop system are obtained via LMIs. In
105 this paper, a compensator-based design problem is considered
106 using the proposed SOF approach. Example from the literature
107 illustrates the efficiency of the method. In Section II, the
108 problem formulation is described. The existence problem is
109 formulated in Section III. In Sections IV and V, stability of
110 the full-order closed-loop system is derived via LMIs, and the
111 reachability problem is presented. Compensator-based design
112 is demonstrated in Section VI.
113
114
II. P ROBLEM F ORMULATION
Consider an uncertain time-delay system
ż(t) = Az(t) + Ad z (t − τ (t)) + B (u(t) + ξ(t, z, u))
y(t) = Cz(t)
(1)
where z ∈ Rn , u ∈ Rm , and y ∈ Rp with m ≤ p ≤ n. The
time-varying delay τ (t) is supposed to be bounded 0 ≤ τ (t) ≤
117 h, and it may be either slowly varying (i.e., differentiable
118 delay with τ̇ (t) ≤ d < 1) or fast varying (piecewise continuous
119 delay). Assume that the nominal linear system (A, Ad , B, C)
120 is known and that the input and output matrices B and C are
121 both of full rank. The unknown function ξ: R+ × Rn × Rm →
122 Rn , which represents the system nonlinearities plus any model
123 uncertainties, is assumed to satisfy the matching condition and
115
116
ξ(t, z, u) < k1 u + a(t, y)
(2)
for some known function a : R+ × Rp → R+ and positive
125 constant k1 < 1. It can be shown that if rank(CB) = m, there
126 exists a coordinate system in which the system (A, Ad , B, C)
127 has the structure
A11 A12
Ad11 Ad12
A=
Ad =
A21 A22
Ad21 Ad22
0
B=
C = [0 T ]
(3)
B2
124
where B2 ∈ Rm×m is nonsingular and T ∈ Rp×p is orthogo- 128
nal. The system can be represented as
129
ż1 (t) = A11 z1 (t) + Ad11 z1 (t − τ (t))
+ A12 z2 (t) + Ad12 z2 (t − τ (t))
2
ż2 (t) =
(A2i zi (t) + Ad2i zi (t − τ (t)))
i=1
+ B2 (u(t) + ξ(t, z, u))
y(t) = Cz(t).
(4)
Consider the following switching function:
130 AQ1
S = {z(t) ∈ Rn : F Cz(t) = 0}
(5)
for some selected matrix F ∈ Rm×p where by design 131
det(F CB) = 0. Let
132
[F1
F2 ] = F T
(6)
where F1 ∈ Rp−m and F2 ∈ Rm . As a result
IE
E
Pr E
oo
f
85
F C = [F1 C1
F2 ]
133
(7)
where
C1 = 0(p−m)×(n−p)
I(p−m) .
134
(8)
Therefore, F CB = F2 B2 and the square matrix F2 is 135
nonsingular. By assumption, the uncertainty is matched, and 136
therefore the sliding motion is independent of the uncertainty 137
represented by ξ(·). In addition, because the canonical form in 138
(3), where it is necessary that the pair (A11 , A12 ) is controllable 139
and (A11 , C1 ) is observable, can be viewed as a special case 140
of the regular form normally used in sliding-mode controller 141
design, the switching function can also be expressed as
142
s(t) = z2 (t) + KC1 z1 (t)
(9)
143
where K ∈ Rm×(p−m) and is defined as K = F2−1 F1 .
Then, a simple SMC law, depending on the output informa- 144
tion F y(t) can be defined by
145
u(t) = −γF y(t) − v(t)
where
v(t) =
F y(t)
ρ(t, y) F
y(t) ,
0,
if F y(t) = 0
otherwise
(10)
146
(11)
where ρ(t, y) is some positive scalar function of the outputs
147
ρ(t, y) = (k1 γ F y(t) + α(t, y) + γ2 ) /(1 − k1 )
where γ and γ2 are positive design scalars [18]. The closed- 148
loops system (4) and (10) can be described by the following 149
equations:
150
ż1 (t) = (A11 − A12 KC1 )z1 (t)
+ (Ad11 − Ad12 KC1 )z1 (t − τ (t))
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
ṡ(t) = (A21 − γB2 KC1 )z1 (t)
3
t−τ
(t)
1
ż1T (s)Rż1 (s)ds ≥
h
+ (Ad21 − γB2 KC1 )z1 (t − τ (t))
+ (A22 − γB2 )z2 (t) + (Ad22 − γB2 )z2 (t − τ (t))
t−h
t−τ
(t)
t−τ
(t)
ż1T (s)dsR
t−h
ż1 (s)ds.
t−h
(18)
+ B2 (ξ(t, z, u) − v(t))
y(t) = Cz(t).
(12)
Then,
162
V̇ (t) ≤ 2z1T (t)P ż1T (t) + h2 ż1T (t)Rż1 (t)
− (z1 (t) − z1 (t − τ (t)))T R (z1 (t) − z1 (t − τ (t)))
151
III. E XISTENCE P ROBLEM
152
153
154
On the sliding manifold s(t) = 0, it is well known [19] that
the reduced-order sliding motion is governed by a free motion
with system matrix
× R (z1 (t − τ (t)) − z1 (t − h))
ż1 (t) = (A11 − A12 KC1 )z1 (t)
− (1 − d)z1T (t − τ (t)) Sz1 (t − τ (t)) .
+(Ad11 − Ad12 KC1 )z1 (t − τ (t)) . (13)
155
Consider a Lyapunov–Krasovskii functional
t
z1T (s)Ez1 (s)ds
+
t−τ (t)
0 t
ż1T (s)Rż1 (s)dsdθ
+h
Using the descriptor method as in [33] and the free-weighting 163
matrices technique from [34], the right-hand side of the 164
expression
165
0 ≡ 2 z1T (t)P2T + ż1T (t)P3T
+(Ad11 − Ad12 KC1 )z1 (t − τ (t))]
z1T (s)Sz1 (s)ds
+
(14)
V̇ (t) ≤ η T (t)Θη(t) ≤ 0
where the symmetric matrices P > 0 and E, S, R ≥ 0.
The condition V̇ (t) < 0 guarantees asymptotic stability of
the reduced order system as in [32]. Differentiating V (t)
159 along (13)
156
V̇ (t) = 2z1T (t)P ż1 (t) + h2 ż1T (t)Rż1 (t)
t
ż1T (s)Rż1 (s)ds + z1T (t)(E + S)z1 (t)
−h
t−h
z1T (t
−
− h)Ez1 (t − h)
− (1 − τ̇ (t)) z1T (t − τ (t)) Sz1 (t − τ (t)) .
160
Further using the identity
t
−h
(15)
θ12 = P − P2T + '(A11 − A12 KC1 )T P2
θ33 = − (E + R)
ż1T (s)Rż1 (s)ds
(16)
t
t
ż1T (s)dsR
t−τ (t)
θ34 = R
θ44 = − 2R − (1 − d)S.
and applying Jensen’s inequality
t−τ (t)
170
θ24 = 'P2T (Ad11 − Ad12 KC1 )
t−τ (t)
1
≥
h
(22)
θ22 = − 'P2 − 'P2T + h2 R
ż1T (s)Rż1 (s)ds
−h
ż1T (s)Rż1 (s)ds
θ14
θ24 
<0
θ34
θ44
+ (A11 − A12 KC1 )T P2 + E + S − R
t−h
t
0
0
θ33
∗
θ11 = P2T (A11 − A12 KC1 )
t
161
θ12
θ22
∗
∗
169

θ14 = P2T (Ad11 − Ad12 KC1 ) + R
= −h
t−h
if the matrix inequality

θ11
 ∗
Θ=
∗
∗
(21)
is feasible, where
t−τ
(t)
ż1T (s)Rż1 (s)ds
(20)
with matrix parameters P2 , P3 = 'P2 ∈ Rn−m is 166
added into the right-hand side of (19). Setting η(t) = 167
col{z1 (t), z˙1 (t), z1 (t − h), z1 (t − τ (t))}, it follows that
168
−h t+θ
157
158
(19)
× [−ż1 (t) + (A11 − A12 KC1 )z1 (t)
t−h
t
+ z1T (t)(E + S)z1 (t) − z1T (t − h)Ez1 (t − h)
IE
E
Pr E
oo
f
V (t)
= z1T (t)P z1 (t)
− (z1 (t − τ (t)) − z1 (t − h))T
ż1 (s)ds
t−τ (t)
(17)
(23)
Multiplying matrix Θ from the right and the left by 171
diag{P2−1 , P2−1 , P2−1 , P2−1 } and its transpose, respectively, and 172
denoting
173
Q2 = P2−1
P = QT
2 P Q2
= QT
E
2 EQ2
= QT RQ2
R
2
S = QT
2 SQ2
4
174
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
< 0, where
it follows Θ < 0 ⇔ Θ


θ11 θ12 0 θ14


=  ∗ θ22 0 θ24  < 0
Θ
 ∗
∗ θ33 θ34 
∗
∗
∗ θ44
(24)
θ11 = (A11 − A12 KC1 )Q2
T
+ QT
2 (A11 − A12 KC1 ) + E + S − R
T
θ12 = P − Q2 + 'QT
2 (A11 − A12 KC1 )
θ14 = (Ad11 − Ad12 KC1 )Q2 + R
2
θ22 = − 'Q2 − 'QT
2 +h R
θ24 = '(Ad11 − Ad12 KC1 )Q2
−R
θ33 = − E
θ34 = R
− (1 − d)S.
θ44 = − 2R
176
177
178
Define the variable Q2 in the following form:
Q11
Q12
Q2 =
Q22 M δQ22
(26)
where Q22 is a (p − m) × (p − m) matrix, and M ∈
R(p−m)×(n−p) and δ ∈ R are a priori selected tuning parameters. It follows that
KC1 Q2 = [KQ22 M δKQ22 ].
179
Defining
Y = KQ22
180
it follows that
KC1 Q2 = [Y M δY ].
181
(27)
To construct K, substitute (27) into (25) to yield
θ11 = A11 Q2 − A12 [Y
207
208
209
then the matrix P̄ satisfies the structural constraint
211
Ā0 = Ā − γ B̄F C̄
= Ā − γ B̄[0 F2 ]
θ14 = Ad11 Q2 − Ad12 [Y M δY ] + R
2
θ22 = − 'Q2 − 'QT
2 +h R
210
(31)
(32)
for sufficiently large γ [18]. In the new coordinate system, the 214
uncertain system (1) can be written as
215
θ24 = 'Ad11 Q2 − 'Ad12 [Y M δY ]
ż(t) = Āz(t) + A¯d z(t − τ ) + B̄ (u(t) + ξ(t, z, u)) .
θ34 = R
− (1 − d)S
θ44 = − 2R
where Ā11 = A11 − A12 KC1 and Ād11 = Ad11 − Ad12 KC1
and F2 is a design parameter. Let P̄ be a symmetric positive
definite matrix partitioned conformably with the matrices in
(29) so that
P̄1 0
P̄ =
(30)
0 P̄2
if the design matrix F2 = B2T P̄2 . The matrix P̄ can be shown 212
to be a Lyapunov matrix for
213
T
T T
θ12 = P − Q2 + 'QT
2 A11 − '[Y M δY ] A12
(33)
The closed-loop system will have the form
(28)
with the tuning parameters δ, ', and M. The following Lemma
may now be stated.
184
Lemma 1: Given a priori selected tuning parameters ', δ,
185 and M ∈ R(p−m)×(n−p) , then (24) is an LMI in the decision
> 0, E
≥ 0, S ≥ 0, R
≥ 0 and matrices Q22 ∈
186 variables P
182
183
203
204
It can be shown [18] that there exist a coordinate system in 205
which the system triple (Ā, Ād , B̄, F C̄) has the property that 206
Ā11 Ā12
Ād11 Ād12
Ād =
Ā =
Ā21 Ā22
Ād21 Ād22
0
B̄ =
(29)
F C̄ = [0 F2 ]
B2
P̄ B̄ = C̄ T F T
T
δY ] + QT
2 A11
− [Y M δY ]T AT
12 + E + S − R
−R
θ33 = − E
IV. S TABILITY OF THE F ULL -O RDER
C LOSED -L OOP S YSTEM
IE
E
Pr E
oo
f
175
(25)
R(p−m)×(p−m) , Q11 ∈ R(n−p)×(n−p) , Q12 ∈ R(n−p)×(p−m) , 187
Y ∈ Rm×(p−m) . If a solution to (24) exists, which may be read- 188
ily obtained from available LMI tools, then the reduced order 189
system (13) is asymptotically stable for all differentiable delays 190
0 ≤ τ (t) ≤ h, τ̇ (t) ≤ d < 1. Moreover, (13) is asymptotically 191
stable for all piecewise-continuous delays 0 ≤ τ (t) ≤ h, if the 192
LMI (24) is feasible with S = 0.
193
Remark: The proposed method is suitable for SOF sliding- 194
mode controller design where Kimura–Davison conditions, 195
written as n ≤ m + p − 1, are not satisfied as in [20]–[22]. No 196
constraints are imposed on the dimensions of the reduced-order 197
triple A11 , A12 , C1 . This represents a constructive and efficient 198
approach to output-feedback-based design for a relatively broad 199
class of systems, which is less conservative than existing results 200
[20]–[22]; an example to illustrate the advantages of the method 201
for systems without time-delay is presented in [17].
202
ż(t) = Ā0 z(t) + Ād z(t − τ ) + B̄ (ξ(t, yt ) − vy (t)) .
216
(34)
For large enough γ > 0, these conditions are delay independent 217
with respect to the delay in z2 .} However, for derivation of 218
this condition using Lyapunov–Krasovskii techniques, it is 219
necessary to consider the case where τ̇ ≤ d < 1. A stability 220
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
condition for the full-order closed-loop system can be derived
222 using the following Lyapunov–Krasovskii functional:
221
t
T
z T (s)Ēz(s)ds
V (t) = z (t)P̄ z(t) +
t−h
t
0 t
T
+
z (s)S̄z(s)ds + h
ż T (s)R̄ż(s)dsdθ
(35)
−h t+θ
t−τ (t)
R̄1 0
223 where the matrix Ē, S̄ ≥ 0 and R̄ =
≥ 0 as it is
0 0
224 desired to determine a stability condition for the time delay
225 system which is delay independent with respect to delay in
226 z2 (t). Differentiating V (t) along the closed-loop trajectories
V̇ (t) ≤ 2z (t)P̄ ż (t) + h ż (t)R̄ż(t)
T
T
Inequality (41) is an LMI in the decision variables P̄1 > 0, Ē ≥ 235
0, S̄ ≥ 0 and R̄1 ≥ 0. Equation (37) is valid if (41) is satisfied 236
and given
237
2z T P̄ B̄ (ξ(t, z, u) − v(t))
= 2y T F T (ξ(t, z, u) − v(t))
× R̄ (z (t − τ (t)) − z(t − h))
< −2 F y(t) (ρ(t, y) − k1 u(t) − α(t, y)) .
(36)
Substitute the right-hand side of (34) into (36). Setting ς(t) =
col{z(t), z(t − h), z(t − τ (t))}, it follows that
is satisfied if ς T (t)Φh ς(t) + h2 ż T (t)R̄ż(t) < 0
2z T P̄ B̄(ξ(t, z, u) − v(t)) < 0, where


φ11 − R̄
0
P̄ A¯d + R̄

Φh = 
∗
−(Ē + R̄)
R̄
∗
∗
−2R̄ − (1 − d)S̄
φ11 = ĀT
0 P̄ + P̄ Ā0 + S̄ + Ē.
and
and so by rearranging
239
≥ k1 (v(t) + γ F y(t)) + α(t, y) + γ2
≥ k1 u(t) + α(t, y) + γ2 .
(43)
240
(38)
V̇ (t) < −2γ2 F y(t) < 0 if z(t) = 0
241
242
243
(39)
and therefore the system is asymptotically stable.
Lemma 2: Given large enough γ, let there exist n × n matrices P̄1 > 0, Ē ≥ 0, S̄ ≥ 0, R̄1 ≥ 0 from the LMI solver
such that LMI (41) holds. Given that the design parameters
k1 , α(t, y), γ2 , and P̄2 have been selected so that (44) holds,
the closed-loop system (33) is asymptotically stable for all
differentiable delays 0 ≤ τ (t) ≤ h, τ̇ (t) ≤ d ≤ 1.
V. F INITE -T IME R EACHABILITY TO THE
S LIDING M ANIFOLD
248
249
h2 ż T (t)R̄ż(t)
T
= h2 z T (t)Ā0 + z T (t − τ )A¯d
+ ξ(t, z, u) − v(t)T B̄ T R̄
238
ρ(t, y) = (k1 γ F y(t) + α(t, y) + γ2 ) /(1 − k1 )
From (37), if (41) is valid, then from (42) and (43),
Setting 1(t) = col{z(t), z(t − h), z(t − τ ), ξ(t, z, u) − v(t)}
However, by definition
ρ(t, y) = k1 ρ(t, y) + k1 γ F y(t) + α(t, y) + γ2
+ 2z T P̄ B̄ (ξ(t, z, u) − v(t)) < 0 (37)
with
(42)
IE
E
Pr E
oo
f
= −2ρ(t, y) F y(t) + 2 F y(t) ξ(t, z, u)
V̇ (t) ≤ ς(t)T Φh ς(t) + h2 ż T (t)R̄ż(t)
232
−R̄1
− (z (t − τ (t)) − z(t − h))T
− (1 − d)z T (t − τ (t)) S̄z (t − τ (t)) .
231
∗∗∗
≤ −2y T F T v(t) + 2 F y(t) ξ(t, z, u)
+ z T (t)(Ē + S̄)z(t) − z T (t − h)Ēz(t − h)
229
230
Using the Schur complement, ξ T (t)Φh ξ(t)+h2 ż T (t)R̄ż(t) < 0 233
holds if
234


I(n−m)
R̄1
hĀT
0
0




0(n,n−m)  Φh

(41)

 < 0.
I
T


(n−m)
R̄1 
hA¯d

0
2 T
− (z(t) − z (t − τ (t)))T R̄ (z(t) − (t − τ (t)))
227
228
5
× Ā0 z + A¯d z(t − τ ) + B̄ (ξ(t, z, u) − v(t))
 T 
Ā0  0n  I(n−m) 2
T
= 1 (t)  ¯ T 
h R̄1
Ad
0
T
B̄
 T T
T Ā0
I
 0n 
× (n−qm)
(40)
 ¯ T  1(t).
Ad
0
T
B̄
(44)
244
245
246
247
Corollary: An ideal sliding motion takes place on the sur- 250
face S if
251
−1 L
−1 L
B Ā0 z(t) + B Ād z(t − τ ) < γ2 − η
(45)
2
2
L
where the matrices ĀL
0 and Ād represent the last m rows of 252
Ā0 and Ād , respectively, and η is a small scalar satisfying 253
0 < η < γ2 .
254
Proof:
255
ṡ(t) = F C̄ Ā0 z̄(t) + F C̄ Ād z(t−τ )+F2 B2 (ξ(t, z, u)−v(t)) .
(46)
6
256
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
Let Vc : Rm → R be defined by
T
Vc (s) = sT (t) F2−1 P̄2 F2−1 s(t).
257
Then, using the fact that F2T = P̄2 B2 it follows that
T
T
F2−1
258
(47)
F2−1
P̄2 F2−1 F C̄ Ā0 = B2−1 ĀL
0
P̄2 F2−1 F C̄ Ād = B2−1 ĀL
d.
(48)
Then, it can be verified that
T
−1 L
V̇c = 2sT (t)B2−1 ĀL
0 z(t) + 2s (t)B2 Ād z(t − τ )
+ 2sT (t) (ξ(t, z, u) − v(t))
≤ 2 s(t) B2−1 ĀL
0 z(t) + 2 s(t)
Fig. 1.
Output against time.
< − 2η s(t)
IE
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× B2−1 ĀL
d z(t − τ ) − 2γ2 s(t)
(49)
if z(t) and z(t − τ ) ∈ Ω. It follows that there exists a t0 such
that z(t) and z(t − τ ) ∈ Ω for all t > t0 . Consequently, (49)
261 holds for all t > t0 . A sliding motion will thus be attained in
262 finite time.
263
Example 1: The following model of a liquid monopropel264 lant rocket motor has been considered in [35]. It is assumed
265 that the variable κ = 0.8 in this case, where Ad (1, 1) = −κ
266 and A(1, 1) = κ − 1. The outputs have been chosen to be the
267 second and fourth states so that in (1)
259
260

−0.2
 0
A=
−1
0
 
0
1
B= 
0
0
0 0
0 0
0 −1
1 1
C=

0
−1 

1
0

−0.8
 0
Ad = 
0
0
0 1
0 0
0 0
0 0

0
0

0
0
0 1 0 0
.
0 0 0 1
(50)
Clearly, the Kimura–Davison conditions are not met. Here,
the rate at which the delay varies with time has been examined
with d = 0, a slow varying delay. The gain from the LMI tool
271 solver with δ = 50, ' = 0.5, h = 0.45s, P̄2 = 1, γ = 2.9, and
272 M = [5 0.2], yields F = [−1 − 2.0754]. The poles of the
273 sliding-mode dynamics are {−2.67, −0.4, −0.2}. A simu274 lation was performed with the initial state values [1 1 1 1].
275 As shown, the LMI solver gave a feasible result for stability
276 for h ≤ 0.45s, but in simulation, the closed-loop system only
277 became unstable for h ≥ 1s (Fig. 1); this is due to the conserv278 ativeness of the method. The LMI solver gave feasible closed279 loop stability results for controller gain γ ≥ 2.9 while a choice
280 of γ ≥ 1 in simulation was able to stabilize the system with the
281 compensation of longer settling time. The switching function
282 for h = 0.45s, γ = 2.9 is shown in Fig. 2.
268
269
270
Fig. 2.
Switching function.
VI. C OMPENSATOR -BASED E XISTENCE P ROBLEM
283
For certain system triples (A11 , A12 , C1 ), LMI (24) is known 284
to be infeasible. In this case, consider a dynamic compensator 285
similar to that of El-Khazali and DeCarlo [36]
286
żc (t) = Hzc (t) + Dy(t)
(51)
where the matrices H ∈ Rq×q and D ∈ Rq×p are to be deter- 287
mined. Define a new hyperplane in the augmented state space, 288
formed from the plant and compensator state spaces, as
289
Sc = (z(t), zc (t)) ∈ Rn+q : Fc zc (t) + F Cz(t) = 0
(52)
where Fc ∈ Rm×q and F ∈ Rm×p . Define D1 ∈ Rq×(p−m) 290
and D2 ∈ Rq×m as
291
[D1
D2 ] = DT
(53)
then the compensator can be written as
żc (t) = Hzc (t) + D1 C1 z1 (t) + D2 z2 (t)
292
(54)
HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
293
294
7
where C1 is defined in (8). The sliding motion, obtained by
eliminating the coordinates z2 (t), can be written as
ż1 (t) = (A11 − A12 KC1 )z1 (t) − A12 Kc zc (t)
+ (Ad11 − Ad12 KC1 )z1 (t − τ ) − Ad12 Kc zc (t − τ )
żc (t) = (D1 − D2 K)C1 z1 (t) + (H − D2 Kc )zc (t)
(55)
where K = F2−1 F1 and Kc = F2−1 Fc , then similar to [37], the
296 design problem becomes one of selecting a compensator, re297 presented by the matrices D1 , D2 , and H, and a hyperplane,
298 represented by the matrices K and Kc , so that the system
ż1 (t)
z1 (t)
A11 − A12 KC1
−A12 Kc
=
(D1 − D2 K)C1 H − D2 Kc
żc (t)
zc (t)
295
Ac
Ad11 − Ad12 KC1
+
0
−Ad12 Kc
0
z1 (t − τ )
zc (t − τ )
It follows that
(56)
is stable. To obtain the compensator gains this problem can be
shown to be a new output-feedback problem with
A11 0
A12
K Kc
C1 0
0
Ac =
−
D1 H
0
0
D2 −Iq
0 Iq
Acd =
Aq
Bq
Kq
Aqd
Bqd
Cq
Ad11 0
Ad12 0
K Kc
C1 0
−
.
D1 H
0
0
0
0
0 Iq
Kq
Cq
(57)
The existence problem represented by system (56), where Ac
and Acd are partitioned as in (57) and D2 is a tuning parameter,
303 can be solved as for the noncompensated case (13). Similarly
304 to (27),
Q11
Q12
Kq Cq Q2 = Kq 0(p−m+q)×(n−p) Ip−m+q
Q22 M δQ22
301
302
= [ Kq Q22 M
= [Y M
δY ]
δKq Q22 ]
(58)
where Y = Kq Q22 , M ∈ R(p−m+q)×(n−p) is a tuning matrix.
306
Example 2: Consider the delay system




0 25 −1
0.1 0.1
0
A= 1
0
0  Ad =  0 0.3 −0.1 
−5 0
1
0 0.2
0
 
0
0 1 0
B = 0 C =
(59)
0 0 1
1
305
307
from which
A11 =
0
1
25
0
308
λ(A11 − A12 KC1 ) = ± (25 + K 2 )
and so (24) is infeasible. Now, consider designing a first-order 309
compensator. Choosing D2 = 1, it follows that
310




0 25 0
0.1 0.1 0
Aq =  1 0 0  Aqd =  0 0.3 0 
0 0 0
0
0 0




−1 0
−0.1 0
Bq =  0
0  Bqd =  0
0
1 −1
0
0
0 1 0
Cq =
.
0 0 1
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Acd
299
300
Fig. 3. Compensator-based controller design h = 2.5s.
A12 =
−1
0
Choosing δ = 50, ' = 5, M = [10 4] , and d = 0 (slowly 311
varying delay) with the maximum allowable delay h = 0.25s, 312
the LMITOOL solver returns
313
−25.38 −0.37
Kq =
.
−25.32 −5.03
The augmented system with compensator given by
H DC
0 0
Aa =
Ada =
0
A
0 Ad
0
I
0
Ba =
Ca = q
B
0 C
314
is asymptotically stablized by the controller
315
[Fc
F ] = [−0.369
− 25.38
Taking the controller in the form of (10) and (11) where γ = 10, 316
ρ = 10. Simulation results with the switching gain and initial 317
conditions [0.5, 0, 0], as shown in Figs. 3 and 4.
318
VII. C ONCLUSION
C1 = [1
0].
1].
319
A descriptor Lyapunov–Krasovskii functional method has 320
been introduced for SOF switching function design for systems 321
8
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
Fig. 4. Compensator-based controller design h = 2.5s.
with state time-varying delays. The delay is assumed bounded
with a known upper bound, either slowly or fast varying. In
addition, a novel stability analysis of the full-order closed325 loop discontinuous time-delay system has been performed via
326 the Krasovskii method, which is delay independent in z2 (t)
327 (and thus the delay is restricted to be slowly varying) and
328 delay dependent in z1 (t), i.e., in the state of the reduced-order
329 system. The proposed SOF design approach also applies to
330 compensator-based design. Examples show the effectiveness of
331 the method. For future work, the results can be extended to the
332 interval delay case, where the lower bound on the delay is taken
333 into account. The Razumikin approach can be employed for the
334 stability analysis of the full-order closed-loop system with fast
335 varying delay.
AQ2
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X. R. Han received the degree in electrical engineering from Fushun Mining College, Fushun, China, in
2002, and the M.Sc. degree in electrical and electronic engineering from the University of Leicester,
Leicester, U.K., in 2006. He is currently working
toward the Ph.D. degree at the University of Kent,
Canterbury, U.K.
His research interests include sliding-mode control and time-delay systems.
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Emilia Fridman received the M.Sc. degree in
mathematics from Kuibyshev State University,
Kuibyshev, U.S.S.R., in 1981, and the Ph.D. degree in mathematics from Voroneg State University,
Voroneg, U.S.S.R., in 1986.
From 1986 to 1992, she was an Assistant and then
an Associate Professor in the Department of Mathematics, Kuibyshev Institute of Railway Engineers,
U.S.S.R. Since 1993, she has been with Tel Aviv
University, Tel Aviv, Israel, where she is currently
Professor of electrical engineering—systems. Her
research interests include time-delay systems, distributed parameter systems,
H∞ control, singular perturbations, nonlinear control, and asymptotic methods. She has published about 80 articles in international scientific journals.
She serves as an Associate Editor for Automatica and the IMA Journal of
Mathematical Control and Information.
9
Sarah K. Spurgeon received the B.Sc. and D.Phil.
degrees from the University of York, York, U.K., in
1985 and 1988, respectively.
She is currently a Professor of control engineering and the Director of the Electronics Department,
University of Kent, Canterbury, U.K. Her research
interests are in the area of robust nonlinear control
and estimation, particularly via sliding-mode techniques in which area she has published in excess of
200 refereed papers.
Dr. Spurgeon is a Fellow of the Institution of
Engineering and Technology, a Fellow of the Institute for Mathematics and
Applications, a Fellow of the Institute of Measurement and Control, and was
elected a Fellow of the Royal Academy of Engineering in 2008. She received
the IEEE Millennium Medal in 2000.
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Chris Edwards was born in Swansea, U.K. He
received the B.Sc. degree in mathematics from the
University of Warwick, Coventry, U.K., in 1987,
and the Ph.D. degree from Leicester University,
Leicester, U.K., in 1995.
From 1987 to 1991, he was a Research Officer
for British Steel Technical, Port Talbot, U.K., where
he was involved with mathematical modeling of
rolling and finishing processes. Since 1991, he has
been with Leicester University, where he was first a
Ph.D. student supported by a British Gas Research
Scholarship, appointed as a Lecturer with the Control Systems Research Group
in 1996, and promoted to Senior Lecturer in 2004 and Reader in 2007. He is a
coauthor of over 200 refereed papers and two books on sliding-mode control.
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HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY
AUTHOR QUERIES
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AQ2 = Please provide publication update in Ref. [9].
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AQ6 = Please provide issue number and month of publication in Ref. [16].
AQ7 = Please provide page range in Ref. [17].
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