In the previous work, the authors studied the problem of robust discretization of linear timeinvariant systems with polytopic uncertainties, where linear matrix inequality (LMI) conditions were developed to find an approximate discretetime (DT) model of a continuoustime (CT) system with uncertainties in polytopic domain. The system matrices of obtained DT model preserved the polytopic structures of the original CT system. In this paper, we extend the previous approach to solve the problem of robust discretization of polytopic uncertain systems with aperiodic sampling. In contrast with the previous work, the sampling period is assumed to be unknown, timevarying, but contained within a known interval. The solution procedures are presented in terms of unidimensional optimizations subject to LMI constraints which are numerically tractable via LMI solvers. Finally, an example is given to show the validity of the proposed techniques.
1. Introduction
Continuoustime (CT) systems controlled by digital controllers are referred to as sampleddata (SD) control systems, which are composed of CT systems to be controlled, discretetime (DT) controllers controlling them, and the ideal sampler and zeroorder holder to convert the CT signals into DT ones and vice versa
[4]
. When the digital controller is implemented on an actual CT plant, the control action through the zeroorder holder appears as a piecewise constant signal in time, which is termed a SD controller. Significant research efforts on the SD control design have been made in the literature, and they can be divided into several categories. For instance, the socalled the direct DT design
[17]
is a design method based on the discretization of the CT system, where a DT controller is designed in DT domain directly. In the socalled lifting techniques
[1
,
4
,
27]
, the SD controller design problem is transformed into an equivalent finitedimensional discrete problem. The socalled jump systembased method
[14
,
25]
is based on the representation of the system in the form of hybrid discrete/continuous model. The input delay approach
[10
,
11
,
20]
treats the SD systems as a CT system with uncertain but bounded timevarying delay in the control input.
Among the promising results, this paper focuses on the direct DT design method, in which the computation of an exact DT model of the original CT plant is required. While for LTI systems, the exact DT model is available in principle, this is not the case for nonlinear systems
[15
,
16]
or uncertain LTI systems
[3
,
17]
. Rather, an approximate DT model can be used in replacement of the exact DT model for the SD control design. A major drawback of the approximation technique is that they can suffer from degradation in performance or even lead to instability of the resulting SD control system when the approximation error is relatively large
[18]
.
Especially for DT LTI systems with poyltopic uncertainties, substantial LMIbased results on robust control problems have been made up to date (e.g.,
[5

9
,
13
,
19
,
28

38]
), and most of them implicitly assumed that either exact or approximate polytopic DT model of the original CT plant is available. In order to apply the linear matrix inequality (LMI) methods for control design of DT systems, it is essential for the obtained approximate DT model to preserve the polytopic structure of the original CT system. A widely used simplest method is to take an approximation via the firstorder Taylor series of the exact DT model under the assumption of fast sampling/fast hold
[18]
. This strategy usually works well under fast sampling, but the approximation error may become prohibitively large if the sampling period is relatively long. To alleviate this problem, in the previous work
[18]
, we developed new LMIbased techniques to search for more exact approximation of the exact DT models of the original CT polytopic uncertain LTI systems, in which the discrepancy between the exact and the approximate DT models was minimized. To this end, we exploited higherorder truncated Taylor series of the exact DT model so that the truncation error of the approximate DT model can be reduced.
Although the proposed method was successful in reducing the approximation error, there was still an unsolved problem: it can be applied only to the case that the sampling period is constant in time. To resolve this problem, in this paper, we investigate the robust discretization problem under aperiodic sampling. Specifically, it is assumed that the sampling period is timevarying and unknown, but lies within a known interval. Similarly to
[18]
, this problem is tackled by minimizing the norm distances between the system matrices of the approximate and exact DT models. To obtain numerically tractable method to compute the approximation, the truncated Taylor series of the exact DT model is used similarly to
[18]
. The solution procedures are given in terms of unidimensional optimizations subject to LMIs, which can be readily tractable via convex optimizations
[2]
. To derive the LMI constraints, the socalled matrixdilation technique
[9]
,
[22

24]
is applied. A sufficient LMI condition to design a state feedback SD controller for the computed DT models is also studied briefly as one of applications of the proposed robust discretization strategy. Finally, an illustrative example is given to demonstrate the potential of the developed method.
2. Preliminaries
 2.1 Notations
The adopted notation is as follows:
and
: sets of nonnegative real numbers and nonnegative integer, respectively;
and
: the
n
dimensional Euclidean space and the set of all
m
×
n
real matrices, respectively;
A^{T}
: transpose of matrix
A
;
A
>0 (
A
<0,
A
≥ 0 , and
A
≤ 0 , respectively): symmetric positive definite (negative definite, positive semidefinite, and negative semidefinite, respectively) matrix
A
;
A
⊗
B
: Kronecker’s product of matrices
A
and
B
; He{
A
} : a shorthand notion for
A
+
A^{T}
;
I_{n}
:
n
×
n
identity matrix; 0
_{n}
and 0
_{m×n}
:
n
×
1
zero vector and
m
×
n
zero matrix, respectively
:
; * inside a matrix: transpose of its symmetric term;  ⋅  : Euclidean vector norm for vectors or the matrix twonorm for matrices;
A
_{⊥}
: any matrices whose columns form bases of the right nullspace of matrix
: unit vector of dimension
j
with a 1 in the
i
th component and 0 ’s elsewhere.
 2.2 Problem formulation
Consider the CT LTI polytopic uncertain system
where
,
is the state,
is the control input, and matrices
and
are not precisely known but assumed to belong to the convex set
where Δ
_{N}
is the unit simplex given by
It is assumed that the system is controlled by the SD controller
where 0 1 {
t
_{0}
,
t
_{1}
,…} represents an unbounded monotonously increasing sequence of sampling instants with elements in
, i.e.,
. We assume that the sampling interval, denoted by
θ_{k}
=
t
_{k+1}
−
t_{k}
, is timevarying and unknown but lies in a known compact set,
θ_{k}
∈ [
θ_{min}
,
θ_{max}
] where 0 <
θ
_{min}
<
θ
_{max}
< ∞ . The closedloop SD control system composed of (1) and (2) is given by
The state at time
t
_{k+1}
is
Introducing notation
Ad
(
α
,
θ_{k}
) :=
e^{Ac(α)θk}
u_{d}
(
k
) :=
u_{c}
(
t_{k}
) , system (3) can be converted to the uncertain DT linear timevarying (LTV) system
where
. DT LTV system (4) can be viewed as the exact discretization of (3) in the sense that 
x_{c}
(
t_{k}
) −
x_{d}
(
k
)  = 0,
,
α
∈ Δ
_{N}
is satisfied with
x_{c}
(0) =
x_{d}
(0) and any control input sequence {
u_{d}
(0),
u_{d}
(1),…}. Note that (4) is the exact DT model of the SD control system (3) (or CT system (1)). As indicated in
[18]
, due to the nonlinear and infinite dimensional nature of
A_{d}
(
α
,
θ_{k}
)
B_{d}
(
α
,
θ_{k}
)with respect to the uncertain parameters and sampling period
θ_{k}
, it may be difficult to find their exact representations that preserve the polytopic structures of
A_{c}
(
α
) and
B_{c}
(
α
) . To simplify the problem, let us consider uniform sampling period
θ
=
θ
_{0}
=
θ
_{1}
=⋯. In this case, most researches addressing the robust control of DT polytopic uncertain LTI systems approximate
A_{d}
(
α
,
θ_{k}
) and
B_{d}
(
α
,
θ_{k}
) to their firstorder power series with the assumption that the sampling period
θ
is sufficiently small. However, when
θ
is relatively large, the approximations become inaccurate. To alleviate this problem, the concept of the robust discretization was suggested in the previous work
[18]
. Roughly speaking, the robust discretization problem is finding approximations
G
(
α
) and
H
(
α
) of matrices
A_{d}
(
α
,
θ_{k}
) and
B_{d}
(
α
,
θ_{k}
) , respectively, such that both
G
(
α
) and
H
(
α
) preserve the polytopic structures of
A_{c}
(
α
) and
B_{c}
(
α
) . In other words, it is required that the approximations can be expressed as convex combinations of given vertices. Specifically, a simplified robust discretization problem addressed in
[18]
can be expressed as finding matrices
G_{i}
,
H_{i}
,
i
∈ {1,2, … ,
N
} that solve the optimizations
where
and
are the
h
order Taylor series approximations of matrices
A_{d}
(
α
,
θ_{k}
)and
B_{d}
(
α
,
θ_{k}
) , respectively. As mentioned in the introduction, the research in
[18]
only considered the case of the uniform sampling period. If the sampling period is timevarying within a known bound, the problem becomes more complicated. In this paper, we cope with the robust discretization problem under aperiodic sampling. The robust discretization problem considered in
[18]
is modified as follows.
Problem (Robust discretization under aperiodic sampling)
. Let integer
h
≥ 1 be given. Compute matrices
G_{ij}
,
H_{ij}
, (
i
,
j
)) ∈ {1, 2, …,
N
} × {1, 2}that solve the following optimizations:
where
Note that
G
(
α
,
θ_{k}
) and
H
(
α
,
θ_{k}
) depend on
θ_{k}
and have poyltopic structures with respect to
θ_{k}
.
3. Main Result
In this section, LMI solutions to the robust discretization with aperiodic sampling are presented. As in
[18]
, optimizations (5) and (6) can be rewritten by
Alternative expressions are
which are equivalent to (9) and (10), respectively. We will use expressions (11) and (12) rather than (9) and (10) since (11) and (12) are more suitable to be converted into LMI conditions. The following results can be viewed as the main results of this paper. They establish sufficient LMI conditions that ensure constraints (11) and (12).
Theorem 1
: Let
h
≥ 1 be given. If there exist matrices
and a scalar
γ_{A}
≥ 0 such that
where (
θ
_{1}
,
θ
_{2}
) = (
θ
_{min}
,
θ
_{max}
) , then constraint in (11) is satisfied.
Proof
. First of all, multiplying (13) by
α_{i}β_{j}
(
θ_{k}
) and summing for (
i
,
j
)∈{1, 2,…,
N
}×{1,2} , we obtain
where
G
(
α
,
θ_{k}
) and
H
(
α
,
θ_{k}
) are defined in (8). Applying the Schur complement to the above inequalities yields
Pre and postmultiplying the last inequality by Π
^{[h]T}
and its transpose, where
and using relation (
L_{h}
⊗
θA_{c}
(
α
)
^{T}
−
R_{h}
⊗
I_{n}
)Π
^{[h]}
= 0
_{nh×n}
we can obtain the constraint in (11). This completes the proof.
Similarly to Theorem 1, an LMI condition that ensures constraint (12) can be obtained.
Theorem 2
: Let
h
≥ 1 be given. If there exist matrices
and a scalar
γ _{B}
≥ 0 such that
where (
θ
_{1}
,
θ
_{2}
) = (
θ
_{min}
,
θ
_{max}
) , then constraint (12) is satisfied.
Proof
. The proof is straightforwardly extended from the proof of Theorem 1 so omitted for brevity.
In this regard, the optimizations in (11) and (12) can be replaced by the following optimizations subject to LMI constraints:
Remark.
Optimizations (16) and (17) are singleparameter minimization problems subject to LMI constraints, and hence, can be solved by means of a sequence of LMI optimizations, i.e. a line search or a bisection process over
γ_{A}
and
γ_{B}
, respectively, or solved by the eigenvalue problem (EVP)
[2]
, which is convex optimization, and hence, can be directly treated with LMI solvers
[12
,
21
,
26]
.
4. Application
Although the proposed strategy provides only approximate solutions to the robust discretization problem with aperiodic sampling, it may be at least more precise than the firstorder Taylor series approximation. Moreover, the proposed technique would be effective from the practical point of view since as stated in
[18]
, once a discretized model of a CT system is obtained, then it can be stored in database and used repeatedly for various SD control design purposes through existing LMIbased DT control design techniques (e.g.,
[5

7
,
9]
to name a few) in the literature. For instance, let us assume that matrices
,
(
i
,
j
)∈{1, 2,…,
N
}×{1, 2} are solutions to optimizations (16) and (17), respectively. Instead of considering exact discretization (4) of the original CT system (3), consider the following DT system which is an approximate discretization of (3) under aperiodic sampling:
where
is the state and
is the control input. Note that DT system (18) can be viewed as an approximate DT model of the exact DT model (4). In addition, let us consider the following statefeedback control law:
π (k) = Fξ (k).
The closedloop system is
Based on the LMI design approach developed in
[6]
, we can readily establish the following LMIbased statefeedback design condition.
Proposition 1.
If there exist matrices
,
and
such that LMIs
hold, then statefeedback gain given by
F
=
KS
^{−1}
stabilizes closedloop system (19) for all
α
∈ Δ
_{N}
and for all timevarying sampling period
θ_{k}
∈ [
θ
_{min}
,
θ
_{max}
]
Proof.
Multiplying (20) by
α_{i} β_{j}
(
θ_{k}
)
β_{l}
(
θ
_{k+1}
) and summing for (
i, j,l
)∈{1, 2,…,
N
}×{1, 2}
^{2}
, we obtain
where
and
G
(
α
,
θ_{k}
) and
H
(
α
,
θ_{k}
) are defined in (8). Next, by pre and postmultiplying the last inequality by
and its transpose, and by applying the extended Schur complement in
[6]
, it follows that
(G( α, θ_{k}) +H( α, θ_{k})F)^{T} X( α, θ_{k+1})
×(G( α, θ_{k}) +H( α, θ_{k})F) − X( α, θ_{k}) < 0,
∀(α, θ_{k}, θ_{k+1} ∈ Δ_{N} ×[θ_{min} ,θ_{max}]^{2}
where
F
=
KS
^{−1}
and
X
(
α
,
θ_{k}
) =
S
^{− T}
P
(
α
,
θ_{k}
)
S
^{−1}
. By means of the Lyapunov theory, one concludes that (19) is asymptotically stable for all
α
∈ Δ
_{N}
and for all timevarying sampling period
θ_{k}
∈ [
θ
_{min}
,
θ
_{max}
]. This completes the proof.
On the other hand, let us consider the SD statefeedback controller
for system (3). The closedloop SD control system is
If
G
(
α
,
θ_{k}
) =
A_{d}
(
α
,
θ_{k}
) and
H
(
α
,
θ_{k}
) =
B_{d}
(
α
,
θ_{k}
) for all
α
∈ Δ
_{N}
and
θ_{k}
∈ [
θ
_{min}
,
θ
_{max}
] , then one can expect that 
x_{c}
(
t_{k}
) −
ξ
(
k
)  = 0,
α ∈Δ
_{N}
is satisfied with
x_{c}
(0) =
ξ
(0) and any control input sequence {
π
(0),
π
(1),…} . Although the idealistic case may not occur in reality, we can still expect that if
G
(
α
,
θ_{k}
) ≅
A_{d}
(
α
,
θ_{k}
) and
H
(
α
,
θ_{k}
) ≅
B_{d}
(
α
,
θ_{k}
), then the solution
x_{c}
(
t
) to (22) closely matches the solution
ξ
(
k
) to (19) at every sampling instants {
t
_{0}
,
t
_{1}
,…} . In this respect, the proposed robust discretization under aperiodic sampling can be viewed as a practically useful and simple approach to deal with various SD control problems.
All numerical examples in the sequel were treated with the help of MATLAB R2012b running on a PC with Intel Core i73770 3.4GHz CPU, 32GB RAM. The LMI problems were solved with SeDuMi 1.3
[26]
and Yalmip
[21]
.
Example 1
. Let us consider the linearized model of the inverted pendulum system taken from
[3]
. Its statespace realization is given by
x_{c,1}
(
t
) is the position of the cart,
,
x
_{c,3}
(
t
) is the angle of the pendulum from the vertical,
,
is the mass of the pendulum,
is the mass of the cart,
is the length of the pendulum, and
u_{c}
(
t
) is the horizontal force applied to the cart. We assume
and
. Then, the system can be described by (1) with two vertices. By applying Theorems 1 and 2 with min max (
θ
_{min}
,
θ
_{max}
,
h
) = (0.01s,0.1s,7) , we obtain the approximate DT system (18) with (
γ_{A}
,
γ_{B}
) = (0.1112,4.8251×10
^{−8}
) and
By using Proposition 1, the statefeedback gain is calculated as follows:
F = [37.8 79.5 1148.3 210.5].
The simulation results with
x
_{0}
= [5 −3 2 −3]
^{T}
and
α
= [0.5,0.5]
^{T}
are depicted in
Figs. 1(a)(d)
, where
x_{c}
(
t
) (solid line) is the solution to the SD closedloop system (22) and
ξ
(
k
) at each sampling instant (dot) is the solution to the DT closedloop system (19). In other words, the dotted lines in
Figs. 1(a)(d)
can be viewed as the state trajectories
ξ
(
k
) of the approximately discretized model of the original CT system (1) and the solid lines indicate the state trajectories
x_{c}
(
t
) of the CT plant (1). The closeness of the two trajectories implies that the robust discretization approach proposed in this paper is an exact approximation of the exact discritization of the CT plant (1). From the figure, we confirm that the trajectory of
ξ
(
k
) closely matches the trajectory of
x_{c}
(
t
) at sampling instants {
t
_{0}
,
t
_{1}
,…} .
The solid line is the solution to the SD closedloop system (22) x_{c,1}(t) and the dotted line is the solution to the DT closedloop system (19) ξ_{1} (k) at each sampling instant.
The solid line is the solution to the SD closedloop system (22) x_{c,2}(t) and the dotted line is the solution to the DT closedloop system (19) ξ_{2} (k) at each sampling instant.
The solid line is the solution to the SD closedloop system (22) x_{c,3}(t) and the dotted line is the solution to the DT closedloop system (19) ξ_{3} (k) at each sampling instant.
The solid line is the solution to the SD closedloop system (22) x_{c,4}(t) and the dotted line is the solution to the DT closedloop system (19) ξ_{4} (k) at each sampling instant.
5. Conclusion
In this paper, our previous work on the robust discretization problem has been extended to deal with the same problem with aperiodic sampling. LMI conditions to compute approximate DT models of the original CT plants have been developed. Finally, an example has been given to illustrate the developed method.
Acknowledgements
This work was supported by National Research Foundation of Korea(NRF) grant by the Korea government (MEST) (NRF2012R1A2A2A01014088).
BIO
Dong Hwan Lee received the B.S. degree in Electronic Engineering from Konkuk University, Seoul, Korea, in 2008 and M.S. degree in Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2010. Since 2014, he is working toward a Ph.D. degree in the Department of Electrical and Computer Engineering, Purdue University, USA. His current research interests include stability analysis in fuzzy systems, fuzzymodelbased control, and robust control of uncertain linear systems.
Jin Bae Park received his B.S. degree in Electrical Engineering from Yonsei University, Seoul, Korea, and his M.S. and Ph.D. degrees in Electrical Engineering from Kansas State University, Manhattan, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, where he is currently a professor. His major interest is mainly in the field of robust control and filtering, nonlinear control, intelligent mobile robot, fuzzy logic control, neural networks, Hadamard transform, chaos theory, and genetic algorithms. He served as EditorinChief for the International Journal of Control, Automation, and Systems (IJCAS) (20062010) and President for the Institute of Control, Robot, and Systems Engineers (ICROS) (2013), and is serving as the VicePresident for Yonsei University (2014present)
Young Hoon Joo received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a project manager. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a visiting professor in the Department of Electrical and Computer Engineering. He is currently a professor in the Department of Control and Robotics Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent robot, intelligent control, humanrobot interaction, wind farm control, and intelligent surveillance systems. He served as President for Korea Institute of Intelligent Systems (KIIS) (20082009) and is serving as EditorinChief for the Intelligent Journal of Control, Automation, and Systems (IJCAS) (2014present) and is serving as the Vice President for the Korean Institute of Electrical Engineers (KIEE) (2013present).
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