NANZAN-TR-2009-01

Stability and Stabilization of Aperiodic Sampled-Data Control
Systems Using Robust Linear Matrix Inequalities
Yasuaki Oishi and Hisaya Fujioka

May 2009

Technical Report of the Nanzan Academic Society
Information Sciences and EngineeringStability and Stabilization of Aperiodic Sampled-Data Control
Systems Using Robust Linear Matrix Inequalities∗
Yasuaki Oishi† and Hisaya Fujioka‡
Stability analysis of an aperiodic sampled-data control system is discussed for application to network and embedded control. The stability condition is described in a linear matrix inequality to
be satisﬁed for all possible sampling intervals. Although this condition is numerically intractable,a tractable suﬃcient condition can be constructed with the mean value theorem. Special care ispaid on tightness of the suﬃcient condition for less conservative stability analysis. Asymptotic
exactness of the approach is discussed and a technique of adaptive division is presented for computational eﬃciency. Extension to stabilization is also discussed. Examples show the eﬃcacy of
the approach.
Keywords: sampled-data control, robust linear matrix inequality, semideﬁnite programming,
conservatism, asymptotic exactness, adaptive division.

1. Introduction
Sampled-data control is a matured research area and established methodology is available both foranalysis and design [3]. However, most of the existing results assume constant sampling interval andcannot be applied to network and embedded control systems, whose sampling interval is uncertain
and varying with time.
For analysis and design of such an aperiodic sampled-data control system, several approaches havebeen proposed. Some of them are based on the continuous-time or hybrid framework [6, 14, 15].
The stability conditions presented there are rather conservative though applicable to general systems.
Recent approaches such as [7, 8, 10, 22] provide less conservative stability conditions in the discretetime framework. Hetel et al. [10] gave a stability condition by approximately evaluating the eﬀect ofaperiodic sampling with a polynomial. Increase of the degree of the polynomial reduces conservatismof the result. On the other hand, Fujioka [7, 8] and Suh [22] gave a stability condition based on divisionof the region where the uncertain sampling interval takes a value. Here, increase of the resolution of
the division leads us to a less conservative result. Skaf and Boyd [20] used such division to evaluatedegradation of the optimal quadratic performance of an aperiodic sampled-data control system.
The approach to be presented in this paper inherits some ideas from Fujioka [7, 8] and Suh [22]but incorporates the following three techniques for reduction of conservatism. First, the stability
condition to be used converges to the continuous-time stability condition as the sampling intervalgoes to zero. Due to this property, less conservative stability analysis is possible without numericaldiﬃculty even when the sampling interval can be small. Second, the eﬀect of aperiodic samplingis modeled as parametric uncertainty rather than matrix uncertainty. Third, the region where the∗

May 6, 2009
Department of Systems Design and Engineering, Nanzan University, Seireicho 27, Seto 489-0863, Japan; email:oishi@nanzan-u.ac.jp‡
Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, Kyoto 606-8501, Japan; email:fujioka@i.kyoto-u.ac.jp†

sampling interval varies is divided into several subregions. Asymptotic exactness of the approach is
discussed and adaptive division is considered for computational eﬃciency. In this paper, the stability
condition is in the form of a linear matrix inequality (LMI, in short) to be satisﬁed for all possible
sampling intervals. In general, a parameter-dependent LMI to be satisﬁed for all possible parametervalues is called a robust LMI. Recently, intensive investigation has been made on a robust LMI whoseparameter dependence is polynomial or rational ([1, 5, 12, 18, 19] for example). Since our stability
condition exponentially depends on the sampling interval, we took a diﬀerent approach based on themean value theorem. This is an adaptation of the technique of Chesi–Hung [4].
This paper is organized as follows. In Section 2, our problem is provided. In Section 3, the stability
condition is presented in a robust LMI and its tractable suﬃcient condition is constructed. Section 4
introduces a region-dividing technique for less conservative stability analysis. Asymptotic exactness
as well as adaptive division is discussed there. Section 5 provides extensions of the approach. After
illustrating examples are presented in Section 6, the paper is concluded in Section 7.
Notation is standard. The symbols O and I denote the zero matrix and the identity matrix ofappropriate size. For a matrix A, the symbol σ(A) expresses its maximum singular value. For asymmetric matrix A, the symbols λ(A) and λ(A) stand for its maximum and minimum eigenvalues,
respectively. For symmetric matrices A and B, the inequalities A Â B and A º B mean λ(A − B) > 0
and λ(A − B) ≥ 0, respectively.

2. Problem
We consider a continuous-time linear systemẋ(t) = Ax(t) + Bu(t)with the n-dimensional state x(t) and its stabilization by state-feedback control with a constant gain F .
The state is measured only at discrete time instants 0 = t0 < t1 < t2 < · · · such that limk→∞ tk = ∞and the input is a piecewise signal determined as(tk ≤ t < tk+1 )u(t) = F x(tk )for each k = 0, 1, 2, . . .. We refer to this control system by S henceforth.
The control system S is diﬀerent from a conventional sampled-data control system in that the
sampling interval tk+1 − tk is not necessarily constant but may vary with k. We assume availabilityof its bounds h and h such thath ≤ tk+1 − tk ≤ h(k = 0, 1, 2, . . .).

Our problem is to verify stability of the control system S. We present an approach using a robust
LMI in the following sections.

23. Proposed Stability Condition
3.1. Formulation into a robust LMI
We use the criterion of quadratic stability for stability analysis of the control system S. Note ﬁrst in
S that the states at two adjacent sampling instants are related byx(tk+1 ) = Φ(tk+1 − tk )x(tk )with(k = 0, 1, 2, . . .)Z hΦ(h) = e

Ah

Z h
At

+

eAt dt(A + BF ).

e dt BF = I +00Hence, the exponential stability of S follows if there exists a symmetric matrix Q such thatQ Â O,Q − Φ(h)QΦ(h)T Â O(h ≤ h ≤ h).

This stability condition has two drawbacks to be utilized for stability analysis. First, the condition
is in the form of a robust LMI and we need to ﬁnd Q that satisﬁes the inequality for inﬁnitely many
values of h. Second, the matrix Φ(h) is close to identity when h is small, which makes the conditiondiﬃcult to handle numerically. In the following, we consider the second drawback ﬁrst and then theﬁrst.
In order to avoid the numerical drawback of Φ(h), we use the following stability condition equivalent
to the previous one.
Proposition 1. The control system S is exponentially stable if there exists a symmetric matrix Q
such thatÃ!
√−Ψ (h)Q − QΨ (h)ThΨ (h)Q√ÂO(h ≤ h ≤ h),(1)hQΨ (h)TQ
whereΨ (h) =

11
(Φ(h) − I) =hh

Z heAt dt(A + BF ).

(2)

0

Proof. Decompose (1) into Q Â O and its Schur complement. Substitution of the deﬁnition ofΨ (h) shows the equivalence to the quadratic stability condition given above. The proposition hence
follows.
¤
In the limit of h → 0, the matrix Ψ (h) converges to A + BF , which is the system matrix ofthe continuous-time control system where a continuous-time state-feedback control u(t) = F x(t) is
applied to ẋ(t) = Ax(t) + Bu(t). In the same limit, the stability condition (1) assures stability of thiscontinuous-time control system. Hence, this condition has no numerical drawback discussed above
even for small h.
The condition (1) is again a robust LMI to be satisﬁed for inﬁnitely many values of h. Since it isdiﬃcult to ﬁnd Q satisfying the robust LMI, we consider its suﬃcient condition expressed by ﬁnitely
many LMIs. We can solve those LMIs using a standard interior-point method. Once a solution Q isfound, the same Q serves as a solution of the original robust LMI. In this approach, it is critical touse a tight suﬃcient condition. To this aim, we assume availability of the real Jordan canonical formof A and consider a suﬃcient condition based on it.
3We consider the case of h > 0 in Section 3.2 and the case of h = 0 in Section 3.3. The equality
h = 0 means that a positive lower bound is not available for the sampling interval. This latter casecan be considered because our stability condition (1) is usable with small h.

3.2. The case of h > 0
The function Ψ (h) in the condition (1) has exponential dependence on the uncertain parameter h.
Since exponential parameter dependence is diﬃcult to handle, we replace Ψ (h) with a new function
dependent on not only h but also additional uncertain parameters in a multi-aﬃne way. This multiaﬃnity leads us to the desired suﬃcient condition expressed by ﬁnitely many LMIs.
Deﬁnition of the new function uses the real Jordan canonical form of A [11, Section 3.4]. For thereal n × n matrix A, there exists a real nonsingular matrix T such that

 (1)
A


A(2)

 −1
T

A=T
..

.


A(r)
(the elements not presented are equal to zero). Here, each A(i) is a matrix having one of the following
two forms:λ(i) 1
P (i) I2








(i) . . .
(i) . . .
λ
P




,
(3)



,
..
..




.
.
1
I
2 



λ(i)
P (i)
Ã
!
p(i) q (i)
(i)
(i)
(i)
where λ is some real number, P is a 2 × 2 real matrix of the form P =, and I2 is−q (i) p(i)the 2 × 2 identity matrix.
The function for replacement of Ψ (h) is now deﬁned as
Z bi
1 h h AtΨbh (h, θ) =e dt + (h − bh)T E(θ)T −1 (A + BF )(4)h 0with bh being any ﬁxed number such that h ≤ b
h ≤ h. Here, E(θ) is an aﬃne function of a newly
introduced n-dimensional uncertain parameter θ and has the block-diagonal form

 (1)
E (θ)


E (2) (θ)
E(θ) = ..
.

E (r) (θ)
consistent with the real Jordan canonical form of A. More precisely, when A(i) is an n(i) × n(i) matrixof the left form of (3), the corresponding E (i) (θ) is (i) (i)(i)
θ1 θ2 · · · θn(i)

.. 
..
(i)

.
θ
. 


1


..
(i) 

.
θ

2 
(i)
θ1
4

(i)

with n(i) uncertain parameters θj (j = 1, 2, . . . , n(i) ), which are elements of θ. When A(i) is a2m(i) × 2m(i) matrix of the right form of (3), the corresponding E (i) (θ) is
 (i)

(i)
(i)
Ξ1
Ξ2
· · · Ξm(i)

.. 
..
(i)

.
Ξ1
. 



,
..
(i) 

.
Ξ2 

(i)
Ξ1
Ã
(i)
where Ξj =

ξj

(i)

ηj

(i)

(i)
−ηj

(i)
ξj

!
(i)

(i)

and ξj , ηj (j = 1, 2, . . . , m(i) ) are 2m(i) uncertain parameters contained
(i)

(i)

(i)

again in θ. Hence, θ is a vector consisting of the parameters θj , ξj , and ηj , whose dimension sums
(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

up to n. The domain of the parameters θj ≤ θj ≤ θj , ξ (i)≤ ξj ≤ ξ j , and η j ≤ ηj ≤ η j arejdeﬁned as(i)hj−1 λ(i) he,h≤h≤h (j − 1)!

θj = max

hj−1 p(i) hecos q (i) h,h≤h≤h (j − 1)!

ξ j = max

hj−1 p(i) hesin q (i) h,
(j
−
1)!
h≤h≤h

η j = max(i)θj = min

(i)

ξ (i)
= min
j

(i)

η (i)
= min
j

hj−1 λ(i) he,h≤h≤h (j − 1)!
hj−1 p(i) hecos q (i) h,h≤h≤h (j − 1)!
hj−1 p(i) hesin q (i) h
(j
−
1)!
h≤h≤h

with the convention 0! = 1. The domain of θ is hence a box-shaped set, which will be denoted by Θ.
Example 2. When A is 2 × 2, its real Jordan canonical form is one of the three forms:
Ã
!
Ã
!
Ã
!
(1)
(1)
(1)
λ(1)
0
λ
1
p
q
T
T −1 ,
T
T −1 ,
T
T −1 .
0 λ(2)0 λ(1)−q (1) p(1)Correspondingly, the function E(θ) in (4) is deﬁned as
Ã (1)!
Ã (1) (1) !
θ1
0
θ1
θ2
(2) ,
(1) ,
0 θ1
0 θ1

Ã

(1)

η1

(1)

ξ1

ξ1

−η1

(1)

(1)

(5)

!
.

In each case, the number of the new parameters is two and thus θ is a 2-dimensional vector.

¤

Properties of Ψbh (h, θ) and Θ are summarized in the next lemma.
Lemma 3. Let bh be any number such that h ≤ b
h ≤ h. The function Ψbh (h, θ) and the domain Θ
deﬁned above has the following properties: (i) for any h ≤ h ≤ h, there exists θ ∈ Θ such that
Ψ (h) = Ψbh (h, θ); (ii) hΨbh (h, θ) is multi-aﬃne in h and θ; (iii) Ψbh (h, θ) is independent of θ at h = bh.
Proof. We prove the property (i) in the special case that A is a 2 × 2 matrix of the leftmost formin (5). The proof in the general case is similar.
For notational convenience, we write!
Ã
λ1 0
T −1 .
A=T
0 λ2
5

From the deﬁnition (2) of Ψ (h) it follows thatZ
1 h Ate dt(A + BF )
Ψ (h) =
h 0
ÃR h
!
Z b
i
λ1 t dte0
1 h h AtbhRh λ t=e dt + T
T −1 (A + BF )
2 dt
h 0
0
b
h e
Ã
!
Z b
i
(h − bh)eλ1 h10
1 h h At−1=e dt + T
T(A + BF ).
h 0
0(h − bh)eλ2 h2

(6)

The last equality is implied by the mean value theorem with h1 and h2 being some numbers betweenbh and h. Since h1 , h2 ∈ [h, h], we havemin eλi h ≤ eλi hi ≤ max eλi hh≤h≤h(i = 1, 2).

h≤h≤h

The property (i) hence follows.
The properties (ii) and (iii) are obvious from the deﬁnition of Ψbh (h, θ).

¤We replace Ψ (h) by Ψbh (h, θ) in the robust LMI (1). We will see below that the resulting LMI has
the vertex property due to the multi-aﬃnity of hΨbh (h, θ) (Lemma 3 (ii)). That is, the LMI holds for
all h ≤ h ≤ h and θ ∈ Θ if and only if the same LMI holds only at the vertices. We hence obtain a
suﬃcient condition consisting of a ﬁnite number of LMIs, which makes the stability analysis tractable.
Let ver Θ denote the set of the vertices of the box-shaped set Θ.
Theorem 4. Suppose h > 0. Let bh be any number such that h ≤ b
h ≤ h. The control system S is
exponentially stable if there exists a symmetric matrix Q such thatÃ!
√−Ψbh (h, θ)Q − QΨbh (h, θ)ThΨbh (h, θ)Q√ÂO(h = h, h; θ ∈ ver Θ).
(7)hQΨbh (h, θ)TQ√
Proof. Multiplication of h to the ﬁrst row and to the ﬁrst column of the matrix in (7) gives anequivalent inequalityÃ!
−hΨbh (h, θ)Q − hQΨbh (h, θ)T hΨbh (h, θ)QÂ O.
hQΨbh (h, θ)T
Q
Thanks to the multi-aﬃnity of hΨbh (h, θ), this inequality is multiconvex in h and θ. Hence, theinequality holds for all h ≤ h ≤ h and θ ∈ Θ if and only if the same inequality holds only at thevertices h = h, h and θ ∈ ver Θ. When the inequality holds for all h ≤ h ≤ h and θ ∈ Θ, the desired
exponential stability follows by Lemma 3 (i) together with Proposition 1.
¤
Some remarks are to follow.
Remark 5. The choice of b
h is up to the user. In particular, the choice b
h = h or b
h = h is computationally attractive. Indeed, by Lemma 3 (iii), the inequality (7) is independent of θ either at h = h
or h = h with this choice, which decreases the number of LMIs. When h is close to zero, the choicebh = h is preferable because the choice b
h = h makes (h − bh)/h large at h = h, which results in large
eﬀect of θ in Ψbh (h, θ).
¤
6

(i)

(i)

Remark 6. The coordinates of θ ∈ ver Θ consist of θj , θj , ξ (i), and others, whose computationjneeds minimization and maximization of some functions. When it is diﬃcult, we can use a boxshaped set larger than Θ by computing lower and upper bounds of the functions. This simpliﬁcation
however introduces additional conservatism into the stability analysis.
¤
Remark 7. The number of LMIs in the stability condition (7) is 2n + 1 with b
h = h or b
h = h. When
this number is large, it may be better to use its further suﬃcient condition consisting of 2n + 2 LMIs.
Such a suﬃcient condition is given by Ben-Tal–Nemirovski [2]. The associated conservatism has been
evaluated in their work.
¤

3.3. The case of h = 0
We next consider the case of h = 0. Although we could take the limit of the condition (7), we follow
a diﬀerent path with higher-order expansion of Ψ (h).
The function for the replacement of Ψ (h) is in this case
Ψ0 (θ) = [I + T E(θ)T −1 A](A + BF )with the same E(θ) as the previous subsection. The parameter θ is again an n-dimensional vector
(i)
(i)
(i)
consisting of θj , ξj , ηj . The lower and upper bounds of these parameters are
(i)

(i)

θj = 0,hj−1 λ(i) he,0≤h≤h (j − 1)!

θj = h max

hj−1 p(i) hecos q (i) h,0≤h≤h (j − 1)!

ξ j = h max

hj−1 p(i) hesin q (i) h,0≤h≤h (j − 1)!

η j = h max

(i)

ξ (i)
= h minj(i)= h minη (i)jhj−1 p(i) hecos q (i) h,0≤h≤h (j − 1)!
hj−1 p(i) hesin q (i) h.
0≤h≤h (j − 1)!
(i)

(i)

(i)

The domain of θ is hence a box-shaped set, denoted by Θ0 . Note that θj , ξ j , η j in this section
are diﬀerent from their counterparts in the previous subsection by factor of h. Hence, the parameterset Θ0 tends to be small when h is close to zero. In such a case, the following stability condition is
expected to give a less conservative result.
Lemma 8. The function Ψ0 (θ) and the domain Θ0 has the following properties: (i) for any 0 ≤ h ≤ h,there exists θ ∈ Θ0 such that Ψ (h) = Ψ0 (θ); (ii) Ψ0 (θ) is aﬃne in θ.
Proof. We again consider the special case that
!
Ã
λ1 0
T −1
A=T
0 λ2
to prove the property (i). Proof in the general case is similar.
In the deﬁnition (2) of Ψ (h), the integral satisﬁes
Z h

Z h
e

0

At

dt = hI +(eAt− I) dt = hI +0

Z hZ teAu du dt A0

70ÃR h R t!
λ1 u du dt
e0
0 0RhRt λ u
T −1 A.
= hI + T
2 du dt0e
0 0Application of the mean value theorem twice shows that it is equal toÃ!
0hh1 eλ1 h1
0hI + TT −1 A00hh2 eλ2 h2for some 0 ≤ h01 ≤ h1 ≤ h and 0 ≤ h02 ≤ h2 ≤ h. Here,00 ≤ hi eλi hi ≤ h max eλi h(i = 1, 2).

0≤h≤hUsing these results in the deﬁnition of Ψ (h), we have the property (i).
The property (ii) is obvious from the deﬁnition.

¤

We replace Ψ (h) by Ψ0 (θ) in the condition (1). The aﬃnity of Ψ0 (θ) implies the vertex property.
Theorem 9. Suppose h = 0. The control system S is exponentially stable if there exists a symmetricmatrix Q such thatÃ!
√−Ψ0 (θ)Q − QΨ0 (θ)ThΨ0 (θ)Q√(8)ÂO(h = 0, h; θ ∈ ver Θ0 ).
hQΨ0 (θ)TQ√
Proof. Aﬃnity of Ψ0 (θ) implies multiconvexity of the inequality (8) in θ and h, which meansthat this inequality holds for all 0 ≤ h ≤ h and θ ∈ Θ0 if and only if it holds only at the vertices.
Hence, under the condition of the theorem, exponential stability of S follows from Lemma 8 (i) and
Proposition 1.
¤

4. A Region-Dividing Technique4.1. Division of the region of the sampling interval
The proposed condition can be considerably conservative when the region of the sampling interval,[h, h], is large because the condition is based on the mean value theorem there. In such a case, we canreduce conservatism of the condition by division of the region of the sampling interval.
[j]
In the sequel, we mean by a division a set of subregions ∆ = {[h[j] , h ] | j = 1, 2, . . . , J} such thath = h[1] < h

[1]

[2]

= h[2] < h[J]= h[3] < · · · < h= h.

For a given division ∆, the following stability condition is considered.
[j]Theorem 10. Let ∆ = {[h[j] , h ] | j = 1, 2, . . . , J} be a division of [h, h] and bh[j] be any number in[j]the subregion [h[j] , h ] for each j. For each j with h[j] > 0, let Θ[j] be Θ with their h, h, b
h replaced[j] b [j]
[j]
[j]
[j]
by h , h , h , respectively. For j with h = 0, if any, let Θ0 be Θ0 with the same replacement.
Then, the control system S is exponentially stable if there exists a symmetric matrix Q such thatÃ!
√−Ψbh[j] (h, θ)Q − QΨbh[j] (h, θ)ThΨbh[j] (h, θ)Q[j]√ÂO(h = h[j] , h ; θ ∈ ver Θ[j] )(9)ThQΨbh[j] (h, θ)Q8for each j with h[j] > 0 andÃ!
√−Ψ0 (θ)Q − QΨ0 (θ)ThΨ0 (θ)Q√ÂO
hQΨ0 (θ)T
Q

[j]

[j]

(h = 0, h ; θ ∈ ver Θ0 )(10)

for j with h[j] = 0, if any.
[j]

Proof. For each subregion [h[j] , h ], the discussion in the proof of Theorem 4 or 9 is applicabledepending on whether h[j] > 0 or h[j] = 0. Consequently, the inequality (1) holds for any h in[j]¤∪j=1,2,...,J [h[j] , h ] = [h, h]. Proposition 1 then implies the exponential stability of S.

4.2. Asymptotic exactness
The stability condition of Theorem 10 is asymptotically exact when the choice bh[j] = h[j] is adopted for
all the subregions. Namely, if there exists Q satisfying the original intractable stability condition (1),the same Q satisﬁes the condition of Theorem 10 for a suﬃciently ﬁne division ∆. Hence, conservatismof our tractable stability condition can be reduced to any degree at the cost of increased computational[j][j]complexity. For more precise statement, we call |h − h[j] | the radius of the subregion [h[j] , h ] and[j][j]
maxj=1,2,...,J |h − h[j] | the maximum radius of the division ∆ = {[h[j] , h ] | j = 1, 2, . . . , J}, which isdenoted by rad ∆.
Theorem 11. Suppose that there exists Q satisfying the original stability condition (1). Then, thesame Q satisﬁes the condition of Theorem 10 for a division ∆ having suﬃciently small rad ∆ when
the choice bh[j] = h[j] is adopted for any j.
¤

Proof. See Appendix A.

4.3. Formulation into an SDP problem
To prepare for the next subsection, we show here that our stability condition can be restated equivalently in a semideﬁnite programming (SDP, in short) problem. Here, an SDP problem is an optimization problem having LMIs as its constraints. This fact is important also because it enables us to test
our stability condition with the softwares for an SDP problem.
[j]

Theorem 12. Let ∆ = {[h[j] , h ] | j = 1, 2, . . . , J} be a division of [h, h]. The control system S is
exponentially stable if the following SDP problem is feasible with a positive x:
maximizex

subject to Q º I,− Ψbh[j] (h, θ)Q − QΨbh[j] (h, θ)T − hΨbh[j] (h, θ)QΨbh[j] (h, θ)T º xI[j](h = h[j] , h ; θ ∈ ver Θ[j] )for each j with h[j] > 0, and− Ψ0 (θ)Q − QΨ0 (θ)T − hΨ0 (θ)QΨ0 (θ)T º xI
[j]

[j]

(h = 0, h ; θ ∈ ver Θ0 )for j with h[j] = 0, if any.
9

Proof. Suppose that the SDP problem above is feasible with a positive x. Then, all the left-handside matrices of the inequalities are positive deﬁnite. This implies the existence of Q that satisﬁes thecondition of Theorem 10. Hence, the exponential stability of S follows.
¤
Some remarks are necessary on the SDP problem above. First, its LMI constraints have been
reduced in size from those in Theorem 10 by considering the Schur complements. This is beneﬁcial
for computational eﬃciency. Second, when the SDP problem is feasible for a positive x, its maximum
value is unbounded because its LMI constraints are linear in x and Q. Finally, no conservatism isintroduced with the restatement as an SDP problem. To see this, assume the existence of Q satisfying
the condition of Theorem 10. Then, decompose each inequality (9) or (10) into Q Â O and its Schurcomplement. Since Q Â O, there exists a positive number c such that cQ º I. Then, with this cQ
and some positive x, the SDP problem in Theorem 12 is feasible.

4.4. Adaptive division
While ﬁne division gives a less conservative stability condition, it increases the number of LMIs andthen computational cost. It is hence desirable that ﬁne division is made only in an important subregionof h. Although such an important subregion is diﬃcult to ﬁnd a priori, the following technique ofadaptive division is often eﬀective. The corresponding technique is used in [16, 17].
Suppose that we construct the SDP problem of Theorem 12 for some division and obtain the
nonpositive maximum value. Since the stability of the control system S is not assured in this case,
we are to reﬁne the division. We here notice an active constraint, which is an LMI constraint suchthat the discrepancy between its two sides has a zero eigenvalue with the obtained maximum solution.
Since an active constraint prevents the maximum value from being improved, we may be able to make
improvement by subdividing a subregion having an active constraint. Based on this idea, we have thefollowing algorithm for adaptive division, which is expected to produce an eﬃcient division, that is, adivision that gives a less conservative result with small amount of computation. We here mean by anactive subregion a subregion having an active constraint. When the maximum value is not attained,
an active constraint or an active subregion is not deﬁned.
Algorithm 13.
0. Prepare a coarse division.
1. Solve the SDP problem of Theorem 12 corresponding to the current division.
2. Stop if the problem is feasible with a positive x.
3. If the maximum value is attained, ﬁnd and subdivide an active subregion. Otherwise, ﬁnd andsubdivide a subregion of the maximum radius.
4. Go back to Step 1 unless the number of subregions exceeds the prescribed number.

¤

Algorithm 13 appears contradictory with Theorem 11 because the produced non-uniform division
is not eﬃcient for reduction of the maximum radius of a division. This contradiction is resolved by thetheorem below, which says that reduction of the maximum active radius is no worse than reduction of10

the maximum radius. Here, the maximum active radius of a division ∆, denoted by a-rad ∆, means
the maximum radius over all active subregions in ∆ when the maximum value is attained in the SDP
problem for ∆. Note that the maximum active radius depends on the maximum solution. When the
maximum solution is not unique, the maximum active radius is deﬁned as the minimum among thepossible values. It is obvious that a-rad ∆ ≤ rad ∆. The theorem below says that, when some divisione∆ has the SDP problem that attains the nonpositive maximum value, there exists a ﬁner division ∆e = a-rad ∆ but is not superior to ∆ in the sense that the maximum value of thethat satisﬁes rad ∆corresponding SDP problem is not larger than that for ∆. Hence, nothing is lost with reduction of
the maximum active radius, which is what Algorithm 13 aims at.
Theorem 14. Suppose that the SDP problem of Theorem 12 attains the nonpositive maximum value
e such that the maximum radius of ∆e is equal to thefor a division ∆. Then, there exists a division ∆e has the maximum value not larger thanmaximum active radius of ∆ and the SDP problem for this ∆that for ∆.
Proof. The idea of the proof is essentially the same as Theorem 8 of [16].
Consider the SDP problem for ∆ and let (x∗ , Q∗ ) be its maximum solution for which the maximum
active radius a-rad ∆ is deﬁned. We subdivide each inactive subregion, if necessary, so that each ofethe created subregion has the radius smaller than or equal to a-rad ∆. The resulting new division ∆e = a-rad ∆.
satisﬁes rad ∆e contains the LMI constraints corresponding to the newly created subThe SDP problem for ∆e is equal to that for ∆.
regions. If (x∗ , Q∗ ) satisﬁes these constraints, the maximum value for ∆Otherwise, it is equal or smaller. In any case, the conclusion of the theorem follows.
¤

5. Extensions
We extend our approach to stability analysis with matrix uncertainty as well as design of a statefeedback gain.

5.1. Approach with matrix uncertainty
We have assumed so far availability of the real Jordan canonical form of the system matrix A in orderto reduce the problem to a robust LMI with parametric uncertainty. This assumption may not bepractical when the system dimension n is large. For such a case, we will consider below an approachwith matrix uncertainty. This approach does not require the real Jordan canonical form though often
more conservative than the previous approach.
For simpliﬁcation of the description, we present the result only in the case that [h, h] is not divided.
It is, however, straightforward to use the region-dividing technique for reduction of conservatism. We
begin with the case of h > 0. We can show the following lemma, which is a counterpart of Lemma 3.
Lemma 15. Let bh be any number such that h ≤ b
h ≤ h. For any h ≤ h ≤ h, there exists Ω such thatm
Ψ (h) = Ψb (h, Ω) and σ(Ω) ≤ ω, whereh1Ψbhm (h, Ω) =
h

hZ bh

i
b
eAt dt + (h − bh)ΩeAh (A + BF )

0

11

and

n
h ¡ A + AT ¢i
ω = max max exp λ(h − b
h) ,
2
h≤h≤b
h

h ¡ A + AT ¢iomax exp λ(h − b
h) .
2
b
h≤h≤h

(11)

The symbols λ and λ stand for the minimum and the maximum eigenvalues, respectively.
¤

Proof. See Appendix B.

We use in the proof a speciﬁc bound on the norm of a matrix exponential function, which has beenused in [7, 8]. Similarly to those papers, it is possible to use diﬀerent bounds instead.
We replace Ψ (h) with Ψbm (h, Ω) in the condition (1) to obtain the following suﬃcient condition.
hTheorem 16. Suppose h > 0. Let bh be any number such that h ≤ b
h ≤ h. Deﬁne ω as (11). The
control system S is exponentially stable if there exist a symmetric matrix Q and a nonnegative number
sh , dependent on h, such that


√
ω(h−b
h)
h
T
E
Q
−
QE
− h1 E1 Q − h1 QE1T − sh I
1
2
h
√ h
√


ω h(h−bh)hT(h = h, h),Q
QE2T 

ÂO
h QE1
h
√
b
b
h)
ω h(h−h)− ω(h−
E2 Q
E2 Q
sh I
h
h
where
E1 =

Z bh

eAt dt(A + BF ),b

E2 = eAh (A + BF ).

0

Proof. It is known in general [23] that, for real matrices M1 , M2 , and M3 of appropriate size, theinequality M1 + M2 ΩM3 + M3T Ω T M2T Â O holds for any matrix Ω such that σ(Ω) ≤ ω if and only ifthere exists a nonnegative number s such thatÃ!
M1 − sM2 M2T ωM3T
Â O.
ωM3
sI
We use this result withÃ− h1 E1 Q − h1 QE1T
√
M1 =
h
T
h QE1

!

√

h
h E1 Q

Q

,

Ã !
I
M2 =
,
O

³
b
h)
M3 = − (h−h E2 Q

√

´

h(h−b
h)
E2 Q
h

.

Then, the assumption of the theorem guarantees!
Ã√ m−Ψbm (h, Ω)Q − QΨbm (h, Ω)ThΨb (h, Ω)Q
h √
h
h
ÂO
hQΨbm (h, Ω)TQhfor h = h, h and for any matrix Ω with σ(Ω) ≤ ω. Similarly to the proof of Theorem 4, the aﬃnity ofhΨbm (h, Ω) in h implies that the same inequality holds for any h ≤ h ≤ h and any Ω with σ(Ω) ≤ ω.
hLemma 15 implies the theorem.
¤
On the choice of bh, Remark 5 again applies. That is, bh = h or b
h = h is a good choice forbcomputational reason. When h is close to zero, h = h is preferable.
In the case of h = 0, a suﬃcient condition can be constructed in a similar way. We present only
the results because their derivation is straightforward. The ﬁrst step is preparation of a function for
replacement of Ψ (h).
12Table 1. Stability analysis with adaptive division
dividing points
0, 1.7294
0, 0.8647, 1.7294
0, 0.8647, 1.2971, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.6214, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.6214, 1.6754, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.6214, 1.6754, 1.7024, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.6214, 1.6754, 1.7024, 1.7159, 1.7294
0, 0.8647, 1.2971, 1.5133, 1.6214, 1.6754, 1.7024, 1.7159, 1.7227, 1.7294

comp. time (s)
0.374
0.493
0.540
0.624
0.658
0.708
0.708
0.727
0.518

max. value−0.805−0.147−0.0353−0.00870−0.00214−5.17 × 10−4
−1.11 × 10−4
−9.81 × 10−6+∞

Lemma 17. For any 0 ≤ h ≤ h, there exists Ω such that Ψ (h) = Ψ0m (Ω) and σ(Ω) ≤ ω0 , whereΨ0m (Ω) = (I + ΩA)(A + BF )andh ¡ A + AT ¢ ih .
ω0 = h max exp λ
2
0≤h≤h

(12)

We replace Ψ (h) with Ψ0m (Ω) in the condition (1) and remove the matrix uncertainty Ω similarly
to the case of h > 0. The resulting condition is the following.
Theorem 18. Suppose h = 0 and deﬁne ω0 as (12). The control system S is exponentially stable if
there exists a symmetric matrix Q and a nonnegative number sh , dependent on h, such that√−(A + BF )Q − Q(A + BF )T − sh Ih(A + BF )Q−ωQ(A + BF )T AT√√(h = 0, h).
hQ(A + BF )TQ
ω hQ(A + BF )T AT  Â O
√−ωA(A + BF )Q
ω hA(A + BF )Q
sh I

5.2. Design of a state-feedback gain
Our approach can be generalized to design of a state-feedback gain. The stability conditions (7) and(8) contain the products Ψbh (h, θ)Q and Ψ0 (θ)Q, which include the factor (A + BF )Q. If we replace
it by AQ + BG and solve the inequalities for Q and G, we can obtain a stabilizing feedback gain
by F = GQ−1 . The region-dividing technique is again eﬀective. Similar extension is possible on theapproach with matrix uncertainty.

6. Examples
The proposed approach is applied to the sampled-data control system with
Ã
!
Ã !
0
1
0
A=
, B=
, F = (−3.75 − 11.5).
0 −0.10.1
Stability of this control system has been analyzed with various approaches. For example, the approachof [15] veriﬁed stability with the region of the sampling interval h being (0, 1.1137]; the approach of13

[14] for h ∈ (0, 1.3659]; the approach of [7] for h ∈ [0.01, 1.72]; the approach of [22] for h ∈ [0.5, 1.729].
This system is known to be unstable for the constant sampling interval h = 1.7295.
We chose the region of the sampling interval with h = 0 and h = 1.7294 (i.e., h ∈ (0, 1.7294]) andsuccessfully veriﬁed the stability with the approach of Section 4. This shows eﬃcacy of the present
approach because the existing approaches do not allow such a large region of sampling interval. The
division was adaptively constructed with Algorithm 13 and consisted of 9 subregions when it assured
the stability. The process of the construction is summarized in Table 1. The region near to 1.7294
was divided ﬁnely, which was considered important for less conservative analysis. Here, we chosebh[j] = h[j] for all the subregions. The SDP problems were solved with the SDP solver SeDuMi [21]and the modeling language YALMIP [13]. The used computer was equipped with Intel Core 2 Duo
U7500 (1.06 GHz) and memory of 2 GBytes.
[j]
The result became even better when we chose bh[j] = h for all the subregions except for the one
including the origin. Namely, the stability was assured for a division consisting of only two subregions[0, 0.8647] and [0.8647, 1.7294]. The computational time was 0.306 s.
We next took the approach in Section 5.1 using matrix uncertainty. The bounds h and h werechosen the same as before and the numbers bh[j] equal to h[j] for all the subregions. The stability was
veriﬁed with a division consisting of 17 subregions, which was constructed adaptively. The computational time was 0.911 s. The increase of the number of subregions shows conservatism of this approach.
[j]
The choice bh[j] = h was ineﬃcient in this case requiring 31 subregions to assure stability.
Finally, we designed the state-feedback gain F for the A and B above. We took the approachin Section 5.2 with h = 0 and h = 10. We chose bh[j] = h[j] for all the subregions. As a result, astabilizing gain F = (−0.238 − 1.674) was obtained with the division consisting of [0, 5] and [5, 10].
The computational time was 0.406 s. In fact, stabilization was possible for a larger region of samplinginterval. In this case, however, the adaptive division did not work well due to numerical diﬃculty andan eﬃcient division had to be found with an ad hoc method.

7. Conclusion
Stability analysis is considered for a sampled-data control system with uncertain sampling interval.
Stability condition is presented in a robust LMI and its tractable suﬃcient condition is obtained withthe mean value theorem. The experimental result is satisfactory and shows the eﬃcacy of the proposed
approach.

A. Proof of Theorem 11
For the proof of the theorem, we need the following lemma, which states that the maximum error
between Ψbh (h, θ) and Ψ (h) in some subregion can be bounded by a linear function of the radius of thesubregion.
Lemma 19. There exists a positive number C such that the inequality0

max 0 max0 σ[Ψbh0 (h, θ) − Ψ (h)] ≤ C|h − h0 |h0 ≤h≤h θ∈Θ14

0
holds for any subregion [h0 , h ] ⊆ [h, h] with h0 > 0 when the choice b
h0 = h0 is adopted. Similarly, inthe case of h = 0, there exists a positive number C0 such that the inequality0

max 0 max0 σ[Ψ0 (θ) − Ψ (h)] ≤ C0 h

0≤h≤h θ∈Θ00holds for any subregion [0, h ] ⊆ [0, h]. Here, the set Θ0 is an adaptation of Θ whose h, h, b
h are
0 b0
0
0
0
replaced by h , h , h = h , respectively. Similarly for Θ0 .
Proof. We prove the ﬁrst statement of the lemma in the special case that
Ã
!
λ1 0
A=T
T −1 .
0 λ2
The proof of the remaining part is similar.
The equations (4) and (6) implyh−bh0Ψbh0 (h, θ) − Ψ (h) =ThÃθ1 − eλ1 h100θ2 − eλ2 h2!
T −1 (A + BF ).

Since θ1 belongs to [minh0 ≤h≤h0 eλ1 h , maxh0 ≤h≤h0 eλ1 h ] and so does eλ1 h1 , the discrepancy |θ1 − eλ1 h1 |0

is bounded from above by maxh0 ≤h≤h0 eλ1 h − minh0 ≤h≤h0 eλ1 h , which is further bounded by c1 |h − h0 |if c1 is larger than maxh≤h≤h |λ1 eλ1 h |. Note that c1 can be chosen independently of the subregion0[h0 , h ]. Similar discussion is possible with |θ2 − eλ2 h2 |. On the other hand, |(h − bh0 )/h| ≤ 1 because0¤h0 = bh0 ≤ h ≤ h . Hence, the ﬁrst statement of the lemma follows.
[j]

Proof of Theorem 11. Let [h[j] , h ] be any subregion in the division ∆ with h[j] > 0. From theleft-hand side of the inequality (9) with respect to this subregion, subtract the left-hand side of (1) to
have
Ã
!
Ã
!
´
√ ´
Ψbh[j] (h, θ) − Ψ (h) ³−Q ³−Q
hQ + √
Ψbh[j] (h, θ)T − Ψ (h)T O .
OhQDue to Lemma 19, the maximum singular value of this matrix can be bounded from above by a number[j]proportional to |h −h[j] |. Hence, if the maximum radius rad ∆ is small enough, the maximum singularvalue of this matrix is small for any subregion in ∆, which establishes the theorem.
¤

B. Proof of Lemma 15
The claim is obvious for h = b
h. We hence consider the case of h 6= b
h in the sequel.
By the deﬁnition of Ψ (h), we haveZ
1 h AtΨ (h) =e dt(A + BF )h 0
Z b
Z h−bhi
1 h h Atb=e dt +eAt dt eAh (A + BF ).
h 00Hence, we are done if the maximum singular value ofZ h−bh1
eAt dth−b
h 015

is less than or equal to ω.
Suppose h − b
h > 0 ﬁrst. Since σ(eAt ) ≤ exp[λ((A + AT )/2)t] for t ≥ 0 [9, p. 577], we have³
σ1

Z h−bhh−bh0´eAt dt ≤h ¡ A + AT ¢ imax exp λt ,20≤t≤h−bhwhich shows the claim. Proof in the case of h − b
h < 0 is similar.

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17