                    
   !"#  $   %  & '    
%       (  

Passivity Degradation under the Discretization with the Zero-Order
Hold and the Ideal Sampler∗
Yasuaki Oishi†
Quantitative study is made on how much passivity is lost when a continuous-time nonlinear
system is discretized for digital implementation of a controller. The standard discretization
with the zero-order hold and the ideal sampler is mainly considered though other types of
discretization are also considered. The result can be used for the choice of the sampling
period. Namely, the sampling period has to be small when the original continuous-timesystem is oscillatory. The general input-output framework is used for the discussion.
Keywords: passivity, discretization, sampling period, sampled-data systems.

1. Introduction
Passivity is an important notion in systems analysis and control especially in the case of nonlinear
systems [1, 2, 3]. It is well-known however that this notion is not compatible with discretization,
which is required for digital implementation of a controller. Namely, even if the original continuoustime system is passive, the discretized system may not be passive any more.
Various authors have analyzed this phenomenon. Laila–Nešić–Teel [4] and de la Sen [5] gave
conditions for passivity to be preserved with a small enough sampling period. Although these authors
used standard discretization with the zero-order hold and the ideal sampler, Costa-Castelló–Fossas[6] considered a diﬀerent type of discretization with the average of the output of the system. They
showed that this discretization preserves passivity for any sampling period. Stramigioli–Secchi–van
der Schaft–Fantuzzi [7] considered the same discretization for port-Hamiltonian systems. A drawbackof this discretization is that the average of the output requires the future value of the output, which
is diﬃcult to compute for a nonlinear system.
In this paper, we quantitatively investigate how much passivity is lost due to discretization. In
particular, we express the passivity degradation as a function of the sampling period. Our maininterest is on the standard discretization with the zero-order hold and the ideal sampler though othertypes of discretization are considered as well. The result gives an insight for the choice of the sampling
period. Namely, a small sampling period is required for an oscillatory system. We use the input-output
framework for the sake of generality.
This paper is organized as follows. In Section 2, the notion of passivity and two types of discretization are presented. Passivity loss under discretization is also discussed there. Section 3 is themain section and evaluates the passivity degradation under the standard discretization. Finally in
Section 4, extension of the result is discussed.
Notation is standard. The symbol Rm stands for the space of the m-dimensional real vectors. The√
symbol T expresses the transpose of a vector. For a vector v ∈ Rm , its norm kvk is deﬁned as v T v.
∗

In Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, USA, December 2010, pp. 7613-7617.
Department of Systems Design and Engineering, Nanzan University, Seireicho 27, Seto 489-0863, Japan; email:oishi@nanzan-u.ac.jp†

2. Passivity and Discretization
The continuous-time system S is an operator mapping a piecewise continuous signal u(t) to a continuous signal y(t), each of which takes a value in Rm . It is assumed to map the zero input u(t) ≡ 0 to
the zero output y(t) ≡ 0.
Suppose that the system S satisﬁes the passivity inequality
Z T
Z T
Z T
T
2
u(t) y(t) dt ≥ ²
ku(t)k dt + δky(t)k2 dt
(1)
0

0

0

for any T ≥ 0 and any piecewise continuous input u(t), where ² and δ are real numbers. Although² and δ can be negative, we are interested in the case that they are nonnegative. In particular, S iscalled (i) passive when ² = δ = 0; (ii) input strictly passive when ² > 0 and δ = 0; (iii) output strictly
passive when ² = 0 and δ > 0; (iv) very strictly passive when ² > 0 and δ > 0 [1, 2, 3].
We next consider discretization of S. With h being some positive number called the sampling
period, we deﬁne the discrete-time input ud (k), k = 1, 2, . . ., and the discrete-time output yd (k),k = 1, 2, . . ., so as to be related to their continuous-time counterparts in a way described later. The
values ud (k) and yd (k) are considered as the signals at the time t = kh. Now the operator from{ud (k)} to {yd (k)} is regarded as a discrete-time system. This procedure as well as the resulting
discrete-time system is called the discretization of S.
Passivity can be considered also on a discrete-time system. Suppose that the discrete-time system
considered above satisﬁes the discrete-time passivity inequality
h

K−1
X

ud (k)T yd (k) ≥ ²d h

k=0

K−1
X

kud (k)k2 + δd h

K−1Xkyd (k)k2

(2)

k=0

k=0

for any positive integer K and any input ud (k), where ²d and δd are real numbers. Then this system
is called (i) passive when ²d = δd = 0; (ii) input strictly passive when ²d > 0 and δd = 0; (iii) output
strictly passive when ²d = 0 and δd > 0; (iv) very strictly passive when ²d > 0 and δd > 0. The factor
h appears in (2) for making its dimension in time consistent with the continuous-time inequality (1).
The most commonly used discretization is the following. The discrete-time input ud (k) is related
to the continuous-time input u(t) by the zero-order hold, that is,u(t) = ud (k),kh ≤ t < (k + 1)h,k = 0, 1, . . . ;(3)

the discrete-time output yd (k) is corresponded to the continuous-time output y(t) by the ideal sampler,that is,yd (k) = y(kh), k = 0, 1, . . . .
(4)
We call this procedure standard discretization and write the resulting discrete-time system as Sdsta .
Important observation here is that, even if S is passive, its standard discretization Sdsta may not be
so. See Example 2 below. The approach of [7, 6] to avoid this problem is to deﬁne the discrete-time
output byZ
1 (k+1)hyd (k) =y(t) dt, k = 0, 1, . . . ,(5)h kh
in place of (4). We call this discretization average discretization and write the resulting discrete-timesystem as Sdave . This discretization preserves passivity. We state the result in a stronger form than2that in [6]. Namely, we consider preservation not only of (nonstrict) passivity but also of the three
types of strict passivity. Moreover, the result is given in a quantitative way. The real number δ isassumed to be nonnegative here.
Theorem 1. Suppose that the original continuous-time system S satisﬁes the continuous-time passivity inequality (1) with some real numbers ² and δ ≥ 0. Then its average discretization Sdave satisﬁes
the discrete-time passivity inequality (2) with ²d = ² and δd = δ. In particular, if S is passive, input
strictly passive, output strictly passive, and very strictly passive, so is its average discretization Sdave ,respectively.
Proof. Assuming the relationships (3) and (5) and T = Kh, we compare the three integrals in thecontinuous-time passivity inequality (1) with the corresponding summations in (2), respectively.
The ﬁrst integral in (1) is equal to the corresponding summation in (2) because
Z Kh
u(t)T y(t) dt =
0

K−1
X

Z (k+1)hud (k)Ty(t) dt = h
kh

k=0

K−1
X

ud (k)T yd (k).

(6)

k=0

As easily seen, the second integral is again equal to the corresponding summation, i.e.,
Z Kh

ku(t)k2 dt = h

0

K−1
X

kud (k)k2 .

(7)

k=0

Finally, the third integral is rewritten as
Z Khky(t)k dt =20m Z (k+1)hK−1XXyi (t)2 dt

kh

k=0 i=1

with y(t) = (y1 (t) y2 (t) · · · ym (t))T . Here, we haves¯Z (k+1)h¯Z (k+1)h√¯¯
2hyi (t) dt ≥ ¯¯yi (t) dt¯¯khkh

by the Cauchy–Schwarz inequality. It hence follows that
Z Kh

ky(t)k dt ≥
2

0

m
K−1
XX
k=0 i=11hµZ (k+1)h¶2
yi (t) dt

=h

kh

K−1
X

kyd (k)k2 .

(8)

k=0

Using (6)–(8) with the passivity inequality (1), we have the desired result. Note that δ has to benonnegative because (8) is an inequality.
¤
Example 2. Let S be a linear systemẋ(t) = −x(t) + u(t),x(0) = 0,y(t) = x(t).
This system satisﬁes ẏ(t)y(t) = −y(t)2 + u(t)y(t) and then
Z T
Z T
Z T
1
2
2
0 ≤ y(T ) =ẏ(t)y(t) dt = −y(t) dt +u(t)y(t) dt,
2
0
0
0
3

which implies that S is output strictly passive with ² = 0 and δ = 1.
This system is not input strictly passive. To see this, let a be a positive integer and consider theRTinput u(t) equal to sin at for 0 ≤ t < 2π and to 0 otherwise. The integral 0 u(t)2 dt = π for any
RTT ≥ 2π irrespective of a. On the other hand, 0 u(t)y(t) dt approaches zero as a → ∞. This meansthat the continuous-time passivity inequality (1) does not hold with ² > 0 and δ = 0.
Its standard discretization Sdsta can be written in the state-space form asx((k + 1)h) = e−h x(kh) + (1 − e−h )ud (k),x(0) = 0,yd (k) = x(kh).
This system is however not passive irrespectively of the sampling period h. To see this, let the input
ud (k) be equal to 1 for k = 0, to −1 for k = 1, and to 0 otherwise. This input gives yd (0) = 0
P−h ) < 0 for K ≥ 2. Hence, theand yd (1) = 1 − e−h , resulting in h K−1
k=0 ud (k)yd (k) = −h(1 − ediscrete-time passivity inequality (2) does not hold with ²d = δd = 0.
On the other hand, the average discretization Sdave remains output strictly passive due to Theorem 1.
3
The average discretization is theoretically attractive because it makes the passivity-based methods
available after the discretization. A drawback of this discretization is that computation of the average
R (k+1)hsample yd (k) = (1/h) khy(t) dt may not be easy at the time t = kh because it requires the future
output value y(t), kh < t < (k + 1)h. When the original system S is linear, computation of yd (k) is
possible at t = kh because the future value is explicitly obtained as a function of the present statex(kh) and input ud (k). When S is nonlinear, however, such explicit expression is diﬃcult usually.
Motivated by this fact, we go back to the standard discretization and investigate how much passivity is lost with it.

3. Passivity Degradation under the Standard Discretization
This is the main section of the paper. We evaluate how much passivity is lost under the standard
discretization. The result is summarized in the following theorem. Nonnegativity of δ is not required
this time.
Theorem 3. Suppose that the original continuous-time system S satisﬁes the continuous-time passivity inequality (1) with some real numbers ² and δ. Suppose also that the operator from u(t) to ẏ(t)has the ﬁnite L2 -gain γ > 0, that is,
Z T
Z T
2
2
kẏ(t)k dt ≤ γku(t)k2 dt
00

for any T ≥ 0 and admissible u(t). Then its standard discretization Sdsta satisﬁes the discrete-time
passivity inequality (2) with
²d = ² − hγ − hγ|δ| − h2 γ 2 |δ|,δd = δ − hγ|δ|.
In particular, if S is input strictly passive and very strictly passive, so is its standard discretization
Sdsta , respectively, for a small enough sampling period h.
4This theorem tells importance of input strict passivity for preservation of passivity. Indeed, if
the original system S lacks input strict passivity, its passivity or output strict passivity may not bepreserved however small the sampling period would be. This is what we observed in Example 2. The
result is consistent with the results of [4, 5].
It is notable that Theorem 3 presents the degradation of passivity as a function of the sampling
period h. In order to make the degradation small, we need to choose h small. It is especially so when
γ is large, that is, when the operator from u(t) to ẏ(t) has a high gain. This is understandable becausesuch a system should be oscillatory and require fast enough sampling for its behavior to be captured.
Proof. We again compare the three integrals in the continuous-time passivity inequality (1) withthe corresponding summations but assuming (3) and (4) this time.
The discrepancy between the ﬁrst integral and the corresponding summation is
¯Z Kh
¯
K−1
X
¯
¯
T
T
¯
D1 := ¯u(t) y(t) dt − hud (k) yd (k)¯¯.
0

k=0Rewriting this with (3) and (4), we obtain¯K−1¯ ¯K−1¯
Z (k+1)hZ (k+1)h Z t
¯X
¯ ¯X
¯
T
T
D1 = ¯¯
ud (k)
y(t) − y(kh) dt¯¯ = ¯¯ud (k)
ẏ(s) ds dt¯¯
≤

kh

k=0
K−1
X

kud (k)kZ (k+1)h Z tkhkẏ(s)k ds dt.

kh

kh

k=0

kh

k=0

In the last expression, the integral with respect to s can be bounded from above ass
Z t
Z (k+1)hZ (k+1)h√kẏ(s)k ds ≤kẏ(s)k ds ≤ hkẏ(s)k2 ds.
kh

kh

kh

We hence have√
D1 ≤ h h

K−1
X
k=0

s
kud (k)kZ (k+1)hvv
uK−1 Z (k+1)h
uK−1
uX
√ uX
t
2
2
kẏ(s)k ds ≤ h hkud (k)k tkẏ(s)k2 ds.

kh

k=0

k=0

kh

Since the L2 -gain from u(t) to ẏ(t) is γ, the second summation in the last expression can be bounded
R Kh
PK−1
from above by γ 2 0 ku(t)k2 dt, which is equal to γ 2 h k=0
kud (k)k2 . This gives
D1 ≤ hγ · h

K−1Xkud (k)k2 .

k=0

On the second integral in (1), the equation (7) remains to hold. On the third integral, we canmake comparison in a similar way to the ﬁrst one and obtain
¯Z Kh
¯
K−1
K−1
K−1
X
X
X
¯
¯
2
2¯
2 2
2
¯
ky(t)k
dt
−
h
ky
(k)k
≤
(hγ
+
h
γ
)h
ku
(k)k
+
hγ
·
h
kyd (k)k2 .
d
d
¯
¯
0

k=0

k=0

k=0

Using the obtained relationships with (1) and arranging the terms, we have the theorem.

5¤

sampling period h110closed-loop pole−1

10

passivity theorem
−3

10

0

10

10

1

2

10

gain gFigure 1. The stability bounds on the sampling period h obtained from the passivity theorem (solidline) and the closed-loop pole location (dashed line)
Example 4. We continue Example 2. Recall that the original system S is output strictly passive andits standard discretization Sdsta is not passive for any sampling period h > 0. This phenomenon is
explained by Theorem 3. Indeed, the lack of input strict passivity of S leads to the loss of passivity.
Quantitative discussion is possible. The operator from u(t) to ẏ(t) is in this caseẋ(t) = −x(t) + u(t),x(0) = 0,ẏ(t) = −x(t) + u(t),which has the L2 -gain γ = 1. Theorem 3 states that the standard discretization Sdsta satisﬁes thepassivity inequality with ²d = −2h − h2 and δd = 1 − h.
This result can be used for stability analysis. Let us recall the passivity theorem [1, p. 350]. There,
we consider a closed-loop system connecting two continuous-time systems by the negative feedback
and assume it well-posed. Then, the closed-loop system is L2 -stable if the two systems satisfy thepassivity inequality (1) with ², δ and ²0 , δ 0 , respectively, and they have ² + δ 0 > 0 and ²0 + δ > 0. The
same result holds on discrete-time systems, too.
We now consider the closed-loop system connecting S and a static linear gain g > 0 by the negative
feedback. It is well-posed due to the strict properness of S. The system S has ² = 0 and δ = 1 as in
Example 2 while the gain g has ²0 = 0 and δ 0 = 1/g as seen easily. The passivity theorem says thatthe closed-loop system is L2 -stable for any g > 0. This can be conﬁrmed also with the location of theclosed-loop pole.
The situation is diﬀerent for the discrete-time closed-loop system consisting of Sdsta and g. Since²d = −2h − h2 , δd = 1 − h, ²0d = 0, and δd0 = 1/g, the stability condition isn ro1
0 < h < min 1, 1 + − 1 .
g
It claims that, when the gain g is large, the sampling period has to be chosen adequately small.
We can obtain a stability condition from the location of the closed-loop pole, which turns out
0 < h < ln

g+1
g−1
6

(g > 1).

The two stability conditions are compared in Figure 1. They show similar behavior for a large g. The
condition from the passivity theorem looks more conservative than the other. This conservatism isconsidered to come from Theorem 3 and, possibly, from the passivity theorem as well. We need to
notice, however, that the condition from the passivity theorem guarantees stability not only for theparticular linear gain g but also all the static nonlinear gain belonging to the sector [0, g].
3

4. Discussion
In this paper, we investigate how much passivity is lost under the standard discretization. Ourinvestigation is quantitative and can be used, for example, for the choice of the sampling period asseen in the example. This is considered to be useful in practice.
An extension of the result is possible. The standard discretization can be improved with theinformation available at the time t = kh as
yd (k) = y(kh) +hẏ(kh),
2

k = 0, 1, . . . .

This is obtained by substitution of the ﬁrst-order Taylor expansion y(t) ≈ y(kh) + (t − kh)ẏ(kh) into
the deﬁnition of the average discretization (5). Let us call it the ﬁrst-order discretization and write
the resulting system as Sdﬁr . Passivity degradation under this discretization can be evaluated similarly
to the case of the standard discretization.
Theorem 5. Suppose that the original continuous-time system S satisﬁes the continuous-time passivity inequality (1) with some real numbers ² and δ. Suppose also that the operator from u(t) to ẏ(t)has the ﬁnite L2 -gain γ > 0 and the operator from u(t) to ÿ(t) has the ﬁnite L2 -gain β > 0. Then its
ﬁrst-order discretization Sdﬁr satisﬁes the discrete-time passivity inequality (2) with
³
´
1
1
1
²d = ² − h2 β − h2 γ 2 + h2 β + h4 β 2 |δ|,
2
2
4
1 2
δd = δ − h β|δ|.
2
In particular, if S is input strictly passive and very strictly passive, so is its ﬁrst-order discretization
Sdﬁr , respectively, for a small enough sampling period h.
¤

Proof. See Appendix.

Passivity degradation under the ﬁrst-order discretization is of order h2 while it is of order h under
the standard discretization. This reduction is a consequence of the improvement of discretization. On
the other hand, it is still not possible to preserve passivity without input strict passivity. Indeed, when
the original system S is not input strictly passive, its discretization Sdﬁr may not be passive however
small the sampling period would be.
In order to quantify the passivity degradation, we need γ and β, which are the L2 -gain from u(t)to ẏ(t) and that from u(t) to ÿ(t), respectively. Their evaluation is easy for a linear S as seen in
the example. It is however diﬃcult for a nonlinear S in general. It is even possible that S does nothave any ﬁnite γ or β. This diﬃculty arises because the present framework allows the input u(t) to7

be inﬁnitely large. In this sense, a local framework of passivity might be useful for extending thediscussion here.
Kawakami–Fujioka [8] analyzed stability of a sampled-data Hamiltonian system by extending thescaled small-gain theorem, which is more general than the passivity theorem. This direction of theresearch should be fruitful for tight stability analysis.

Acknowledgments
This research is supported in part by the Grant-in-Aid for Scientiﬁc Research of the Japan Society for
the Promotion of Science and the Nanzan University Pache Research Subsidy I-A-2 for the academic
year 2010. This research is performed during the sabbatical stay of the author at the MAC group,
LAAS-CNRS, Toulouse, France.

Appendix: Proof of Theorem 5
The idea of the proof is the same as in Theorems 1 and 3. We compare the three integrals in (1) withthe corresponding summations in (2). To do this, the following relationship is useful:hy(t) − yd (k) = y(t) − y(kh) − ẏ(kh)2
Z (k+1)h hi
1=y(t) − y(kh) − (s − kh)ẏ(kh) ds
h kh
Z
Z
Z s
i
1 (k+1)h h tẏ(r) dr −ẏ(kh) dr ds
=
h kh
kh
kh
Z
Z
Z s
i
1 (k+1)h h t=ẏ(r) dr +ẏ(r) − ẏ(kh) dr ds
h kh
s
kh
Z
Z Z
Z
Z
1 (k+1)h s r
1 (k+1)h tÿ(q) dq dr ds.
ẏ(r) dr ds +
=
h kh
h kh
s
kh kh

(9)

We compare the ﬁrst integral in (1) with the corresponding summation in (2). Their discrepancy
is written as¯Z Kh¯ ¯K−1¯
Z (k+1)h
K−1
X
¯
¯ ¯X
¯
0
T
T
T
¯
¯
¯
D1 := ¯u(t) y(t) dt − hud (k) yd (k)¯ = ¯ud (k)
y(t) − yd (k) dt¯¯.
0

k=0

k=0

kh

We substitute (9) into the last expression. Noting that the ﬁrst term of (9) vanishes after the integration with t, we have
¯K−1
¯X

¯
Z
Z Z
¯
1 (k+1)h s r
ud (k)
ÿ(q) dq dr ds dt¯¯
h kh
kh
kh kh
k=0
Z
Z
Z
K−1
(k+1)hsrX≤
kud (k)kkÿ(q)k dq dr ds

D10 = ¯¯

≤

k=0
K−1
X
k=0

Z (k+1)hTkhkud (k)kh22kh

Z (k+1)hkhkÿ(q)k dq

kh

8

≤

K−1
X

kud (k)k

k=0

≤

h2 √
2

h2 √
2

s

Z (k+1)hhkÿ(q)k2 dq

kh

v
v
uK−1
uK−1 Z (k+1)h
uX
uX
t
2
h
kud (k)k tkÿ(q)k2 dq.
k=0

k=0

kh

R Kh
PK−1
The second summation in the last expression is bounded by β 2 0 ku(t)k2 dt = β 2 h k=0
kud (k)k2 .
We hence have
K−1
X
h2 β
D10 ≤
·h
kud (k)k2 .
2
k=0

The second integral in (1) is equal to the corresponding summation in (2). On the third integral, its
discrepancy with the corresponding summation can be evaluated similarly to the ﬁrst one. Speciﬁcally
we have
¯Z Kh
¯ ³
K−1
K−1
K−1
X
X
¯
¯
1 4 2´ X
1 2
1 2
2
2¯
2
2 2
¯
ky(t)k
dt
−
h
ky
(k)k
h
β
+
h
β
h
ku
(k)k
+
h
β
·
h
kyd (k)k2 .
≤
h
γ
+
d
d
¯
¯
2
4
2
0
k=0

k=0

k=0

Using the obtained relationships with (1) and arranging the terms, we have the desired inequality.

References[1] M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed. Englewood Cliﬀs, USA: Prentice-Hall, 1993.
[2] A. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control. London, UK: Springer,
2000.
[3] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland, Dissipative Systems Analysis and Control:
Theory and Applications, 2nd ed. London, UK: Springer, 2007.
[4] D. S. Laila, D. Nešić, and A. R. Teel, “Open and closed loop dissipation inequalities under sampling
and controller emulation,” European Journal of Control, vol. 8, no. 2, pp. 109–125, 2002.
[5] M. de la Sen, “Preserving positive realness through discretization,” Positivity, vol. 6, no. 1, pp. 31–
45, 2002.
[6] R. Costa-Castelló and E. Fossas, “On preserving passivity in sampled-data linear systems,” inProceedings of the 2006 American Control Conference, Minneapolis, USA, June 2006, pp. 4373–
4378.
[7] S. Stramigioli, C. Secchi, A. J. van der Schaft, and C. Fantuzzi, “Sampled data systems passivity
and discrete port-Hamiltonian systems,” IEEE Transactions on Robotics, vol. 21, no. 4, pp. 574–
587, 2005.
[8] S. Kawakami and H. Fujioka, “Stability analysis for a class of Hamiltonian systems with digital
control,” in Proceedings of the 2010 SICE Annual Conference, Taipei, Taiwan, August 2010.

9