A class of discrete-time models for a
continuous-time system
T. Mori, PhD
P.N. Nikiforuk, DSc
M.M. Gupta, PhD
N. Hori, PhD
Indexing terms: Mathematical techniques, Modelling, Control systems
Abstract: An equivalence class, called the T equivalence class in this study, is introduced for
discrete time models of a continuous-time system,
such that any member of this class has the same
input/output characteristics when the discretetime interval approaches zero. This concept provides a systematic way of viewing the relationship
between the discrete and continuous-time systems.
A discrete-time function: A discrete-time function f *(t) is
a function which is defined only at discrete instants
t = kT, where k is an integer and T is a discrete-time
interval.
The Z-transform off *(t): Let f * ( t ) = 0 for t < 0. Then,
the Z-transform off *(t),denoted by Z [ f *(t)] or simply
f (z),is defined 181 as
m
f (z)= Z [ f * ( t ) ] =
1
Introduction
A number of techniques exist for obtaining a discretetime model for a continuous time system [l-31. They
permit the selection of the model that is most suitable for
a specific design purpose. Among these, the step invariant
model, which is obtained by inserting a zero-order hold
and a sampler into a continuous-time system, has been
widely used for the purpose of digital control. However,
there has been little study reported regarding relationships between the discrete-time models and continuoustime systems. The discrepancies between a continuoustime system and its discrete-time model have been recognised mainly as the unstable zero problem [ 4 ] . To
overcome this inherent problem, various techniques have
been proposed [S-71 for the step invariant model case.
In this study, a T-equivalence class of discrete-time
systems is introduced and a discrete-time model of a
continuous-time system is defined. It is shown then that
all the discrete-time models of a continuous-time system
form an equivalence class in which its members have
practically the same input/output characteristics for a
sufficiently small discrete-time interval. Several important
properties of the models pertaining to this class are presented as well. Using the results presented in this paper,
many techniques, those in References [S-71 for instance,
can readily be extended to a wider class of discrete-time
models.
2
Preliminaries
The notations used in this paper are first defined so as to
avoid confusion.
Paper 6557D (CS), first received 12th February and in revised form 3rd
November 1988
Dr. Mori is with the Department of Control and Information Engineering, Toyota Technological Institute, Nagoya 468, Japan
Drs. Nikiforuk and Gupta are with the Department of Mechanical
Engineering, University of Saskatchewan, Saskatoon, Sask, Canada
Dr. Hori is with the Department of Mechanical Engineering, McGill
University, 817 Sherbrooke Street West, Montreal, PQ, Canada H3A
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IEE PROCEEDINGS, Vol. 136, Pt. D, N o . 2, M A R C H 1989
C f *(kT)Tz-k
k=O
(1)
This definition differs slightly from the conventional one
in that the magnitude is scaled by T . Iff(z) is obtained by
sampling a continuous-time functionf(t), it follows that
where f (s) is the Laplace transform of f ( t ) . This result is
attributed to the definition of Z-transform by eqn. 1. One
other advantage of using this definition is given by eqn.
28.
A discrete-time system and its transfer function: A
discrete-time system is one whose input and output are
both discrete-time functions. A discrete-time system
whose output is given by y*(t) and input by u*(t) is
denoted by S*. In addition, a continuous-time system
whose output is given by f i t ) and input by u(t) is denoted
by S. The transfer function of S*, denoted by G(z), is
defined by
ZCY*(t)l = G(z)ZCu*(t)I
3
(3)
The class of discrete-time models
Let it be assumed that both S* and S are linear time
invariant and have the proper rational transfer functions
G(z)and G(s),respectively.
Definition 1: The discrete time systems Sf and S; are T equivalent if
(4)
and
(5)
79
< n, where G , ( z ) = ql.(z)/pl(z) and G2(z).=
q2(z)/p2(z)with pl(z) and p2(z) being monic polynomials
for 0 <j
the conditions of theorem 1 imply that
(coefficient of a maximum degree term is one) of degree n.
Dejinition 2 : Assume that the inputs u*(t) and u(t) are
zero for t < 0 and finite for a finite t. A discrete-time
system S* is then said to be a discrete-time model of
a continuous-time system S if the following condition
is satisfied: limT+olu(t) - u*(kT)I = 0, for each fixed t,
which implies that lim,,, I y(t) - y*(kT) I = 0, for each
fixed t, where k is an integer such that kT < t < (k + 1)T.
Since the T-equivalence clearly satisfies an equivalence
relation, it partitions the set of all discrete-time systems
into disjointed equivalence classes. Using definitions 1
and 2, theorem 1 is obtained.
Theorem I: Let G(z) = q(z)/p(z)and G(s) = b(s)/a(s)be the
transfer functions of S* and S, respectively, with p(z) and
a(s) being monic polynomials of degree n. Any discretetime system S* which belongs to the T-equivalent class in
which
G(z)=
where
limT+oA a j = O
for
O<j<n- 1
and
limT+oAbj = 0 for 0 <j < n.
This means that if G(s) is realised by the observable
canonical form
+ bu(t)
y(t) = c'x(t) + du(t)
i(t) =
Ax(t)
(15)
(16)
then G(z)can be realised by
+ ( A + AA)T)x*(kT)
+ (b + Ab)Tu*(kT)
y*(kT) = c'x*(kT) + (d + Ad)u*(kT)
~ * ( (+
k 1)T) = ( I
where
(7)
where limT-o A A
= 0,
(17)
(18)
limT-o Ab = 0 and limT+oAd
=
0. Therefore,
and
k- 1
y*(kT) =
1 [c'(Z + ( A + A A ) T ) k - l - j
j=O
x ( b + Ab)Tu*(jT)]
where
(9)
for 0 <j < n, is a discrete-time model of a continuoustime system S.
The proof of this theorem follows:
Let ii(t) be a function such that ii(t) = u*(kT) for
kT < t < (k + 1)T, and fit) be the output of S subject to
the input ii(t). For the inputs u*(t) and u(t) which satisfy
limT+o I u(t) - u*(kT)I = 0 for each fixed t, consider the
following equation :
I Y ( t ) - Y*(kT)I < I f i t ) - Y(k7-1 I
+ I Y ( k T )- j ( k T )I + I j ( k T ) - Y * ( W I (10)
j ( k T ) = c' rTeA(kT-r)bG(r)
dz
C
= j = O cf
cyeAT)k-l-j
for some constants M,, M , and M,. The proof of these
equations is a straightforward but tedious procedure and
it is not given here for the purpose of brevity.
In order to derive limT-o\y(t) - y*(kT)( = 0, it
remains to be shown that limT+oI j ( k T )- y*(kT)I = 0
for each fixed t. Since G(s) can be expressed as
[ e A r dzbu*(jT)
+ du*(kT)
(20)
Thus,
lim I j ( k T ) - y*(kT) I
T-0
k- 1
c'[I
1
< lim T
+
k- 1
I 1
+ ( A + AA)T]j(b + Ab)T + I Ad I
T-0
(12)
eA(lrTPr)
dzbu *( jT ) +du*(kT)
j=O
-
0,cr<t
Jji
k-1
=
and
I y(kT) - j ( k T )I < M , max I U(T) - i4z) I
+ dtr(kT)
r(i+l)T
k-1
OGrGt
+ M , I u(t) - u(kT)I (11)
(19)
On the other hand, j ( k T )is obtained by
For each fixed t , the first and the second terms in the
right-hand side of the above equation approach zero as
T goes to zero. This can be observed from
I YO) - Y ( W 1 G MI max I u(z) I ( t - kT)
+ (d + Ad)u*(kT)
[;r
c'(eATy -
eAr dzb
U,,,
- ( b + Ab)
I c'[(eATy- ( I + ( A + AA)T)j](b+ Ab) 1
j=O
1
U,,,
(21)
where
U,,,
is a constant that satisfies I u*(jT)I < U,,, for
lim
[f
[ e A r dzb - ( b
0 <j
< k. Since
T+O
1
+ Ab)
=0
and
lim [(eATY - (I
+ ( A + AA)T)j] = 0,
(23)
T-0
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1EE PROCEEDINGS, Vol. 136, P t . D, No. 2, M A R C H 1989
eqn. 2 1 becomes
Some of the well known models are:
lim I j ( k T ) - y * ( k T )I = 0
z-1
(24)
T-0
a forward difference model f ( z ) = T
z-1
b backwards difference model f ( z ) = Tz
2(x - 1)
c Tustin's modelflz) =
T ( z 1)
which leads to
lim I y(t) - y * ( k T )I = 0
25)
T-0
for each fixed t .
A discrete-time model given in definition 2 is well
defined. To show this, let it be assumed that a
continuous-time system S has a general state space representation, not necessarily in observable form, as
+ bu(t)
y(t) = c'x(t) + du(t)
i(t)=
Ax(t)
(26)
(26)
~
Besides these models, an infinite number of mapping
models can be derived using theorem 1.
3.4 Matched Z-transform model
This model is obtained by mapping the poles ui and zeros
pi of G(s)to e-uiTand epSiT,respectively; that is,
Its transfer function is
G(s) = ~ ' ( sl A)-'b
+d
(27)
It is demonstrated in the following that all discrete-time
models proposed in the literature [l-31 belong to the
same T-equivalence class.
+
G(s) = K
fi (s +
pi)
JJ (s + Ui)
i= 1
to
m
3.1 Impulse invariant model
This model is usually obtained as the conventional 2transform of G(s); that is, G(z)= c'(z1 - eAT)-'bz + d,
which does not satisfy the conditions of theorem 1.
However, if the 2-transform of eqn. 1 is used, G(z) is
modified to
G(z)= c'(z1 - eAT)-'bTz
+d
(28)
Rewriting G(z)as
x b[ 1
+T
( q ) ]
+d
(29)
eAT- I
~
- A,
T+O
it can be seen that the conditions of theorem 1 are satisfied.
3.2 Step invariant model
This model is obtained by inserting a zero-order hold
and a sampler into a continuous-time system S , and is
given by
G(z) = c'(z1- eAT)Since
lim
T-0
$ [eAr
c
eArdzb
+d
(30)
dz = I ,
it can be shown in the same way as for the impulseinvariant model case that G(z) satisfies the conditions of
theorem 1 .
3.3 Mapping models
Mapping models are obtained by substituting s = f ( z ) in
G(s);that is,
G(z) = G(s)
s=
/(a
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4
Properties of the discrete-time model
The discrete-time model obtained using our definition
has several desirable properties, which are described in
the following theorems.
and noting that
lim
where m < n.
Since the stability of poles and zeros of G(s) is preserved in this G(z),the minimum phase property of a continuous time system is carried over to the corresponding
discrete-time system.
It is evident that the mapping models and the matched
Z-transform model satisfy the conditions of theorem 1.
Theorem 2: Let S: and S t be discrete-time models of S ,
and S , , respectively. Then the series, parallel or feedback
connection of S y and S t are, respectively, discrete-time
models of the series, parallel, or feedback connected
systems of S , and S , . The proof follows:
Let Gi(s),a transfer function of Si, be given for i = 1 , 2
by
- bi,nis"i+ bi,ni-lsni-l+ . . . + b i , , S + bi,0
-
+
ai,nis"i
+
Qi,ni-lsni-l
" '
+ aiJs + ai.0
(ai,ni = 1) (31)
Gxz), the transfer function of ST, can be represented using
theorem 1 for i = 1,2 as
where
and
The coefficient of (z - l)kin the polynomial pl(z)pz(z)
is given, therefore, by
1 pl,ipz,j
i+j=k
for 0 < k
< nl + nz,
where
Since the zeros of P(w) approach the poles of S as T + 0
and the left half w-plane corresponds to the unit disc of
the z-plane, an asymptotic stability and an instability of S
implies, respectively, those of S*, for a sufficiently small
T.
It follows from theorems 2 and 3 that for a sufficiently
small T the stability of the overall discrete-time systems
is not affected by replacing a part of the system with
another discrete-time system which belongs to the same
T-equivalence class. Furthermore, it can be seen that if
two systems are discrete-time models of the same asymptotically stable continuous-time system, their input/
output behaviours can be made identical by letting
T + 0.
Theorem 4 : Let SY and S: be discrete-time models of an
asymptotically
stable continuous-time system S. For all
(33)
lim ~nl+nl-k
pl,ipz,j=
C a1,iaz.j
E > 0 and all bounded inputs u*(t), there exists a 6 > 0
T+O
i+j=k
i+j=k
which is the coefficient of sk in the polynomial al(s)az(s). such that 0 < T < 6 implies I yr(t) - y f ( t ) I < E for all
t = kT, where y r ( t ) and y t ( t ) are the outputs, respectively,
The coefficient of (z - l)k in the polynomial ql(z)qz(z)is
of S: and Sf subject to the same input u*(t). The proof
follows:
q l , i q z . j for 0 < k < n, n2,
i+j=k
Since S: and S y are discrete-time models of thcsame
continuous-time system, the transfer function C(z) =
where
C,(z) - G,(z) can be expressed as
1
41,iqZ,j=
b1,ibz.j
(34)
1
1
+
1
T+O
1
i+j=k
i+j=k
which is the coefficient of sk in the polynomial bl(s)b2(s).
Hence, the series connection of G,(z) and G2(z), which is
4 1(z)qz(z)
Pl(Z)PZ(Z)'
is the transfer function of a discrete-time model of the
series connected system of S, and S , , whose transfer
function is
where
iimT+oAi+ = 0 for
0 < i < 2n - 1
and
limT+oAb, = 0 for 0 < i < 2n. This system is realised,
then, in the following state space form:
i * ( ( k + 1 ) ~ =) ( I
+ ( A+ A A ) T ) ~ * ( ~ T )
+A~Tu*(~T)
j*(kT) = E'i*(kT) + Adu*(kT)
The proofs for the parallel and the feedback connections can be made in the same manner as described here.
Theorem 3: If S* is a discrete-time model of an asymptotically stable or unstable continuous-time system S,
then there exists a 6 == 0 such that 0 < T < 6 implies,
respectively, an asymptotic stability or instability of S*.
The proof is as follows:
As in the proof of theorem 1, the denominator of a
transfer function of S* can be represented by
x
( 9 Y - l
+ . . . + (ao + Aao)]
where limr+o Aaj = 0 for 0 < j
formed now by
Z =
into
+
1 (T/2)w
1 -(T/~)w
< n - 1.
(38)
(39)
where limT+oAA = 0, limT+oA6 = 0, and limT+oAd =
0.
Hence, for any bounded input u*(t) which satisfies
I u*(t) I < u M , the following relation is obtained:
IY W T )- Y f ( W I = Ij*(kT)I
(35)
p(z) is trans-
Since theorems 2 and 3 aksure that there exists a 6 > 0
such that, for 0 < T < 6, C(z) is asymptotically stable, it
follows that
I?[I
+ ( A+ A&T]'A6T 1 + I Ad1
which proves the theorem.
5
1
=0
(41)
Conclusions
In this study, a T-equivalence class was introduced for
discrete-time models of a continuous-time system. When
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IEE PROCEEDINGS, Vol. 136, Pt. D, N o . 2, M A R C H 1989
the discrete-time interval is selected to be sufficiently
small, any discrete-time model which belongs to the same
T-equivalence class has practically the same input/output
behaviour. One of the interesting consequences of this
concept is that various discrete-time models proposed in
the literature pertain to the same equivalence class. It was
also pointed out that the conventional impulse-invariant
model is not a discrete-time model. The impulseinvariant model defined with the slightly different Ztransform is, however, a discrete-time model. These and
other results presented in this paper provide one with a
better perspective of the relationship between the discrete
and continuous-time systems.
6
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pp. 12&125
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system analysis’ (Academic Press, 1969)
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