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called sampled-data systems). The input u[k] and output y[k] of this system are discrete with u[k] =
⋅
⋅
u(kT ) and y[k] = y(kT ) for k = 0,1,2,…, where T is the constant sampling period. The discrete input
u[k] is applied to the continuous system from a digital computer or microprocessor and is held constant
during the time interval T (zero-order hold). A sampler acquires the output of the continuous system at
each time instant T yielding the discrete output y[k]. The discrete system is between the input u[k] and
14
the output y[k] [11, Chapter 1]. The equivalent discrete-time state-space representation of the contin-
uous-time state-space model given by Eqs. (23.107) and (23.108) is given by (the details of the formulation
can be found in [11, Chapter 5, section 5])
[
xk + 1] = A D xk[] + B D uk[] (23.142)
yk[] = C D xk[] + D D uk[] (23.143)

where


A D = e , B D =  ∫  T e d λ B, C D = C, and D D = D (23.144)
AT
Al

0
15
and matrices C D and D D are not changed by the sampling. This discrete model (Eqs. (23.142) and
(23.143)) is the representation of the sampled-data system shown in Fig. 23.24.
Solutions to the Discrete-Time State-Space Equations
The solution to the discrete model (Eqs. (23.142) and (23.143)) is given by

k−1
xk[] = A D x 0[] + ∑ A D B D uj[] (23.145)
k
k−j−1
j=0
k−1
k
yk[] = CA D x 0[] + C ∑ A D k−j−1 B D uj[] + Du k[] (23.146)
j=0
for each sampling step k. Details of the formulation can be found in [11, Chapter 5, section 3]. The state
response x[k] to an applied input u[k] is characterized by the system matrices (A D , B D , C D , D D ). In
particular, the output y[k] will be bounded for any bounded input u[k] if the system is stable. A system
in the form given by Eq. (23.142) is stable if the magnitude of all the eigenvalues of A D are less than unity,
i.e., lie within the unit circle center at the origin of the z-plane [11, Chapter 5, section 6].

The z-Transform and Relationship with the State-Space
The input-to-output relationship in the frequency-domain for a discrete-time system is represented by
a discrete transfer-function called the z-transform, written in terms of the variable z [12, Chapter 4].
Analogous to the continuous-time case, the model of a dynamic system in discrete transfer-function
form can be useful in the design and control of systems [12, Chapter 7]. If the system model is available
in discrete transfer-function form, then a state-space realization can be found as follows. Given a discrete
system described by the following z-transform G(z):

1
–
d 0 + d 1 z + … + d n z – n
Gz() = ----------------------------------------------------- (23.147)
1 + c 1 z + … + c n z – n
–
1
14
We do not discuss quantizing and quantization error. See [11, Chapter 1, section 3] for details.
15
Given a continuous-time state-space model (A, B, C, D), the MATLAB command c2d, gives the discrete time
equivalent for a specified sampling period T.
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