Page 775 - The Mechatronics Handbook

0066_Frame_C24 Page 23 Thursday, January 10, 2002 3:45 PM

We can then think of the states variables x m+1 [t], …, x 1 [t] as the temperature at equally spaced points,
between the heat source and the temperature sensor.
When the time delay t is not a multiple of the sampling period ∆, an additional pole at the origin and
an additional zero appear in the discrete transfer function. The details can be found elsewhere, e.g., in [7].

State Similarity Transformation
The idea of transforming the state via a similarity transformation equally applies to discrete-time systems.
The system properties also remain unchanged.

State Space and Transfer Functions
For discrete-time systems the relation between state space and transfer function models is basically the
same as in the continuous-time case (see section “State Space and Transfer Functions”). As we said then,
the state space description of linear time invariant systems is an alternative description to that provided
by transfer functions, although in some situations it provides more information on the system.
m
p
For a linear discrete-time invariant system with input u[t] ∈ and output y[t] ∈ , the transfer
function, H[z]∈ C p×m , is defined by the equation
Y i z[]
Y z[] = H z[]U z[], where [ H z[]] ij = ------------- (24.133)
U j z[]

th
i.e., the (i, j) element in matrix H[z] is the Zeta transformation of the response in the i output when
th
a unit Kronecker’s delta is applied at the j input, with zero initial conditions and with the remaining
inputs equal to zero for all t ≥ 0.
On the other hand, if we apply Zeta transform to the discrete time state space model (24.100) and
(24.101), with zero initial conditions, we have

X z[] = ( zIA d ) B d U z[] (24.134)
1
–
–
Y z[] = C d X z[] + D d U z[] (24.135)
leading to

(
C d zIA d ) B d + D d = H z[] (24.136)
–
1
–
T
In the following analysis, we will focus on the class of scalar systems, i.e., m = p = 1, B d , C d are column
vectors, and D d = H[∞]. We can then see that H[z] is a quotient of polynomials in z, i.e.,
C d Adj zIA d )B d + D d det zIA d )
(
(
–
–
Hz[] = --------------------------------------------------------------------------------------- (24.137)
det zIA d )
(
–
where Adj(o) denotes the adjoint matrix of (o).
We have again, paralleling the continuous-time case, that the transfer function poles are eigenvalues
of A d . However, it is not true in general that the set of transfer function poles is identical to the set of
eigenvalues of the matrix. It is important to realize that transfer function models can hide cancellations
between poles and zeros, with the consequences described in subsections “Controllability, Reachability
and Stabilizability” and “Observability, Reconstructability and Detectability.”
A key result for discrete-time system is the same for continuous-time systems: the transfer function
may not provide the same amount of information than the state space model for the same system.

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