User`s guide

Identifying State-Space Models
Discrete-Time Representation
Discrete-time state-space models provide the same type of linear difference
relationship between the inputs and the outputs as the linear ARX model, but
are rearranged such that there is only one delay in the expressions. The
discrete-time state-space model structure is often written in the innovations
form that describes noise:
x kT T Ax kT Bu kT Ke kT
ykT CxkT DukT ekT
x
( ) () () ()
() () () ()
()
+= + +
=+ +
=0 xx0
where T is the sampling interval, u(kT) is the input at time instant kT,and
y(kT) is the output at time instant kT.
Note K=0 gives the state -space representation of an Output-Error m odel.
For more information about Output-Error models, see “What Are Black-Box
Polynom ial M odels?” on page 3-41.
Relationship Between Continuo us-Time and Discrete-Time
State Matrices
The relationship s betw ee n the discrete state-space matrices A, B, C, D,andK
and the continuous-time state-space matrices F, G, H, D,and

K
are giv en for
piece-wise-constant input, as follows:
Ae
BeGd
CH
FT
F
T
=
=
=
∫
τ
τ
0
These relationships assume that the input is piece-wise-constant over time
intervals
kT t k T≤< +()1
.
3-75