Statespace representation of internal delays
[H,tau]
= getDelayModel(sys)
[A,B1,B2,C1,C2,D11,D12,D21,D22,E,tau]
= getDelayModel(sys)
[
decomposes a
statespace model H
,tau
]
= getDelayModel(sys
)sys
with internal delays into
a delayfree statespace model, H
, and a vector
of internal delays, tau
. The relationship among sys
, H
,
and tau
is shown in the following diagram.
[
returns the set
of statespace matrices and internal delay vector, A,B1,B2,C1,C2,D11,D12,D21,D22,E
,tau
]
= getDelayModel(sys
)tau
,
that explicitly describe the statespace model sys
.
These statespace matrices are defined by the statespace equations:
Continuoustime sys
:
$$\begin{array}{c}E\frac{dx\left(t\right)}{dt}=Ax\left(t\right)+{B}_{1}u\left(t\right)+{B}_{2}w\left(t\right)\\ y\left(t\right)={C}_{1}x\left(t\right)+{D}_{11}u\left(t\right)+{D}_{12}w\left(t\right)\\ z\left(t\right)={C}_{2}x\left(t\right)+{D}_{21}u\left(t\right)+{D}_{22}w\left(t\right)\\ w\left(t\right)=z\left(t\tau \right)\end{array}$$
Discretetime sys
:
$$\begin{array}{c}Ex\left[k+1\right]=Ax\left[k\right]+{B}_{1}u\left[k\right]+{B}_{2}w\left[k\right]\\ y\left[k\right]={C}_{1}x\left[k\right]+{D}_{11}u\left[k\right]+{D}_{12}w\left[k\right]\\ z\left[k\right]={C}_{2}x\left[k\right]+{D}_{21}u\left[k\right]+{D}_{22}w\left[k\right]\\ w\left[k\right]=z\left[k\tau \right]\end{array}$$

Any statespace ( 

Delayfree statespace model ( If 

Vector of internal delays of If 

Set of statespace matrices that, with the internal delay vector For explicit statespace models (E = I,
or If 