-->//list defining a linear system -->A=[0 -1;1 -3];B=[0;1];C=[-1 0]; -->Sys=syslin('c',A,B,C) Sys = Sys(1) (state-space system:) !lss A B C D X0 dt ! Sys(2) = A matrix = ! 0. - 1. ! ! 1. - 3. ! Sys(3) = B matrix = ! 0. ! ! 1. ! Sys(4) = C matrix = ! - 1. 0. ! Sys(5) = D matrix = 0. Sys(6) = X0 (initial state) = ! 0. ! ! 0. ! Sys(7) = Time domain = c -->//conversion from state-space form to transfer form -->Sys.A //The A-matrix ans = ! 0. - 1. ! ! 1. - 3. ! -->Sys.B ans = ! 0. ! ! 1. ! -->hs=ss2tf(Sys) hs = 1 --------- 2 1 + 3s + s -->size(hs) ans = ! 1. 1. ! -->hs.num ans = 1 -->hs.den ans = 2 1 + 3s + s -->typeof(hs) ans = rational -->//inversion of transfer matrix -->inv(hs) ans = 2 1 + 3s + s ---------- 1 -->//inversion of state-space form -->inv(Sys) ans = ans(1) (state-space system:) !lss A B C D X0 dt ! ans(2) = A matrix = [] ans(3) = B matrix = [] ans(4) = C matrix = [] ans(5) = D matrix = 2 1 + 3s + s ans(6) = X0 (initial state) = [] ans(7) = Time domain = c -->//converting this inverse to transfer representation -->ss2tf(ans) ans = 2 1 + 3s + s

The list representation allows manipulating linear systems as
abstract data objects. For example, the linear system can be combined
with other linear systems or the transfer function representation of
the linear system can be obtained as was done above using `ss2tf`.
Note that the transfer function representation of the linear system
is itself a tlist.
A very useful aspect of the manipulation of systems
is that a system can be handled as a data object.
Linear systems can be
inter-connected,
their representation
can easily be changed from state-space to transfer function
and vice versa.

The inter-connection of linear systems can be made as illustrated in Figure 2.1.

For each of the possible inter-connections of two systems`S1/.S2`

.
The representation of linear systems can be in state-space
form or in transfer function form. These two representations can
be interchanged by using the functions
`tf2ss` and
`ss2tf`
which change the representations of systems from transfer function
to state-space and from state-space to transfer function, respectively.
An example of the creation, the change in representation, and the
inter-connection of linear systems is demonstrated in the following
Scilab session.

-->//system connecting -->s=poly(0,'s'); -->S1=1/(s-1) S1 = 1 ----- - 1 + s -->S2=1/(s-2) S2 = 1 ----- - 2 + s -->S1=syslin('c',S1); -->S2=syslin('c',S2); -->Gls=tf2ss(S2); -->ssprint(Gls) . x = | 2 |x + | 1 |u y = | 1 |x -->hls=Gls*S1; -->ssprint(hls) . | 2 1 | | 0 | x = | 0 1 |x + | 1 |u y = | 1 0 |x -->ht=ss2tf(hls) ht = 1 --------- 2 2 - 3s + s -->S2*S1 ans = 1 --------- 2 2 - 3s + s -->S1+S2 ans = - 3 + 2s ---------- 2 2 - 3s + s -->[S1,S2] ans = ! 1 1 ! ! ----- ----- ! ! - 1 + s - 2 + s ! -->[S1;S2] ans = ! 1 ! ! ----- ! ! - 1 + s ! ! ! ! 1 ! ! ----- ! ! - 2 + s ! -->S1/.S2 ans = - 2 + s --------- 2 3 - 3s + s -->S1./(2*S2) ans = - 2 + s ----- - 2 + 2s

The above session is a bit long but illustrates some very important
aspects of the handling of linear systems. First, two linear systems
are created in transfer function form using the function called
`syslin`.
This function was used to label the systems in this example
as being continuous (as opposed to discrete).
The primitive `tf2ss` is used to convert one of the
two transfer functions to its equivalent state-space representation
which is in list form (note that the function `ssprint` creates a more
readable format for the state-space linear system).
The following multiplication of the two systems yields their
series inter-connection. Notice that the inter-connection
of the two systems is effected even though one of the systems is
in state-space form and the other is in transfer function form.
The resulting inter-connection is given in state-space form.
Finally, the function `ss2tf` is used to convert the resulting
inter-connected systems to the equivalent transfer function representation.