%% Numerical Do's and Don'ts Demo -- Model Interconnections

%   Copyright 1986-2004 The MathWorks, Inc.
%   $Revision: 1.1.6.1 $  $Date: 2004/08/17 21:33:15 $


%% Model Interconnections
% You can connect LTI models using +, *, [ , ], [ ; ], and the commands
% SERIES, PARALLEL, and FEEDBACK.  To ensure that the resulting model has
% minimal order, it is important to follow some simple rules:
% 
% * Convert all models to state-space form before connecting them
% * Respect the block diagram structure and do not use aggregate expressions
%
% As an example, try to compute a model for the block diagram sketched
% above, with

G = [1 , tf(1,[1 0]) , 5];
Fa = tf([1 1] , [1 2 5]);
Fb = tf([1 2] , [1 3 7]);

%%
NDDModelInterconnections_aux; % Draw Diagram


%% "Optimal" Method
% Let's start with the "optimal" way to connect these three blocks:

H1 = [ss(Fa) ; ss(Fb)] * G;

%%
% Note that (a) Fa and Fb are converted to state space, and (b) this
% operation mirrors the block diagram structure.  The resulting model H1
% has order 5, which is minimal:

size(H1,'order')


%% "Worst" Method
% Let's now look at the "worst" way to derive a state-space model for this
% block diagram.  Observing that the overall transfer function is 
% H = [Fa * G ; Fb * G], you could compute H as

H2 = ss( [Fa * G ; Fb * G] );
size(H2,'order')

%%
% The order of the resulting model H2 is 14, almost three times higher than
% H1!
% 
% While H2 is a valid model, it contains a lot of duplicated dynamics
% because (a) G appears twice in the expression, and (b) the state-space
% conversion is performed on a 2x3 MIMO transfer matrix.
%
% Here are a couple additional combinations with their respective
% fortunes:
  
H3 = ss( [Fa ; Fb] * G );      % order = 14
H4 = ss( [Fa ; Fb] ) * G;      % order = 5, lucky...
 
%% Moral
% *Moral:*  Use the state-space form for model interconnections, and stay
% true to the block diagram structure.