bloc2exp - block-diagram to symbolic expression
given a block-diagram representation of a linear system bloc2exp returns its symbolic evaluation. The first element of the list blocd must be the string 'blocd'. Each other element of this list (blocd(2),blocd(3),...) is itself a list of one the following types :
list('transfer','name_of_linear_system')
list('link','name_of_link',
[number_of_upstream_box,upstream_box_port],
[downstream_box_1,downstream_box_1_portnumber],
[downstream_box_2,downstream_box_2_portnumber],
...)
The strings 'transfer' and 'links' are keywords which indicate the type of element in the block diagram.
Case 1 : the second parameter of the list is a character string which may refer (for a possible further evaluation) to the Scilab name of a linear system given in state-space representation (syslin list) or in transfer form (matrix of rationals).
To each transfer block is associated an integer. To each input and output of a transfer block is also associated its number, an integer (see examples)
Case 2 : the second kind of element in a block-diagram representation is a link. A link links one output of a block represented by the pair [number_of_upstream_box,upstream_box_port], to different inputs of other blocks. Each such input is represented by the pair [downstream_box_i,downstream_box_i_portnumber].
The different elements of a block-diagram can be defined in an arbitrary order.
For example
[1] S1*S2 with unit feedback.
There are 3 transfers S1 (number n_s1=2) , S2 (number n_s2=3) and an adder (number n_add=4) with symbolic transfer function ['1','1'].
There are 4 links. The first one (named 'U') links the input (port 0 of fictitious block -1, omitted) to port 1 of the adder. The second and third one link respectively (output)port 1 of the adder to (input)port 1 of system S1, and (output)port 1 of S1 to (input)port 1 of S2. The fourth link (named 'Y') links (output)port 1 of S2 to the output (port 0 of fictitious block -1, omitted) and to (input)port 2 of the adder.
//Initialization
syst=list('blocd'); l=1;
//
//Systems
l=l+1;n_s1=l;syst(l)=list('transfer','S1'); //System 1
l=l+1;n_s2=l;syst(l)=list('transfer','S2'); //System 2
l=l+1;n_adder=l;syst(l)=list('transfer',['1','1']); //adder
//
//Links
// Inputs -1 --> input 1
l=l+1;syst(l)=list('link','U',[-1],[n_adder,1]);
// Internal
l=l+1;syst(l)=list('link',' ',[n_adder,1],[n_s1,1]);
l=l+1;syst(l)=list('link',' ',[n_s1,1],[n_s2,1]);
// Outputs // -1 -> output 1
l=l+1;syst(l)=list('link','Y',[n_s2,1],[-1],[n_adder,2]);
//Evaluation call
w=bloc2exp(syst);
The result is the character string: w=-(s2*s1-eye())\s2*s1.
Note that invoked with two output arguments, [str,names]= blocd(syst) returns in names the list of symbolic names of named links. This is useful to set names to inputs and outputs.
[2] second example
//Initialization
syst=list('blocd'); l=1;
//
//System (2x2 blocks plant)
l=l+1;n_s=l;syst(l)=list('transfer',['P11','P12';'P21','P22']);
//
//Controller
l=l+1;n_k=l;syst(l)=list('transfer','k');
//
//Links
l=l+1;syst(l)=list('link','w',[-1],[n_s,1]);
l=l+1;syst(l)=list('link','z',[n_s,1],[-1]);
l=l+1;syst(l)=list('link','u',[n_k,1],[n_s,2]);
l=l+1;syst(l)=list('link','y',[n_s,2],[n_k,1]);
//Evaluation call
w=bloc2exp(syst);
In this case the result is a formula equivalent to the usual one:
P11+P12*invr(eye()-K*P22)*K*P21;