bloc2exp - Block-diagram to symbolic expression (original) (raw)

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Description

given a block-diagram representation of a linear systembloc2exp 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 (numbern_s1=2) , S2 (numbern_s2=3) and an adder (numbern_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 ofS2 to the output (port 0 of fictitious block -1, omitted) and to (input)port 2 of the adder.

syst=list('blocd'); l=1;

l=l+1;n_s1=l;syst(l)=list('transfer','S1');
l=l+1;n_s2=l;syst(l)=list('transfer','S2');
l=l+1;n_adder=l;syst(l)=list('transfer',['1','1']);

l=l+1;syst(l)=list('link','U',[-1],[n_adder,1]);

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]);

l=l+1;syst(l)=list('link','Y',[n_s2,1],[-1],[n_adder,2]);

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

syst=list('blocd'); l=1;

l=l+1;n_s=l;syst(l)=list('transfer',['P11','P12';'P21','P22']);

l=l+1;n_k=l;syst(l)=list('transfer','k');

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]);

w=bloc2exp(syst);

In this case the result is a formula equivalent to the usual one:

P11+P12*invr(eye()-K*P22)*K*P21;