minreal.m

本书是电子通信类的本科、研究生辅助教材

function [am,bm,cm,dm] = minreal(a,b,c,d,tol)
%MINREAL Minimal realization and pole-zero cancellation.
%	[Am,Bm,Cm,Dm] = MINREAL(A,B,C,D) returns a minimal realization
%	of the state-space system {A,B,C,D}.  A message is displayed 
%	indicating the number of states removed.
%	[Am,Bm,Cm,Dm] = MINREAL(A,B,C,D,TOL) uses the tolerance TOL
%	in deciding which states to eliminate.
%
%	[Zm,Pm] = MINREAL(Z,P), where Z and P are column vectors
%	containing poles and zeros, cancels the common roots that
%	are within TOL = 10*SQRT(EPS)*ABS(Z(i)) of each other.
%	[Zm,Pm] = MINREAL(Z,P,TOL) uses tolerance TOL.
%
%	For transfer functions, [NUMm,DENm] = MINREAL(NUM,DEN), where
%	NUM and DEN are row vectors of polynomial coefficients, cancels
%	the common roots in the polynomials.
%	[NUMm,DENm] = MINREAL(NUM,DEN,TOL) uses tolerance TOL.

%	J.N. Little 7-17-86
%	Revised A.C.W.Grace 12-1-89
%	Copyright (c) 1986-93 by The MathWorks, Inc.

disp(' ')

if nargin == 2 | nargin == 3
	z = a;
	p = b;
	[mz,nz] = size(z);
	[mp,np] = size(p);
	if (mz == 1) & (mp == 1)
		% If transfer function, convert to zero-pole:
		[z,p,k] = tf2zp(z,p);
	end

	% Strip infinities from zeros and throw away.
	z = z(finite(z));

	mz = max(size(z));
	mp = max(size(p));
	iz = ones(mz,1);

	% Loop through zeros, looking for matching poles:
	for i=1:mz
		zi = z(i);
		if (nargin == 2)
			tol = 10*abs(zi)*sqrt(eps);
		else
			tol=c;
		end
		kk = find(abs(p-zi) <= tol);
		if all(size(kk))
			p(kk(1)) = [];
			iz(i) = 0;
		end
	end

	% Eliminate matches in zeros:
	z = z(iz);

	% If transfer function, convert back.
	if (nz > 1) | (np > 1)
		[z,p] = zp2tf(z,p,k);
	end
	am = z;
	bm = p;
	disp([int2str(mz-sum(iz)),' pole-zeros cancelled'])
	return
end

% Do state-space case
[ns,nu] = size(b);
if nargin == 4
	tol = 10*ns*norm(a,1)*eps;
end
[am,bm,cm,t,k] = ctrbf(a,b,c,tol);
kk = sum(k);
nu = ns - kk;
nn = nu;
am = am(nu+1:ns,nu+1:ns);
bm = bm(nu+1:ns,:);
cm = cm(:,nu+1:ns);
ns = ns - nu;
if ns
	[am,bm,cm,t,k] = obsvf(am,bm,cm,tol);
	kk = sum(k);
	nu = ns - kk;
	nn = nn + nu;
	am = am(nu+1:ns,nu+1:ns);
	bm = bm(nu+1:ns,:);
	cm = cm(:,nu+1:ns);
end
disp([int2str(nn),' states removed'])
dm = d;