75 lines
3.3 KiB
Matlab
75 lines
3.3 KiB
Matlab
% Impulse response invariant discretization of fractional order
|
|
% integrators/differentiators
|
|
%
|
|
% irid_fod function is prepared to compute a discrete-time finite dimensional
|
|
% (z) transfer function to approximate a continuous irrational transfer
|
|
% function s^r, where "s" is the Laplace transform variable, and "r" is a
|
|
% real number in the range of (-1,1). s^r is called a fractional order
|
|
% differentiator if 0 < r < 1 and a fractional order integrator if -1 < r < 0.
|
|
%
|
|
% The proposed approximation keeps the impulse response "invariant"
|
|
%
|
|
% IN:
|
|
% r: the fractional order
|
|
% Ts: the sampling period
|
|
% norder: the finite order of the approximate z-transfer function
|
|
% (the orders of denominator and numerator z-polynomial are the same)
|
|
% OUT:
|
|
% sr: returns the LTI object that approximates the s^r in the sense
|
|
% of impulse response.
|
|
% TEST CODE
|
|
% dfod=irid_fod(-.5,.01,5);figure;pzmap(dfod)
|
|
%
|
|
% Reference: YangQuan Chen. "Impulse-invariant and step-invariant
|
|
% discretization of fractional order integrators and differentiators".
|
|
% August 2008. CSOIS AFC (Applied Fractional Calculus) Seminar.
|
|
% http://fractionalcalculus.googlepages.com/
|
|
% --------------------------------------------------------------------
|
|
% YangQuan Chen, Ph.D, Associate Professor and Graduate Coordinator
|
|
% Department of Electrical and Computer Engineering,
|
|
% Director, Center for Self-Organizing and Intelligent Systems (CSOIS)
|
|
% Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA
|
|
% E: yqchen@ece.usu.edu or yqchen@ieee.org, T/F: 1(435)797-0148/3054;
|
|
% W: http://www.csois.usu.edu or http://yangquan.chen.googlepages.com
|
|
% --------------------------------------------------------------------
|
|
%
|
|
% 9/5/2009
|
|
%
|
|
function [sr]=irid_fod(r,Ts,norder)
|
|
if nargin<3; norder=5; end
|
|
% if Ts < 0 , sprintf('%s','Sampling period has to be positive'), return, end
|
|
% if abs(r)>=1, sprintf('%s','The fractional order should be less than 1.'), return, end
|
|
% if norder<2, sprintf('%s','The order of the approximate transfer function has to be greater than 1'), return, end
|
|
%
|
|
L=200; %number of points of the impulse response function h(n)
|
|
Taxis=[0:L-1]*Ts; r0=r; r=abs(r);
|
|
ha0=(7*Ts/8)^r; n=1:L-1; h=[ha0, (Ts^r)*(n.^(r-1))/gamma(r)];
|
|
[b,a]=prony(h,norder,norder);
|
|
sr=tf(b,a,Ts);if r0>0, sr=1/sr; r=-r0; end
|
|
|
|
if 1 % change this to 0 if you do not want to see plots
|
|
% approximated h()
|
|
wmax0=2*pi/Ts/2; % rad./sec. Nyquist frequency
|
|
if r0>0, hhat=impulse(1/sr,Taxis);
|
|
else, hhat=impulse(sr,Taxis); end
|
|
% figure;plot(Taxis,hhat,'r');hold on;plot(Taxis,h,'ok')
|
|
% xlabel('time');ylabel('impulse response'); legend(['true for 1/s^{',num2str(abs(r)),'}'],'approximated')
|
|
% figure;
|
|
wmax=floor(1+ log10(wmax0) ); wmin=wmax-5;
|
|
w=logspace(wmin,wmax,1000);
|
|
srfr=(j*w).^(-r);
|
|
% subplot(2,1,1)
|
|
% semilogx(w,20*log10(abs(srfr)),'r');grid on
|
|
% hold on;
|
|
srfrhat=freqresp(sr,w); %semilogx(w,20*log10(abs(reshape(srfrhat, 1000, 1))),'k');grid on
|
|
% xlabel('frequency in Hz');ylabel('dB');
|
|
% legend('true mag. Bode','approximated mag. Bode')
|
|
% subplot(2,1,2)
|
|
% semilogx(w,(180/pi) * (angle(srfr)),'r');grid on;hold on
|
|
% semilogx(w,(180/pi) * (angle(reshape(srfrhat, 1000, 1))),'k');grid on
|
|
% xlabel('frequency in Hz');ylabel('degree');
|
|
% legend('true phase Bode','approximated Phase Bode')
|
|
end % if 1
|
|
|
|
|