70 lines
3.2 KiB
Matlab
70 lines
3.2 KiB
Matlab
% Impulse response invariant discretization of fractional order
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% low-pass filters
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%
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% irid_folpf function is prepared to compute a discrete-time finite
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% dimensional (z) transfer function to approximate a continuous-time
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% fractional order low-pass filter (LPF) [1/(\tau s +1)]^r, where "s" is
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% the Laplace transform variable, and "r" is a real number in the range of
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% (0,1), \tau is the time constant of LPF [1/(\tau s +1)].
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%
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% The proposed approximation keeps the impulse response "invariant"
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%
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% IN:
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% tau: the time constant of (the first order) LPF
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% r: the fractional order \in (0,1)
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% Ts: the sampling period
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% norder: the finite order of the approximate z-transfer function
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% (the orders of denominator and numerator z-polynomial are the same)
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% OUT:
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% sr: returns the LTI object that approximates the [1/(\tau s +1)]^r
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% in the sense of invariant impulse response.
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% TEST CODE
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% dfod=irid_folpf(.01,0.5,.001,5);figure;pzmap(dfod)
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%
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% Reference: YangQuan Chen. "Impulse-invariant discretization of fractional
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% order low-pass filters".
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% Sept. 2008. CSOIS AFC (Applied Fractional Calculus) Seminar.
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% http://fractionalcalculus.googlepages.com/
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% --------------------------------------------------------------------
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% YangQuan Chen, Ph.D, Associate Professor and Graduate Coordinator
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% Department of Electrical and Computer Engineering,
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% Director, Center for Self-Organizing and Intelligent Systems (CSOIS)
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% Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA
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% E: yqchen@ece.usu.edu or yqchen@ieee.org, T/F: 1(435)797-0148/3054;
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% W: http://www.csois.usu.edu or http://yangquan.chen.googlepages.com
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% --------------------------------------------------------------------
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%
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% 9/7/2009
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% Only supports when r in (0,1). That is fractional order low pass filter.
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% HOWEVER, if r is in (-1,0), we call this is a "fractional order
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% (proportional and derivative controller)" - we call it FO(PD).
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% Note: it may be needed to make FO-LPF discretization minimum phase first.
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%
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% See also irid_fod.m
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% at http://www.mathworks.com/matlabcentral/files/21342/irid_fod.m
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% See also srid_fod.m
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% (See how the nonminimum phase zeros are handled)
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% See also gml_fun.m
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% at http://www.mathworks.com/matlabcentral/files/20849/gml_fun.m
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function [sr]=irid_folpf(tau,r,Ts,norder)
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if nargin<4; norder=5; end
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% if tau < 0 , sprintf('%s','Time constant has to be positive'), return, end
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% if Ts < 0 , sprintf('%s','Sampling period has to be positive'), return, end
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% if r>=1 | r<= 0, sprintf('%s','The fractional order should be in (0,1)'), return, end
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% if norder<2, sprintf('%s','The order of the approximate transfer function has to be greater than 1'), return, end
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wmax0=2*pi/Ts/2; % rad./sec. Nyquist frequency
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L=abs(tau)*4/Ts; % decides the number of points of the impulse response function h(n)
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Taxis=[0:L-1]*Ts;
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ha0=(7.0*Ts/8)^r;
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n=1:L-1; n=n*Ts;
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h1=gml_fun(1,r,r,-1/tau*n); % http://www.mathworks.com/matlabcentral/files/20849/gml_fun.m
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h2=(1/tau)^r*h1.*(n.^(r-1));
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h0= ha0*(1/(tau+ (7.0*Ts/8)))^r;
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h=[h0,h2*Ts]; %% [ha0, (Ts^r)*(n.^(r-1))/gamma(r)];
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q=norder;p=norder;
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[b,a]=prony((h),q,p);
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sr=tf(b,a,Ts);
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