Exercícios = + + Pretende-se estimar, por análise espectral, a função de transferência deste sistema.

Transcrição

1 8 Exercícios Considere o seguinte sistema, amostrado a 2Hz: = + + Pretende-se estimar, por análise espectral, a função de transferência deste sistema. Utilize como sinais de entrada, sequências PRBS, ruído branco e chirp e observe as saídas correspondentes. Utilize a função de análise espectral do matlab (spa) para obter as estimativas. Tenha atenção que o sistema contínuo é instável (tem dois polos na origem)!

2 9 Resolução % spa_ident.m % System identification with % spectral analysis methods % for systems with known components % % Number of points for input signal N = 255; % the sampling period T =.5; % the time vector t=:t:t*n; % the sampling frequency ws = 2*pi/T; % frequency vector w =.:.:ws/2; % generating a prbs input u_prbs = idinput(n,'prbs',[ ]); % generation of a chirp signal wstart =.2; wend = 3.; u_chirp = chirp(t,wstart/2/pi,t(end),wend/2/pi)'; % another chirp signal with higher frequencies wstart = 2.; wend = 6.; u_chirp2 = chirp(t,wstart/2/pi,t(end),wend/2/pi)'; % generation of a white gaussian noise input u_noise = randn(size(t))'; %system model - second order + double integrator sys = tf([],[.25 ]); % convertion to discrete time with zero-order-hold dsys = c2d(sys,t,'zoh');

3 % simulate system with all inputs y_prbs = lsim(dsys,u_prbs,[]); y_chirp = lsim(dsys,u_chirp,[]); y_chirp2 = lsim(dsys,u_chirp2,[]); y_noise = lsim(dsys,u_noise,[]); % preprocessing data two unstable poles yf_prbs = filter([ -2 ],,y_prbs); yf_chirp = filter([ -2 ],,y_chirp); yf_chirp2 = filter([ -2 ],,y_chirp2); yf_noise = filter([ -2 ],,y_noise); %creating the data structures for id funcs data_prbs = iddata(yf_prbs, uf_prbs, T); data_chirp = iddata(yf_chirp,uf_chirp, T); data_chirp2 = iddata(yf_chirp2, uf_chirp2, T); data_noise = iddata(yf_noise, uf_noise, T); %displaying data figure(); idplot(data_prbs); figure(2); idplot(data_chirp); figure(3); idplot(data_chirp2); figure(4); idplot(data_noise); %transfer function estimation with spec. analysis g_prbs_spa = spa(data_prbs,3,w); g_chirp_spa = spa(data_chirp,3,w); g_chirp2_spa = spa(data_chirp2,3,w); g_noise_spa = spa(data_noise, 3,w); %must model the known part of the tf dsysknown = tf([ ],[ -2 ],T); %magnitude and phase vectors for plotting [mag_true, phase_true] = bode(dsys,w); [mag_known, phase_known] = bode(dsysknown,w); [mag_prbs_spa, phase_prbs_spa] = bode(g_prbs_spa); [mag_chirp_spa, phase_chirp_spa] = bode(g_chirp_spa); [mag_chirp2_spa, phase_chirp2_spa] = bode(g_chirp2_spa); [mag_noise_spa, phase_noise_spa] = bode(g_noise_spa); % display figure(5); subplot(2); loglog(w,mag_prbs_spa(:).*mag_known(:),w,mag_true(:),'--'); title('spectral Analysis - PRBS Input'); ylabel(''); xlabel(''); subplot(22); semilogx(w,phase_prbs_spa(:)+phase_known(:),w,phase_true(:)-36,'--'); ylabel(''); xlabel('');

4 y y CHIRP u PRBS u y.6 y CHIRP u 4 2 u WHITE NOISE

5 2 5 Spectral Analysis PRBS Input 5 Spectral Analysis LOW FREQ CHIRP Input Spectral Analysis HIGH FREQ CHIRP Input 5 Spectral Analysis GAUSSIAN WHITE NOISE Input

6 Without Filtering (raw data) 3 5 Spectral Analysis - PRBS Input 5 Spectral Analysis - LOW FREQ CHIRP Input Spectral Analysis - HIGH FREQ CHIRP Input 5 Spectral Analysis - GAUSSIAN WHITE NOISE Input