Discrete-time equivalent systems example from matlab: the c2d command

Transcription

1 Discrete-time equivalent systems example from matlab: the cd command Create the -time system using the tf command and then generate the discrete-time equivalents using the cd command. We use this command to generate the Tustin (linear transformation) equivalent discrete-time equivalent of the -time system preceded by a zero-order hold. We generate three such discrete-time equivalents with sampling times T =.5,.5,.5 seconds.that is sampling at, and Hertz. >> csys = tf(,[ ]) csys = s + Continuous-time transfer function. >> dsys_slow = cd(csys,.5, ) dsys_slow =. z z -.6 Sample time:.5 seconds >> dsys_mid = cd(csys,.5, ) dsys_mid =.49 z z -.95 Sample time:.5 seconds >> dsys_fast = cd(csys,.5, ) dsys_fast =.494 z z Sample time:.5 seconds

2 Step responses Next we generate five-second step responses for each system, recognizing that matlab itself is computing a discretized solution for the -time response. We use the step function generate the discrete time-vectors using the sampling time parameters from tf. >> tcts = [:.:5.] ; >> ycts = step(csys,tcts); >> [yslow,tslow] = step(dsys_slow,5.); >> [ymid,tmid] = step(dsys_mid,5.); >> [yfast,tfast] = step(dsys_fast,5.); >> plot(tcts,ycts,tslow,yslow, o,tmid,ymid, *,tfast,yfast, + );shg >> title( Step responses ) >> xlabel( );ylabel( ); >> legend(, Hz sampling, Hz sampling, Hz sampling ) >> print -dpdf Steps.pdf >> print -dpdf StepsZoom.pdf Step responses Step responses Hz sampling Hz sampling Hz sampling Hz sampling Hz sampling Hz sampling Sinusoidal responses at differing frequencies Frequency 4π radians per second,. Hz We have the following discrete-time equivalent frequencies. Sample rate discrete frequency (per sample) π Hz =.π Hz =.π Hz =. >> sin_slow = sin(4*pi/*tslow);

3 >> sin_mid = sin(4*pi/*tmid); >> sin_fast = sin(4*pi/*tfast); >> sin_cts = sin(4*pi/*tcts); >> ys_slow = lsim(dsys_slow,sin_slow,tslow); >> ys_mid = lsim(dsys_mid,sin_mid,tmid); >> ys_fast = lsim(dsys_fast,sin_fast,tfast); >> ys_cts = lsim(csys,sin_cts,tcts); >> plot(tcts,ys_cts,tslow,ys_slow, o,tmid,ys_mid, *,tfast,ys_fast, + );shg >> title( Slow sine responses ) >> xlabel( );ylabel( ); >> legend(, Hz sampling, Hz sampling, Hz sampling ) >> print -dpdf SlowSine.pdf >> print -dpdf SlowSineZoom.pdf.4. Slow sine responses Hz sampling Hz sampling Hz sampling.. Slow sine responses Hz sampling Hz sampling Hz sampling Frequency 4π radians per second,.hz We have the following discrete-time equivalent frequencies. Sample rate discrete frequency (per sample) Hz aliased π Hz =.π Hz = >> sinf_mid = sin(4*pi/*tmid); >> sinf_fast = sin(4*pi/*tfast); >> sinf_cts = sin(4*pi/*tcts); >> ysf_mid = lsim(dsys_mid,sinf_mid,tmid); >> ysf_fast = lsim(dsys_fast,sinf_fast,tfast); >> ysf_cts = lsim(csys,sinf_cts,tcts); >> plot(tcts,ysf_cts,tmid,ysf_mid, *,tfast,ysf_fast, + );shg >> title( Midrange sine responses ) >> xlabel( );ylabel( ); >> legend(, Hz sampling, Hz sampling ) >> print -dpdf MidSine.pdf

4 >> print -dpdf MidSineZoom.pdf.5.4 Midrange sine responses Hz sampling Hz sampling.5. Midrange sine responses Hz sampling Hz sampling Frequency 4π radians per second,.hz We have the following discrete-time equivalent frequencies. Sample rate discrete frequency (per sample) Hz aliased Hz aliased.π Hz = >> sins_fast=sin(4*pi/*tfast); >> sins_cts=sin(4*pi/*tcts); >> yss_fast=lsim(dsys_fast,sins_fast,tfast); >> yss_cts=lsim(csys,sins_cts,tcts); >> plot(tcts,yss_cts,tfast,yss_fast, + );shg >> title( Fast sine responses ) >> xlabel( );ylabel( ); >> legend(, Hz sampling ) >> print -dpdf FastSine.pdf >> print -dpdf FastSineZoom.pdf Frequency 4π radians per second,.hz; with frequency pre-warping Let us try to match the responses at this high frequency π/ c per sample by using Tustin s method with frequency pre-warping. This is done by scaling the s-plane by a factor f so that the scaled s-domain point, s = jfω, maps to the correct z-domain point, z = e jωt. The bilinear transformation of Tustin has s T z z + and thus z + T s T s.

5 # Fast sine responses Hz sampling # -.5 Fast sine responses Hz sampling If we use the above elements, this means solving This has the solution f = [ cos ωt ] ωt sin ωt. z = e jωt = + jf T ω jf T ω. We find the pre-warped system by substituting for s as above into s+. dsys warp(z) = = = = z z+ +, z + z + z +, ( + + z + ) z + ( z + z + / +/. ), >> omt = *pi/ % omega T value in z-domain omt =.944 >> f=*(-cos(omt))/omt/sin(omt) % f-value for pre-warping the s-domain f =.654 >> tft = /f/.5 % / value tft =

6 4.899 >> dsys_warp = tf([ ]/(+tft),[ (-tft)/(+tft)],.5) % as is the formula above dsys_warp =.48 z z Sample time:.5 seconds Now try the fast sinusoid input signal again. We see that the pre-warping matches very well. # # My mistake in interpreting the figure in class was to fail to recognize the period-three nature of the fast sinusoid it really only takes on three distinct value. If we alter the frequency a little we see a different plot where the intercept points move to different parts of the input sinusoid. >> *pi/ ans =.944 >> sinsd_cts = sin(4*tcts); >> sinsd_fast = sin(4*tfast); >> yssd_cts=lsim(csys,sinsd_cts,tcts); >> yssd_fast=lsim(dsys_fast,sinsd_fast,tfast); >> yssd_warp=lsim(dsys_warp,sinsd_fast,tfast); >> plot(tcts,yssd_cts,tfast,yssd_fast, o,tfast,yssd_warp, k+ );shg But what happens to the step response of this pre-warped system? It all goes down the toilet at dc ( c per sample) because it fits at π/ c per sample and we can only pre-warp to fit at one point on the unit circle.

7 # # Step Response pre-warped Amplitude Time (seconds)