Solution for Take Home Exam 2

Problem 1: For the following closed-loop transfer functions find the maximum overshoot. Show the poles & zeros graphically & manually. Show the approximate & exact phase margin. Convert the transfer functions to state space & prove them by simulation. Use matlab as appropriate. a) to e) are:

Part 1: find the maximum overshoot. The approximation formula for Mp is equation 2.18

on p. 21 of the text.

The matlab code is:

function Mp = getMp(num,den,Ts,i);sys = tf(num,den,Ts);[wn,z] = damp(sys);Mp = 100*exp(-pi*z(1)/sqrt(1-z(1)*z(1)));figure(i)y = dstep(num,den);dstep(num,den),grid on,hold onplot([0:length(y)-1]',(1+Mp/100*ones(1,length(y))),'r'),hold offtitle(['step response, Mp = ',num2str(Mp),'%'])

In the above num is the discrete transfer function numerator, den is the denominator, Ts is the sampling interval (arbitrary in this case), and ‘i’ is the figure number.

The matlab driver code is:

num = [1 0.5]; den = 3*[1 -1 0.5]; Mp = getMp(num,den,10,1) num = [0.5 0]; den = [1 -1 0.5]; Mp = getMp(num,den,10,2) num = [1 0.5]; den = 3*[1 -1 0.5 0 0]; Mp = getMp(num,den,10,3)

num = [0.5]; den = [1 -1 0.5 0 0]; Mp = getMp(num,den,10,4) num = conv(0.316*[1 0.002],[1 0.5]); den = conv([1 -1 0.5],[1 -0.05]); Mp = getMp(num,den,10,5)

The plots are:

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1.4step response, Mp = 25%

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In all cases the Mp is 25%.

Part 2): Show the poles & zeros graphically & manually. The matlab code is:

function getpolz(num,den,Ts,i)figure(i) sys = tf(num,den,Ts);[p,z] = pzmap(sys);pzmap(sys)title(['poles = ',num2str(p','%7.3f')])xlabel(['real axis. zeros = ',num2str(z','%7.3f')])

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1poles = 0.500-0.500i 0.500+0.500i

real axis. zeros = -0.500

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1poles = 0.500-0.500i 0.500+0.500i

real axis. zeros = 0.000

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1poles = 0.000+0.000i 0.000+0.000i 0.500-0.500i 0.500+0.500i

real axis. zeros = -0.500

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1poles = 0.000+0.000i 0.000+0.000i 0.500-0.500i 0.500+0.500i

real axis. zeros =

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1poles = 0.500-0.500i 0.500+0.500i 0.050+0.000i

real axis. zeros = -0.500 -0.002

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Part 3): Show the approximate & exact phase margin. The matlab code is:

function getPM(num,den,Ts,i)figure(i) sys = tf(num,den,Ts);margin(sys)[wn,z] = damp(sys);xlabel(['the approximate PM = ',num2str(100*z(1))])

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Bode DiagramGm = 9.54 dB (at 1.57 rad/sec) , Pm = 50.1 deg (at 0.982 rad/sec)

the approximate PM = 40.3713 (rad/sec)

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Bode DiagramGm = 14 dB (at 3.14 rad/sec) , Pm = 60 deg (at 1.05 rad/sec)

the approximate PM = 40.3713 (rad/sec)

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Bode DiagramGm = -2.36 dB (at 0.77 rad/sec) , Pm = -62.4 deg (at 0.982 rad/sec)

the approximate PM = 40.3713 (rad/sec)

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Bode DiagramGm = -2.96 dB (at 0.683 rad/sec) , Pm = -120 deg (at 1.05 rad/sec)

the approximate PM = 40.3713 (rad/sec)

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Bode DiagramGm = 9.11 dB (at 1.51 rad/sec) , Pm = 49.7 deg (at 0.968 rad/sec)

the approximate PM = 40.3713 (rad/sec)

Note that the approximate PM doesn’t vary, whereas the exact PM does vary significantly.

Part 4): Convert the transfer functions to state space & prove them by simulation.

The matlab code is:

function getss(num,den,Ts,i)figure(i) sys = tf(num,den,Ts);[phi,gam,c,d] = ssdata(sys);sysd = ss(phi,gam,c,d,Ts);step(sys,sysd)legend('tf','ss') The following plots show that they are plotted on top of each other. In this case, Ts = 1 sec.

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4.11 Consider the system described by the transfer function .

a) Draw the block diagram corresponding to this system in control canonical form, define the state vector, and give the corresponding description matrices F, G, H, J. b) Write G(s) in partial fractions and draw the corresponding parallel block diagram with each component part in control canonical form. Define the state ξ and give the corresponding state description matrices A, B, C, D.

a)

b) G(s) in partial fractions

4.18 Draw out each block of Fig. 4.10 in a) control and b) observer canonical form. Write out the state-description matrices in each case.

One of the 1st steps is to convolve the numerator and denominator as follows:

conv([1 1],[1 -0.5 0.25])conv([1 -1 0],[1 -1.6 0.81]) which gives: [1.0000 0.5000 -0.2500 0.2500] for the numerator and,

[1.0000 -2.6000 2.4100 -0.8100 0] for the denominator.

5.15 Find the transform of the output Y(s) and its samples Y*(s) for the block diagrams shown in Fig. 5.23. Indicate whether a transfer function exists in each case.

5.16 Assume the following transfer functions are preceded by a sampler and zero-order hold and followed by a sampler. Compute the resulting discrete transfer functions.

The governing equation is , which is equation (2.39) of the text.

The table look up for a) or G1(s) is:

The table look up for c) or G3(s) is:

The table look up for e) or G5(s) is:

The matlab code is:

Ts = 1;c2d(tf(1,[1 0 0]),Ts)c2d(tf(1,[1 1],'inputdelay',1.5),Ts)c2d(tf(1,[1 1 0]),Ts)c2d(tf(1,[1 1 0],'inputdelay',1.5),Ts)c2d(tf(1,[1 0 -1]),Ts) The assumed sample interval is 1 second.

The results are:

Transfer function: 0.5 z + 0.5-------------z^2 - 2 z + 1 Sampling time: 1 Transfer function: 0.3935 z + 0.2387z^(-1) * -----------------

z^2 - 0.3679 z Sampling time: 1 Transfer function: 0.3679 z + 0.2642----------------------z^2 - 1.368 z + 0.3679 Sampling time: 1 Transfer function: 0.1065 z^2 + 0.4709 z + 0.05471z^(-1) * ------------------------------- z^3 - 1.368 z^2 + 0.3679 z Sampling time: 1 Transfer function:0.5431 z + 0.5431-----------------z^2 - 3.086 z + 1 Sampling time: 1

6.3 The following transfer function is a lead network designed to add about 60 degrees

phase lead at w1 = 3 rad/s .

a) For each of the following design methods compute and plot in the z-plane the pole and zero locations and compute the amount of phase lead given by the equivalent network at z1 = if T = 0.25 sec and the design is via1) Forward rectangular rule 2) Backward rectangular rule3) Bilinear rule4) Bilinear with prewarping (use w1 as the warping frequency)5) Zero-pole mapping6) Zero-order-hold equivalent7) Triangular-hold equivalent

b) Plot over the frequency range wl = 0.1 wh = 100 the amplitude and phase Bode plots for each of the above equivalents.

Compute the phase @ 3 rad/s for the lead network H(s) by the following matlab code:

w1 = 3;H = tf([1 1],[0.1 1]);

[mag,ph] = bode(H,w1)

The result for the continuous lead is:mag = 3.0289, ph = 54.8658

The result for the forward rectangular rule is:mag = 2.9222, ph = 74.5542

The result for the backward rectangular rule is:mag = 3.0036, ph = 38.9183

The result for the bilinear rule is:mag = 3.1514, ph = 54.9030

The result for the bilinear rule with prewarp @ 3 rad/s is:mag = 3.0289, ph = 54.8658

The result for the matched zero-pole rule is:mag = 3.0112, ph = 47.5753

The result for the zero-order hold rule is:mag = 7.4779, ph = 58.1401

The result for the first-order hold rule is:mag = 3.1262, ph = 48.2400

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1H(s) continuous

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1Backw ard rectangular rule, phase @ 3 rad/s = 38.9183

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1Trapezoid (bilinear) rule, phase @ 3 rad/s = 54.903

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1Trapezoid w ith prew arp @ 3 rule, phase @ 3 rad/s = 54.8658

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1Zero-pole matched rule, phase @ 3 rad/s = 47.5753

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1Zero-order hold rule, phase @ 3 rad/s = 58.1401

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1First-order hold rule, phase @ 3 rad/s = 48.24

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Bode Diagram

Frequency (rad/sec)

Hbackforbilinprew arpzpzohfoh

The complete matlab code is:

function Prob_6_3()%close allw1 = 3;H = tf([1 1],[0.1 1]);figure(1)pzmap(H)title('H(s) continuous')[mag,ph] = bode(H,w1);T = 0.25;[a,b,c,d] = ssdata(H);[phif,gamf,hf,jf] = get_for(a,b,c,d,T);Hf = ss(phif,gamf,hf,jf,T);[mag,ph] = bode(Hf,w1);pzmap(Hf)title('Forward rectangular rule')[phib,gamb,hb,jb] = get_back(a,b,c,d,T);Hb = ss(phib,gamb,hb,jb,T);[mag,ph] = bode(Hb,w1);pzmap(Hb)title(['Backward rectangular rule, phase @ 3 rad/s = ',num2str(ph)])[phibil,gambil,hbil,jbil] = get_bil(a,b,c,d,T);Hbil = ss(phibil,gambil,hbil,jbil,T);[mag,ph] = bode(Hbil,w1);pzmap(Hbil)

title(['Trapezoid (bilinear) rule, phase @ 3 rad/s = ',num2str(ph)])Hpre = c2d(H,T,'prewarp',w1);[mag,ph] = bode(Hpre,w1);pzmap(Hpre)title(['Trapezoid with prewarp @ 3 rule, phase @ 3 rad/s = ',num2str(ph)])Hmat = c2d(H,T,'matched');[mag,ph] = bode(Hmat,w1);pzmap(Hmat)title(['Zero-pole matched rule, phase @ 3 rad/s = ',num2str(ph)])Hzoh = c2d(H,T,'zoh');[mag,ph] = bode(Hzoh,w1);pzmap(Hzoh)title(['Zero-order hold rule, phase @ 3 rad/s = ',num2str(ph)])Hfoh = c2d(H,T,'foh');[mag,ph] = bode(Hfoh,w1);pzmap(Hfoh)title(['First-order hold rule, phase @ 3 rad/s = ',num2str(ph)])bode(H,Hf,Hb,Hbil,Hpre,Hmat,Hzoh,Hfoh,[0.1:.1:25])legend('H','back','for','bilin','prewarp','zp','zoh','foh') function [phif,gamf,hf,jf] = get_for(a,b,c,d,T);phif = eye(size(a)) + a*T;gamf = b*T;hf = c;jf = d; function [phib,gamb,hb,jb] = get_back(a,b,c,d,T);phib = inv(eye(size(a)) - a*T);gamb = phib*b*T;hb = c*phib;jb = d + c*gamb; function [phibil,gambil,hbil,jbil] = get_bil(a,b,c,d,T);temp = inv(eye(size(a)) - a*T/2);phibil = (eye(size(a)) + a*T/2)*temp;gambil = temp*b*sqrt(T);hbil = sqrt(T)*c*temp;jbil = d + c*temp*b*T/2;