Question

Also write the MATLAB cosde of the answers

Objects smaller than the wavelengths of visible light are a staple of contemporary science and technology. Biologists study single molecules of protein or DNA; materials scientists examine atomic-scale flaws in crystals; microelectronics engineers lay out circuit patterns only a few tenths of atoms thick. Until recently, this minute world could be seen only by cumbersome, often destructive methods, such as electron microscopy and X-ray diffraction. It lay beyond the reach of any instrument as simple and direct as the familiar light microscope. New microscopes, typified by the scanning tunneling microscope (STM), are now available. The precision of position control required is in the order of nanometers. The STM relies on piezoelectric sensors that change size when an electric voltage across the material is changed. The "aperture" in the STM is a tiny tungsten probe, its tip ground so fine that it may consist of only a single atom and measure just 0.2 nanometer in width. Piezoelectric controls maneuver the tip to within a nanometer or two of the surface of a conducting specimen so close that the electron clouds of the atom at the probe tip and of the nearest atom of the specimen overlap. A feedback mechanism senses the variations in tunneling current and varies the voltage applied to a third, 2- axis control. The z-axis piezoelectric moves the probe vertically to stabilize the current and to maintain a constant gap between the microscope's tip and the surface. The control system is shown in Fig. 1a, and the block diagram is shown in Fig. 1b. Display Pierockctric control V-axis Controller x-axis Reference signal Sensor voltage Fig. la

Answer 1

MATLAB CODE

syms s t1 t2 K Gp % treating t1 t2 K Gp as varibles

% K= % if value is known, just remove the % sign at starting

% t1=

% t2=

% Gp=

% PART -A

G = 17640/((s)*(s^2+59.4*s+1764));

Gc = K*(t1*s+1)*(t2*s+1)/s;

T = G*Gc/(1+G*Gc);

R = 1/s;

Rp = Gp/s;

y = ilaplace(R*T);

yp = ilaplace(Rp*T);

display(y,'step response without prefilter')

display(yp,'step response with prefilter')

% PART B

Td = 1/s;

Tfd = G/1+G*Gc;

yd = ilaplace(Td*Tfd);

display(yd,'step disturbance response')

% PART C

G2 = 16000/((s)*(s^2+4*s+1600));

T2 = G2*Gc/(1+G2*Gc);

y2 = ilaplace(R*T2);

yp2 = ilaplace(Rp*T2);

display(y2,'step response without prefilter')

display(yp2,'step response with prefilter')

MATLAB OUTPUT

>> c_18

step response without prefilter =

1 - 297*symsum((exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t)*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2)/(88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4) - 5*symsum((exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t)*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3)/(88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4) - 8820*symsum((root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t))/(17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4)

step response with prefilter =

Gp - 8820*Gp*symsum((root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t))/(17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4) - 297*Gp*symsum((exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t)*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2)/(88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4) - 5*Gp*symsum((exp(root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t)*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3)/(88200*K*t1 + 88200*K*t2 + 891*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^2 + 20*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)^3 + 17640*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k) + 176400*K*root(s3^4 + (297*s3^3)/5 + 17640*K*s3^2*t1*t2 + 1764*s3^2 + 17640*K*s3*t2 + 17640*K*s3*t1 + 17640*K, s3, k)*t1*t2), k, 1, 4)

step disturbance response =

(4901*K)/864360 + t*(10*K*t1 - (33*K)/98 + 10*K*t2 + 10) - (33*K*t1)/98 - (33*K*t2)/98 + 5*K*t^2 + exp(-(297*t)/10)*(cos((3*9799^(1/2)*t)/10) + (10*9799^(1/2)*sin((3*9799^(1/2)*t)/10)*((43226820*K*t1 - 297*K + 43226820*K*t2 - 2567149200*K*t1*t2 + 43226820)/(1455300*K*t1 - 24505*K + 1455300*K*t2 - 43218000*K*t1*t2 + 1455300) - 297/10))/29397)*((33*K*t1)/98 - (4901*K)/864360 + (33*K*t2)/98 - 10*K*t1*t2 + 33/98) + 10*K*t1*t2 - 33/98

step response without prefilter =

1 - 4*symsum((exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t)*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2)/(4*(4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4) - symsum((exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t)*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3)/(4*(4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4) - 1600*symsum((root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t))/(4*(800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4)

step response with prefilter =

Gp - 4*Gp*symsum((exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t)*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2)/(4*(4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4) - Gp*symsum((exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t)*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3)/(4*(4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4) - 1600*Gp*symsum((root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*exp(root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t))/(4*(800*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k) + 4000*K*t1 + 4000*K*t2 + 3*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^2 + root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)^3 + 8000*K*root(s4^4 + 4*s4^3 + 16000*K*s4^2*t1*t2 + 1600*s4^2 + 16000*K*s4*t2 + 16000*K*s4*t1 + 16000*K, s4, k)*t1*t2)), k, 1, 4)

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