Untitled

Assignment

We would like to answer the questions using the Symbolic Toolbox or the Control Toolbox functions.

clear variables

Solving by Using the Symbolic Math Toolbox

syms s

syms omega real

assume(omega >=0)

G(s) = 10/((1+10*s)*(1+2*s))

G(s) =

G_om(omega ) = subs(G, s, 1i*omega)

G_om(omega) =

First Question: Magnitude and Phase of the Frequency Response at a Given Angular Frequency

G10 = G_om(1/10)

G10 =

G2 = G_om(1/2)

G2 =

The magnitude of the frequency response at

rad/sec :

magG10sym = abs(G10)

magG10sym =

and the computation of the phase

angle(G10)

ans =

The magnitude of the frequency response at

rad/sec :

magG2sym = abs(G2)

magG2sym =

and the computation of the phase

angle(G2)

ans =

Second Question: Solving a Trigonometric Equation

phaseG(omega) = simplify(angle(G_om))

phaseG(omega) =

Foolish Approach

Let's rely on the MATLAB function solve:

EQ1 = phaseG ==-pi/2;

sol = solve(EQ1,omega,"ReturnConditions",true)

sol = struct with fields:
         omega: [2×1 sym]
    parameters: x
    conditions: [2×1 sym]

sol.omega

ans =

sol.conditions

ans =

Note: the provided solution is nonsense. We must rewrite the equation to be solved more clearly and efficiently.

Yet Another Foolish Approach

Let rewrite the equation describing the problem, by explicitly determining real and imaginary part of

imagG_om = simplify(imag(G_om))

imagG_om(omega) =

realG_om = simplify(real(G_om))

realG_om(omega) =

Now, let's write the computation of

by explicitly computing the arctan function, according to Part 8, pages 36-42 of the course material (in particular, refer to page 37)

phaseG2 = atan(simplify(imagG_om/realG_om))-pi

phaseG2(omega) =

Now, let's write the equation:

EQ2 = phaseG2 == -pi/2

EQ2(omega) =

solve(EQ2, omega)

ans =
Empty sym: 0-by-1

Note: now it should be clear why the previously presented approach was nonsense, and why there is no solution with the current approach:

is undefined when
we need a different approach to solve the equation.

An Effective Approach

Recall that (Part 8, page 40 of the course material)

Thus, the straightforward way to solve the equation is to put the real part of the frequency response equal to zero

OMsol = solve(realG_om==0)

OMsol =

barOM = sym(1)/sym(sqrt(20))

barOM =

double(barOM)

ans = 0.2236

Refer to Part 8, page 59 of the course material.

Solving by Using the Control Toolbox

clear s
 
s = tf('s')
s =
 
  s
 
Continuous-time transfer function.
Model Properties
Gs = 10/((1+10*s)*(1+2*s))
Gs =
 
         10
  -----------------
  20 s^2 + 12 s + 1
 
Continuous-time transfer function.
Model Properties

First Question: Magnitude and Phase of the Frequency Response at a Given Angular Frequency

s10 = 1i*0.1; % <<-- NB
s2 = 1i*0.5; 
 
FreqResp_Gs10 = evalfr(Gs, s10)
FreqResp_Gs10 = 3.8462 - 5.7692i
FreqResp_Gs2 = evalfr(Gs, s2)
FreqResp_Gs2 = -0.7692 - 1.1538i

 
magG10 = abs(FreqResp_Gs10) 
magG10 = 6.9338
double(magG10sym)
ans = 6.9338
 
magG2 = abs(FreqResp_Gs2) 
magG2 = 1.3868
double(magG2sym)
ans = 1.3868

rad2deg(angle(FreqResp_Gs10))
ans = -56.3099
 
rad2deg(angle(FreqResp_Gs2))
ans = -123.6901

Second Question: Solving a Trigonometric Equation

There is no direct way to solve this equation. You can only find an approximate solution, by "exploring" the Bode diagram of the phase of the frequency response.

Put the mouse cursor on the phase diagram, and move it, still staying on the curve, until you find a point close enough to the desired phase

figure;

bodeplot(Gs)