Analysis of TF and SS Models

Question 1

Purpose of Analyzing Models

Answer 1

1. Predict response in time/frequency domain
2. Stability
3. Speed of Response
4. Steady State Error
5. Oscillation Prevention

Question 2

Time Response of System

Answer 2

1. Transient Response: depends on initial conditions, corresponds to homogenous solution of D.E.
2. Steady State Response:
   depends on system inputs, corresponds to the particular solution of D.E.

Question 3

Frequency Response of System

Answer 3

1. For LTI systems with sinusoidal input, the steady state output is also sinusoidal at the same frequency
2. Output has different magnitude and phase
3. Frequency response is found from transfer function:
   G(s), s = j*w.

Question 4

Bode Plot

1. Definition
2. Q-Factor

Answer 4

1. Plot of magnitude/phase of frequency response on Log graph
2. Q = 1/(2*d): size and sharpness of peaks

Question 5

Nyquist Diagram

Answer 5

Plot of magnitude/phase of frequency response on complex plane

Question 6

BIBO Stability

Answer 6

1. Given bounded input, output is bounded
2. All poles of transfer function have negative real parts. Any pole has a positive real part system is unstable.
3. Poles must be in left-half plane for stability.
4. If any pair of poles is on imaginary axis system is marginally stable (oscillations)
5. Purely imaginary poles is not BIBO stable.

Question 7

First Order System

1. ODE
2. Transfer Function
3. DC Gain
4. Time Constant
5. Stability

Answer 7

1. y’ + ay = bu or Ty’ + y = ku
2. G(s) = b/(s+a) = k/(T*s + 1)
3. k = b/a
4. T = 1/a
5. s = -a: a>0 stable, a<0 unstable

Question 8

Second Order System
1. ODE
2. Transfer Function
3. DC Gain
4. Damping Ratio
5 Natural Frequency
6 Poles/zeros

Answer 8

my’‘+by’+ky = f(t) or y’’ + 2dwy’ + w^2y = kDCw^2u

2. G(s) = 1/(ms^2 + bs + k) = (kDCw^2)/(s^2 + 2dws + w^2)
3. kDC = 1/k
4. d = b/(2sqrt(km))
5. w = sqrt(k/m)
6. s_p = -dw + or - jw*sqrt(1-d^2)

Question 9

Underdamped Second Order System

Answer 9

1. d < 1
2. Poles are complex valued with negative real parts
3. System oscillates while approaching steady state

Question 10

Overdamped Second Order System

Answer 10

1. d > 1
2. Poles are real and negative
3. Stable

Question 11

Critically Damped Second Order System

Answer 11

1. d = 1
2. Both poles are real and have same magnitude
3. Stable System

Question 12

Undamped Second Order System

Answer 12

1. d = 0
2. poles are purely imaginary
3. Marginally Stable

Question 13

Settling Time

1. Definition
2. Equation for 2nd Order Underdamped System

Answer 13

1. Time required for the system to fall within a certain percentage of steady state value for a step input.
2. Ts = -ln(tolerance fraction)/(d*w)

Question 14

Percent Overshoot

1. Definition
2. Equation for 2nd Order Underdamped System

Answer 14

1. Percent by which system’s step response exceeds final steady state value.
2. Mp = exp(-d/(sqrt(1-d^2)))