MAE488 Final Exam Prep

Choose the best completion to this statement:

A causal element is an element

a) where the current output or state depends ONLY on the current input.

b) where the current output does not depend on future input.

c) where the current output depends on future input.

b) where the current output does not depend on future input

An exponential function (a function of the form y(x) = be^mx) results in a straight line when plotted on

a) rectilinear axes

b) log-log axes

c) semi-log axes with the dependent variable on the logarithmic axis

d) semi-log axes with the dependent variable on the linear axis

c) Semi-log axes with the dependent variable on the logarithmic axis

The superposition principle can be broken up into two components:

a) the additive property

b) the distributive property

c) the commutative

d) the homogeneous property

e) the associative property

a) the additive property

d) the homogeneous property

An power function (a function of the form y(x) = be^mx) results in a straight line when plotted on

a) rectilinear axes

b) log-log axes

c) semi-log axes with the dependent variable on the logarithmic axis

d) semi-log axes with the dependent variable on the linear axis

b) log-log axes

Choose the best completion to this statement:

A static element is an element

a) where the current output or state depends ONLY on the current input.

b) where the current output does not depend on future input.

c) where the current output depends on future input.

a) where the current output or state depends ONLY on the current input.

Suppose that you have a non-linear system. What can you do to obtain a linear model?

a) Linearize the model using partial fraction expansion.

b) Linearize the model using the bode plot method.

c) Linearize the model using Taylor Series.

d) Nothing.

c) Linearize the model using Taylor Series.

Consider the following differential equation

3d^3x/dt^3 + 5tdx/dt = sin(t)

This is a ________________ Equation

a) Homogeneous

b) NonHomogeneous

b) NonHomogeneous

Consider the following system of differential equations

d^2x/dt^2 + 3d^3x/dt^3 - 7dx/dt = sin(t)

d^2x/dt^2 + 5y = 10u(t)

This is a ____ order system of differential equations

5th

Consider the following differential equation.

3d^3x/dt^3 + 5tdx/dt = sin(t)

The independent variable is

a) t

b) x

a) t

(T/F) The initial value theorem will always provide the correct initial condition, regardless of the input or transfer function of the system.

False

(T/F) Since electrons are negatively charged, a positive current flows in the opposite direction of the electrons

True

Which of the following is the equation for Ohm's Law?

1. V = i/R

2. V = iR

3. i = VR

4. None of these

2. V = iR

Impedance is a transfer function with

a) Voltage output and current input

b) Current output and voltage input

a) Voltage output and current input

(T/F) In a DC motor, the torque constant and the back emf constant are numerically equal assuming they are written with the same units.

True

To adjust the speed of an armature controlled DC motor, you adjust the ________________.

a) armature voltage

b) field voltage

c) None of these

d) torque constant

e) back emf

a) armature voltage

In a DC motor:

The ___________ changes the direction of the armature current so that the force is always in the same direction.

The _______________ is a coil of wire upon which the force is applied.

_____________________ is the law specifying the magnitude and direction of this force.

1. Commutator

2. Armature

3. Lorentz Force Law

A 5 input, 4 state, 2 output system can be represented in a transfer function form. The size of the matrix is ___ rows by ___ columns.

1. Two

2. Five

A system with 2 inputs, 4 states, and 3 outputs is represented in state space form.

The size of the A matrix is __ rows by __ columns

The size of the B matrix is __ rows by __ columns

The size of the C matrix is __ rows by __ columns

The size of the D matrix is __ rows by __ columns

A(SS)

B(SI)

C(OS)

D(OI)

Carefully read the following definition of a transfer function. Is the definition correct?

"The transfer function is the ratio of the Laplace transform of the input over the Laplace transform of the output, with zero initial conditions."

No

Consider two systems: system 1 with a time constant of tau1 = 0.5 and system 2 with a time constant of tau2 = 0.05. System 3 is constructed from the serial connection of system 1 and 2. Which system dominates the response of system 3?

a) System 1

b) System 2

c) Neither system 1 or system 2

d) Not possible to determine

a) System 1

If all of the stable poles of a system lie on a vertical line in the complex plane, the poles have the same

a) damping ratio

b) natural frequency

c) damped natural frequency

d) time constant

d) time constant

(T/F) Consider the unit step response for a standard 2nd order system. For a fixed damping ratio, zeta, the natural frequency, omega_n, has very little effect on the settling time.

False

Consider the free response of a standard 2nd order stable system. As the imaginary part of the poles decreases,

a) the settling time increases

b) the settling time decreases

c) the frequency increases

d) the frequency decreases

d) the frequency decreases

If all of the stable poles of a system lie on an oblique line )not vertical or horizontal) in the complex plane, the poles have the same

a) damping ratio

b) natural frequency

c) damped natural frequency

d) time constant

a) damping ratio

Match the values for the damping ratio, zeta, on the left to the corresponding response type (sinusoidal, underdamped, overdamped, critically damped, unstable)

<0 _______________

=0 _______________

between 0 and 1 _______________

=1 _______________

>1 _______________

<0 unstable

=0 sinusoidal

between 0 and 1 underdamped

=1 critically damped

>1 overdamped

(T/F) The frequency transfer function is obtained by setting s = jomega in the transfer function.

True

(T/F) The resonant frequency is always the same as the natural frequency.

False

(T/F) The frequency response of a system describes how the system responds to a sinusoidal input as the frequency of the input changes from 0 to infinity.

True

(T/F) A tuned mass spring damper is used to maintain (not increase or decrease) the magnitude of oscillations in a system.

False

(T/F) A bode plot consists of the magnitude and phase plotted versus damping ratio.

False

If the input to a stable, linear, time-invariant system is a sinusoid then the steady state output will be a sinusoid with the _______ frequency, ______ magnitude, and ______ phase.

1. Same

2. Different

3. Different

(T/F) One if the reasons we use control is to compensate for a disturbance.

True

(T/F) Pole placement is a controller design method in which the values of adjustable parameters in the controller are chosen so that a desired closed loop characteristic is obtained.

True

(T/F) A gain controller is a type of PID controller with only the proportional term. You can think of this type of controller as "pushing harder on the plant" as the gain is increased to make the plant respond faster thus reducing the error.

True

The reference input to a control system is a unit ramp function. If the control system is perfect, what is the output?

a) a unit step function

b) a decaying sinusoid

c) a unit ramp function

d) Not enough information

c) a unit ramp function

(T/F) Caution must be used when designing a PID controller to avoid amplifying noise.

True

Consider the control system shown below with C(s) = K and P(s) = 1/(s + a) where a = 17. What should the gain K be so that the closed loop pole is s = -45?

IMAGE

T(s) = G(s)/(1+G(s)) = (K/(s+17))/(1+(K/(s+17))) = K/(s + 17 + K)

s = -(17 + K)

-45 = -(17 + 45) -> K = 28

Long

Long

Consider the root locus plot below obtained for the plant transfer function.

(s^2 + 22s + 40)/(s^3 - 5s^2 - 175s + 1875)

Is it possible to stabilize this system with gain K > 0

IMAGE

No

(T/F) A "root locus" is the path taken by all of the open loop poles while satisfying a particular condition.

False

For positive values of K, the root locus plot starts at the ________ ________ and ends at the ________ ________.

1. Open loop (plant)

2. Poles

3. Open loop (plant)

4. Zeros

(T/F) When using pole placement and full state feedback to design a controller, the desired closed loop eigenvalues are obtained by adjusting the values in the gain matrix

True

Consider the state space model given by

xdot = Ax + Bu

y = Cx + Du

You are to design a controller using pole placement and full state feedback. The input to the closed-loop system is an arbitrary input r(t). What is the control law? Note that in Law 1 and Law 2 there is a minus sign between the first and second term that may not show up.

Law 1: u(t) = A - BK

Law 2: u(t) = r(t) − Kx(t)

Law 3: u(t) = Kx(t) + r(t)

Law 4: y(t) = Kx(t) + r(t)

Law 2

Consider the state space model given by

xdot = Ax + Bu

y = Cx + Du

Suppose we want to design a full state feedback controller. Which of the following must be true?

a) The system must be observable

b) A matrix must be invertible

c) C = I and D = 0

d) None of the above

c) C = I and D = 0

Is the following block diagram a correct representation of an arbitrary full state feedback control system?

IMAGE

No

Consider the state space model given by

xdot = Ax + Bu

y = Cx + Du

In order to independently place all of the poles using full state feedback, the system must be

a) state observable

b) stable

c) state controllable

d) static

c) state controllable

Consider the state space model given by

xdot = Ax + Bu

y = Cx + Du

You are going to design a full state feedback controller using pole placement. The plant has 6 inputs, 6 states, and 3 outputs. How many elements are in the gain matrix, K? For example, if the K matrix is 1 x 5, it has 1 * 5 = 5 elements.

input x states = 6*6 = 36

Given the following transfer function, obtain the differential equation.

Y(s)/X(s) = (s + b)/((s + a1)(s + a2))

with b = 5, a1 = 1, a2 = 4.

What is the coefficient of y_dot(t) if the coefficient of y(dotdot)(t) = 1? Enter your answer as an integer.

a1 + a2 = 1 + 4 = 5

Consider the free body diagram below.

IMAGE

with k1 = 43, k2 = 5, c2 = 9, m2 = 16.11

Write the corresponding equation of motion assuming that downward is positive. Write the equation with the highest derivative isolated on the left hand side of the equation. What is the coefficient of x2?

k2 - k1 = 5 - 43 = -38

A linear system has a frequency response given by

M(omega) = 1 - omega^2 (magnitude)

PHI(omega) = 90 - omega^2 + 2omega (phase)

If the input to the system is x(t) = A1*sin(omegahat + phi1) where A1 = 9.5, and omegahat = 5.2 rad/sec, and phi1 = 26, what is thephase of the output, phi2, where the output is given by

y(t) = A2sin(omegahatt + phi2)

phi2 = phi1 + PHI(omegahat)

PHI(5.2) = 90 - (5.2)^2 + 2(5.2) = 90 - 27.04 + 10.4 = 73.36

phi2 = 26 + 73.36 = 99.36

99 (rounded to nearest integer)

Obtain the impulse response of the following ODE assuming zero initial conditions.

0.25xdotdot(t) + 2.75xdot(t) + 6x(t) = f(t)

Calculate the vale of the response at t = 0.125, in other words, calculate x(0.125)

x(0.125) = 0.2555

Which of the following is the correct free body diagram?

a)

[^_v_^]

[^___v___^]