Engineering Control Theory

Question 1

The set of real numbers are denoted by…

Answer 1

ℝ

Question 2

Complex numbers are denoted by…

Answer 2

ℂ

Question 3

Using the engineering convention, what is Euler’s formula?

Answer 3

e^(jθ) = cosθ + jsinθ

Question 4

What does it mean if a rational function is said to be ‘proper’?

Answer 4

A rational function is said to be proper if the denominator has a degree greater than or equal to that of the numerator (n ≥ m).

Question 5

What is a ‘black box’?

Answer 5

A black box is an input-output system which converts an input function u = u(t) into an output function y = y(t).

u (t) → y (t)

Question 6

What does it mean for a black box to be ‘casual’?

Answer 6

The black box is causal if

the output, y (t), at any time t depends at most on the values u (τ) of the input up to that time (τ ≤ t)

• but not on the future values!

Question 7

What is the ‘homogeneity property’ of black boxes?

Answer 7

An input-output system is said to have the homogeneity property if,

whenever u (t) →y (t), we have a u (t) → a y (t) for any scalar a

Question 8

What is the ‘superposition property’ of black boxes?

Answer 8

An input-output system is said to have the superposition property if whenever we have

u1 (t) →y1 (t) and u2 (t) → y2 (t)

then u1 (t) + u2 (t) → y1 (t) + y2 (t).

Question 9

What does it mean if a black box is ‘linear’?

Answer 9

A black box is linear if it satisfies both homogeneity and superposition properties.

Question 10

What does it mean if a black box is ‘time-invariant’?

Answer 10

An input-output system is said to be time-invariant if shifting time does not affect the input-output relations.

That is, if u (t) → y (t) then we have u (t − τ ) → y (t − τ )

Question 11

What is a ‘proportional’ black box?

Answer 11

For each constant k, a proportional black box
turns u (t) into k u (t),
where k is called the gain.

If |k| > 1 it is called an amplifier.

If |k| < 1 it is called an attenuator.

Question 12

What is a ‘differentiator’ black box?

Answer 12

A differentiator is the black box that takes input u (t) and gives output ̇u (t) = du/dt

Question 13

What is an ‘integrator’ black box?

Answer 13

An integrator is the black box that takes input u (t) and gives output ∫ (t,0) u (τ ) dτ

Question 14

What is a ‘delay’ black box?

Answer 14

A delay takes input u (t) and gives the output u (t − T ) for a fixed time delay T > 0

Question 15

What is a ‘squarer’ black box?

Answer 15

A squarer takes input u (t) and outputs u (t)^2

Question 16

What is the ‘inverse’ of a black box?

Answer 16

If we have a black box u → y then its inverse is the black box y → u.

Question 17

Define the term ‘pole’ of a transfer function G(s).

Answer 17

A complex number p in the s-plane is called a pole of a transfer function G if we have:

lim|G(s)| = ∞
s→p

(Bottom of transfer function)

Question 18

Define the term ‘zero’ of a transfer function G(s).

Answer 18

A complex number z in the s-plane is called a zero of a transfer function G if we have:

lim |G(s)| = 0.
s→z

(Top)

Question 19

What is the Laplace transform of f (t)?

Answer 19

The Laplace transform of f (t) is:

F (s) =∫(∞,0) e^−st f (t) dt

where s is allowed to be complex.

Question 20

What is the Initial Value Theorem?

Answer 20

IVT:

f (0) =

lim sF (s)
s→∞

Question 21

State the (Hurwitz) condition for a transfer function to describe a stable linear time invariant system.

Answer 21

The Hurwitz condition for stability of a transfer function of a linear time invariant system

is that all the poles have strictly negative real part

(i.e. lie in the left hand complex plane).

Question 22

What is the Final Value Theorem?

Answer 22

FVT:

f (∞) =

lim sF (s)
s→0+

Question 23

What is the power of a signal proportional to?

Answer 23

The amplitude squared.

Question 24

What is the equation for decibels?

Answer 24

D(ω) = 20log(10) g(ω) dB

Where g(ω) is the amplitude

Question 25

What is impedance?

Answer 25

Impedance, represented by the symbol Z, is a measure of the opposition to electrical flow. It is measured in ohms.

Question 26

A circuit consists of a capacitor and an inductor in series. What is its impedance function Z(s)?

Answer 26

Z(s) = R, 1/sC, sL respectively

Question 27

State the form of the amplitude gain for the steady state output if we input a harmonic signal of frequency ω.

Answer 27

a(ω) = |G(jω)|

Question 28

State the form of the phase change for the steady state output if we input a harmonic signal of frequency ω.

Answer 28

θ(ω) = arg G(jω)

Question 29

What is BIBO stability?

Answer 29

An input-output model is stable if

every bounded input leads to a bounded output.

Question 30

Let f(t) be defined for t ≥ 0, and let F(s) be its Laplace transform. What is the Laplace transforms of its derivative ˙f(t)?

Answer 30

L{˙f(t) } = sF(s) - f(0)

Question 31

Let f(t) be defined for t ≥ 0, and let F(s) be its Laplace transform. What is the Laplace transforms of its integral ∫(t,0) f(τ)dτ ?

Answer 31

L{ ∫(t,0) f(τ) } = F(s) / s