Engineering Control Theory Flashcards
The set of real numbers are denoted by…
ℝ
Complex numbers are denoted by…
ℂ
Using the engineering convention, what is Euler’s formula?
e^(jθ) = cosθ + jsinθ
What does it mean if a rational function is said to be ‘proper’?
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).
What is a ‘black box’?
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)
What does it mean for a black box to be ‘casual’?
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!
What is the ‘homogeneity property’ of black boxes?
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
What is the ‘superposition property’ of black boxes?
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).
What does it mean if a black box is ‘linear’?
A black box is linear if it satisfies both homogeneity and superposition properties.
What does it mean if a black box is ‘time-invariant’?
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 )
What is a ‘proportional’ black box?
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.
What is a ‘differentiator’ black box?
A differentiator is the black box that takes input u (t) and gives output ̇u (t) = du/dt
What is an ‘integrator’ black box?
An integrator is the black box that takes input u (t) and gives output ∫ (t,0) u (τ ) dτ
What is a ‘delay’ black box?
A delay takes input u (t) and gives the output u (t − T ) for a fixed time delay T > 0
What is a ‘squarer’ black box?
A squarer takes input u (t) and outputs u (t)^2
What is the ‘inverse’ of a black box?
If we have a black box u → y then its inverse is the black box y → u.
Define the term ‘pole’ of a transfer function G(s).
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)
Define the term ‘zero’ of a transfer function G(s).
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)
What is the Laplace transform of f (t)?
The Laplace transform of f (t) is:
F (s) =∫(∞,0) e^−st f (t) dt
where s is allowed to be complex.
What is the Initial Value Theorem?
IVT:
f (0) =
lim sF (s)
s→∞
State the (Hurwitz) condition for a transfer function to describe a stable linear time invariant system.
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).
What is the Final Value Theorem?
FVT:
f (∞) =
lim sF (s)
s→0+
What is the power of a signal proportional to?
The amplitude squared.
What is the equation for decibels?
D(ω) = 20log(10) g(ω) dB
Where g(ω) is the amplitude
What is impedance?
Impedance, represented by the symbol Z, is a measure of the opposition to electrical flow. It is measured in ohms.
A circuit consists of a capacitor and an inductor in series. What is its impedance function Z(s)?
Z(s) = R, 1/sC, sL respectively
State the form of the amplitude gain for the steady state output if we input a harmonic signal of frequency ω.
a(ω) = |G(jω)|
State the form of the phase change for the steady state output if we input a harmonic signal of frequency ω.
θ(ω) = arg G(jω)
What is BIBO stability?
An input-output model is stable if
every bounded input leads to a bounded output.
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)?
L{˙f(t) } = sF(s) - f(0)
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 ?
L{ ∫(t,0) f(τ) } = F(s) / s
