What is a transfer function?

Ratio of the output variable to the input variable when laplaced transformed.

What is the force displacement equation for a spring

What is the force displacement equation for a damper?

What is the force displacement equation for a mass?

What is the torque angular displacement equation for a spring?

What is the torque angular displacement equation for a damper?

What is the torque angular displacement equation for interia?

How do you draw a block diagram?

- Block diagrams are formed by blocks (devices) and directional lines (signals/parameters)
- First, identify the output of the control system
- Then identify devices and parameters from the devices
- Finally draw a block diagram to show how the system work.

What is the Laplace function of a 1st order differential, f’(t)?

What is the Laplace transfer function of a second order differential, f’‘(t)?

What is a useful rule with impedances and gears?

What is Kirchhoff’s current law?

The total current flowing towards a junction is equal to the total current flowing from that junction

What is Kirchoff’s voltage law?

In a closed circuit or loop, the algebraic sum of the voltages across each part of the circuit is equal to the applied voltage

What are the voltage current and current voltage characteristics of a capacitor?

What are the voltage current and current voltage characteristics of a resistor?

What are the voltage current and current voltage characteristics of an inductor?

How can you eliminate the feedback loop with transfer functions?

How would you combine these?

Genrally, what is the transfer function of a first order system?

What is the time constant for a first order system?

1/a

What is T_{r} of a first order system?

2.2/a

What is the form of a second order system?

T=G/(1+G)

What is the natural frequency of a second order system?

ω_{n}=k^{0.5}

What is the damping ratio of a second order system?

ζ=p/2ω_{n}

p is from the characteristic equation. s^{2}+ps+k=0

What is the damped frequency of a second order system?

ω_{d}=ω_{n}(1-ζ^{2})^{0.5}

What do the different values of the damping ratio tell you?

ζ<1 means underdamped

ζ=1 means criticaly damped

ζ>1 means over damped.

What is the laplace transform of an integral function?

What is the settling time, T_{s}?

The time to stay within 2% of the final value

T_{s}=4/ςω_{n}

How would you find the transfer function of this diagram?

T=G/(1+GH)

What is the laplace input of an impluse function?

R(s)=1/s

What is the final value theorem?

How do you find the steady state error?

Genrally E(s)=R(s)-Y(s)

For a unity feedback system, E(s)=R(s)/1+G(s)

How is system type defined?

By the number of zero poles

If the denomerator is (s+8)(s-3) it is 0 type

s(s+8)(s-3) is 1 type

s^{2}(s+8)(s-3) is two type

What are the static error constants?

K_{p}=lim G(s)

Kv=lim sG(s)

Ka=lim s^{2}G(s)

as s approaches 0

What is the ramp input?

r(t)=t

The LaPlace transform of t is s^{-2}

What are the steady state erros for step, ramp and parabloic inputs in terms of the static error constants?

step: (1+k_{p})^{-1}

ramp: k_{v}^{-1}

Parabola: k_{a}^{-1}

What is Tr?

The rise time. The time needed to go from 0.1 to 0.9 of the original value

How do you find the transfer function of second order system?

How can you find the position of a pole on the pole plot plane?

cos(θ)=ς

Always a distance ω_{n} from the origin.

What does a phase lag compensator look like and what is its function?

Increasing K reduces steady state error

Increasing the phase margin reduces the overshoot

When desiging a phase lag compensator, what is the formula for α?

α=z/p

What equation relates damping ratio and phase margin?

ζ=0.01 Φ_{mp}

When designing a phase lag compensator, what is the cross over frequency?

ω_{c}’ is read off a bode diagram. It is the frequency where the top line should cross 0

When designing a phase lag compensator, what is the attenuation?

What is the relalation to the phase margin?

The amount the top graph must move to meet 0 at the desired cross over frequency. Read off a bode diagram.

M_{a}=20log_{10}(α)

α is the phase margin.

When designing a phase lag compenstor, how do you find z?

z=ω_{c}‘/10

What does a nyquist plot look like and how is it used to determine stability?

let N be the number of clockwise encirclement of (-1,0) by Nyquist plot, and P the number of poles of GH(s) on the right half plane.

The closed system T(s) is stable if Z = N + P = 0.

What are the step, ramp and parabola inputs?

Hoe do you draw a Routh array?

When using a Routh array, what are the required conditions for stability?

Every element in the first column must be positive.

Otherwise, the number of sign changes in the first column is equal to the number of poles in the right hand side.

How do you find phase margin and gain margin?

What is the special case with routh diagrams and a 0 in the first column?

What is the special case with a routh array and an entire row of 0’s?

What is the gain margin?

The gain margin is the change in open-loop gain, expressed in decibels (dB), required at 1800 of phase shift to make the closed-loop system unstable.

Suggest all the possible ways a compensator can be physically implemented.

It can be implemented in one of the following ways: (a) an electric circuit with capacitors, resistors and inductors; (b) a circuit with a microprocessor or a computer, ADC and DAC. Power amplifier may be required.

What are dominant poles?

The dominate poles of a system are the poles which influence the system responses most heavily, i.e. dominate the system response. They are often complex poles.

Does the transient response always peak?

No. It only will if there are complex poles. Critcaly and overdamped systems don’t peak and only have real roots.

If two complex poles and a third pole exist in a third order system, how will the third pole affect the system response?

The extent of the influence from the third pole is related to the distance it is from the complex poles.

- If the third pole is close to the complex poles, it will influence the system response heavily.
- If the real part of the third pole is a few times of the real part of the complex poles, the influence will decay rapidly.
- If it is 5 or 10 times, the influence of the third pole can be ignored.

Describe in your words what Nyquist absolute stability criteria is;

Nyquist (absolute) stability criteria: if the Nyquist contour of GH(s) encircles (-1,0) in clockwise direction the same times as the number of poles of GH(s), then the system is stable. Otherwise, it is not stable.

What is relative Nyquist stability?

How does a phase lag compensator work?