First and second order systems

Question 1

What is linearisation? Why is it useful?

Answer 1

Approximating a curve by its slope at a point. Useful for turning non-linear systems into linear systems for analysis. Good approximation for close values, terrible for further values.

Question 2

What is stability?

Answer 2

A system is stable if it converges to a finite value when there is no control input (i.e. u(t) = 0)

Question 3

How can you tell if a second order system is stable?

Answer 3

The system is stable if a and b are both positive (in s^2 + as +b)

Question 4

What do the poles of a second order system tell us about stability?

Answer 4

Stable if the poles have negative real parts.
Unstable if the poles have positive real parts (or if they are two repeated zero roots).
Marginally stable if the two poles are purely imaginary.

Question 5

What is rise time? (+ rule of thumb for wn and a)?

Answer 5

The time taken to move from 10% to 90% of the steady state response.
tr = 1.8/wn
tr = 2.2/a

Question 6

What is settling time?

Answer 6

The time it takes for the system response to reach a 2% band around its steady state and stay there.

Question 7

In the complex plane, what does the natural frequency represent?

Answer 7

The magnitude of the distance to the pole locations.

Question 8

In the complex plane, what does the damping factor represent?

Answer 8

The angle between the pole location and the imaginary axis is the inverse sine of the damping factor.

Question 9

How is the damping factor related to the overshoot of the system?

Answer 9

Overshoot is ONLY related to the damping factor! Formula is Mp = e^(-pi*zeta)/(sqr(1-zeta^2))

Question 10

What are some rules of thumb relating zeta to the overshoot of a system?

Answer 10

Mp = 16% -- zeta = 0.5
Mp = 10% -- zeta = 0.6
Mp = 5% -- zeta = 0.7

Question 11

What is the formula for settling time?

Answer 11

ts = 4/wn*zeta

Question 12

What is overshoot?

Answer 12

The maximum amount that the system overshoots its final value in terms of ratio, usually expressed by percentages.

Question 13

What is the peak time (and formula)?

Answer 13

The peak time is the time when the response reaches maximum value. tp = pi/wd

Question 14

How do wd and sigma relate to the complex poles?

Answer 14

The equation:
x_ddot + ax_dot + bx = cu
yields two stable complex poles:
p1 = -sigma + jwd
p2 = -sigma -jwd 
Where sigma = a/2 and wd = sqrt(4b-a^2)/2

Question 15

How does sigma/wd relate to wn/zeta?

Answer 15

wn = sqrt(sigma^2+wd^2)
zeta = sigma/wn

Question 16

How would you reduce ts?

Answer 16

Increase sigma

Move poles further left in complex plane

Question 17

How would you reduce tr?

Answer 17

Increase wn

Question 18

How would you reduce Mp?

Answer 18

Increase zeta

Question 19

How would you reduce tp?

Answer 19

Increase wd

Question 20

What do each of the components of PD control mean?

Answer 20

Proportion of the error

Derivative of the error

Question 21

What is the time domain representation of PD control?

Answer 21

u = Kp(r - x) + Kd(r_dot - x_dot)

Question 22

What is the frequency domain representation of PD control?

Answer 22

u = Kp + Kds

Question 23

Why does PD control introduce a zero into the system?

Answer 23

Kp + Kds on numerator of transfer function

Question 24

What is a system zero?

Answer 24

The value at which G(s) = 0.

Question 25

What is the effect of a system zero?

Answer 25

For the zero b_1s + b_0; if b_1 is greater than zero and large compared to b_0 > 0, then any sudden changes in u will be amplified by the u_dot term - the system will receive an additional kick when it changes from 0 to 1 (zero in LHP). Now if b_1 < 0 and b_0 > 0, then the initial kick is in the opposite direction to the steady state response (zero in RHP)

Question 26

What is a non-minimum phase system?

Answer 26

A system with a zero in the right half of the complex plane.

Question 27

What is the general solution of bx_dot + ax = u?

Answer 27

ae^-kt + b ???

Question 28

How can you make an estimate for % overshoot from a transfer function?

Answer 28

Using 2nd order assumptions to say that the % overshoot will be determined by the dominant complex poles - i.e. those that are closest to the Im axis (since smaller zeta = larger overshoot).

Question 29

What is the small angle theorem?

Answer 29

For small angles sin(theta) = theta, cos(theta) = 1, tan(theta) = theta.

Question 30

What is the quadratic formula?

Answer 30

-b+/-sqrt(b^2 - 4ac) all over 2a

Question 31

What are the 5 steps in linearisation?

Answer 31

1. Coordinate change - move coordinates so equilibrium point is at zero 2. Write ODE in new coordinates (*) 3. Approximate something using first order Taylor expansion 4. Sub back into * 5. At equilibrium all deltas are 0