Exam 2

Question 1

The zeros of the characteristic equation change as ________ changes

Answer 1

K

Lecture 13: Routh Stability Criterion

Question 2

Routh Stability Criterion (definition)

Answer 2

A method of obtaining information about the location of the roots of a polynomial without actually solving for the roots

Lecture 13: Routh Stability Criterion

Question 3

What are the tests for Routh Stability?

Answer 3

1. Are all of the coefficients of the characteristic polynomial positive (and present)?
2. Are all the elements in the 1st column of the Routh array POSITIVE?

Lecture 13: Routh Stability Criterion

Question 4

In a Routh Array, ________________ gives the number of poles of the characteristic equation that are on the right side of the S-plane.

Answer 4

The number of sign changes of the elements in the 1st column

Lecture 13: Routh Stability Criterion

Question 5

Steps to analyzing system using a Routh Array

Answer 5

1. Find the transfer function of the closed loop system
2. Re-arrange characteristic equation to the form:
   1s^n + a_1s^(n-1) + … + a_n-1s^1+a_ns^0
3. Apply test 1
4. Form Routh Array
5. Apply test 2

Lecture 13: Routh Stability Criterion

Question 6

Two assumptions for this class

Answer 6

1. The plant and controller are LTI systems
2. Single input - single output (SISO) systems

Lecture 14: Basic Eqn.s of Control

Question 7

Basic concerns of controls engineers

Answer 7

1. Stability
2. Tracking
3. Regulation
4. Sensitivity

Our mission is to create an appropriate control signal to drive the system

Lecture 14: Basic Eqn.s of Control

Question 8

Types of control

Answer 8

1. Tracking: cause the output to follow the reference as closely as possible
2. Regulation: keep the error as small as possible

Lecture 15: Conflicting Tradeoffs

Question 9

What is the purpose of the Root Locus?

Answer 9

Determines

1. The values of K at which the system is stable
2. How the time response changes as K changes

AKA it is a guide to how poles change as K changes

Lecture 15: Conflicting Tradeoffs

Question 10

The Root Locus can be thought of as…

Answer 10

A method for inferring the dynamic properties of the closed loop system a K changes

Question 11

The value of s is a closed-loop pole of a closed loop system if:

Answer 11

1. | KG(s)H(s) | = 1
2. < KG(s)H(s) = (2k+1)180deg

Lecture 15: Conflicting Tradeoffs

Question 12

Characteristic polynomial form for using Root Locus

Answer 12

1+KL(s)
where L(s) = b(s)/a(s) for a(s)+K*b(s)

Lecture 16: Root Locus

Question 13

Root Locus Rules

Answer 13

1. The n branches of the locus start at the poles of L(s) when K=0, and the m branches end on the zeros of L(s) when K = infinity.
2. The loci are on the real axis to the left of an odd number of poles and zeros
3. For large s and K, n-m branches of the loci are asymptotic to lines at angles p radiating out from the center point s = alpha on the real axis
   p = (180 + 360(l-1)) / (n-m), l = 1, 2, …, n-m
   anchor point = alpha = [sum(pi) - sum(zi)] / (n-m)

Lecture 16: Root Locus

Question 14

For the Root Locus, what does changing K mean?

Answer 14

1. The poles change
2. Consequently, the step response (i.e. time response) changes

Lecture 16: Root Locus

Question 15

Integral Controller Expression

Answer 15

Ki/s

Lecture 17: PI Controller

Question 16

What is the purpose of an integral controller?

Answer 16

Minimize the steady-state tracking error and the steady-state output response to disturbances
AKA: Remove the steady state error

Lecture 17: PI Controller

Question 17

When should you use a PD controller?

Answer 17

When the root locus does not fit the time domain response you need

Lecture 18: PD Controller

Question 18

Derivative Controller Expression

Answer 18

Kd*s

Lecture 18: PD Controller

Question 19

What is the goal of derivative feedback?

3 things

Answer 19

1. Improve closed loop system stability
2. Speed up the transient response
3. Reduce overshoot

Lecture 18: PD Controller

Question 20

Control Law for PD Controller

Answer 20

Trend of error:
U(t) = Kd*de(t)/dt
Increasing control amplifies error/noise

Lecture 18: PD Controller

Question 21

An extra zero may….

Answer 21

Increase the overshoot of the step response by a little

Lecture 18: PD Controller

Question 22

To reduce the steady state error to constant reference and disturbance inputs use a(n) ___________ controller

Answer 22

Integral

Question 23

To speed up the transient response and reduce overshoot use this type of controller:

Answer 23

Derivative

Question 24

“anticipatory term”

Answer 24

Derivative

Question 25

Gain Margin, GM

Answer 25

The change in the open-loop gain required at 180deg phase shift to make the closed loop system unstable
Expressed in dB

Question 26

Phase Margin, PM

Answer 26

The change in open-loop phase shift required at unity gain to make the closed-loop system unstable

Question 27

Stability: Typically an open-loop system is stable in closed-loop if…

Answer 27

The open-loop magnitude frequency response has a gain of less than 0dB at the frequency where the phase frequency response is 180

Question 28

Percent overshoot is reduced by…

Answer 28

Increasing the phase margin

Question 29

Speed of Response is increased by….

Answer 29

Increasing the bandwidth

Question 30

Steady-state error is improved by

Answer 30

Increasing the low-frequency magnitude response (even if the high-f response is attenuated)

Question 31

A _____ PM is required for stability

Answer 31

positive

Question 32

How to find GM

Answer 32

1. Find where phase crosses -180deg
2. Follow the frequency up to magnitude plot
3. GM = magnitude at that point-0dB; log(w)