Control Systems

Question 1

transfer function

Answer 1

used to represent systems with one input function and one output function

Question 2

feedback system

Answer 2

• output signal is returned as input in a feedback loop
• basic system consists of
• -> dynamic unit
• -> feedback unit
• -> pick-off pint
• -> summing point

Question 3

Closed- Loop Feedback Systems

Answer 3

X(s) is the input transfer function
Y(s) is the output transfer function
The overall transfer function (also called closed-loop gain or control ratio)

T(s) = Y(s)/X(s) is the transfer function of the overall feedback system

Question 4

Closed Loop Feedback Systems continued

Answer 4

Each unit has an associated transfer function:
G(s) is the forward transfer function
H(s) is the reverse transfer function (or feedback gain). When H(s)=1 it is called unity feedback

The loop transfer function (also called loop gain or open loop gain) +G(s)H(s) is the gain after going around the loop one time

Question 5

block diagram

Answer 5

• can represent a physical system
• with transfer functions, can be used to describe and clarify complex relationships of causes and effects throughout a dynamic system
• can be simplified into a single block operation

In a block diagram,

• interconnected black boxes (cascaded blocks) represent functions
• each block represents and describes a portion of the system

Question 6

block diagram of a linear system

Answer 6

typically consists of four components

• signal
• block (with transfer function)
• summing point
• pick-off-point

Question 7

simplifying block diagrams

Answer 7

a block diagram can be simplified using a mathematical system of rules known as a block diagram algebra

Question 8

unity feedback system

Answer 8

• feedback loop has value of H(s) = 1
• open-loop transfer function is G(s)H(s)=G(s)
• unstable if feedback is positive

Question 9

for a step input

Answer 9

• often called DC input, because it is like switching a DC source on or off
• steady-state response obtained by substituting zero for s everywhere in transfer function, then multiplying by magnitude of step

Question 10

for a sinusoidal input

Answer 10

• steady-state response obtained by substituting jw for s everywhere in transfer function T(s) <–> T(jw)
• output has same frequency as input
• transforms Laplace transform transfer function into the frequency domain, which is physically meaningful
• frequency domain analyzed with Bode diagrams

Question 11

steady-state error, ess

Answer 11

• determine long-term performance of a system
• values will always be zero, infinity or a constant
• ideally errors e(t) and ess both equal zero
• always nonzero for pure gain sytems
• can be near zero for integrating
• depends on type of input R(s) and system type
• found from the final value theorem

Question 12

pole of a transfer function

Answer 12

• a value of s that makes the denominator of G(s) zero (pole values are system eigenvalues)
• therefore makes G(s) infinite

Question 13

zero of a transfer function

Answer 13

• a value of s that makes the numerator of G(s) zero

- therefore makes G(s) zero

Question 14

poles and zeros

Answer 14

• need not be real or unique

- can be imaginary and repeated within function

Question 15

pole-zero diagram

Answer 15

a plot of poles and zeros in the s-plane, a rectangular coordinate system with real and imaginary axes

Question 16

Bode Plot

Answer 16

• a graphical representation of the response of a control system in the frequency domain
• gives a visual way to assess control system stability
• if the input is a steady-state sinusoid at frequency w, the gain plot shows the gain of the system

Question 17

horizontal axis

Answer 17

logarithmic, allowing many orders of magnitude in a single plot

Question 18

magnitude (vertical axis)

Answer 18

plotted in decibels, allowing many orders of magnitude in a single plot
–> terms that are products can be added, and terms that are quotients can be subtracted

Question 19

single zero plot

Answer 19

this is normalized plot of log (w/z) versus 20 log I 1+jw/z I

Question 20

root-locus diagram

Answer 20

a plot in the s-plane that shows the location of the poles of the closed-loop transfer function for all values of the system gain constant, K, if the open-loop transfer function G(s) H(s) is known

Question 21

asymptotes

Answer 21

• the number of asymptotes is the number of poles, n, minus the number of zeros, m

Question 22

position-integral-derivative (PID) controller

Answer 22

can control the output a system based on a combination of

• position of the input
• integral of the input
• derivative of the input

Question 23

lead or lag compensator

Answer 23

• common control component placed in a feedback circuit
• improves frequency response of control system
• also used to
• -> reduce stead-state error
• -> reduce resonant peaks
• -> improve system response by reducing rise time

Question 24

Routh criterion

Answer 24

• the poles of a transfer function determine whether the control system is stable
• for the system to be stable, poles must be in the negative real section of the s-plane

Question 25

Routh test

Answer 25

an efficient method of testing the denominator of a closed-loop transfer function for real positive poles

Question 26

state variable models

Answer 26

- a system with an nth order differential equation can be represented as n first-order differential equations in a matrix-vector form - state variable models use linear algebra to solve differential equations and so are useful for computer solutions - a state variable can be used for a control system, expressing input and output as separate equations using a dummy variable