Control Systems Flashcards
transfer function
used to represent systems with one input function and one output function
feedback system
- output signal is returned as input in a feedback loop
- basic system consists of
- -> dynamic unit
- -> feedback unit
- -> pick-off pint
- -> summing point
Closed- Loop Feedback Systems
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
Closed Loop Feedback Systems continued
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
block diagram
- 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
block diagram of a linear system
typically consists of four components
- signal
- block (with transfer function)
- summing point
- pick-off-point
simplifying block diagrams
a block diagram can be simplified using a mathematical system of rules known as a block diagram algebra
unity feedback system
- feedback loop has value of H(s) = 1
- open-loop transfer function is G(s)H(s)=G(s)
- unstable if feedback is positive
for a step input
- 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
for a sinusoidal input
- 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
steady-state error, ess
- 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
pole of a transfer function
- a value of s that makes the denominator of G(s) zero (pole values are system eigenvalues)
- therefore makes G(s) infinite
zero of a transfer function
- a value of s that makes the numerator of G(s) zero
- therefore makes G(s) zero
poles and zeros
- need not be real or unique
- can be imaginary and repeated within function
pole-zero diagram
a plot of poles and zeros in the s-plane, a rectangular coordinate system with real and imaginary axes
Bode Plot
- 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
horizontal axis
logarithmic, allowing many orders of magnitude in a single plot
magnitude (vertical axis)
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
single zero plot
this is normalized plot of log (w/z) versus 20 log I 1+jw/z I
root-locus diagram
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
asymptotes
- the number of asymptotes is the number of poles, n, minus the number of zeros, m
position-integral-derivative (PID) controller
can control the output a system based on a combination of
- position of the input
- integral of the input
- derivative of the input
lead or lag compensator
- 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
Routh criterion
- 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
Routh test
an efficient method of testing the denominator of a closed-loop transfer function for real positive poles
state variable models
- 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
