UNIT-1 System Variables The system’s boundary dep ends upon the defined ob jective function of the syste m. • The system’s function is expressed in terms of measured output variables. • The system’s operation is ma nipulated through the control input variables. • The system’s operation is also affected in an uncontrolled manner through the disturbance input variables. Open-loop control system A system that utilizes a device to control the process without feedback. Thus the output has no effect upon the signal to the process Closed-loop feedback control system A system that uses a measurement of the output and compares it with the desired output Types of feedback control systems – Negative feedback - the output signal is fed back so that it subtracts (-) from input signal * – Positive feedbackExample 1: Car and Driver Objective function: to control the direction and speed of the car• Outputs: actual direction and speed of the car• Control inputs: road markings and speed signs • Disturbances: road surface and grade, wind, obstacles • Possible subsystems: the car alone, power steering system, braking system, . . . Control System Components System, plant or process (to be controlled) • Actuators (converts the control signal to a power signal) • Sensors (provides measurement of the system output) • Reference input (represents the desired output) • Error detection (forms the control error) • Controller(operates on the control error to form the
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
The system’s boundary depends upon the defined objective function of the system.• The system’s function is expressed in terms of measured output variables.• The system’s operation is manipulated through the control input variables.
• The system’s operation is also affected in an uncontrolled manner through the disturbanceinput variables.Open-loop control system
A system that utilizes a device to control the process without feedback. Thus the output hasno effect upon the signal to the processClosed-loop feedback control system
A system that uses a measurement of the output and compares it with the desired outputTypes of feedback control systems
– Negative feedback - the output signal is fed back sothat it subtracts (-) from input signal * – Positive feedback
Example 1: Car and Driver
Objective function: to control the direction and speed of the car • Outputs: actual direction and speed of the car • Control inputs: road markings and speed signs
• Disturbances: road surface and grade, wind, obstacles• Possible subsystems: the car alone, power steering system, braking system, . . .Control System Components
System, plant or process (to be controlled)• Actuators (converts the control signal to a power signal)• Sensors (provides measurement of the system output)• Reference input (represents the desired output)• Error detection (forms the control error)• Controller (operates on the control error to form the
Control System TerminologyDesired Response – the idealised instantaneous behaviour that we
would like from the system.Transient Response – the gradual change in the system as itapproaches its approximation of the desired response.Steady-State Response – the response of the system once it hasfinished changing and is now approximating the desired response.Error – the difference between the input and the output of the system.Steady-State Error – the difference between the steady-stateresponse and the desired response.Stability – the ability of the system to settle into a steady-stateresponse.Controller – the part of the system that generates the input to the
plant or process being controlled.Open-Loop – a system that does not monitor its output. Open-loopsystems can not correct for disturbances.Closed-Loop – a system that monitors its output and makescorrections to reduce error. By monitoring the output the system cancorrect for disturbances.Disturbance – a signal that is not modelled or calibrated in thesystem leading to corruption of the expected behaviour.Compensator – a system inserted into the controller to improveperformance.Feedback – a path that allows signals from the output of some sub-
system to flow back and affect the input of some sub-system earlier inthe system signal path.Robust – a system that will still work as expected with changes to thesystem parameters, as might be caused by wear of components, or achange in behaviour with temperature.
• In the s-domain a function is plotted on the imaginary axis (y) and the real axis (x)
• Roots of q(s) give poles• Roots of p(s) give zeros
-6 -5 -4 -3 -2 -1 0 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Pole-ZeroMap
Real Axis
I m a g i n a r y A x i s
Steady-state error is defined as the difference between the input and output of a systemin the limit as time goes to infinity (i.e. when the response has reached the steady state).The steady-state error will depend on the type of input (step, ramp, etc) as well as thesystem type (0, I, or II).
Note: Steady-state error analysis is only useful for stable systems. It is your
responsibility to check the system for stability before performing a steady-state
error analysis. Many of the techniques that we present will give an answer even if
the system is unstable; obviously this answer is meaningless for an unstable system.
Calculating steady-state errors
Before talking about the relationships between steady-state error and system type, we willshow how to calculate error regardless of system type or input. Then, we will startderiving formulas we will apply when we perform a steady state-error analysis. Steady-state error can be calculated from the open or closed-loop transfer function for unityfeedback systems. For example, let's say that we have the following system:
We can calculate the steady state error for this system from either the open or closed-looptransfer function using the final value theorem (remember that this theorem can only beapplied if the denominator has no poles in the right-half plane):
Now, let's plug in the Laplace transforms for different inputs and find equations tocalculate steady-state errors from open-loop transfer functions given different inputs:
• Step Input (R(s) = 1/s):
• Ramp Input (R(s) = 1/s^2):
• Parabolic Input (R(s) = 1/s^3):
When we design a controller, we usually want to compensate for disturbances to asystem. Let's say that we have the following system with a disturbance:
we can find the steady-state error for a step disturbance input with the followingequation:
Lastly, we can calculate steady-state error for non-unity feedback systems:
By manipulating the blocks, we can model the system as follows:
Now, simply apply the equations we talked about above.
System type and steady-state error
If you refer back to the equations for calculating steady-state errors for unity feedback systems, you will find that we have defined certain constants ( known as the static error constants). These constants are the position constant (Kp), the velocity constant (Kv), andthe acceleration constant (Ka). Knowing the value of these constants as well as thesystem type, we can predict if our system is going to have a finite steady-state error.
First, let's talk about system type. The system type is defined as the number of pureintegrators in a system. That is, the system type is equal to the value of n when the systemis represented as in the following figure:
UNIT-3• The steady-state response of a linear system to sinusoidal excitation may be
determined from the system transfer function T(s) by replacing s by jω
• The resulting frequency response function T(jω ) has the polar form :
• For any value of ω , T(jω ) reduces to a complex number whose amplitude,A(jω ) gives , and φ (ω ) gives .
\
Steady-state vs. general response• In control work. the response of the system to general input functions, e.g. step
input, impulse input, are of most interest (rather than steady state sinusoidalreponse)
• However, with Fourier analysis, any time function can be represented as a seriessum of sinusoids :
Therefore it is possible to predict the nature of a response, y(t) to any given x(t)from a knowledge of how the gain, A, and phase shift, φ , vary with angular frequency
• In particular, the closed-loop behaviour of a feedback system may be usefully predicted from the open-loop gain and phase-shift characteristics
Open loop response : Bode plots
T(s)
X(s ) Y(s)
)()()( ω φ ω ω j
e A jT =
)(
)(
s X
sY
)(
)(arg
s X
sY
22
2)(2
++
=
s s sT
......)2()()( 22110 +−+−+= φ ω α φ ω α α t Sint Sint x
• Amplitude is plotted in dB• Phase is plotted in degrees• Used to identify system order, corner frequencies, etc
Open loop response : Polar plots
• Polar plots are more useful in control system design work – open-loop polar plot indicates whether the closed-loop system is stable– also gives a measure of how stable the system is
• Represent T(jw) on an Argand diagram :
• Polar plot is traced out by the trajectory of T(jw) as w increases progressivelyfrom zero