PID Controller -A PID controller is an instrument used in industrial control applications to regulate temperature, flow, pressure, speed and other process variables.PID control uses closed-loop control feedback to keep the actual output from a process as close to the target or setpoint output as possible.
The PI, or Proportional and Integral, controller is a commonly used method in control systems to correct for error between the commanded set point and the actual value based on some type of feedback.
Bode Plot is method in which logarithmic value of magnitude are to be plotted against logarithmic values of frequencies. Logarithmic plots makes it possible to plot wide range of variations on a single paper
Stability from Bode Plot - Marginally Stable system - GM PM
Gain Cross over frequency(ωgc) < Phase cross over frequency (ωpc)
ωgc < ωpc, Gain margin and Phase margin are positive, stable system
ωgc > ωpc, G.M. and P.M. negative, unstable system
ωgc = ωpc, G.M. and P.M. zero, marginally stable system
Gain Cross over frequency(ωgc): The frequency at which magnitude of |G(jω)H(jω)| is unity or 0dB
Phase cross over frequency (ωpc): The frequency at which the phase angle of G(jω)H(jω) is −180◦
Gain Margin(G.M.): It is the gain reciprocal of the magnitude |G(jω)H(jω)| at the frequency at which the phase
angle is −180◦ or at phase cross over frequency ωpc.
the gain margin indicates how much the gain can be increased before the system becomes unstable.
G.M. = −20 log10 |G(jω)H(jω)|
ω=ωpc
phase margin (P.M.): The amount of phase lag at gain cross over frequency,which can be introduced in the system till system reaches on the verge of instability
P.M. = 180◦ + ∠(G(jω)H(jω))
ω=ωgc
Stability? A system is said to be stable, if its output is under control. Otherwise, it is said to be unstable. A stable system produces a bounded output for a given bounded input
A system is said to be marginally stable, if it produces the output with the constant amplitude and constant frequency of oscillations.
Frequency response is the steady-state response of a system to a sinusoidal input. In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response
Difference between time domain and frequency domain
time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies
The Frequency Domain refers to the analytic space in which mathematical functions or signals are conveyed in terms of frequency, rather than time.
Parameters for frequency domain specifications are
Resonant Peak Mr
Resonant frequency ωr
Cut Off frequency ωc
Bandwidth BW
Frequency response of second order system shows a peak called resonant peak Mr and corresponding frequency is
called resonant frequency ωr
Resonant peak exists only for ζ < 0.707
cut off frequency (ωc) : The frequency at which the magnitude of the closed loop frequency response is 3 dB below .its zero frequency value is called the cut off frequency (ωc)
The closed loop system filters out the signal components whose frequencies are greater then the cut off frequency and
transmits those signal components with frequencies lower than the cut off frequency
Bandwidth: The range of frequency ( 0 ≤ ω ≤ ωc) in which magnitude of C(jω)/R(jω) is greater then -3dB is called the
bandwidth of the system
Control System as the term implies is a system that relates inputs and outputs of the sytem to maintain a required balance in the system
Eg. space vehicle, aircraft, robot, process control plants like refinaries, pharmaceutical inudstry,
How to proceed to design control system?
1 Mathematical model of the system
2 Analysis of the system
3 Simulation of the system
4 Realization of the system
Open Loop Control System
If in a physical system there is no automatic correction of the variation in its output, it is called an open loop control system. That is, in this type of system, sensing of the actual output and comparing of this output (through feedback) with the desired input does not take place.
Eg Automatic washing machine
traffic signal system
home heating system
Closed Loop Control System
A closed loop control system is a system where the output has an effect upon the input quantity in such a manner as to maintain the desired output value as it includes a feedback. This feedback will automatically correct the change in output due to disturbances.
Laplace Transform
The best way to convert differential equations into algebraic equations is the use of Laplace transformation.
Inverse Laplace Transform
Partial fraction
Case I - Distinct Poles
Case II - Poles of multiplicity
Case III - complex pole
SFG - Signal Flow Graph - the representation which is obtained from the equations, which shows how the signal
flows in the system or the graphical representation of the variables of a set of
linear algebraic equations representing the system is called the
signal flow graph representation
Properties
1 SFG is applicable only to linear time invariant system (Time-invariant systems are systems where the output does not depend on when an input was applied. )
2 The signal in the system flows along the branches and along the arrow heads associated with the branches
3 The signal get multiplied by the branch gain or branch transmittance when it travels along it
4 The value of variable represented by any node is an algebraic
sum of all the signals entering at the node
Methods of obtaining sfg
From the system equation
From the given block diagram
From the system
Represent each variable by a separate node
Coefficients of the variables in the equations are to be
represented as the branch gains, joining the nodes in signal
flow graph
4 Show the input and output variables separately to complete
signal flow graph
From block diagram
Name all the summing points and take off points in the block diagram
2 Represent each summing and take off point by a separate node in SFG
3 Connect them by the branches instead of blocks, indicating block transfer functions as the gains of the corresponding branches
4 Show the input and output nodes separately if required, to
complete signal flow graph
Mason’s Gain Formula
Transient and Steady State Response
Need for analysis of Response of a system
1 In analyzing and designing control system, we must have a basis of comparison of performance of various control system
2 This basis may be set up by specifying test input signals and by comparing the responses of various systems to these input signals
3 Commonly used input test signal are impulse function, step function, ramp function, acceleration function, sinusoidal function
PID Controller -A PID controller is an instrument used in industrial control applications to regulate temperature, flow, pressure, speed and other process variables.PID control uses closed-loop control feedback to keep the actual output from a process as close to the target or setpoint output as possible.
Time-domain specifications ( TDS ) include the lower and/or upper bounds of the quantities of the time response such as the first peak time, maximum peak time, rise time, maximum overshoot, maximum undershoot, setting time, and steady-state error.
The Frequency Domain refers to the analytic space in which mathematical functions or signals are conveyed in terms of frequency, rather than time.
Control loops are systems applied in various industrial applications to maintain process variables (PVs) at a desired value or set point (SP). Control loops are important for maintaining the stability of a system, and for consistently producing the desired outcome of a process.
Control loop diagram shows all elements of the process measurement and control based on a process flow diagram
The transfer function of a control system is the ratio of Laplace transform of output to that of the input while taking the initial conditions, as 0. Basically it provides a relationship between input and output of the system.
The time response of control system consists of two parts:
1 Transient response - is the response which goes from the initial state to the final state
2 Steady State Response - is the response in which the system output behaves as t approaches infinity
Transient response analysis is the most general method for computing forced dynamic response. The purpose of a transient response analysis is to determine the behavior of a structure subjected to time-varying excitation
The system response c(t) may be written as
c(t) = ctr(t) + css (t)
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively.
we can directly find the order of the transfer function by just determining the highest power of 's' in the denominator of the transfer function.
First Order System and its responses to various inputs
Tf= Y (s)/R(s) = 1/ τ s + 1
Systems with only one pole are called first order system
Examples are RC circuit, RL circuit etc.
τ is the system time constant
τ characterizes the speed of response of a system to an input
Higher the time constant, slower is the response and vice-versa
The time constant can be defined as the time it takes for the step response to rise up to 63% or 0.63 of its final value. We refer to this as t = 1/a.
Second Order System Transfer function
Y (s)/R(s) = ωn^2 / s^2 + 2ζωns + ω^2
ωn is the system natural frequency
ζ is system damping ratio
System damping ratio ζ: is a dimensionless quantity describing
the decay of oscillations during a transient response. It is the ratio of actual damping to the critical damping of the system
ζ = R/2√C/L
2 System Natural frequency ωn: it is angular frequency at which
system tends to oscillate in the absence of damping force
ωn=√1/LC
3 System Damped frequency ωd : It is anugular frequency at which the system tends to oscillate in the presence of damping force.
ωd = ωn √1 − ζ^2
Second system responses can be classified into four types
depending on value of ζ
1 Case I: ζ > 1 - Overdamped System
2 Case II: ζ = 1 - Critically Damped System
3 Case III: < ζ < 1 - Underdamped System
4 Case IV: ζ = 0 - Undamped System
1 Overdamped System - Transient in the system exponentially decays to steady state without any oscillation
2 Critically Damped System - Transients in the system decay to steady state without any oscillations in shortest possible time
3 Underdamped System - System transients oscillates with the amplitude of oscillations gradually decreasing to zero
4 Undamped System - system keeps on oscillating at its natural frequency with constant amplitude of oscillation
Need for Time Response Specifications of a system
● Time response specifications refer to the performance indices of step response of a system
● In general these indices are specified as part of design requirement of a control system
Time Response Specifications
Time-domain specifications ( TDS ) include the lower and/or upper bounds of the quantities of the time response such as the first peak time, maximum peak time, rise time, maximum overshoot, maximum undershoot, setting time, and steady-state error.
Difference between time domain and frequency domain
time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies
Delay time td: Time required for the response to reach 50%/ half of the final value at first instance
td =1 + 0.7ζ / ωn
Rise time tr: Time required for the response to rise from 10% to 90% of the final value for overdamped systems and 0% to 100% of the final value for underdamped systems, at first
instance
Tr= π - θ /ωd
θ = tan−1(√1 − ζ^2 / ζ
Peak time tp: Time required for the response to reach the peak value of time response
tp = π/ ωd
Peak overshoot Mp: It is the normalised difference between the peak value of time response and the steady state value
Mp = e^ − √πζ /1−ζ^2
final value is not unity, then
Mp =y(tp) − y(∞)/ y(∞)∗ 100%
Settling time ts: Time required for the response to reach and stay within a specified tolerance band of its final value or steady state value.
Usually the tolerance band is 2% or 5%
Note: tp and Mp are not defined for overdamped and critically damped systems
ts=4/ ζ ωn. For 2%
ts=3/ζ ωn. For 5%
Steady state error
Steady state error is the difference between the actual output
and the desired output as t → ∞
ess = lim sR(s) / 1 + G(s)
s→0
For Unit Step Input R(s) = 1/s
Steady state error for step input
ess = lim 1/ 1 + G(s) = 1 / 1 + Kp
s→0
where Kp = lim
s→0
G(s) is called position error constant
For Unit Ramp Input R(s) = 1/s^2
Steady state error for ramp input
ess = lim. 1/ sG(s) = 1 / Kv
s→0
where Kv = lim. sG(s)
s→0
is called velocity error constant
For Unit parabolic Input R(s) = 1/ s^3
Steady state error for parabolic input
ess = lim. 1/ s^2G(s) = 1 / Ka
s→0
where Ka = lim. s^2G(s)
s→0
is called acceleration error constant
The error constants Kp,Kv and Ka describe the ability of a system to reduce or eliminate steady state errors
These values mostly depend on the type of the system
As the type of the system becomes higher, more steady-state errors are eliminated
Features
It means the steady state error should be as low as possible
and hence it is an important performance measure
Steady state errors depend on two factors:
1 Type of the reference input R(s) - step, ramp or parabolic
2 Type of the system G(s)
Steady state errors are calculated only for closed loop stable systems
Type of a System
Consider the unity feedback control system with the following open
loop transfer function G(s)
G(s) = K(s + z1)(s + z2)...(s + zm)/ sN (s + p1)(s + p2)...(s + pk)
Terms
N in the denominator denotes the number of the poles (N) at origin
System with N poles at origin is called Type-N system
A system is called Type 0 , Type 1, Type 2, ... if N = 0,1,2,...
Routh’s Stability Criterion
1 A system is stable if its poles lie on the left half of the S Plane.
2 To find the poles we have to factorize the denominator of the transfer function
3 Routh’s stability criterion tells whether the system is stable or unstable without actually finding the roots of the
characteristics equation
Criterion
Routh’s Stability criterion states that the number of roots of with positive real parts is equal to the number of
changes in sign of the coefficients of the first column of the
array
The necessary and sufficient condition that all the roots of lie in the left half s-plane is that all the coefficients of
be positive and all terms in the first column of the Routh’s array have positive sign
Root locus
The root locus of a feedback system is the graphical representation in the complex s-plane of the possible locations of its closed-loop poles for varying values of a certain system parameter.
Use of root locus
● By using the root locus method the designer can predict the effects on the location of the closed loop poles by varying the gain values or adding open loop poles and/or open loop zeros
● Root locus method indicates the manner in which the open loop poles and zeros should be modified so that the response meets the performance specifications of system
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively.
.
we can directly find the order of the transfer function by just determining the highest power of 's' in the denominator of the transfer function. To determine the TYPE of the system, just count the number of poles lying at origin i.e at 0 in the 's-plane'. So, the no. of poles at origin gives the type of the system.
the order is the highest exponent in the transfer function.
If all the poles lie in the left half of the s-plane, then the system is stable. If the system has two or more poles in the same location on the imaginary axis, then the system is unstable. If the system has one or more non-repeated poles on the imaginary axis, then the system is marginally stable.
For the closed loop control system shown above
C(s)/R(s) = G(s) / 1 + KG(s)H(s)
characteristics equation of the closed loop system is given
as
1 + KG(s)H(s) = 0
A unity feedback system has open loop transfer function
G(s) = K/s
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