 First order system
 Step response for first order system
τ = time constant (sec) is defined as ratio of amplitude to initial rate of response
t  y(t)  
τ  63.2% of its ultimate value  
2τ  86.5% of its ultimate value  
3τ  95% of its ultimate value  
4τ  98% of its ultimate value 
 First order response
The unit step response of a first order system with time constant τ and steady state gain k_{p}
 Second order system
 Overshoot of second order function
 For second order process control system
 Stability of a system
 Stability at cross over frequency
For stability at cross over frequency AR ≤ 1
 Process involving series of transfer function
Calculations involving bode stability criteria – for process involving series of transfer functions
 Laplace Transformation
Y(t) = y(t) – y(s) and X(t) = x(t) – x(s)
 Laplace transformation of few common functions
Laplace transformation of ∆s, sin(at), cosh(at), t^{(1/2)}, t^{1/2}, e^{u2}, tf(t), e^{at}
 Laplace transformation of periodic function
Laplace transfer function f(t) of periodic function with period T
 Unit delayed function
Laplace of unit delayed function (tto)
 Transfer function of proportional derivative controller
 Amplitude ratio
 Temperature sensing through thermowell
Temperature sensing through thermowell is a first order process
 Transfer function of feed forward controller
Transfer function of feed forward controller for perfect disturbance rejection
 Time constant
 Time constant for thermometer
 Time constant for flow of liquid through container
Time constant for flow of liquids through container of volume v and flow rate of liquid q
 System in series
Equivalent time constant for two systems in series
 Offset
Offset is defined as difference between setpoint at t tending to infinity and output at t tending to infinity
 Initial value theoram
 Final value theoram
 Steady state gain
To calculate steady state gain of closed loop, put s=0.
 Half life of first order system
 Linearization
Linearization / linearization of tank model of process flow
 Output of an impulse function
 Integral time and proportional gain
 Resistor
Equation for resistor
 Amplitude ratio and gain margin
Amplitude ratio is inverse of gain margin
 Equation for control valves
 Equal percentage
 Linear
 Transfer function of pure dead time system
Transfer function of pure dead time system with dead time τ_{D} is
 Transfer function of first order system
Consider first order system
 Transfer function of feedback control system
In feedback control system G and H denote open loop and closed loop transfer function then overall transfer function of feedback control system is
 Descartes rule of sign change for transfer function
Maximum number of positive roots is equal to number of sign changes in f(x).
 Process Dynamics and Control
 Stability and its criteria



Bode Stability criteria


Bode diagrams are generated from output response of system subjected to ramp input.




a. Phase cross over frequency: Phase cross over frequency, w_{pc}, is the frequency at which phase shift is equal to 180^{o}.
b. Gain cross over frequency: Gain cross over frequency, w_{gc}, is the frequency at where the amplitude ratio is one or where log modulus is equal to zero. 
Stability criteria: If at phase cross over frequency, the corresponding log modulus of G(iw_{pc}) is less than 0dB, then the feedback system is stable.

Stability margin

Gain margin: let x = G(iw_{pc}), then gain margin, GM=1/x
Phase margin: q = arg (G(iw_{gc}), then phase margin PM = 180^{o} + q
Positive margin means indicates there is safety margin before instability. 
Finding Gain margin

a. Find the frequency at which phase becomes 180^{o}
b. Find the Gain G (in dB) or AR at this same frequency.
c. The gain margin = 0 – G dB
d. In term if G is in ratio, dB = 20log_{10}AR
e. Gain Margin = 1/AR. 
Finding Phase margin

a. Find frequency where Gain is 0dB.
b. Find phase P at same frequency.
c. Phase margin = +P + 180^{o} 
At cross over frequency 180^{o},

Argument G(jw) = 180^{o}, AR>1 => unstable
Argument G(jw) = 180^{o}, AR< 1 => stable


Poles at origin implies integrating response and zero at right half of plane implies inverse response.

For finding minimum k_{c}, use Routh’s test. For finding maximum k_{c} use Bode stability criteria.

The closed loop poles of a stable second order system could be both real and negative.


ZieglerNicholas controller settings



Control system application

Pressure control in high pressure system is most often benefit from derivative control.
Distillation column level is controlled with bottom flow P – I control.
Distillation column pressure to be controlled by manipulating vapour flow from top plate using P – I – D control.
Flow control of a liquid from a pump by positioning the valve in the line using P – I control.
It needs quick control action against fluctuation.
Control of temperature of a CSTR with coolant flow in the jacket using P – I – D control.


First order system

Time constant for first order process with resistance R and capacitance C is RC.

For a first order process, if initial rate of k is maintained, the response would have been completed in τ seconds.

Pure gain system


A pure gain system is a first order system with zero time constant. Its principal property is that it has instantaneous response
For example
a. Flow in a capillary of an incompressible fluid
b. Error e(t) for control action c(t)
Response of a pure gain system
a. Step input response
b. Pulse input response
c. Ramp input response



Pure capacity system


Example: flow pumped from a tank
Response
a. Step response
Inverse in time domain is ramp function.
b. Pulse input response
Inverse in time domain is ramp function.
c. Sinusoidal input response
The amplitude of pure capacity process is inversely proportional to frequency.



The frequency of a first order system has a phase shift with lower and upper bounds given by (𝜋/2, 0).


Second order system

The protective sheath and thermocouple together behave as second order system (two first order system in series) with ξ > 1. Therefore, the response is slower and non – oscillatory.

For nonoscillatory response of a second order system, imaginary part of roots should vanish.

The sheath and thermocouple will behave like a second order systems, two first order in series. The damping coefficient will be greater than unity. Therefore, the system will be slower and nonoscillatory.

U – Tube manometer is an open loop second order system.


Sinusoidal Input

For sinusoidal input
The nonsinusoidal term predominates initially up to time t = 3τ, when the response in nonsinusoidal. It is only after the lapse of time t = 3τ, the contribution of the first term becomes negligible and second term (sinusoidal) becomes dominating and the response is sinusoidal.


Multiple tank system

Time constant for tank is “AR”.

Response of two tanks of same size and resistance in series is overdamped for interacting system and critically damped for noninteracting system.


Controller and control system

Proportional plus reset control mode combines the immediate output characteristics of a proportional control mode with zero residual offset characteristics. E.g., the integral mode.

Integral controller is also known as reset controller.

Feed forward control scheme is sensitive to modelling errors. It cannot cope with unmeasured disturbances. Control action is taken before the effect of disturbance has been felt by the system. It requires good knowledge of process model and identification of all possible disturbances and their direct measurement.

The control valve characteristics is selected such that the product of process gain and valve gain remains constant as the value of manipulated variable changes.

Cascade control has more than one measurement and one manipulated variable.

Controller output is product of error, gain and bias.


Instrumentation and its control

Piezoelectricity is electricity generated in some element upon application of mechanical stress.

Liquid level is measured by displacer devices and composition is measured by infrared analyzer.

In orsat analysis is done by bringing the combustion product to STP and hence water is not considered in analysis.

Bimetallic strip is used to convert a temperature change into mechanical displacement. It is a type of thermal expansion sensor.

A thermistor is a type of resistor where electrical resistance is dependent on temperature, more so than standard resistor.

Strain Gauge


Strain gauge pressure transducer converts mechanical energy to electrical energy. A strain gauge is a long length of conductor arranged in a zig zag pattern on membrane. When it is stretched its resistance increases. Strain gauges are mounted in the same direction as the strain and often in fours to form a full “Wheatstone bridge”. A downward bend stretches the gauge on top and compresses those on the bottom. A pressure transducer contains a diaphragm which is deformed by the pressure which can be measured by a strain gauged element.



Thermister


Thermister is an electrical resistor for whose resistance is greatly increased/reduced by heating. It is used for measurement and control. It is of two types. Negative temperature coefficient (NTC) and Positive temperature coefficient (PTC). In NTC resistance decreases to protect against over voltage conditions. It is installed in series in circuit. In PTC, resistance rises to protect against over current conditions. It is installed in series circuit.



For a thermocouple, ∆emf ∝ ∆temperature irrespective of initial and final temperatures.

Bourdon tube element is used to measure pressure. Infrared analyzer is used to measure composition and displacer devices are used to measure liquid level.

Bimetallic thermometer


bimetallic thermometer are made up of bimetallic strips formed by joining two different metals having different thermal expansion coefficient. They work best at high temperature since their accuracy tends to reduce at low temperature.