Process control notes

  • First order system
  • Step response for first order system

step response of 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
  86.5% of its ultimate value
  95% of its ultimate value
  98% of its ultimate value
  • First order response

first order response

The unit step response of a first order system with time constant τ and steady state gain kp

unit step response of first order system

  • Second order system
  • Overshoot of second order function

  • For second order process control system

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

bode criteria of stability at cross over frequency

  • Laplace Transformation

Y(t) = y(t) – y(s) and X(t) = x(t) – x(s)

laplace transformation

  • Laplace transformation of few common functions

Laplace transformation of ∆s, sin(at), cosh(at), t-(1/2), t1/2, e-u2, tf(t), eat

laplace transformation of delta S, sin(at) cosh(at) t to power - 1/2 t to power 1/2 tf(t) e to power at

  • Laplace transformation of periodic function

Laplace transfer function f(t) of periodic function with period T

laplace transformation of periodic function

  • Unit delayed function

Laplace of unit delayed function (t-to)

laplace transformation of unit delayed function

  • Transfer function of proportional derivative controller

transfer function of potential derivative controller

  • Amplitude ratio

amplitude ratio of control function

  • Temperature sensing through thermowell

Temperature sensing through thermowell is a first order process

thermowell is a first order process

  • Transfer function of feed forward controller

Transfer function of feed forward controller for perfect disturbance rejection

feed forward controller

  • Time constant
  • Time constant for thermometer

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

time constant for flow of liquid through container

  • System in series

Equivalent time constant for two systems 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

offset value of a controller

  • Initial value theoram

initial value theoram

  • Final 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

half life of first order system

  • Linearization

Linearization / linearization of tank model of process flow

linearization of a function

  • Output of an impulse function

output of a impulse function

  • Integral time and proportional gain

integral time and proportional gain

  • Resistor

Equation for resistor

equation for resistor

  • Amplitude ratio and gain margin

Amplitude ratio is inverse of gain margin

amplitude ratio and gain margin

  • Equation for control valves

equation for control valves

  • Equal percentage

equation for equal percentage control valves

  • Linear

  • Transfer function of pure dead time system

Transfer function of pure dead time system with dead time τD is

transfer function of pure dead time system

  • Transfer function of first order system

Consider first order system

transfer function of 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

transfer function of feedback control system

  • Descartes rule of sign change for transfer function

descartes rule

Maximum number of positive roots is equal to number of sign changes in f(x).

  • Process Dynamics and Control
  • Stability and its criteria
      1. 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, wpc, is the frequency at which phase shift is equal to -180o.
          b. Gain cross over frequency: Gain cross over frequency, wgc, 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(iwpc) is less than 0dB, then the feedback system is stable.

        • Stability margin

        • Gain margin: let x = G(iwpc), then gain margin, GM=1/x
          Phase margin: q = arg (G(iwgc), then phase margin PM = 180o + q
          Positive margin means indicates there is safety margin before instability.

        • Finding Gain margin

        • a. Find the frequency at which phase becomes -180o
          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 = 20log10AR
          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 + 180o

        • At cross over frequency -180o,

        • Argument G(jw) = -180o, AR>1 => unstable 
          Argument G(jw) = -180o, AR< 1 => stable

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

      2. For finding minimum kc, use Routh’s test. For finding maximum kc use Bode stability criteria.

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

    1. Ziegler-Nicholas controller settings

    1. 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.

    2. 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


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).

    1. 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 non-oscillatory 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 non-oscillatory.

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

    2. Sinusoidal Input

      • For sinusoidal input 

        The non-sinusoidal term predominates initially up to time t = 3τ, when the response in non-sinusoidal. 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.

    3. 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 non-interacting system.

    4. 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.

    5. 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.