Fractional PI controller in liquid level

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Fractional PI controller in liquid level flow
controlExperimental study on Coupled Tank System

• Varsha Bhambhani
• Graduate Research Assistant
• Center of Self Organizing and Intelligent Systems
  (CSOIS)
• Electrical Engineering Department,
• Utah State University, Logan, USA

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Liquid Level Flow Control

• Common control problem in Water-treatment
  plants, Petro-chemical industries and other
  process industries.
• Human blood circulating system is another
  classic example of liquid level flow control.
• Coupled Tank System
• An important laboratory and teaching level
  instrument present in many universities which
  helps in study of design, operation and
  applications of common controllers.
• Benefits- Useful in system modeling based on
  static and dynamic control study, steady state
• and transient behavior
  analysis, controller design and controller tuning
  method
• study.
• Working- a) compact , bench top instrument
  consisting of two water tanks made of perplex
• seated on a water
  reservoir which stores water.
• b) A baffle plate can be
  slided up down to vary interaction or coupling
  dynamics
• between two tanks.
• c) Two PWM operated motor
  pumps use either 0-5V analog voltage (internal
  signal
• conditioning system
  covert analog to PWM (digital) signals) or
  external PWM
• sources for their
  operation. Flow rates of water into tanks can be
  varied by
• change of these pump
  voltages.

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Coupled Tank System Specifications contd.

• d)Two capacitive probes, one in each tank
  ,provided to measure water level.
  Output signals from these probes are
  conditioned to give 0-5V DC analog output.
• e) A water outlet at side near base of each tank
  connected by a flexible tube returns water to
  reservoir.
• f) Two potentiometers at back side of CTS are
  provided for manual operation of motors.
  

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Cases Studied

• A comparison of Integer and fractional
  Proportional Integral (PI) Controller is made.
• In general a PI controller has following effects
• add damping
• improve the steady-state error
• the rise time and settling time are penalized
• For this three configurations of Coupled Tank
  System are studied
• First order Single Input Single Output (SISO)
  plants.
• A second Order SISO plant.
• Cascaded control plant.
• Each case involves
• Brief description of system modeling.
• Real time System identification.
• Controller design.
• Experiments simulations.
• Comments.

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Case I - First order Single Input Single Output
(SISO) plants.

• System modeling
• When baffle plate is lowered completely, two
  tanks operate independently as first order single
  input single output (SISO) systems.
• Relation between water entering and leaving tank
  is expressed as-
• Where,
• is water flow in tank
• Is water flow out of tank
• A is cross-sectional area of tank
• H is height of water in tank
• The output flow through the valve is related to
  the water level in the tank by the relation-

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Case I contd.

• Where, C is discharge coefficient of the valve.
• And a is the area of cross section of the
  orifice.
• And g is gravitational constant 9.8 m/s2.
• Summarizing we get,
• Above non-linear equation describes the system
  behavior of first order SISO system.
• Real time system identification
• In terms of transfer function, in real time, the
  manipulated variable/plant input is pump input
  voltage and process variable/plant output is
  water level in the tank.
• The transfer function of first order SISO system
  is given by

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Case I contd.

• Where, K is the gain of the system, is the
  time constant and L is the delay of the system.
• One can either do frequency response analysis or
  step response analysis to identify transfer
  function.
• Frequency response Analysis of first order SISO
  plant.
• Procedure Apply sinusoidal input at different
  frequencies to the open loop control plant and
  observe the gain and the phase shift at steady
  state.
• Sinusoidal input-

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Real Time Experimental Setup

• A typical feedback control system will have the
  following components the Plant, Sensors,
  Actuators and a Controller.
• In a digital real-time control application the
  analog controller is replaced by the digital
  computer (PC). They give the flexibility of
  changing the program according to the change in
  design requirements or dynamics of the system.
• The digital controller needs feedback from the
  plant in the appropriate format which is ensured
• by the DACB (Data Acquisition and Control boards)
  provided by Quanser.

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Q4 Hardware In Loop Board from Quanser

• Key Features
• 4 x 14-bit analog inputs
• 4x 12-bit D/A voltage outputs
• 4 quadrature encoder inputs
• 16 programmable digital I/O channels
• Simultaneous sampling of both analog and encoder
  sections
• 2x 32-bit dedicated counter/timers
• 4x 24-bit reconfigurable encoder counter/timers
• 2x on-board PWM outputs
• 32-bit, 33 MHz PCI bus interface
• Supports Quanser real-time control software
  WinCon (2000/XP)
• Totem Pole digital I/O for high speed
• Easy synchronization of multiple Q4 boards

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Real time control of hardware in loop

• These boards accept the sensor signals from the
  plant and convert them to digital signals which
  is then sent back to the computer.
• The code which emulates the controller computes
  on this data and decides the next set of control
  signals and sends digital data to the DACB which
  converts it to an analog signal, sent to the
  actuators.
• All these operations are performed in real-time
  and this is achieved by the real-time Windows2000
• /XP application WinCon which runs in
  real-time the C code generated for the control
  law imple--mented in MATLAB/ Simulink Real Time
  Workshop.

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Case I - Frequency response Analysis of first
order system block settings
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Results of frequency response analysis of first
order system (diff freq considered)
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System identification by frequency response
Frequency (radians/sec) Magnitude (Decibels) Angle (Degrees)
0.001 7.224 -0
0.005 6.7408 -8
0.024 6.4208 -45
0.07 1.8852 -80
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Extracted Data (from Bode Plot) Extracted Data (from Bode Plot) Experimental Data (from Frequency Response Experiments) Experimental Data (from Frequency Response Experiments)
Frequency (Rad/Sec) Magnitude (Decibels) Angle (Degrees) Magnitude (Decibels) Angle (Degrees)

0.001 7.22 -2.45 7.224 -0
0.005 7.03 -12 6.7408 -8
0.024 4.19 -45 6.4208 -45
0.07 -2.51 -71 1.8852 -78
Inputs that are not manipulated, classified as
disturbance or load variable result in
differences between extracted bode plot data
experimental data
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Case I - closed loop system response first order
system block settings/Controller Design
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Controller design/ Tuning rules

• Integer Order Proportional Integral Controller
  (PI) tuned by
• Ziegler Nichol s Method
• Modified Ziegler Nichols Method
• MethodKc,Pm,wc,wp margin(G)
• Tc 2pi/wc
• Fractional Order Proportional Integral Controller
  (FOPI) tuned by
• Fractional Ms constrained Integral Gain
  (FMIGO) Method

P I
PI-ZN 0.4Kc 0.8Tc
PI-MZN Kc r cos(a) Tc/2pi/tan(a)
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Simulated results of First Order Coupled Tank
SISO System
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Real Time results of First Order Coupled Tank
SISO System
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Comments on First Order SISO Coupled Tank
Experiments

• Real time system results are in accordance with
  the simulation results with little difference
  which can be accounted due to uncontrollable real
  time environmental disturbances.
• Both simulated and real time results confirm that
  the percentage overshoot is minimum in FOPI
  (FMIGO) method, though this is at the expense of
  slow response time when compared with integer
  order PI-ZN and PI-MZN methods.
• Thus results of first order SISO coupled tank
  experiments clearly shows that a fraction order
  controller is a promising controller for water
  level and flow control.

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Case II - Second order Single Input Single
Output (SISO) plants.

• System modeling
• When baffle plate is raised, water flows from one
  tank to another. The target is to maintain a
  fixed level of water in second tank by varying
  voltage input to first tank.
• The control system has two states, levels in two
  tanks, i.e.
• in tank 1 and in tank2.
• , pump flow to tank 1, is the control
  input.
• , water level in tank 2, is output.
• Valve B and C account for load disturbances.
• The equation of water flow balance in tank 1 is
  given by
• is water flow from tank 1 to tank 2.

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Cases II contd.

• The water flow balance equation for tank 2 is
  given by
• is flow of water out of tank 2 through
  valve C.
• Assuming orifices to be ideal, the non- linear
  ties are computed by square root law in
  substituted in above equations as-
• Above non-linear equations are linear zed
  further to obtain state equation of the coupled
  tank system.
• The transfer equation for second order SISO
  system is given by

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Case II contd.

• Where, K is the process gain of the system,
• is the damping ratio and is defined as
  degree of oscillation in the process response
  after a perturbation.
• is the natural frequency of the system
  is the inverse of time constant which
  determines the speed of response of the system.
• Analyzing the denominator, we get
• The roots of the characteristic equation are

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Case II contd.

• Three cases arise as shown in table above
• One can either do frequency response analysis or
  step response analysis to identify transfer
  function.
• Frequency response Analysis of first order SISO
  plant.
• Procedure Apply sinusoidal input at different
  frequencies to the open loop control plant and
  observe the gain and the phase shift at steady
  state.
• The real time experimental setup is shown in next
  slide.
• The block diagram and the different block
  settings for frequency analysis is shown in next
  to next slide.

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Case II- Frequency response Analysis of Second
order system block settings
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Results of frequency response analysis of second
order system (diff freq considered)
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System identification by frequency response
Frequency (radians/sec) Magnitude (Decibels) Angle (Degrees)
0.01 6.06 -23.8
0.02 5.3 -48
0.04 1.83 -90
0.05 -0.283 -108
0.1 -10.5 -143
0.8 -30.0063 -180
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Extracted Data (from Bode Plot) Extracted Data (from Bode Plot) Experimental Data (from Frequency Response Experiments) Experimental Data (from Frequency Response Experiments)
Frequency (Rad/Sec) Magnitude (Decibels) Angle (Degrees) Magnitude (Decibels) Angle (Degrees)

0.01 6.06 -23.8 8.1434 -39
0.02 5.3 -48 6.2773 -45
0.04 1.83 -90 1.9382 -90
0.05 -0.283 -108 -1.9861 -110
0.1 -10.5 -143 -8.2055 -120
0.8 -45.6 -175 -30.0063 -180
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Approximation of second order transfer function
by first order transfer function using getfod
file
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Case I I - closed loop system response second
order system block settings/Controller Design
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Simulated results of second Order Coupled Tank
SISO System
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Real time results of second Order Coupled Tank
SISO System
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Comments on second Order SISO Coupled Tank
Experiments

• Real time system results are in accordance with
  the simulation results with little difference
  which can be accounted due to uncontrollable real
  time environmental disturbances.
• It is seen that in case of second order SISO
  system, FOPI (FMIGO) results in large overshoot
  when compared with integer order PI-ZN and PI-MZN
  method but then the response time is much less
  when compared to other controllers.
• Thus results of second order SISO coupled tank
  experiments clearly shows that a fraction order
  controller is a promising controller for water
  level and flow control.

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Case III - Cascaded control plant system.

• System modeling
• This type of control system has two cascaded
  controllers namely primary and secondary
  controllers.
• The controlled variable is water flow to tank1.
  The master controller decides the set point of
  the slave controller. The slave controller tries
  to track the set point.
• The master controller uses water level in tank 2
  as process variable by varying water level in
  tank1.
• Suitable baffle opening between two tanks
  introduces significant time separation between
  the two controllers which minimizes the effect of
  disturbance in water level of tank 1 to water
  level of tank 2.

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Case III contd.

• Both cascaded plants are configured as first
  order transfer function namely primary and
  secondary plants having transfer function in
  general form as
• Where, K is the gain of the system, is the
  time constant and L is the delay of the system.
• One can either do frequency response analysis or
  step response analysis to identify transfer
  function.
• Instead of doing frequency response twice for
  each plant ( which is time consuming), one can do
  step response in which step input is applied to
  the plant and the response recorded .
• Step Input-

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Case III- Step response Analysis of Cascaded
plants block settings
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Results of step response analysis of two cascaded
plants

• An input step of 2V is applied and step response
  of tank1(secondary tank) and tank2(primary tank)
  recorded in real time.
• The transfer function two plants computed from
  step responses are

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Case III - closed loop system response cascaded
control plants block settings/Controller Design
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Simulated results of Cascaded control plants
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Real time results of Cascaded control plants
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Comments on Cascaded control plant Coupled Tank
Experiments

• Real time system results are in accordance with
  the simulation results with little difference
  which can be accounted due to uncontrollable real
  time environmental disturbances.
• It is seen that in case of real time system, FOPI
  (FMIGO) controller in time interval 600-1000
  seconds gives no overshoot and is an ideal
  controller when compared with integer order
  PI-ZN,PI-MZN method.
• Thus results of cascaded control plants coupled
  tank experiments clearly shows that a fraction
  order controller is a promising controller for
  water level and flow control.

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Results of Experimental study

• 1_) A very intensive study showing system
  identification, controller design of coupled tank
  system was performed.
• 2) A thorough comparison between the three
  controllers were made.
• Results show that FOPI(FMIGO) is a promising
  controller in process industries and can even
  perform better at some point when compared with
  integer order PI controllers.
• 4) Major problems in real time controller design
  seen are due to
• Transient response design is hard
• a) Robustness is always an issue
• - Modeling uncertainty.
• - Parameter variations.
• - Disturbances.
• b) Lack of theory (design uncertainty)
• - Relation between pole/zero
  locations and transient response.
• - Relation between Q/R weighting
  matrices in optimal control and
• transient response.