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Automatic Optimisation of Controller Gains in a Heated Stirred Tank Samir Khan Samir.Khan@adeptscience.co.uk

A cold oil stream is heated in a stirred tank. The dynamic change in temperature in the tank is modelled by an ordinary differential equation, derived via a heat balance The temperature of the liquid in the tank is driven to a set point via a PI controller, which alters the heat added to the system. The optimium controller gains are found by minimising an objective function that describes how far the actual temperature profile is from a user-defined profile. Additionally, overshoot is attenuated via a penalty multiplier to the objective function. This application highlights two Maple computational features Numeric solution of differential equations Non-linear optimisation 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Mass of oil in tank QyQ+SSJNRzYiIiM1ISIi Specific heat capacity of oil QyQ+SSNDcEc2IiIiIyEiIg== Mass flowrate of feed stream QyQ+SSJXRzYiIiIjISIi Temperature of feed stream QyQ+SSNUY0c2IiIjNSEiIg== Temperature set point QyQ+SSVUc2V0RzYiIiQrIiEiIg== Time at which to terminate analysis QyQ+SSVUZW5kRzYiIiM1ISIi

A heat balanace across the tank gives QyQ+SSRlcTFHNiIvKihJIk1HRiUiIiJJI0NwR0YlRiktSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiVEdGJTYjSSJ0R0YlRjJGKSwmKihJIldHRiVGKUYqRiksJkkjVGNHRiVGKUYvISIiRilGKUkiUUdGJUYpRjg= Q is the heat added by the PI controller... QyQ+SSJRRzYiKiZJI0twR0YlIiIiLChJJVRzZXRHRiVGKC1JIlRHRiU2I0kidEdGJSEiIiomSSNLaUdGJUYoLUkpZXJyb3JpbnRHRiVGLUYoRihGKEYv where Kp and Ki are the proportional and integral controller gains, and... QyQ+SSRlcTJHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JKWVycm9yaW50R0YlNiNJInRHRiVGLiwmSSVUc2V0R0YlIiIiLUkiVEdGJUYtISIiRjQ= Initial conditions QyQ+SSNpY0c2IjYkLy1JIlRHRiU2IyIiIUkjVGNHRiUvLUkpZXJyb3JpbnRHRiVGKkYrISIi

QyQ+SSRyZXNHNiItSSh1bmFwcGx5R0YlNiUtLkknZHNvbHZlR0YlNiY8JUkkZXEyR0YlLUklc3Vic0clKnByb3RlY3RlZEc2JS9JI0twR0YlSSNrcEdGJS9JI0tpR0YlSSNraUdGJUkkZXExR0YlSSNpY0dGJTwkLUkiVEdGJTYjSSJ0R0YlLUkpZXJyb3JpbnRHRiVGPkkobnVtZXJpY0dGMS9JJ291dHB1dEdGJUkubGlzdHByb2NlZHVyZUdGJUY1RjghIiI= QyQ+SSl0ZW1wa3BraUc2ImYqNiRJI2twR0YlSSNraUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3Vic0clKnByb3RlY3RlZEc2JC1JJHJlc0dGJUYnLUkiVEdGJTYjSSJ0R0YlRiVGJUYlISIi Hence the temperature in the tank as a function of the controller gains and the time is QyQ+SSV0ZW1wRzYiZio2JUkja3BHRiVJI2tpR0YlSSJ0R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtLUkpdGVtcGtwa2lHRiU2JEYoRik2I0YqRiVGJUYlISIi

Define a function that gives the desired temperature profile QyQ+SSd0YXJnZXRHNiJmKjYjSSJ0R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJkkjVGNHRiUiIiIqJiwmSSVUc2V0R0YlRi5GLSEiIkYuLUkkZXhwR0YlNiMqKEYyRi4kIiIkRjJGLkYoRjJGLkYuRiVGJUYlRjI=

Overshoot penalty for optimisation process. QyQ+STFvdmVyc2hvb3RQZW5hbHR5RzYiIiIiISIi The following function is the the instantaneous squared difference between the actual and target temperature. If the actual temperature is greater than the target temperature, the difference is multiplied by an overshoot penalty (this acts to attenuate overshoot). QyQ+SS1lcnJvckluc3RhbnRHNiJmKjYlSSNrcEdGJUkja2lHRiVJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjIiIiEsJi1JJ3RhcmdldEdGJTYjOSYiIiItSSV0ZW1wR0YlNiU5JDklRjghIiIqJEY0IiIjMUY0RjMqJkkxb3ZlcnNob290UGVuYWx0eUdGJUZBRjRGQUYlRiVGJUY/ Hence the total error is the instantaneous error integrated over the analysis period. This is the objective function. QyQ+SStlcnJvclRvdGFsRzYiZio2JEkja3BHRiVJI2tpR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJEludEdGJTYmLS5JLWVycm9ySW5zdGFudEdGJTYlRihGKUkidEdGJS9GODsiIiFJJVRlbmRHRiUvSShlcHNpbG9uR0YlJCIiIiEiIy9JJ21ldGhvZEdGJUkmX0RleHBHRiVGJUYlRiUhIiI=

Minimize the objective function by finding the optimimum values of kp and ki. QyQ+SSV2YWxzRzYiLSZJLU9wdGltaXphdGlvbkdGJTYjSSlOTFBTb2x2ZUdGJTYmLS5JK2Vycm9yVG90YWxHRiU2JEkja3BHRiVJI2tpR0YlL0knbWV0aG9kR0YlSTFub25saW5lYXJzaW1wbGV4R0YlL0kwZXZhbHVhdGlvbmxpbWl0R0YlIiUrNS9JNG9wdGltYWxpdHl0b2xlcmFuY2VHRiUkIiImISIiRjw= Hence the optimised values of kp and ki are QyQ+SSNLcEc2Ii1JJHJoc0clKnByb3RlY3RlZEc2IyYtSSdzZWxlY3RHRig2JUkkaGFzR0YoJkkldmFsc0dGJTYjIiIjSSNrcEdGJTYjIiIiRjU=QyQ+SSNLaUc2Ii1JJHJoc0clKnByb3RlY3RlZEc2IyYtSSdzZWxlY3RHRig2JUkkaGFzR0YoJkkldmFsc0dGJTYjIiIjSSNraUdGJTYjIiIiRjU=

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