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Linearizing ODEs of a PID

Controller

Anchored by:

Rob ChockleyandScott Dombrowski

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Basis for Linearizationdx

dt f(x)

dx

dt f(x) f(a) f' (a) * (x a)

Linear Approximation

Taylor Series Expansion

Ordinary Differential Equation

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Real Life Example

For our example, wecreated a real world

situation.

In: 3 mol/s (const)

Out: Controlled by PID

We are using a PID

controller to maintain the

tank pressure at a constant

pressure of 8 atm.

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Linearizing Our Model

Our system is being controlled by aPID controller.

The first equation models thedifferential pressure change.

The second equation is thecontroller output.

The third is the function is the firststep of linearization. F(P) is equalto the combined model of thedifferential pressure change.

Finally, the four equation is the

derivative of the function f(P). Thisis used in the linearization model atthe steady state pressure of 8 atm.

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Model for the

System

The graph to the right is themodel for the system. The

pressure within the tank is

being controlled by the valve.

The valve is being throttled

according to the output from

the sensor.

The pressure fluctuates a great

deal at the beginning of the run

but eventually reaches steady

state at our desired pressure.

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

0 50 100 150 200 250

Pressure

Time

Pressure

Gas Flow into a Tank: Controlling Pressure using a PID

Controller

timestep 1s

V 100L tank volume

T 300K tank temperature

R 0.08206Latm/molK gas constant

nin 3mol/s flow rate in

F 5mol/s max flow rate out

ni 40mol initial tank contents

Pinitial 9.8472atm initial tank pressure

Pset 8atm tank set point

bias 0.6

Kc 0.1

Ti 10

Td 0.1

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Linearization of dP/dt

This graph shows the plotof

dP/dt vs. P and the linearapproximation from atruncated linear expansion.

Because the pressureoscillated there are multiplevalues of dP/dt for onepressure.

The linearization is not a

good approximation for thebehavior of the differentialequation.

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Linear Approximation

behaving Like

Nonlinear Function

We change the coefficientswithin the model to create a

new behavior.

Here, the ratio of change in the

output to the change in input is

three time what it waspreviously , and the Integral

time is 10 times than before.

Here the desired pressure is

reached more quickly.

Gas Flow into a Tank: Controlling Pressure using a PID Controller

timestep 1 s

V 100 L tank volume

T 300 K tank temperature

R 0.08206 Latm/molK gas constant

nin 3 mol/s flow rate in

F 5 mol/s max flow rate out

ni 40 mol initial tank contentsPinitial 9.8472 atm initial tank pressure

Pset 8 atm tank set point

bias 0.6

Kc 0.3

Ti 100

Td 0.1

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Linearization of dP/dt

Again

For this trial themodel had no

oscillation so the

nonlinear equations

that govern thechange in pressure

are more easily

linearized.

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Conclusion

This walkthrough and model shows that linearizing

nonlinear equations is not always the best idea. The PID

controller creates instances where the differential of

pressure is not an independent function of one variable.Under certain conditions, however, The model can be

use. These models may not occur in real life.