linearizing odes of a pid controller
TRANSCRIPT
-
8/11/2019 Linearizing ODEs of a PID Controller
1/9
Linearizing ODEs of a PID
Controller
Anchored by:
Rob ChockleyandScott Dombrowski
-
8/11/2019 Linearizing ODEs of a PID Controller
2/9
Basis for Linearizationdx
dt f(x)
dx
dt f(x) f(a) f' (a) * (x a)
Linear Approximation
Taylor Series Expansion
Ordinary Differential Equation
-
8/11/2019 Linearizing ODEs of a PID Controller
3/9
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.
-
8/11/2019 Linearizing ODEs of a PID Controller
4/9
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.
-
8/11/2019 Linearizing ODEs of a PID Controller
5/9
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
-
8/11/2019 Linearizing ODEs of a PID Controller
6/9
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.
-
8/11/2019 Linearizing ODEs of a PID Controller
7/9
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
-
8/11/2019 Linearizing ODEs of a PID Controller
8/9
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.
-
8/11/2019 Linearizing ODEs of a PID Controller
9/9
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.