Simulink Control Design™ - MathWorks

∂q

∂A

=Cd

2

μ

p

p^2 +pcr^2

0.25

The linearized mass flow rate equation is:

q=Cd

2

μ

p

p^2 +pcr^2

0.25

A+

∂q

∂μμ

+

∂q

∂pcr

∂pcr

∂pa

+

∂q

∂p

pa+

∂q

∂pcr

∂pcr

∂pb

−

∂q

∂p

pb

where · represents a deviation from the nominal variable.

In the linearized equation, if the nominal pressure drop p across the orifice is zero, then A

has no influence on q. That is, if the instantaneous pressure drop across the orifice is

zero, the orifice area has no influence on the mass flow rate. Therefore, you cannot

control the piston position using the orifice area control variable.

To avoid this condition, linearize the model about an operating point where the pressure

drop over the orifice is nonzero (pa ≠ pb).

Troubleshooting Tips

To fix linearization problems caused by poor initial conditions of network states, you can:

1 Linearize the system at a snapshot operating point or trimmed operating point. When

possible, this approach is recommended.

(^2) Find and modify the problematic states of the operating point. This option can be
difficult for models with many states.
Using the first approach, you can ensure that the model states are consistent via the
Simulink and Simscape simulation engine. Simscape initial conditions are not necessarily
in a consistent state. The Simscape engine places them in a consistent state during
simulation and for trimming using the Simscape trim solvers.
A common workflow is to simulate your model, observe at what time the model satisfies
the operating condition at which you want to linearize, then take a simulation snapshot.
Alternatively, you can trim the model about the condition you are interested in. In either
case, the network states are in a consistent condition, which solves most poor
linearization issues.
2 Linearization