Status of a variable for the linearization

Linear Analysis Status

LA status

We define state variable in its broadest sense to include explicit variable, implicit variables and constraint variables.

All state variables have an status which is one of the following:

• Free state Free state.

• Fixed state Fixed state.

• State observer State observer.

A fixed state is a state which is excluded from the linearization process. If a state is not fixed, it is free. If there are N states, M of which are fixed, the order of the Jacobian produced is N-M. A free state that is also an observer variable is called a state observer.

All variables other than state variables, also have an LA status which is one of the following:

• Clear Clear variable

• Control Control variable

• Observer Observer variable

No state variable can be made a control variable.

Simcenter Amesim can be used to linearize a system around an operating point producing the standard A, B, C and D matrices for the state space equations with:

where x, u and y are respectively the state, control and observer variables. Strictly speaking this is true only for classical ordinary differential equations (O.D.E.s).

It is possible to apply the linearization process to a reduced subset of state variables. To do this, the excluded states must be frozen, or fixed. These states are then set to fixed states. We can also say that their linear analysis status is set to fixed status.

Since these states are excluded from linearization, the size of the A matrix is reduced, as the number of eigenvalues.

If a state is not fixed, it is included in the linearization, at least as a free state. However, it can also be part of the observer vector by setting its status to State observer.

Here is a brief summary of the possible linear analysis status and their repercussion on the x and y vectors:

• any state variable set to Fixed state is excluded from linearization.

• any state variable set to Free state is included in x vector.

• any state variable set to State observer is included in x and y vectors.

• any variable set to Control is included in the u vector.

• any variable set to Observer is included in the y vector.

• any variable set to Clear will not be included in any vector.