I am working with model order reduction for electrical circuits. I am looking to implement Balanced Truncation as a form of model reduction.
The systems I work with are definitely stable, but not necessarily passive. They are formed through Modified Nodal Admittance(MNA).
$$ C \dot{x} = -G x + B u $$ $$ y = L x $$
Naturally, they are exactly in the standard form for LTI Continuous Differential Algebraic Equations (DAEs). When looked at as a state space system, they are considered as being in Generalized State Space form.
$$ E \dot{x} = A x + B u $$ $$ y = C x + D u $$
The important part is the descriptor matrix. Its a trivial case if E is invertible, since I can just multiply the rest of the equation by it and treat it as regular state space system.
$$ \dot{x} = A x + B u $$ $$ y = C x + D u $$
Now I am increasingly confused over the connection between various types of systems.
I know that (please correct me if I'm wrong) :
- A system is called minimal if it's the lowest order system giving EXACTLY that response.
- A system is minimal iff its completely observable and controllable.
- The controllability (observability) grammian is positive definite iff the system is completely controllable (observable).
- Only minimal systems can be balanced.
What I don't know is, how does the invertibility of E affect these properties?
