# Are descriptor systems necessarily non-minimal?

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) :

1. A system is called minimal if it's the lowest order system giving EXACTLY that response.
2. A system is minimal iff its completely observable and controllable.
3. The controllability (observability) grammian is positive definite iff the system is completely controllable (observable).
4. Only minimal systems can be balanced.

What I don't know is, how does the invertibility of E affect these properties?

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