Q2. What are the future works mentioned in the paper "Theoretical analysis of balanced truncation for linear switched systems" ?
Another open question is how to extend balanced truncation to systems which do not admit a common Lyapunov function.
Q3. What is the procedure for reducing r to an observable LSS?
Observability reduction: Assume that kerO(Σ) = n − no and let b1, . . . , bn be a basis in Rn such that bno+1, . . . , bn span kerO(Σ).
Q4. What is the proof of the second part of the theorem?
The proof of the second part of the theorem follows by an easy computation and by recalling that if Σ1 and Σ2 are two equivalent and minimal LSSs, then they are related by an LSS isomorphism.
Q5. What is the result of applying Procedure 1 to T?
(2) If Σrt is the result of applying Procedure 1 to ΣT ,then ΣTrt = Σo, where Σo is the result of application of Procedure 2 to Σ.As a consequence, K(Σ) 6= ∅ if and only if K(ΣT ) 6= ∅. Since Procedure 1 preserves non-emptiness of K(ΣT ), wehave that K(Σrt) 6= ∅, which implies that K(ΣTrt) = K(Σo) 6= ∅.
Q6. What is the observability grammian of the system?
Q : ATq Q+QAq + CTq Cq < 0. (2) Likewise, define a controllability grammian of the system as a strictly positive definite P > 0 which satisfies.∀q ∈ Q : AqP + PATq +BqBTq < 0. (3) By applying a suitable state-space isomorphism, the system can be brought into a form where P = Q = Λ = diag(σ1, . . . , σn) are diagonal matrices and σ1 ≥ . . . ≥ σn > 0.
Q7. What is the definition of a minimal LSS?
The LSS Σm is said to be a minimal realization of an input-output map f , if Σm is a realization of f and if for any other LSS Σ which is a realization of f , dim Σm ≤ dim Σ.
Q8. What is the definition of linear switched systems?
A linear switched system with external switching (abbreviated as LSS) is a tuple Σ = (n,Q, {(Aq, Bq, Cq) | q ∈ Q}), where for each q ∈ Q, (Aq, Bq, Cq) ∈ Rn×n × Rn×m × Rp×n.
Q9. what is the observability of the linear subsystem?
Observability: The LSS Σ is observable if and only if rank O(Σ) = n, whereO(Σ) = [ (C̃Av1) T , (C̃Av2) T , . . . , (C̃AvM ) T ]T ∈ Rp|Q|M×n.where C̃ = [ CT1 C T 2 , . . . , C T D ]T ∈ Rp|Q|×n.
Q10. What is the definition of a quadratic stability?
Q : ATq P + PAq < 0. (8)It is well-known Liberzon (2003) that quadratic stability implies exponential stability for all switching signals.
Q11. What is the effect of a quadratically stable system?
As a byproduct, the authors also show that if an inputoutput map can be realized by a quadratically stable system 1 , then any minimal realization of this map will be quadratically stable.
Q12. What is the purpose of this paper?
In this paper the authors address certain theoretical problems which arise in balanced truncation of continuous-time linear switched systems using balanced truncation.