5. Analysis of LDIs: State Properties
Society for Industrial and Applied Mathematics eBooks · 1994
In this chapter we consider properties of the state x of the LDI x ˙ =A (t) x,A (t) ∈Ω, 5.1 where has one of four forms: • LTI systems: LTI systems are described by . • Polytopic LDIs: PLDIs are described by , . • Norm-bound LDIs: NLDIs are described by x ˙ =Ax+ Bp p,q= Cq x+ Dqp p,p=Δ (t)q, ‖Δ (t) ‖ ≤1, which we will rewrite as x ˙ =Ax+ Bp p, pT p≤ ( Cq x+ Dqp p)T ( Cq x+ Dqp p) . 5.2 We assume well-posedness, i.e., . • Diagonal Norm-bound LDIs: DNLDIs are described by x ˙ =Ax+ Bp p,q= Cq x+ Dqp p, pi = δi (t) qi , | δi (t) |≤1, i=1,…, nq . 5.3 which can be rewritten as x ˙ =Ax+ Bp p,q= Cq x+ Dqp p, | pi | ≤ | qi |, i=1,…, nq . Again, we assume well-posedness.