Continuous-time Markov jump linear systems by Oswaldo Luiz do Valle Costa

By Oswaldo Luiz do Valle Costa

1.Introduction.- 2.A Few instruments and Notations.- 3.Mean sq. Stability.- 4.Quadratic optimum keep watch over with entire Observations.- 5.H2 optimum keep watch over With whole Observations.- 6.Quadratic and H2 optimum keep an eye on with Partial Observations.- 7.Best Linear filter out with Unknown (x(t), theta(t)).- 8.H_$infty$ Control.- 9.Design Techniques.- 10.Some Numerical Examples.- A.Coupled Differential and Algebraic Riccati Equations.- B.The Adjoint Operator and a few Auxiliary Results.- References.- Notation and Conventions.- Index

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The next result is proved in Chap. 3 (see also Sect. 2 of [242]). 21 Let A ∈ B(Rn ). The following conditions are equivalent: (i) Re{λ(A)} < 0. (ii) There are constants k > 0 and b > 0 such that eAt ≤ ke−bt (iii) (iv) for all t ≥ 0. eAt x0 → 0 as t → ∞ for every x0 ∈ Cn . ∞ eAt x0 dt < ∞ for every x0 ∈ Cn . 7 Auxiliary Results 27 The result also holds replacing Cn by Rn in (iii) and (iv). The next proposition, adapted from [242], will be useful in deriving some stability results in Chap. 3. 22 Let A ∈ B(Rn ) and {f (t); t ∈ R+ } be a continuous function in Rn such that limt→∞ f (t) = f0 .

40) for the general case). 25) we have, for any Q ∈ HnC , that: 2 2 2 (a) ϕ(L(Q)) ˆ = Aϕ(Q), ˆ (b) ϕ(T ˆ (Q)) = A∗ ϕ(Q), ˆ (c) ϕ(F(Q)) ˆ = B ϕ(Q). ˆ Proof It follows from the definition of ϕˆ in Sect. 25). 27). 25), and consider the homogeneous system y(t) ˙ = Ay(t), t ∈ R+ , with initial condition y(0) = ϕ(Q), ˆ n Q ∈ HC . Then, y(t) = eAt y(0) = ϕˆ eLt (Q) . 28) The result also holds replacing A and L by A∗ and T , respectively. Proof We begin by noticing that the solution of the above differential equation is 2 given by y(t) = eAt y(0).

GN ) ∈ Hn such that L(G) + S = 0. 42) Moreover, (a) (b) (c) (d) −1 ˆ Gi = −ϕˆ −1 i (A ϕ(S)); ∞ At ϕ(G) ˆ = 0 e ϕ(S) ˆ dt ; S ∈ Hn∗ iff G ∈ Hn∗ ; S ∈ Hn+ implies G ∈ Hn+ . 42) reads as T (G) + S = 0. 42) is equivalent to Aϕ(G) ˆ = −ϕ(S). 44) The expression for Gi follows immediately from the assumption on A and the ¯ = (G ¯ 1, . . , G ¯ N ) ∈ Hn such that definition of ϕ. ˆ Assume now that there exists G ¯ ¯ − G) = 0, or Li (G) + Si = 0. 44), we have Aϕ( ˆ G ¯ ¯ ϕ( ˆ G − G) = 0, which implies that G − G = 0, and the uniqueness follows.

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