System Theory

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

By Oswaldo Luiz do Valle Costa

1.Introduction.- 2.A Few instruments and Notations.- 3.Mean sq. Stability.- 4.Quadratic optimum keep an eye on with whole Observations.- 5.H2 optimum keep an eye on With whole Observations.- 6.Quadratic and H2 optimum regulate with Partial Observations.- 7.Best Linear clear 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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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.

Then, according to [91], Sect. 10, by relabeling the states appropriately, Π and Π κ can be written as Π= 0 = Π22 Π11 Π21 ρκ Π κ = κ=1 ρκ κ=1 κ Π11 κ Π21 κ Π12 , κ Π22 κ where Π11 is a square matrix. But since ρ κ ≥ 0 and each element of the matrix Π12 κ κ is nonnegative, we must have for κ such that ρ > 0 that Π12 = 0, in contradiction with the hypothesis that Π κ is irreducible. We next present a useful result that will be employed in Chap. 4 to get Dynkin’s formula for the MJLS. First, we recall the following definition.

In addition, we assume for this case that {θ (t); t ∈ R+ } is an irreducible Markov chain. We recall that, as seen in Sect. 21)), in this case there exists a limiting probability {πi ; i ∈ S} that does not depend on the initial distribution, with { i∈S πi = 1}, and satisfies, for some positive constants α > 0 and β > 0, max pj (t) − πj ≤ αe−βt . 27), we will use along this book the notation ϑt = x(t), θ (t) , and when we say arbitrary initial condition ϑ0 = (x0 , θ0 ), we mean any distribution ν = {νi ; i ∈ S} for θ0 (P (θ0 = i) = νi , i ∈ S) and any distribution for x0 satisfying E( x0 2 ) < ∞.

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