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 an eye on with whole Observations.- 5.H2 optimum regulate With entire Observations.- 6.Quadratic and H2 optimum keep watch over 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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GN ) and G = (G1 , . . , GN ) in H . 4 Mean-Square Stability for the Homogeneous Case 51 transpose we have (bearing in mind that the entries of Ai are real) √ √ Li GR + −1Li GI < 0, Li GR − −1Li GI < 0, i ∈ S, so that, summing both expressions, we obtain the real version of (b). The proof for (c) is analogous if we keep in mind that the entries of A are real. 22 Note that the Lyapunov operator L works on a Hilbert space of dimension N n(n+1) rather than N n2 , the dimension of the matrix A.

1). 16 (Each mode is unstable, but the overall system is stable) Consider an MJLS with A1 = 1 2 0 −1 , −2 A2 = −2 0 −1 1 2 , Π= −β β β , −β so that each mode is unstable. However, depending on the value of β, the overall system will be mean-square stable. , Re{λ(A)} < 0 if and only if β > 4/3. This shows that as the number of jumps per unit of time increases, the effect of switching between the unstable modes makes the overall system mean-square stable. 17 (Each mode is stable, but the overall system is unstable) Consider now an MJLS with A1 = −1 0 10 , −1 A2 = −1 10 0 , −1 Π= −β β β , −β whose modes are both individually stable, so that Re{λ(A)} < 0 for β = 0.

A. a. t means almost all t , since for those t where A is discontinuous, the lefthand side of the differential equation is not defined in the usual sense). 9) is given by x(t) = Φ(t, t0 )x0 + t Φ(t, τ )B(τ )u(τ ) dτ. 6) that the state transition matrix is given by Φ(t, τ ) = eA(t−τ ) . 5 Continuous-Time Markov Chains For a positive integer number N , we define S := {1, . . , N }. Let (Ω, F, P ) be a complete probability space equipped with a filtration {Ft ; t ∈ R+ } satisfying the usual hypotheses, that is, a right-continuous filtration augmented by all null sets in the P -completion of F .

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