Read e-book online A structure utilizing inexact primal-dual interior-point PDF

By Janne Harju Johansson.

ISBN-10: 9173938718

ISBN-13: 9789173938716

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If stability holds for a quadratic V (P, x) when P ∈ Sn++ , the system is said to be quadratically stable. 2 Polytopic Linear Differential Inclusions The presentation in this section follows the description given in Boyd et al. (1994). Now the systems that are to be analyzed are defined. 5) where F is a set-valued function on Rn × R+ . 5) is found, it is denoted a trajectory. Note that nothing is said about uniqueness of a solution. If a system is initiated with different starting conditions x(0) the trajectories will not be identical.

These methods have good convergence properties and is still an evolving area with active research. A non-stationary method has an update scheme that is varying for every iterate of the algorithm. , the methods operate in the space called the Krylov subspace. 1. The Krylov subspace for a matrix A ∈ Rn×n and a vector v ∈ Rn is Km (A, v) = span{v, Av, . . , Am−1 v}. , 2006, p. 160) and Arnoldi (1951). An interesting point is that the Gram-Schmidt procedure is a special case of the Arnoldi algorithm when choosing v(1) = v/ v 2 .

29) 2 ∞ where the L2 -norm of a signal u is u 22 = 0 uT u dt. 28). 30) is satisfied. 30) implies that the L2 -gain is less than γ, integrate it from 0 to T with x(0) = 0 resulting in T V (x(T )) + 0 (z T z − γ 2 wT w) dt ≤ 0. 31) Since V (x(T )) ≥ 0 the L2 -gain is less than γ. 30) is T ATi P + P Ai + Cz,i Cz,i T Bw,i P P Bw,i −γ 2 Inu 0, i = 1, . . 32) n where P ∈ S++ . 32). 32). 28) with nz = nw is said to have dissipation η if T 0 (wT z − ηwT w) dt ≤ 0 holds for all trajectories, when x(0) = 0 and T ≥ 0.

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A structure utilizing inexact primal-dual interior-point method for analysis of linear differential inclusions by Janne Harju Johansson.

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