The Riccati Equation by Sergio Bittanti (auth.), Sergio Bittanti, Alan J. Laub, Jan

By Sergio Bittanti (auth.), Sergio Bittanti, Alan J. Laub, Jan C. Willems (eds.)

Conceived by means of count number Jacopo Francesco Riccati greater than 1 / 4 of a millennium in the past, the Riccati equation has been commonly studied within the next centuries. because its advent up to the mark concept within the sixties, the matrix Riccati equation has recognized a powerful variety of purposes, reminiscent of optimum keep an eye on, H? optimization and powerful stabilization, stochastic attention, synthesis of linear passive networks, to call yet a couple of. This e-book comprises eleven chapters surveying the most thoughts and effects on the topic of the matrix Riccati equation, either in non-stop and discrete time. idea, purposes and numerical algorithms are widely offered in an expository method. As a foreword, the background and prehistory of the Riccati equation is concisely presented.

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45». The following result (proved in [13» describes all minimal unitary completions in terms of hermitian solutions of the associated Riccati equation. 29. 43) t*] is the state characteristic matrix. 44) D* D)I/2; BD*)(1 - DD*)-1/2; Y = (I - D* D)-1/2(iB* H- l - D*C), is a minimal unitary completion ofW(,,\). 43). \) with fixed value at infinity. \). P. Lancaster and L. 11 J-Spectral Factorization Recently, the Hoc optimization problems attracted the attention of many mathematicians and engineers, and the continuous algebraic Riccati equation and its hermitian solutions play a significant role in development of the theory.

49) are necessarily real. 49), and by the maximality property we have X+ ~ X+. This implies X+ = X+, so X+ is real. The same argument applies to X_as well. 4). It is easy to see that X is a real special solution if and only if the graph G(X) (considered as a subspace in R2n) is Mr-invariant and H-nonnegative. 32. Let A, C, D be real with C = C T , D ~ 0 and (A, D) controllable. Assume further that the matrix H is invertible and has signature zero. 49). Proof. 7, we have to demonstrate existence of an n-dimensional Mr-invariant H -non-negative subspace in R2n.

HAUTUS. • Controllability and observability conditions 0/ linear autonomous systems. Ned. Akad. wet. Proc•• Ser. A 12(1969). 443-448. WONHAM. • Linear Multivariable Control, Springer-Verlag. Berlin. 1970 and 1979. RAN. M. and RODMAN. • Stability of invariant maximal semidefinite subspaces II. Applications: selfadjoint rational matrix functions, algebraic Riccati equation. Linear Algebra and Appl. 63(1984). 133-173. PAIGE. C. and VAN LOAN. • A Schur decomposition for Hamiltonian matrices. Linear Algebra and Appl.

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