الفهرس 
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Abstract
1.1 Historical Overview and Motivation
The design of a stabilizing feedback laws for unstable systems has been studied extensively in
the past and still attracts many researchers. In this thesis we analyze one of this family, namely
the optimal output feedback design problem. TIlls problem was considered first by Levine and
Athans [39]. They introduced a first order algorithm to solve the underlying problem by tack-
ling the nonlinear system of matrix equations obtained from the first order necessary optimality
conditions. This algorithm will be discussed briefly in chapter 5. Moreover, we develope a first
order algorithm by using the Lagrangian method, and we show that the Lagrangian algorithm is
identical to the algorithm of Levine and Athans. The numerical results has shown that the La-
grangian method perofonns better than the Levine-Athans algorithm. But, as a disdvantage of
both algorithms, if an in stabilizing feedback gain is encountered at any iteration, both algorithms
breakdown. Anderson and Moore [2] introduced an algorithm which can be easily implemented.
In their method the positive definite part of the second order information is used while the pos-
sibly indefinite part of the Hessian is rejected. Only linear rate of convergence can be expected
from their method. Moreover, similar to the Levine-Athans and the Lagrangian methods, the
algorithm breaks down if at any iteration in stabilizing feedback gain is reached. Toivonen and
Maki1a [69] solved the problem by Newton’s method with linesearch as a globalization tech-
nique. Moreover, Rautert and Sachs [55] introduced the structured quasi-Newton method as a
comptetive method for solving the optimal output feedback design problem. In addition, they
Ihowoc1 that the Anc1craon-More alaorithm QonVcracl at Q,uadratig rate if thc inclefinite part of
the Hessian vanishes at the solution. On the other hand, in [55] and [69] the condition on stabil-
ity was fulfilled by checking an eigenvalues-condition, in which the stabilizing feedback gain
F E RPxr is to be chosen from the set of stabilizing feedback gains
Ds:= {F E RPxr : Re(vi(AF)) < 0, i = 1,2, ... ,n},
(1.1)
where Re(lIj(AF)) are the real parts of the eigenvalues of the closed-loop system matrix AF :=
[At BFC1, with A E IRnxn, B E Rnxp, and C E ]Rrxn. Moreover, in [55] and [69] this condi-
tion on stability is embedded in the Annijo step size reduction rule. But, this way for checking
stabilizing feedback gains every iteration is practically ineffecient. Leibfritz [40], [41] used a.