By Geir E. Dullerud
In the course of the 90s powerful keep watch over idea has obvious significant advances and completed a brand new adulthood, headquartered round the idea of convexity. The target of this booklet is to provide a graduate-level direction in this concept that emphasizes those new advancements, yet while conveys the most ideas and ubiquitous instruments on the center of the topic. Its pedagogical pursuits are to introduce a coherent and unified framework for learning the speculation, to supply scholars with the control-theoretic historical past required to learn and give a contribution to the examine literature, and to provide the most principles and demonstrations of the most important effects. The publication might be of worth to mathematical researchers and laptop scientists, graduate scholars planning on doing learn within the sector, and engineering practitioners requiring complicated regulate ideas.
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Additional info for A Course In Robust Control Theory
If H is a real matrix, that is in H n , then all the matrices in the construction above are also real, proving the latter part of the theorem. We remark in addition that the (real) eigenvalues of H can be arranged in decreasing order in the diagonal of . This follows directly from the above induction argument: just take 1 to be the largest eigenvalue. We now focus on the case where these eigenvalues have a de nite sign. Given Q 2 H n , we say it is positive de nite, denoted Q > 0, if x Qx > 0 for all nonzero x 2 C n .
Matrix Theory 43 Notice that since U = U ;1 for a unitary U , the above expression is a similarity transformation. Therefore the theorem says that a self-adjoint matrix can be diagonalized by a unitary similarity transformation. Thus the columns of U are all eigenvectors of H . Since the proof of this result assembles a number of concepts from this chapter we provide it below. Proof . We will use an induction argument. Clearly the result is true if H is simply a scalar, and it is therefore su cient to show that if the result holds for matrices in H n;1 then it holds for H 2 H n .
7, for the case X = xx1 2 R2 : 2 The convex set depicted in the gure represents the feasibility set C = fX : F (X ) Qg for the linear objective problem while we have drawn a bounded set, we remark that this is not necessarily the case. 7. 7. Also the picture suggests that there are no other local minima for the function in the set, namely for every other point there is a \descent" direction. This property, fundamental to convex optimization problems, is now stated precisely. 13. Suppose X0 is a local minimum of the linear objective problem, that is c(X0 ) c(X ) for every X 2 N (X0 ) \ C .
A Course In Robust Control Theory by Geir E. Dullerud