By Geir E. Dullerud

Throughout the 90s strong keep watch over conception has visible significant advances and accomplished a brand new adulthood, based round the idea of convexity. The aim of this e-book is to provide a graduate-level path in this conception that emphasizes those new advancements, yet whilst conveys the most ideas and ubiquitous instruments on the center of the topic. Its pedagogical targets are to introduce a coherent and unified framework for learning the idea, to supply scholars with the control-theoretic historical past required to learn and give a contribution to the study literature, and to give the most rules and demonstrations of the main effects. The booklet can be of worth to mathematical researchers and computing device scientists, graduate scholars planning on doing study within the zone, and engineering practitioners requiring complex keep watch over options.

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**Additional info for A Course in Robust Control Theory**

**Example text**

We next consider two explicit examples which will be key to us later. Examples: Here we will consider representing hyperplanes for two explicit vector spaces, the spaces Rn and the symmetric matrices Sn. Let us rst consider Rn . Given a linear functional F : Rn ! R we see that it is completely de ned by the numbers F (e1 ) : : : F (en ) where fe1 : : : en g is the canonical basis for Rn . Thus given any vector x 2 Rn we have F (x) = x1 F (e1 ) + + xn F (en ) Conversely given any y 2 Rn a linear functional F is de ned on Rn by F (x) = y1 x1 + : : : + yn xn , which we express more concisely using matrix multiplication by F (x) = y x: Thus we see that any hyperplane in Rn is characterized by the equation y x=a n for some y 2 R and a 2 R.

A rst observation, which will be used extensively below is that (AB ) = B A for matrices A, B of compatible dimensions this follows directly by de nition. Another basic concept closely related to the adjoint is the Euclidean length of a vector x 2 C n , de ned by p jxj = x x This extends the usual de nition of magnitude of a complex number, so our notation will not cause any ambiguity. In particular, jxj2 = x x = n X i=1 jxi j2 : Clearly jxj is never negative, and is zero only when the vector x = 0.

As a further step, consider the space C m n of complex m n matrices of the form 2 a11 a1 n 3 A = 64 ... . ... 75 : am1 amn Using once again componentwise addition and scalar multiplication, C m n is a (real or complex) vector space. We now de ne two vector spaces of matrices which will be central in our course. First, we de ne the Hermitian conjugate or adjoint of the above matrix A 2 C m n by 2 a11 am 1 3 A = 64 ... . ... 75 2 C n m a1n amn where we use a to denote the complex conjugate of a number a 2 C .