By Geir E. Dullerud
Throughout the 90s strong keep watch over thought has obvious significant advances and completed a brand new adulthood, situated round the suggestion of convexity. The target of this booklet is to offer a graduate-level path in this concept that emphasizes those new advancements, yet while conveys the most rules and ubiquitous instruments on the center of the topic. Its pedagogical ambitions 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 offer the most principles and demonstrations of the most important effects. The booklet might be of price to mathematical researchers and laptop scientists, graduate scholars planning on doing learn within the quarter, and engineering practitioners requiring complex keep watch over thoughts.
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Extra info for A Course in Robust Control Theory
This too is an LMI in the variables X and Y . To see this explicitly let Z := (X Y ) 2 Sn Rm n and de ne F (Z ) = A XA + BY + Y B . Then F : Sn Rm n ! H n is a linear map and the LMI can be written compactly as F (Z ) < ;T: With these examples and de nition in hand, we will easily be able to recognize an LMI. Here we have formulated LMIs in terms of the Hermitian matrices, which is the most general situation for our later analysis. In some problems LMIs are written over the space of symmetric matrices Sn, and this is the usual form employed for computation.
We illustrate this by a simple example. Example: Let A 2 Rn n , b c 2 Rn , and d 2 R. The inequality (Ax + b) (Ax + b) ; c x ; d < 0 is not an LMI, since the expression is quadratic in the variable x. However the Schur complement formula implies it is equivalent to c x + d (Ax + b) > 0: Ax + b I Since the left hand side matrix now depends a nely on x, the latter is an LMI and can easily be expressed in the form F (x) < Q. As our course progresses we will nd that many control problems can be formulated in terms of nding solutions to LMIs.
Then F : Sn Rm n ! H n is a linear map and the LMI can be written compactly as F (Z ) < ;T: With these examples and de nition in hand, we will easily be able to recognize an LMI. Here we have formulated LMIs in terms of the Hermitian matrices, which is the most general situation for our later analysis. In some problems LMIs are written over the space of symmetric matrices Sn, and this is the usual form employed for computation. 4. Linear Matrix Inequalities 49 to the following discussion, and furthermore in the exercises we will see that the Hermitian form can always be converted to the symmetric form.
A Course in Robust Control Theory by Geir E. Dullerud