By Dullerud G.E., Paganini F.

Throughout the 90s strong keep an eye on conception has obvious significant advances and completed a brand new adulthood, founded round the suggestion of convexity. The target of this booklet is to provide a graduate-level direction in this thought that emphasizes those new advancements, yet while conveys the most rules and ubiquitous instruments on the middle 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 history required to learn and give a contribution to the learn literature, and to provide the most rules and demonstrations of the foremost effects. The publication can be of worth to mathematical researchers and machine scientists, graduate scholars planning on doing examine within the quarter, and engineering practitioners requiring complicated keep watch over suggestions.

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**Sample text**

Linear Matrix Inequalities 51 The latter is a linear objective problem in the variables t 2 R and X 2 X , and the corresponding LMI constraint is automatically (strictly) feasible. It is an easy exercise to see that J < 0 if and only if the LMI F (X ) < Q is feasible. Given these relationships, we focus for the rest of the section on the linear objective problem, with the assumption that the strict LMI F (X ) < Q is feasible. 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.

Given these relationships, we focus for the rest of the section on the linear objective problem, with the assumption that the strict LMI F (X ) < Q is feasible. 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.

Example: Let X = R2 nd the in mum of x1 subject to ;x1 0 0 1 : 0 ;x2 1 0 It is easy to show that the answer is zero, but this value is not achieved by any matrix satisfying the constraint notice this happens even in the case of non-strict LMI constraints. Clearly the linear objective problem makes sense only if the LMI is feasible, which seems to imply that such problems are of higher di culty than feasibility. In fact, both are of very similar nature: In the exercises you will show how the linear objective problem can be tackled by a family of feasibility questions.