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Control of Singular Systems with Random Abrupt Changes by El-Kébir Boukas

By El-Kébir Boukas

This ebook offers with the category of singular platforms with random abrupt alterations sometimes called singular Markovian leap platforms. Many difficulties like stochastic balance, stochastic stabilization utilizing kingdom suggestions keep watch over and static output keep an eye on, Hinfinity regulate, filtering, assured expense keep watch over and combined H2/Hinfinity keep an eye on and their robustness are tackled. regulate of singular structures with abrupt alterations examines either the theoretical and useful points of the regulate difficulties handled within the quantity from the perspective of the structural houses of linear systems.

The thought awarded within the varied chapters of the amount are utilized to examples to teach the usefulness of the theoretical effects. keep watch over of singular platforms with abrupt adjustments is a superb textbook for graduate scholars in powerful keep watch over conception and as a reference for educational researchers on top of things or arithmetic with curiosity up to speed idea. The reader must have accomplished first-year graduate classes in likelihood, linear algebra, and linear platforms. it is going to even be of serious price to engineers training in fields the place the structures might be modeled by way of singular platforms with random abrupt changes.

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Let us now pre- and post-multiply the left hand side term by X (i) and X(i) respectively. This gives: X (i)A (i) + A(i)X(i) + X (i)K (i)B (i) + B(i)K(i)X(i) N εP λi j X (i) X −1 ( j) + X − ( j) X(i) < 0 . 2, we get: X −1 ( j) + X − ( j) ≤ I + X −1 ( j)X − ( j) = I + X ( j)X( j) −1 , that gives in turn: X (i)A (i) + A(i)X(i) + X (i)K (i)B (i) + B(i)K(i)X(i) N + λii X (i)E (i) + εP λi j X (i) I + X ( j)X( j) j=1, j i −1 X(i) < 0 . 2 Design of State Feedback Stabilization Notice that N εP λi j X (i)X(i) = Zi (X)Zi (X) , j=1, j i N εP λi j X (i) X ( j)X( j) −1 X(i) = Si (X)V −1 (X)Si (X) , j=1, j i with −1 V(X) = diag ε−1 P X (1)X(1), · · · , εP X (i − 1)X(i − 1), −1 ε−1 P X (i + 1)X(i + 1), · · · , εP X (N)X(N) , Si (X) = λi1 X (i), · · · , ··· , Zi (X) = λii+1 X (i), λiN X (i) , εP λi1 X (i), · · · , ··· , λii−1 X (i), εP λii−1 X (i), εP λii+1 X (i), εP λiN X (i) .

29) where J(i) = P (i)A(i) + A (i)P(i) + εA (i)E (i)E A (i) N + λii E (i)P(i) + λi j E ( j)P( j) . 30) As we did previously for the nominal case with the different assumptions on how to get an upper bound of the term E (i)P(i), we can easily establish the results of the following corollaries. 1 Let εP = (εP (1), · · · , εP (N)) be a given set of positive scalars. 31) 34 2 Stability with the following constraints: εP (i) P(i) + P (i) ≥ E (i)P(i) = P (i)E(i) ≥ 0 . 32) Now if we use the fact that E (i)P(i) ≤ ε(i)P (i)P(i), for any ε(i) > 0, we get the following results.

J=1, j i Using this and Schur complement we get the following set of LMIs: ⎡ ⎤ ⎢⎢⎢ J(i) P (i)DA (i) Si (P) ⎥⎥⎥ ⎢⎢⎢ ⎥ 0 ⎥⎥⎥⎥ < 0 , ⎢⎢⎣ DA (i)P(i) −εA (i)I ⎦ Si (P) 0 −Xi (ε) where J(i) = A (i)P(i)+P (i)A(i)+λii E (i)P(i)+εA (i)E A (i)E A (i)+ which combined with the following constraints: N 1 j=1, j i 4 λi j ε( j)I , E (i)P(i) = P (i)E(i) ≥ 0, ∀i ∈ S , give the required conditions to check if the system is piecewise regular, impulse-free and stochastically stable. This result is given by the following corollary.

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