By Magdi S. Mahmoud

Switched Time-Delay platforms provides an introductory, but entire, remedy of switched time-delay structures via together combining primary attributes: the process dynamics possesses an inherent time-delay and the approach operational mode undergoes switching between various modes. whereas each one characteristic has been tested separately in numerous texts, the combination of either attributes is kind of specified and merits detailed attention.

The fabric contained during this booklet isn't just geared up to target the recent advancements within the research and regulate methodologies for such switched time-delay platforms, however it additionally integrates the influence of the hold up issue on matters equivalent to asymptotic/exponential balance, suggestions stabilization and keep an eye on layout.

Switched Time-Delay structures consists of chapters outfitted with illustrative examples and correct notes. will probably be necessary to engineers and researchers within the box of switched time-delay systems.

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**Extra resources for Switched Time-Delay Systems: Stability and Control**

**Example text**

7 Partitioned Matrices Given a partitioned matrix (matrix of matrices) of the form ⎡ M =⎣ A B ⎤ ⎦ C D where A, B, C, and D are of compatible dimensions. Then (1) if A−1 exists, a Schur complement of M is defined as D − C A−1 B, and (2) if D −1 exists, a Schur complement of M is defined as A − B D −1 C. When A, B, C, and D are all n × n matrices, then ⎡ a) det ⎣ ⎡ b) det ⎣ A B C D A B ⎤ ⎦ = det(A) det(D − C A−1 B), det(A) = 0 ⎤ ⎦ = det(D) det(A − B D −1 C), det(D) = 0 C D In the special case, we have ⎡ det ⎣ A B C 0 ⎤ ⎦ = det(A) det(C) 30 2 Mathematical Foundations where A and C are square.

X j . . 0]t An important application arises in the Taylor’s series expansion of f (x) about xo Δ in terms of δx = x − xo . The first three terms are 26 2 Mathematical Foundations f (x) = f (xo ) + t ∂ f (x) ∂x ∂ 2 f (x) δx ∂x2 1 δx + δx t 2 where ⎡ ∂ f (x) ⎤ ⎢ ⎢ ∂ f (x) ⎢ =⎢ ⎢ ∂x ⎢ ⎣ ⎡ ∂ 2 f (x) ∂ = ∂x ∂x2 ∂ f (x) ∂x t ⎢ ⎢ =⎢ ⎣ ∂ x1 .. ∂ f (x) ∂ xn ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∂ 2 f (x) ∂ x12 .. ∂ 2 f (x) ∂ xn ∂ x1 ··· .. ··· ∂ 2 f (x) ∂ x1 ∂ xn ⎤ ⎥ ⎥ ··· ⎥ ⎦ 2 ∂ f (x) ∂ xn2 The derivative of a real scalar-valued function f (A) with respect to a matrix ⎡ ⎤ A11 · · · A1n ⎢ ⎥ A = ⎣ ...

The first three terms are 26 2 Mathematical Foundations f (x) = f (xo ) + t ∂ f (x) ∂x ∂ 2 f (x) δx ∂x2 1 δx + δx t 2 where ⎡ ∂ f (x) ⎤ ⎢ ⎢ ∂ f (x) ⎢ =⎢ ⎢ ∂x ⎢ ⎣ ⎡ ∂ 2 f (x) ∂ = ∂x ∂x2 ∂ f (x) ∂x t ⎢ ⎢ =⎢ ⎣ ∂ x1 .. ∂ f (x) ∂ xn ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∂ 2 f (x) ∂ x12 .. ∂ 2 f (x) ∂ xn ∂ x1 ··· .. ··· ∂ 2 f (x) ∂ x1 ∂ xn ⎤ ⎥ ⎥ ··· ⎥ ⎦ 2 ∂ f (x) ∂ xn2 The derivative of a real scalar-valued function f (A) with respect to a matrix ⎡ ⎤ A11 · · · A1n ⎢ ⎥ A = ⎣ ... . · · · ⎦ ∈ Ren×n An1 · · · Ann is given by ⎡ ∂ f (A) ∂ A11 ∂ f (A) ⎢ .