A Cage of Butterflies by Brian Caswell

By Brian Caswell

Mikki and the others dwell at 'the farm' - a complicated studying facility, a think-tank for a host of teens with very excessive IQs. yet what's fairly occurring on the farm? And what concerning the 5 a lot more youthful teenagers often called the infants, frail as butterflies?

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12) then ˜ z(k + 1) = Az(k). 13) The following result will present an easy way to design the controller matrix gain K. 7. 11) is stable if there exist a positive definite maQ11 0 trices Q = with Q11 ∈ Rn×n , Q22 ∈ Rτ n×τ n and a matrix 0 Q22 Y ∈ Rm×n such that the following inequalities holds ¯T −Q QA¯T + I1T Y T B ¯ ¯ AQ + BY I1 −Q < 0. 14) Then, the state feedback can be chosen as K = Y Q−1 11 with which the resulting closed-loop system is stable. Proof. 11) is stable if and only if there exists positive definite matrix P such that A˜T P A˜ − P < 0.

The existence condition of a linear sliding surface for systems with mismatched uncertainties was given in [25], [26], [100], but their methods cannot be applied to systems with time-delay. In [161], a new robust stability criterion for uncertain time-delay systems is given and the SMC is proved to be applicable. There, due to use of matrix norm, the result is more or less conservative and complicated which leads to inconvenience in designing the sliding surface for uncertain time-delay systems.

0 0 .. ··· ··· .. 0 0 .. 0 0 ··· I ⎥ 0⎥ ⎥ 0⎥ ⎥ .. 8) 0⎥ ⎥ 0⎥ ⎥ .. ⎦ 0 for j = 1, 2, · · · , τ¯. 9) where σ is the piecewise constant switching signal taking value from the finite index set F = {0, 1, · · · , τ¯}. According to the theory of switched systems [172], we have the following result. 5. 1) with time-varying delay τ (k) is stable if there exists a positive definite matrix P ∈ R(¯τ +1)n×(¯τ +1)n such that the following linear inequalities hold for any i ∈ {0, 1, · · · , τ¯}: A¯Ti P A¯i − P < 0.

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