where 1 is an appropriately dimensioned unit vector.
Using the Newton-Leibniz formula, for any appropriately dimensioned matrices N i, M i, the following equations are true: (14).
A useful property may now be derived by performing the following parameter transformation on the system in Eq. (27) such that (29) (30)where I are appropriately dimensioned identity matrices.
if there exit exist symmetric matrix X = X T > 0, and any appropriately dimensioned matrices V 1, V 2, such that the above LMI holds, ?
(8) where (x_{c} k)inmathbb{R}^{n}) is the state estimate of system (1), and (A_{ci}, B_{ci}, C_{ci}) are appropriately dimensioned parameters matrices to be determined.
provided that the scalar ?, symmetric matrices P ˆ 11, P ˆ 22 and appropriately dimensioned matrices P ˆ 12, X, Y, U, A ˆ, B ˆ, C ˆ, D ˆ satisfy the following LMIs, (15) (15).
end{aligned} (mathcal{P}) is the appropriately dimensioned square permutation matrix such that mathcal{P} [a_{1} quad a_{2} quad cdots quad a_{2m} ] = [ a_{1} quad a_{3} quad cdots quad a_{2m-1} qua_{2_{2} quad a_{4} quad cdots quad a_{2m} ].
Denote 〈 x, H y 〉 T = ∫ 0 T x T ( t ) H y ( t ) d t, in this article, we focus our attention on the quadratic supply rate E ( v, e, T ) = 〈 e, Q e 〉 T + 2 〈 e, S v 〉 T + 〈 v, R v 〉 T, where Q, S, R are appropriately dimensioned, and Q, R are symmetric matrices.
if there exist symmetric matrix P = P T > 0, any appropriately dimensioned matix V and a given scalar ?, such that the following inequality hold: [ - V A T - A V T - B P + V T - ? V A T V C T * - ? 2 I - ? B T D T * * ? V + ? V T 0 * * * - I ] < 0. (7).
end{aligned} (25) On the other hand, for any appropriately dimensioned invertible matrix M, the following zero equality holds: begin{aligned} 0=&2 bigl[e^{T} k)M+eta^{T} k)M bigr] Biggl[ A-KC_{1}-I)e(k)-KC_{2}e Biggl[ A-KC_{ bigr)+B_{1}-I e k -KC_{2}er) &+B_{2}gbigl(e bigl(k-tau(k) bigr)bigr)+B_{3}sum _{i=1}-I e k -KC_{2}e(i)g bigl(x(k-i) bigr)+(D_{1}-I e k -KC_{2}ek)-eta(k) bigl k-tau
