Bounds for the solution of the Lyapunov matrix equation — A unified approach

A remark on the stabilization of partially linear composite systems

“…Applying Theorem 3.8, we obtain 1 (P ) 4: However, the results in [1], [2], [5], [8], and [9] cannot be used for this example. It can be seen that using the modified Lyapunov equation (10), most of the known results in the current literature can be generalized, and better estimates can be obtained in this way.…”

“…Proof: First of all, if the linear subsystem (2) satisfies H1)-H3), then it is invertible, weakly minimum phase, and with CB symmetric positive definite, and it is possible to choose the matrix K 2 M k; p (IR) and the symmetric positive definite matrix P 2 M p; p (IR) such that y T Qy = 0 ) Cy = 0: (8) Indeed, as done in [6], one can assume, without loss of generality, that (2) is in the special coordinate basis (see [7]) _ y 01 = A 01 y 01 + A 11 y 1 _ y 02 = A 02 y 02 + A 12 y 1 _ y1 = D01y01 + D02y02 + D1y1 + CBũ y = y 1 with A 01 Hurwitz, A 02 + A T 02 = 0, and take K = (K 01 ; K 02 ; K 1 ) Hence, y T Q y = ky 01 k 2 + ky 1 k 2 and so y T Q y = 0 ) Cy = y 1 = 0: Assume now that (5) is of L-T, and set X(x) = f(x; 0); x 2 IR n .…”

“…In order to prove Theorem 1, let us show that is the origin of IR n+p . By (8), onẼ the vector field Z is given by…”

Matrix bounds of the solutions of the continuous and discrete Riccati equations--a unified approach

“…Those bounds unify the results for solutions of the continuous and discrete Riccati and Lyapunov equations. However, Mrabti and Hmamed (1992) and Kwon et al (1996) proposed only the eigenvalue bounds. As mentioned in the above descriptions, matrix bounds are the most general ones.…”

“…Hence they are the most general findings. Recently, by making use of the delta operator developed by Middleton and Goodwin (1990), unified approaches were presented to obtain eigenvalue bounds of the solutions of the Riccati and Lyapunov equations (Mrabti andHmamed 1992, Kwon et al 1996). Those bounds unify the results for solutions of the continuous and discrete Riccati and Lyapunov equations.…”