On existence of solutions to structured lyapunov inequalities

Abstract

In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist. The motivation for the problem comes from a recently proposed basis pursuit algorithm. In particular, this algorithm can provide approximate solutions to optimisation programmes with constraints involving Lyapunov inequalities using linear or second order cone programming. This algorithm requires an initial feasible point, which we aim to provide in this paper. Our existence conditions are based on the so-called H matrices. We also establish a link between H matrices and an application of a small gain theorem to the drift matrix. We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality.

On the Existence of Block-Diagonal Solutions to Lyapunov and ${\mathcal {H}_\infty }$ Riccati Inequalities

“…Proof. According to Proposition 3 there exist positive scalars e i , d i , g i , f i such that (7,8) hold. Let…”

“…Consider the network of the stable subsystems G i = C i (sI − A i ) −1 B i (the matrices A i are Hurwitz) interconnected through matrices M , N and K as in Figure 2. Consider the comparison system (5) with the state-space matrices defined in (34) and let there exist positive vectors e, d, g, f , and a scalar δ satisfying (7,8). Then (i) the network also satisfies conditions (32, 33), with Y i…”

“…In this paper, we focus on a generalisation of positive systems based on diagonally dominant matrices since it is known that for this class of systems separable Lyapunov functions exist [7] and can be computed using linear programming [8]. Recently, the diagonal dominant approach was applied to block-partitioned matrices, which lead to conditions for the existence of block-diagonal solutions to Lyapunov inequalities [9].…”

Block-diagonal solutions to Lyapunov inequalities and generalisations of diagonal dominance

“…For example, if A is Metzler or A is lower triangular (a ij = 0, for all i < j) then A is Hurwitz if and only if M(A) is Hurwitz. More generally, if M(A) is Hurwitz, then A is Hurwitz and A admits a diagonal solution to (2) [5], which can be constructed using linear algebra, linear or second order cone programming [6]. In the proof of these results Proposition 2 is applied to a Hurwitz Metzler matrix M(A), which leads to existence of positive d i , e i such that (4) and (5) hold, that is A is strictly row and column scaled diagonally dominant.…”

Block Factor-Width-Two Matrices in Semidefinite Programming

“…The key idea of this extension is to partition a matrix into a set of nonintersecting blocks of entries and enforce the SDD constraints on these blocks instead of the individual entries [21]. We introduce the class of block factor-width-two matrices based on the block SDD definitions from [22], [23]. A block factorwidth-two matrix is also PSD and the constraint "the matrix is block factor-width-two" can be enforced using a number of PSD constraints whose size is determined by the size of the blocks.…”