Hosoo Lee

Invariant metrics, contractions and nonlinear matrix equations

In this paper we consider the semigroup generated by the self-maps on the open convex cone of positive definite matrices of translations, congruence transformations and matrix inversion that includes symplectic Hamiltonians and show that every member of the semigroup contracts any invariant metric distance inherited from a symmetric gauge function. This extends the results of Bougerol for the Riemannian metric and of Liverani–Wojtkowski for the Thompson part metric. A uniform upper bound of the Lipschitz contraction constant for a member of the semigroup is given in terms of the minimum eigenvalues of its determining matrices. We apply this result to a variety of nonlinear equations including Stein and Riccati equations for uniqueness and existence of positive definite solutions and find a new convergence analysis of iterative algorithms for the positive definite solution depending only on the least contraction coefficient for the invariant metric from the spectral norm.

Weighted geometric means

Abstract. Taking a weighted version of Bini–Meini–Poloni symmetrization procedure for a multivariable geometric mean, we propose a definition for a weighted geometric mean of n positive definite matrices, where the weights vary over all n-dimensional positive probability vectors. We show that the weighted mean satisfies multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean. Significant portions of the derivation can be and are carried out in general convex metric spaces, which means that the results have broader application than the setting of positive definite matrices.

Finite-time leaderless consensus of uncertain multi-agent systems against time-varying actuator faults

Abstract This paper addresses the finite-time leaderless consensus problem for a class of continuous-time multi-agent systems subject to linear fractional transformation uncertain parameters using an observer-based fault-tolerant controller. Here, it is considered that the network of the system is described by an undirected graph subject to fixed topology and the aforementioned controller is impacted by time-varying actuator faults. Then, the desired consensus protocols are proposed in such a way that the effects of possible uncertainties and actuator faults are compensated efficiently within a prescribed finite-time period. More precisely, the leaderless consensus analysis is carried out in the framework of Lyapunov-Krasovskii functional and the required conditions for the existence of proposed fault-tolerant controller are derived in terms of linear matrix inequalities. Moreover, the proposed consensus design parameters can be computed by solving a set of linear matrix inequality constraints. Finally, two examples including a formation flying satellites model are provided to show the efficiency and usefulness of the proposed control scheme.

Weighted Carlson Mean of Positive Definite Matrices

Taking the weighted geometric mean (11) on the cone of positive definite ma- trix, we propose an iterative mean algorithm involving weighted arithmetic and geometric means of n positive definite matrices which is a weighted version of Carlson mean pre- sented by Lee and Lim (13). We show that each sequence of the weigthed Carlson iterative mean algorithm has a common limit and the common limit of satisfies weighted multidi- mensional versions of all properties like permutation symmetry, concavity, monotonicity, homogeneity, congruence invariancy, duality, mean inequalities.