Application of the transfer matrix and by solving the coupled field equations, solutions to the mechanical displacements are obtained for the film and elastic substrate, respectively.
G is the (q × p) lead field matrix obtained by solving the forward problem i.e., by estimating the contribution of each unit dipolar source on the sensors (Hallez et al. 2007).
Note that the intrinsic timescale of node j is absorbed into the matrix W. By solving Equation 9, ϕ j (t) can be expressed in terms of the eigenvectors of the connection matrix W, yielding (10) ϕ j (t ) = ∑ λ A λ (t ) v λ (j ) (Rugh, 1995).
Stochastic analysis is conducted, in combination with the Lyapunov functional approach, to derive sufficient conditions for the existence of the desired estimators whose gain matrices are obtained by solving a set of matrix inequalities.
In addition, by using cone complementarity algorithm, the solutions to the observer gain matrices are obtained by solving a set of linear matrix inequalities (LMIs).
The state feedback matrices are computed by solving LMIs in terms of the matrices defining the individual agent dynamics.
The robust Kalman filter gain matrices are designed by solving two algebraic Riccati equations (AREs) that are expressed as two linear matrix inequality (LMI) feasibility conditions.
The relay matrices are found by solving an optimization problem.
The optimum relay gain matrices are obtained by solving the optimization problem using Lagrange dual variables method.
