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This study presents a linear programming (LP)-based multivariable root locus following technique for coordinating the gain settings of power system stabilisers (PSSs). The stabiliser robustness is accounted for in the design problem by simultaneously considering the state-space representations and multivariable root loci corresponding to different operating scenarios. The proposed technique computes a curve in the PSS gain parameter space such that when the PSS gains move along this curve to their optimal values, the branches of the corresponding multivariable root loci terminate at satisfactory points in the complex plane. The curve in the gain parameter space is computed via a linear program that successively minimises the Euclidean distance between the unsatisfactory and satisfactory eigenvalue locations. The design method is demonstrated on a 39-bus test system with 14 operating scenarios. A comparison is carried out between the coordination results of two PSS structures, one involving two phase-lead blocks and the other comprised of two phase-lead blocks and a phase-lag block.