Abstract:
A lot of control tasks, such as, for example, the synthesis of static output feedback, are expressed in the form of bilinear matrix inequalities. Solving such tasks based on iterative algorithms attended with considerable time, especially in the case of large-scale systems. If no solution is found for the initial values, then repeating calculations with different initial values does not guarantee success. The reason is the non-convexity of the feasible sets. The article investigates the possibility of bilinear matrix inequalities reducing to linear matrix inequalities by replacing of Lyapunov function matrix arbitrary to block-diagonal matrix. A sufficient condition for such a replacing is Kimura condition feasibility. It is proved that the necessary conditions for the linear matrix inequality feasibility are satisfied by two linear non-generate transformations of the system basis. To the synthesis problem investigating within the framework of linear matrix inequalities computational experiments were performed, in which 1000 linear systems were randomly generated. Based on the results of computational experiments, a hypothesis is proposed according to which the Kimura condition is a sufficient condition for bilinear matrix inequalities reducing to linear matrix inequalities with nonempty feasible sets.
Keywords:static output feedback, Kimura condition, linear matrix inequalities, Hurwitz matrix.
UDC:517.977 BBK:
22.18
Received: June 2, 2025 Published: September 30, 2025
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