Explicit Expression For An Upper Bound On The H₂ Norm And Parametrization Of Control Gains In Structural Systems

This paper presents an explicit expression for an upper bound on the H₂ norm for structural systems, and derive an explicit parameterization of controller gains .By employing simple algebraic tools, we improve the classical methods which are time-consuming and intractable.First, we review the history and development of the control theory, and introduce the latest results of structural control theory. Then, we introduce the fundamental control theory in terms of mathematics, including the state space system (the controllability, the stabilizability, the observability, minimal realizations, and transfer functions), linear analysis (normed and inner product spaces, operators, frequency domain spaces: signals), Lyapunov equations and inequalities, observability and controllability operator and gramian, stabilizing controllers, H₂ optimal control. Finally,given an H₂ upper bound, we study the H₂ control synthesis for the input-output feedback. By employing simple algebraic tools, we derive an explicit expression for an upper bound on the H₂ norm and an explicit parameterization of controller gains which guarantee a prescribed H₂ norm performance of the closed-loop system. The effectiveness of the proposed bound and control design method becomes apparent, especially in very large scale structural systems where using classical methods of solving Lyapunov and Riccati equations are time-consuming or intractable.