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Unformatted text preview: 17 MODAL ANALYSIS 17.1 Introduction The evolution of states in a linear system occurs through independent modes, which can be driven by external inputs, and observed through plant output. This section provides the basis for modal analysis of systems. Throughout, we use the state-space description of a system with D = 0: γx ˙ = Aγx + Bγu y x. γ = Cγ 17.2 Matrix Exponential 17.2.1 Definition In the instance of an unforced response to initial conditions, consider the system γ ˙ x = Aγx, γx ( t = 0) = γν. In the scalar case, the response is x ( t ) = νe at , giving a decaying exponential if a < 0. The same notation holds for the case of a vector γx , and matrix A : At γ γx ( t ) = e ν, where ( At ) 2 At e = I + At + + e 2! · · · At is usually called the matrix exponential. 84 17 MODAL ANALYSIS 17.2.2 Modal Canonical Form Introductory material on the eigenvalue problem and modal decomposition can be found in the MATH FACTS section. This modal decomposition of A leads to a very useful state-space representation. Namely, since A = V Φ V − 1 , a transformation of state variables can be made, γx = V γ z , leading to...
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- Fall '04
- WI, i'th mode