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Unformatted text preview: ECE521 Linear Systems Fall 2009 Homework 1 Solutions Problem 1 : The example of a mechanical system given in lecture notes for Lecture 1 can be generalized to M ¨ q ( t ) + L ˙ q ( t ) + Kq ( t ) = f ( t ) , where q ∈ R k is a vector of positions, f ∈ R k is a vector of forces, M ∈ R k × k is an invertible mass matrix, L ∈ R k × k is a damping matrix, and K ∈ R k × k is a stiffness matrix. Choose a state vector x ( t ) for this system and derive a state-space model for this system in the standard, that is a model of the form braceleftbigg ˙ x = A ( t ) x ( t ) + B ( t ) u ( t ) y = C ( t ) x ( t ) + D ( t ) u ( t ) Assume that the output is y ( t ) = q ( t ) . Specify matrices A,B,C,D and the input u(t). The general form can be rewritten as ¨ q = − M- 1 L ˙ ( q − M- 1 Kq + M- 1 f ( t ) By defining x ( t ) as x ( t ) = parenleftbigg q ˙ q parenrightbigg we arrive at ˙ x = parenleftbigg k I k − M- 1 L − M- 1 K parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright A x + parenleftbigg k M- 1 parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright B f ( t ) The output equation is y ( t ) = ( I k k ) bracehtipupleft bracehtipdownrightbracehtipdownleft...
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This note was uploaded on 02/09/2010 for the course MAE 123 taught by Professor 123 during the Spring '10 term at École Normale Supérieure.
- Spring '10