ME 343 Control Systems Fall 2009Solution of State Space Equation426We consider the linear timeinvariant systemAnd we solve to obtainSolution by the Laplace TransformWe Laplace transform the state equ ID: 871734
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1 ME 343 Control Systems Fall 2009425ME
ME 343 Control Systems Fall 2009425ME 343 Control Systems ME 343 Control Systems Fall 2009Solution of State Space Equation426 We consider the linear, time-invariant systemAnd we solve to obtainSolution by the Laplace
2 Transform: We Laplace transform the sta
Transform: We Laplace transform the state equation to obtain )(0)(11sBUAsIxAsIsX tuLsUtxLsX )(,)( )(xssXtxL ME 343 Control Systems Fall 2009Solution of State Space Equation427We inverse Laplace transform to obtain
3 where we have used that )(0)(11sBUAsIxA
where we have used that )(0)(11sBUAsIxAsIsX ME 343 Control Systems Fall 2009State Transformation428We consider the linear, time-invariant system We define the state transformationThen w
4 e can writeto obtain ~ + ~ =DDCTCBTBATTA
e can writeto obtain ~ + ~ =DDCTCBTBATTA ~ , ~ , ~ , ~ 11 ME 343 Control Systems Fall 2009431 1p 2p 3p 4p U sY ME 343 Control Systems Fall 2009432Problem Definition:A system is said to be controllable if and only if
5 it is possible, by means of the input, t
it is possible, by means of the input, to transfer the system from anyinitial state (0) to any other state ) in a finite time Theorem:A system is controllable if and only if the matrix is full-rank. Controllability Matr