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Unformatted text preview: System State and Differential Equations: Finding Zero Input Response EE 313 Linear Systems and Signals Fall 2008 Some Parts Adapted by Courtesy of Prof. Brian Evan s Prof. Adnan Kavak Dept. of Electrical and Computer Engineering The University of Texas at Austin 3  2 f y a y a y a y = + + + 1 2 y x y x y x = = = 3 2 1 ( 29 ( 29 ( 29 f x a x a x a x y x x y x x y + = = = 3 2 2 1 1 3 3 2 2 1 System State Example: Reformulate N thorder differential equation into N simultaneous firstorder differential equations Define three state variables After substituting the state variables, we obtain a statespace description as Lathi, 2 nd ed, Section 1.10 3  3 f x x x a a a x x x +  = 1 1 1 3 2 1 2 1 3 2 1 f b x A x + = System State By putting the state variables in vector x , If A is not a function of time, then solution is exp( M ) for matrix M yields a matrix dt t f e e t t t + = ) ( ) ( ) ( b x x A A 3  4 System State General form of statespace description Statespace descriptions can Describe timevarying and nonlinear systems Be simulated by computer (e.g. Spice for circuit simulation) Knowledge of state variables allows one to determine every possible output of the system Statespace descriptions covered in controls courses, e.g. EE 362K Intro. to Automatic Control f b x A x + = 3  5 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t f b t f dt d b t f dt d b t f dt d b t y a t y dt d a t y dt d a t y dt d N N M M M N M M M N N N N N N N + + + + = + + + + + 1 1 1 1 1 1 1 1 k k k dt d D dt d D dt d D = = = 2 2 2 ( 29 ( 29 ( 29 ( 29 t f b...
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This note was uploaded on 02/10/2009 for the course EE 313 taught by Professor Cardwell during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Cardwell

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