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EE313_system_zero_input_response

# EE313_system_zero_input_response - EE 313 Linear Systems...

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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

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3 - 2 f y a y a y a y = + + + 0 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 0 3 3 2 2 1 System State Example: Reformulate N th-order differential equation into N simultaneous first-order differential equations Define three state variables After substituting the state variables, we obtain a state-space description as Lathi, 2 nd ed, Section 1.10
3 - 3 f x x x a a a x x x + - - - = 1 0 0 1 0 0 0 1 0 3 2 1 2 1 0 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 + = 0 ) ( ) 0 ( ) ( b x x A A

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3 - 4 System State General form of state-space description State-space descriptions can Describe time-varying 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 State-space 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 D b D b D b t y a D a D a D D P N N M M N M M N D Q N N N N

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EE313_system_zero_input_response - EE 313 Linear Systems...

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