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lecture3 - EE 313 Linear Systems and Signals Spring 2009...

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System State and Differential Equations EE 313 Linear Systems and Signals Spring 2009 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans 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

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lecture3 - EE 313 Linear Systems and Signals Spring 2009...

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