lecture 21

lecture 21 - k x ) 1 ( ) 1 ( ) ( ) ( ) 1 ( Let , kT t = + =...

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1 Analysis digital control systems in state space Block diagram of state space model representation ) ( ) ( ) ( ) ( ) ( ) ( t Du t Cx t y t Bu t Ax t x + = + =
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2 ) ( ) ( ) ( ) ( ) ( ) 1 ( k Du k Cx k y k Hu k Gx k x + = + = + Example 1. Solving discrete-time state-space equation Recursive approach ) ( ) ( ) ( ) ( ) ( ) 1 ( k Du k Cx k y k Hu k Gx k x + = + = + ...... ) 2 ( ) 1 ( ) 0 ( ) 0 ( ) 2 ( ) 2 ( ) 3 ( ) 1 ( ) 0 ( ) 0 ( ) 1 ( ) 1 ( ) 2 ( ) 0 ( ) 0 ( ) 1 ( 2 3 2 Hu GHu Hu G x G Hu Gx x Hu GHu x G Hu Gx x Hu Gx x + + + = + = + + = + = + =
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3 = = + + = + = 1 0 1 1 0 1 ) ( ) ( ) 0 ( ) ( ) ( ) 0 ( ) ( k j j k k k j j k k k Du j Hu G C x CG k y j Hu G x G k x ) ( k G k ψ = is defined as state transition matrix. z transform approach ) ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) 1 ( 1 1 z HU G zI zx G zI z X z HU zx z X G zI z HU z GX zx z zX k Hu k Gx k x + = + = + = + = + Example 2.
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4 Pulse transfer function matrix ) ( ) ( ) ( ) ( ) ( ) 1 ( k Du k Cx k y k Hu k Gx k x + = + = + ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( z DU z CX z Y z HU z GX zx z zX + = + = , Let , 0 ) 0 ( = x ) ( ] ) ( [ ) ( ) ( ) ( ) ( 1 1 z U D H G zI C z Y z HU G zI z X + = = ) ( ) ( ) ( ) ( 1 z F D H G zI C z U z Y = + =
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5 Since , G zI G zI adj G zI = ) ( ) ( 1 G zI H G zI Cadj z F = ) ( ) ( The characteristic equation of the discrete-time system is: 0 = G zI Discretisation of continuous-time state space equations + = + = t t t A t t A d Bu e t x e t x t Bu t Ax t x 0 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( τ Let : kT t T k t = + = 0 , ) 1 ( + + + = + T k kT A T k A AT d Bu e e k x e k x ) 1 ( ) 1 ( ) ( ) ( ) 1 ( Assuming the zero order hold circuit is used, t cons k u u tan ) ( ) ( = =
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6 + + + = + T k kT A T k A AT d kT Bu e e k x e
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Unformatted text preview: k x ) 1 ( ) 1 ( ) ( ) ( ) 1 ( Let , kT t = + = + T At AT AT dt kT Bu e e k x e k x ) ( ) ( ) 1 ( Let , t T = + = + T A AT d kT Bu e k x e k x ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( k Hu k Gx k x + = + B d e T H e T G T A AT = = ) ( ) ( .... ! 3 1 ! 2 1 3 3 2 2 + + + + = T A T A AT I e AT TB T A AT I B I e A B d e T H T A AT I e T G AT T A AT ...) ! 3 1 ! 2 1 ( ) ( ) ( ! 2 1 ) ( 2 2 1 2 2 + + + = = = + + = = Example 3....
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lecture 21 - k x ) 1 ( ) 1 ( ) ( ) ( ) 1 ( Let , kT t = + =...

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