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lecture10_small

# lecture10_small - Discrete-time equivalents Continuous to...

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ZOH equivalence in state-space ZOH Equivalence P ( s ) ZOH a a T A A A A u ( k ) u ( t ) y ( t ) y ( k ) P ( s ) dx ( t ) dt = Ax ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) . We would like to get a description of the form, P ( z ) x ( k + 1) = A d x ( k ) + B d u ( k ) y ( k ) = C d x ( k ) + D d u ( k ) . Approach: Solve the state equation over one sample period. x ( t ) = e At x (0) + i t 0 e A ( t - τ ) Bu ( τ ) dτ, And over a single sample period ( kT to kT + T ) this is, x ( kT + T ) = e AT x ( kT ) + i kT + T kT e A ( kT + T - τ ) Bu ( τ ) dτ, Roy Smith: ECE 147b 10 : 2 Discrete-time equivalents Continuous to discrete transforms in state-space We have several ways of calculating a discrete-time transfer function from a continuous-time one, depending on the application. ZOH Equivalence P ( s ) ZOH a a T A A A A u ( k ) u ( t ) y ( t ) y ( k ) P ( z ) = ( 1 - z - 1 ) Z b P ( s ) s B . Controller approximation z - 1 a a x ( k + 1) x ( k ) 1 s A A dx ( t ) dt x ( t ) Forward di±erence: C ( z ) = C ( s ) | s = z - 1 T Backward di±erence: C ( z ) = C ( s ) | s = z - 1 Tz Tustin/bilinear: C ( z ) = C ( s ) | s = 2 T ( z - 1) ( z +1) Roy Smith: ECE 147b 10 : 1

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ZOH equivalence in state-space ZOH equivalent So far we have calculated x ( k + 1) as a linear function of x ( k ) and u ( k ). What about y ( k )? y ( kT ) = C x ( kT ) + D u ( kT ) . By deFnition, y ( k ) = y ( kT ), and as u ( t ) is constant over the sample period, u ( k ) = u ( kT ). y ( k ) = C x ( k ) + D u ( k ) . Clearly then, C d = C and D d = D . b A B C D B ZOH = e AT i T 0 e B dη C D A d and B d are calculated via Matlab commands c2d or zohequiv . Roy Smith: ECE 147b 10 : 4 ZOH equivalence in state-space Key observation The integration involves u ( τ ) from τ = kT to τ = kT + T . But u ( τ ) is constant over this time period. It is the output of a ZOH . So, u ( τ ) = u ( k ) for kT τ < kT + T . Therefore, x ( kT + T ) = e AT x ( kT ) + bi kT + T kT e A ( kT + T - τ ) B dτ, B u ( k ) .
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lecture10_small - Discrete-time equivalents Continuous to...

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