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Unformatted text preview: Design by approximation Approach: A transfer function, C ( s ), can be realised with integrators, gains, and summation blocks. C ( s ) = y ( s ) u ( s ) = 1 s 3 + a 2 s 2 + a 1 s + a . is equivalent to: a 1 /s a 1 1 /s a 2 1 /s a108 + a108 + a108 + a116 a116 a116 a27 a27 a27 a27 a27 a45 a54 a45 a45 a54 a45 a45 a54 u ( s ) y ( s ) Now replace the integrators (1 /s blocks) with a discretetime approximation to integration. Roy Smith: ECE 147b 4 : 2 Design by approximation Approximating C ( s ) with C ( z ) P ( s ) C ( s ) C ( z ) P ( z ) Approximation of C ( s ) with C ( z ) Model P ( s ), and sample/hold as P ( z ) Continuoustime design Discretetime design Design a continuoustime controller, C ( s ), for P ( s ). Approximate C ( s ) with a discretetime controller, C ( z ). (Franklin & Powell refer to this procedure as emulation.) Roy Smith: ECE 147b 4 : 1 Forward difference approximation Forward difference approximation: y f ( kT + T ) = y f ( kT ) + T x ( kT ) . By taking ztransforms, zy f ( z ) = y f ( z ) + T x ( z ) , or, y f ( z ) x ( z ) = T z 1 . kT kT+T t x(t) x(kT+T) x(kT) y f (kT+T) y f (kT) So, the approximation is: 1 s T z 1 . This is equivalent to the substitution: s = z 1 T . This approximation is also known as an Euler approximation. Roy Smith: ECE 147b 4 : 4 Approximations to integration Integration: 1 /s a27 a27 y ( s ) x ( s ) y ( t ) = y (0) + integraldisplay t x ( ) d, The output, y ( t ), over a single sample period of T seconds, is given by y ( kT + T ) = y ( kT ) + integraldisplay kT + T kT x ( ) d. kT kT+T t x(t) x(kT+T) x(kT) y(kT+T) y(kT) Objective: Find a discretetime approximation, F ( z ), to the inputoutput relationship of the integrator. Find F ( z ) 1 /s , then, s F 1 ( z ), and C ( z ) = C ( s )  s = F 1 ( z ) . F ( z ) a27 a27 y ( z ) x ( z ) Roy Smith: ECE 147b 4 : 3 Trapezoidal approximation Trapezoidal approximation: y bl ( kT + T ) = y bl ( kT ) + T x ( kT ) + ( x ( kT + T ) x ( kT )) T/ 2 ....
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 Spring '07
 RODWELL

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