chapter
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## z e =

, where ω is an equivalent s-plane frequency. Thus, the following equation is obtained from Eq. (4.18): Equating the exponents yields

The pre-warping approach for the Tustin approximation takes the s-plane imaginary axis and folds it back to π/2 to −π/2 as seen from Fig. 4.2. The spectrum of the input must also be taken into consideration when selecting an approximation procedure with or without pre-warping. It should be noted that in the previous discussion only the frequency has been pre-warped due to the interest in the controller frequency response. The real part of the s-plane pole influences such parameters as rise time, overshoot, and settling time. Thus, consideration of the warping of the real pole component is now analyzed as a fine-tuning approach. Proceeding in the same manner as used in deriving Eq. (4.22) substitute

2/ˆ1

2/ˆ1

T

T e

σσ

+ = (4.23)

denominator in Eq. (4.23) results in the expression

/ˆ1

ˆ 11

σ

σσ σ

+=+++ m (4.24)

4-3 NON-MINIMUM PHASE ANALOG PLANT

 τ

τ

τ (4.27)

s

s sA

τ

τ

+

≡′

1 )( (4.28)

1 )(1

+

+ =′

τω

τω A (4.31)

DNA j φφωττωωφ −=−−−−=′∠=′ )1/(tan)1/(tan)( 11A (4.32)

mo 130)( −=−∠ γφωjL

o0)( <′∠= ′

sA

= (4.35)

5. Form )()()( sLsAsL moo ′= in order to obtain

)(

)( )(

sP

sL sG

o = (4.37)

4-4 DISCRETE MISO MODEL WITH PLANT UNCERTAINTY

=−==

=+==

zL

zPD zR

zL

zFzL zY DR +=

+ +

+ = (4.38)

where )()()( 1 zGzPGzL zo= (4.39)

)()1( )(

)1()()( 11 zPz s

sP zzPGzP eZOz

 Z (4.40)

sP zP ee ZZ =≡ 

s

sP sPe

)]()([)( sDsPzPD Z= (4.43)

)()( )(1

)()( )( zTzF

zL

zLzF zT RR ′=

+ = (4.44)

)(1

)( )(

zL

zPD zYD

+ = (4.46)

( ) )()()(1)()( 111 zPzGzePzzGzL z=−= − (4.47) Note, for a unit step disturbance input function, D(s) = 1/s, that

ssPsDsPsPe /)()()()( == (4.48)

)()( zPDzPe = (4.49)

4-5 QFT w-DOMAIN DIG DESIGN

ωσωσ jjs spsp +=+= (4.50)

 +

z

z

T jvujw wpwp ωσ (4.52)

==

T v

= (4.54)

TTee TsT ωωσ ∠=∠== zz (4.55)

297.0 2

2 2

and T spsp ωσ

α (4.56)

are both satisfied then s ≈ w. If for the range of parameter uncertainty and for a specified value of T, both conditions of Eq. (4.56) are satisfied in the low frequency range, then the approximation

)()( ws PP ℑ≈ℑ (4.57)

is valid in the low frequency range. The w-domain QFT design procedure presented in this chapter is based upon a stable uncertain plant.