Page 64 of 92 ty pe is equivalent to a condition for the smallest singular value g( Lo(jw)) of the open loop transfer function at the output to be large at low frequencies g(Lo Gw))>wp(u) The uncertainties of physical Systems are often largest at high frequencies. If det ailed knowl edge of the uncert aunties of the process is not available, it would be natural to specify a multiplicative un cert ainty mo del for the output (6.5) (6.6) where Wu(s)is a scalar transfer function with high p ass characteristics, such that Wu gw) has a value corresponding to the dc gain at low frequencies and a value at high frequencies of more than unity. Requirements of this type lead to the condition 7(T0(j) Vu gw) (6.7) This is equivalent to requiring the largest singular value of the open loop transfer matrix o( Lo Gu)) to be small at high frequencies (Lo gu)) (6.8) Wuga) In this way, the specification of weight functions b ecomes a matter of trade-off between a good disturbance attenuation and robustness, and the interesting choice is the frequen cy where the curves intersect. At this frequency, it should be ensured that either transfer function is smaller than unity. Otherwise, it will be imp ossible to meet the requirements. Figure 6. 1 shows an example of how such requirement s could manifest. The method does not give an expli loop singular values should proceed close to the cross- over frequencies, i.e. where the family of curves for the open loop singular values intersect unity(0 dB). It can be seen, though, that it is convenient if the singular values can be made Other transfer functions than So(s) and To(s) could be of interest to the loop shaping ap proach. Ensuring that the control signals u(s) remain reasonably bounded, can be obtained y bounding the control sensitivity M()=(I+ k(sG(s)K(s. Moreover, in the case where the uncertainties can be described in terms of an additive uncert ainty description, this also leads to upper bounds for the control sensitivity M(s) 2 Modeling individual Channels In the loop shaping approach, the interest is mainly focused on the size of the sensitivity and the complentary sensitivity functions, which for multivariable Sy stems leads to requirements for the largest and smallest singular values of the open loop transfer matrix. This has the￾
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