3.6. H2AND H∞ANALYSISAND SYNTHE SIS 97
equation solution procedure often becomes poorly conditioned. Displaying intermediate
calculation results allows the user to fine tune several tolerancesif necessar y. The
intermediate Hamiltonian and Riccati solution details are displayed as the bisection
proceeds. The bisection stopping tolerance, Riccati solution tolerances and the Ricca ti
solution method are specified via keywords. Details on the meaning of these toler ances
are given in Section 2.3.5.
Wenow give a simple example illustrating the use of these functions. An oscillatory
non-minimum phase SISO system is to be controlled in a unity gain negativefeedback
configuration. This example is for pedagogical purposes only and does not illustrate the
generality of the approach with respect to MIMO systems and more general control
configurations. The weighting functions have been chosen to be appropriate for an H∞
design. The performance weight, Wperf,iscloseto1/sgiving good tracking. The
actuator weight,Wact, increases at high frequency, penalizing fast actuator action. A
sensitivity function and step response comparison have been included to illustrate
typical function calls for these procedures. Inappropriate weight choices make this
comparison blantantly unfair to the H2approach;the usershould not drawnany
conclusions about the relativemerits of either approach from this example.
# Set up a simple closed loop problem.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Examine the plant frequency response.
omega = logspace(0.001,100,200)
plantg = freq(plant,omega)
g0 = ctrlplot(plantg,{bode});
g0 = plot(g0,{title="Open loop plant"})?