for all T ≥ 0. Constraining the I/O trajectories of G is equivalent to restricting the output
trajectories z(t) of the system H = [G;I] to the sector defined by:
∫ 0
T
ztTQztdt< 0.(See “About Sector Bounds and Sector Indices” (Control System Toolbox) for more details
about this equivalence.) To specify a constraint of this type using Conic Sector Goal,
specify u as the input signal, and specify y and u as output signals. When you specify u as
both input and output, Conic Sector Goal sets the corresponding transfer function to the
identity. Therefore, the transfer function that the goal constrains is H = [G;I] as intended.
This treatment is specific to Conic Sector Goal. For other tuning goals, when the same
signal appears in both inputs and outputs, the resulting transfer function is zero in the
absence of feedback loops, or the complementary sensitivity at that location otherwise.
This result occurs because when the software processes analysis points, it assumes that
the input is injected after the output. See “Mark Signals of Interest for Control System
Analysis and Design” on page 2-53 for more information about how analysis points work.
Algorithms
Let
Q=W 1 W 1 T−W 2 W 2 T
be an indefinite factorization of Q, where W 1 TW 2 = 0. If W 2 THs is square and minimum
phase, then the time-domain sector bound
∫ 0
T
zt
T
Qztdt< 0,is equivalent to the frequency-domain sector condition,
H−jωQHjω < 0for all frequencies. Conic Sector Goal uses this equivalence to convert the time-domain
characterization into a frequency-domain condition that Control System Tuner can handle
in the same way it handles gain constraints. To secure this equivalence, Conic Sector Goal
also makes W 2 THs minimum phase by making all its zeros stable. The transmission zeros
affected by this minimum-phase condition are the stabilized dynamics for this tuning goal.
The Minimum decay rate and Maximum natural frequency tuning options control the
Conic Sector Goal