D is a diagonal scaling (for MIMO loops). S is the sensitivity function at Location. WS is
a frequency-weighting function derived from the minimum loop gain profile you specify.
The gain of this function roughly matches the specified loop gain for values ranging from
–20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this
range, unless the specified gain profile changes slope outside this range. This adjustment
is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor
numeric conditioning for tuning, it is not recommended to specify gain profiles with very
low-frequency or very high-frequency dynamics. For more information about
regularization and its effects, see “Visualize Tuning Goals” on page 10-187.
Although S is a closed-loop transfer function, driving f(x) < 1 is equivalent to enforcing a
lower bound on the open-loop transfer function, L, in a frequency band where the gain of
L is greater than 1. To see why, note that S = 1/(1 + L). For SISO loops, when |L| >> 1, |S
| ≈ 1/|L|. Therefore, enforcing the open-loop minimum gain requirement, |L| > |WS|, is
roughly equivalent to enforcing |WsS| < 1. For MIMO loops, similar reasoning applies,
with ||S|| ≈ 1/σmin(L), where σmin is the smallest singular value.
Implicit Constraints
This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity
function measured at the specified, evaluated with loops opened at the specified loop-
opening locations. The dynamics affected by this implicit constraint are the stabilized
dynamics for this tuning goal. The Minimum decay rate and Maximum natural
frequency tuning options control the lower and upper bounds on these implicitly
constrained dynamics. If the optimization fails to meet the default bounds, or if the
default bounds conflict with other requirements, on the Tuning tab, use Tuning Options
to change the defaults.
See Also
Related Examples
- “Specify Goals for Interactive Tuning” on page 10-39
- “Manage Tuning Goals” on page 10-177
- “Visualize Tuning Goals” on page 10-187
See Also