Package: TuningGoal
Minimum loop gain constraint for control system tuning
Use the TuningGoal.MinLoopGain
object to enforce a minimum
loop gain in a particular frequency band. Use this tuning goal with control system tuning
commands such as systune
or looptune
.
This tuning goal imposes a minimum gain on the openloop frequency response (L) at a specified location in your control system. You specify the minimum openloop gain as a function of frequency (a minimum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
When you tune a control system, the minimum gain profile is converted to a minimum gain constraint on the inverse of the sensitivity function, inv(S) = (I + L).
The following figure shows a typical specified minimum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The shaded region represents gain profile values that are forbidden by this tuning goal. The figure shows that when L is much larger than 1, imposing a minimum gain on inv(S) is a good proxy for a minimum openloop gain.
TuningGoal.MinLoopGain
and TuningGoal.MaxLoopGain
specify
only lowgain or highgain constraints in certain frequency bands.
When you use these tuning goals, systune
and looptune
determine
the best loop shape near crossover. When the loop shape near crossover
is simple or well understood (such as integral action), you can use TuningGoal.LoopShape
to
specify that target loop shape.
creates
a tuning goal for boosting the gain of a SISO or MIMO feedback loop.
The tuning goal specifies that the openloop frequency response (L)
measured at the specified locations exceeds the minimum gain profile
specified by Req
= TuningGoal.MinLoopGain(location
,loopgain
)loopgain
.
You can specify the minimum gain profile as a smooth transfer
function or sketch a piecewise error profile using an frd
model
or the makeweight
(Robust Control Toolbox) command.
Only gain values greater than 1 are enforced.
For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
specifies
a minimum gain profile of the form Req
= TuningGoal.MinLoopGain(location
,fmin
,gmin
)loopgain = K/s
(integral
action). The software chooses K
such that the gain
value is gmin
at the specified frequency, fmin
.

Location at which the maximum openloop gain is constrained, specified as a character vector or cell array of character vectors that identify one or more locations in the control system to tune. What loopopening locations are available depends on what kind of system you are tuning:
If 

Minimum openloop gain as a function of frequency. You can specify loopgain = frd([100 100 10],[0 1e1 1]); When you use an Only gain values larger than 1 are enforced. For multiinput,
multioutput (MIMO) feedback loops, the gain profile is interpreted
as a lower bound on the smallest singular value of If you are tuning in discrete time (that is, using a 

Frequency of minimum gain Use this argument to specify a minimum gain profile of the form 

Value of minimum gain occurring at Use this argument to specify a minimum gain profile of the form 

Minimum openloop gain as a function of frequency, specified
as a SISO The software automatically maps the input argument Use 

Frequency band in which tuning goal is enforced, specified as
a row vector of the form Set the Req.Focus = [1,100]; Default: 

Stability requirement on closedloop dynamics, specified as
1 ( When Default: 1 ( 

Toggle for automatically scaling loop signals, specified as In multiloop or MIMO control systems, the feedback channels
are automatically rescaled to equalize the offdiagonal terms in the
openloop transfer function (loop interaction terms). Set Default: 

Location at which minimum loop gain is constrained, specified
as a cell array of character vectors that identify one or more analysis
points in the control system to tune. For example, if The value of the 

Models to which the tuning goal applies, specified as a vector of indices. Use the Req.Models = 2:4; When Default: 

Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loopopening locations. The tuning goal is evaluated against the openloop configuration created by opening feedback loops at the locations you identify. If you are using the tuning goal to tune a Simulink model
of a control system, then If you are using the tuning goal to tune a generalized statespace
( For example, if Default: 

Name of the tuning goal, specified as a character vector. For example, if Req.Name = 'LoopReq'; Default: 
This tuning goal imposes an implicit stability constraint on
the closedloop sensitivity function measured at Location
,
evaluated with loops opened at the points identified in Openings
.
The dynamics affected by this implicit constraint are the stabilized
dynamics for this tuning goal. The MinDecay
and MaxRadius
options
of systuneOptions
control the bounds on these
implicitly constrained dynamics. If the optimization fails to meet
the default bounds, or if the default bounds conflict with other requirements,
use systuneOptions
to change
these defaults.
When you tune a control system using a TuningGoal
,
the software converts the tuning goal into a normalized scalar value
f(x). Here, x is
the vector of free (tunable) parameters in the control system. The
software then adjusts the parameter values to minimize f(x)
or to drive f(x) below 1 if
the tuning goal is a hard constraint.
For TuningGoal.MinLoopGain
, f(x)
is given by:
$$f\left(x\right)={\Vert {W}_{S}\left({D}^{1}SD\right)\Vert}_{\infty}.$$
Here, D is a diagonal scaling (for MIMO loops). S is
the sensitivity function at Location
. W_{S} is
a frequencyweighting function derived from the minimum loop gain
profile, MinGain
. The gain of this function roughly
matches MaxGain
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 W_{S} close
to s = 0 or s = Inf
might
lead to poor numeric conditioning of the systune
optimization
problem, it is not recommended to specify gain profiles with very
lowfrequency or very highfrequency dynamics.
To obtain W_{S}, use:
WS = getWeight(Req,Ts)
where Req
is the tuning goal, and Ts
is
the sample time at which you are tuning (Ts = 0
for
continuous time). For more information about regularization and its
effects, see Visualize Tuning Goals.
Although S is a closedloop transfer function, driving f(x) < 1 is equivalent to enforcing a lower bound on the openloop 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 openloop minimum gain requirement, L > W_{S}, is roughly equivalent to enforcing W_{s}S < 1. For MIMO loops, similar reasoning applies, with S ≈ 1/σ_{min}(L), where σ_{min} is the smallest singular value.
For an example illustrating the constraint on S, see Minimum Loop Gain as Constraint on Sensitivity Function.
looptune
 systune
 systune
(for slTuner)
(Simulink Control Design)  looptune (for slTuner)
(Simulink Control Design)  viewGoal
 evalGoal
 TuningGoal.Gain
 TuningGoal.LoopShape
 TuningGoal.MaxLoopGain
 TuningGoal.Margins
 slTuner
(Simulink Control Design)  sigma