Package: TuningGoal
Maximum loop gain constraint for control system tuning
Use the TuningGoal.MaxLoopGain
object to enforce
a maximum loop gain and desired rolloff in a particular frequency
band. Use this requirement with control system tuning commands such
as systune
or looptune
.
This requirement imposes a maximum gain on the openloop frequency response (L) at a specified location in your control system. You specify the maximum openloop gain as a function of frequency (a maximum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as an upper bound on the largest singular value of L.
When you tune a control system, the maximum gain profile is converted to a maximum gain constraint on the complementary sensitivity function, T) = L/(I + L).
The following figure shows a typical specified maximum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The pink region represents gain profile values that are forbidden by this requirement. The figure shows that when L is much smaller than 1, imposing a maximum gain on T is a good proxy for a maximum openloop gain.
TuningGoal.MaxLoopGain
and TuningGoal.MinLoopGain
specify
only highgain or lowgain constraints in certain frequency bands.
When you use these requirements, 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 requirement for limiting the gain of a SISO or MIMO feedback
loop. The requirement limits the openloop frequency response measured
at the specified locations to the maximum gain profile specified by Req
= TuningGoal.MaxLoopGain(location
,loopgain
)loopgain
.
You can specify the maximum gain profile as a smooth transfer function
or sketch a piecewise error profile using an frd
model
or the makeweight
command.
Only gain values smaller than 1 are enforced.
specifies
a maximum gain profile of the form Req
= TuningGoal.MaxLoopGain(location
,fmax
,gmax
)loopgain = K/s
(integral
action). The software chooses K
such that the gain
value is gmax
at the specified frequency, fmax
.

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

Maximum openloop gain as a function of frequency. You can specify loopgain = frd([1 1e1 1e3],[1 10 100]); bodemag(loopgain)
When you use an Only gain values smaller than 1 are enforced. For multiinput,
multioutput (MIMO) feedback loops, the gain profile is interpreted
as a minimum rolloff requirement, which is an upper bound on the
largest singular value of 

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

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

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

Frequency band in which tuning requirement 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 maximum loop gain is constrained, specified as a string or cell array of strings. These strings identify one or more loopopening locations in the control system to tune. The value of the 

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

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

Name of the requirement object, specified as a string. For example, if Req.Name = 'LoopReq'; Default: 
When you tune a control system using a TuningGoal
object
to specify a tuning requirement, the software converts the requirement
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 requirement is a hard constraint.
For the TuningGoal.MaxLoopGain
requirement, f(x)
is given by:
$$f\left(x\right)={\Vert {W}_{T}\left({D}^{1}TD\right)\Vert}_{\infty}.$$
W_{T} is the reciprocal
of the maximum loop gain profile, MaxGain
. D is
a diagonal scaling (for MIMO loops). T is the complementary
sensitivity function at Location
.
Although T is a closedloop transfer function, driving f(x) < 1 is equivalent to enforcing an upper bound on the openloop transfer, L, in a frequency band where the gain of L is less than one. To see why, note that T = L/(I + L). For SISO loops, when L << 1, T ≈ L. Therefore, enforcing the openloop maximum gain requirement, L < 1/W_{T}, is roughly equivalent to enforcing W_{T}T < 1. For MIMO loops, similar reasoning applies, with T ≈ σ_{max}(L), where σ_{max} is the largest singular value.
Create a requirement that limits the maximum openloop gain of a feedback loop to a specified profile.
Suppose that you are tuning a control system that has a loopopening location identified by PILoop
. Limit the openloop gain measured at that location to 1 (0 dB) at 1 rad/s, rolling off at a rate of 20 dB/dec up to 10 rad/s, and a rate of 40 dB/dec at higher frequencies. Use an frd
model to sketch this gain profile.
loopgain = frd([1 1e1 1e3],[1 10 100]);
Req = TuningGoal.MaxLoopGain('PILoop',loopgain);
The software converts loopgain
into a smooth function of frequency that approximates the piecewisespecified requirement. Display the requirement using viewSpec
.
viewSpec(Req)
The yellow region indicates where the requirement is violated, except that gain values greater than 1 are not enforced. Therefore, this requirement only specifies minimum rolloff rates at frequencies above 1 rad/s.
You can use Req
as an input to looptune
or systune
when tuning the control system.
Create a requirement that specifies a maximum loop gain of the form L = K / s . The maximum gain attains the value of 20 dB (0.01) at 100 rad/s.
Req = TuningGoal.MaxLoopGain('X',100,0.01);
viewSpec(Req)
viewSpec
confirms that the requirement is correctly specified. You can use this requirement to tune a control system that has a loopopening location identified as 'X'
. Since loop gain values greater than 1 are ignored, this requirement specifies a rolloff of 20 dB/decade above 1 rad/s, with no restriction on loop gain below that frequency.
Create a requirement that specifies a maximum loop gain of –20 dB (0.01) at 100 rad/s on the inner loop of the following control system.
Req = TuningGoal.MaxLoopGain('X2',100,0.01);
Configure the requirement to apply to the loop gain of the inner loop measured with the outer loop open.
Req.Openings = 'X2';
Setting Req.Openings
tells the tuning algorithm
to enforce this requirement with loops open at the specified location.
By default, tuning using TuningGoal.MaxLoopGain
imposes
a stability requirement as well as the maximum loop gain requirement.
Practically, in some control systems it is not possible to achieve
a stable inner loop. When this occurs, remove the stability requirement
for the inner loop by setting the Stabilize
property
to false
.
Req.Stabilize = false;
The tuning algorithm still imposes a stability requirement on the overall tuned control system, but not on the inner loop alone.
evalSpec
 looptune
 looptune
(for slTuner)
 sigma
 slTuner
 systune
 systune (for slTuner)
 TuningGoal.Gain
 TuningGoal.LoopShape
 TuningGoal.Margins
 TuningGoal.MinLoopGain
 viewSpec