# TuningGoal.MinLoopGain

Minimum loop gain constraint for control system tuning

## Description

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 open-loop frequency response
(*L*) at a specified location in your control system. You specify the minimum
open-loop 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 open-loop gain.

`TuningGoal.MinLoopGain`

and `TuningGoal.MaxLoopGain`

specify only low-gain or high-gain 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.

## Creation

### Syntax

### Description

creates a tuning goal for boosting the gain of a SISO or MIMO feedback loop. The tuning goal
specifies that the open-loop frequency response (`Req`

= TuningGoal.MinLoopGain(`location`

,`loopgain`

)*L*) measured at the
specified locations exceeds the minimum gain profile specified by
`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*.

### Input Arguments

## Properties

## Examples

## Tips

This tuning goal imposes an implicit stability constraint on the closed-loop 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.

## Algorithms

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 frequency-weighting 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*close to

_{S}*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 low-frequency or very high-frequency 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 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*| >
|*W _{S}*|, is roughly equivalent to enforcing |

*W*| < 1. For MIMO loops, similar reasoning applies, with ||

_{s}S*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.

## Version History

**Introduced in R2016a**

## See Also

`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`