## Minimum Loop Gain Goal

### Purpose

Boost gain of feedback loops at low frequency when using Control System Tuner.

### Description

Minimum Loop Gain Goal enforces a minimum loop gain in a particular frequency band. This tuning goal is useful, for example, for improving disturbance rejection at a particular location.

Minimum Loop Gain 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 green region represents gain profile values that are forbidden
by this requirement. 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.

Minimum Loop Gain Goal is a constraint on the open-loop gain
of the specified control loop. Thus, the loop gain is computed with
the loop open at the specified location. To compute the gain with
loop openings at other points in the control system, use the **Compute
response with the following loops open** option in the **Open-Loop
Response Selection** section of the dialog box.

Minimum Loop Gain Goal and Maximum Loop Gain Goal specify only low-gain or high-gain constraints in certain frequency bands. When you use these requirements, the software determines the best loop shape near crossover. When the loop shape near crossover is simple or well understood (such as integral action), you can use Loop Shape Goal to specify that target loop shape.

#### Creation

In the **Tuning** tab of Control System Tuner, select **New Goal** > **Minimum gain for open-loop response** to create a Minimum Gain Goal.

#### Command-Line Equivalent

When tuning control systems at the command line, use `TuningGoal.MinLoopGain`

to
specify a minimum loop gain goal.

### Open-Loop Response Selection

Use this section of the dialog box to specify the signal locations at which to compute the open-loop gain. You can also specify additional loop-opening locations for evaluating the tuning goal.

**Shape open-loop response at the following locations**Select one or more signal locations in your model at which to compute and constrain the open-loop gain. To constrain a SISO response, select a single-valued location. For example, to constrain the open-loop gain at a location named

`'y'`

, click**Add signal to list**and select`'y'`

. To constrain a MIMO response, select multiple signals or a vector-valued signal.**Compute response with the following loops open**Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named

`'x'`

, click**Add signal to list**and select`'x'`

.

**Tip**

To highlight any selected signal in the Simulink^{®} model, click . To remove a signal from the input or output list, click . When you have selected multiple signals, you can reorder
them using and . For more information on how to specify signal locations
for a tuning goal, see
Specify Goals for Interactive Tuning.

### Desired Loop Gain

Use this section of the dialog box to specify the target minimum loop gain.

**Pure integrator K/s**Check to specify a pure integrator shape for the target minimum loop gain. The software chooses the integrator constant,

*K*, based on the values you specify for a target minimum gain and frequency. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 1 in the**Choose K to keep gain above**text box. Then, enter 10 in the**at the frequency**text box. The software chooses the integrator constant such that the minimum loop gain is 1 at 10 rad/s.**Other gain profile**Check to specify the minimum gain profile as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (

`tf`

,`zpk`

, or`ss`

model). Alternatively, you can sketch a piecewise target loop gain using an`frd`

model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired minimum loop gain. For example, to specify minimum gain of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/dec at higher frequencies, enter`frd([100 100 10],[0 1e-1 1])`

.If you are tuning in discrete time, you can specify the minimum gain profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the gain profile in continuous time, the tuning software discretizes it. Specifying the profile in discrete time gives you more control over the profile near the Nyquist frequency.

### Options

Use this section of the dialog box to specify additional characteristics of the minimum loop gain goal.

**Enforce goal in frequency range**Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form

`[min,max]`

, expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter`[1,100]`

. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time.**Stabilize closed loop system**By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select

`No`

to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select`No`

.**Equalize loop interactions**For multi-loop or MIMO loop gain constraints, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Select

`Off`

to disable such scaling and shape the unscaled open-loop response.**Apply goal to**Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or block-parameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select

**Only Models**. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter`2:4`

in the**Only Models**text box.For more information about tuning for multiple models, see Robust Tuning Approaches (Robust Control Toolbox).

### Algorithms

#### Evaluating Tuning Goals

When you tune a control system, the software converts each 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 **Minimum Loop Gain Goal**, *f*(*x*)
is given by:

$$f\left(x\right)={\Vert {W}_{S}\left({D}^{-1}SD\right)\Vert}_{\infty}.$$

*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 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

*W*close to

_{S}*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.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.

#### 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.