# TuningGoal.Gain class

Package: TuningGoal

Gain constraint for control system tuning

## Description

Use the `TuningGoal.Gain` object to specify a constraint that limits the gain from a specified input to a specified output. Use this requirement for control system tuning with tuning commands such as `systune` or `looptune`.

When you use a `TuningGoal.Gain` requirement, the software attempts to tune the system so that the gain from the specified input to the specified output does not exceed the specified value. By default, the constraint is applied with the loop closed. To apply the constraint to an open-loop response, use the `Openings` property of the `TuningGoal.Gain` object.

You can use a gain constraint to:

• Enforce a design requirement of disturbance rejection across a particular input/output pair, by constraining the gain to be less than 1

• Enforce a custom roll-off rate in a particular frequency band, by specifying a gain profile in that band

## Construction

```Req = TuningGoal.Gain(inputname,outputname,gainvalue)``` creates a tuning requirement `Req`. This requirement constrains the gain from `inputname` to `outputname` to remain below the value `gainvalue`.

You can specify the `inputname` or `outputname` as cell arrays (vector-valued signals). If you do so, then the tuning requirement constrains the largest singular value of the transfer matrix from `inputname` to `outputname`. See `sigma` for more information about singular values.

`Req = TuningGoal.Gain(inputname,outputname,gainprofile)` specifies the maximum gain as a function of frequency. You can specify the target gain profile (maximum gain across the I/O pair) as a smooth transfer function. Alternatively, you can sketch a piecewise error profile using an `frd` model.

### Input Arguments

 `inputname` Input signals for the requirement, specified as a string or as a cell array of strings, for multiple-input requirements. If you are using the requirement to tune a Simulink® model of a control system, then `inputname` can include: Any model input.Any linear analysis point marked in the model.Any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. If you are using the requirement to tune a generalized state-space (`genss`) model of a control system, then `inputname` can include: Any input of the `genss` model Any `AnalysisPoint` location in the control system model For example, if you are tuning a control system model, `T`, then `inputname` can be a string contained in `T.InputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_u`, then `inputname` can include `'AP_u'`. Use `getPoints` to get a list of analysis points available in a `genss` model. If `inputname` is an `AnalysisPoint` location of a generalized model, the input signal for the requirement is the implied input associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Marking Signals of Interest for Control System Analysis and Design. `outputname` Output signals for the requirement, specified as a string or as a cell array of strings, for multiple-output requirements. If you are using the requirement to tune a Simulink model of a control system, then `outputname` can include: Any model output.Any linear analysis point marked in the model.Any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. If you are using the requirement to tune a generalized state-space (`genss`) model of a control system, then `outputname` can include: Any output of the `genss` model Any `AnalysisPoint` location in the control system model For example, if you are tuning a control system model, `T`, then `inputname` can be a string contained in `T.OutputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_y`, then `inputname` can include `'AP_y'`. Use `getPoints` to get a list of analysis points available in a `genss` model. If `outputname` is an `AnalysisPoint` location of a generalized model, the output signal for the requirement is the implied output associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Marking Signals of Interest for Control System Analysis and Design. `gainvalue` Maximum gain (linear). The gain constraint `Req` specifies that the gain from `inputname` to `outputname` is less than `gainvalue`. `gainvalue` is a scalar value. If the signals `inputname` or `outputname` are vector-valued signals, then `gainvalue` constrains the largest singular value of the transfer matrix from `inputname` to `outputname`. See `sigma` for more information about singular values. `gainprofile` Gain profile as a function of frequency. The gain constraint `Req` specifies that the gain from `inputname` to `outputname` at a particular frequency is less than `gainprofile`. You can specify `gainprofile` as a smooth transfer function (`tf` , `zpk`, or `ss` model). Alternatively, you can sketch a piecewise gain profile using a `frd` model or the `makeweight` function. When you do so, the software automatically maps the gain profile onto a `zpk` model. The magnitude of this `zpk` model approximates the desired gain profile. Use `viewSpec(Req)` to plot the magnitude of the `zpk` model. `gainprofile` is a SISO transfer function. If `inputname` or `outputname` are cell arrays, `gainprofile` applies to all I/O pairs from `inputname` to `outputname`

## Properties

 `MaxGain` Maximum gain as a function of frequency, expressed as a SISO `zpk` model. The software automatically maps the `gainvalue` or `gainprofile` input arguments to a `zpk` model. The magnitude of this `zpk` model approximates the desired gain profile, and is stored in the `MaxGain` property. Use `viewSpec(Req)` to plot the magnitude of `MaxGain`. `Focus` Frequency band in which tuning requirement is enforced, specified as a row vector of the form `[min,max]`. Set the `Focus` property to limit enforcement of the requirement to a particular frequency band. Express this value in the frequency units of the control system model you are tuning (rad/`TimeUnit`). For example, suppose `Req` is a requirement that you want to apply only between 1 and 100 rad/s. To restrict the requirement to this band, use the following command:`Req.Focus = [1,100];` Default: `[0,Inf]` for continuous time; `[0,pi/Ts]` for discrete time, where `Ts` is the model sample time. `Stabilize` Stability requirement on closed-loop dynamics, specified as 1 (`true`) or 0 (`false`). By default, `TuningGoal.Gain` imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain requirement. If stability is not required or cannot be achieved, set `Stabilize` to `false` to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, set `Stabilize` to `false`. Default: 1(`true`) `InputScaling` Input signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vector-valued input signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from `Input` to `Output` when the tuning requirement is evaluated. Suppose T(s) is the closed-loop transfer function from `Input` to `Output`. The requirement is evaluated for the scaled transfer function Do–1T(s)Di. The diagonal matrices Do and Di have the `OutputScaling` and `InputScaling` values on the diagonal, respectively. The default value, `[]` , means no scaling. Default: `[]` `OutputScaling` Output signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vector-valued output signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from `Input` to `Output` when the tuning requirement is evaluated. Suppose T(s) is the closed-loop transfer function from `Input` to `Output`. The requirement is evaluated for the scaled transfer function Do–1T(s)Di. The diagonal matrices Do and Di have the `OutputScaling` and `InputScaling` values on the diagonal, respectively. The default value, `[]` , means no scaling. Default: `[]` `Input` Input signal names, specified as a cell array of strings. These strings specify the names of the inputs of the transfer function that the tuning requirement constrains. The initial value of the `Input` property is set by the `inputname` input argument when you construct the requirement object. `Output` Output signal names, specified as a cell array of strings. These strings specify the names of the outputs of the transfer function that the tuning requirement constrains. The initial value of the `Output` property is set by the `outputname` input argument when you construct the requirement object. `Models` Models to which the tuning requirement applies, specified as a vector of indices. Use the `Models` property when tuning an array of control system models with `systune`, to enforce a tuning requirement for a subset of models in the array. For example, suppose you want to apply the tuning requirement, `Req`, to the second, third, and fourth models in a model array passed to `systune`. To restrict enforcement of the requirement, use the following command: `Req.Models = 2:4;` When `Models = NaN`, the tuning requirement applies to all models. Default: `NaN` `Openings` Feedback loops to open when evaluating the requirement, specified as a cell array of strings that identify loop-opening locations. The tuning requirement is evaluated against the open-loop 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 `Openings` can include any linear analysis point marked in the model, or any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points and loop openings to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. If you are using the requirement to tune a generalized state-space (`genss`) model of a control system, then `Openings` can include any `AnalysisPoint` location in the control system model. Use `getPoints` to get the list of analysis points available in the `genss` model. Default: `{}` `Name` Name of the requirement object, specified as a string. For example, if `Req` is a requirement: `Req.Name = 'LoopReq';` Default: `[]`

## Algorithms

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), where 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.Gain` requirement, f(x) is given by:

$f\left(x\right)={‖\frac{1}{\text{MaxGain}}{D}_{o}^{-1}T\left(s,x\right){D}_{i}‖}_{\infty }.$

T(s,x) is the closed-loop transfer function from `Input` to `Output`. Do and Di are diagonal matrices with the `OutputScaling` and `InputScaling` property values on the diagonal, respectively. ${‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{\infty }$ denotes the H norm (see `norm`).

## Examples

### Disturbance rejection

Create a gain constraint that enforces a disturbance rejection requirement from a signal `'du'` to a signal `'u'`.

`Req = TuningGoal.Gain('du','u',1);`

This requirement specifies that the maximum gain of the response from `'du'` to `'u'` not exceed 1 (0 dB).

### Custom roll-off specification

Create a gain constraint that constrains the response from a signal `'du'` to a signal `'u'` to roll off at 20 dB/decade at frequencies greater than 1. The gain constraint also specifies disturbance rejection (maximum gain of 1) in the frequency range [0,1].

```gmax = frd([1 1 0.01],[0 1 100]); Req = TuningGoal.Gain('du','u',gmax); ```

These commands use a `frd` model to specify the gain profile as a function of frequency. The maximum gain of 1 dB at the frequency 1 rad/s, together with the maximum gain of 0.01 dB at the frequency 100 rad/s, specifies the desired rolloff of 20 dB/decade.

The software converts `gmax` into a smooth function of frequency that approximates the piecewise specified requirement. Display the error requirement using `viewSpec`.

```viewSpec(Req) ```

The yellow region indicates where the requirement is violated.

### Disturbance rejection

Create a gain constraint that enforces a disturbance rejection requirement from a signal `'du'` to a signal `'u'`.

`Req = TuningGoal.Gain('du','u',1);`

This requirement specifies that the maximum gain of the response from `'du'` to `'u'` not exceed 1 (0 dB).