# 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.

## 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).