# 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 tuning goal for control system tuning with tuning commands such as systune or looptune.

When you use TuningGoal.Gain, 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 goal that 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 goal 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. The tuning goal derives and is stored in the MaxGain property. Use viewGoal(Req) to plot the magnitude of MaxGain. Focus Frequency band in which tuning goal is enforced, specified as a row vector of the form [min,max]. Set the Focus property to limit enforcement of the tuning goal 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 tuning goal that you want to apply only between 1 and 100 rad/s. To restrict the tuning goal 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 goal is evaluated. Suppose T(s) is the closed-loop transfer function from Input to Output. The tuning goal 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 goal is evaluated. Suppose T(s) is the closed-loop transfer function from Input to Output. The tuning goal 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 character vectors that identify the inputs of the transfer function that the tuning goal constrains. The initial value of the Input property is set by the inputname input argument when you construct the tuning goal. Output Output signal names, specified as a cell array of character vectors that identify the outputs of the transfer function that the tuning goal constrains. The initial value of the Output property is set by the outputname input argument when you construct the tuning goal. Models Models to which the tuning goal 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 goal for a subset of models in the array. For example, suppose you want to apply the tuning goal, Req, to the second, third, and fourth models in a model array passed to systune. To restrict enforcement of the tuning goal, use the following command: Req.Models = 2:4; When Models = NaN, the tuning goal applies to all models. Default: NaN Openings Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loop-opening locations. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. If you are using the tuning goal 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 tuning goal 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. For example, if Openings = {'u1','u2'}, then the tuning goal is evaluated with loops open at analysis points u1 and u2. Default: {} Name Name of the tuning goal, specified as a character vector. For example, if Req is a tuning goal: Req.Name = 'LoopReq'; Default: []

## Examples

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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).

Create a tuning goal 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 tuning goal 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 gain profile using viewGoal.

viewGoal(Req)

The dashed line shows the gain profile, and the region indicates where the requirement is violated.

## Tips

• This tuning goal imposes an implicit stability constraint on the closed-loop transfer function from Input to Output, 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 object, the software converts the tuning goal 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 goal is a hard constraint.

For TuningGoal.Gain, f(x) is given by:

$f\left(x\right)={‖{W}_{F}\left(s\right){D}_{o}^{-1}T\left(s,x\right){D}_{i}‖}_{\infty },$

or its discrete-time equivalent, for discrete-time tuning. Here, 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 getPeakGain).

The frequency weighting function WF is the regularized gain profile, derived from the maximum gain profile you specify. The gains of WF and 1/MaxGain roughly match inside the frequency band Focus. WF is always stable and proper. Because poles of WF close to s = 0 or s = Inf might lead to poor numeric conditioning of the systune optimization problem, it is not recommended to specify maximum gain profiles with very low-frequency or very high-frequency dynamics.

To obtain WF, use:

WF = 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.

## Compatibility Considerations

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Behavior changed in R2016a