# 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.

## Examples

collapse all

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

expand all

Behavior changed in R2016a