## Step Rejection Goal

### Purpose

Set a minimum standard for rejecting step disturbances, when using Control System Tuner.

### Description

Use Step Rejection Goal to specify how a step disturbance injected at a specified location in your control system affects the signal at a specified output location.

You can specify the desired response in time-domain terms of peak value, settling time, and damping ratio. Control System Tuner attempts to make the actual rejection at least as good as the desired response. Alternatively, you can specify the response as a stable reference model having DC-gain. In that case, the tuning goal is to reject the disturbance as well as or better than the reference model.

To specify disturbance rejection in terms of a frequency-domain attenuation profile, use Disturbance Rejection Goal.

When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the target step response you specify. The solid line is the current corresponding response of your system.

#### Creation

In the Tuning tab of Control System Tuner, select New Goal > Rejection of step disturbance to create a Step Rejection Goal.

#### Command-Line Equivalent

When tuning control systems at the command line, use `TuningGoal.StepRejection` to specify a step response goal.

### Step Disturbance Response Selection

Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal.

• Specify step disturbance inputs

Select one or more signal locations in your model at which to apply the input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the step-disturbance response from a location named `'u'` to a location named `'y'`, click Add signal to list and select `'u'`. To constrain a MIMO response, select multiple signals or a vector-valued signal.

• Specify step response outputs

Select one or more signal locations in your model at which to measure the response to the step disturbance. To constrain a SISO response, select a single-valued output signal. For example, to constrain the transient response from a location named `'u'` to 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. For MIMO systems, the number of outputs must equal the number of outputs.

• Compute the 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 Response to Step Disturbance

Use this section of the dialog box to specify the shape of the desired response to the step disturbance. Control System Tuner attempts to make the actual response at least as good as the desired response.

• Response Characteristics

Specify the desired response in terms of time-domain characteristics. Enter the maximum amplitude, maximum settling time, and minimum damping constant in the text boxes.

• Reference Model

Specify the desired response in terms of a reference model.

Enter the name of the reference model in the MATLAB® workspace in the Reference Model text field. Alternatively, enter a command to create a suitable reference model, such as ```tf([1 0],[1 1.414 1])```.

The reference model must be stable and must have zero DC gain. The model can be continuous or discrete. If the model is discrete, it can include time delays which are treated as poles at `z` = 0.

For best results, the reference model and the open-loop response from the disturbance to the output should have similar gains at the frequency where the reference model gain peaks.

### Options

Use this section of the dialog box to specify additional characteristics of the step rejection goal.

• Adjust for amplitude of input signals and Adjust for amplitude of output signals

For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued signals. This information is used to scale the off-diagonal terms in the transfer function from the tuning goal inputs to outputs. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal.

When these options are set to `No`, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to `Yes` to provide the relative amplitudes of the input signals and output signals of your transfer function.

For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 100 times greater than the first signal. In that case, select `Yes` and enter `[1,100]` in the Amplitudes of input signals text box.

Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitudes of output signals and Amplitudes of input signals values on the diagonal, respectively.

The default value, `No`, means no scaling is applied.

• 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.

### 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 requirement is a hard constraint.

Step Rejection Goal aims to keep the gain from disturbance to output below the gain of the reference model. The scalar value of the requirement f(x) is given by:

`$f\left(x\right)={‖{W}_{F}\left(s\right){T}_{dy}\left(s,x\right)‖}_{\infty },$`

or its discrete-time equivalent. Here, Tdy(s,x) is the closed-loop transfer function of the constrained response, and ${‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{\infty }$ denotes the H norm (see `norm`). WF is a frequency weighting function derived from the step-rejection profile you specify in the tuning goal. The gain of WF roughly matches the inverse of the reference model for gain values within 60 dB of the peak gain. For numerical reasons, the weighting function levels off outside this range, unless you specify a reference model that changes slope outside this range. This adjustment is called regularization. Because poles of WF close to s = 0 or s = `Inf` might lead to poor numeric conditioning for tuning, it is not recommended to specify reference models with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see Visualize Tuning Goals.

#### Implicit Constraints

This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, 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.