Documentation

# TuningGoal.Tracking class

Package: TuningGoal

Tracking requirement for control system tuning

## Description

Use TuningGoal.Tracking to specify a frequency-domain tracking requirement between specified inputs and outputs. This tuning goal specifies the maximum relative error (gain from reference input to tracking error) as a function of frequency. Use this tuning goal for control system tuning with tuning commands such as systune or looptune.

You can specify the maximum error profile directly by providing a transfer function. Alternatively, you can specify a target DC error, peak error, and response time. These parameters are converted to the following transfer function that describes the maximum frequency-domain tracking error:

$\text{MaxError}=\frac{\left(\text{PeakError}\right)s+{\omega }_{c}\left(\text{DCError}\right)}{s+{\omega }_{c}}.$

Here, ωc is 2/(response time). The following plot illustrates these relationships for an example set of values.

## Construction

Req = TuningGoal.Tracking(inputname,outputname,responsetime,dcerror,peakerror) creates a tuning goal Req that constrains the tracking performance from inputname to outputname in the frequency domain. This tuning goal specifies a maximum error profile as a function of frequency given by:

$\text{MaxError}=\frac{\left(\text{PeakError}\right)s+{\omega }_{c}\left(\text{DCError}\right)}{s+{\omega }_{c}}.$

The tracking bandwidth ωc = 2/responsetime. The maximum relative steady-state error is given by dcerror, and peakerror gives the peak relative error across all frequencies.

You can specify a MIMO tracking requirement by specifying signal names or a cell array of multiple signal names for inputname or outputname. For MIMO tracking requirements, use the InputScaling property to help limit cross-coupling. See Properties.

Req = TuningGoal.Tracking(inputname,outputname,maxerror) specifies the maximum relative error as a function of frequency. You can specify the target error profile (maximum gain from reference signal to tracking error signal) as a smooth transfer function. Alternatively, you can sketch a piecewise error profile using an frd model.

## Properties

 MaxError Maximum error as a function of frequency, expressed as a SISO zpk model. This property stores the maximum tracking error as a function of frequency (maximum gain from reference signal to tracking error signal). If you use the syntax Req = TuningGoal.Tracking(inputname,outputname,maxerror), then the MaxError property is the zpk equivalent or approximation of the LTI model you supplied as the maxerror input argument. If you use the syntax Req = TuningGoal.Tracking(inputname,outputname,resptime,dcerror,peakerror), then the MaxError is a zpk transfer function given by: $\text{MaxError}=\frac{\left(\text{PeakError}\right)s+{\omega }_{c}\left(\text{DCError}\right)}{s+{\omega }_{c}}.$ MaxError is a SISO LTI model. If inputname or outputname are cell arrays, MaxError applies to all I/O pairs from inputname to outputname. Use show(Req) to plot the magnitude of MaxError. 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. InputScaling Reference signal scaling, specified as a vector of positive real values. For a MIMO tracking requirement, when the choice of units results in a mix of small and large signals in different channels of the response, use this property to specify the relative amplitude of each entry in the vector-valued step input. This information is used to scale the off-diagonal terms in the transfer function from reference to tracking error. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. For example, suppose that Req is a tuning goal that signals {'y1','y2'} track reference signals {'r1','r2'}. Suppose further that you require the outputs to track the references with less than 10% cross-coupling. If r1 and r2 have comparable amplitudes, then it is sufficient to keep the gains from r1 to y2 and r2 and y1 below 0.1. However, if r1 is 100 times larger than r2, the gain from r1 to y2 must be less than 0.001 to ensure that r1 changes y2 by less than 10% of the r2 target. To ensure this result, set the InputScaling property as follows. Req.InputScaling = [100,1]; This tells the software to take into account that the first reference signal is 100 times greater than the second reference signal. The default value, [] , means no scaling. Default: [] Input Reference signal names, specified as a character vector or cell array of character vectors specifying the names of the signals to be tracked, populated by the inputname argument. Output Output signal names, specified as a character vector or cell array of character vectors specifying the names of the signals that must track the reference signals, populated by the outputname argument. 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

### Tracking Goal With Response Time and Maximum Steady-State Tracking Error

Create a tracking goal specifying that a signal 'theta' track a signal 'theta_ref'. The required response time is 2, in the time units of the control system you are tuning. The maximum steady-state error is 0.1%.

Req = TuningGoal.Tracking('theta_ref','theta',2,0.001);

Since peakerror is unspecified, this tuning goal uses the default value, 1.

### Tracking Goal With Maximum Tracking Error as a Function of Frequency

Create a tracking goal specifying that a signal 'theta' track a signal 'theta_ref'. The maximum relative error is 0.01 (1%) in the frequency range [0,1]. The relative error increases to 1 (100%) at the frequency 100.

Use an frd model to specify the error profile as a function of frequency.

err = frd([0.01 0.01 1],[0 1 100]);
Req = TuningGoal.Tracking('theta_ref','theta',err);

The software converts err into a smooth function of frequency that approximates the piecewise specified profile. Display this function using viewGoal.

viewGoal(Req)

The dashed line is the target error profile stored in MaxError, and the shaded region indicates where the tuning goal 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, 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.Tracking, f(x) is given by:

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

or its discrete-time equivalent. Here, T(s,x) is the closed-loop transfer function from Input to Output, and ${‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{\infty }$ denotes the H norm (see getPeakGain). WF is a frequency weighting function derived from the error profile you specify in the tuning goal. The gains of WF and 1/MaxError roughly match for gain values between –20 dB and 60 dB. 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 of the systune optimization problem, it is not recommended to specify error 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