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TuningGoal.Tracking class

Package: TuningGoal

Tracking requirement for control system tuning

Description

Use the TuningGoal.Tracking object to specify a frequency-domain tracking requirement between specified inputs and outputs. This requirement specifies the maximum relative error (gain from reference input to tracking error) as a function of frequency. Use this requirement 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 requirement Req that constrains the tracking performance from inputname to outputname in the frequency domain. This tuning requirement 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 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 sampling time. InputScaling Reference signal scaling, specified as a vector of positive real values. For a MIMO tuning 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 requirement 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. String or cell array of strings specifying the names of the signals to be tracked, populated by the inputname argument. Output Output signal names. String or cell array of strings specifying the names of the signals that must track the reference signals, populated by the outputname argument. 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.Tracking requirement, f(x) is given by:

$f\left(x\right)={‖\frac{1}{\text{MaxError}}\left(T\left(s,x\right)-I\right)‖}_{\infty }.$

T(s,x) is the closed-loop transfer function from Input to Output.${‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{\infty }$ denotes the H norm (see norm).

Examples

Tracking Requirement With Response Time and Maximum Steady-State Tracking Error

Create a tracking requirement 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 requirement uses the default value, 1.

Tracking Requirement With Maximum Tracking Error as a Function of Frequency

Create a tracking requirement 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 a frd model 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 requirement. Display the error requirement using viewSpec.

```viewSpec(Req)
```

The yellow region indicates where the requirement is violated.