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:

$MaxError = \frac{(PeakError) s + ω_{c} (DCError)}{s + ω_{c}} .$

Here, ω_c 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:

$MaxError = \frac{(PeakError) s + ω_{c} (DCError)}{s + ω_{c}} .$

The tracking bandwidth ω_c = 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.

Input Arguments

`inputname`	Input signals for the tuning goal, specified as a character vector or, for multiple-input tuning goals, a cell array of character vectors. If you are using the tuning goal to tune a Simulink^® model of a control system, then `inputname` can include: Any model input. Any linear analysis point marked in the model. Any linear analysis point in an `slTuner` (Simulink Control Design) interface associated with the Simulink model. Use `addPoint` (Simulink Control Design) to add analysis points to the `slTuner` interface. Use `getPoints` (Simulink Control Design) to get the list of analysis points available in an `slTuner` interface to your model. For example, suppose that the `slTuner` interface contains analysis points `u1` and `u2`. Use `'u1'` to designate that point as an input signal when creating tuning goals. Use `{'u1','u2'}` to designate a two-channel input. If you are using the tuning goal to tune a generalized state-space (`genss`) model of a control system, then `inputname` can include: Any input of the `genss` model Any `AnalysisPoint` location in the control system model For example, if you are tuning a control system model, `T`, then `inputname` can be any input name in `T.InputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_u`, then `inputname` can include `'AP_u'`. Use `getPoints` to get a list of analysis points available in a `genss` model. If `inputname` is an `AnalysisPoint` location of a generalized model, the input signal for the tuning goal is the implied input associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design.
`outputname`	Output signals for the tuning goal, specified as a character vector or, for multiple-output tuning goals, a cell array of character vectors. If you are using the tuning goal to tune a Simulink model of a control system, then `outputname` can include: Any model output. Any linear analysis point marked in the model. Any linear analysis point in an `slTuner` (Simulink Control Design) interface associated with the Simulink model. Use `addPoint` (Simulink Control Design) to add analysis points to the `slTuner` interface. Use `getPoints` (Simulink Control Design) to get the list of analysis points available in an `slTuner` interface to your model. For example, suppose that the `slTuner` interface contains analysis points `y1` and `y2`. Use `'y1'` to designate that point as an output signal when creating tuning goals. Use `{'y1','y2'}` to designate a two-channel output. If you are using the tuning goal to tune a generalized state-space (`genss`) model of a control system, then `outputname` can include: Any output of the `genss` model Any `AnalysisPoint` location in the control system model For example, if you are tuning a control system model, `T`, then `outputname` can be any output name in `T.OutputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_u`, then `outputname` can include `'AP_u'`. Use `getPoints` to get a list of analysis points available in a `genss` model. If `outputname` is an `AnalysisPoint` location of a generalized model, the output signal for the tuning goal is the implied output associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design.
`responsetime`	Target response time, specified as a positive scalar value. The tracking bandwidth is given by ω_c = 2/`responsetime`.Express the target response time in the time units of the models to be tuned. For example, when tuning a model `T`, if `T.TimeUnit` is `'minutes'`, then express the target response time in minutes.
`dcerror`	Maximum steady-state fractional tracking error, specified as a positive scalar value. For example, `dcerror` = 0.01 sets a maximum steady-state error of 1%. If `inputname` or `outputname` are vector-valued, `dcerror` applies to all I/O pairs from `inputname` to `outputname`. Default: 0.001
`peakerror`	Maximum fractional tracking error across all frequencies, specified as a positive scalar value greater than 1. Default: 1
`maxerror`	Target tracking error profile as a function of frequency, specified as a SISO numeric LTI model. `maxerror` is the maximum gain from reference signal to tracking error signal. You can specify `maxerror` as a smooth transfer function (`tf`, `zpk`, or `ss` model). Alternatively, you can sketch a piecewise error profile using an `frd` model. When you do so, the software automatically maps the error profile to a `zpk` model. The magnitude of the `zpk` model approximates the desired error profile. Use `show(Req)` to plot the magnitude of the `zpk` model. `maxerror` must be a SISO LTI model. If `inputname` or `outputname` are cell arrays, `maxerror` applies to all I/O pairs from `inputname` to `outputname`. If you are tuning in discrete time (that is, using a `genss` model or `slTuner` interface with nonzero `Ts`), you can specify `maxerror` as a discrete-time model with the same `Ts`. If you specify `maxerror` in continuous time, the tuning software discretizes it. Specifying the error profile in discrete time gives you more control over the error profile near the Nyquist frequency.

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: $MaxError = \frac{(PeakError) s + ω_{c} (DCError)}{s + ω_{c}} .$ `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` (Simulink Control Design) interface associated with the Simulink model. Use `addPoint` (Simulink Control Design) to add analysis points and loop openings to the `slTuner` interface. Use `getPoints` (Simulink Control Design) 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)

Figure contains an axes object. The axes object contains an object of type line. These objects represent Max error, Effective bound.

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:

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

or its discrete-time equivalent. Here, T(s,x) is the closed-loop transfer function from Input to Output, and ${‖ \cdot ‖}_{\infty}$ denotes the H_∞ norm (see getPeakGain). W_F is a frequency weighting function derived from the error profile you specify in the tuning goal. The gains of W_F 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 W_F 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 W_F, 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.