Package: TuningGoal
Tracking requirement for control system tuning
Use TuningGoal.Tracking
to specify a frequencydomain
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 frequencydomain tracking error:
$$\text{MaxError}=\frac{\left(\text{PeakError}\right)s+{\omega}_{c}\left(\text{DCError}\right)}{s+{\omega}_{c}}.$$
Here, ω_{c} is 2/(response time). The following plot illustrates these relationships for an example set of values.
creates
a tuning goal Req
=
TuningGoal.Tracking(inputname
,outputname
,responsetime
,dcerror
,peakerror
)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 ω_{c} =
2/responsetime
. The maximum relative steadystate
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 crosscoupling. See Properties.
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 Req
= TuningGoal.Tracking(inputname
,outputname
,maxerror
)frd
model.

Input signals for the tuning goal, specified as a character vector or, for multipleinput tuning goals, a cell array of character vectors.
For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. 

Output signals for the tuning goal, specified as a character vector or, for multipleoutput tuning goals, a cell array of character vectors.
For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. 

Target response time, specified as a positive scalar value. The tracking bandwidth is given by ω_{c} =
2/ 

Maximum steadystate fractional tracking error, specified as
a positive scalar value. For example, If Default: 0.001 

Maximum fractional tracking error across all frequencies, specified as a positive scalar value greater than 1. Default: 1 

Target tracking error profile as a function of frequency, specified as a SISO numeric LTI model.
If you are tuning in discrete time (that is, using a 

Maximum error as a function of frequency, expressed as a SISO If you use the syntax If you use the syntax $$\text{MaxError}=\frac{\left(\text{PeakError}\right)s+{\omega}_{c}\left(\text{DCError}\right)}{s+{\omega}_{c}}.$$
Use 

Frequency band in which tuning goal is enforced, specified as
a row vector of the form Set the Req.Focus = [1,100]; Default: 

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 vectorvalued step input. This information is used to scale the offdiagonal terms in the transfer function from reference to tracking error. This scaling ensures that crosscouplings are measured relative to the amplitude of each reference signal. For example, suppose that 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, Default: 

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 

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 

Models to which the tuning goal applies, specified as a vector of indices. Use the Req.Models = 2:4; When Default: 

Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loopopening locations. The tuning goal is evaluated against the openloop 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 If you are using the tuning goal to tune a generalized statespace
( For example, if Default: 

Name of the tuning goal, specified as a character vector. For example, if Req.Name = 'LoopReq'; Default: 
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 steadystate 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.
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.
This tuning goal imposes an implicit stability
constraint on the closedloop 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.
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:
$$f\left(x\right)={\Vert {W}_{F}\left(s\right)\left(T\left(s,x\right)I\right)\Vert}_{\infty},$$
or its discretetime equivalent. Here, T(s,x)
is the closedloop transfer function from Input
to Output
,
and $${\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert}_{\infty}$$ 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
lowfrequency or very highfrequency 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.
looptune
 systune
 systune
(for slTuner)
(Simulink Control Design)  looptune (for slTuner)
(Simulink Control Design)  viewGoal
 evalGoal
 TuningGoal.Gain
 TuningGoal.LoopShape
 slTuner
(Simulink Control Design)