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Package: TuningGoal
Gain constraint for control system tuning
Use the TuningGoal.Gain object to specify a constraint that limits the gain from a specified input to a specified output. Use this requirement for control system tuning with tuning commands such as systune or looptune.
When you use a TuningGoal.Gain requirement, the software attempts to tune the system so that the gain from the specified input to the specified output does not exceed the specified value. By default, the constraint is applied with the loop closed. To apply the constraint to an openloop response, use the Openings property of the TuningGoal.Gain object.
You can use a gain constraint to:
Enforce a design requirement of disturbance rejection across a particular input/output pair, by constraining the gain to be less than 1
Enforce a custom rolloff rate in a particular frequency band, by specifying a gain profile in that band
Req = TuningGoal.Gain(inputname,outputname,gainvalue) creates a tuning requirement Req. This requirement constrains the gain from inputname to outputname to remain below the value gainvalue.
You can specify the inputname or outputname as cell arrays (vectorvalued signals). If you do so, then the tuning requirement constrains the largest singular value of the transfer matrix from inputname to outputname. See sigma for more information about singular values.
Req = TuningGoal.Gain(inputname,outputname,gainprofile) specifies the maximum gain as a function of frequency. You can specify the target gain profile (maximum gain across the I/O pair) as a smooth transfer function. Alternatively, you can sketch a piecewise error profile using an frd model.
inputname 
Input signal for requirement, specified as a string or a cell array of strings for vectorvalued signals. The signals available to designate as input signals for the tuning requirement are as follows.
If inputname is a loopswitch channel of a generalized model, the input signal for the requirement is the implied input associated with the switch:

outputname 
Output signal for requirement, specified as a string or a cell array of strings for vectorvalued signals. The signals available to designate as output signals for the tuning requirement are as follows.
If outputname is a loopswitch channel of a generalized model, the output signal for the requirement is the implied output associated with the switch:

gainvalue 
Maximum gain (linear). The gain constraint Req specifies that the gain from inputname to outputname is less than gainvalue. gainvalue is a scalar value. If the signals inputname or outputname are vectorvalued signals, then gainvalue constrains the largest singular value of the transfer matrix from inputname to outputname. See sigma for more information about singular values. 
gainprofile 
Gain profile as a function of frequency. The gain constraint Req specifies that the gain from inputname to outputname at a particular frequency is less than gainprofile. You can specify gainprofile as a smooth transfer function (tf , zpk, or ss model). Alternatively, you can sketch a piecewise gain profile using a frd model. When you do so, the software automatically maps the gain profile onto a zpk model. The magnitude of this zpk model approximates the desired gain profile. Use viewSpec(Req) to plot the magnitude of the zpk model. gainprofile is a SISO transfer function. If inputname or outputname are cell arrays, gainprofile applies to all I/O pairs from inputname to outputname 
Input 
Input signal names, specified as a cell array of strings. These strings specify the names of the inputs of the transfer function that the tuning requirement constrains. The initial value of the Input property is set by the inputname input argument when you construct the requirement object. 
InputScaling 
Input signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vectorvalued input signals when the choice of units results in a mix of small and large signals. This information is used to scale the closedloop transfer function from Input to Output when the tuning requirement is evaluated. Suppose T(s) is the closedloop transfer function from Input to Output. The requirement is evaluated for the scaled transfer function D_{o}^{–1}T(s)D_{i}. D_{o} and D_{i} are diagonal matrices with the OutputScaling and InputScaling values on the diagonal, respectively. The default value, [] , means no scaling. Default: [] 
Output 
Output signal names, specified as a cell array of strings. These strings specify the names of the outputs of the transfer function that the tuning requirement constrains. The initial value of the Output property is set by the outputname input argument when you construct the requirement object. 
OutputScaling 
Output signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vectorvalued output signals when the choice of units results in a mix of small and large signals. This information is used to scale the closedloop transfer function from Input to Output when the tuning requirement is evaluated. Suppose T(s) is the closedloop transfer function from Input to Output. The requirement is evaluated for the scaled transfer function D_{o}^{–1}T(s)D_{i}. D_{o} and D_{i} are diagonal matrices with the OutputScaling and InputScaling values on the diagonal, respectively. The default value, [] , means no scaling. Default: [] 
MaxGain 
Maximum gain as a function of frequency, expressed as a SISO zpk model. The software automatically maps the gainvalue or gainprofile input arguments to a zpk model. The magnitude of this zpk model approximates the desired gain profile, and is stored in the MaxGain property. Use viewSpec(Req) to plot the magnitude of MaxGain. 
Stabilize 
Stability requirement on closedloop dynamics, specified as 1 (true) or 0 (false). By default, TuningGoal.Gain imposes a stability requirement on the closedloop transfer function from the specified inputs to outputs, in addition to the gain requirement. If stability is not required or cannot be achieved, set Stabilize to false to remove the stability requirement. For example, if the gain constraint applies to an unstable openloop transfer function, set Stabilize to false. Default: 1(true) 
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. 
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 loopopening locations. The available loopopening locations depend on what kind of system you are tuning:
All feedback loops are closed by default, except where there is a permanent loopopening defined in an slTuner interface. Default: {} 
Name 
Name of the requirement object, specified as a string. For example, if Req is a requirement: Req.Name = 'LoopReq'; Default: [] 
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.Gain requirement, f(x) is given by:
T(s,x) is the closedloop transfer function from Input to Output. D_{o} and D_{i} are diagonal matrices with the OutputScaling and InputScaling property values on the diagonal, respectively. denotes the H_{∞} norm (see norm).
Create a gain constraint that enforces a disturbance rejection requirement from a signal 'du' to a signal 'u'.
Req = TuningGoal.Gain('du','u',1);
This requirement specifies that the maximum gain of the response from 'du' to 'u' not exceed 1 (0 dB).
Create a gain constraint that constrains the response from a signal 'du' to a signal 'u' to roll off at 20 dB/decade at frequencies greater than 1. The gain constraint also specifies disturbance rejection (maximum gain of 1) in the frequency range [0,1].
gmax = frd([1 1 0.01],[0 1 100]); Req = TuningGoal.Gain('du','u',gmax);
These commands use a frd model to specify the gain profile as a function of frequency. The maximum gain of 1 dB at the frequency 1 rad/s, together with the maximum gain of 0.01 dB at the frequency 100 rad/s, specifies the desired rolloff of 20 dB/decade.
The software converts gmax 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.
Create a gain constraint that enforces a disturbance rejection requirement from a signal 'du' to a signal 'u'.
Req = TuningGoal.Gain('du','u',1);
This requirement specifies that the maximum gain of the response from 'du' to 'u' not exceed 1 (0 dB).
looptune  looptune (for slTuner)  slTuner  systune  systune (for slTuner)  TuningGoal.LoopShape  TuningGoal.Tracking  viewSpec