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 open-loop 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 roll-off rate in a particular frequency band, by specifying a gain profile in that band
You can specify the inputname or outputname as cell arrays (vector-valued 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.
Input signal for requirement, specified as a string or a cell array of strings for vector-valued 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:
Output signal for requirement, specified as a string or a cell array of strings for vector-valued 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 is a scalar value. If the signals inputname or outputname are vector-valued signals, then gainvalue constrains the largest singular value of the transfer matrix from inputname to outputname. See sigma for more information about singular values.
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 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.
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.
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.
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 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.
Feedback loops to open when evaluating the requirement, specified as a cell array of strings that identify loop-opening locations. The available loop-opening locations depend on what kind of system you are tuning:
All feedback loops are closed by default.
Name of the requirement object, specified as a string.
For example, if Req is a requirement:
Req.Name = 'LoopReq';
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 closed-loop transfer function from Input to Output. 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).
Custom roll-off specification
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.
The yellow region indicates where the requirement is violated.
Constrain Open-Loop Gain
Create a gain constraint that limits to 100 the open-loop gain from a signal 'u' to a signal 'y'. Set the Openings property to the name of signal at which you want to open the loop.
Req = TuningGoal.Gain('u','y',100); Req.Openings = 'u';