SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
PIDTune
PIDTune[lsys]
gives a feedback PID controller for the linear timeinvariant system lsys.
PIDTune[lsys, "carch"]
gives a controller of architecture (, , , etc).
PIDTune[lsys, {"carch", "trule"}]
gives a controller using the tuning rule .
PIDTune[lsys, ..., "prop"]
returns the value for the property .
Details and OptionsDetails and Options
 PIDTune will produce a PID controller such that the closedloop system csys is good at rejecting disturbances as well as follow changes in the reference .
 The ideal PID controller effectively computes the control signal through with proportional, integral, and derivative terms.
 By default, the transfer function representing is returned.
 The system lsys can be a continuoustime linear SISO system, such as TransferFunctionModel, StateSpaceModel, etc.
 The following controller architectures can be used:

"P" proportional "PI" proportional integral "PD" proportional derivative "PID" proportional integral derivative "PFD" with filtered derivative part "PIFD" with filtered derivative part  By default, a controller is chosen.
 The tuning rule is chosen automatically, unless explicitly specified, based on stability properties of the system lsys.
 The following tuning rules can be used:

"AMIGO" approximate constrained integral gain optimization "AMIGOFrequency" from frequency response "ChienHronesReswick" uses rules based on step response "CohenCoon" dominant pole placement for amplitude decay ratio 0.25 "DisturbanceRejection" minimizes infinity norm of disturbance response "ErrorIntegral" minimizes an integral of error "KappaTau" dominant pole placement "KappaTauFrequency" dominant pole placement from frequency response "LambdaTuning" uses internal model control "LoopShaping" exact "SkogestadIMC" uses internal model control based on halfrule reduction "TyreusLuyben" uses rules based on frequency response "ZieglerNichols" uses rules based on step response "ZieglerNicholsFrequency" uses rules based on frequency response  PIDTune[lsys, ..., "PIDData"] returns a PIDData object pid that can be used to extract additional properties using the form pid["property"].
 PIDTune[lsys, ..., "property"] can be used to directly give the value of pid["property"].
 PIDTune can optionally compute a feedforward filter to improve tracking of the reference independently of disturbance rejection capabilities of the feedback controller .
 The feedforward weights b and c affect what fraction of the reference signal is provided to the different terms of the PID controller. The resulting controller is effectively given by , but provided as a separate feedforward filter having same effect when combined with the feedback controller .
 Properties related to different transfer functions include:

"Feedback" feedback "Feedforward" feedforward "OpenLoop" series connection of and lsys "DisturbanceOutput" transfer function from to "DisturbanceControl" transfer function from to "ReferenceOutput" transfer function from to "ReferenceControl" transfer function from to "SensorOutput" transfer function from to "SensorControl" transfer function from to "ISA" transfer function from and to  Properties related to parametrizations of transfer functions:

"FeedbackIdealParameters" in "FeedbackSeriesParameters" in "FeedbackParallelParameters" in "FeedforwardParameters" in the "DerivativeFilterParameter" n controlling derivative filters in and  The derivative filter replaces the direct derivative with its filtered version , effectively series connecting a lowpass filter with the derivative.
 Properties related to tuning rule and internal tuning model for lsys:

"TuningRule" chosen tuning rule "TuningModel" model and parameters  The possible tuning models and parameters:

{"FOPTD2",{k_{v},l}} first order plus time delay {"FOPTD3",{k_{s},l,t_{c}} first order plus time delay {"Frequency2",{k_{u},t_{u}}} ultimate gain and ultimate period {"Frequency3",{k_{u},t_{u},k_{s}} ultimate gain, ultimate period, and static gain {"SOPTD",{k_{s},l,t_{c1},t_{c2}}} secondorder plus time delay  The parameters are static gain , velocity constant , delay , time constants , ultimate gain and ultimate period .
 The following options can be given:

PIDFeedforward None reference weights for PIDDerivativeFilter None filtered derivative Method Automatic methods to use  The settings for Method control the parameter estimation method used to derive the tuning models.
 Possible settings for Method include:

Automatic automatically choose the estimation method "CharacteristicArea" use the characteristic area of the step response "GainMargins" use the phase crossover frequency and gain margin "InflectionPoint" use the inflection point of the step response "MethodOfMoments" match moments "TwoPointSampling" use the 28% and 63% points of the step response
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