WOLFRAM SYSTEMMODELER
LimPIDP, PI, PD, and PID controller with limited output, antiwindup compensation and setpoint weighting 
SystemModel["Modelica.Blocks.Continuous.LimPID"]
This information is part of the Modelica Standard Library maintained by the Modelica Association.
Via parameter controllerType either P, PI, PD, or PID can be selected. If, e.g., PI is selected, all components belonging to the Dpart are removed from the block (via conditional declarations). The example model Modelica.Blocks.Examples.PID_Controller demonstrates the usage of this controller. Several practical aspects of PID controller design are incorporated according to chapter 3 of the book:
Besides the additive proportional, integral and derivative part of this controller, the following features are present:
The parameters of the controller can be manually adjusted by performing simulations of the closed loop system (= controller + plant connected together) and using the following strategy:
Initialization
This block can be initialized in different ways controlled by parameter initType. The possible values of initType are defined in Modelica.Blocks.Types.InitPID. This type is identical to Types.Init, with the only exception that the additional option DoNotUse_InitialIntegratorState is added for backward compatibility reasons (= integrator is initialized with InitialState whereas differential part is initialized with NoInit which was the initialization in version 2.2 of the Modelica standard library).
Based on the setting of initType, the integrator (I) and derivative (D) blocks inside the PID controller are initialized according to the following table:
initType  I.initType  D.initType 
NoInit  NoInit  NoInit 
SteadyState  SteadyState  SteadyState 
InitialState  InitialState  InitialState 
InitialOutput and initial equation: y = y_start 
NoInit  SteadyState 
DoNotUse_InitialIntegratorState  InitialState  NoInit 
In many cases, the most useful initial condition is SteadyState because initial transients are then no longer present. If initType = InitPID.SteadyState, then in some cases difficulties might occur. The reason is the equation of the integrator:
der(y) = k*u;
The steady state equation "der(x)=0" leads to the condition that the input u to the integrator is zero. If the input u is already (directly or indirectly) defined by another initial condition, then the initialization problem is singular (has none or infinitely many solutions). This situation occurs often for mechanical systems, where, e.g., u = desiredSpeed  measuredSpeed and since speed is both a state and a derivative, it is natural to initialize it with zero. As sketched this is, however, not possible. The solution is to not initialize u_m or the variable that is used to compute u_m by an algebraic equation.
If parameter limitAtInit = false, the limits at the output of this controller block are removed from the initialization problem which leads to a much simpler equation system. After initialization has been performed, it is checked via an assert whether the output is in the defined limits. For backward compatibility reasons limitAtInit = true. In most cases it is best to use limitAtInit = false.
u_s 
Type: RealInput Description: Connector of setpoint input signal 


u_m 
Type: RealInput Description: Connector of measurement input signal 

y 
Type: RealOutput Description: Connector of actuator output signal 
controllerType 
Value: .Modelica.Blocks.Types.SimpleController.PID Type: SimpleController Description: Type of controller 

k 
Value: 1 Type: Real () Description: Gain of controller 
Ti 
Value: 0.5 Type: Time (s) Description: Time constant of Integrator block 
Td 
Value: 0.1 Type: Time (s) Description: Time constant of Derivative block 
yMax 
Value: Type: Real Description: Upper limit of output 
yMin 
Value: yMax Type: Real Description: Lower limit of output 
wp 
Value: 1 Type: Real Description: Setpoint weight for Proportional block (0..1) 
wd 
Value: 0 Type: Real Description: Setpoint weight for Derivative block (0..1) 
Ni 
Value: 0.9 Type: Real Description: Ni*Ti is time constant of antiwindup compensation 
Nd 
Value: 10 Type: Real Description: The higher Nd, the more ideal the derivative block 
initType 
Value: .Modelica.Blocks.Types.InitPID.DoNotUse_InitialIntegratorState Type: InitPID Description: Type of initialization (1: no init, 2: steady state, 3: initial state, 4: initial output) 
limitsAtInit 
Value: true Type: Boolean Description: = false, if limits are ignored during initialization 
xi_start 
Value: 0 Type: Real Description: Initial or guess value value for integrator output (= integrator state) 
xd_start 
Value: 0 Type: Real Description: Initial or guess value for state of derivative block 
y_start 
Value: 0 Type: Real Description: Initial value of output 
strict 
Value: false Type: Boolean Description: = true, if strict limits with noEvent(..) 
with_I 
Value: controllerType == SimpleController.PI or controllerType == SimpleController.PID Type: Boolean Description: 
with_D 
Value: controllerType == SimpleController.PD or controllerType == SimpleController.PID Type: Boolean Description: 
addP 
Type: Add Description: 


addD 
Type: Add Description: 

P 
Type: Gain Description: 

I 
Type: Integrator Description: 

D 
Type: Derivative Description: 

gainPID 
Type: Gain Description: 

addPID 
Type: Add3 Description: 

addI 
Type: Add3 Description: 

addSat 
Type: Add Description: 

gainTrack 
Type: Gain Description: 

limiter 
Type: Limiter Description: 

Dzero 
Type: Constant Description: 

Izero 
Type: Constant Description: 
Demonstrates the usage of a Continuous.LimPID controller 

Test example: ElectricalExcitedSynchronousInductionMachine with voltage controller 

Test example: ElectricalExcitedSynchronousInductionMachine with rectifier 