Documentation 
Create PID controller in parallel form, convert to parallelform PID controller
C = pid(Kp,Ki,Kd,Tf)
C = pid(Kp,Ki,Kd,Tf,Ts)
C = pid(sys)
C = pid(Kp)
C = pid(Kp,Ki)
C = pid(Kp,Ki,Kd)
C = pid(...,Name,Value)
C = pid
C = pid(Kp,Ki,Kd,Tf) creates a continuoustime PID controller with proportional, integral, and derivative gains Kp, Ki, and Kd and firstorder derivative filter time constant Tf:
$$C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}.$$
This representation is in parallel form. If all of Kp, Ki, Kd, and Tf are real, then the resulting C is a pid controller object. If one or more of these coefficients is tunable (realp or genmat), then C is a tunable generalized statespace (genss) model object.
C = pid(Kp,Ki,Kd,Tf,Ts) creates a discretetime PID controller with sampling time Ts. The controller is:
$$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$$
IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter. By default, IF(z) = DF(z) = T_{s}z/(z – 1). To choose different discrete integrator formulas, use the IFormula and DFormula properties. (See Properties for more information about IFormula and DFormula). If DFormula = 'ForwardEuler' (the default value) and Tf ≠ 0, then Ts and Tf must satisfy Tf > Ts/2. This requirement ensures a stable derivative filter pole.
C = pid(sys) converts the dynamic system sys to a parallel form pid controller object.
C = pid(Kp) creates a continuoustime proportional (P) controller with Ki = 0, Kd = 0, and Tf = 0.
C = pid(Kp,Ki) creates a proportional and integral (PI) controller with Kd = 0 and Tf = 0.
C = pid(Kp,Ki,Kd) creates a proportional, integral, and derivative (PID) controller with Tf = 0.
C = pid(...,Name,Value) creates a controller or converts a dynamic system to a pid controller object with additional options specified by one or more Name,Value pair arguments.
Specify optional commaseparated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Use Name,Value syntax to set the numerical integration formulas IFormula and DFormula of a discretetime pid controller, or to set other object properties such as InputName and OutputName. For information about available properties of pid controller objects, see Properties.
C 
PID controller, represented as a pid controller object, an array of pid controller objects, a genss object, or a genss array.

Kp, Ki, Kd 
PID controller gains. The Kp, Ki, and Kd properties store the proportional, integral, and derivative gains, respectively. Kp, Ki, and Kd values are real and finite. 
Tf 
Derivative filter time constant. The Tf property stores the derivative filter time constant of the pid controller object. Tf are real, finite, and greater than or equal to zero. 
IFormula 
Discrete integrator formula IF(z) for the integrator of the discretetime pid controller C: $$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$$ IFormula can take the following values:
When C is a continuoustime controller, IFormula is ''. Default: 'ForwardEuler' 
DFormula 
Discrete integrator formula DF(z) for the derivative filter of the discretetime pid controller C: $$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$$ DFormula can take the following values:
When C is a continuoustime controller, DFormula is ''. Default: 'ForwardEuler' 
InputDelay 
Time delay on the system input. InputDelay is always 0 for a pid controller object. 
OutputDelay 
Time delay on the system Output. OutputDelay is always 0 for a pid controller object. 
Ts 
Sampling time. For continuoustime models, Ts = 0. For discretetime models, Ts is a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model. To denote a discretetime model with unspecified sampling time, set Ts = 1. Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous and discretetime representations. Use d2d to change the sampling time of a discretetime system. Default: 0 (continuous time) 
TimeUnit 
String representing the unit of the time variable. This property specifies the units for the time variable, the sampling time Ts, and any time delays in the model. Use any of the following values:
Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior. Default: 'seconds' 
InputName 
Input channel names. Set InputName to a string for singleinput model. For a multiinput model, set InputName to a cell array of strings. Alternatively, use automatic vector expansion to assign input names for multiinput models. For example, if sys is a twoinput model, enter: sys.InputName = 'controls'; The input names automatically expand to {'controls(1)';'controls(2)'}. You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName. Input channel names have several uses, including:
Default: Empty string '' for all input channels 
InputUnit 
Input channel units. Use InputUnit to keep track of input signal units. For a singleinput model, set InputUnit to a string. For a multiinput model, set InputUnit to a cell array of strings. InputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
InputGroup 
Input channel groups. The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example: sys.InputGroup.controls = [1 2]; sys.InputGroup.noise = [3 5]; creates input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the controls inputs to all outputs using: sys(:,'controls') Default: Struct with no fields 
OutputName 
Output channel names. Set OutputName to a string for singleoutput model. For a multioutput model, set OutputName to a cell array of strings. Alternatively, use automatic vector expansion to assign output names for multioutput models. For example, if sys is a twooutput model, enter: sys.OutputName = 'measurements'; The output names to automatically expand to {'measurements(1)';'measurements(2)'}. You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName. Output channel names have several uses, including:
Default: Empty string '' for all input channels 
OutputUnit 
Output channel units. Use OutputUnit to keep track of output signal units. For a singleoutput model, set OutputUnit to a string. For a multioutput model, set OutputUnit to a cell array of strings. OutputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
OutputGroup 
Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example: sys.OutputGroup.temperature = [1]; sys.InputGroup.measurement = [3 5]; creates output groups named temperature and measurement that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the measurement outputs using: sys('measurement',:) Default: Struct with no fields 
Name 
System name. Set Name to a string to label the system. Default: '' 
Notes 
Any text that you want to associate with the system. Set Notes to a string or a cell array of strings. Default: {} 
UserData 
Any type of data you wish to associate with system. Set UserData to any MATLAB^{®} data type. Default: [] 
SamplingGrid 
Sampling grid for model arrays, specified as a data structure. For model arrays that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model in the array. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. For example, suppose you create a 11by1 array of linear models, sysarr, by taking snapshots of a linear timevarying system at times t = 0:10. The following code stores the time samples with the linear models. sysarr.SamplingGrid = struct('time',0:10)
Similarly, suppose you create a 6by9 model array, M, by independently sampling two variables, zeta and w. The following code attaches the (zeta,w) values to M. [zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>) M.SamplingGrid = struct('zeta',zeta,'w',w) When you display M, each entry in the array includes the corresponding zeta and w values. M M(:,:,1,1) [zeta=0.3, w=5] = 25  s^2 + 3 s + 25 M(:,:,2,1) [zeta=0.35, w=5] = 25  s^2 + 3.5 s + 25 ... Default: [] 
PID Controller with Proportional and Derivative Gains, and Filter Time Constant (PDF Controller)
Create a continuoustime controller with proportional and derivative gains, and filter time constant (PDF controller).
Kp=1; Ki=0; Kd=3; Tf=0.5; C = pid(Kp,Ki,Kd,Tf)
C = s Kp + Kd *  Tf*s+1 with Kp = 1, Kd = 3, Tf = 0.5 Continuoustime PDF controller in parallel form.
The display hows the controller type, formula, and parameter values.
DiscreteTime PI Controller
Create a discretetime PI controller with trapezoidal discretization formula.
To create a discretetime controller, set the value of Ts using Name,Value syntax.
C = pid(5,2.4,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s
This command produces the result:
Discretetime PI controller in parallel form: Ts*(z+1) Kp + Ki *  2*(z1) with Kp = 5, Ki = 2.4, Ts = 0.1
Alternatively, you can create the same discretetime controller by supplying Ts as the fifth argument after all four PID parameters Kp, Ki, Kd, and Tf.
C = pid(5,2.4,0,0,0.1,'IFormula','Trapezoidal');
PID Controller with Custom Input and Output Names
Create a PID controller, and set dynamic system properties InputName and OutputName.
C = pid(1,2,3,'InputName','e','OutputName','u');
Array of PID Controllers
Create a 2by3 grid of PI controllers with proportional gain ranging from 1–2 and integral gain ranging from 5–9.
Create a grid of PI controllers with proportional gain varying row to row and integral gain varying column to column. To do so, start with arrays representing the gains.
Kp = [1 1 1;2 2 2]; Ki = [5:2:9;5:2:9]; pi_array = pid(Kp,Ki,'Ts',0.1,'IFormula','BackwardEuler');
These commands produce a 2by3 array of discretetime pid objects. All pid objects in an array must have the same sample time, discrete integrator formulas, and dynamic system properties (such as InputName and OutputName).
Alternatively, you can use stack to build arrays of pid objects.
C = pid(1,5,0.1) % PID controller Cf = pid(1,5,0.1,0.5) % PID controller with filter pid_array = stack(2,C,Cf); % stack along 2nd array dimension
These commands produce a 1by2 array of controllers. Enter the command:
size(pid_array)
to see the result
1x2 array of PID controller. Each PID has 1 output and 1 input.
Convert PID Controller from Standard to Parallel Form
Convert a standard form pidstd controller to parallel form.
Standard PID form expresses the controller actions in terms of an overall proportional gain K_{p}, integral and derivative times T_{i} and T_{d}, and filter divisor N. You can convert any standard form controller to parallel form using pid.
stdsys = pidstd(2,3,4,5); % Standardform controller parsys = pid(stdsys)
These commands produce a parallelform controller:
Continuoustime PIDF controller in parallel form: 1 s Kp + Ki *  + Kd *  s Tf*s+1 with Kp = 2, Ki = 0.66667, Kd = 8, Tf = 0.8
Convert Dynamic System to ParallelForm PID Controller
Convert a continuoustime dynamic system that represents a PID controller to parallel pid form.
The dynamic system
$$H\left(s\right)=\frac{3\left(s+1\right)\left(s+2\right)}{s}$$
represents a PID controller. Use pid to obtain H(s) to in terms of the PID gains K_{p}, K_{i}, and K_{d}.
H = zpk([1,2],0,3); C = pid(H)
These commands produce the result:
Continuoustime PID controller in parallel form: 1 Kp + Ki *  + Kd * s s with Kp = 9, Ki = 6, Kd = 3
Convert DiscreteTime ZeroPoleGain Model to ParallelForm PID Controller
Convert a discretetime dynamic system that represents a PID controller with derivative filter to parallel pid form.
% PIDF controller expressed in zpk form sys = zpk([0.5,0.6],[1 0.2],3,'Ts',0.1)
The resulting pid object depends upon the discrete integrator formula you specify for IFormula and DFormula. For example, if you use the default ForwardEuler for both formulas:
C = pid(sys)
returns the result
Discretetime PIDF controller in parallel form: Ts 1 Kp + Ki *  + Kd *  z1 Tf+Ts/(z1) with Kp = 2.75, Ki = 60, Kd = 0.020833, Tf = 0.083333, Ts = 0.1
Converting using the Trapezoidal formula returns different parameter values:
C = pid(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')
This command returns the result:
Discretetime PIDF controller in parallel form: Ts*(z+1) 1 Kp + Ki *  + Kd *  2*(z1) Tf+Ts/2*(z+1)/(z1) with Kp = 0.25, Ki = 60, Kd = 0.020833, Tf = 0.033333, Ts = 0.1
For this particular sys, you cannot write sys in parallel PID form using the BackwardEuler formula for DFormula. Doing so would result in Tf < 0, which is not permitted. In that case, pid returns an error.
Discretize a Continuoustime PID Controller
First, discretize the controller using the 'zoh' method of c2d.
Cc = pid(1,2,3,4) % continuoustime pidf controller Cd1 = c2d(Cc,0.1,'zoh')
c2d computes new parameters for the discretetime controller:
Discretetime PIDF controller in parallel form: Ts 1 Kp + Ki *  + Kd *  z1 Tf+Ts/(z1) with Kp = 1, Ki = 2, Kd = 3.0377, Tf = 4.0502, Ts = 0.1
The resulting discretetime controller uses ForwardEuler (T_{s}/(z–1)) for both IFormula and DFormula.
The discrete integrator formulas of the discretized controller depend upon the c2d discretization method, as described in Tips. To use a different IFormula and DFormula, directly set Ts, IFormula, and DFormula to the desired values:
Cd2 = Cc; Cd2.Ts = 0.1; Cd2.IFormula = 'BackwardEuler'; Cd2.DFormula = 'BackwardEuler';
These commands do not compute new parameter values for the discretized controller. To see this, enter:
Cd2
to obtain the result:
Discretetime PIDF controller in parallel form: Ts*z 1 Kp + Ki *  + Kd *  z1 Tf+Ts*z/(z1) with Kp = 1, Ki = 2, Kd = 3, Tf = 4, Ts = 0.1