# pid

PID controller in parallel form

## Description

Use pid to create parallel-form proportional-integral-derivative (PID) controller model objects, or to convert dynamic system models to parallel PID controller form.

The pid controller model object can represent parallel-form PID controllers in continuous time or discrete time.

• Continuous time — $C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}$

• Discrete time — $C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}$

Here:

• Kp is the proportional gain.

• Ki is the integral gain.

• Kd is the derivative gain.

• Tf is the first-order derivative filter time constant.

• IF(z) is the integrator method for computing integral in discrete-time controller.

• DF(z) is the integrator method for computing derivative filter in discrete-time controller.

You can then combine this object with other components of a control architecture, such as the plant, actuators, and sensors to represent your control system. For more information, see Control System Modeling with Model Objects.

You can create a PID controller model object by either specifying the controller parameters directly, or by converting a model of another type (such as a transfer function model tf) to PID controller form.

You can also use pid to create generalized state-space (genss) models or uncertain state-space (uss (Robust Control Toolbox)) models.

## Creation

You can obtain pid controller models in one of the following ways.

• Create a model using the pid function.

• Use pidtune function to tune PID controllers for a plant model. Specify a 1-DOF PID controller type in the type argument of the pidtune function to obtain a parallel-form PID controller. For example:

sys = zpk([],[-1 -1 -1],1);
C = pidtune(sys,'PID');
• Interactively tune PID controller for a plant model using:

### Description

example

C = pid(Kp,Ki,Kd,Tf) creates a continuous-time parallel-form PID controller model and sets the properties Kp, Ki, Kd, and Tf. The remaining properties have default values.

example

C = pid(Kp,Ki,Kd,Tf,Ts) creates a discrete-time PID controller model with sample time Ts.

example

C = pid(Kp) creates a continuous-time proportional (P) controller.

example

C = pid(Kp,Ki) creates a proportional and integral (PI) controller.

example

C = pid(Kp,Ki,Kd) creates a proportional, integral, and derivative (PID) controller.

example

C = pid(___,Name,Value) sets properties of the pid controller object specified using one or more Name,Value arguments for any of the previous input-argument combinations.

example

C = pid creates a controller object with default property values. To modify the property of the controller model, use dot notation.

example

C = pid(sys) converts the dynamic system model sys to a parallel-form pid controller object.

### Input Arguments

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Dynamic system, specified as a SISO dynamic system model or array of SISO dynamic system models. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, ss, or pidstd models.

• Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

The resulting PID controller assumes

• current values of the tunable components for tunable control design blocks.

• nominal model values for uncertain control design blocks.

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), idproc (System Identification Toolbox), idpoly (System Identification Toolbox), and idgrey (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

### Output Arguments

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PID controller model, returned as:

• A parallel-form PID controller (pid) model object, when all the gains have numeric values. When the gains are numeric arrays, C is an array of pid controller objects.

• A generalized state-space model (genss) object, when the numerator or denominator input arguments includes tunable parameters, such as realp parameters or generalized matrices (genmat).

• An uncertain state-space model (uss) object, when the numerator or denominator input arguments includes uncertain parameters. Using uncertain models requires Robust Control Toolbox software.

## Properties

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Proportional gain, specified as a real and finite value or a tunable object.

• To create a pid controller object, use a real and finite scalar value.

• To create an array of pid controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (realp) or generalized matrix (genmat).

• To create a tunable gain-scheduled controller model, use a tunable surface created using tunableSurface.

Integral gain, specified as a real and finite value or a tunable object.

• To create a pid controller object, use a real and finite scalar value.

• To create an array of pid controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (realp) or generalized matrix (genmat).

• To create a tunable gain-scheduled controller model, use a tunable surface created using tunableSurface.

Derivative gain, specified as a real and finite value or a tunable object.

• To create a pid controller object, use a real and finite scalar value.

• To create an array of pid controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (realp) or generalized matrix (genmat).

• To create a tunable gain-scheduled controller model, use a tunable surface created using tunableSurface.

Time constant of the first-order derivative filter, specified as a real and finite value or a tunable object.

• To create a pid controller object, use a real and finite scalar value.

• To create an array of pid controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (realp) or generalized matrix (genmat).

• To create a tunable gain-scheduled controller model, use a tunable surface created using tunableSurface.

Discrete integrator formula IF(z) for the integrator of the discrete-time pid controller:

$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$

Specify IFormula as one of the following:

• 'ForwardEuler'IF(z) = $\frac{{T}_{s}}{z-1}.$

This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the ForwardEuler formula can result in instability, even when discretizing a system that is stable in continuous time.

• 'BackwardEuler'IF(z) = $\frac{{T}_{s}z}{z-1}.$

An advantage of the BackwardEuler formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.

• 'Trapezoidal'IF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$

An advantage of the Trapezoidal formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the Trapezoidal formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.

When C is a continuous-time controller, IFormula is ''.

Discrete integrator formula DF(z) for the derivative filter of the discrete-time pid controller:

$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$

Specify DFormula as one of the following:

• 'ForwardEuler'DF(z) = $\frac{{T}_{s}}{z-1}.$

This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the ForwardEuler formula can result in instability, even when discretizing a system that is stable in continuous time.

• 'BackwardEuler'DF(z) = $\frac{{T}_{s}z}{z-1}.$

An advantage of the BackwardEuler formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.

• 'Trapezoidal'DF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$

An advantage of the Trapezoidal formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the Trapezoidal formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.

The Trapezoidal value for DFormula is not available for a pid controller with no derivative filter (Tf = 0).

When C is a continuous-time controller, DFormula is ''.

Time delay on the system input. InputDelay is always 0 for a pid controller object.

Time delay on the system output. OutputDelay is always 0 for a pid controller object.

Sample time, specified as:

• 0 for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. Specify Ts in the time unit specified by the TimeUnit property.

PID controller models do not support unspecified sample time (Ts = -1).

Note

Changing Ts does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use c2d and d2c. To change the sample time of a discrete-time system, use d2d.

The discrete integrator formulas of the discretized controller depend upon the c2d discretization method you use, as shown in this table.

'zoh'ForwardEulerForwardEuler
'foh'TrapezoidalTrapezoidal
'tustin'TrapezoidalTrapezoidal
'impulse'ForwardEulerForwardEuler
'matched'ForwardEulerForwardEuler

If you require different discrete integrator formulas, you can discretize the controller by directly setting Ts, IFormula, and DFormula to the desired values. However, this method does not compute new gain and filter-constant values for the discretized controller. Therefore, this method might yield a poorer match between the continuous-time and discrete-time PID controllers than using c2d.

Time variable units, specified as one of the following:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing TimeUnit has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

Input channel name, specified as one of the following:

• A character vector.

• '', no name specified.

Alternatively, assign the name error to the input of a controller model C as follows.

C.InputName = 'error';

You can use the shorthand notation u to refer to the InputName property. For example, C.u is equivalent to C.InputName.

Use InputName to:

• Identify channels on model display and plots.

• Specify connection points when interconnecting models.

Input channel units, specified as one of the following:

• A character vector.

• '', no units specified.

Use InputUnit to specify input signal units. InputUnit has no effect on system behavior.

For example, assign the concentration units 'mol/m^3' to the input of a controller model C as follows.

C.InputUnit = 'mol/m^3';

Input channel groups. This property is not needed for PID controller models.

By default, InputGroup is a structure with no fields.

Output channel name, specified as one of the following:

• A character vector.

• '', no name specified.

For example, assign the name 'control' to the output of a controller model C as follows.

C.OutputName = 'control';

You can also use the shorthand notation y to refer to the OutputName property. For example, C.y is equivalent to C.OutputName.

Use OutputName to:

• Identify channels on model display and plots.

• Specify connection points when interconnecting models.

Output channel units, specified as one of the following:

• A character vector.

• '', no units specified.

Use OutputUnit to specify output signal units. OutputUnit has no effect on system behavior.

For example, assign the unit 'volts' to the output of a controller model C as follows.

C.OutputUnit = 'volts';

Output channel groups. This property is not needed for PID controller models.

By default, OutputGroup is a structure with no fields.

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'.

User-specified data that you want to associate with the system, specified as any MATLAB data type.

System name, specified as a character vector. For example, 'system_1'.

Sampling grid for model arrays, specified as a structure array.

Use SamplingGrid to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the 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 must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 11-by-1 array of linear models, sysarr, by taking snapshots of a linear time-varying system at times t = 0:10. The following code stores the time samples with the linear models.

sysarr.SamplingGrid = struct('time',0:10)

Similarly, you can create a 6-by-9 model array, M, by independently sampling two variables, zeta and w. The following code maps 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

...

For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that correspond to each entry in the array. For instance, the Simulink Control Design™ commands linearize (Simulink Control Design) and slLinearizer (Simulink Control Design) populate SamplingGrid automatically.

By default, SamplingGrid is a structure with no fields.

## Object Functions

The following lists contain a representative subset of the functions you can use with pid models. In general, any function applicable to Dynamic System Models is applicable to a pid object.

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 step Step response plot of dynamic system; step response data impulse Impulse response plot of dynamic system; impulse response data lsim Plot simulated time response of dynamic system to arbitrary inputs; simulated response data bode Bode plot of frequency response, or magnitude and phase data nyquist Nyquist plot of frequency response nichols Nichols chart of frequency response bandwidth Frequency response bandwidth
 pole Poles of dynamic system zero Zeros and gain of SISO dynamic system pzplot Pole-zero plot of dynamic system model with additional plot customization options margin Gain margin, phase margin, and crossover frequencies
 zpk Zero-pole-gain model ss State-space model c2d Convert model from continuous to discrete time d2c Convert model from discrete to continuous time d2d Resample discrete-time model
 feedback Feedback connection of multiple models connect Block diagram interconnections of dynamic systems series Series connection of two models parallel Parallel connection of two models
 pidtune PID tuning algorithm for linear plant model rlocus Root locus plot of dynamic system piddata Access coefficients of parallel-form PID controller make2DOF Convert 1-DOF PID controller to 2-DOF controller pidTuner Open PID Tuner for PID tuning tunablePID Tunable PID controller

## Examples

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Create a continuous-time controller with proportional and derivative gains and a filter on the derivative term. To do so, set the integral gain to zero. Set the other gains and the filter time constant to the desired values.

Kp = 1;
Ki = 0;   % No integrator
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

Continuous-time PDF controller in parallel form.

The display shows the controller type, formula, and parameter values, and verifies that the controller has no integrator term.

Create a discrete-time PI controller with trapezoidal discretization formula.

To create a discrete-time PI controller, set the value of Ts and the discretization formula using Name,Value syntax.

C1 = pid(5,2.4,'Ts',0.1,'IFormula','Trapezoidal')    % Ts = 0.1s
C1 =

Ts*(z+1)
Kp + Ki * --------
2*(z-1)

with Kp = 5, Ki = 2.4, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PI controller in parallel form.

Alternatively, you can create the same discrete-time controller by supplying Ts as the fifth input argument after all four PID parameters, Kp, Ki, Kd, and Tf. Since you only want a PI controller, set Kd and Tf to zero.

C2 = pid(5,2.4,0,0,0.1,'IFormula','Trapezoidal')
C2 =

Ts*(z+1)
Kp + Ki * --------
2*(z-1)

with Kp = 5, Ki = 2.4, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PI controller in parallel form.

The display shows that C1 and C2 are the same.

When you create a PID controller, set the dynamic system properties InputName and OutputName. This is useful, for example, when you interconnect the PID controller with other dynamic system models using the connect command.

C = pid(1,2,3,'InputName','e','OutputName','u')
C =

1
Kp + Ki * --- + Kd * s
s

with Kp = 1, Ki = 2, Kd = 3

Continuous-time PID controller in parallel form.

The display does not show the input and output names for the PID controller, but you can examine the property values. For instance, verify the input name of the controller.

C.InputName
ans = 1x1 cell array
{'e'}

Create a 2-by-3 grid of PI controllers with proportional gain ranging from 1–2 across the array rows and integral gain ranging from 5–9 across columns.

To build the array of PID controllers, start with arrays representing the gains.

Kp = [1 1 1;2 2 2];
Ki = [5:2:9;5:2:9];

When you pass these arrays to the pid command, the command returns the array.

pi_array = pid(Kp,Ki,'Ts',0.1,'IFormula','BackwardEuler');
size(pi_array)
2x3 array of PID controller.
Each PID has 1 output and 1 input.

Alternatively, use the stack command to build an array of PID controllers.

C = pid(1,5,0.1)           % PID controller
C =

1
Kp + Ki * --- + Kd * s
s

with Kp = 1, Ki = 5, Kd = 0.1

Continuous-time PID controller in parallel form.

Cf = pid(1,5,0.1,0.5)      % PID controller with filter
Cf =

1            s
Kp + Ki * --- + Kd * --------
s          Tf*s+1

with Kp = 1, Ki = 5, Kd = 0.1, Tf = 0.5

Continuous-time PIDF controller in parallel form.

pid_array = stack(2,C,Cf); % stack along 2nd array dimension

These commands return a 1-by-2 array of controllers.

size(pid_array)
1x2 array of PID controller.
Each PID has 1 output and 1 input.

All PID controllers in an array must have the same sample time, discrete integrator formulas, and dynamic system properties such as InputName and OutputName.

Convert a standard form pidstd controller to parallel form.

Standard PID form expresses the controller actions in terms of an overall proportional gain Kp, integral and derivative time constants Ti and Td, and filter divisor N. You can convert any standard-form controller to parallel form using the pid command. For example, consider the following standard-form controller.

Kp = 2;
Ti = 3;
Td = 4;
N = 50;
C_std = pidstd(Kp,Ti,Td,N)
C_std =

1      1              s
Kp * (1 + ---- * --- + Td * ------------)
Ti     s          (Td/N)*s+1

with Kp = 2, Ti = 3, Td = 4, N = 50

Continuous-time PIDF controller in standard form

Convert this controller to parallel form using pid.

C_par = pid(C_std)
C_par =

1            s
Kp + Ki * --- + Kd * --------
s          Tf*s+1

with Kp = 2, Ki = 0.667, Kd = 8, Tf = 0.08

Continuous-time PIDF controller in parallel form.

Convert a continuous-time dynamic system that represents a PID controller to parallel pid form.

The following dynamic system, with an integrator and two zeros, is equivalent to a PID controller.

$H\left(s\right)=\frac{3\left(s+1\right)\left(s+2\right)}{s}.$

Create a zpk model of H. Then use the pid command to obtain H in terms of the PID gains Kp, Ki, and Kd.

H = zpk([-1,-2],0,3);
C = pid(H)
C =

1
Kp + Ki * --- + Kd * s
s

with Kp = 9, Ki = 6, Kd = 3

Continuous-time PID controller in parallel form.

Convert a discrete-time dynamic system that represents a PID controller with derivative filter to parallel pid form.

Create a discrete-time zpk model that represents a PIDF controller (two zeros and two poles, including the integrator pole at z = 1).

sys = zpk([-0.5,-0.6],[1 -0.2],3,'Ts',0.1);

When you convert sys to PID form, the result depends on which discrete integrator formulas you specify for the conversion. For instance, use the default, ForwardEuler, for both the integrator and the derivative.

Cfe = pid(sys)
Cfe =

Ts               1
Kp + Ki * ------ + Kd * -----------
z-1         Tf+Ts/(z-1)

with Kp = 2.75, Ki = 60, Kd = 0.0208, Tf = 0.0833, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PIDF controller in parallel form.

Now convert using the Trapezoidal formula.

Ctrap = pid(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')
Ctrap =

Ts*(z+1)                 1
Kp + Ki * -------- + Kd * -------------------
2*(z-1)         Tf+Ts/2*(z+1)/(z-1)

with Kp = -0.25, Ki = 60, Kd = 0.0208, Tf = 0.0333, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PIDF controller in parallel form.

The displays show the difference in resulting coefficient values and functional form.

For this particular dynamic system, you cannot write sys in parallel PID form using the BackwardEuler formula for the derivative filter. Doing so would result in Tf < 0, which is not permitted. In that case, pid returns an error.

Discretize a continuous-time PID controller and set integral and derivative filter formulas.

Create a continuous-time controller and discretize it using the zero-order-hold method of the c2d command.

Ccon = pid(1,2,3,4);  % continuous-time PIDF controller
Cdis1 = c2d(Ccon,0.1,'zoh')
Cdis1 =

Ts               1
Kp + Ki * ------ + Kd * -----------
z-1         Tf+Ts/(z-1)

with Kp = 1, Ki = 2, Kd = 3.04, Tf = 4.05, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PIDF controller in parallel form.

The display shows that c2d computes new PID gains for the discrete-time controller.

The discrete integrator formulas of the discretized controller depend on the c2d discretization method. For the zoh method, both IFormula and DFormula are ForwardEuler.

Cdis1.IFormula
ans =
'ForwardEuler'

Cdis1.DFormula
ans =
'ForwardEuler'

If you want to use different formulas from the ones returned by c2d, then you can directly set the Ts, IFormula, and DFormula properties of the controller to the desired values.

Cdis2 = Ccon;
Cdis2.Ts = 0.1;
Cdis2.IFormula = 'BackwardEuler';
Cdis2.DFormula = 'BackwardEuler';

However, these commands do not compute new PID gains for the discretized controller. To see this, examine Cdis2 and compare the coefficients to Ccon and Cdis1.

Cdis2
Cdis2 =

Ts*z               1
Kp + Ki * ------ + Kd * -------------
z-1         Tf+Ts*z/(z-1)

with Kp = 1, Ki = 2, Kd = 3, Tf = 4, Ts = 0.1

Sample time: 0.1 seconds
Discrete-time PIDF controller in parallel form.

## Version History

Introduced in R2010b