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# pidstd

Create a PID controller in standard form, convert to standard-form PID controller

## Syntax

C = pidstd(Kp,Ti,Td,N)
C = pidstd(Kp,Ti,Td,N,Ts)
C = pidstd(sys)
C = pidstd(Kp)
C = pidstd(Kp,Ti)
C = pidstd(Kp,Ti,Td)
C = pidstd(...,Name,Value)
C = pidstd

## Description

C = pidstd(Kp,Ti,Td,N) creates a continuous-time PIDF (PID with first-order derivative filter) controller object in standard form. The controller has proportional gain Kp, integral and derivative times Ti and Td, and first-order derivative filter divisor N:

$C={K}_{p}\left(1+\frac{1}{{T}_{i}}\frac{1}{s}+\frac{{T}_{d}s}{\frac{{T}_{d}}{N}s+1}\right).$

C = pidstd(Kp,Ti,Td,N,Ts) creates a discrete-time controller with sample time Ts. The discrete-time controller is:

$C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$

IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter. By default,

$IF\left(z\right)=DF\left(z\right)=\frac{{T}_{s}}{z-1}.$

To choose different discrete integrator formulas, use the IFormula and DFormula inputs. (See Properties for more information about IFormula and DFormula). If DFormula = 'ForwardEuler' (the default value) and N ≠ Inf, then Ts, Td, and N must satisfy Td/N > Ts/2. This requirement ensures a stable derivative filter pole.

C = pidstd(sys) converts the dynamic system sys to a standard form pidstd controller object.

C = pidstd(Kp) creates a continuous-time proportional (P) controller with Ti = Inf, Td = 0, and N = Inf.

C = pidstd(Kp,Ti) creates a proportional and integral (PI) controller with Td = 0 and N = Inf.

C = pidstd(Kp,Ti,Td) creates a proportional, integral, and derivative (PID) controller with N = Inf.

C = pidstd(...,Name,Value) creates a controller or converts a dynamic system to a pidstd controller object with additional options specified by one or more Name,Value pair arguments.

C = pidstd creates a P controller with Kp = 1.

## Input Arguments

 Kp Proportional gain. Kp can be: A real and finite value.An array of real and finite values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. Default: 1 Ti Integrator time. Ti can be: A real and positive value.An array of real and positive values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. Default: Inf Td Derivative time. Td can be: A real, finite, and nonnegative value.An array of real, finite, and nonnegative values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When Td = 0, the controller has no derivative action. Default: 0 N Derivative filter divisor. N can be: A real and positive value.An array of real and positive values.A tunable parameter (realp) or generalized matrix (genmat).A tunable surface for gain-scheduled tuning, created using tunableSurface. When N = Inf, the controller has no filter on the derivative action. Default: Inf Ts Sample time. To create a discrete-time pidstd controller, provide a positive real value (Ts > 0).pidstd does not support discrete-time controller with undetermined sample time (Ts = -1). Ts must be a scalar value. In an array of pidstd controllers, each controller must have the same Ts. Default: 0 (continuous time) sys SISO dynamic system to convert to standard pidstd form. sys must represent a valid controller that can be written in standard form with Ti > 0, Td ≥ 0, and N > 0. sys can also be an array of SISO dynamic systems.

### Name-Value Pair Arguments

Specify optional comma-separated 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 discrete-time pidstd controller, or to set other object properties such as InputName and OutputName. For information about available properties of pidstd controller objects, see Properties.

## Output Arguments

 C pidstd object representing a single-input, single-output PID controller in standard form. The controller type (P, PI, PD, PDF, PID, PIDF) depends upon the values of Kp, Ti, Td, and N. For example, when Td = Inf and Kp and Ti are finite and nonzero, C is a PI controller. Enter getType(C) to obtain the controller type. When the inputs Kp,Ti, Td, and N or the input sys are arrays, C is an array of pidstd objects.

## Properties

 Kp Proportional gain. Kp must be real and finite. Ti Integral time. Ti must be real, finite, and greater than or equal to zero. Td Derivative time. Td must be real, finite, and greater than or equal to zero. N Derivative filter divisor. N must be real, and greater than or equal to zero. IFormula Discrete integrator formula IF(z) for the integrator of the discrete-time pidstd controller C: $C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$ IFormula can take the following values: '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 ''. Default: 'ForwardEuler' DFormula Discrete integrator formula DF(z) for the derivative filter of the discrete-time pidstd controller C: $C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$ DFormula can take the following values: '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 pidstd controller with no derivative filter (N = Inf). When C is a continuous-time controller, DFormula is ''. Default: 'ForwardEuler' InputDelay Time delay on the system input. InputDelay is always 0 for a pidstd controller object. OutputDelay Time delay on the system Output. OutputDelay is always 0 for a pidstd controller object. Ts Sample time. For continuous-time models, Ts = 0. For discrete-time 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. PID controller models do not support unspecified sample time (Ts = -1). Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous- and discrete-time representations. Use d2d to change the sample time of a discrete-time system. Default: 0 (continuous time) TimeUnit Units for the time variable, the sample time Ts, and any time delays in the model, specified as one of the following values:'nanoseconds''microseconds''milliseconds''seconds' 'minutes''hours''days''weeks''months''years' 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 name, specified as a character vector. Use this property to name the input channel of the controller model. For example, 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. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: Empty character vector, '' InputUnit Input channel units, specified as a character vector. Use this property to track input signal units. For example, assign the concentration units mol/m^3 to the input of a controller model C as follows. C.InputUnit = 'mol/m^3'; InputUnit has no effect on system behavior. Default: Empty character vector, '' InputGroup Input channel groups. This property is not needed for PID controller models. Default: struct with no fields OutputName Output channel name, specified as a character vector. Use this property to name the output channel of the controller model. For example, assign the name control to the output of a controller model C as follows. C.OutputName = 'control'; You can use the shorthand notation y to refer to the OutputName property. For example, C.y is equivalent to C.OutputName. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: Empty character vector, '' OutputUnit Output channel units, specified as a character vector. Use this property to track output signal units. For example, assign the unit Volts to the output of a controller model C as follows. C.OutputUnit = 'Volts'; OutputUnit has no effect on system behavior. Default: Empty character vector, '' OutputGroup Output channel groups. This property is not needed for PID controller models. Default: struct with no fields Name System name, specified as a character vector. For example, 'system_1'. Default: '' Notes Any 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'. Default: {} UserData Any type of data you want to associate with system, specified as 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 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, suppose you create a 6-by-9 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 ... 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 example, the Simulink Control Design™ commands linearize and slLinearizer populate SamplingGrid in this way. Default: []

## Examples

Create a continuous-time standard-form PDF controller with proportional gain 1, derivative time 3, and a filter divisor of 6.

C = pidstd(1,Inf,3,6);
C =

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

with Kp = 1, Td = 3, N = 6

Continuous-time PDF controller in standard form

The display shows the controller type, formula, and coefficient values.

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

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

C = pidstd(1,0.5,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s

This command produces the result:

Discrete-time PI controller in standard form:

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

with Kp = 1, Ti = 0.5, Ts = 0.1

Alternatively, you can create the same discrete-time controller by supplying Ts as the fifth argument after all four PID parameters Kp, Ti, Td, and N.

C = pidstd(5,2.4,0,Inf,0.1,'IFormula','Trapezoidal');

Create a PID controller and set dynamic system properties InputName and OutputName.

C = pidstd(1,0.5,3,'InputName','e','OutputName','u')

Create a 2-by-3 grid of PI controllers with proportional gain ranging from 1–2 and integral time ranging from 5–9.

Create a grid of PI controllers with proportional gain varying row to row and integral time varying column to column. To do so, start with arrays representing the gains.

Kp = [1 1 1;2 2 2];
Ti = [5:2:9;5:2:9];
pi_array = pidstd(Kp,Ti,'Ts',0.1,'IFormula','BackwardEuler');

These commands produce a 2-by-3 array of discrete-time pidstd objects. All pidstd 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 the stack command to build arrays of pidstd objects.

C = pidstd(1,5,0.1)        % PID controller
Cf = pidstd(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 1-by-2 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 a parallel-form pid controller to standard form.

Parallel PID form expresses the controller actions in terms of an proportional, integral, and derivative gains Kp, Ki, and Kd, and a filter time constant Tf. You can convert a parallel form controller parsys to standard form using pidstd, provided that:

• parsys is not a pure integrator (I) controller.

• The gains Kp, Ki, and Kd of parsys all have the same sign.

parsys = pid(2,3,4,5);  % Standard-form controller
stdsys = pidstd(parsys)

These commands produce a parallel-form controller:

Continuous-time PIDF controller in standard form:

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

with Kp = 2, Ti = 0.66667, Td = 2, N = 0.4

Convert a continuous-time dynamic system that represents a PID controller to standard pidstd form.

The dynamic system

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

represents a PID controller. Use pidstd to obtain H(s) to in terms of the standard-form PID parameters Kp, Ti, and Td.

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

These commands produce the result:

Continuous-time PID controller in standard form:

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

with Kp = 9, Ti = 1.5, Td = 0.33333

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

% PIDF controller expressed in zpk form
sys = zpk([-0.5,-0.6],[1 -0.2],3,'Ts',0.1)

The resulting pidstd 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 = pidstd(sys)

you obtain the result:

Discrete-time PIDF controller in standard form:

1       Ts                 1
Kp * (1 + ---- * ------ + Td * ---------------)
Ti      z-1         (Td/N)+Ts/(z-1)

with Kp = 2.75, Ti = 0.045833, Td = 0.0075758, N = 0.090909, Ts = 0.1

For this particular sys, you cannot write sys in standard PID form using the BackwardEuler formula for the DFormula. Doing so would result in N < 0, which is not permitted. In that case, pidstd returns an error.

Similarly, you cannot write sys in standard form using the Trapezoidal formula for both integrators. Doing so would result in negative Ti and Td, which also returns an error.

Discretize a continuous-time pidstd controller.

First, discretize the controller using the 'zoh' method of c2d.

Cc = pidstd(1,2,3,4)  % continuous-time pidf controller
Cd1 = c2d(Cc,0.1,'zoh')

c2d computes new parameters for the discrete-time controller:

Discrete-time PIDF controller in standard form:

1       Ts                 1
Kp * (1 + ---- * ------ + Td * ---------------)
Ti      z-1         (Td/N)+Ts/(z-1)

with Kp = 1, Ti = 2, Td = 3.2044, N = 4, Ts = 0.1

The resulting discrete-time controller uses ForwardEuler (Ts/(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:

Discrete-time PIDF controller in standard form:

1      Ts*z                 1
Kp * (1 + ---- * ------ + Td * -----------------)
Ti      z-1         (Td/N)+Ts*z/(z-1)

with Kp = 1, Ti = 2, Td = 3, N = 4, Ts = 0.1

## Tips

• Use pidstd either to create a pidstd controller object from known PID gain, integral and derivative times, and filter divisor, or to convert a dynamic system model to a pidstd object.

• To tune a PID controller for a particular plant, use pidtune or pidTuner.

• Create arrays of pidstd controllers by:

• Specifying array values for Kp,Ti,Td, and N

• Specifying an array of dynamic systems sys to convert to standard PID form

• Using stack to build arrays from individual controllers or smaller arrays

In an array of pidstd controllers, each controller must have the same sample time Ts and discrete integrator formulas IFormula and DFormula.

• To create or convert to a parallel-form controller, use pid. Parallel form expresses the controller actions in terms of proportional, integral, and derivative gains Kp, Ki and Kd, and a filter time constant Tf:

$C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}.$

• There are two ways to discretize a continuous-time pidstd controller:

• Use the c2d command. c2d computes new parameter values for the discretized controller. The discrete integrator formulas of the discretized controller depend upon the c2d discretization method you use, as shown in the following 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. (See this example.) 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- and discrete-time pidstd controllers than using c2d.