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Contents

Index

• Getting Started

• User's Guide

• Blocks

• Functions

• GUIs

• Numeric Linear Models

• Generalized Linear Models

• Data Extraction

• Conversions

• System Interconnections

• System Gain and Dynamics

• Time Domain Analysis

• Frequency Domain Analysis

• Model Simplification

• Compensator Design

• LQR/LQG Design

• State-Space Models

• Frequency Response Data Models

• Time Delays

• Model Dimensions and Characteristics

• Overloaded and Arithmetic Operators

• Matrix Equation Solvers

• Preferences

• Visualization of Model Dynamics and Responses

• Plot Customization

Examples

• Release Notes

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

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

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 pidtool.
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 sampling 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 K_p, K_i and K_d, and a filter time constant T_f:
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.
  c2d Discretization Method IFormula DFormula
  
  'zoh' ForwardEuler ForwardEuler
  
  'foh' Trapezoidal Trapezoidal
  
  'tustin' Trapezoidal Trapezoidal
  
  'impulse' ForwardEuler ForwardEuler
  
  'matched' ForwardEuler ForwardEuler
  
  For more information about c2d discretization methods, See the c2d reference page. For more information about IFormula and DFormula, see Properties .
- 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.

`c2d` Discretization Method	`IFormula`	`DFormula`
`'zoh'`	`ForwardEuler`	`ForwardEuler`
`'foh'`	`Trapezoidal`	`Trapezoidal`
`'tustin'`	`Trapezoidal`	`Trapezoidal`
`'impulse'`	`ForwardEuler`	`ForwardEuler`
`'matched'`	`ForwardEuler`	`ForwardEuler`

Input Arguments

`Kp`	Proportional gain. `Kp` must be a real and finite value. For an array of `pidstd` controllers, `Kp` must be an array of real and finite values. Default: 1
`Ti`	Integral time. `Ti` must be a real and positive value. When `Ti` = `Inf`, the controller has no integral action. For an array of `pidstd` controllers, `Ti` must be an array of real and positive values. Default: `Inf`
`Td`	Derivative time. `Td` must be a real, finite, and nonnegative value. When `Td` = 0, the controller has no derivative action. For an array of `pidstd` controllers, `Td` must be an array of real, finite, and nonnegative values. Default: 0
`N`	Time constant of the first-order derivative filter. `N` must be a real and positive value. When `N` = `Inf`, the controller has no derivative filter. For an array of `pidstd` controllers, `N` must be an array of real and positive values. Default: `Inf`
`Ts`	Sampling 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`.
`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, where 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

pidstd controller objects have the following 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 time. `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`: `IFormula` can take the following values: `'ForwardEuler'` — IF(z) = This formula is best for small sampling time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sampling time, the `ForwardEuler` formula can result in instability, even when discretizing a system that is stable in continuous time. `'BackwardEuler'` — IF(z) = 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) = 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`: `DFormula` can take the following values: `'ForwardEuler'` — DF(z) = This formula is best for small sampling time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sampling time, the `ForwardEuler` formula can result in instability, even when discretizing a system that is stable in continuous time. `'BackwardEuler'` — DF(z) = 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) = 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`	Sampling time. For continuous-time models, `Ts = 0`. For discrete-time models, `Ts` is a positive scalar representing the sampling period expressed in the unit specified by the `TimeUnit` property of the model. To denote a discrete-time 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 discrete-time representations. Use `d2d` to change the sampling time of a discrete-time system. Default: 0 (continuous time)
`TimeUnit`	String representing the unit of the time variable, any time delays in the model (for continuous-time models), and the sampling time `Ts` (for discrete-time models). `TimeUnit` can take the following values: `'nanoseconds'` `'microseconds'` `'milliseconds'` `'seconds'` `'minutes'` `'hours'` `'days'` `'weeks'` `'months'` `'years'` Changing this property 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 single-input model. For a multi-input model, set `InputName` to a cell array of strings. Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if `sys` is a two-input model, enter: sys.InputName = 'controls'; The software automatically expands the input names 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: Identifying channels on model display and plots Extracting subsystems of MIMO systems Specifying connection points when interconnecting models Default: Empty string `''` for all input channels
`InputUnit`	Input channel units. Use `InputUnit` to keep track of input signal units. Set `InputUnit` to a string for single-input model, or to a cell array of strings for a multi-input model. `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 whose field names are the group names and whose field values are the input channels belong 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 single-output model. For a multi-output model, set `OutputName` to a cell array of strings. Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if `sys` is a two-output model, enter: sys.OutputName = 'measurements'; The software automatically expands the output names 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: Identifying channels on model display and plots Extracting subsystems of MIMO systems Specifying connection points when interconnecting models Default: Empty string `''` for all input channels
`OutputUnit`	Output channel units. Use `OutputUnit` to keep track of output signal units. Set `OutputUnit` to a string for single-input model, or to a cell array of strings for a multi-input model. `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 whose field names are the group names and whose field values are the output channels belong 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 wish 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: `[]`

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);

Confirm the controller type:

getType(C)

This command produces the result:

ans =

PDF

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 standard form pid controller to parallel form.

Parallel PID form expresses the controller actions in terms of an proportional, integral, and derivative gains K_p, K_i, and K_d, and a filter time constant T_f. 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 parallel pid form.

The dynamic system

represents a PID controller. Use pidstd to obtain H(s) to in terms of the standard-form PID parameters K_p, T_i, and T_d.

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 (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:

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