pidstd
PID controller in standard form
Description
Use pidstd
to create standard-form PID controller objects, or
to convert dynamic system models to
standard PID controller form.
The pidstd
controller model object can represent standard-form PID
controllers in continuous time or discrete time.
Continuous time —
Discrete time —
Here:
Kp is the proportional gain.
Ti is the integral time.
Td is the derivative time.
N is the first-order derivative filter divisor.
IF(z) is the integrator method for computing the integral in the discrete-time controller.
DF(z) is the integrator method for computing the derivative filter in the 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 pidstd
to create generalized state-space (genss
) models or uncertain state-space (uss
(Robust Control Toolbox)) models.
Creation
You can obtain pidstd
controller models in one of the following
ways.
Create a model using the
pidstd
function.Use the
pidtune
function to tune PID controllers for a plant model. Specify a baseline standard-form PID controller type using theC0
argument of thepidtune
function. For example:sys = zpk([],[-1 -1 -1],1); C0 = pidstd(1,1,1); C = pidtune(sys,C0)
Interactively tune the PID controller for a plant model using:
The Tune PID Controller Live Editor task.
The PID Tuner app.
Syntax
Description
Input Arguments
Kp
— Proportional gain
1
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
Proportional gain, specified as a real and finite value or a tunable object.
To create a
pidstd
controller object, use a real and finite scalar value.To create an array of
pidstd
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
.
Ti
— Integral time
Inf
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
Integral time, specified as a real and positive value or a tunable object.
To create a
pidstd
controller object, use a real and positive scalar value.To create an array of
pidstd
controller objects, use an array of real and positive 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
.
Td
— Derivative time
0
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
Derivative time, specified as a real, finite, and nonnegative value or a tunable object.
To create a
pidstd
controller object, use a real, finite, and nonnegative scalar value.To create an array of
pidstd
controller objects, use an array of real, finite, and nonnegative 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
.
N
— Derivative filter divisor
Inf
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
First-order derivative filter divisor, specified as a real and positive value or a tunable object.
To create a
pidstd
controller object, use a real and positive scalar value.To create an array of
pidstd
controller objects, use an array of real and positive 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
.
When N
= Inf
, the controller has no filter on the derivative action.
Ts
— Sample time
0
(default) | positive scalar
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 theTimeUnit
property.
In an array of pidstd
controllers, the same
Ts
applies to all controllers.
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.
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 c2d
.
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
.
sys
— Dynamic system
dynamic system model | model array
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
, orpidstd
models.Generalized or uncertain LTI models such as
genss
oruss
(Robust Control Toolbox) models.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), andidgrey
(System Identification Toolbox) models.
Output Arguments
C
— Standard-form PID controller model
pidstd
model object | genss
model object | uss
model object
PID controller model, returned as:
A standard-form PID controller (
pidstd
) model object, when all the gains have numeric values. When the gains are numeric arrays,C
is an array ofpidstd
controller objects.A generalized state-space model (
genss
) object, when thenumerator
ordenominator
input arguments includes tunable parameters, such asrealp
parameters or generalized matrices (genmat
).An uncertain state-space model (
uss
) object, when thenumerator
ordenominator
input arguments includes uncertain parameters. Using uncertain models requires Robust Control Toolbox™ software.
Properties
Kp, Ti, Td, N
— PID controller coefficients
scalars
PID controller coefficients, specified as scalars. When creating a
pidstd
controller object or array of objects, specify these
coefficients in the Kp
, Ti
,
Td
, and N
input arguments.
IFormula
— Method for computing integral in discrete-time controller
'ForwardEuler'
(default) | 'BackwardEuler'
| 'Trapezoidal'
Discrete integrator formula IF(z) for the
integrator of the discrete-time pidstd
controller:
Specify IFormula
as one of the following:
'ForwardEuler'
— IF(z) =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) =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, theTrapezoidal
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 ''
.
DFormula
— Method for computing derivative in discrete-time controller
'ForwardEuler'
(default) | 'BackwardEuler'
| 'Trapezoidal'
Discrete integrator formula DF(z) for the
derivative filter of the discrete-time pidstd
controller:
Specify DFormula
as one of the following:
'ForwardEuler'
— DF(z) =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) =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, theTrapezoidal
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 ''
.
InputDelay
— Input delay
0 (default)
This property is read-only.
Time delay on the system input. InputDelay
is always 0 for a
pidstd
controller object.
OutputDelay
— Output delay
0 (default)
This property is read-only.
Time delay on the system output. OutputDelay
is always 0 for a
pidstd
controller object.
Ts
— Sample time
0
(default) | positive scalar
Sample time, specified as:
0
for continuous-time systems.A positive scalar representing the sampling period of a discrete-time system.
Ts
in specified in the time unit specified by theTimeUnit
property.
If pidstd
is an array of PID controllers, the same Ts
applies to all controllers.
TimeUnit
— Time variable units
'seconds'
(default) | 'nanoseconds'
| 'microseconds'
| 'milliseconds'
| 'minutes'
| 'hours'
| 'days'
| 'weeks'
| 'months'
| 'years'
| ...
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.
InputName
— Input channel name
''
(default) | character vector
Input channel name, specified as one of the following:
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.
InputUnit
— Input channel units
''
(default) | character vector
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';
InputGroup
— Input channel groups
structure
Input channel groups. This property is not needed for PID controller models.
By default, InputGroup
is a structure with no fields.
OutputName
— Output channel names
''
(default) | character vector
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.
OutputUnit
— Output channel units
''
(default) | character vector
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';
OutputGroup
— Output channel groups
structure
Output channel groups. This property is not needed for PID controller models.
By default, OutputGroup
is a structure with no fields.
Notes
— User-specified text
{}
(default) | character vector | cell array of character vectors
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'
.
UserData
— User-specified data
[]
(default) | any MATLAB® data type
User-specified data that you want to associate with the system, specified as any MATLAB data type.
Name
— System name
''
(default) | character vector
System name, specified as a character vector. For example, 'system_1'
.
SamplingGrid
— Sampling grid for model arrays
structure array
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 pidstd
models. In general, any function applicable to Dynamic System Models is applicable to a pidstd
object.
Linear Analysis
step | Step response of dynamic system |
impulse | Impulse response plot of dynamic system; impulse response data |
lsim | Compute time response simulation data of dynamic system to arbitrary inputs |
bode | Bode frequency response of dynamic system |
nyquist | Nyquist response of dynamic system |
nichols | Nichols response of dynamic system |
bandwidth | Frequency response bandwidth |
Stability Analysis
Model Transformation
Model Interconnection
Examples
Create Continuous-Time Standard-Form PDF Controller
Create a continuous-time standard-form PDF controller with proportional and derivative terms, and a filter divisor. To do so, set the integral time to infinite. Set the other gains and the filter divisor constant to the desired values.
Kp = 1; Ti = Inf; Td = 3; N = 6; C = pidstd(Kp,Ti,Td,N)
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 parameter values, and verifies that the controller has no integrator term.
Discrete-Time Standard-Form PI Controller
Create a discrete-time standard-form 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.
C2 = pidstd(1,0.5,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s
C2 = 1 Ts*(z+1) Kp * (1 + ---- * --------) Ti 2*(z-1) with Kp = 1, Ti = 0.5, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PI controller in standard form
Alternatively, you can create the same discrete-time controller by supplying Ts
as the fifth input argument after all four PID parameters, Kp
, Ti
, Td
, and N
. Since you only want a PI controller, set Td
to zero and N
to infinite.
C2 = pidstd(5,2.4,0,Inf,0.1,'IFormula','Trapezoidal')
C2 = 1 Ts*(z+1) Kp * (1 + ---- * --------) Ti 2*(z-1) with Kp = 5, Ti = 2.4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PI controller in standard form
The display shows that C1
and C2
are the same.
Standard-Form PID Controller with Named Input and Output
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 = pidstd(1,2,3,'InputName','e','OutputName','u')
C = 1 1 Kp * (1 + ---- * --- + Td * s) Ti s with Kp = 1, Ti = 2, Td = 3 Continuous-time PID controller in standard 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'}
Array of Standard-Form PID Controllers
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]; Ti = [5:2:9;5:2:9];
When you pass these arrays to the pidstd
command, the command returns an array of controllers.
pi_array = pidstd(Kp,Ti,'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.
Create a PID controller.
C = pidstd(1,5,0.1)
C = 1 1 Kp * (1 + ---- * --- + Td * s) Ti s with Kp = 1, Ti = 5, Td = 0.1 Continuous-time PID controller in standard form
Create a PIDF controller.
Cf = pidstd(1,5,0.1,0.5)
Cf = 1 1 s Kp * (1 + ---- * --- + Td * ------------) Ti s (Td/N)*s+1 with Kp = 1, Ti = 5, Td = 0.1, N = 0.5 Continuous-time PIDF controller in standard form
Stack the controllers along the second array dimension.
pid_array = stack(2,C,Cf);
This command returns 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 PID Controller from Parallel to Standard Form
Convert a parallel-form pid
controller to standard form.
Parallel PID form expresses the controller actions in terms of an overall proportional, integral, and derivative gains Kp
, Ki
and Kd
, and a filter time constant Tf
. You can convert any parallel-form controller to standard form using the pidstd
command, provided that:
The parallel-form controller is not a pure integrator.
The gains
Kp
,Ki
andKd
all have the same sign.
For example, consider the following parallel-form controller.
Kp = 2; Ki = 3; Kd = 4; Tf = 5; C_par = pid(Kp,Ki,Kd,Tf)
C_par = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 2, Ki = 3, Kd = 4, Tf = 5 Continuous-time PIDF controller in parallel form.
Convert this controller to standard form using pidstd
.
C_std = pidstd(C_par)
C_std = 1 1 s Kp * (1 + ---- * --- + Td * ------------) Ti s (Td/N)*s+1 with Kp = 2, Ti = 0.667, Td = 2, N = 0.4 Continuous-time PIDF controller in standard form
Convert Dynamic System to Standard-Form PID Controller
Convert a continuous-time dynamic system that represents a PID controller to standard pidstd
form.
The following dynamic system, with an integrator and two zeros, is equivalent to a PID controller.
Create a zpk
model of H. Then use the pidstd
command to obtain H in terms of the PID gains Kp
, Ti
, and Td
.
H = zpk([-1,-2],0,3); C = pidstd(H)
C = 1 1 Kp * (1 + ---- * --- + Td * s) Ti s with Kp = 9, Ti = 1.5, Td = 0.333 Continuous-time PID controller in standard form
Convert Discrete-Time Dynamic System to Standard-Form PID Controller
Convert a discrete-time dynamic system that represents a PID controller with derivative filter to standard pidstd
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.
C = pidstd(sys)
C = 1 Ts 1 Kp * (1 + ---- * ------ + Td * ---------------) Ti z-1 (Td/N)+Ts/(z-1) with Kp = 2.75, Ti = 0.0458, Td = 0.00758, N = 0.0909, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in standard form
For this particular dynamic system, you cannot write sys
in standard PID form using the BackwardEuler
formula for the derivative filter. 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 PID form using the Trapezoidal
formula. Doing so would result in negative Ti
and Td
, which is also not permitted.
Discretize Continuous-Time Standard-Form PID Controller
Discretize a continuous-time PID controller and set integral and derivative filter formulas.
Create a continuous-time PIDF controller and discretize it using the zero-order-hold method of the c2d
command.
Ccon = pidstd(1,2,3,4);
Cdis1 = c2d(Ccon,0.1,'zoh')
Cdis1 = 1 Ts 1 Kp * (1 + ---- * ------ + Td * ---------------) Ti z-1 (Td/N)+Ts/(z-1) with Kp = 1, Ti = 2, Td = 3.2, N = 4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in standard 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 = 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 Sample time: 0.1 seconds Discrete-time PIDF controller in standard form
Version History
Introduced in R2010b
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