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 —
Discrete time —
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 the integral in discrete-time controller.
DF(z) is the integrator method for computing the 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 the
pidtune
function to tune PID controllers for a plant model. Specify a 1-DOF PID controller type in thetype
argument of thepidtune
function to obtain a parallel-form PID controller. For example:sys = zpk([],[-1 -1 -1],1); C = pidtune(sys,'PID');
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
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
.
Ki
— Integral gain
0
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
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
.
Kd
— Derivative gain
0
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
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
.
Tf
— Derivative filter time constant
0
(default) | scalar | vector | matrix | realp
object | genmat
object | tunableSurface
object
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
.
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 pid
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
— Parallel-form PID controller model
pid
model object | genss
model object | uss
model object
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 ofpid
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, Ki, Kd, Tf
— PID controller coefficients
scalars
PID controller coefficients, specified as scalars. When creating a
pid
controller object or array of objects, specify these coefficients
in the Kp
, Ki
, Kd
, and
Tf
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 pid
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 pid
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 pid
controller with no derivative filter
(Tf = 0
).
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
pid
controller object.
OutputDelay
— Output delay
0 (default)
This property is read-only.
Time delay on the system output. OutputDelay
is always 0 for a
pid
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 pid
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 pid
models. In general, any function applicable to Dynamic System Models is applicable to a pid
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
PDF Controller
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.
Discrete-Time PI Controller
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.
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 = 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'}
Array of 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]; 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 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 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 Dynamic System to Parallel-Form PID Controller
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.
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 Discrete-Time Dynamic System to Parallel-Form PID Controller
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 Continuous-Time PID Controller
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
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