PID tuning algorithm for linear plant model
C = pidtune(sys,type)
C = pidtune(sys,C0)
C = pidtune(sys,type,wc)
C = pidtune(sys,C0,wc)
C = pidtune(sys,...,opts)
[C,info] = pidtune(...)
a PID controller of type
C = pidtune(
type for the plant
type specifies a one-degree-of-freedom (1-DOF)
PID controller, then the controller is designed for the unit feedback
loop as illustrated:
type specifies a two-degree-of-freedom
(2-DOF) PID controller, then
a 2-DOF controller as in the feedback loop of this illustration:
pidtune tunes the parameters of the PID controller
balance performance (response time) and robustness (stability margins).
Single-input, single-output dynamic system model of the plant
for controller design.
If the plant has unstable poles, and
you must use
Controller type of the controller to design, specified as a
string. The term controller type refers to
which terms are present in the controller action. For example, a PI
controller has only a proportional and an integral term, while a PIDF
controller contains proportional, integrator, and filtered derivative
For more information about 2-DOF PID controllers generally, see Two-Degree-of-Freedom PID Controllers.
2-DOF Controllers with Fixed Setpoint Weights
For more detailed information about fixed-setpoint-weight 2-DOF PID controllers, see PID Controller Types for Tuning.
For more information about PID controller forms and formulas, see:
Target value for the 0 dB gain crossover frequency of the tuned
open-loop response. Specify
Option set specifying additional tuning options for the
Controller designed for
In either case, however, where the algorithm can achieve adequate
performance and robustness using a lower-order controller than specified
If you specify
Data structure containing information about performance and
robustness of the tuned PID loop. The fields of
This example shows how to design a PID controller for the plant given by:
As a first pass, create a model of the plant and design a simple PI controller for it.
sys = zpk(,[-1 -1 -1],1); [C_pi,info] = pidtune(sys,'PI')
C_pi = 1 Kp + Ki * --- s with Kp = 1.14, Ki = 0.454 Continuous-time PI controller in parallel form. info = Stable: 1 CrossoverFrequency: 0.5205 PhaseMargin: 60.0000
C_pi is a
pid controller object that represents a PI controller. The fields of
info show that the tuning algorithm chooses an open-loop crossover frequency of about 0.52 rad/s.
Examine the closed-loop step response (reference tracking) of the controlled system.
T_pi = feedback(C_pi*sys, 1); step(T_pi)
To improve the response time, you can set a higher target crossover frequency than the result that
pidtune automatically selects, 0.52. Increase the crossover frequency to 1.0.
[C_pi_fast,info] = pidtune(sys,'PI',1.0)
C_pi_fast = 1 Kp + Ki * --- s with Kp = 2.83, Ki = 0.0495 Continuous-time PI controller in parallel form. info = Stable: 1 CrossoverFrequency: 1 PhaseMargin: 43.9973
The new controller achieves the higher crossover frequency, but at the cost of a reduced phase margin.
Compare the closed-loop step response with the two controllers.
T_pi_fast = feedback(C_pi_fast*sys,1); step(T_pi,T_pi_fast) axis([0 30 0 1.4]) legend('PI','PI,fast')
This reduction in performance results because the PI controller does not have enough degrees of freedom to achieve a good phase margin at a crossover frequency of 1.0 rad/s. Adding a derivative action improves the response.
Design a PIDF controller for
Gc with the target crossover frequency of 1.0 rad/s.
[C_pidf_fast,info] = pidtune(sys,'PIDF',1.0)
C_pidf_fast = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 2.72, Ki = 0.985, Kd = 1.72, Tf = 0.00875 Continuous-time PIDF controller in parallel form. info = Stable: 1 CrossoverFrequency: 1 PhaseMargin: 60.0000
The fields of info show that the derivative action in the controller allows the tuning algorithm to design a more aggressive controller that achieves the target crossover frequency with a good phase margin.
Compare the closed-loop step response and disturbance rejection for the fast PI and PIDF controllers.
T_pidf_fast = feedback(C_pidf_fast*sys,1); step(T_pi_fast, T_pidf_fast); axis([0 30 0 1.4]); legend('PI,fast','PIDF,fast');
You can compare the input (load) disturbance rejection of the controlled system with the fast PI and PIDF controllers. To do so, plot the response of the closed-loop transfer function from the plant input to the plant output.
S_pi_fast = feedback(sys,C_pi_fast); S_pidf_fast = feedback(sys,C_pidf_fast); step(S_pi_fast,S_pidf_fast); axis([0 50 0 0.4]); legend('PI,fast','PIDF,fast');
This plot shows that the PIDF controller also provides faster disturbance rejection.
Design a PID controller in standard form for the plant defined by
To design a controller in standard form, use a standard-form
controller as the
C0 argument to
sys = zpk(,[-1 -1 -1],1); C0 = pidstd(1,1,1); C = pidtune(sys,C0)
C = 1 1 Kp * (1 + ---- * --- + Td * s) Ti s with Kp = 2.18, Ti = 2.36, Td = 0.591 Continuous-time PID controller in standard form
Design a discrete-time PI controller using a specified method to discretize the integrator.
If your plant is in discrete time,
returns a discrete-time controller using the default Forward Euler
integration method. To specify a different integration method, use
create a discrete-time controller having the desired integration method.
sys = c2d(tf([1 1],[1 5 6]),0.1); C0 = pid(1,1,'Ts',0.1,'IFormula','BackwardEuler'); C = pidtune(sys,C0)
C = Ts*z Kp + Ki * ------ z-1 with Kp = -0.518, Ki = 10.4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PI controller in parallel form.
C0 as an input causes
design a controller
C of the same form, type, and
discretization method as
C0. The display shows
that the integral term of
C uses the Backward Euler
Specify a Trapezoidal integrator and compare the resulting controller.
C0_tr = pid(1,1,'Ts',0.1,'IFormula','Trapezoidal'); Ctr = pidtune(sys,C_tr)
Ctr = Ts*(z+1) Ki * -------- 2*(z-1) with Ki = 10.4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time I-only controller.
Design a 2-DOF PID Controller for the plant given by the transfer function:
Use a target bandwidth of 1.5 rad/s.
wc = 1.5; G = tf(1,[1 0.5 0.1]); C2 = pidtune(G,'PID2',wc)
C2 = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s with Kp = 1.26, Ki = 0.255, Kd = 1.38, b = 0.665, c = 0 Continuous-time 2-DOF PID controller in parallel form.
Using the type string
pidtune to generate a 2-DOF controller, represented as a
pid2 object. The display confirms this result. The display also shows that
pidtune tunes all controller coefficients, including the setpoint weights
c, to balance performance and robustness.
The PID Tuner GUI cannot design controllers for multiple plants at once.
Åström, K. J. and Hägglund, T. Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006.