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(...)
pidtune tunes the parameters of the PID controller C to balance performance (response time) and robustness (stability margins).
Single-input, single-output dynamic system model of the plant for controller design. sys can be:
If the plant has unstable poles, and sys is one of the following:
you must use pidtuneOptions to specify the number of unstable poles in the plant, if any.
Controller type (actions) of the controller to design, specified as one of the following strings.
If sys is a discrete-time model with sampling time Ts, pidtune designs a discrete-time controller with the same Ts. The controller has the ForwardEuler discrete integrator formula for both integral and derivative actions. Use the input C0 instead of type if you want to design a controller having a different discrete integrator formula.
Target value for the 0 dB gain crossover frequency of the tuned open-loop response L = sys*C. Specify wc in units of radians/TimeUnit, where TimeUnit is the time unit of sys. The crossover frequency wc roughly sets the control bandwidth. The closed-loop response time is approximately 1/wc.
Increase wc to speed up the response. Decrease wc to improve stability. When you omit wc, pidtune automatically chooses a value, based on the plant dynamics, that achieves a balance between response and stability.
Option set specifying additional tuning options for the pidtune design algorithm, such as target phase margin. Use pidtuneOptions to create opts.
Controller designed for sys. If sys is an array of linear models, pidtune designs a controller for each linear model and returns an array of PID controllers.
In either case, however, where the algorithm can achieve adequate performance and robustness using a lower-order controller than specified with type or C0, pidtune returns a C having fewer actions than specified. For example, C can be a PI controller even though type is 'pidf'.
If you specify C0, C also obtains model properties such as InputName and OutputName from C0. For more information about model properties, see the reference pages for each type of dynamic system model.
Data structure containing information about performance and robustness of the tuned PID loop. The fields of info are:
If sys is an array of plant models, info is an array of data structures containing information about each tuned PID loop.
This example shows how to design a PID controller for the plant
As a first pass, design a simple PI controller:
sys = zpk(,[-1 -1 -1],1); % define the plant [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('C\_pi','C\_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 = 1.03, Kd = 1.76, 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('C\_pi\_fast','C\_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('C\_pi\_fast','C\_pidf\_fast');
This plot shows that the PIDF controller also provides faster disturbance rejection.
This example shows how to 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 pidtune.
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
This example shows how to design a discrete-time PI controller using a specified method to discretize the integrator.
If your plant is in discrete time, pidtune automatically returns a discrete-time controller using the default Forward Euler integration method. To specify a different integration method, use pid or pidstd to 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.
Using C0 as an input causes pidtune to 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 integration method.
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.
The PID Tuner GUI cannot design controllers for multiple plants at once.
Typical PID tuning objectives include:
Closed-loop stability — The closed-loop system output remains bounded for bounded input.
Adequate performance — The closed-loop system tracks reference changes and suppresses disturbances as rapidly as possible. The larger the loop bandwidth (the first frequency at which the open-loop gain is unity), the faster the controller responds to changes in the reference or disturbances in the loop.
Adequate robustness — The loop design has enough phase margin and gain margin to allow for modeling errors or variations in system dynamics.
The MathWorks algorithm for tuning PID controllers helps you meet these objectives by automatically tuning the PID gains to balance performance (response time) and robustness (stability margins).
By default, the algorithm chooses a crossover frequency (loop bandwidth) based upon the plant dynamics, and designs for a target phase margin of 60°. If you specify the crossover frequency using wc or the phase margin using pidtuneOptions, the algorithm computes PID gains that best meet those targets.
Åström, K. J. and Hägglund, T. Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006.