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Linear Control Systems Lectures #5 - PID Controller
Guillaume Drion Academic year 2018-2019
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Outline
PID controller: general form
Effects of the proportional, integral and derivative actions
PID tuning
Integrator windup and setpoint weighting
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A simple controller to control complex systems: PID
Closing the loop: the controller signal enters in the input
SYSTEMInput Output
CONTROLLER
Classical controller: Proportional-Integral-Derivative (PID)where is an error measure between a reference and the output of the system.
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The classical controller: PID controller
PID stands for Proportional-Integral-Derivative.
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The classical controller: PID controller
Proportional term: considers the current value of the error .
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The classical controller: PID controller
Integral term: considers the past values of the error .
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The classical controller: PID controller
Derivative term: “predicts” the future values of the error .
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The classical controller: PID controller
Derivative term: “predicts” the future values of the error .
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!! Most PID controllers do not use derivative action !!
PID controller design: shaping the feedback gains
Controller design: shaping the loop gains to improve the static and dynamic performances of the controller.
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Two types of PID controllers
Controller design: shaping the loop gains to improve the static and dynamic performances of the controller.
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In (a), P, I and D act on control error. In (b), I acts on control error, and P and D act on systems output.
Outline
PID controller: general form
Effects of the proportional, integral and derivative actions
PID tuning
Integrator windup and setpoint weighting
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PID controller with error feedback
The two forms encountered in control systems:
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Ti = integral time constant, Td = derivative time constant.
PID controller design: pure proportional feedback
Pure proportional feedback: steady-state error! Indeed:
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For a pure proportional feedback at steady-state, the error is given by
The error goes to zero as kp goes to infinity, but increasing kp will eventually destabilize the closed-loop system (gain margin).
PID controller design: pure proportional feedback
Pure proportional feedback: steady-state error! Indeed:
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For a pure proportional feedback at steady-state, the error is given by
To avoid steady-state error, we can use a feedforward term:
with
uff us called reset in the PID literature, and has to be adjusted manually.
PID controller design: derivative action
Derivative action: predictive and anticipatory action.
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If is increased, system responses is damped. But derivative action amplifies high frequencies (hence reduces noise rejection). kd
Derivative action should be used with a filter (= lead compensator):
Cd(s) =kds
1 + sTf
PID controller design: integral action
Integral action: no steady-state error.
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PID controller design
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Outline
PID controller: general form
Effects of the proportional, integral and derivative actions
PID tuning
Integrator windup and setpoint weighting
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Ziegler-Nichol’s tuning
Feedback gains are extracted from the dynamical response of the open-loop process.
Two methods: a time-domain method and a frequency-domain method
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Ziegler-Nichol’s tuning - time-domain method
Feedback gains are extracted from the step response of the process.
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is an approximation of the time delay of the system. is the steepest slope of the step response.⌧a/⌧
Ziegler-Nichol’s tuning - frequency-domain method
Start with zero gain, and increase proportional gain until systems start to oscillate.
= critical proportional gain, = period of oscillation, .
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!c =2⇡
TcTckc
Ziegler-Nichol’s tuning - Improvements
Time-domain method: characterize the step response by , and in the model
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T⌧K
Outline
PID controller: general form
Effects of the proportional, integral and derivative actions
PID tuning
Integrator windup and setpoint weighting
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Integrator windup
If the control variable saturates (i.e. reaches the actuator limits), there will be a residual error that will be continuously integrated by the controller. The integral term will build up, and eventually become very large.
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The control signal will then remain saturated even when the error changes, and it may take a long time before the integrator and the controller output come inside the saturation range.
Integrator windup
Integrator windup
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Anti-windup
Anti-windup: avoiding error integration while in saturation
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Integrator windup
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Integrator windup
If the control variable saturates (i.e. reaches the actuator limits), there will be a residual error that will be continuously integrated by the controller. The integral term will build up, and eventually become very large.
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Setpoint weighting
When there is an abrupt change in the reference, the proportional and derivative actions can become very big and lead to a large initial peak.
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To reduce this peak, we can only “show a fraction of the reference” to the proportional and derivative controllers:
Setpoint (or reference) weight ( ) Setpoint (or reference) weight ( )
No setpoint weight on the integral action! This would lead to systematic steady-state error.
2 [0, 1] 2 [0, 1]
Setpoint weighting
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The classical controller: PID controller
PID stands for Proportional-Integral-Derivative.
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