ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems PID Control Dr. Tony E. Grift Dept. of Agricultural & Biological Engineering University of Illinois

PID control is a classic method that does not require in- depth knowledge of the controlled system Judge the behavior of the system Final value theorem to evaluate steady state output and error Step response / Impulse response MatLab: Step / impulse Pole-zero (root locus) maps MatLab: rlocus Bode plot MatLab: bode Nyquist diagram Judge the behavior of the feedback controlled system

Final and initial value theorems

Initial value theorem proof. Interchange the limits

Final value theorem proof. Interchange the limits

Steady state error of a feed forward system with unit step input. Example: 1 st order system with unit step input

Find the steady state error when a unit step function is applied to a feed forward system.

Steady state error in case of a unit feedback system. Actual value Set point Error

Find the steady state error of a P-controlled system with unit step input.

Find the steady state error of a P-controlled system with ramp function input.

Steady state error of a P-controlled system with unit step input.

Steady state error of a PI-controlled system with unit step input.

The steady state error of a P-controlled system does NOT go to zero. That of a PI controlled system does!

A high control value is a good thing, since it reduces the influence of the controlled system itself. However, if the control value becomes too large the system becomes unstable. Transfer function: This is highly desirable since the forward dynamics (system under control G) has no more influence!

Unit step response of a first order system. i

Convert to the Laplace domain: The transfer function now becomes:

Let’s check if this system is dimensionally homogeneous. The transfer function itself has to be unit less.

Unit step response of a second order system. Partial fraction expansion: Leading to the coefficients:

Unit step response of a second order system. The solution in the time domain is now:

Unit step response of a P-controlled first order system.

Partial Fraction Expansion

Unit step response of a P-controlled first order system cont.

Unit step response of a P controlled first order system for Kp values of 1,5 and the feed forward system (FF). FF Kp=1 Kp=5

Unit step response of a P-controlled first order system cont.

Unit step response of a PI controlled first order system.

Unit step response of a PI-controlled first order system. Partial Fraction Expansion

Math recap: Completing the square

Completing the square gives: Unit step response of a PI-controlled first order system cont.

We need the following FORM (this is NOT equivalent to the previous equation): Therefore we do the following: Unit step response of a PI-controlled first order system cont.

The solution in the time domain is now: Unit step response of a PI- controlled first order system cont.

FF system P controlled PI controlled

Pole-zero map of the transfer function TF. The poles of this TF are: The zeros of this TF are:

Bode plot of the transfer function TF.

Nyquist diagram of the FF system.

Nichols chart of the transfer function TF.

Error optimization: How good is the controlled system (not just the controller)? Criteria are 1. Transient response 2. Dynamic bandwidth

Unit step response of second order system. + - i

i Unit step response of a second order system. Convert to the Laplace domain: The transfer function now becomes:

Let’s check if this system is dimensionally homogeneous. The transfer function itself has to be unit less.

Unit step response of a second order system. Partial fraction expansion: Leading to the coefficients:

Unit step response of a second order system. Completing the square gives:

Unit step response of a second order system. The solution in the time domain is now:

Unit step response of a P controlled second order system. + -

+ - Cleanup and applying the step function gives:

Unit step response of a P controlled second order system. Partial fraction expansion: Leading to the coefficients:

Unit step response of a P controlled second order system. Partial fraction expansion gives: Completing the square gives:

Unit step response of a P controlled second order system. The solution in the time domain is now:

Second order system simulations.

Control objective Design a controller that makes the controlled system stable, fast responding, and energy efficient To do this, we need to know the system dynamics of the valve and actuator(s) Everything needs to be translated into the Laplace (s) Domain first

Ziegler-Nichols PID Tuning rules Setting the I (integral) and D (derivative) gains to zero. The "P" (proportional) gain is then increased (from zero) until it reaches the ultimate gain K u, at which the output of the control loop oscillates with a constant amplitude (edge of instability!). K u and the oscillation period T u are used to set the P, I, and D gains depending on the type of controller used:

Z–N tuning creates a "quarter wave decay". This is an acceptable result for some purposes, but not optimal for all applications. The Ziegler-Nichols tuning rule is meant to give PID loops best disturbance rejection performance. This setting typically does not give very good command tracking performance. Z–N yields an aggressive gain and overshoot – some applications wish to instead minimize or eliminate overshoot, and for these Z–N is inappropriate.

ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems PID Control Dr. Tony E. Grift The End Dept. of Agricultural & Biological Engineering University of Illinois