EEN-E1040 Measurement and Control of Energy Systems Control I: Control, processes, PID controllers and PID tuning Nov 3rd 2016 If not marked otherwise, graphs and formulas in the presentation are from Åström & Hägglund: PID Controllers - Theory, Design, and Tuning (2nd Edition)

What is control? Intentional modification of a system’s state via external means Used for bringing a system (close) to a state the user desires Applied to various quantities (temperature, flow, lighting etc.) Can be manual or automatic, discrete (ON/OFF) or continuous (proportional)

Example: manual ON/OFF control ACME Electric Heater

Example: automated ON/OFF control with a setpoint ACME Electric Heater MK2 20.0 set 5.0 Indoor temperature … 30.0 … 15.0 28.7 …

Example: automated PID (proportional integral-derivative) control with a setpoint ACME Electric Heater MK3 PID 20.0 set 5.0 Indoor temperature … 23.0 … 19.0 20.0 …

Process models Let’s go back to the electric heater example and present it in the form of a diagram: Measured temperature (process variable) y Control variable u Setpoint ysp Σ -1 Error e Heating element (process) Controller

Process characterization Static response: if control variable u is fed to a process, what will be the value of the process variable y?

Process characterization Dynamic response: if control variable u is fed to a process, how will the value of the process variable y behave over time?

Process characterization Dead time, time constant, Important aspects for control: instability, nonlinearity

Proportional control Let’s consider our model process from before: Σ Heating element (process) Controller Setpoint ysp Measured temperature (process variable) y Control variable u Σ -1 Error e

Proportional control ON/OFF control  value of u same regardless of magnitude of e Instability and oscillations Proportional control: magnitudes of u and e linked Overreaction to minor errors removed

Proportional control Controllers with only proportional action have a steady-state error For derivation, see Åström’s book, pp. 62-64 To get rid of the error, we need something else…

PID controller - Background (P)roportional-(I)ntegral-(D)erivative By far the most common control algorithm (over 90% of all industrial control loops) When properly tuned, provides accurate control for a wide variety of processes

PID controller – Formulation Let’s consider the controller output u as a function of e – the difference between the setpoint and the measured value: P I D

PID controller – Proportional action With only proportional action, PID controller equation is reduced to ub is controller bias/reset, often zero but can also be something else

PID controller – Static example of proportional action Σ Load disturbance l Process output Measurement noise n Σ x Process Controller ysp Measured temperature (process variable) y u Σ -1 e

PID controller – Static example of proportional action From the relations obtained from the block diagram, we get K = controller gain, Kp = process gain, KKp = loop gain In a system with no measurement noise and zero controller bias, x = y (process variable)  high loop gain results in low steady-state error and resistance to load disturbances Why not always set controller gain to a very high value? In reality noise is always there and a high loop gain amplifies its effect Deciding on an optimal loop gain is not easy, and it is always a tradeoff between different objectives for the control

PID controller – Dynamic processes Almost every real process is dynamic Time-dependent process gain and load disturbance Setting loop gain too high leads to instability How to solve steady-state error?

PID controller – Integral action Always removes the steady-state error of the control Steady state: constant error  As long as error is nonzero, u will vary over time Large Ti  slow convergence but less oscillation

PID controller – Derivative action Dynamic closed loop systems often unstable due to dead time (process reacts to control too slowly) To avoid this, prediction of future error needed Consider a PD controller: First order Taylor series expansion for the error: Thus, control signal ~ linear approximation of error after time Td

PID controller – Derivative action

PID controller – When to use PI and PID Derivative control quite often left out PI sufficient for first order processes (ones that stabilize when control parameter is kept constant) PID used for second order processes that involve oscillations and/or instability Also useful for first order processes with dead time

PID controller – Caveats and limitations Integrator windup Occurs when control variable saturates due to actuator bounds For example control valves can never be more than fully open Typically happens with large sudden variations of setpoint Can be avoided with modified control algorithm or limitations in setpoint variation Higher order processes Typically complicated processes with an order larger than 2 require more than PID for accurate control Processes with large dead time Error in the linear approximation via the D term gets large if the prediction needs to be made far into the future Sophisticated predictive controllers better than PID in these cases

PID tuning – Process reaction curve Open loop method Controller set in manual mode Requires process to be stable How to do: Start logging the process variable Introduce a step change to the control variable Wait until the system reaches a new steady state Plot the logged data and analyse it to find the controller parameters

PID tuning – Process reaction curve Graph and formulas taken from http://www.chem.mtu.edu/~tbco/cm416/cctune.html

PID tuning – Ziegler-Nichols closed loop method Controller active during tuning How to do: Start logging the process variable Remove integral and derivative action from the controller Set the control gain to a constant value Modify the setpoint slightly Observe oscillations in the process variable If dampening, increase gain If amplifying, decrease gain Repeat step 5 until stable oscillations are found Record the gain value and the period of the stable oscillations

PID tuning – Ziegler-Nichols closed loop method Graph and formulas taken from https://controls.engin.umich.edu/wiki/index.php/PIDTuningClassical

PID tuning – Fine-tuning of a system Effects of PID parameters when increasing them: Rise Time Overshoot Settling time Steady-state error Stability K Decrease Increase Very small effect Degrade tI Eliminate tD No effect Improve if tD small

Thank you!