PID Control of Car Position

1. PID Control of Car Position Exercises in Manual Tuning

2. Before We Begin • In this assignment, you will be tuning PID controllers manually. • Proper manual tuning of PID controllers relies on intuitive understanding of PID performance, so we are aiming here to provide such an understanding. • There are MANY excellent resources that provide more rigorous/mathematical analysis. • Here is one: http://lorien.ncl.ac.uk/ming/pid/PID.pdf

3. Introduction of Terminology • Before we discuss PID control, we must first introduce a few terms that occur frequently in feedback control. • Plant • This is the physical system in a control loop. In our example, this is the car. It is often represented with a G. • Sensor • A device that measures the quantity we are trying to control. In our system, this is the optical encoder. • Actuator • A device that enables us to change the state of the system in some way. In our system, this is the DC motor. • Controller • A piece of logic/arithmetic/magic that operates on the error signal to compute actuator effort. It is often represented with a C.

4. Feedback Loop Control/Actuator Effort Error Signal Loop Output Reference Trajectory C G + - Sensor

5. Feedback Loop for Car Position Position Error (in meters) Control Effort (Motor Voltage) Actual Car Position (in meters) Desired Car Position (in meters) • The signals represent actual physical quantities. • Controller computes a voltage from car position error. • Plant responds to this voltage, resulting in a position. C G + - Encoder

6. PID Control • A PID controller computes control effort, u, from error signal, e, as follows: u=KPe+KDe’+KI∫e(τ )∂τ or in the Laplace domain: U=E(KP+sKD+KI/s) • Called “PID” because one of these terms is Proportional to e, one scales with the Integral of e, and another scales with the Derivative of e. • KP,I,Dare coefficients that you must assign. Finding these values is called “tuning” the controller.

7. Proportional Control • If you take KD =KI =0, then we have proportional control: U=KpE • That is, the actuator effort is proportional to the error signal. • KP determines how “hard” the controller pushes in response to an error. • If KP is large, then the actuator will respond strongly to an error. This is called a “firm” controller. • If KP is small, then the actuator will respond weakly to an error. This is called a “loose” controller.

8. PD Control • If you take KI =0, KP,KD≠0: U=E(KP+sKD) • This is called PD control. • KP still determines how “hard” the controller pushes in response to an error. • KD weights the derivative of the error signal.

9. PD Control • In a properly tuned controller, the effect of the derivative term is to slow the plant down as it approaches its end-point. • Reduces overshoot, increases damping. • Differentiation amplifies high frequency sensor noise, which can result in instability.

10. PI Control • If you take KD =0, KP,KI>0: U=E(KP+KI/s) • This is called PI control. • The integral term provides large steady-state gain. • In the presence of even a small steady-state error, the integral of the error signal grows until the error is corrected. • The integral term degrades transient behavior. • Increases overshoot, reduces damping.

11. PID Control • If you take, KP,I,D>0, then you have a full PID controller. • Proportional term dictates controller stiffness. • Integral term provides large steady-state gain but degrades transient response. • Derivative term improves transient performance at the cost of increased susceptibility to high-frequency noise.

12. Assignment 3 • In order to complete this assignment, you will need to install some software. • Download this zipped directory: http://dl.dropbox.com/u/8406152/EE%20141%20PID%20Assignment.rar • You will need to download winrar or something similar to unzip this. • Run setup.exe to install PID_control.exe, which you will use for this assignment. • The download file also contains a configuration file called config.txt. Create a working folder and put config.txt in it. You will be using this folder for all data, controllers, etc that you generate in future assignments. • This is a very big file (200MB) because it includes the LabVIEWRun-Time Engine. Future programs will not include this and will be much smaller in size.

13. Getting Familiar with the Software • Step 1: Provide a path to your working directory. • Step 2: Slide the controls to set KP,I,D. • Step 3: Press the “Done Tuning?” button and watch the resulting step response.

14. What Will You be Turning In? • This assignment will walk you through manual tuning of a PID controller. • You will be asked to provide a document showing the performance resulting from a number of controllers. • Use the PrntScrn function to copy and paste the step response of the controllers into this document (see below). • Make sure that figure shows both the position and voltage traces as well as the gains in the boxes just underneath. • Provide a descriptive title for each plot, but do not waste your time writing anything else. • Additionally, you will be demonstrating your controllers during your group meetings.

15. Problem 1:Proportional Controller • Find the proportional gain that yields the best performance. Provide a plot for this controller. • What happens if you increase KP to much? What happens if KP is too small? Provides plots of these cases.

16. Problem 2:PD Controller • Tune a PD controller to provide the best performance. Provide a plot of this. • One method to do this is to increase KP until you start to see significant oscillation or “ringing” and then increase KD until this is sufficiently damped. • What happens if you double KD? Halve it? Provide plots of the resulting controllers.

17. Problem 3:PID Controller • Tune a PID controller to provide the best performance. Provide plots of the resulting controller. • You might want to start with your PD controller and work from there. • What effect does adding the integral term have on the transient response? Steady-state response? • Can you see the effect of the integral controller in the voltage trace?

18. Problem 4:Ziegler-Nichols Tuning • Ziegler-Nichols tuning is a heuristic method of tuning a PID controller. • It typically yields very stiff controllers with high overshoot, poor damping. • The resulting controller is often used as a starting point for subsequent optimization.

19. Ziegler-Nichols Tuning • Set KI,D=0, and increase KP until you see a purely oscillatory response. • This condition is called marginal stability. • The gain at which this happens is called Ku. • Compute the period of this oscillation. Call this Tu. • The Ziegler-Nichols tuned P,PI, and PID controllers are found using the following table:

20. Ziegler-Nichols Tuning • Generate each of these Z-N controllers (P,PI,PID) • Provide a plot of each. • What can you say about the performance relative to your controllers?

21. What is the take home lesson? • PID controllers are very versatile, highly useful, and ubiquitous. • Used in a vast majority of industrial control systems. • With a bit of skill, they can be tuned manually with little trouble for many systems. • No plant model is required for manual tuning.

22. What is the take home lesson? • There are some very significant shortcomings to this approach. • Guess-and-check approach is possible for this small/cheap/innocuous system, but you wouldn’t want to do this with an expensive or potentially dangerous system. • There are many systems for which PID control is either impossible or highly sub-optimal. • More systematic and mathematically rigorous methods are often required.

23. What is the take home lesson? • A pure derivative controller is not physically realizable (or all that desirable), so PID controllers are typically replaced with lead-lag compensators. • This will be the subject of a future assignment.