ELG 4152 :Modern Control Winter 2007 Printer Belt Drive Design Presented to : Prof: Dr.R.Habash TA: Wei Yang Presented by: Alaa Farhat Mohammed Al-Hashmi Mubarak Al-Subaie April,

Outline  Brief Overview  Design PD Controller  Design PI Controller  Design PID Controller  Proposed Solution  Results  References

References  “Improved Design of VSS Controller for a Linear Belt-Driven Servomechanism”, by Aleˇs Hace,Karel Jezernik, belt al, from IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 4, AUGUST 2005  “Adaptive High-Precision Control of Positioning Tables”, by Weiping Li, and Xu Cheng, from IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY,V OL. 2, NO. 3, SEPTEMBER 1994  “Modern Control Systems”, Richard C. Dorf and Robert H. Bishop, Prentice Hall.  “Development of a linear DC motor drive with robust position control”, by Liaw, C.M., Shue, R.Y. et al, from Electric Power Applications, IEEE Proceedings. Volume 148, Issue 2, March 2001 Page(s):  “High-precision position control of a belt-driven mechanism”, by Pan, J.; Cheung, N.C.; Jinming Yang, from Industrial Electronics, IEEE Proceedings. Volume 52, Issue 6, Dec Page(s):

Goal  To determine the effect of the belt spring flexibility using different controllers in order to improve the system performance v1v1 y T2T2 T1T1 ControllerLight Sensor Motor m k k v2v2 Sha ft θ1θ1 θ2θ2

A Brief Overview ControllerLight sensorPrinting device position DC motors Belt Motor voltage Printer device Massm = 0.2 kg Light sensor Spring Constant K 1 = 1 V/m k = 20 Radius of the pulleyr = 0.15 m Motor InductanceL ≈ 0 Frictionb = 0.25N-ms/rad ResistanceR = 2 Ω ConstantK m = 2 N-m/A InertiaJ = J motor + J pulley : J=0.01 kg- m 2

Work Distribution  Mubarak Research, Paper  Mohammed Research, Paper, Presentation  Alaa Research, Simulation

The motor torque in our case is equivalent to the sum of the torque provided by the DC motor and the undesired load torque caused by the disturbance that will affect the system stability. The forces present are the tensions T1 and T2 by the following equations: Mechanical Model

System Model with D Controller Td(s) 1/J 1/ sr X 1 (s) 2k/m K2K1K2K1 K m /R b/J 2K r /J 1/ s

System Model with D Controller (cont.) Mason’s Rule Td(s) X 1 (s)

Design PD Controller Td(s) 1/J 1/ sr X 1 (s) 2k/m K2K1K2K1 K m /R b/J

Design PD Controller X 1 (s) Td(s)

Design PID Controller  New State Variables

System with PID Controller Td(s) 1/J1/sr X 1 (s) 2k/m K2K1K2K1 K m /R b/J

System with PID Controller X 1 (s) Td(s)

Open Loop System For the open loop system, the sensor output V1 will be equal to zero. Then our three state space variables equal to the first and second derivative of the displacement and the first derivative of, we get the following state space equations, and the open loop transfer function: Td(s) X 1 (s)

Range of Stability:  In order to know the range of the gain of our controller, we apply the Routh Test and obtain the appropriate values of the gains:  D controller: ;K2=0.1

System Response using the D Controller

Implementation of the System with a PD Controller  Similar to the case of the D Controller, in order to know the range of the gain so the system stays stable, we apply the Routh Test. The characteristic equation for our system is:  We then find that and but in order to facilitate our calculation we will assign K2 = 0.1 and K3 =10.

System Response using the PD controller

System Stability with the PID Controller  Similar to the case of the D and PD Controller, in order to know the range of the gain so the system is stable, we planned on applying the Routh Test but the coefficients of the characteristic equation for our system are large. Using MATLAB, and fixing K2 to 0.1, we were able to find that and.  In order to facilitate our calculation we will study the case with K2 equal to 0.1, K3 equal to –5 and K4 equal to 10.

System Response using the PID controller

Open-Loop System Resoponse

Conclusion  If we are more concerned with the speed of the system, we should go with the PI or PID controller since they are much then the open loop or the PD controller.  They decrease the rise time and settling time and eliminate the steady state error.

Questions