*J. Deur*

## Compensation of Torsion and Friction Effects in Servo Systems

*Ph. D. Thesis (in Croatian), University of Zagreb, Croatia*

*, 1999*

Research, design and experimental examination of a digital speed and position control system for electrical drives with elastic transmission and friction are presented in the thesis.
The influence of the transmission elasticity on the behavior of the speed control loop with a PI controller tuned according to the symmetrical optimum is analyzed using an algebraic method based on the damping optimum properties. The analysis is carried out for the cases of motor speed feedback and load speed feedback, and for a wide range of characteristic process parameters: the ratio of the moment of inertia of the load to the motor, the ratio of the mechanical system resonant frequency to the bandwidth of the equivalent lag term of the open-loop system and the mechanical system damping ratio. The results of the analysis are interpreted in the frequency domain by Bode diagrams.
The linear speed control loop is designed with the aim of damping torsional vibrations caused by the transmission elasticity, i.e. achieving fast and well-damped system response with efficient compensation of the load torque effect. Application of different types of speed controller is considered: a PI controller, full and reduced-order state controllers, a polynomial controller utilizing the motor or the load speed feedback and a PI controller combined with inner polynomial torque controller. Optimization and comparative analysis of the speed control loop for a wide range of characteristic process parameters and different sampling times are carried out, taking into account the sensitivity of the speed control loop to modeling errors of the inner current control loop. An analytical procedure of the speed control loop optimization based on the damping optimum is applied
The superimposed position control loop is realized using the proportional position controller independently of the applied speed controller. The gain of the position controller is determined by a simple analytical procedure in order to achieve the critically damped step response of the pointing system, i.e. the quasiaperiodic step response of the tracking system. A feedforward controller is introduced in the reference branch with the aim of decreasing the reference response time of the position control loop, i.e. reducing the position tracking error. The feedforward controller of any order is optimized in the continuous-time or discrete-time domain applying the magnitude optimum.
Friction effects in controlled electrical drives are analyzed in the time-domain by computer simulation. By applying the Karnopp friction model, it is possible to analyze the control system behavior separately for stick and slip intervals in which the mechanical system behavior is linear. The analysis is carried out for drives with stiff and elastic transmission, different static friction models (Coulomb model, static+Coulomb model and generalized Stribeck model) and different control tasks (speed control, positioning and tracking).
In order to reduce friction effects, the linear speed and position controllers are expanded with a nonlinear friction compensator based on the Karnopp static+Coulomb friction model. The compensator acts through the speed reference, so that it can be applied as part of the superimposed control algorithm in drives with a speed controller built in the servounit. The compensator is developed for the electrical drive with stiff transmission, and then it is modified for the drive with elastic transmission realized by different types of speed controller.
An adaptive speed control system utilizing the full-order state controller is designed in order to decrease the sensitivity of the control system to variations of the mechanical system parameters. According to the principle of explicit self-tuning control, the parameters of the speed controller and the friction compensator are calculated on-line based on the estimated parameters of the mechanical system. Parameter estimation is based on the physical model of the mechanical system, assuming that the drive and the load position are measurable.
The developed speed and control system is experimentally examined on a laboratory model of the electrical drive with the capability of independent adjustment of the load torque, inertia ratio, transmission stiffness, friction and backlash. Prior to control experiments, a series of identification experiments is carried out in order to obtain the process parameters.