Fuzzy Control of a Nonlinear Servomotor Model

A nonlinear servomotor model with friction, saturation, backlash and motor starting voltage is presented in this paper. Fuzzy control and nonlinear PID control are compared using numerous computer simulations. A systematical method for tuning a PD-type fuzzy controller with training data is introduced. 1.0 Introduction Actual servomotors always have nonlinearities that have to be considered in careful controller design. Most important of these are amplifier saturation, load friction, backlash and motor starting voltage. Mathematical analysis that takes all these nonlinearities into account at the same time becomes extremely complex and experimental tuning is still frequently required. Complexity is also the main disadvantage of some sophisticated control methods such as adaptive control or friction compensation. With fuzzy inference control, however, it is possible to tune an accurate controller without any complicated mathematical analysis, which provides a large number of servomotor users with the opportunity of applying an advanced control method. An example of this is provided by Li and Lau. Their servo model, as most discussed in the literature, is linear. In this paper, a nonlinear DC-driven servomotor model is implemented with Simulink, a program for simulating dynamic systems with Matlab. There are standard blocks for backlash and saturation in the program, and the friction is easy to connect in the simulation model as a function. The friction model used consists of static friction, kinetic Coulomb friction and exponential viscous friction. Tuning of a fuzzy controller for the nonlinear servomotor model is first discussed. The performance of the fuzzy control is compared with a nonlinear PID controller tuned by using numerical optimization. Finally, a systematical approach to tuning the fuzzy controller is sought by developing an algorithm for optimizing the control surface in the (e,∆e,u) -space. With all controllers, special attention has been paid to the controllers' behavior under model parameter and load variations. Also the responses of different step sizes are issues of interest. 2.0 The Servomotor Model In the literature, characteristics of the total servo system have been discussed. Individual models for the amplifier, motor, gear train and load have been presented and these have been combined to give the block diagram of a complete servomotor model. Inertias of load and motor are separated and the finite gear stiffness is taken into account in the diagram. Inserting nonlinearities of friction, amplifier current saturation, backlash and motor starting voltage completes the servomechanism model. Analytically, the DC-driven motor can be presented with equations (1) , (2) where ia is the current to the amplifier, θm the motor position in radians, La and Ra the inductance and resistance of the magnetizing circuit, K the gain factor of the motor, bm its damping factor, Jm the motor inertia, Tr the torque reduced from the load and Tfm the motor starting torque. These equations do not include the amplifier saturation, but in the simulations it is easy to limit the current ia. For numerical values, see appendix A. With backlash, the torque from load to motor can be written as , if (3) , otherwise (4) where Ks is the gear stiffness factor and θl, θmr the load position and the motor position reduced to load, respectively. The gear ratio is introduced by equations Saturation s 1 1/La Ra