Loading DocumentIn control engineering nonlinearity may occur in the dynamics of the plant to be controlled or in the components used to implement the control, for example a dead zone in a valve, which may be referred to as inherent nonlinearity, because it exists although one would probably prefer this not to be the case. Alternatively, one may have intentional nonlinearities which have been purposely designed into the system to improve the system specifications, either from a technical or economic viewpoint.
All systems have actuator saturation, and in some cases, it might occur for relatively low error signals, for example in rotary position control it is not unusual for a step input of say 10°, or even less, to produce maximum motor drive torque. It simply is the result of good economical design.
Valves used to control fluid or gas flow, apart from having a saturation value also possess a dead zone due to friction and have slightly different behavior when opening compared with closing due to the unidirectional pressure of the fluid.
Friction always occurs in mechanical systems and is very difficult to model, with many quite sophisticated models having been presented in the literature. The simplest is to assume the three components of stiction, an abbreviation for static friction, Coulomb friction and viscous friction. As its name implies stiction is assumed to exist only at zero differential speed between the two contact surfaces. Coulomb friction with a value less than stiction is assumed to be constant at all speeds, and viscous friction is a linear effect being directly proportional to speed. In practice there is often a term proportional to a higher power of speed, and this is also the situation for many shaft loads.
Many mechanical loads are driven through gearing rather than directly. Although geared drives, like all areas of technology, have improved through the years they always have some small backlash. This may be avoided by using anti-backlash gears, which are only available for low torques.
Backlash which is a very complicated phenomenon involving impacts between surfaces is often modeled in a very simplistic manner. For example, the simple approach used in some digital simulation languages, such as Simulink the simulation component of the well-known software package Matlab, consists of an input-output position characteristic of two parallel straight lines with possible horizontal movement between them. This makes two major assumptions, first that the load shaft friction is high enough for contact to be maintained with the drive side of the backlash when the drive slows down to rest. Secondly when the drive reverses the backlash is crossed and the new drive side of the gear ‘picks up’ the load instantaneously with no loss of energy in the impact and both then move at the drive shaft speed. Clearly both these assumptions are never true in practice, but no checks exist in Simulink to determine how good they are or, indeed, when they are completely invalid.
The most widely used intentional nonlinearity is the relay. The on-off type, which can be described mathematically by the signum function, that is switches on if its input exceeds a given value and off if it goes below the value, is widely used normally with some hysteresis between the switching levels. Use of this approach provides a control strategy where the controlled variable oscillates about the desired level. The switching mechanism varies significantly according to the application from electromechanical relays at low speed to fast electronic switches employing transistors or thyristors.
A common usage of the relay is in temperature control of buildings, where typically the switching is provided by a pool of mercury on a metal expansion coil. As the temperature drops the coil contracts and this causes a change in angle of the mercury capsule so that eventually the mercury moves and closes a contact. Electronic switching controllers are being used in many modern electric motor drive systems, for example, to regulate phase currents in stepping motors and switched reluctance motors and to control currents in vector control drives for induction motors. Relays with a dead zone, that is, three position relays giving positive, negative and a zero output are also used. The zero output allows for a steady state position within the dead zone, but this affects the resulting steady state control accuracy.
CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION – Vol. III – Basic Nonlinear Control Systems
D. P. Atherton