Automatic control systems I. Nonlinearities in Control System

Category of nonlinearities Smooth nonlinearities When system includes nonlinearities with no a sharp discontinuities then the nonlinearity can be treated a simple gain whose value equals the slope of the non-linear characteristic at the working point. If the operating conditions don‘t only change small then the nonlinear characteristic can be replace some straight line Piecewise linear function The piecewise linear function is linear over a certain region and then change slope suddenly, or it has an step discontinuity. The most important piecewise linear function are: relay, saturation, dead-zone, hysteresis, speed limiter and combinations thereof. One can always choose the open loop strategy if the performance of controlled variable is on/off. The open loop strategy is always better than closed loop, but this solution often isn’t economical because it needs more transmitter and a lot of measuring on the process for developing an appropriate model. If the parameters of the mathematical model a little bit change in time, than it causes an error. The closed loop control can eliminate this error, but the open loop control not.

The smooth nonlinearities of the plan w(t) y(t) Y In frequency or time domain the dynamic behaviour is describe by more transfer function or differential equitation. The most of case only the parameters of the functions or equations are changing. The changes at the border are overlapping avoiding the too frequently parameters changing. Y2 Y1 W1 W2 W The lower and upper 20% of this characteristic in general isn't used, because most of the time has an additional nonlinearity 3

The types of piecewise linear blocks V W V v -w w -v Saturation Dead-zone W V W V max W -max Relay Hysteresis Speed limiter 4

Cascading nonlinearities block Three-point Dead-zone, relay, hysteresis Relay with hysteresis W V W V 5

Dynamic system with nonlinearities When is a nonlinearity present in a dynamic system it often can be divorce the nonlinear and dynamic effects. The nonlinear effects can be lumped into one block without dynamic. The input and output relationship of the nonlinear block is the static characteristic of the nonlinearity. The Laplace transform can't be used. In generally we can be examining this only in time strictly divorce the effect the nonlinearity in the actual input value in time as a gain. The differentia equation of the dynamic element and this gain are in cascade in the same actual time. 6

Location and type of the smooth nonlinearities in the control loop GA(s) GC(s) GP1(s) GP2(s) GT(s) Smooth nonlinearities of the actuator (frequently). Smooth nonlinearities of the plan (almost always). Smooth nonlinearities of transmitter block (nowadays it is rarely).

Location and type of the piecewise nonlinearities GA(s) GC(s) GP1(s) GP2(s) GT(s) At the end of compensator block (saturation). Inside of the actuator (hysteresis) At the end (saturation, speed limiter) of actuator block Inside of the plant (dead-zone) At the end of transmitter block (saturation).

Intentional piecewise nonlinearities in the controller GC(s) GA(s) GP1(s) GP2(s) GT(s) Dead-zone at after the error detector Relay block instead of the compensator block Relay with hysteresis instead of the compensator block Dead-zone, relay with hysteresis (three point controller)

The saturation limits of the model Creating the model the unit equals 10% and the working point is zero at 50% were assumed. all variable in the model Customary limits in the real word counted u* action signal max 100% 10 2048 20 mA 5 50% D/A conversion 4 mA 0 mA 0% min The exampleLecture4 library contains the model as NonLinSaturation.mdl 10

Without saturation Sometimes the action signal is outside of the range of the validity. The model is built only linear element and yet the response signal behaviour isn't the same in the two working point!

Saturation at the end of the compensator block Nowadays the compensator algorithm is a software. The output of the compensator is the action signal which must be 4 – 20 mA, and so there is a permissible range of the result of the algorithm (u*). The range depends on the number of bit of the D/A converter. In the industrial area these are 11 bit (permissible range 0 - 2048) and 12 bit (range of this 0 – 4096). The permissible range causes a saturation The limits of this saturation depends on the dynamic model of the closed loop system. 12

With saturation at the end of GPID(s) The action signal is stopped at the border of the range of the validity.

Saturation at the end of the actuator or transmitter It is caused by an incorrect selection of the actuator (upper limit) or incorrect settings of actuator (lower limit, for example controller sends 4 – 20 mA range, but actuator receives 0 – 20 mA range). It is caused by an incorrect settings of the transmitter (transmitter sends 4 – 20 mA range, but receives receives 0 – 20 mA range or incorrect settings of upper and lower limit of the sensor). 14

Upper or lower limit error of GA(s) The upper limit of modify variables increases the settling time and close to the limit points an additional steady-state error appears and the ability of the disturbance suppression decreases. The lower limit of modify variables modifies the quality parameters if the plant is self-alignment, but if the plant contains integral effect the closed loop can't work well. The incorrect setting in the transmitter causes changes in the quality parameters, especially close to the wrong limit.

The speed limiter error of GA(s) If only a slowly changes of the modify variables is required it doesn’t cause problem. The speed limiter increases the settling time and overshoot. t T

Dead-zone inside the process or at the end of the actuator It cause an extra delay time. It decrease the modified variables or inside variables of the plant. 17

Relay with hysteresis and saturation Two point controllers Relay with hysteresis Relay with hysteresis and saturation U U Switch off point E R-YM Switch on point 18

Two point controller Not necessary use hysteresis, because the real dead time limits the frequency. If use PT1 feedback block the hysteresis is useful to limit the frequency. Useful to use hysteresis, because it limits the frequency. 19

Feedback blocks The TI, TD, KC are defined by the CHR method using the process parameters. The system is nonlinear and the settings needs tuning! 20

Two point controllers with HPT1 The On-Off control in steady-state has an permanent oscillation with constant amplitude and frequency. These values depend on the time constant of the control loop and the hysteresis of controller. The principle of the superposition isn't available due to the nonlinear nature! The average of steady-state amplitude isn't the same of the required value described by reference signal.

Two point controllers with PTn Without inside feedback the feature is very similar to the previous HPT1. The effect of one PT1 feedback block is similar to a PDT compensation and two parallel feedback block such as a PIDT controller The hysteresis increases the frequency, but the amplitude of the oscillation (fluctuation margin) too!

Three point controllers U U E R Relay with dead-zone and hysteresis Three point with saturation 23

Feedback blocks The feedback blocks arrangement is the same, but the in this case the PDT compensation isn't the preferable. 24

Three point controllers with PTn This feature of the controlled variable is very good! Many times the drive unit a motor, which is controlled only three command: turn left, turn right, stop.

Three point controllers with IT1 The process model isn't perfect. We cant set steady-state at a working point, where the w disturbance be considered as 0. It causes a remaining error.