A CONTROL SYSTEM STUDY a brief look…. by Mahesh Mishra SSCET BHADRAWATI

Automatic Control Systems has following features.

1) It compares output of system with some reference input and uses difference as means of control. This is accomplished by using feedback system.

2) In general Control system has Error Detector, Amplifier, Actuator, and Plant and in feedback there is sensor whose job is to convert output of system in comparable form with reference input.

3) Examples of error detector are Mechanical levers, Pneumatic nozzle flapper device, Input terminals of amplifiers, Pneumatic Bellows with partition and so on.

4) Example of Actuator Electric motor, Pneumatic Valve and hydraulic actuators.

5) Plant is actuated by actuator to produce output which may be desired speed, may be required pressure and temperature of steam or may be any other physical variable.

As far as scope of study is concerned followings should be studied.

1) Transfer function: It is defined as ratio of Laplace Transform of o/p to Laplace transform of i/p assuming all initial condition to be zero. Assumptions are i) System is

linear, ii) System is time invariant iii) system equations are differentiable.

2) Transfer Function representation is the mathematical modeling either by using Block Diagram Algebra or Signal Flow Graph.

3) The Block diagram algebra involves Block Reduction Techniques while Signal flow graph uses Masson’s Gain Formula to find overall Transfer function of System

4) The transfer function opens many doors for further analysis and study. i) Time response: Transient response and Steady State Response in Transient response speed of response, rise time, peak time, and Peak overshoot are studied. In steady state response settling time, steady state errors are studied. ii) This Dynamic (Transient) behavior, Steady state behavior and stability of the system are beautifully studied under root locus.

5) Root locus and Routh Stabily Criteria provides platform to predict stability of the system and System design for desired level of performance

6) Root locus is defined as Locus plotted Verses roots of Characteristic equations when system gain is varied from Zero to infinity. For Stability all the roots of characteristic equation must lie in left half of S plane. If any roots lie in

right half, System is unstable for that particular damping and system gain.

7) Root locus Mathematics is the beautiful mathematics which beautifully correlates exponential term involved in time response expression with Stability, damping and settling time of system. Let’s have a look at second order time response expression for step input. C(t) = 1 – e - r wn sin ( w dt + ѳ) √1- r 2

In above time response expression C (t) is output, Input is Unit Step input, r is damping ratio, wn is undamped natural frequency, wd is damped frequency. Here rwn is negative power of exponential, which clearly tells as time proceeds this term will vanish out. The real beauty of mathematics is here that - rwn+jwd , -- rwn-jwd are the conjugates pair of roots of characteristic equation , and rwn is the real part , i.e. distance from imaginary axis on S Plane decides the settling time of the system. Farther the roots in left half from imaginary axis, quicker it will settle on steady state. For 2% tolerance band settling time is given by ts= 4/ rwn.

Further wd = wn√1- r 2 is the imaginary part and appears in expression as the sin of angle and when roots are plotted on S Plane its orientation tells us damping of the

system as r = cosѳ. Thanks to W R Evans for his great methodology.

8) Bode Plot and Polar Plot is the tools to study Frequency response which is the back bone in designing filters, compensators in control system. It is defined as Steady state response for sinusoidal input.

9) Bode plot and Polar Plots are plotted for open loop transfer function but again beauty of mathematics is that we can study and predict the behavior and hence performance of closed loop system with these open loop plots. Very simple method Laplace Operator s in open loop transfer function is replaced by jw.

10) State Space analysis is the mathematical modeling in direct time domain approach, which provides a basis for modern control theory and system Optimization. Its use is in designing Linear and non-linear, time invariant and time varying, multi input and multi output systems. There is no restriction of zero initial condition as it is with Transfer Function Representation.