Modeling and Simulation of simple dynamic systems
A control system consists of subsystems assembled for the purpose of obtaining a desired output with a desired performance given a specific input.To understand and control complex systems one must obtain a quantitative mathematical models of these systems.A dynamic system consisting of finite number of elements is described by ordinary differential equations in which time is the independent variable.
There are two methods of developing mathematical models from schematic of physical systems namely:
Transfer function ( frequency domain)
State equations (Time domain)
Using Laplace transforms,the transfer functions are derived from the linear time invariant differential equations. The state space model of representation turns an nth order differential equation into n simultaneous 1st order differential equations. It is used for systems that cannot be described by linear differential equations and also for systems that require simulations in a computer.
The state space representation can handle multiple input multiple output systems.Systems with non-zero initial conditions and non – linear systems.It is for this reason that it is exensively used in modern control theory.
On the other hand,frequency domain methods are used for analysis Linear time invariant single input single output systems example those governed by a constant coefficient diffential equations
When finding transfer functions we assume initial conditions are zero.Let Y(s) be output of transfer function and U(s) be the input.Then equate the ratio to the differential equations we have
Y(s)/U(s) =( bmSm + bm-1sm-1 +...+b1s + bo)/(ansn + an-1sn-1 +...+ a1s + ao)
It is useful to factor the numerator and the denominator of the Transfer function into a zero -pole gain form.Therefore,the equation becomes
Y(s)/U(s) = N(s)/D(s) = k[(s-z1)(s-z2)...(s-zm)]/[(s-p1)(s-p2)...(s-pm)]
The zeros of the Transfer function are the roots of the numerator polynomial(I.e the values of s such that N(s) is zero).The poles of the Transfer functions are the roots of the denominator polynomial I.e the valuesof s such that D(s) is zero.Both zeros and poles may be complex values (may have real and imaginary parts.The system gain k is given by bm/an
Example of input components and devices used in dynamic systems:switches,potentiometer,microphone,keyboards and sensors e.t.c
Examples of output components and devices:displays,heat,light and sound.
There are also plant components and devices such as the gear trains,ampilifiers and actuators that are also used in this systems.
Sensors are devices used to measure and detect physical variables like temperature,light,flow rate,velocity.They produce digital signals that indicate physical value they measure.these signals are always uploaded in computers and analysed for differential applications.
For temperature sensors we can write the heat flow to sensor and heat content of sensor as follows
Rate of change of heat content of sensor = Rate of heat flow to sensor.
CsdTs(t)/dt = [Ta(t) – Ts(t)]/CsRth
These are systems that use fluid for transmission of signals thus controlling machinery.They convert energy of compressed gases or liquids into mechanical energy.
This are systems used where extensive force is required to control the motion of devices. They are useful when continuous control of motion of devices is required.They are preferred to Pneumatic systems because of various factors: Accuracy and precision, simplicity of operation and flexibility.
Other systems include the mechanical and electrical systems .