In practice, the input signal to an automatic control system is not known ahead of time but usually random in nature. Thus, in the analysis and design of control systems, engineers usually employ test input signals as the basis for comparison of system performance.

MATLAB Control System Toolbox contain the functions step and impulse which allows the simulation of the response of a control system to these test signals. Given the numerator and denominator of a system transfer function and simulation time duration t given as a vector, these functions will return the response y(t) that determines the system performance [See MATLAB Documentation for more information].

>> clc
>> num = 1; den = [2 1]; t = [0:0.2:1];
>> y = step(num,den,t)
 y = 

                   0
   0.095162581964040
   0.181269246922018
   0.259181779318282
   0.329679953964361
   0.393469340287367
>> y = impulse(num,den,t)
y =

   0.500000000000000
   0.452418709017980
   0.409365376538991
   0.370409110340859
   0.335160023017820
   0.303265329856317

Unfortunately, these useful toolboox functions are not available here on Cody.

Your task is to write a program that performs the same function for first order control systems. We will consider three important aperiodic test signals: (1.) Unit impulse (2.) Unit step (3.) Unit ramp. This will be specified as an additional parameter to your program. Your program will then return:

y (t) – the system response during time interval t (exactly the same with MATLAB toolbox for impulse and step)

HINT: This is a simple problem in classical control involving the application of Laplace Transform and partial fractions.