The tool DC-01 plots time response of a first-order system subject to input and initial disturbance. Three types of inputs are considered: impulse, step and ramp inputs.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Example: A first-order system**

The input in the tool is

After clicking "Run", the results will be shown in another page. The responses of the first-order system are

The analytical results are also shown as follows.

Time Constant, τ = 0.5

Free Response with Initial Condition y(0) = 0.2

y_{Free} = y_{0}*e^{-t/τ}

where

y_{0} = 0.2;
τ = 0.5

Forced Response with Initial Condition y(0) = 0

y_{Forced} = Term 1 + Term 2

where

Term 1 = -2*e^{-2*t}

Term 2 = 2*δ(t), with δ(t) being the delta function

First-order dynamic systems are commonly-seen models in various engineering applications. Examples are diverse, including rotating shaft-bearing systems, RC circuits, DC motors, liquid-level systems and thermal systems. First-order systems are usually described by transfer functions in two forms: standard form and general form.

The standard form of transfer function representation of a first-order system is

where R(s) and Y(s) are the input and output of the system, respectively; s is the Laplace transform parameter; and T is the time constant of the system.
The transfer function G_{0}(s) has a pole at s = -1/T, and it does not have any zeros at all. By Eq. (1), the governing differential equation of the system is

In the previous equations,

The free response of the system is described by

where y_{0} is the initial value of the system output. The free response of the system thus is obtained by solving the previous equation and it is given by

The forced response of the system is described by

where zero initial disturbance has been prescribed. The solution of Eq. (5) can be obtained by several methods. One method is Laplace transform method

Listed below is forced response in three cases of inputs.

(a) Impulse response: given an impulsive input

where δ(t) is the Dirac delta function, the system response is

(b) Step response: given a step (constant) input

the system response is given by

The steady state response of the system (the final value of the system output) in this input case is

By Eqs. (8) and (9)

which indicates that the system response at t = T is 63.2% of its final value. Equation (10) can be used to identify time constant T.

(c) Ramp response: given a ramp input

the system response is

Define an error function as the difference between the input and output

It can be shown that the steady-state error of the system subject to a ramp input is

The transfer function representation for general first-order systems is

The transfer function has a pole at s = -a_{0}/a_{1}, and a zero at s = -b_{0}/b_{1} if b_{1} is nonzero.
For a_{0} being nonzero, the above transfer function can be written as

where the time constant

The free response of the system is described by

and it has the same form as Eq. (4), with the time constant given in Eq. (16).

Assume that b_{1} is nonzero. The forced response of the system in the s domain (Laplace transform domain) can be expressed as

which indicates that the output of a first-order system in general form can be expressed by that of
the corresponding system in standard form. For instance, given a step input r(t) = r_{0}, the forced response in the s domain is

By Eqs. (7) and (8), inverse Laplace transform of Y(s) given in Eq. (19) yields the time response

which is a linear combination of impulse response and step response of the standard system (1).

For a first-order system subject to an input and initial disturbance, its total response is the summation of the free response and forced response. For instance, consider a system subject to a ramp input and initial disturbance, which is described by

The total response, by Eqs. (4) and (11) is given by

1. R.C. Dorf and R.H. Bishop, 2010, Modern Control Systems, 12th edition, Prentice Hall.

2. K. Ogata, 2009, Modern Control Engineering, 5th Edition, Prentice Hall.

3. N.S. Nise, 2010, Control Systems Engineering, 5th Edition, Wiley.