Question
In the study of aircraft control system dynamics at certain air speeds, an impulse function with a weight of eight is input to the system and the response is measured and found to be approximately as indicated in the diagram. Assume the initial conditions to be zero and find: a) the system transfer function. b) the differential equation governing the system.
The diagram is a simple input/output box,with y(t) entering the box, G(s) inside the box, and x(t) exiting... y(t)=8sigma(t)---> G(s)=?--->x(t)=[4e^(-2t)]sin3t
I understand that G(s)=x(s)/y(s)...I'm just not sure how to proceed with the sigma(t) in the y(t) value...
Explanation / Answer
A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions and equilibrium point to be zero. If we have an input function of X(s), and an output function Y(s), we define the transfer function H(s) to be: [Transfer Function] H(s) = {Y(s) over X(s)} Readers who have read the Circuit Theory book will recognize the transfer function as being the Laplace transform of a circuit's impulse response. Laplace Block.svg For comparison, we will consider the time-domain equivalent to the above input/output relationship. In the time domain, we generally denote the input to a system as x(t), and the output of the system as y(t). The relationship between the input and the output is denoted as the impulse response, h(t). We define the impulse response as being the relationship between the system output to its input. We can use the following equation to define the impulse response: h(t) = rac{y(t)}{x(t)} Impulse Function It would be handy at this point to define precisely what an "impulse" is. The Impulse Function, denoted with d(t) is a special function defined piece-wise as follows: [Impulse Function] delta(t) = left{ egin{matrix} 0, & t < 0 \ mbox{undefined}, & t = 0 \ 0, & t > 0 end{matrix} ight. The impulse function is also known as the delta function because it's denoted with the Greek lower-case letter d. The delta function is typically graphed as an arrow towards infinity, as shown below: Delta Function.svg It is drawn as an arrow because it is difficult to show a single point at infinity in any other graphing method. Notice how the arrow only exists at location 0, and does not exist for any other time t. The delta function works with regular time shifts just like any other function. For instance, we can graph the function d(t - N) by shifting the function d(t) to the right, as such: DeltaN Function.svg An examination of the impulse function will show that it is related to the unit-step function as follows: delta(t) = rac{du(t)}{dt} and u(t) = int delta(t) dt The impulse function is not defined at point t = 0, but the impulse response must always satisfy the following condition, or else it is not a true impulse function: int_{-infty}^infty delta(t)dt = 1 The response of a system to an impulse input is called the impulse response. Now, to get the Laplace Transform of the impulse function, we take the derivative of the unit step function, which means we multiply the transform of the unit step function by s: mathcal{L}[u(t)] = U(s) = rac{1}{s} mathcal{L}[delta(t)] = sU(s) = rac{s}{s} = 1