Linear Systems 101 - Part 5 - Impulse response

So, let us use the following input \begin{equation}v_i(t) = \delta(t)\end{equation}

where δ(t) is the ever elusive and mysterious, but ubiquitous, Dirac delta.  Therefore, \begin{equation}h_c(t) \triangleq v_c(t)=\frac{1}{RC}\int_{0}^{t}e^{-(t-\tau)/RC}\delta(\tau)d\tau  =\frac{1}{RC}e^{-t/RC}, t \geq 0.\end{equation}

Ok........ Let us pause a little bit and pose some questions here:

Why use an impulse as an input to a dynamic system? What is the physical meaning?

It turns out that the impulse is a very important entity for  Linear Systems and Control Theory. Let us have a closer look at the forced response, \begin{equation} v_c(t)=\frac{1}{RC}\int_{0}^{t}e^{-(t-\tau)/RC}v_i(\tau)d\tau = \int_{0}^{t}h_c(t-\tau)v_i(\tau)d\tau = (h_c * v_i) (t), \;\; t \geq 0\end{equation}

where * is the convolution operator.  Wait, what? Are you saying that for obtaining the voltage across the cap, we just have to calculate the convolution of the input with its impulse response? Yes, precisely, and this is a fact for all linear time-invariant (LTI) systems. This is our case since (1) there is no nonlinear term involving vc; (2) the system's coefficients are constant, as we assumed at the beginning. This fact explains the reason why it is important to obtain the impulse response of the system since, through it, we can calculate its forced response. This also explains the fancy notation hc. Indeed, for the more general system, we would equivalently define,

\begin{equation}h(t) \triangleq \int_{0}^{t}e^{-a(t-\tau)}b\delta(\tau)d\tau  =e^{-at}b, t \geq 0.\end{equation}

as well as

 \begin{equation} x(t)=\int_{0}^{t}e^{-a(t-\tau)}bu(\tau)d\tau = \int_{0}^{t}h(t-\tau)u(\tau)d\tau = (h * u) (t), \;\; t \geq 0.\end{equation}

Cool... But what about the physical meaning of an impulse applied to the system? Well, it is a common impression (I believe) that the Dirac Delta is quite a strange entity. It has infinity amplitude and zero duration (!?), so could we argue that physically it would describe a perfect burst of infinity force (or voltage or whatever) applied at once to the system? Well, this is quite a stretch. Instead of going through this way, let us instead analyse the impulse response of the general system, \begin{equation}h(t) = e^{-at}b,\;\; t \geq 0\end{equation}and compare it to the unforced response \begin{equation}x(t) = e^{-at}x_0, \;\; t \geq 0.\end{equation} The similarity is uncanny. It is almost the same thing, with the only difference being x0 and b. Let us be brave and take this leap! We can definitely say that these two things are equivalent:

1. The unforced response of the system with initial condition x0
2. The impulse response of the system with the input parameter being equal to b = x0

This fact is widely used in some Optimal Control circles, it is quite useful, and hopefully, we will see it sometime in the future in this blog.

Finally (as this post is already quite long), a brief remark. In this series, we are not going to study Classic Control, this is not my goal here. My final goal is reaching system norms to give way to Optimal, Stochastic, and Robust Control, going through a time-based analysis instead of a frequency analysis. But this discussion would not be complete without this brief addendum. 

Recall the Laplace Transform of a given continuous-time function. Define the transfer function of our system as follows\begin{equation}H(s) \triangleq \mathcal L\{h(t)\} = \frac{b}{s+a}\end{equation}Note also that \begin{equation}X(s) \triangleq \mathcal L\{x(t)\} = \mathcal L\{(h * u) (t)\} = H(s)U(s) \implies H(s) = \frac{X(s)}{U(s)},\end{equation}where we used the convolution property. Therefore, the transfer function can be defined as the Laplace transform of the impulse response of the system, or equivalently, the system's input-output relationship (initial conditions set to zero). This entity is a complete, albeit different, description of our system. It eventually leads us to all the tools of Classic Control Systems such as the Routh-Hurwitz stability criterion, Root Locus,  Nyquist Stability Criterion, Frequency Response Analysis, Bode Plot, and so on. For the interested reader, the website Linear Physical Systems Analysis covers all those topics in a very intuitive and clear manner.