Is that a constant? Or is it a delta?

This might be a stupid question.. but oh well. So the inverse laplace of a constant is the dirac delta function times the constant.

With a proportional controller, K(s) = Kp, the inverse laplace of the controller would be the delta function. Is the delta function asymptotically stable?


As it turns out, this is not a stupid question at all! It hits right at the heart of what is and how to represent a dynamic system.

The question is motivated by the fact that a proportional controller is simply

$$u(t) = K_p e(t)$$

and its Laplace transform is therefore

$$U(s) = K(s) E(s), \quad K(s) = K_p$$

Everything looks good in that direction. However when you take the Laplace inverse of $K(s) = K_p$ you get

$$k(t) = K_p \delta(t)$$

and you’re thinking “how do I make sense of the delta function?”

The best way to make sense of this is to complicate. Say that you have a generic controller in the frequency domain

$$U(s) = K(s) E(s)$$

and that you take the Laplace inverse of $K(s)$ to obtain the controller impulse response

$$k(t) = \mathcal{L}\{K(s)\}.$$

What would be corresponding $u(t)$ in terms of $k(t)$? The answer is

$$u(t) = \int_{0}^{\infty} k(t – \tau) e(\tau) \, d\tau.$$

This is a convolution, from Table 3.2. See Chapter 3 for details. Now try that with the proportional controller. In this case $k(t) = K_p \delta(t)$ and 

$$u(t) = \int_{0}^{\infty} k(t – \tau) e(\tau) \, d\tau = \int_{0}^{\infty} K_p \delta(t – \tau) e(\tau) \, d\tau = K_p e(t).$$

See? All is fine! No reason for panic.

As for stability, of course a constant gain is asymptotically stable since
multiplying a bounded signal by a bounded constant is sure to produce a bounded output signal. Nevertheless, if you need to be reassured of the stability of the impulse response, use the definition from Section 3.6 to calculate

$$\|k\|_1 = \int_{0}^{\infty} |k(\tau)| \, d\tau = \int_{0}^{\infty} k(\tau) \, d\tau = 1 < \infty$$

to conclude that yes, the constant is indeed asymptotically stable.

What is at work here is that we can go back and forth on representing a time-invariant linear system in the frequency domain by its transfer-function or in the time domain by its impulse response. Ironically, looking back at Chapter 1, when we were working with static gains in the time domain, maybe we should have added a delta to each constant in the block-diagrams…

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