In order to make up for the post I missed, I'll deliver two by this sunday, of which this will be the first. The topic of this blog will be a really easy topic but also something which is totally essentially to SISO control: transfer functions. I'll also leave you guys with an open question I've been mulling over from a book on "open problems in control theory"
1. The Laplace Transform
To talk about transfer functions we need to understand a few Laplace transforms. Laplace transforms are a specific case of the more general idea of integral transforms, which are essentially any linear transformation of the form
$$F(s) = \int_{x\in X} k(s,x) f(x)dx,$$
where \(f\) is the input \(F\) is the output and \(k(s,x)\) is a function called the kernel of the transformation. The kernel, along with the selection of the set the integral is taken over, are the elements which define the specific transformation. While the theory on general integral transforms is extensive, control theorists are most concerned with either the Laplace or Fourier transforms, and of these two mostly the Laplace transform. The Laplace transform is given by
$$F(s) = \mathcal{L}f(t) = \int_{0}^{\infty}e^{-st} f(t)dt,$$
and itself has a long and interesting history in theory of functions, but for our purposes is simply a way of solving differential equations by turning them into algebraic equations. It's actually easy to see how this happens. Let's suppose \(y(t)\) is a time-domain function whose Laplace transformation is denoted \(Y(s)\). We want to find the Laplace transform of \(\dot{y}(t)\). This is
$$\int_{0}^{\infty}e^{-st} \dot{y}(t)dt = -se^{-st}y(t)|_0^\infty - \int_{0}^{\infty}(-se^{-st}) y(t)dt = sY(s),$$
assuming the surface term vanishes. Applying this argument recursively yields the important result
$$\mathcal{L}y^{n}(t) = s^nY(s),$$
which we shall use in the next section.
2. Transfer Functions
Control theory has been said to have emerged from two strains of engineering heritage: electrical engineering and mechanics. Electrical engineering is formulated in terms of input-output relationships for black-box systems. A signal \(u\) is fed into the box and a response \(y\) is output. To the electrical engineer the objective of feedback control is to change the input signal to achieve the desired output. Mechanics on the other hand is formulated in terms of differential equations. To a mechanical engineer, the objective of feedback control is to find a forcing term for the equation which produces the desired solution. The Laplace transform gives us a way to represent the differential equation for a system as an input-output relation--so long that the equation is linear (you can look at Blog 1 to find out how to approximate a nonlinear system by a linear one). Let
$$a_ny^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_1\dot{y}+a_{0}y = b_mu^{(m)}+b_{m-1}u^{(m-1)}+\cdots+b_1\dot{u}+b_{0}u$$
be our model of the system. Applying the Laplace transformation we have
$$\begin{aligned}\mathcal{L}a_ny^{(n)}+\mathcal{L}a_{n-1}y^{(n-1)}+\cdots+&\mathcal{L}a_1\dot{y}+\mathcal{L}a_{0}y\\&= \mathcal{L}b_mu^{(m)}+\mathcal{L}b_{m-1}u^{(m-1)}+\cdots+\mathcal{L}b_1\dot{u}+\mathcal{L}b_{0}u.\\\end{aligned}$$
Whose LHS is
$$\begin{aligned}a_ns^nY(s) + a_{n-1}s^{n-1}Y(s) +\cdots &+ a_{1}sY(s) + a_{0}Y(s)\\ = &(a_ns^n+a_{n-1}s^{n-1}+\cdots +a_1s +a_0)Y(s),\\\end{aligned}$$
and whose RHS is
$$\begin{aligned}b_ms^mU(s) + b_{m-1}s^{m-1}U(s) +\cdots &+ b_{1}sU(s) + b_{0}U(s)\\ = &(b_ns^n+b_{n-1}s^{n-1}+\cdots +b_1s +b_0)U(s).\\\end{aligned}$$
Putting these together we have
$$\frac{Y(s)}{U(s)} = \frac{b_ms^m+b_{m-1}s^{m-1}+\cdots +b_1s +b_0}{a_ns^n+a_{n-1}s^{n-1}+\cdots +a_1s +a_0}.$$
We typically denote the fraction \(Y(s)/U(s)\) as a single function, something like \(H(s)\). The transfer function can be used to determine virtually every significant thing about the controller and system, from stability to rise/settling times, overshoots, gain and phase margins, etc. In fact, without using transfer functions there's no way of easily understanding what is known as "classical" control theory.3. An Open Question
So now that I've described transfer functions, I'll leave you with an open question. Supposing we have a transfer function \(G\), then find transfer functions \(G_0\) and \(H\) for which
$$ G = G_0\circ H.$$
It has been shown by Fernandez and Martinez-Garcia (G. Fernandez, "Preservation of SPR functions and stabilization by substitutions in SISO plants," IEEE Transaction on Automatic Control, vol. 44, no. 11, pp. 2171-2174, 1999.; G. Fernandez and J. Alvarez, “On the preservation of stability in families of polynomials via substitutions,” Int. J. of Robust and Nonlinear Control, vol. 10, no. 8, pp. 671-685, 2000.) that controlling \(G\) is equivalent to controlling \(G_0\) by substituting \(K(s)\) by \(K(H(s))\). This is one of those interesting problems in classical control that peaks my interest. I've been working on it a bit and might be announcing a few results soon ;-)
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