(Late) Weekly Blog 2: Transfer functions and Their Compositions

First: only one week into my project and I have already failed to reach my intended goal as the post slated for last week never happened. I apologize for this but also am undeterred in my commitment to continue this blog!

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 ;-)

Weekly Blog 1: Linearization and the Informal Perturbation Approach

I've decided to devote the first weekly blog to looking at linearization--an incredibly powerful and pervasive technique in control. I will also look at how many informal approaches connect linearization to perturbation theory; on a side note I've always wanted to look at how a formal perturbation approach would stack up, if formal perturbation theory is even relevant at all, but wont have time for that this week.

1. Introduction to Linearization

Consider the following plant model:
$$\ddot{x} + \frac{K}{1-x/r} = u(t),\ x(0) = x_0,\ \dot{x}(0) = V_0 $$
Suppose the control objective is to regulate the solution to approach a setpoint of \(x_{c}\) within a specified settling time and overshoot range, reject disturbances, etc. This is equivalent to finding a compensator function for \(u\) and the required gains to achieve these objectives, but unlike what you might have seen in a linear controls course, the plant is nonlinear, hence no simple transfer function can be obtained. We could, of course, try to control the nonlinear plant, but with this approach would have to throw out all of the wonderful linear theory that your course developed. You might think that we could just replace it with an equally robust and practical nonlinear theory, but sadly no such theory exists for general nonlinear systems, despite years of effort. Indeed, even proving the stability of most nonlinear systems is quite a Herculean task.

A second approach is to find a way to make the linear theory fit the nonlinear model--viz. to make the nonlinear model look linear. Although this might seem crazy, it will actually work so long as the system state--in this case \(x\)--does not deviate too much from a specified point at which we linearize the model. How exactly do we linearize the model? In this case we observe
$$ \frac{1}{1-x/r} = 1 + \frac{x}{r} + \left(\frac{x}{r}\right)^2 +  \left(\frac{x}{r}\right)^3 + \dots$$
which converges so long as \(-r < x< r\). If \(|x|\ll r\) we can argue that all the terms beyond the first-order linear term are too small to matter and can be discarded. This truncation of the series to first-order allows us to make the approximation
$$\frac{1}{1-x/r} \approx 1 + \frac{x}{r} $$
which we place into the original plant model to obtain
$$\ddot{x} + K\left(1+\frac{x}{r}\right) = u(t).$$
The offset can be compensated for by defining \(\tilde{u} = u - K\) and we may now write the system as
$$\ddot{x} + K\frac{x}{r} = \tilde{u}$$
which is treatable by the linear techniques so long as \(x\ll r\). Of course if this limit is exceeded, higher order terms in the expansion become relevant and the linear approximation breaks down, leaving us at square one with the nonlinear problem.

2. A General Approach to Linearization 

So at this point you have seen how a system can be linearized by expanding the nonlinear part in a Taylor series and dropping the higher order terms. Unfortunately from the last example you might be of the impression that this can only be done with particularly nice models which have an obvious series expansion. Not so! Indeed, any smooth function \(f(x)\) can be expanded about an arbitrary point \(c\) via
$$f(x) = f(c) + f'(c)(x-c) + \frac{1}{2}f''(c)(x-c)^2 + \dots$$
so that the general first-order ODE \(\dot{x} = f(x,u)\) can be linearized about a trim point \(x=c,u=u_0\) by
$$\dot{x} = f(x,u) \approx f(c,u_0) + \frac{\partial f}{\partial x}|_{c,u_0}(x-c) + \frac{\partial f}{\partial u}|_{c,u_0}(u-u_0) $$
And you may have been told that any system of ODEs can be put into first-order form by defining mulligan states for the derivative terms (e.g. \(v = \dot{x}\) so that \(\ddot{x} = \dot{v}\)). So expanding the linearization to the case of an arbitrary number of plant and controller states \(x_1,x_2,\dots,x_n\) and \(u_1\dots,u_m\), we simply linearize about a set of conditions which uniquely specifies the trim point and sum over all the partials. This is
$$\begin{aligned}
\dot{x}_i = &f_i(x_1,\dots,x_n;u_1,\dots,u_m) \\
&\approx f_i(\text{Trim}) + \sum_{j=1}^n\frac{\partial f_i}{\partial x_j}|_{\text{Trim}}(x_j-x(\text{Trim})_j)  + \sum_{k=1}^m\frac{\partial f_i}{\partial u_k}|_{\text{Trim}}(u_k-u(\text{Trim})_k).\\
\end{aligned}$$
It is of course easier to write this as a matrix equation
$$\dot{x} = f(x,u) + A(x-x(\text{Trim})) + B(u-u(\text{Trim}))$$
where
$$ A_{ij} = \frac{\partial f_i}{\partial x_j}|_{\text{Trim}},\ B_{ik} =  \frac{\partial f_i}{\partial u_k}|_{\text{Trim}}$$
e.g. the state and controller vector Jacobians evaluated at the trim point. Note that we have assumed that the plant model is smooth not only in the dynamical variable but also in all of the state variables--viz. the EOMs are at least \(C^1\) for the linearization to be consistent, but hopefully \(C^{\infty}\) so that the Taylor expansion is justified. The deviations from Trim, calculated by \(x_j-x(\text{Trim})_j\) and discussed further below, are assumed small enough so that the linearized system is an accurate enough approximation of the true plant dynamics. The limits of the linear region may sometimes be obtained by a rigorous proof, but more often than not the applicability of the linearized model is checked by simulating the nonlinear plant with the linear controller.

3. Informal Perturbation Approach

There's two things you notice about the general linearization formula which prevent a typical transfer function from being obtained. The first is the fact that the trim is subtracted off of the state variables, which means the system does not look like the typical linear system
$$\dot{x} = A(t)x(t) + B(t)u(t).$$
The second is the constant left over from evaluating the EOMs at trim. The informal perturbation approach (as opposed to a rigorous, formal perturbation approach) is a method for dealing with these two nonidealities. To apply this approach we first separate the solution into two parts: the homogenous or known part \(x_h(t)\) and the perturbation \(\delta_x(t)\). The total solution is \(x(t) = x_h(t) + \delta_x(t)\). The homogenous part is typically called the reference solution, and need not be time-invariant as many texts would imply. It could, for instance, be computed numerically and perturbations made about it. The important part of the reference solution is it is a known function, thus so too is \(\dot{x}_h\), and furthermore we let
$$\dot{x}_h = f(\text{Trim}).$$
Note that this implies trim is not necessarily constant and indeed it doesn't need to be in general. The perturbation formalism assumes that \(\text{Trim}(t)\) is known since this is equivalent to \(x_h(t)\), so a time-varying trim simply results in a time-varying linear state model. Next we introduce the perturbation quantities as
$$\delta_{x,i} = x_i - x_i(\text{Trim}),\ \delta_{u,i} = u_i - u_i(\text{Trim})$$
and we finally write
$$\dot{\delta}_x = A(\text{Trim}(t))\delta_{x} + B(\text{Trim}(t))\delta_{u},$$
a linearized form of the solution which is valid for all model and controller states which are "close enough" to the trim, and which can be treated using linear control techniques.

Now, go forth and linearize! The first week's post is finished--and with a whole 23 hours to spare!

An Early Summer's New Year's Resolution

I've never really been that fascinated with New Year's resolutions. To me, the resolutions you keep tend not to be those you've cooked up because of some arbitrary tradition, but rather the ones you were planning on getting to but never remembered to get started on. Thus, why wait for the new year when, as the aphorism goes, "tomorrow is the first day of the rest of your life"?

This is the spirit in which I've dedicated myself to the following project: to write a new post on some aspect of control theory or practice every week until the new year. Depending on how I am feeling at that point, I may abandon this project or continue on, but regardless of which of these fates I consign my little endeavor to, the larger goal--of more thoroughly exploring my own field of study and expertise and sharing this exploration with the public--will be attained. The rules are as follows:

1. The post will be due Sunday, at 11:59 p.m. sharp, every week.
2. There will not be a word requirement or limit, nor a required topic, other than control of course.
3. Topics cannot be repeated, but may be expanded upon.

With these modest goals, I will set about this new project. Feedback from anyone is welcome but none of the information will be guaranteed to be accurate ;-)