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!