Why we want to write all linear system into form of first order differential equations?

Question

It seems to be taken granted that when we deal with some control system, the first thing we want to do is to linearize the system and put it in form of first order differential equations like $\dot x = Ax + Bu$.

But why do we want to do this? And most importantly, why it is always first order? Why we don't want to write the dynamics like $\ddot x = A\dot x + Bx + Cu$?

Answer 1

I will use Mr. Richard Feynman's quote to answer your first question:

Finally, we make some remarks on why linear systems are so important. The answer is simple: because we can solve them!

Yes. That's all there is to it. It is well understood how to deal with linear systems. It is possible to generalize any notion about linear systems, but not about nonlinear systems (i.e. nonlinear systems need to be thoroughly studied). There are some systems, however, that cannot be linearized, so nonlinear methods exist to deal with these nonlinear systems (e.g., a very fast pendulum cannot be modeled accurately with a linearized model).

The second question involves why differential equations are converted into first order systems. Once again, solving them is the answer. First order differential equations are much simpler and easier to handle than higher order differential equations. In control systems, this representation is called space-state representation. You can read more about this technique.

Answer 2

:EDIT:

I was writing a comment that I don't think there's going to be a definitive answer on this question, that the question is akin to "Why do we draw free body diagrams for an engineering analysis?" or, "Why do we factor polynomials?"

I then realized that this question is very much like, "Why do we factor polynomials?" Factoring a polynomial is useful because it reduces a higher order expression into products of first order expressions.

Translating a system's dynamics equations into a series of coupled first order linear ODEs does the same thing. This still isn't an answer more than the original answer below, just a reinforcing concept that we generally choose to start with the simplest representation available because it affords the simplest means of solving the problem.

:Original post:

Dynamic systems are usually represented that way because it's a "canonical form" for control problems.

A canonical form is advantageous because, if you can express your problem in a canonical format, you get to use all of the preexisting tools that have been developed to analyze and solve that problem.

Consider standard interfaces in other settings - I can connect my watch, a printer, an Arduino, a game controller, etc. to my computer because they all use the USB standard interface. I don't need to keep a kit full of proprietary cables, or have a laptop that has a bank of proprietary connectors, thanks to the standard interface.

Similarly, I don't need a "toolkit" of methods to evaluate or solve controls problems if I can get to the canonical control form. In particular, some important "standard tools" include:

1. Eigenvalues of $\bf{A}$ reveal system stability,
2. Standard methods to check for observability and controllability,
3. The separation principle, which shows that you can design the controller and observer independently,
4. Standard method for modifying system response (pole placement),
5. Scalability from single input, single output (SISO) systems to multi-input, multi-output (MIMO) systems, and the associated linear quadratic regulator,
6. Relatively easy to construct a Kalman filter from that representation (if the Kalman filter has $\hat{x}_{k|k-1} = F_k \hat{x}_{k-1|k-1} + B_k u_k$, then you can use $F_k = (I + A\Delta t)$ and $B_k = B \Delta t$, and then $H_k = C$).

This is a list of things that rely on your system to be in the form:

$$ \dot{x} = Ax + Bu \\ y = Cx + Du \\ $$

If you want (or need) to express your problem differently then the tools I've mentioned above are no longer valid and you need to find/make equivalents for your application. The hard part there isn't just finding something that "looks like" the tools mentioned, or that has "similar looking step responses," but proving things like BIBO stability, etc. This is the hard work that has been done for you already if you (can) use the existing tools.

Answer 3

I believe this is done for generality. You can always see a $n^{th}$ order differential equation as a system of $n$ coupled first order differential equations. Therefore it is a good practice to first bring back your higher order system to a first order system so that it can then be solved in the standard way.

The first order is the lowest order that you can use to describe your higher order differential equation. Being the lowest order it is also the most compact (the resulting matrices are more dense).

If the scientific community decided to use second order diff. eq. as a new standard form then the resulting solution matrices would be very sparse (lots of null coefficients) and we could not use that procedure for solving first order diff. equations anyway.

Answer 4

Using the first order differential equations allows you to have unique state variables in x in . This prevents redundancy that would otherwise exists if it were to be written in higher order differential equations.

Imagine having it written in 2nd order differential equations. It would have this form: . If it was represented in this form, there will be a redundancy in how each of the state equation is expressed in terms of the state variables.

Answer 5

The reason we always look to linearize system is that we have a lot of results at hand for linear systems, and they are given for a "canonical form". We can assess stability, controllability and a lot of other property as well as robustness through proven results.

Then why choose a first order system ? Well because a second order system can be written as a first order, hence first order system are chosen as canonical form.

Consider $\ddot{x} = A \dot{x} + B x + C u$, it is equivalent to

$ \dot{ \begin{bmatrix} \dot{x} \\ x \end{bmatrix} } = \begin{bmatrix} A & B \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \dot{x} \\ x \end{bmatrix} + \begin{bmatrix} C \\ 0 \end{bmatrix} u$

which is a first order system. So every linear system can be brought to a first-order system for which results have been proven.