# Is state space representation useful for nonlinear control systems?

I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding the eigenvalues of a matrix. However, for nonlinear systems, the transfer function can only represent a linear approximation, while the state space form can represent the full system. But what's the advantage of using state space form for nonlinear systems, if we can't generally solve them by matrix methods? How does state space representation help us analyze or design nonlinear control systems any better than we could by sticking with the original DE representation?

Some background: My impression was that the state space form of linear systems is essentially just syntactic sugar for the final result of transforming a nth order DE into a system of n first order DE's, and writing that system as a single matrix equation. It "hides" the derivatives under the extra parameterization variables. But for nonlinear systems, we can't just get a system of linear equations and write it as a single matrix equation that doesn't explicitly involve derivatives. So I don't see how the state space form simplifies anything for nonlinear systems.

So you have correctly identified the purpose as,

transforming a nth order DE into a system of n first order DE's, and writing that system as a single matrix equation

and this is not insignificant. It greatly simplifies (1) the statement of any theorems regarding a control system as we need only consider first-order DEs and (2) the statement of any algorithm and procedure as we need only consider 1st-order DEs. Everything we can claim to do with a first-order [vector] ODE we can apply generally to higher-order scalar ODE. As a very simple example, consider the following theorem,

Theorem: Let $$x(t)\in\mathbb{R}^n.$$ Consider the system $$\dot{x}(t) = f(x(t))$$ and suppose $$f(0) = 0,$$ $$f$$ is continuously differentiable and define, $$A = \left.\frac{\partial f}{\partial x}\right|_{x = 0}.$$ If $$A$$ is Hurwitz then the system is locally exponentially stable at $$0.$$

This tells us that we can ascertain the stability of a first-order nonlinear system near an equilibrium by just looking at the Jacobian at the equilibrium point. Since this is written in state-space form and applies to the vector equivalent, it means we have a way to ascertain stability of any higher-order nonlinear ODE: just write it in state-space form and use this result. Imagine, for a moment: how would you write this theorem if you had to only work with the original, possibly higher-order, nonlinear ODE? How would it be described? (It is in principle doable)

It gets even more interesting when we want to do control design. A very commonly studied class of nonlinear systems are those of the control-affine variant, i.e.

$$\dot{x}(t) = f(x(t)) + g(x(t))\,u(t).$$

How do we control these systems? One method is that of feedback linearization, which asks whether or not we can apply feedback to cancel out all the nonlinearities and make the system behave exactly like a linear one (in possibly different coordinates $$z(t) = \Phi(x(t))$$). How do we know when its possible? There are conditions on $$f$$ and $$g$$ that tell us when this problem is solvable. Since these are conditions on the state-space formulation and apply even when $$x$$ is a vector, the results apply equally well to higher-order nonlinear control systems. We just rewrite them in state-space form.

There are, of course, exceptions. If you have a singular ODE then it may not have a state-space formulation. But, most systems are not singular in practice, so they are often not studied nor presented in most literature on control theory (although there are those that study such systems).

The upshot here is that by analyzing the state-space formulations, we can describe techniques that apply equally well to higher-order systems. It is to make our lives easier by giving a standard presentation that covers both lower and higher order systems equally. By the way, it isn't just state-space formulations that are used in the nonlinear case. We also work with input-output formulations for nonlinear systems; for linear systems the most commonly used input-output formulation is the transfer function! Systems are almost always rewritten in the input-output (transfer functions, operators, etc.) or state-space formulation.

As a side note, try to design a controller meeting specific settling-time requirements for a 4th-order linear ODE with some zero dynamics. Do not go into Laplace domain nor into state-space form. I think you'll find yourself having a hard time. I sure would. The difficulty only ramps up when you have multiple-inputs and multiple-outputs. It is doable but I think the procedures are more cleanly presented in state-space/input-output than in the original ODE. That itself to me makes it worth the while. I wouldn't even bother working with the nonlinear ODE without transforming it into an input-output representation (operator of some kind) or state-space representation.

• I guess I should add a shout-out to those who work in [differential] flatness theory. That is one example that comes to mind where nonlinear control theorists work with the original ODE. Commented Jul 5, 2023 at 2:07
• So I think the part I don't get is: If the system is nonlinear, why would we even be able to write it in matrix form at all? The matrix form seems like major benefit of the ss form for linear systems, but I don't see how we could meaningfully represent a system of nonlinear DE's with matrices, since matrices are typically just alternative notation for linear transformations. Commented Jul 5, 2023 at 4:01
• @MikaylaEckelCifrese As implied by my answer, it isn't a matrix form. State space models aren't all linear State space models. In the nonlinear case you have a nonlinear map f from Rn to Rn. Commented Jul 5, 2023 at 4:10
• state space models are not only those with Ax + Bu. those are just linear state space models. state space models (for ODEs) are more general than that. But possibly in a course context in linear control you may learn only the linear kind. As discussed in your other question...state is a general concept that applies to many systems. Commented Jul 5, 2023 at 4:26