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Context:

Control Theory question. It's really your standard, boiler-plate setup: Say I've got some non-linear system, and I want to build a model with autonomous feedback control, that I'll later embed on a microcontroller or use to run simulations. There should then be a function $f$ describing the time rate of change of the system before any feedback control is added, such that

$$ \dot{x} = f(x) $$

Additionally, the textbooks tell us that we should convert our non-linear system into a linear one, in part by finding the jacobian of $f$, so that

$$ \dot{x} = f(x) \Longrightarrow \ \dot{x} = Ax $$

Question:

Why bother transforming $f(x) \Longrightarrow A$?

Like, $f(x)$ is computable, I can't see any reason for why we couldn't stick it in a loop and recompute it over and over. In some cases, it seems like it might even be more performant to compute $f(x)$ than to compute $A$.

My intuition is that the correct answer should be "because $Ax$ can produce a control term $Bu$ with ease so that

$$ \dot{x} = Ax - Bu = Ax - BKx $$

where as $f(x)$ cannot", but I'm wondering if that's the correct motivation.

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    $\begingroup$ Hi: Often linearization is done in order to make the problem more appropriate for a kalman filter ( the standard KF assumes linearity ) but I don't if that's the case in optimal control. $\endgroup$
    – mark leeds
    Aug 6 at 4:35

2 Answers 2

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If the motivation of your model is the accurate simulation of the process, then the model should not be linearized and the nonlinear model should be simulated directly. Linear systems have a rather limited diversity in their behavior and most of the interesting phenomena arise from nonlinearities. If our world were linear, we would not have anything, let alone fundamental physical forces. Nonlinear effects are fundamental to our world.

Linearization arises very often in the context of the local analysis of the stability of the equilibrium points. This is due to the fact that, in the hyperbolic case, the local stability of the equilibrium point is equivalent to the stability of the linearized dynamics at that equilibrium point. A basin/region of attraction may also be computed for that equilibrium point.

The quantitative validity of the linearized model is very local and may fail to represent the actual system outside of a small neighborhood of the equilibrium point. In this regard, it is not a good idea to consider the linearized dynamics for the modeling and the simulation of a nonlinear system. Moreover, it is not difficult to simulate forward a nonlinear system.

However, when it comes to control, the linearization plays a more important role. Why designing a controller on local linearized dynamics is good idea while this linear model is only valid (very) locally? One reason is that we have a lot of tools for the stabilization of linear systems but we do also have some methods for the control of nonlinear systems. But another reason is that it turns out that the domain of validity of the linearized dynamics in the stabilization problem can be much bigger than the domain of validity of the linearized dynamics evaluated in terms of the accuracy of the local representation. This can be proven using the so-called $\nu$-gap metric (and its extensions) which was introduced by G. Vinnicombe (see e.g. the book "Uncertainty And Feedback, $H_\infty$ Loop Shaping And The $\nu$-Gap Metric") and is a metric that considers the distance between systems in the stabilization/stabilizability viewpoint. This is nicely illustrated in the paper "Understanding Neighborhood of Linearization in Undergraduate Control Education [Focus on Education]" by D. Qian, J. Yi, and S. Tong (https://doi.org/10.1109/MCS.2013.2258767) where they show this phenomenon in the case of the inverted pendulum.

Finally, regarding the control of nonlinear systems, we do have methods to control systems of the form $\dot{x}=f(x,u)$ or, more specifically control-affine systems of the form $\dot{x}=f(x)+g(x)u$. So, the linearization is not necessarily about allowing to control the system in an easier way. Linearization does, however, simplify the control design problem as we have way more tools for linear systems than for nonlinear ones, but one has to face other issues such as robustness, region of convergence/global stability issues, etc. This is the reason why some hybrid control methods consisting of two distinct control laws have been introduced. The first control law is designed to be globally stabilizing (but may behave poorly in a neighborhood of the equilibrium point) whereas the second one is made to have only valid locally but in such a way that it optimizes the behavior of the process in a neighborhood of the equilibrium point (e.g. using an LQ criterion). Then, a heuristic is used to switch between the different control laws depending on the current state of the process in order to have benefit from the best of both worlds. This was notably studied in "Uniting Local and Global Output Feedback Controllers" by C. Prieur and A. R. Teel (https://doi.org/10.1109/TAC.2010.2091436).

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There are number of reasons for linearizing the system and then using the linear controller on it. One as you mentioned is the state feedback controller, which is very simple and elegant. The state feedback can also be used in the LQR setting which also brings in some form of optimality into your equation. In addition, linear system theory is well-known so it's easy to analyze the system, prove its stability and design the controller with some properties like convergence. Stability is not defined as easily as a linear system. It can only be defined in some settings like Lyapunov stability, which makes it more challenging.

Finally, many systems are underactuated in nature, which limits the controller for the non-linear system. For example, the wheel-inverted pendulum is underactuated and if you try to use a non-linear controller like feedback linearization to balance both the cart position and the pendulum upright position, you will fail because zero dynamics become unstable. So you need to switch to a linearized controller as soon you enter the region of convergence of unstable equilibrium. The exception can be model predictive controllers but they are more computationally challenging as compared to the linearized controllers.

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