Hi Pravesh,

If we choose other points, then cannot assess its stability. A system always move around equilibrium. If an equilibrium point is unstable then the system will move away from the point, and if it is stable turn tgen the system trajectory will move towards it.

For eg, inverted pendulum has two equilibrium point, one vertically upward and one downward. Do you think linearizing on some other random points will give you the picture of the stability of the system?

Dear Mr. Kumar,

It is also important to define these equilibrium points.

A nonlinear differential equation cannot always be solved. The recommended approach is the Phase Plane method. With its help, the equilibrium points, as well as the type of stability/instability, are determined without solving the equation.

I would recommend the article:

https://www.researchgate.net/publication/326199562_Phase_Behaviour_of_First_-_Order_Systems

I am currently preparing two more articles on phase trajectories, which I hope will be published by the end of the year.

Best regards

K.Yankov

The linear stability of any nonlinear dynamical system is seen with the help of linearization relative to a defined equilibrium. The linearization may be relaxed for investigation of strongly nonlinear stability. The equilibrium or the fixed points are dictated by the system itself.

For details, please look at

[1] H. González-Aguilar and E. Ugalde (eds), Nonlinear Dynamics New Directions: Theoretical Aspects (Springer, Switzerland, 2015).

[2] M. Lakshamanan and S. Rajasekhar, Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer, New York, 2002).

If you control a system, you often want to operate it around an equilibrium, so it makes sense to use this point. However, there might be cases for which you want shift this operating point via (nonlinear) control and generate a new equilibrium. The simple idea is given here

https://www.cds.caltech.edu/~murray/courses/cds101/fa02/faq/02-10-09_linearization.html.

Then you could linearize about the new equilibrium point of the closed loop.

Generally, for control purposes you also can linearize along a known trajectory. Think of a rocket or a robot arm. In this case your system is time varying. To use well established linear tools, you can check (closed loop) stability on these operating points (i.e. which are not equilibrium points) along the trajectory, or design controllers along the trajectory. This will give you the behavior close to this point. However, stability does not mean that it will return to this point.

However, for equil. points you know by the Hartman-Grobman Theorem that the behavior close to this point is surely close to the nonlinear one. This might not be the case for arbitrary points.

Hi Pravesh,

The general answer is that analysis of nonlinear systems is not simple.

For example, let’s assume that you compute the equilibrium points of xdot=f(x) and you got 3 equilibrium points. In principle, if the system is at any one of those equilibrium points, it should stay there forever.

However, one observes that, while at some equilibrium point the situation is indeed so, from others the trajectories just run away.

First, this let people to divide the equilibrium points into stable and unstable equilibrium points.

Second, one learns that one should not check stability of the equilibrium point AT the equilibrium point, but rather in its close neighborhood.

If this is an unstable equilibrium point, even an infinitesimally small disturbance that moves the trajectory just a bit away, makes itthen run away from the equilibrium point.

Instead, if this is a stable equilibrium point, a small disturbance again moves it away, yet the trajectory then runs back towards the equilibrium point.

So, for checking what happens in the small neighborhood, the linear approximation is both efficient and sufficient.

Hope it helps.

The term “equilibrium point” is used usually for a system where a set of equilibrium points may be existing especially for nonlinear systems. For a nonlinear control system and with the existence of the control input u, the equilibrium point becomes function to the control input u, or more specific to the steady-state control input uss. In this case, it is named as the operating point. Designing a stabilizing controller based on the nonlinear system model may be a difficult task; so, the reason for linearizing the nonlinear system about a certain operating point (desired point) is, of course, to be able to stabilize the selected operating point using the linear control theory.

Your question can be answered with two viewpoints:

1- The aim of designing a controller, especially the desired input tracking controller is to change the equilibrium point of the system to that desired point, so examination of the equilibrium points of the system where there are no active dynamics in the system gives the designer an overview of the system space, wherein all the physical systems there are physical limitations, for example, spring has more than one operating mode and also has a physical limit where compressing it will not change its elasticity any more, hence choosing random points to linearize the system around it may cause the designer to choose a point out the physical space for that system.

2- The examination of the equilibrium points and linearization of the system at these points is to create a space or neighborhood around one of them to validate the useability of linear control theory to design a controller that drives the system to operate in the region of that linearization neighborhood so random points may not exist in this neighborhood and so the system will show a dynamical disfunction.

Although linearization around an equilibrium point is by far the most frequently encountered case, it is not the only possibility. An equilibrium point is simply a particular solution of a dynamical system i.e. a constant solution. It is possible to linearize a dynamical system around any particular solution. However, we linearize only around solutions of interest, and often these are equilibrium points. There are situations when other non-constant solutions are of interest. For example, in case of a rotational motor, it is possible that a periodic solution, that is a cycle, is chosen for linearization.

The linearization about an equilibrium point is used to approximate the nonlinear system with a small neighbourhood around the equilibrium point [ Khalil ch4].

We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking at the qualitative dynamics of a system. Equilibrium points– steady states of the system– are an important feature that we look for. Many systems settle into a equilibrium state after some time, so they might tell us about the long-term behavior of the system.

Equilibrium points can be stable or unstable: put loosely, if you start near an equilibrium you might, over time, move closer (stable equilibrium) or away (unstable equilibrium) from the equilibrium. Physicists often draw pictures that look like hills and valleys: if you were to put a ball on a hill top and give it a push, it would roll down either side of the hill. If you were to put a ball at the bottom of a valley and push it, it would fall back to the bottom of the valley.

We linearize around an equilibrium point because any nonlinear system linearized around that point gives a minimum workable system for control systems design and Analysis.

Good question! But let's take flight dynamics. I have developed the linearized state-space equations for and aircraft in a pull-up maneuver, which is an unsteady non-equilibrium reference, or the linear equations of of an air-to-air missile in a 20 g turn. These equations can be used to build controllers that that include unsteady terms. (You can see the derivation of these equations in my textbook "Modeling and Simulation of Aerospace Vehicle Dynamics")

I think the point of linearization is simply because for a linear system, with have a large library of mathematical tools that enable us to systematically and more easily design a controller that guarantees the desired output(s). Also these tools allow you to analyze the performance your output(s) say how fast it converges, how much energy it requires and If you don't linearize your equations, then you'll either have to design your controller based on trial and error which is very time consuming and you wouldn't know how stable your system would be, or use complication nonlinear controller design methods that require high level math and don't offer too much information regarding performance of your system.

Nonlinear Control systems are available and well developed may be as much as Linear control systems. However, the reasons behind linearization varies. For example, in academic environment, the fully parametrized non-linear model may not be available for public use as it is the proprietary of the manufacturing company. Thus, for teaching of controller design, a linearized model is used where several controllers are designed at various operating points. Additionally, switching among controllers via what is called gain-scheduling may be applied for further exploration.

One thing to be noticed that having several linearized models may not give good representation as a high-fidelity nonlinear model.

As an example of usage of linearized model, in flight dynamics and control, an operating point may be picked to be as the aircraft at cruise phase of the flight envelope. Knowing the gap between nonlinear model and linearized models, the outcome of the analysis of such controller is quite helpful to carry further studies pertaining to e.g. flying qualities.

Stability of any system is property of its equilibrium operating conditions. Aircraft aerodynamic properties vary linearly with operating conditions such as Angle of attack and sideslip angle before hitting stall. Infinitesimal small external disturbances lead to mostly linear variations of equilibrium operating flight conditions. Thus, we can linearize mathematical equations of rigid body dynamics and aerodynamics models to analyze flight dynamic stability and design control laws for operating equilibrium flight conditions. Moreover, linearized dynamic analysis and control design is well established science in itself as compared to developing nonlinear analysis and control design science which is computationally heavy too.

You may want to check my post elsewhere at this forum ― as a case study related to transient mass balance analysis:

https://www.researchgate.net/post/Hello-everyone-can-you-please-help-me-answer-this-questionWhat-is-the-steady-state-concentration-of-the-toluene-in-the-individuals-vascular-system

The system can be linearized around the equilibrium point, because the dependencies of the variables are quasilinear and therefore the loss of generality in the solutions are minimized. The important thing is not whether it is linearized or not but rather the interval around the equilibrium point in which it is linearized, because if the interval is very large the dependencies are no longer linearizable and the loss of generality and accuracy increases.