Mathematicians often classify equations according to their difficulty in solving. Compared to nonlinear problems, linear equations are easier to solve because their variables are represented as a straight line.

Scientists at Tsukuba University have developed a new technique for constructing approximate linear equations for challenging nonlinear problems. They demonstrate through simulation findings that the responses produced by the model created using their proposed pseudo-linearization strategy are more similar to those produced by the well-known alternative method.

This work can help scientists and engineers predict and, more precisely, apply feedback control over mechanical systems described by nonlinear equations.

Once the physicist Stanislaw Ulam said: *“Using a term like ‘non-linear science’ is like referring to most of zoology as the study of ‘non-elephant animals’. The world we live in is complex and variables often interact confusingly. Predictions become significantly more difficult due to the possibility of feedback loops and even chaotic instabilities that result from these couplings.*

Therefore, limiting scientists to exclusively using linear equations would prevent them from simulating a wide variety of significant events, such as mechanical systems they are trying to control.

Unfortunately, much of the developed math only works for linear equations. Therefore, the ability to convert nonlinear dynamical systems into corresponding approximate linear versions would be of great value.

A theoretical representation of all possible solutions to the problem for which the system would be in equilibrium, called “equilibrium space”, has now been defined by scientists at Tsukuba University’s Department of Intelligent and Mechanical Interface Systems.

This work is considered a bridge between the abstract mathematics of nonlinear dynamical systems with infinite equilibria and the real world of system control problems.

Senior author Professor Triet Nguyen-Van said: *“Although the basic concept has been proposed by scientists before, our definition of an abstract ‘equilibrium space’ brings it closer to applications in engineering. Then pseudo-linearization can be performed, calculating approximate linear equations that have the same equilibrium states as the original problem.”*

Scientists demonstrated the value of their method using a gyroscope simulation that can rotate freely on gimbals around all three axes. Their method proved more accurate in determining steady state behavior based on a given input torque.

Scientists noted, *“These findings can be applied to the design of non-linear control systems in many situations. Some of these applications include preventing machines with many degrees of freedom from becoming unstable, improving performance and safety.”*

**Magazine reference:**

- Ryotaro Sakata et al., Equilibrium space and a pseudolinearization of nonlinear systems, Scientific Reports (2022). DOI: 10.1038/s41598-022-25616-1