Conditions for stability and a numerical algorithm to achieve stabilization of nonlinear systems

What is it about?

Many technical processes are modelled with the help of differential equations. And a successful control of the process can be achieved by successfully controlling the model. The more complex the process is, the more complicated the model becomes. On the level of the differential equations, which we refer to as the system, this basically means nonlinear. We first observe that most such nonlinear models can be rewritten in a form that looks like a linear equation. Then we employ and extend a theory that has been used for linear differential equations to determine whether the system is stable. Here, stable is used in the sense, that the system tends towards a desired state. If the system is not stable, then it will not attain the desired state voluntarily. In this case, one needs to put some force onto the system by means of a so called control input. We exploit the previous theoretical considerations about stability, to determine control inputs that enforce stabilization. Finally, we provide a numerical algorithm that compute such stabilizing control inputs. For that, an initial control input law is computed by solving a nonlinear equation (a matrix Riccati equation) once. Then comes the twist: although the system is nonlinear, it appears to be sufficient to continuously update the input only through the solutions of linear system (which is a more or less easy task for standard mathematical software).

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Why is it important?

Our work is one of, possibly very few, really nonlinear but general approaches to the stabilization of nonlinear systems. It can work out of the box for many systems that otherwise would require a thorough a-priori analysis or particular expertise or heuristics. Also, since at the core only linear equations are solved, despite its generality it is efficiently implemented in particular for large-scale systems.

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