Nonlinear Controller Design Using Methods from Algebraic Geometry

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Before designing a control system, the first question is whether the respective control objectives can be achieved at all with the given actuators and available sensors. This question leads to concepts like controllability and observability or stability and stabilizability. For linear time-invariant systems, such system properties are well-defined and can be verified with known criteria. In contrast, there are numerous variations for nonlinear systems and usually only limited possibilities to check the respective property algorithmically. If the corresponding requirements are fulfilled, there are considerable degrees of freedom with respect to the design of the control law. This means, however, that even the mathematical description of the existing degrees of freedom is not straightforward. In principle, these degrees of freedom can be used to design the controller with respect to the given requirements and desired properties. In the case of a given ansatz for a controller, the question arises again whether the control objectives can be achieved with it. Problems from algebraic geometry can theoretically be formulated by means of quantifier elimination problems. However, algorithms for quantifier elimination are usually very computationally expensive. Many problems can be reformulated to obtain algebraic conditions that are easier to verify thanks to adapted algorithms. Therefore, it is essential to transform the control engineering problems into a convenient form. For this purpose, particularly suitable formulations are to be identified within the scope of the research project. Additionally, more efficient algorithms for solving algebraic problems in the control engineering context are to be developed. In order to treat systems with algebraic methods, a description using polynomial functions is necessary. Many models can be converted to this form, although an embedding in a higher-dimensional space may be necessary. However, this works only partly for mechanical systems, often resulting in an implicit formulation. For this class of systems, the methods developed so far have to be extended.