Nonlinear Controller Design Using Methods from Algebraic Geometry

在设计控制系统之前，第一个问题是，在给定的执行器和可用的传感器的情况下，是否能够实现各自的控制目标。这个问题引出了可控性和可观测性或稳定性和稳定度等概念。对于线性时不变系统，这样的系统性质是定义良好的，并且可以用已知的准则来验证。相比之下，非线性系统有许多变化，通常只有有限的可能性来检查各自的算法性质。如果满足相应的要求，则在控制律的设计方面有相当大的自由度。然而，这意味着，即使是对现有自由度的数学描述也不是直接的。原则上，这些自由度可以用来根据给定的要求和期望的特性来设计控制器。对于给定的控制器，是否能达到控制目标的问题又出现了。从理论上讲，代数几何中的问题可以通过量词消去问题来表示。然而，用于量词消除的算法通常在计算上非常昂贵。许多问题可以重新表述，以获得更容易验证的代数条件，这要归功于改进的算法。因此，将控制工程问题转化为一种方便的形式是十分必要的。为此，应在研究项目范围内确定特别合适的配方。此外，在控制工程背景下，将开发更有效的算法来解决代数问题。为了用代数方法处理系统，需要用多项式函数来描述。许多模型可以转换为这种形式，尽管可能需要嵌入到更高维的空间中。然而，这只适用于机械系统的一部分，经常导致隐含的公式。对于这类系统，到目前为止开发的方法必须扩展。

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.