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下是否可以实现各自的控制目标。这个问题引出了可控性和可观察性或稳定性和稳定性等概念。对于线性定常系统，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性质算法。如果满足了相应的要求，那么控制律的设计就有相当大的自由度。然而，这意味着即使是对现有自由度的数学描述也不是直截了当的。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。在给定控制器的ansatz的情况下，问题再次出现，是否可以实现控制目标。代数几何问题在理论上可以用量词消去问题来表述。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，这要归功于适应算法。因此，将控制工程问题转化为方便的形式是十分必要的。为此目的，应在研究项目范围内确定特别合适的配方。此外，更有效的算法，以解决代数问题，在控制工程的背景下，将被开发。为了用代数方法处理系统，必须使用多项式函数进行描述。许多模型可以转换为这种形式，尽管可能需要在高维空间中嵌入。然而，这只部分适用于机械系统，经常导致隐式公式。对于这类系统，迄今为止开发的方法必须加以扩展。

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.