Nonlinear Controller Design Using Methods from Algebraic Geometry

在设计控制系统之前，首先要考虑的问题是，在给定的执行器和可用的传感器的情况下，是否能够实现相应的控制目标。这个问题引出了可控性和可观测性或稳定性和可稳定性等概念。对于线性时不变系统，这样的系统属性是明确定义的，并且可以用已知的标准来验证。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属性算法。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，这些自由度可以用于设计关于给定要求和期望特性的控制器。在给定一个控制器参数的情况下，问题再次出现，即控制目标是否可以用它来实现。代数几何中的问题理论上可以通过量词消去问题来表述。然而，量词消除算法通常是非常昂贵的计算。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，将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，解决代数问题的控制工程方面的发展。 为了用代数方法来处理系统，使用多项式函数的描述是必要的。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，这仅部分适用于机械系统，通常导致隐式公式化。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.