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Design parametrization and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules

机译：通过内射热发电机商信号模块设计参数化和稳定输出反馈补偿器的极点位置

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Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective cogenerator signal module ℱ over the polynomial algebra D = F[s] (F an infinite field), a saturated multiplicatively closed set T of stable polynomials and its quotient ring DT of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called T-stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module DTFT and of quotient behaviors BT of DF-behaviors B. (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case, and that of the compensator enables its realization by Kalman state space equations or elementary building blocks. We note that every behavior admits an IO decomposition with proper transfer matrix, but that most of these decompositions do not have this property, and therefore we do not assume the properness of the plant. (4) The new technique can also be applied to more general control interconnections according to Willems, in particular to two-parameter feedback compensators and to the recent tracking framework of Fiaz/Takaba/Trentelman. In contrast to these authors, however, we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms.
机译：控制设计属于控制工程最重要和最困难的任务，因此已被许多著名的研究人员和许多教科书所采用，该系统通常用其传递矩阵或Rosenbrock方程来描述，最近也被描述为行为。除了前人关于传递矩阵的互质分解和稳定补偿器的参数化的思想外，我们的控制器设计方法还使用了一种新的数学技术，该技术使设计更简单，并且尽管有许多出色的结果，但也有了新的定理。文献：（1）我们在多项式代数D = F [s]（F为无穷大）上使用内射余热信号模块ℱ，稳定多项式的饱和乘积闭集T和稳定有理函数的商环DT。这使得可以同时处理连续和离散系统以及所有稳定性概念，即T稳定性。我们通过输入/输出（IO）行为的输出反馈来研究稳定控制设计，并研究完整的IO反馈行为，尤其是其自治部分，而不仅是其传递矩阵。 （2）这项新技术的特点是永久性使用内射热发生器商信号模块DTFT和DF行为 B 的商行为BT。 （3）对于跟踪，扰动抑制，模型匹配和去耦的控制任务（不一定是适当的设备），我们得出了具有适当和稳定的闭环行为的适当稳定补偿器存在的必要条件，这些条件使所有此类补偿器（如IO）参数化行为，而不仅仅是它们的传递矩阵，并为其构造提供了新的算法。此外，我们为完整的反馈行为解决了极点放置或频谱可分配性的问题。完全反馈行为的适当性可确保在连续情况下不存在脉冲解，而补偿器的适当性则可通过卡尔曼状态空间方程或基本构造块实现。我们注意到，每种行为都允许使用适当的传递矩阵进行IO分解，但是这些分解大多数都不具有此属性，因此我们不假定植物的正确性。 （4）根据Willems，新技术还可以应用于更通用的控制互连，尤其是两参数反馈补偿器以及Fiaz / Takaba / Trentelman的最新跟踪框架。但是，与这些作者相比，我们特别注意所有构造的转移矩阵的正确性，这需要更精细的算法。

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