System Theory and Control of Model Sets

通过这项研究，寻求了一种基于不确定对象模型的系统控制器设计方法。作为我们研究的起点，我们认识到被控对象的建模不应该只使用一个模型，而应该使用一组模型，因为一个被控对象模型不可避免地存在一些不确定性。从这个角度出发，我们将不确定性量化为模型集的大小，将不确定系统的控制视为模型集的控制。1995年，我们得到了不确定性的量化过程和模型集的数学公式。具体地说，在给定对象增益和有限长度输入输出数据的上界的情况下，模型集被定义为与这些先验数据一致的所有传递函数的集合。此外，我们还利用复变函数论开发了一个算法来计算这个集合。在控制方面，我们扩展了二次镇定技术，使其适用于一类时变系统的模型集。1996年，我们对模型集辨识有了更深的理解，并寻求将其与鲁棒控制技术相结合。即提出了一种基于欧几里得范数的模型集辨识方法，并研究了传递函数为倒数的系统结构奇异值的性质。此外，我们还证明了通过扩展前一年得到的二次镇定方法，可以使不确定控制系统估计相关的Lyapunov函数的值。

Through this research, methodology for a systematic controller design based on an uncertain plant model has been sought. As a starting point of our research, we recognized that controlled plant should be modeled not using only one model but using a set of models because it is unavoidable that a plant model has some uncertainty. From this viewpoint, we considered several problems concerned with quantification of uncertainty as a size of a model set and control of uncertain system as control of a model set.In 1995, we obtained a procedure to quantify uncertainty and a mathematical formulation of a model set. Specifically, provided an upper bound of a plant gain and finite-length input-output data, a model set is defined as the set of all transfer functions consistent to these a priori data. Moreover, we developed an algorithm to calculate this set using a complex function theory. As for control, on the other hand, we extend a technique of quadratic stabilization so that it can be applied for a class of model sets of time-varying systems.In 1996, we achieved deeper understanding of model set identification and seeked to combine it with robust control techniques. namely, a model set identification based on Euclid norms is proposed and properties of structured singular values are investigated regarding systems whose transfer functions are reciprocal. Furthermore, we showed that we can make an uncertain control system estimate a value of an associated Lyapunov function by extending a quadratic stabilization method obtained in the previous year.