System Theory and Control of Model Sets

通过这项研究，我们找到了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，我们认识到受控对象不应仅使用一个模型进行建模，而应使用一组模型，因为对象模型不可避免地存在一些不确定性。从这个角度来看，我们考虑了与将不确定性量化为模型集大小和将不确定系统控制为模型集控制有关的几个问题。1995年，我们获得了量化不确定性的程序和模型集的数学公式。具体来说，提供了设备增益的上限和有限长度的输入输出数据，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，我们开发了一种使用复函数理论来计算该集合的算法。另一方面，对于控制，我们扩展了二次稳定技术，使其可以应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入的理解，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识别，并研究了传递函数为倒数的系统的结构化奇异值性质。此外，我们还表明，通过扩展前一年获得的二次稳定方法，我们可以使不确定控制系统估计相关李雅普诺夫函数的值。

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.