Extended-Space Control Design with Parameter-Dependent Lyapunov Functions

本研究试图建立一个新的控制设计框架，允许参数依赖的李雅普诺夫函数。在本研究中，我们特别考虑多目标控制，多面体不确定性的鲁棒控制，以及线性变参数系统的增益调度控制。通过逆消元引理，引入虚参数，将给定的双线性矩阵不等式问题转化为扩展的线性矩阵不等式问题。在这种转换中，原始的BMI变量被分成两个不同的LMI项的代价是产生一些新的BMI项的虚拟参数，我们可以解决问题的不同的李雅普诺夫解。换句话说，我们可以用参数依赖的李雅普诺夫函数来解决混合问题或矩阵多面体问题。基于这一思想，我们发展了一个详细的理论描述，并通过一些数值设计实例验证了所提出的方法的适用性。 ...更多信息 es.在2002年，我们制定了扩展的LMI表示的李雅普诺夫稳定性，H2控制，H∞控制。结合它们，我们已经成功地解决了多目标控制设计问题与不同的李雅普诺夫解决方案。在2003年，我们已经制定了扩展的LMI公式的多面体不确定性和增益调度控制LPV系统的鲁棒控制。在每一种情况下，过去的保守性的共同李雅普诺夫解决方案已大大改善。在矩阵多面体问题的鲁棒H2控制中，我们无法事先知道哪个顶点达到最差的H2代价。在2004年，我们已经得出了一个很好的公式，其中最坏的H2成本最小化，即使我们不能提前实现它。此外，关于dP/dt的条件应包括在增益调度控制的公式中，其中李雅普诺夫函数由下式给出：在2004年，我们已经成功地包括了dP/dt的条件，并将其简化为一组顶点条件。少

This research attempts to establish a new framework of control design that allows for parameter-dependent Lyapunov functions. In this research, we particularly consider multiobjective control, robust control for polytopic uncertainties, and gain-scheduled control for LPV (Linear Parameter-Varying) systems. Through inverse use of the elimination lemma, we translate a given BMI(Bilinear Matrix Inequality) problem into an extended LMI(Linear Matrix Inequality) problem while introducing a virtual parameter. In this translation, the original BMI variables are split into two different LMI terms at the expense of generating some new BMI terms with the virtual parameter, and we can solve the problem with distinct Lyapunov solutions. In other words, we can solve mixed problems or matrix polytope problems with parameter-dependent Lyapunov functions. Based on this idea, we develop a theory described in detail and verify the applicability of the proposed method through some numerical design exampl … More es. In 2002, we have formulated extended LMI representations of Lyapunov stability, H2 control, and H∞ control. Combining them, we have successfully solved multiobjective control design problems with distinct Lyapunov solutions. In 2003, we have formulated extended LMI formulations of both robust control for polytopic uncertainties and gain-scheduled control for LPV systems. In every case, the past conservatism of common Lyapunov solutions has been considerably improved. In robust H2 control in matrix polytope problems, we cannot know which vertex achieves the worst H2 cost in advance. In 2004, we have derived an excellent formulation in which the worst H2 cost is minimized even though we cannot which one achieves it in advance. Moreover, a condition regarding dP/dt should be included in the formulation for gain-scheduled control, where the Lyapunov function is given by Ψ=x'Px. In 2004, we have successfully included a condition of dP/dt and reduced it to a set of conditions at vertices. Less