LMI-based multiobjective controller design with non-common Lyapunov variables
LMI-based multiobjective controller design with non-common Lyapunov variables
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基于 LMI 的非公共 Lyapunov 变量多目标控制器设计
DOI：
10.14989/doctor.k9568
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Y. Ebihara
中科院分区：
文献类型：
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作者：
Y. Ebihara
本文研究了线性矩阵不等式（LMI）方法在多目标控制器设计中的应用.多目标控制器设计问题是指混合了Hoo性能、H2性能、区域极点配置约束等设计指标的控制器设计问题，近年来的研究表明，这些设计指标可以用矩阵不等式表示，其中包含了控制器变量和双线性形式的所谓的李雅普诺夫变量。当我们处理一个单一的设计规格，矩阵不等式对应的设计规格可以成功地减少到一个LMI，因此，我们可以得到一个所需的控制器很容易通过公认的凸优化技术。另一方面，当我们处理多个设计规范时，这不再是真的。也就是说，耦合矩阵不等式，反映了多个设计规格被认为是本质上是双线性矩阵不等式（BMI的）。BMI的求解是一个非凸优化问题，从数值计算的角度来看是相当困难的。为了避免在处理这样的BMI的困难，一个所谓的公共李雅普诺夫变量已被迫为所有的设计规范，使BMI的可以转换成LMI的。然而，一个共同的李雅普诺夫变量的限制是相当有限的，这种方法带来了一些保守的设计。本论文的目的是避开保守性，并针对非共同李雅普诺夫变数的多目标控制器设计问题。本论文针对非共同李雅普诺夫变数的多目标控制器设计问题，提出三种方法，其中第一种与第二种方法是针对状态反馈问题，而第三种方法则同时针对状态与输出反馈问题。在第一种方法中，我们施加了一些额外的约束的李雅普诺夫变量，使我们凸化的问题，并获得线性矩阵不等式特征，同时保持状态反馈增益直接作为变量之一。由于在约束条件下的李雅普诺夫变量的自由度，我们的。制定原来给一组LMI特征，允许非共同的李雅普诺夫变量。另一方面，在第二种方法中，我们执行一个标准的程序，称为变量的变化，并表示作为一组仿射函数的新变量的结果变量。这些仿射函数被选择为具有一个重要的特性，即不管新变量如何，都能满足麻烦的非凸约束。有了这些仿射函数，我们很容易得到一组LMI特征，允许非共同的李雅普诺夫变量。我们还表明，这第二种方法与上述第一种方法的一个简单的组合，导致一个有效的迭代算法，我们可以绕过保守性相当。我们提出的第三种方法与上述两种方法截然不同。在这种方法中，我们得到新的扩张矩阵不等式的设计规格，其中控制器变量和李雅普诺夫变量之间的解耦已经实现，因此它们之间的双线性项消失。这是通过引入新的辅助变量来实现的，这些辅助变量与控制器变量而不是李雅普诺夫变量形成乘积。这些新的扩张矩阵不等式使我们得到了一种新的方法，该方法使具有非公共李雅普诺夫变量但具有公共辅助变量的问题凸化。结果表明，我们可以保证这种方法，以实现更好的性能比传统的方法。虽然我们在这篇论文中的主要兴趣是多目标控制器的设计问题，事实证明，新的扩张的设计指标的特征有另一个潜在的处理鲁棒性能分析和综合问题的真实的多面体不确定性。粗略地说，这些问题的传统方法是这样的，他们寻求一个共同的李雅普诺夫变量在整个不确定性域，从而达到保守的结果。另一方面，新的扩张特征使我们能够采用所谓的参数依赖的李雅普诺夫变量，因此传统的方法的保守性可以成功地规避。矩阵不等式中的李雅普诺夫变量与控制器变量解耦的思想在处理多目标控制器设计、真实的多面体不确定性的鲁棒性能分析与综合等问题中具有重要意义，本文的研究为解决这些问题提供了一种有趣的方法，并保证了性能的改善。
This thesis studies linear matrix inequality (LMI) approaches to multiobjective controller design problems. By multiobjective controller design problems, we mean design problems with a mixture of different design specifications such as the Hoo performance, the H2 performance, the regional pole placement constraints and so on. Recent studies show that these design specifications are characterized as matriX' inequalities that include controller variables and so-called Lyapunov variables in their bilinear forms. When we deal with a single design specification, the matrix inequality corresponding to the design specification can be reduced successfully to an LMI and hence we can obtain a desired controller easily via well-established convex optimization techniques. On the other hand, when we deal with multiple design specifications, this is no longer true. Namely, the coupled matrix inequalities that reflect multiple design specifications are considered to be essentially bilinear matrix inequalities (BMI's). Solving BMI's is a non-convex optimization problem and quite hard from the viewpoint of numerical computation. In order to avoid the difficulties in dealing with such BMI's, a so-called common Lyapunov variable has been forced for all design specifications so that the BMI's can be converted into LMI's. However, the restriction to a common Lyapunov variable is quite confining and this approach brings some conservatism into the design. The goal of this thesis is to get around the conservatism, and we tackle the multiobjective controller design problems with non-common Lyapunov variables. This thesis proposes three approaches to the multiobjective controller design problems with non-common Lyapunov variables, where the first and second ones deal with the statefeedback problems, while the third one deals with both stateand output-feedback problems. In the first approach, we impose some additional constraints on the Lyapunov variables so that we convexify the problem and obtain LMI characterizations while keeping the statefeedback gain directly as one of the variables. Because of the freedom left in the Lyapunov variables under the constraints, our .formulation turns out to give a set of LMI characterizations that allow non-common Lyapunov variables. On the other hand, in the second approach, we perform a standard procedure called change of variables, and represent the resulting variables as a set of affine functions of yet new variables. These affine functions are chosen to have a crucial characteristic that troublesome non-convex constraints are satisfied regardless of the new variables. With these affine functions, we readily derive a set of LMI characterizations that allow non-common Lyapunov variables. We also show that a simple combination of this second approach with the above first approach leads to an effective iterative algorithm, with which we can get around the conservatism considerably. The third approach we propose is quite distinct from the above two. In this approach, we derive new dilated matrix inequality characterizations for the design specifications, where the decoupling between the controller variables and the Lyapunov variables has been achieved and hence the bilinear terms between them disappear. This is achieved by the introduction of new auxiliary variables that form product with the controller variables instead of the Lyapunov variables. These new dilated matrix inequalities lead us to a new approach which convexifies the problems with non-common Lyapunov variables but with a common auxiliary variable. It is shown that we can guarantee this approach to achieve better performance than that with the conventional approach. Although our main interest in this thesis is the multiobjective controller design problems, it turns out that the new dilated characterizations for the design specifications have another potential in dealing with robust performance analysis and synthesis problems for real polytopic uncertainty. Roughly speaking, the conventional approach to these problems is such that they seek a common Lyapunov variable over the whole uncertainty domain and hence arrives at conservative results. On the other hand, the new dilated characterizations enable us to employ a so-called parameter-dependent Lyapunov variable, and hence the conservatism of the conventional approach can be circumvented successfully. The idea to decouple the Lyapunov variables and the controller variables in the matrix inequality characterizations is quite important in dealing with such involved problems as the multiobjective controller design problems, robust performance analysis and synthesis for real polytopic uncertainty and so on. This thesis offers an intriguing methodology that could cover such involved problems, and ensures improvement of the performance over the conventional approach.
DOI：
10.1109/cdc.2000.914226
发表时间：
2000-12
期刊：
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
影响因子：
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作者：
T. Shimomura;T. Fujii
通讯作者：
T. Shimomura;T. Fujii
具有参数相关李亚普诺夫函数的扩展空间控制设计
DOI：
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发表时间：
2005
期刊：
Transactions of the Society of Instrument and Control Engineers(SICE) vol.41, no.5
影响因子：
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作者：
T.Shimomura;T.Fujii;T.Shimomura
通讯作者：
T.Shimomura