LQ-optimal boundary control of infinite-dimensional systems with Yosida-type approximate boundary observation
LQ-optimal boundary control of infinite-dimensional systems with Yosida-type approximate boundary observation
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DOI：
10.1016/j.automatica.2015.12.033
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发表时间：
2016-05
期刊：
影响因子：
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通讯作者：
Jérémy R. Dehaye;J. Winkin
中科院分区：
文献类型：
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作者：
Jérémy R. Dehaye;J. Winkin
考虑了一类具有边界观测的边界控制系统，其中的无界算子常常导致技术上的困难。描述和分析了这类系统的一个扩展模型，该模型除了动态生成器外，不包含任何无界算子。描述了一种求解该模型的LQ最优控制问题的方法，该解为具有无界算子的标称系统提供了一个稳定的反馈，即在闭环系统中，状态轨迹以指数速度收敛到零。该模型由扩展的抽象微分方程组成，其状态分量为边界输入、状态(直至仿射变换)和标称系统输出的Yosida型近似。证明了在适当的条件下，该模型是适定的，特别是动力学算子是C0-半群的生成元。此外，该模型被证明是可观测的，并且具有标称系统的可控性、镇定性和可检测性。为了解决该模型的LQ最优控制问题，描述了一种基于多维算子值谱密度的谱分解问题的一般分解方法。这种方法有望在建模成本和解决此类系统控制问题的方法的效率之间取得良好的平衡。
A class of boundary control systems with boundary observation is considered, for which the unbounded operators often lead to technical difficulties. An extended model for this class of systems is described and analyzed, which involves no unbounded operator except for the dynamics generator. A method for the resolution of the LQ-optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators, in the sense that, in closed-loop, the state trajectories converge to zero exponentially fast. The model consists of an extended abstract differential equation whose state components are the boundary input, the state (up to an affine transformation) and a Yosida-type approximation of the output of the nominal system. It is shown that, under suitable conditions, the model is well-posed and, in particular, that the dynamics operator is the generator of a C 0-semigroup. Moreover, the model is shown to be observable and to carry controllability, stabilizability and detectability properties from the nominal system. A general method of resolution based on the problem of spectral factorization of a multi-dimensional operator-valued spectral density is described in order to solve a LQ-optimal control problem for this model. It is expected that this approach will lead hopefully to a good trade-off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems.