arXiv 1802.085202018-02-23math.OC

寻优控制中稳态解的非唯一性

Non-Uniqueness of Stationary Solutions in Extremum Seeking Control

Olle Trollberg, Elling W. Jacobsen

摘要：

寻轨控制是一种经典的适应控制方法，它完全基于输出反馈，用于稳态优化。极值搜索控制回路在控制器的某些温和条件下，在被控对象的稳态输入输出映射的极值点附近存在稳定的平稳周期解。然而，这是一个局部的结果，本文研究是否这个解决方案是唯一的，基本的最优化问题是凸的。我们首先推导出ESC回路的任何定态解必须满足的一个必要条件。对于极值点是由于纯静态非线性的设备，如Hammerstein或Wiener设备，该条件涉及稳态梯度。然而，对于更一般的植物的必要条件涉及的相位滞后的局部线性化的植物，表明可能存在的解决方案没有任何关系的最优性。结合导出的必要条件和局部解的存在性，我们利用分歧理论来追踪稳定解的分支。我们推导出这些分支分叉时，在多个固定的解决方案。结果表明，该系统可能存在循环折叠分支，导致存在多个稳定周期1解，其中只有一个解与最优性条件有关。我们通过一个例子说明了结果，其中在化学反应器中的转换使用ESC进行优化。我们表明，至少有五个固定的解决方案可能同时存在现实的控制参数，这些解决方案中的几个是稳定的。非唯一性的一个后果是，一般需要从接近最优开始，以确保收敛到接近最优解。

英文摘要：

Extremum seeking control (ESC) is a classical adaptive control method for steady-state optimization, purely based on output feedback. It is well known that the extremum seeking control loop, under certain mild conditions on the controller, has a stable stationary periodic solution in the vicinity of an extremum point of the steady-state input-output map of the plant. However, this is a local result only and this paper investigates whether this solution is necessarily unique given that the underlying optimization problem is convex. We first derive a necessary condition that any stationary solution of the ESC loop must satisfy. For plants in which the extremum point is due to a purely static nonlinearity, such as in Hammerstein or Wiener plants, the condition involves the steady-state gradient. However, for more general plants the necessary condition involves the phase lag of the locally linearized plant, indicating the possible existence of solutions without any relationship to optimality. Combining the derived necessary condition with the existence of a local solution close to the optimum, we employ bifurcation theory to trace out branches of stationary solutions. We derive conditions on when these branches bifurcate, resulting in multiple stationary solutions. The results show that cyclic fold bifurcations may exist, resulting in the existence of multiple stationary period one solutions, of which only one is related to the optimality conditions. We illustrate the results through an example in which the conversion in a chemical reactor is optimized using ESC. We show that at least five stationary solutions may exist simultaneously for realistic control parameters, and that several of these solutions are stable. One consequence of the non-uniqueness is that one in general needs to start close to the optimum to ensure convergence to the near-optimal solution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文