The stabilization problem of the upright position of the controlled spherical pendulum is treated in detail. This problem is reduced to the stabilization of the stable manifold /spl Omega//sub st/ of the upright position of the unforced spherical pendulum. It is shown that for any smooth feedback control derived by the speed-gradient algorithm with the objective to stabilize /spl Omega//sub st/ the closed loop system has a limit cycle /spl Gamma/, which does not belong to the desired attractor /spl Omega//sub st/. It is shown that /spl Gamma/ is hyperbolic.
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机译:详细讨论了受控球形摆的直立位置的稳定问题。该问题减少到使未受力的球摆的竖立位置的稳定歧管稳定。结果表明,对于由速度梯度算法得出的,旨在稳定/ spl Omega // subst /的任何平滑反馈控制,闭环系统都有一个极限周期/ spl Gamma /,这不属于所需的吸引子/ spl Omega // sub st /。显示/ spl Gamma /是双曲线的。
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