This article deals with stabilization and optimal control of an articulated flexible arm by a passive approach. This approach is based on the boundary control of the Euler–Bernoulli beam by means of wave-absorbing feedback. Due to the specific propagative properties of the beam, such controls involve long-memory, non-rational convolution operators. Diffusive realizations of these operators are introduced and used for elaborating an original and efficient wave-absorbing feedback control. The globally passive nature of the closed-loop system gives it the unconditional robustness property, even with the parameters uncertainties of the system. This is not the case in active control, where the system is unstable, because the energy of high frequencies is practically uncontrollable. Our contribution comes in the achievement of optimal control by the diffusion equation. The proposed approach is original in considering a non-zero initial condition of the diffusion as an optimization variable. The optimal arm evolution, in a closed loop, is fixed in an open loop by optimizing a criterion whose variable is the initial diffusion condition. The obtained simulation results clearly illustrate the effectiveness and robustness of the optimal diffusive control.

本文通过被动方法来处理关节柔性臂的稳定性和最佳控制。该方法基于通过吸收波反馈对Euler–Bernoulli光束进行边界控制。由于光束的特定传播特性，此类控制涉及长内存，非理性卷积算符。引入了这些算子的扩散实现，并将其用于阐述原始有效的波吸收反馈控制。闭环系统的全局被动性质使其具有无条件的鲁棒性，即使系统的参数不确定。在系统不稳定的主动控制中，情况并非如此，因为高频能量实际上是不可控制的。我们的贡献在于通过扩散方程实现了最佳控制。所提出的方法最初是将扩散的非零初始条件视为优化变量。通过优化一个变量为初始扩散条件的准则，可以在闭环中将最佳的臂演化固定在开环中。获得的仿真结果清楚地说明了最佳扩散控制的有效性和鲁棒性。