Tracking and Anti-sway Control for Double-pendulum Rotary Cranes Using Novel Sliding Mode Algorithm
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摘要: 本文针对旋转起重机系统中旋臂定位和两级摆动抑制问题，提出一种含有非线性滑模面的滑模控制算法.不同于传统的线性滑模面，非线性滑模面可以使闭环系统的阻尼比从开始的较小值变化为最终的较大值.较小的阻尼比可以为系统提供较快的响应速度而较大的阻尼比则可减小超调量从而使得旋臂更加精确地跟踪给定轨迹.通过李雅普诺夫定理验证系统稳定性.比较仿真结果表明，该方法在实现摆角抑制的同时，起伏角和旋转角的跟踪误差分别降低了大约40%和52%.Abstract: This paper proposes a novel sliding mode controller with nonlinear sliding surface for achieving both high-precision tracking control and double-pendulum load sway suppression. Unlike the traditional linear sliding surface, the nonlinear one can change the damping ratio of the closed-loop system from its initial low value to final high value. The low value can provide a quick response whereas the high one can eliminate overshoot to make boom track the given trajectory more precisely. The stability of the whole system is proven by the Lyapunov technique. Comparative simulations indicate that the tracking error for vertical and horizontal boom motions can be reduced about 40% and 52%, respectively, the double-pendulum load sways can be suppressed as well.
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Key words:
- Rotary crane /
- double-pendulum effect /
- tracking control /
- anti-sway control /
- nonlinear sliding surface
1) 本文责任编委 穆朝絮 -
表 1 起重机系统模型参数
Table 1 Parameters of crane system
$M_0$ (kg) $m_1$ (kg) $m_2$ (kg) $L$ (m) $l_{1}$ (m) $l_{2}$ (m) $J_5$ (kg${\rm{m}}^{2}$) $J_6$ (kg${\rm{m}}^{2}$) $g$ (m/${\rm{s}}^{2}$) 0.86 2.00 0.56 0.65 0.50 0.20 0.52 0.52 9.80 
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表 2 起伏子系统控制器参数
Table 2 Parameters of controller in vertical subsystem
$\Gamma_1$ $P_1$ $K_1$ $Q_1$ $\lambda_{i1}$ NLSS ${\rm{diag}}\{1.0, 1.0, 0.5\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ ${\rm{diag}}\{0.5, 0.5, 0.8\}$ ${\rm{diag}}\{1.0, 1.0, 1.5\}$ ${\rm{diag}}\{1.2, 1.5, 0.7\}$ LSS ${\rm{diag}}\{1.0, 1.0, 0.5\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ ${\rm{diag}}\{0.5, 0.5, 0.8\}$ ${\rm{diag}}\{1.0, 1.0, 1.5\}$ —
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表 3 旋转子系统控制器参数
Table 3 Parameters of controller in horizontal subsystem
$\Gamma_2$ $P_2$ $K_2$ $Q_2$ $\lambda_{i2}$ NLSS ${\rm{diag}}\{1.0, 1.0, 1.5\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ ${\rm{diag}}\{0.5, 0.5, 0.9\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ ${\rm{diag}}\{1.2, 1.5, 0.8\}$ LSS ${\rm{diag}}\{1.0, 1.0, 1.5\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ ${\rm{diag}}\{0.5, 0.5, 0.9\}$ ${\rm{diag}}\{1.0, 1.0, 1.4\}$ —
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表 4 定量分析($l_2=0.1$ m)
Table 4 Quantitative analysis ($l_2=0.1$ m)
最大起伏角误差 最大旋转角误差 最大摆角 最大摆角 最大摆角 最大摆角 $e_{5\max}$ (deg) $e_{6\max}$ (deg) $\theta_{1\max}$ (deg) $\theta_{2\max}$ (deg) $\theta_{3\max}$ (deg) $\theta_{4\max}$ (deg) NLSS 1.21 1.91 2.91 2.75 3.15 3.21 LSS 1.68 2.91 2.77 2.81 3.11 3.18
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表 5 定量分析($l_2=0.2$ m)
Table 5 Quantitative analysis ($l_2=0.2$ m)
最大起伏角误差 最大旋转角误差 最大摆角 最大摆角 最大摆角 最大摆角 $e_{5\max}$ (deg) $e_{6\max}$ (deg) $\theta_{1\max}$ (deg) $\theta_{2\max}$ (deg) $\theta_{3\max}$ (deg) $\theta_{4\max}$ (deg) NLSS 1.21 1.81 3.03 2.94 3.69 4.21 LSS 1.61 2.81 2.79 2.83 3.41 4.11
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表 6 定量分析($l_2=0.3$ m)
Table 6 Quantitative analysis ($l_2=0.3$ m)
最大起伏角误差 最大旋转角误差 最大摆角 最大摆角 最大摆角 最大摆角 $e_{5\max}$ (deg) $e_{6\max}$ (deg) $\theta_{1\max}$ (deg) $\theta_{2\max} $(deg) $\theta_{3\max}$ (deg) $\theta_{4\max}$ (deg) NLSS 1.41 1.99 2.93 2.94 2.93 5.22 LSS 1.81 2.99 2.52 2.92 4.27 5.02
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