Fractional-order Terminal Sliding-mode Control of MIMO Systems With Unmatched Uncertainties
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摘要: 针对一类非匹配不确定多输入多输出(Multi-input multi-output, MIMO)系统提出一种分数阶终端滑模控制(Fractional-order terminal sliding-mode, FOTSM)策略, 使系统输出收敛到零而非其邻域. 该方法解除传统反步法控制律设计中, 虚拟控制增益右伪逆矩阵必须存在的严苛限制; 对系统不确定性的假设不局限于慢时变和H2范数有界型扰动, 分析控制增益存在摄动情况下系统的控制问题. 分数阶终端滑模面及其控制律的设计使得虚拟和实际控制信号连续, 削弱抖振现象, 利用自适应滑模切换增益技术解决由控制增益矩阵摄动引起的代数环问题. 最后, 仿真分析验证所提方法的正确性和优越性.Abstract: This paper proposes a fractional-order terminal sliding-mode control (FOTSM) for multi-input multi-output (MIMO) systems with unmatched uncertainties to force the outputs to converge to the equilibrium point rather than their neighborhoods. The control method breaks through the limitation in traditional back-stepping control that the right Moore-Penrose pseudo inverse of the gain matrix of the virtual control signal must exist. The proposed control method can deal with uncertainties, which are not limited to being slow time-varying or H2 norm-bounded. Besides, the method can apply to the system with uncertain control gain. Meanwhile, an adaptive sliding mode control law is designed to compensate for the uncertainties caused by the gain matrix variation and attenuate the switching gain＇s amplitude. The smooth virtual and actual control signals can be obtained thanks to the fractional-order sliding mode control method, which eliminates the chattering in the MIMO systems with unmatched uncertainties. Finally, simulations demonstrate the correctness and superiority of the proposed control method.
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图 2 分数阶滑模控制算法框图
Fig. 2 Block diagram of the fractional-order sliding-mode control method
图 3 五种不同控制方法的系统输出相量${{\boldsymbol{x}}}_1$
Fig. 3 System output ${{\boldsymbol{x}}}_1$ under the five control methods
图 4 五种不同控制方法的系统输出相量${x}_{11}$
Fig. 4 System output ${x}_{11}$ under the five control methods
图 5 五种不同控制方法的系统输出相量${x}_{12}$
Fig. 5 System output ${x}_{12}$ under the five control methods
图 6 五种不同控制方法的系统输出相量${x}_{13}$
Fig. 6 System output ${x}_{13}$ under the five control methods
图 7 五种不同控制方法的系统输出的2范数$\left\| {{\boldsymbol{x}}}_1\right\|$
Fig. 7 2-norm of system output $\left\| {{\boldsymbol{x}}}_1\right\|$ under five methods
图 8 分数阶终端滑模控制下状态${\boldsymbol{x}}_{2}$和虚拟控制信号${\boldsymbol{x}}_{2ref}$
Fig. 8 States ${\boldsymbol{x}}_{2}$ and virtual control ${\boldsymbol{x}}_{2ref}$ under FOTSM
图 9 五种不同控制方法的实际控制信号${\boldsymbol{u}}$
Fig. 9 Actual control ${\boldsymbol{u}}$ under the five control methods
表 1 控制器主要设计参数
Table 1 The design parameters of the controllers
控制方法 控制器参数 DOBSM $c_1=c_2=10, k_1=k_2=700$, $l_1=l_2=10$ SOSM $c=60, k=2\;000$ FSM ${\boldsymbol{C} }_{1}={\rm{diag}}\{60,60\}$, ${\boldsymbol{C} }_{2}={\rm{diag}}\{150,170\}$, $p=5, q=3$ FSM-Sat ${\boldsymbol{C} }_{1}={\rm{diag}}\{60,60\}$, ${\boldsymbol{C} }_{2}={\rm{diag}}\{150,170\}$, $\delta=1/8$, $p=5,q=3$ 
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表 2 不同控制方法性能对比
Table 2 Performance comparison of the five methods
控制方法 收敛速度 控制精度 控制信号 DOBSM 较快 $\leq0.05$ 不连续, 抖振 SOSM 较快 $\leq0.04$ 不连续, 抖振 FSM 快 $\leq 1\times10^{-3}$ 不连续, 抖振 FSM-Sat 快 $\leq0.06$ 连续 FOTSM 快 $\leq4\times10^{-3}$ 连续
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