Control system design using non-smooth feedback

1.我们得到了不连续系统的李雅普诺夫逆定理的一个简单表达式。建立了具有膨胀的不连续齐次系统的逆Lyapunov定理，证明了每一个具有膨胀的不连续齐次强稳定系统都有一个齐次Lyapunov函数。结果表明，齐次度表示收敛速度，即具有负齐次度的齐次强稳定系统具有有限时间收敛性。利用所得到的定理2证明了高阶滑模控制系统的有限时间收敛。利用有限时间收敛的概念，得到了典型非完整控制系统链式系统的一个新的齐次不连续控制律。控制律是局部有界的，其被控系统是局部指数稳定和全局渐近稳定的。我们给出了一种通过将齐次范数与控制律相乘来设定收敛速度的方法，以避免抖振现象。将该方法推广到高阶链式系统控制中，使状态在有限时间内收敛到最终滑动超曲面。此外，我们还提出了另一种非完整系统的有限时间稳定控制方法，即带有非光滑Lyapunov函数的控制-Lyapunov函数方法。将上述控制律应用于一个汽车停放问题。我们用一个小型移动机器人对停车问题进行了实验，并证明了所提出方法的有效性。提出了一种利用具有不连续动力学特性的精确微分器进行机器人角速度估计的新方法。在实际机械臂上的实验结果表明，该方法具有良好的降噪性能。此外，我们利用所提出的估计量构造了一个真实机器人的自适应控制系统。

1.We obtained a simple expression of the converse Lyapunov theorem for discontinuous systems. Also, the converse Lyapunov theorem for discontinuous homogeneous systems with dilations was developed, which shows that each discontinuous homogeneous strongly stable system with dilation has a homogeneous Lyapunov function. It was shown that the degree of homogeneity indicates the speed of the convergence, i.e. a homogeneous strongly stable system with a negative degree of homogeneity has a finite-time convergence property. The proof of the finite time convergence of high-order sliding mode control system follows the obtained theorem.2.We obtained a new homogeneous discontinuous control law for chained system, a typical nonholonomic control system, by using a concept of finite-time convergence. The control law is locally bounded, and its controlled system is locally exponentially stable and globally asymptotically stable. We showed a method for setting the speed of convergence by multiplying the homogeneous norm to the control law to avoid a chattering phenomenon. This method was extended to the high-order chained system control so that the state converges to a final sliding hyper-surface in finite time. Moreover, we proposed another finite-time stabilizing control method for nonholonomic system, which is a control-Lyapunov-function approach with non-smooth Lyapunov function.3.The control law mentioned above was applied to a car parking problem. We experimented on the parking problem with a small mobile robot, and showed the effectiveness of the proposed method.4.We proposed a new angular-velocity estimation method for robot manipulators using an exact differentiator that has discontinuous dynamics. Experimental results on a real robot manipulator showed that the proposed method has a good noise-reduction performance. Moreover, we constructed an adaptive control system for a real robot manipulator with the proposed estimator.