Control system design using non-smooth feedback

1.给出了不连续系统逆Lyapunov定理的一个简单表达式。给出了带扩张的不连续齐次系统的逆Lyapunov定理，证明了每个带扩张的不连续齐次强稳定系统都有齐次Lyapunov函数。结果表明，齐次度表示收敛的速度，即齐次度为负的齐次强稳定系统具有有限时间收敛性质。证明了高阶滑模控制系统的有限时间收敛。2.利用有限时间收敛的概念，针对链式系统这一典型的非完整控制系统，得到了一种新的齐次间断控制律。控制律是局部有界的，其受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乘以控制律来设置收敛速度的方法，以避免抖振现象。将该方法推广到高阶链式系统控制，使得状态在有限时间内收敛到最终的滑动超曲面。此外，我们还提出了另一种非完整系统的有限时间镇定控制方法，即具有非光滑Lyapunov函数的控制-Lyapunov函数方法。3.将上述控制律应用于一个停车问题。通过对一个小型移动机器人的停车问题进行实验，验证了该方法的有效性。4.提出了一种基于具有不连续动力学的精确微分器的机器人操作手的角速度估计方法。在实际机器人机械手上的实验结果表明，该方法具有良好的降噪性能。此外，我们还利用所提出的估计器构造了一个实际机器人机械手的自适应控制系统。

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.