Control system design using non-smooth feedback

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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.