Stability is usually in the sense of Lyapunov′s asymptotical stability,thus the solutions starting from points close to a stable equilibrium may have a very long transient.In the applications of time-delayed feedback controls,it is important not only to determine the stable regions in the gain plane or gain space,but also to find out the abscissa that can be used as an index of stability.Based on the D-subdivision method,this paper proposes a simple algorithm for finding and labeling the stable regions in feedback gain plane with abscissa.The labeled sub-regions with smaller abscissa are better in applications.The main results are presented for the controlled pendulum or inverted pendulum under a delayed feedback,and are illustrated with two case studies.
Despite the intensive studies on neurons, the control mechanism in real interactions of neurons is still unclear. This paper presents an understanding of this kind of control mechanism, controlling a neuron by stimulating another coupled neuron, with the uncertainties taken into consideration for both neurons. Two observers and a differentiator, which comprise the first-order low-pass filters, are first designed for estimating the uncertainties. Then, with the estimated values combined, a robust nonlinear controller with a saturation function is presented to track the desired membrane potential. Finally,two typical bursters of neurons with the desired membrane potentials are proposed in the simulation, and the numerical results show that they are tracked very well by the proposed controller.
In the dynamics analysis and synthesis of a controlled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a systematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost characteristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial values around the origin are fixed. The main results are checked by numerical simulation.
