In this brief, a novel tracking differentiator (TD) based on discrete time optimal control (DTOC) is presented. In particular, using the state back-stepping method, a DTOC law for a discrete-time, double-integral system is determined by linearized criterion, which equips the TD with a simple structure. The analysis of the proposed TD reveals its filtering mechanism. Simulation results show that it performs well in signal tracking and differentiation acquisition, and reduces the computational resources needed. Experiments conducted on the speed and position detection system for a maglev train demonstrate that the proposed TD group, with moving average algorithm, can filter noises, amend distortion signals effectively, and compensate for phase delays when the train is passing over track joints.

Time optimal control (TOC) based tracking differentiator (TD) was first proposed by Han \cite{c1} as a practical solution to avoid setpoint jump in active disturbance rejection control (ADRC). In practice, the discrete-time optimal control (DTOC) is implemented in the form of state feedback for a double-integral system, which is widely used to design controllers, observers and noise-tolerant differentiators. The convergence of the DTOC-TD, however, has not been fully understood. This paper provides a rigorous full convergence analysis of the DTOC-TD. It then illustrates the frequency-domain characteristics of this DTOC-TD in signal-tracking filtering and differentiation acquisition, giving a rule of thumb for regulating the parameters. Finally, the case studies including comparison simulations and experiments on processing gap sensor' signals in the suspension system of maglev train are carried out to verify the effectiveness of the DTOC-TD.

Cascading failures in power systems normally occur as a result of initial disturbances or faults on electrical elements, closely followed by situational awareness errors of human operators. It remains a great challenge to systematically trace the source of cascading failures in power systems. In this paper, we develop a mathematical model to describe the cascading outages of transmission lines in power networks. In particular, the direct current (DC) power flow equation is employed to calculate the power flow on the branches. By regarding the disturbances on branches as the control inputs, the problem of identifying the initial disruptive disturbances is formulated with optimal control theory, which provides a systematic approach to explore the most disruptive disturbances that give rise to changes of branch admittance in addition to direct branch outages. Moreover, an iterative search algorithm is proposed to look for the optimal solution leading to the worst-case cascading failures. Theoretical analysis guarantees the asymptotic convergence of the iterative search algorithm. Finally, numerical simulations are carried out on the IEEE test systems to validate the