Stabilization of LTI systems via derivative feedback and time delay effects

Abstract

This study investigates the output derivative feedback control issue for linear time-invariant (LTI) systems and introduces the novel concept of derivative-induced stability. By imposing structural assumptions on the system, we establish a category of systems that fail to be stabilised by conventional output feedback control. Subsequently, to achieve stability under derivative feedback, two parameterization methods for designing feedback gains are proposed, involving algebraic analysis and Lyapunov theory. Furthermore, to overcome the practical challenges of measuring derivatives and reduce noise sensitivity, a difference feedback control framework based on finite-difference approximation was introduced. It was shown that by incorporating a properly selected delay, the system not only achieves effective derivative estimation but also benefits from improved stability, this phenomenon aligns with the theory of delay-induced stability. Using Lyapunov functional analysis, sufficient conditions are established to ensure asymptotic stability under difference feedback control. Finally, the proposed method is validated through two numerical examples, which also highlight the beneficial impact of output derivatives and time delays on system stability.