<p>The distributed observer (DO) serves as an effective tool for estimating the leader’s state in the coordination of multi-agent systems. This paper investigates the convergence-time constrained DOs for a leader with unknown nonlinearity and general high-order dynamics. In particular, the communication networks among agents are allowed to be directed and dynamically switched. First, two novel lemmas are proposed to establish the finite-time and prescribed-time stability of the systems, respectively. Different from the existing finite-time and prescribed-time stability criteria that require continuous Lyapunov functions, the proposed lemmas allow the Lyapunov functions being piecewise and discontinuous at certain instants. Then, an innovative finite-time DO is proposed to estimate the leader’s state under directed switching networks, which is robust to leader’s unknown nonlinear dynamics. To construct the finite-time convergence, the topology-dependent multiple Lyapunov functions are wisely constructed and the jump magnitude of Lyapunov functions is carefully estimated. Subsequently, a sufficient condition on the dwell time for ensuring finite-time convergence is derived. To eliminate the dependence of the convergence time on the initial error, a robust prescribed-time DO is further designed utilizing time-varying gains, which realizes that the convergence time of the estimation errors can be predefined by the user explicitly independent of the Initial errors, and the estimation laws are guaranteed to be bounded. Finally, simulation examples are performed to show the effectiveness and the practicality of the proposed finite-time and prescribed-time DOs.</p>