This article focuses on a class of nonlinear time-delay systems with Hamiltonian structure of different powers. Under the presence of external disturbances, parameter uncertainties, and state delays, the predefined-time observer-basedH∞control and adaptive robust stabilization problems are investigated. First, to study general nonlinear time-delay system, we develop a novel predefined-time stability criterion. Moreover, to overcome the limitation of the traditional same-power Hamiltonian function method, this article attempts to study the cases of Hamiltonian function with different powers. In addition, for applying the obtained criterion in the delay system, we construct a new Lyapunov function form with exponential invariance. Based on these and designing the predefined-time observer, we give the predefined-time observer-basedH∞control and adaptive robust stabilization results. Several simulation experiments with infinite-time, finite-time and predefined-time observer methods verified that the proposed method has better anti-interference and convergence performance. It is worth noting that, unlike existing observer-based control methods (which rely on linearized differencing approaches), this paper studies systems in a nonlinear form and uses dimensional expansion technique to overcome the limitation of differencing method. Furthermore, different from existing Hamiltonian method, this paper adopts Hamiltonian form of different powers, further reducing the conservativeness of the results. The main innovations and contributions of this work have: 1) Developing a novel predefined-time stability criterion on nonlinear time-delay systems that does not contain the time-varying gain. 2) Adopting Hamiltonian function form of different powers, which breaks the limitation of conventional same-power structure. 3) Constructing a new exponentially invariant Lyapunov functional form, overcoming the difficulty that traditional Lyapunov functional cannot be applied directly to study the predefined-time control of time-delay systems. 4) Taking into account various factors that affect the actual system such as time delays, external disturbances, and parameter uncertainties, which broadens the scope of the results.