Final answer:
To design a direct digital controller for a system with the transfer function G(s)=1/(s² +2s+1) to meet 25% overshoot and 4 seconds settling time, we use standard formulas for second-order systems to calculate the damping ratio and natural frequency needed, while tuning the controller to account for feedback delay and discretization effects.
Step-by-step explanation:
To design a direct digital controller for the given system G(s)=1/(s² +2s+1), which corresponds to a standard second-order system, we need to meet the performance metrics of 25% overshoot and a settling time of 4 seconds. The overshoot is influenced by the damping ratio (ζ), whereas the settling time is influenced by the natural frequency (ωn) of the system. To achieve these performance metrics, we can use the standard second-order system formulas for overshoot (Overshoot = e^{(-πζ/√(1-ζ²))}) and settling time (Settling Time = 4/(ζωn)). By solving these equations, we can find the necessary parameters for the controller.
Additionally, it is important to consider that the presence of feedback delay can lead to increased overshoot and oscillations, which may affect the actual performance of the system compared to the nominal no-delay scenario. Tuning the controller to account for potential delays and achieve desired performance is crucial in the design process. The controller's digital nature will also need to account for sampling time and discretization effects.