Question

For a unity feedback system with loop transfer function KG(s) = K/(s+1)(s+4), design a PID controller such that the compensated system response exhibits a peak time of 1.047s, a damping ration of 0.8, and zero error to a unit step input.

Answer 1

The task is to design a PID controller for a unity feedback system to achieve a peak time of 1.047s, damping ratio of 0.8, and zero steady-state error for a step input, incorporating control system design techniques and the dynamics of a damped harmonic oscillator.

Step-by-step explanation:

The question involves designing a PID (Proportional-Integral-Derivative) controller for a unity feedback system with a specific loop transfer function so that the compensated system meets given performance specifications. These specifications include a peak time of 1.047s, a damping ratio of 0.8, and zero steady-state error to a unit step input. The solution will entail using the knowledge of control system design, particularly regarding the behavior of damped harmonic oscillators and the effect of PID controllers in modifying system responses.

To meet these specifications, one must first use the relationship between the system's natural frequency and damping ratio to the desired peak time and overshoot. The PID controller can then be designed by adjusting the controller's parameters—proportional (Kp), integral (Ki), and derivative (Kd) gains—to achieve the desired natural frequency and damping ratio, thereby ensuring that the controlled system exhibits the specified transient response characteristics. The integral action of the PID controller will aid in achieving zero steady-state error for a step input, as required.