Final answer:
The BIBO stability for the system requires that the proportional gain Kp and integral gain Ki are selected such that the closed-loop transfer function has no poles with positive real parts. Determining the range of these gain values involves using stability criteria like the Routh-Hurwitz criterion or root locus methods. Exact values require a detailed stability analysis using system parameters.
Step-by-step explanation:
To determine the range of values for Kp (proportional gain) and Ki (integral gain) that ensures Bounded Input, Bounded Output (BIBO) stability of the system T(s), we must ensure the closed-loop transfer function does not have any poles with positive real parts. Given T(s) = G(s) / (1 + G(s)H(s)) and G(s) = 10 / s(s + 5) and H(s) = (Kp + Ki/s), we need to find the values of Kp and Ki that will not make the denominator zero in the right-half s-plane, which would indicate instability.
Firstly, the characteristic equation derived from the denominator of T(s) is 1 + G(s)H(s) = 0. After substituting G(s) and H(s) and simplifying, we can solve for conditions on Kp and Ki based on the roots of the characteristic equation. Typically, a Routh-Hurwitz criterion or root locus approach is used to determine stability and the range of the gain values for stability. Unfortunately, without the specific criteria of system stability for this system, we cannot provide exact numerical values.
In practice, the selection of Kp and Ki often involves iterative tuning, balancing between overshoot, response time, and steady-state error, which is why tools like the root locus, Bode plots, or Nyquist plots are essential in determining these parameters. For actual BIBO stability ranges, a detailed analysis using the system's parameters and stability criteria needs to be conducted.