ANS:D - all 'a', 'b' & 'c'

The characteristic equation of a control system, which is given by 1+G(s)=01 + G(s) = 01+G(s)=0, where G(s)G(s)G(s) is the transfer function of the system, plays a critical role in determining the stability of the system. Here's how it relates to the options provided:

1. Depends only upon the open loop transfer function:
   • The characteristic equation is derived from the closed-loop transfer function G(s)1+G(s)\frac{G(s)}{1 + G(s)}1+G(s)G(s)​. However, the stability of the system is primarily influenced by the poles of the open-loop transfer function G(s)G(s)G(s). These poles dictate the behavior of the closed-loop system, including its stability.
2. Determines its stability:
   • The roots (or poles) of the characteristic equation 1+G(s)=01 + G(s) = 01+G(s)=0 directly determine whether the closed-loop system is stable, marginally stable, or unstable. Stable systems have all poles with negative real parts, while unstable systems have at least one pole with a positive real part.
3. Is the same for set point or load changes:
   • The characteristic equation remains unchanged regardless of whether there are changes in the set point or load conditions. It solely depends on the system's transfer function G(s)G(s)G(s).
4. Conclusion:
   • Therefore, the characteristic equation for a control system primarily determines its stability and depends on the open-loop transfer function G(s)G(s)G(s). It encapsulates critical information about the system's behavior and stability properties, making it essential for analysis and design considerations in control engineering.