Final answer:
The Routh-Hurwitz criterion is used to find control gains k and i that ensure BIBO stability by avoiding sign changes or zeros in the first column of the Routh array.
Step-by-step explanation:
The Routh-Hurwitz criterion is used to determine the stability of a linear system based on its characteristic equation's coefficients. BIBO stability, or Bounded Input Bounded Output stability, implies that for any given bounded input to the system, the output will also be bounded. In the context of your question, we are looking to find the range of control gains k and i that ensure the system's BIBO stability.The question does not provide specific system equations, so exact ranges cannot be calculated. Generally, negative feedback loops are associated with stability, and the Routh-Hurwitz criterion helps in finding appropriate ranges for k and i to maintain stability.
Without the specific system equations or characteristic equation provided, we cannot calculate the exact ranges of k and i. However, generally, positive feedback loops lead to system instability, whereas negative feedback loops are employed to maintain system stability. The Routh-Hurwitz criterion will provide a systematic way to determine these ranges by constructing the Routh array and ensuring that the first column does not contain any sign changes or zeros.
The ranges of stability will depend on how the control gains affect the coefficients in the characteristic equation. Typically, one would construct the Routh array with the characteristic equation's coefficients expressed in terms of k and i, and determine the conditions under which all elements of the first column of the Routh array are positive, which in turn indicates that all roots of the characteristic equation have negative real parts—a requirement for stability.