Final answer:
The student's question pertains to the stability of a system described by a characteristic polynomial, requiring the use of the Routh-Hurwitz criterion to find the conditions for stability (K > 0) and marginal stability by identifying the value of K that results in a Routh array with a row of zeros.
Step-by-step explanation:
The student is asking about the stability of a system given the characteristic equation s⁴+6s³+11s²+6s+K. Stability in this context is determined by the location of the roots of the characteristic polynomial, which must all have negative real parts for the system to be stable. the Routh-Hurwitz criterion is a method used to determine stability without explicitly calculating the roots of the polynomial. to ensure stability, all of the coefficients of the characteristic polynomial must be positive, and we must also have no change in sign in the first column of the Routh array. Since the coefficients of the characteristic equation are already positive for the terms independent of K, we must ensure that K is also positive for stability. Hence, for a stable system K > 0.
A marginally stable system occurs when the system has at least one pair of purely imaginary roots and all other roots with negative real parts. For marginal stability, typically, one row of the Routh array will be all zeros, leading to a row of coefficients with at least one element being zero. The value of K that causes this scenario would lead to a marginally stable system. Computing this would require forming the Routh array and then determining the value of K that leads to a row of zeros.