A single-loop unity-feedback architecture is defined as follows

A single-loop unity-feedback architecture is defined as follows. Time \(t\) is measured in seconds. Capital letters denote unilateral Laplace transforms, and every initial condition is zero. The reference \(R(s)\) and the plant output \(Y(s)\) enter a summing junction that forms the actuating error \[ E(s)=R(s)-Y(s) \] (negative feedback). A series compensator \[ C(s)=\frac{3(s+4)}{s+12} \] maps \(E\) to the plant input \(U\). The plant \[ P(s)=\frac{5}{s(s+6)} \] maps \(U\) to \(Y\). All blocks are linear, time-invariant, and rational. The loop transfer function is \(L(s):=C(s)P(s)\). The complementary sensitivity is \(T(s):=Y(s)/R(s)\) and the sensitivity is \(S(s):=E(s)/R(s)\). (a) Write \(L(s)\) as a ratio of coprime polynomials in \(s\) with monic denominator. (b) Derive \(T(s)\) and \(S(s)\) as ratios of coprime polynomials with a common monic denominator. Prove that \(S(s)+T(s)=1\) as an identity of rational functions. (c) State the closed-loop characteristic polynomial, defined as that common monic denominator of \(T\) and \(S\). State the relative degree of \(T\). (d) Evaluate the finite values \(T(0)\) and \(S(0)\). Interpret \(T(0)\) as the zero-frequency gain from a constant reference to a constant output, assuming the closed-loop signals admit a finite steady state.

** In unity negative feedback the three maps \(L\), \(T=L/(1+L)\), and \(S=1/(1+L)\) share a single algebraic denominator \(1+L\).