A control system has forward-path transfer function G(s) and a feedback element H(s) = 2. What is the closed-loop transfer function C(s)/R(s)?
AG(s) / (1 + G(s))
BG(s) / (1 + 2G(s))
C2G(s) / (1 + G(s))
DG(s) / (1 + G(s)²)
For a negative-feedback loop with forward gain G(s) and feedback element H(s), the closed-loop transfer function is T(s) = G(s) / (1 + G(s)H(s)). With H(s) = 2, this gives T(s) = G(s) / (1 + 2G(s)). Option A is the unity-feedback formula (H = 1) — the most common error, arising from memorizing T = G/(1+G) and forgetting that H must be included when feedback is non-unity. The open-loop transfer function is G(s)H(s) = 2G(s), and this product appears in the denominator.
Question 2 Multiple Choice
In block diagram reduction, the roots of the equation 1 + G(s)H(s) = 0 are best described as:
AThe open-loop poles — the values of s where G(s)H(s) goes to infinity
BThe closed-loop zeros — the values of s where the output is zero for any input
CThe closed-loop poles — the values of s that determine stability and transient response
DThe gain crossover frequencies — relevant only for frequency-domain stability analysis
Setting the denominator of the closed-loop transfer function to zero — 1 + G(s)H(s) = 0 — defines the characteristic equation whose roots are the closed-loop poles. These poles determine everything about closed-loop behavior: stability (poles in the left half-plane = stable; right half-plane = unstable), transient response (damping, natural frequency), and sensitivity to disturbances. They are distinct from the open-loop poles (poles of G(s)H(s) alone), which are the starting points for root locus analysis. The denominator 1 + G(s)H(s) is called the characteristic polynomial precisely because it characterizes the closed-loop system.
Question 3 True / False
When two ideal transfer function blocks G₁(s) and G₂(s) are connected in series (the output of G₁ feeds directly into the input of G₂), the combined transfer function is G₁(s) · G₂(s).
TTrue
FFalse
Answer: True
True. For ideal blocks (where the downstream block does not load the upstream one — the standard infinite input impedance assumption), signals cascade multiplicatively: U₂(s) = G₁(s)·U₁(s) and Y(s) = G₂(s)·U₂(s) = G₁(s)·G₂(s)·U₁(s). The combined transfer function is the product. This series multiplication rule is one of the three fundamental reduction operations, alongside the additive rule for parallel blocks and the feedback formula G/(1+GH).
Question 4 True / False
For a closed-loop system with unity feedback (H = 1) and forward gain G(s), the closed-loop transfer function is G(s) / (1 + G(s)²).
TTrue
FFalse
Answer: False
False. The correct closed-loop transfer function for unity feedback is T(s) = G(s) / (1 + G(s)), not G(s) / (1 + G(s)²). The characteristic denominator is 1 + G(s)H(s), and with H = 1 this is 1 + G(s). The squared term has no algebraic basis — it may arise from incorrectly multiplying the forward gain by the feedback gain rather than forming the product G·H that appears in the denominator. A common check: as G(s) → ∞, T(s) should approach 1 (perfect tracking), which works for G/(1+G) but not G/(1+G²).
Question 5 Short Answer
What is the 'characteristic equation' of a closed-loop control system, and why is it central to determining whether the system is stable?
Think about your answer, then reveal below.
Model answer: The characteristic equation is 1 + G(s)H(s) = 0, formed by setting the denominator of the closed-loop transfer function to zero. Its roots (in the complex s-plane) are the closed-loop poles. Stability of a linear time-invariant system is determined entirely by closed-loop pole locations: if all poles have negative real parts (left half s-plane), the system is stable; if any pole has a positive real part, the system is unstable. The characteristic equation encodes how the feedback loop modifies the open-loop dynamics into a single polynomial whose roots reveal closed-loop behavior.
This is why control design focuses on shaping the characteristic equation: root locus plots show how closed-loop poles move as gain varies; Bode and Nyquist methods assess stability margins by examining G(s)H(s) without explicitly solving 1 + GH = 0. The characteristic equation is distinct from the open-loop transfer function poles — feedback fundamentally changes where the poles are, and the denominator 1 + GH encodes this transformation. A well-designed controller shapes 1 + GH so all roots lie safely in the left half-plane with desired damping and natural frequency.