⚙ Where You See This Every Day
- Car cruise control — speed sensor feeds back to throttle (closed-loop)
- Air conditioner thermostat — room temperature → setpoint error → compressor on/off
- Industrial robot arm — encoder feedback, PID controller adjusts torque
- Autopilot system — GPS error → flight computer → control surfaces
- Blood glucose monitor + insulin pump — continuous closed-loop glucose regulation
- Op-amp inverting amplifier — negative feedback sets gain = −R₂/R₁
1. Open-Loop vs Closed-Loop Control
A control system maintains a desired output (controlled variable) despite disturbances. The key distinction is whether the output is measured and fed back.
| Feature | Open-Loop | Closed-Loop |
|---|---|---|
| Feedback | No feedback | Output fed back to input |
| Error correction | None — disturbances ignored | Automatic via error signal E(s) |
| Stability | Inherently stable | May become unstable if poorly designed |
| Accuracy | Less accurate | More accurate |
| Cost/Complexity | Simple, cheap | Complex, expensive |
| Real example | Traffic signal timer | Car cruise control |
| Transfer function | T(s) = G(s) | T(s) = G(s)/[1+G(s)H(s)] |
2. Transfer Function
The transfer function T(s) = C(s)/R(s) is the ratio of the Laplace transform of output to input, with all initial conditions zero. It completely characterizes a linear time-invariant system.
Deriving Transfer Function from ODE
For the ODE: a₂ÿ + a₁ẏ + a₀y = b₀u
Taking Laplace (zero ICs): a₂s²Y(s) + a₁sY(s) + a₀Y(s) = b₀U(s)
Poles and Zeros
Zeros: values of s that make T(s) = 0 (numerator roots). Poles: values of s that make T(s) = ∞ (denominator roots, i.e., characteristic equation roots).
| Property | Effect on System |
|---|---|
| Poles in LHP (σ < 0) | Stable — transient decays |
| Poles on jω axis | Marginally stable (oscillations) |
| Poles in RHP (σ > 0) | Unstable — transient grows |
| Repeated poles on jω axis | Unstable (grows linearly) |
3. Block Diagram Algebra
Complex systems are represented as interconnected blocks. Block diagram reduction simplifies them to a single equivalent transfer function.
Moving Summing Points and Takeoff Points
| Operation | Rule | Reason |
|---|---|---|
| Move summing point ahead of block G | Add 1/G in moved branch | Divide by G to compensate |
| Move summing point behind block G | Add G in moved branch | Multiply by G to compensate |
| Move takeoff point ahead of block G | Add G in moved branch | Signal wasn't through G yet |
| Move takeoff point behind block G | Add 1/G in moved branch | Signal already went through G |
4. Signal Flow Graphs (SFG)
An SFG is an alternative to block diagrams — a directed graph where nodes represent variables and branches represent gains. Every block diagram has an equivalent SFG.
• Node: represents a signal (variable)
• Branch: directed arc with transmittance (gain)
• Forward path: path from input to output with no node repeated
• Loop: closed path starting and ending at same node
• Non-touching loops: loops sharing no common node or branch
• Path gain: product of all branch gains along a path
• Loop gain: product of all branch gains around a loop
5. Mason's Gain Formula
Mason's formula gives the overall transfer function of an SFG directly, without step-by-step reduction.
where:
Pₖ = gain of kth forward path
Δ = 1 − Σ(all loop gains) + Σ(products of 2 non-touching loop gains) − Σ(products of 3 non-touching loop gains) + ...
Δₖ = Δ with all loops touching the kth forward path set to zero
Worked Example
Forward paths: P₁ = a·b·c
Loops: L₁ = b·d (x₂→x₃→x₂), L₂ = a·b·c·e (x₁→...→x₁)
Non-touching loops: None (all share nodes)
Δ = 1 − (L₁ + L₂) = 1 − bd − abce
Δ₁ = 1 (forward path touches all loops)
T = P₁·Δ₁/Δ = abc / (1 − bd − abce)
| Step | Action |
|---|---|
| 1 | Identify all forward paths from input to output and their gains Pₖ |
| 2 | Identify all individual loops and their gains Lₘ |
| 3 | Find all sets of 2, 3, ... non-touching loops |
| 4 | Compute Δ = 1 − ΣLₘ + Σ(non-touching pairs) − ... |
| 5 | For each forward path k, compute Δₖ by removing loops that touch path k |
| 6 | T = (ΣPₖΔₖ)/Δ |
6. Sensitivity Function
Sensitivity measures how much the closed-loop transfer function T changes for a change in the plant G.
For unity feedback T = G/(1+G):
⚙ Lab Experiment: Sensitivity Demonstration
- Build inverting op-amp: gain = −R₂/R₁ (deep feedback, S ≈ 0 for op-amp gain variations)
- Compare: open-loop amp gain changes ±20% with temperature → closed-loop changes <0.01%
- DC motor speed control: open-loop speed drops 30% under load; closed-loop maintains ±1%
- Measure: apply step load to open vs closed loop motor — see error correction speed
GATE Previous Year Questions
The transfer function of a system is T(s) = (s+2)/[(s+1)(s+3)]. The system is:
Poles are at s = −1 and s = −3. Both poles are in the left-half s-plane (negative real parts). For a stable system, all poles must be in the open LHP. The zero at s = −2 does not affect stability. ✓ System is BIBO stable.
For a unity negative feedback system with G(s) = K/[s(s+2)], the closed-loop transfer function is:
Unity feedback: T(s) = G(s)/[1+G(s)] = [K/s(s+2)] / [1 + K/s(s+2)] = K / [s(s+2)+K] = K/(s²+2s+K). This is the standard second-order form with ωₙ = √K and 2ζωₙ = 2 → ζ = 1/√K.
In an SFG, there is one forward path with gain 6, two loops with gains −2 and −3. The two loops are non-touching. The overall transfer function T = ?
Δ = 1 − (L₁+L₂) + L₁·L₂ (non-touching pair) = 1 − (−2−3) + (−2)(−3) = 1+5+6 = 12
Forward path touches both loops → Δ₁ = 1
T = P₁·Δ₁/Δ = 6·1/12 = 0.5
A system has forward path G(s) = 10/(s+1) and feedback path H(s) = 0.1. The DC gain of the closed-loop system is:
Closed-loop T(s) = G/(1+GH). At s=0 (DC): G(0) = 10/(0+1) = 10, H(0) = 0.1
T(0) = 10/(1+10×0.1) = 10/(1+1) = 10/2 = 5
For a block diagram with two blocks G₁ and G₂ in series inside a unity feedback loop, the closed-loop sensitivity S_G1^T at s=0, given G₁(0)=5, G₂(0)=2 is:
Forward gain G = G₁G₂. At s=0: G(0)=5×2=10. Unity feedback H=1.
Sensitivity S = 1/(1+GH) = 1/(1+10×1) = 1/11
A block diagram has: input → G₁ → summing point → G₂ → output; with feedback H from output to the summing point (negative). G₂'s output also feeds positively back to the summing point via G₃. The overall closed-loop TF is:
Summing junction: E = G₁R − G₂H·E + G₂G₃·E (positive feedback through G₃ adds, negative feedback through H subtracts).
G₂·E → C. So C = G₂·E, and E(1 + G₂H − G₂G₃) = G₁R → T = G₁G₂/(1+G₂H−G₂G₃).
Which of the following is NOT a feature of closed-loop control systems?
Closed-loop systems can become unstable if the controller gain is too high. High gain increases the loop gain GH, which can cause the characteristic equation 1+GH=0 to have roots in the RHP. Stability is a design challenge unique to closed-loop systems — not guaranteed for any gain value.
An SFG has: nodes x₁,x₂,x₃,x₄. Branches: x₁→x₂ (gain=1), x₂→x₃ (gain=2), x₃→x₄ (gain=3), x₃→x₂ (gain=−0.5). T = x₄/x₁ = ?
Forward path: P₁ = 1×2×3 = 6
Loop: L₁ = 2×(−0.5) = −1 (x₂→x₃→x₂)
Δ = 1−L₁ = 1−(−1) = 2. Path P₁ touches loop → Δ₁=1.
T = 6×1/2 = 3. [Note: if x₃→x₂ gain is −0.25, Δ=1.5, T=4.]
The transfer function of a system is T(s) = 1/(s+1)². The system has:
(s+1)² = s²+2s+1. Roots: s = −1 (double pole). Both in LHP → system is stable, but the step response will have no overshoot (critically damped). The repeated pole causes a t·e⁻ᵗ term in the impulse response.
A unity feedback system has G(s) = K/s². For what value of K does the closed-loop characteristic equation have roots on the jω axis?
Closed-loop: 1+G(s) = 0 → s²+K = 0 → s² = −K → s = ±j√K for any K>0.
The roots are always on the jω axis for K>0 → system is marginally stable for all positive K. No real damping → pure oscillation. This is why a double integrator plant (s²) is very difficult to stabilize.