Control Systems
How machines keep themselves on target — from a room thermostat to a solar inverter's MPPT. Transfer functions, time response, stability and PID control, with worked examples, a step-response calculator and a quiz. Useful for B.Tech and GATE preparation.
Open-loop and closed-loop systems
A control system makes an output follow a desired input (the reference or set point).
Open loop
- No feedback; the output is not measured
- Simple and cheap, but cannot correct disturbances
- Examples: a timer-based washing machine, a toaster, traffic lights on a fixed cycle
Closed loop (feedback)
- Measures the output and corrects the error
- Accurate, rejects disturbances, less sensitive to parameter changes
- Can become unstable if designed badly
- Examples: AC thermostat, cruise control, solar tracker, inverter voltage control
Transfer function, poles and zeros
Definition
The transfer function is the Laplace transform of the output divided by the Laplace transform of the input, with all initial conditions zero.
G(s) = C(s) ÷ R(s)Poles and zeros
Zeros are values of s that make G(s) = 0 (roots of the numerator). Poles make G(s) infinite (roots of the denominator). The poles decide stability and the shape of the response.
Example: G(s) = (s + 2) ÷ (s² + 4s + 3) has a zero at −2 and poles at −1 and −3.
Order and type
The order is the highest power of s in the denominator. The type is the number of poles at the origin (pure integrators, 1/s) in the open-loop G(s)H(s).
Common Laplace pairs
unit step → 1/s ramp t → 1/s² e−at → 1/(s + a) sin ωt → ω/(s² + ω²)
Block diagram reduction
| Connection | Equivalent |
|---|---|
| Series (cascade) | G₁ · G₂ |
| Parallel | G₁ ± G₂ |
| Negative feedback | G ÷ (1 + G·H) |
| Positive feedback | G ÷ (1 − G·H) |
| Unity negative feedback | G ÷ (1 + G) |
The characteristic equation is 1 + G(s)H(s) = 0; its roots are the closed-loop poles. For complex diagrams, Mason's gain formula gives the overall gain from a signal flow graph.
Time response of first- and second-order systems
First order
Reaches 63.2% of the final value in one time constant τ and settles (within 2%) in about 4τ. No overshoot.
Second order
ωn = natural frequency, ζ = damping ratio. ζ decides the shape of the response.
| Damping ratio ζ | Type | Step response |
|---|---|---|
| ζ = 0 | Undamped | Oscillates forever |
| 0 < ζ < 1 | Underdamped | Overshoots, then settles with decaying oscillation |
| ζ = 1 | Critically damped | Fastest response with no overshoot |
| ζ > 1 | Overdamped | Slow, no overshoot |
Underdamped specifications
| Specification | Formula |
|---|---|
| Damped frequency | ωd = ωn√(1 − ζ²) |
| Rise time (0–100%) | tr = (π − θ) ÷ ωd, where θ = cos⁻¹ζ |
| Peak time | tp = π ÷ ωd |
| Peak overshoot | Mp = e−ζπ/√(1−ζ²) × 100% |
| Settling time | ts ≈ 4 ÷ (ζωn) (2% band), 3 ÷ (ζωn) (5% band) |
Worked example
Unity feedback with G(s) = 25 ÷ (s(s + 6)). Closed loop: 25 ÷ (s² + 6s + 25), so ωn = 5 rad/s and 2ζωn = 6, giving ζ = 0.6.
ωd = 5 × 0.8 = 4 rad/s · Mp = e−0.6π/0.8 ≈ 9.5% · tp = π ÷ 4 ≈ 0.785 s · tr = (π − 0.927) ÷ 4 ≈ 0.554 s · ts (2%) = 4 ÷ 3 ≈ 1.33 s
Steady-state error and error constants
The error left after the transient dies out. It depends on the input type and the system type (number of integrators).
| Input | Error constant | Type 0 | Type 1 | Type 2 |
|---|---|---|---|---|
| Step | Kp = lims→0 GH | 1 ÷ (1 + Kp) | 0 | 0 |
| Ramp | Kv = lims→0 s·GH | ∞ | 1 ÷ Kv | 0 |
| Parabolic | Ka = lims→0 s²·GH | ∞ | ∞ | 1 ÷ Ka |
Example: G(s) = 25 ÷ (s(s + 6)) is type 1, so the step error is 0. Kv = 25 ÷ 6 ≈ 4.17, so the ramp error is 1 ÷ 4.17 ≈ 0.24.
Stability and the Routh–Hurwitz criterion
What stability means
A system is stable if every bounded input gives a bounded output. For a linear system this means all closed-loop poles lie in the left half of the s-plane (negative real parts). Poles on the imaginary axis give sustained oscillation (marginally stable).
Routh–Hurwitz test
Build the Routh array from the coefficients of the characteristic equation. The number of sign changes in the first column equals the number of poles in the right half-plane. No sign changes means stable.
Example: s³ + 6s² + 11s + 6 = 0
s³ | 1 | 11 |
s² | 6 | 6 |
s¹ | (6×11 − 1×6) ÷ 6 = 10 | 0 |
s⁰ | 6 |
First column 1, 6, 10, 6: all positive, no sign changes, so the system is stable. (The roots are −1, −2 and −3.)
Root locus shows how the closed-loop poles move as the gain K increases from 0 to ∞. The gain at which the locus crosses the imaginary axis is the limit of stability.
Frequency response, Bode plots and margins
Bode plot
Two graphs against log frequency: gain in dB (20 log|G(jω)|) and phase in degrees. Each pole adds a −20 dB/decade slope and up to −90° of phase; each zero adds +20 dB/decade and up to +90°.
Gain margin (GM)
How much the gain can rise before instability, measured at the phase crossover frequency where the phase is −180°.
GM (dB) = −20 log |G(jωpc)H(jωpc)|Phase margin (PM)
How much extra phase lag can be tolerated, measured at the gain crossover frequency where the gain is 0 dB.
PM = 180° + ∠G(jωgc)H(jωgc)Rule of thumb
A closed-loop system is stable when both GM and PM are positive. Good designs aim for GM ≥ 6 dB and PM of about 45–60°. Nyquist plots give the same information for more complex cases.
PID controllers
The most widely used controller in industry. It combines three actions on the error e(t).
| Action | Effect | Drawback |
|---|---|---|
| Proportional (P) | Faster response, smaller error | Leaves a steady-state error; too much causes overshoot |
| Integral (I) | Removes steady-state error | Slower, more overshoot; can cause "windup" |
| Derivative (D) | Reduces overshoot, adds damping | Amplifies sensor noise |
Tuning in practice
Start with P only and raise Kp until the response is quick with modest overshoot. Add I to remove the remaining error, then a little D if the overshoot is too large. The Ziegler–Nichols method gives starting values from the gain and period at which the loop just oscillates.
Control systems inside a solar plant
MPPT
The inverter continuously adjusts the panel voltage to stay at the maximum power point as sunlight and temperature change. Perturb-and-observe is a simple feedback loop.
Grid synchronisation (PLL)
A phase-locked loop tracks the grid's voltage, frequency and phase so the inverter can inject current in step with the grid.
Current and voltage control
Inner PI loops regulate the inverter output current; outer loops control DC-link voltage and power export limits.
Battery and sun tracking
Charge controllers regulate battery charging stages. Single-axis trackers use closed-loop position control to follow the sun.
Second-order step response calculator
Change ωn and ζ to see the specifications and the response curve update instantly.
Quick formula sheet
Revise before your exam. Use "Print / save notes as PDF" at the top to keep a copy.
| Topic | Formula / rule |
|---|---|
| Transfer function | G(s) = C(s) ÷ R(s), zero initial conditions |
| Negative feedback | T(s) = G ÷ (1 + GH) |
| Characteristic equation | 1 + G(s)H(s) = 0 |
| First order | c(t) = 1 − e−t/τ, ts ≈ 4τ |
| Second order | ωn² ÷ (s² + 2ζωns + ωn²) |
| Damped frequency | ωd = ωn√(1 − ζ²) |
| Peak overshoot | Mp = e−ζπ/√(1−ζ²) |
| Peak / settling time | tp = π/ωd, ts ≈ 4/(ζωn) |
| Error constants | Kp = lim GH, Kv = lim sGH, Ka = lim s²GH (s → 0) |
| Steady-state error | step 1/(1 + Kp), ramp 1/Kv, parabola 1/Ka |
| Stability | All closed-loop poles in the left half-plane; no sign changes in the Routh first column |
| Margins | GM at −180° phase, PM at 0 dB gain; aim GM ≥ 6 dB, PM 45–60° |
| PID | C(s) = Kp + Ki/s + Kds |
Control systems topics checklist
Control systems quiz
1. Which of these is a closed-loop system?
2. The closed-loop transfer function of G with negative feedback H is…
3. A system has s² + 4s + 16 as its denominator. What is ζ?
4. A damping ratio of ζ = 1 gives a response that is…
5. A type 1 system with a step input has a steady-state error of…
6. In a Routh array, the first column is 1, 3, −2, 5. How many poles are in the right half-plane?
7. Which PID action removes steady-state error?
8. Phase margin is measured at the frequency where…