Control systems basics

In short

A control system is a set of components arranged so that one quantity (the output — speed, position, temperature, voltage, level) follows a desired reference in spite of disturbances and parameter changes. The basics cover how a physical system is modelled as a differential equation, converted to a transfer function through the Laplace transform, simplified by block-diagram or signal-flow-graph reduction, and then judged by its time response to standard test inputs. This is the foundation on which stability analysis, PID tuning, state space and digital control are built, so it carries a large share of the marks in the Control Systems paper of Indian university syllabi and typically 8–12 marks in GATE EE/EC/IN. It is examined in placement tests of BHEL, NTPC, Powergrid, L&T, Siemens, ABB, Honeywell, Rockwell, Yokogawa and the ISRO/DRDO/BARC technical rounds, and used daily by instrumentation, drives and automation engineers.

Explained

Open loop, closed loop and basic terminology

• Plant / process: the equipment to be controlled. Controller: generates the actuating signal. Actuator: final control element (motor, valve, thyristor bridge). Sensor / feedback element: measures the output.
• Open loop (no feedback): output has no influence on the control action. Example: a domestic toaster, a stepper motor positioning without encoder, a water pump run on a timer. Simple, cheap, and stable provided the plant itself is stable — but inaccurate, and cannot correct disturbances or ageing.
• Closed loop (feedback): the measured output is compared with the reference and the error drives the controller. Accurate, rejects disturbances, reduces sensitivity to parameter change — at the cost of complexity, extra hardware and the risk of instability/oscillation.
• Reference input r(t), feedback signal b(t), with B(s) = H(s)C(s), error / actuating signal e(t) = r − b, controlled output c(t).
• Negative feedback subtracts the feedback signal (normal practice); positive feedback adds it and is used mainly in oscillators, latches and regenerative circuits.
• Servomechanism: a feedback control system whose controlled variable is mechanical position, velocity or acceleration (antenna positioner, CNC axis, robot joint).
• Regulator vs tracking system: a regulator holds a constant reference against disturbances (boiler drum level, grid frequency); a tracking/follow-up system makes the output follow a changing reference (radar tracker, CNC contour).
• Classifications: linear/non-linear, time-invariant/time-varying, continuous/discrete, SISO/MIMO, deterministic/stochastic, lumped/distributed.
• Linearity needs both superposition and homogeneity; real systems are linearised about an operating point by a Taylor expansion, keeping only the first-order term.

Mathematical modelling and transfer functions

• Electrical analogy: write KVL/KCL with v = L di/dt, i = C dv/dt, v = Ri. Mechanical (translational): f = M d²x/dt² + B dx/dt + Kx. Rotational: T = J d²θ/dt² + B dθ/dt + Kθ.
• Force–voltage (direct) analogy: M↔L, B↔R, K↔1/C, x↔q, f↔v. Force–current analogy: M↔C, B↔1/R, K↔1/L, f↔i. Used to convert a mechanical problem into a circuit and solve it by mesh/node methods.
• Transfer function: G(s) = L{output}/L{input} with all initial conditions zero. Defined only for linear time-invariant systems; it is the Laplace transform of the impulse response.
• Poles = roots of the denominator (characteristic polynomial) — they fix the nature of the transient. Zeros = roots of the numerator — they shape the magnitude of the response. Order = number of poles; type = number of poles at the origin.
• A transfer function is proper if degree of numerator ≤ degree of denominator; most physical systems are strictly proper.
• D.C. gain = G(0) for a type-0 system; it decides the final value of the step response.
• Limitation: transfer functions hide initial conditions and internal variables, and apply only to single-input single-output LTI systems — which is why state space exists.

Block diagram reduction and signal flow graphs

Reduction rules, applied in order:

1. Blocks in cascade multiply: G₁G₂. Blocks in parallel add: G₁ ± G₂.
2. Negative feedback loop: C/R = G/(1 + GH); positive feedback: G/(1 − GH). With H = 1 (unity feedback), C/R = G/(1 + G).
3. Move a summing point before a block: divide the incoming signal by that block. Move a take-off point after a block: divide the branch by the block (and vice versa — multiply).
4. Interchange adjacent summing points freely; never interchange a summing point with a take-off point.
5. For multi-input systems use superposition: set all other inputs to zero, find each output contribution, and add. This is how the disturbance transfer function C/D = G₂/(1 + G₁G₂H) is obtained.
6. Signal flow graph (SFG): nodes are variables, directed branches carry gains. Forward path: source to sink with no node repeated. Loop: closed path. Non-touching loops share no node.
7. Mason's gain formula: T = Σ Pₖ Δₖ / Δ, where Δ = 1 − (Σ all loop gains) + (Σ products of two non-touching loop gains) − (Σ products of three non-touching loops) + …, and Δₖ is Δ with all loops touching path k removed. Faster than block reduction for interconnected loops.

Standard test signals and time-response specifications

• Impulse δ(t) ↔ 1; step u(t) ↔ 1/s; ramp t ↔ 1/s²; parabola t²/2 ↔ 1/s³; sinusoid → frequency response. Each later signal is the integral of the previous one.
• Transient response dies out with time; steady-state response is what remains as t → ∞. Total response = transient + steady state.
• First-order system C/R = 1/(1 + sT): unit-step response c(t) = 1 − e^−t/T. Time constant T is the time to reach 63.2% of final value; 95% in 3T, 98.2% in 4T; initial slope = 1/T. No overshoot, never unstable for T > 0.
• Standard second-order system: C/R = ωₙ²/(s² + 2ζωₙs + ωₙ²). ζ = damping ratio, ωₙ = undamped natural frequency (rad/s).
• Damping cases: ζ = 0 undamped (sustained oscillation at ωₙ); 0 < ζ < 1 underdamped (damped oscillation at ω_d = ωₙ√(1−ζ²)); ζ = 1 critically damped (fastest response without overshoot); ζ > 1 overdamped (sluggish, no overshoot).
• Specifications for a unit step, underdamped case: delay time t_d (to 50%), rise time t_r (10%→90%, or 0→100% for second order), peak time t_p, peak overshoot M_p, settling time t_s (to stay within ±2% or ±5% band), steady-state error e_ss.
• Design trade-off: increasing ζ reduces overshoot but increases rise time; increasing ωₙ speeds up the response for the same overshoot. Practical servo designs use ζ ≈ 0.4–0.8, commonly 0.707 (overshoot ≈ 4.3%, flat closed-loop frequency response).

Steady-state error, system type and error constants

• For a unity-feedback system, E(s) = R(s)/[1 + G(s)], and by the final value theorem e_ss = lim(s→0) sR(s)/[1 + G(s)].
• Static error constants: position K_p = lim(s→0) G(s); velocity K_v = lim(s→0) sG(s); acceleration K_a = lim(s→0) s²G(s).
• Type of a system = number of open-loop poles at s = 0. Each integrator removes the steady-state error to one more class of input but reduces the phase margin (worsens stability and increases overshoot).

• Error constants are defined only for unity feedback; for non-unity H, either reduce to an equivalent unity-feedback form or compute e_ss directly from the error expression.
• Generalised error coefficients (dynamic error series) are used when the input is arbitrary rather than a standard test signal.

Effects of feedback, and controller/compensator basics

• Feedback reduces gain by (1 + GH), and reduces sensitivity to plant variation by the same factor: S = (∂T/T)/(∂G/G) = 1/(1 + GH).
• It increases bandwidth, reduces the effect of disturbance and noise entering after the controller, and changes the pole locations — hence it can stabilise or destabilise.
• Proportional (P) action raises loop gain, reduces e_ss, increases overshoot. Integral (I) action raises the type by one and removes offset but adds −90° phase (slower, more oscillatory). Derivative (D) action adds damping and predicts error trend, but amplifies measurement noise; it is never used alone since it gives no output for a constant error. (Detailed tuning belongs to the PID topic.)
• Lead compensator (zero nearer origin than the pole) improves transient response and phase margin; lag compensator improves steady-state accuracy; lag-lead does both.
• Non-linearities to remember: saturation of the amplifier/actuator, dead zone, backlash in gears, static friction (stiction), hysteresis, and integral windup when the actuator saturates.
• Common devices: potentiometer and synchro (position error detector), tachogenerator (velocity feedback), armature-controlled and field-controlled DC servomotor, AC servomotor (high rotor resistance for linear, negative-slope torque–speed curve), stepper motor, gear train (inertia reflected to the motor shaft J_eq = J + n²J_L with n = N₁/N₂), amplidyne.
• Indian practice and standards: instrumentation and control signals follow the 4–20 mA / 1–5 V convention with HART; P&ID symbols and tag letters follow ISA-5.1, adopted in India through IS/ISO documents and used in every DCS/PLC specification. Functional safety of control loops is governed by IEC 61508 and IEC 61511 (SIL levels), issued in India as IS/IEC equivalents; grid-side controls (governor droop, AGC) follow the CEA Grid Standards and the CERC Indian Electricity Grid Code — exact clause numbers are revised periodically, so verify against the current notification.

Key points

• Open loop - no feedback; cheap and stable if the plant is, but inaccurate and cannot reject disturbances.
• Closed loop - error-driven; accurate and disturbance-tolerant but can become unstable.
• Transfer function - ratio of Laplace transforms of output to input with zero initial conditions; valid only for LTI SISO systems.
• Poles - decide the form of the transient; zeros - shape its magnitude; order = number of poles.
• Characteristic equation - 1 + G(s)H(s) = 0; its roots are the closed-loop poles.
• Mason's formula - T = ΣPₖΔₖ/Δ; the fastest route through a multi-loop signal flow graph.
• Force–voltage analogy - M↔L, B↔R, K↔1/C; converts a mechanical model into an equivalent circuit.
• Time constant T - 63.2% of the final value in one T, ≈98% in 4T; initial slope 1/T.
• ζ (damping ratio) - fixes the percentage overshoot; ωₙ - scales the speed of response.
• Damped frequency - ω_d = ωₙ√(1 − ζ²), always less than ωₙ for an underdamped system.
• Peak overshoot - M_p = exp(−πζ/√(1−ζ²)); 0% at ζ = 1, 100% at ζ = 0, ≈4.3% at ζ = 0.707.
• Settling time - t_s ≈ 4/(ζωₙ) for the 2% band, 3/(ζωₙ) for 5%; depends on the product ζωₙ, i.e. on the real part of the pole.
• System type - number of open-loop poles at the origin; each integrator kills the error to one further input class.
• Error constants - K_p, K_v, K_a; defined only for unity feedback.
• Sensitivity - 1/(1 + GH); feedback trades gain for accuracy, bandwidth and robustness.
• Dominant poles - the pair closest to the jω axis; a higher-order system is approximated by them when other poles are ≥5 times farther left.
• Critically damped - ζ = 1 gives the quickest response with no overshoot; practical servos use ζ ≈ 0.4–0.8.
• Derivative action - adds damping, amplifies noise, gives nothing for a constant error; never used alone.

Formulas and examples

Example 1 — Second-order specifications. A unity-feedback system has G(s) = 100/[s(s + 8)]. Find ζ, ωₙ, M_p, t_p and t_s (2%).

• Closed loop: C/R = 100/(s² + 8s + 100). Comparing: ωₙ² = 100 → ωₙ = 10 rad/s; 2ζωₙ = 8 → ζ = 0.4.
• ω_d = 10√(1 − 0.16) = 10 × 0.9165 = 9.165 rad/s.
• t_p = π/9.165 = 0.343 s.
• M_p = exp(−π × 0.4/0.9165) = exp(−1.3711) = 0.2538 → 25.4% overshoot.
• t_s = 4/(0.4 × 10) = 1.0 s; φ = cos⁻¹0.4 = 1.1593 rad → t_r ≈ (3.1416 − 1.1593)/9.165 = 0.216 s.

Example 2 — Steady-state error and system type. G(s) = 20(s + 2)/[s(s + 1)(s + 5)], unity feedback. Input r(t) = 4 + 6t.

• Type 1 (one pole at origin) → step error = 0.
• K_v = lim(s→0) s·G(s) = 20 × 2/(1 × 5) = 8 s⁻¹.
• Error to the ramp component 6t = 6/K_v = 6/8 = 0.75. Total e_ss = 0 + 0.75 = 0.75 units.
• To a parabolic input the error would be infinite, since K_a = 0 for a type-1 system.

Example 3 — Designing for a given overshoot. A servo must have ≤ 10% overshoot and settling time (2%) ≤ 0.5 s. Find the required ζ and ωₙ.

• M_p = 0.10 → πζ/√(1−ζ²) = ln10 = 2.3026 → ζ/√(1−ζ²) = 0.7329 → ζ² = 0.5371/1.5371 = 0.3494 → ζ = 0.591 (take 0.6).
• t_s = 4/(ζωₙ) ≤ 0.5 → ζωₙ ≥ 8 → ωₙ ≥ 8/0.6 = 13.3 rad/s. Choose ωₙ = 14 rad/s, giving the target C/R = 196/(s² + 16.8s + 196).

Example 4 — First-order thermal system. An oven behaves as 1/(1 + 120s) °C per unit input, step input giving a final rise of 250 °C from ambient.

• At t = 120 s: rise = 250(1 − e⁻¹) = 250 × 0.6321 = 158.0 °C.
• At t = 300 s: 250(1 − e^−2.5) = 250 × 0.9179 = 229.5 °C.
• Time to reach 200 °C: 200/250 = 0.8 → e^−t/120 = 0.2 → t = 120 × ln5 = 120 × 1.6094 = 193.1 s.

Example 5 — Block diagram and Mason's formula. A signal flow graph has one forward path P₁ = G₁G₂G₃ and two non-touching loops L₁ = −G₁H₁ and L₂ = −G₃H₂.

• Δ = 1 − (L₁ + L₂) + L₁L₂ = 1 + G₁H₁ + G₃H₂ + G₁G₃H₁H₂.
• Both loops touch the forward path → Δ₁ = 1.
• T = G₁G₂G₃/(1 + G₁H₁ + G₃H₂ + G₁G₃H₁H₂). With G₁ = 5, G₂ = 2, G₃ = 4, H₁ = 0.1, H₂ = 0.25: numerator = 40; denominator = 1 + 0.5 + 1 + 0.5 = 3 → T = 13.33.

Where it is used

• Power plants and grid operation: boiler drum level and combustion control in NTPC/state-sector stations, turbine governor droop control, AVR and excitation loops, automatic generation control at Load Despatch Centres.
• Industrial drives and automation: VFD speed/torque loops (ABB, Siemens, Danfoss, Delta), cascade current-speed-position loops in CNC and robotics at Bharat Fritz Werner, HMT and Addverb-type integrators.
• Process industry: flow, pressure, temperature and level loops in refineries (IOCL, Reliance, BPCL), fertiliser and cement plants, run on DCS from Honeywell, Yokogawa and Emerson — design and commissioning roles for instrumentation engineers.
• Aerospace and defence: ISRO launch-vehicle attitude control and gimballed engine actuators, DRDO missile autopilots and radar antenna servos, HAL flight-control actuation.
• Consumer and mobility products: inverter air-conditioners and refrigerators, washing-machine drum speed, EV motor control and battery thermal management at Tata Motors, Mahindra, Ola and Ather.
• Job roles: control systems engineer, instrumentation and C&I engineer, drives application engineer, automation/PLC-DCS engineer, model-based design engineer using MATLAB/Simulink or Scilab/Xcos.

Interview questions

• Q: Why is negative feedback preferred over positive feedback in control systems? A: Negative feedback reduces error, lowers sensitivity to parameter changes and attenuates disturbances; positive feedback increases gain and sensitivity and usually drives the system to instability.
• Q: What assumptions underlie a transfer function? A: The system must be linear and time-invariant, single-input single-output, and all initial conditions are taken as zero.
• Q: What does the damping ratio physically tell you? A: How oscillatory the response is — it fixes the percentage overshoot and the number of oscillations, independent of how fast the system is, which is set by ωₙ.
• Q: Two systems have the same ζ but different ωₙ. How do their step responses differ? A: Both overshoot by the same percentage, but the one with the larger ωₙ rises and settles proportionally faster.
• Q: Why does adding an integrator remove steady-state error but worsen stability? A: It raises the system type so the error to that input class becomes zero, but it also adds a constant −90° phase lag, cutting the phase margin and increasing overshoot.
• Q: What is the difference between type and order of a system? A: Order is the total number of poles (degree of the characteristic polynomial); type is only the number of poles at the origin.
• Q: Can a zero make a response undershoot? A: Yes — a zero in the right half plane makes the system non-minimum-phase, and the step response initially moves the wrong way.
• Q: When is Mason's gain formula preferred to block diagram reduction? A: When loops are many and interlinked, since it gives the overall gain in a single calculation without redrawing the diagram.
• Q: What is the use of a tachogenerator in a position servo? A: It provides velocity feedback, which acts like derivative action on position — it adds damping and reduces overshoot without the noise of differentiating the position signal.
• Q: Why is derivative control never used alone? A: Its output depends only on the rate of change of error, so a constant error produces no corrective action and the loop cannot hold the setpoint.
• Q: What is a dominant pole, and why does it matter? A: The pole (or pole pair) nearest the imaginary axis; it decays slowest and so governs the transient, letting a higher-order system be approximated by a second-order model.
• Q: Give one open-loop and one closed-loop example from the same application. A: A fan regulator on a fixed voltage tap is open loop; an inverter air-conditioner that measures room temperature and modulates compressor speed is closed loop.

Common mistakes

• Writing the closed-loop transfer function as G/(1 + G) when H ≠ 1, instead of G/(1 + GH).
• Computing K_p, K_v, K_a or e_ss for a non-unity-feedback system without first converting to an equivalent unity-feedback form.
• Confusing ωₙ with the actual oscillation frequency ω_d, or quoting rad/s figures as Hz (divide by 2π).
• Treating the number of poles at the origin (type) as the order of the system.
• Forgetting to set initial conditions to zero, or to subtract the operating point when linearising a non-linear plant.
• Mishandling the take-off/summing point shift rule — multiplying where division is required when a block is moved across.
• Assuming a higher controller gain always improves performance; beyond a limit it raises overshoot and drives the loop unstable, and actuator saturation invalidates the linear model.

Related topics

Stability: Routh, root locus and Bode · PID controllers and tuning · State space analysis · Digital control systems · Transducers and sensors basics · Electrical machines and drives.