Control systems

What to remember

• Transfer function G(s) = Y(s)/R(s) is the Laplace-transform ratio of output to input with zero initial conditions; the poles of the closed loop decide stability.
• A system is stable only if all roots of the characteristic equation lie in the left half of the s-plane; Routh's test checks this without solving for the roots.
• Gain margin and phase margin measure relative stability; a higher phase margin means a better damped response.

Basics and modelling

A control system keeps an output at a desired value. An open-loop system has no feedback (a washing timer, a traffic light). A closed-loop system feeds back the output to compare with the reference (a thermostat, a speed governor). Feedback reduces sensitivity to parameter change and disturbance, but can cause instability and reduces overall gain.

Closed-loop transfer function with negative feedback: T(s) = G(s)/(1 + G(s)H(s)). The characteristic equation is 1 + G(s)H(s) = 0. Poles are the roots of the denominator; zeros are the roots of the numerator.

Electrical-mechanical analogies: Force-voltage analogy maps mass to inductance, damper to resistance, spring to the inverse of capacitance. Force-current analogy maps mass to capacitance, damper to conductance, spring to the inverse of inductance. DC servomotors and AC servomotors are common actuators; a tachogenerator measures speed; a potentiometer or synchro measures position; a synchro pair acts as an error detector for angular position.

Block diagram rules: blocks in cascade multiply; blocks in parallel add; a feedback loop has gain G/(1 ± GH). Mason's gain formula for signal flow graphs: T = (Σ Pk Δk)/Δ, where Pk is a forward-path gain, Δ is 1 minus the sum of loop gains plus the sum of products of non-touching loop pairs, and so on, and Δk is Δ with loops touching path k removed.

Time response and error

Standard second-order system: G(s) = ωn²/(s² + 2ζωn s + ωn²). Here ζ is the damping ratio and ωn is the natural frequency.

Time-domain specifications for an underdamped system: damped frequency ωd = ωn√(1 − ζ²); rise time, peak time tp = π/ωd; percentage overshoot Mp = 100 e^(−πζ/√(1 − ζ²)); settling time ts ≈ 4/(ζωn) for 2 % tolerance (3/(ζωn) for 5 %). Larger ζ means smaller overshoot. Overshoot depends only on ζ.

First-order system: G(s) = 1/(Ts + 1). Time constant T; the step response reaches 63.2 % of the final value at t = T and settles (2 %) at about 4T.

Steady-state error and system type: the type is the number of poles at the origin in the open-loop transfer function.

Error constants: Kp = lim G(s)H(s) as s→0; Kv = lim s·G(s)H(s); Ka = lim s²·G(s)H(s).

Stability: Routh and root locus

For the characteristic polynomial, all coefficients must be positive and present (necessary condition). In the Routh array, the number of sign changes in the first column equals the number of right-half-plane roots. A zero in the first column is replaced by a small ε. A row of all zeros indicates roots symmetric about the origin (auxiliary equation is used). A system with simple poles on the imaginary axis is marginally stable.

Example: s³ + 6s² + 11s + 6 = 0. Roots −1, −2, −3, stable. For s³ + 2s² + 3s + K, the condition for stability is 0 < K < 6 (from 2 × 3 > K).

Root locus plots the closed-loop poles as gain K varies from 0 to ∞. Rules: the locus starts at open-loop poles (K = 0) and ends at open-loop zeros (or at infinity). Number of branches = number of poles (if poles ≥ zeros). Number of asymptotes = P − Z. Centroid = (Σ poles − Σ zeros)/(P − Z). A point on the real axis is on the locus if the number of poles and zeros to its right is odd. The locus is symmetric about the real axis. Adding a pole pushes the locus to the right (less stable); adding a zero pulls it left (more stable).

Frequency response

Bode plot: magnitude in dB versus log frequency, and phase versus log frequency. A simple pole at ω = 1/T gives −20 dB/decade slope after the corner frequency; a simple zero gives +20 dB/decade; a pole at the origin gives −20 dB/decade through the whole range. For each pole, the phase falls by 90° overall (−45° at the corner).

Gain crossover frequency: where |GH| = 1 (0 dB). Phase crossover frequency: where phase = −180°. Gain margin = 1/|GH| at phase crossover (in dB, −20 log|GH|). Phase margin = 180° + phase at gain crossover. A stable system has both margins positive. Bandwidth is the frequency where the closed-loop gain falls 3 dB below its low-frequency value; a larger bandwidth gives a faster response.

Nyquist criterion: Z = N + P, where Z is the number of closed-loop poles in the right half-plane, P is the number of open-loop right-half-plane poles, and N is the number of clockwise encirclements of −1 + j0. For stability, Z = 0. Resonant peak Mr = 1/(2ζ√(1 − ζ²)) for ζ < 0.707.

Compensation and controllers

A lead compensator adds phase lead, improves the transient response and increases bandwidth. A lag compensator improves steady-state accuracy but reduces bandwidth. A lag-lead compensator combines both. A PID controller: P gives speed, I removes steady-state error, D improves damping and reduces overshoot. Controller form: Gc(s) = Kp + Ki/s + Kd s.

State-space model: x' = Ax + Bu, y = Cx + Du. The eigenvalues of A are the poles. The transfer function is C(sI − A)⁻¹B + D. A system is controllable if the controllability matrix [B AB A²B…] has full rank; it is observable if the observability matrix has full rank.

Sensors, actuators and common systems

A synchro transmitter and control transformer together form an error detector for shaft position. A servo system follows a changing reference (position control of a radar antenna); a regulator holds a constant output despite disturbance (a voltage regulator); a process control system regulates variables such as temperature, level or flow, often using a PID controller. An AC servomotor is a two-phase induction motor with high-resistance rotor, giving a falling torque-speed curve and good damping. A DC servomotor may be armature-controlled or field-controlled; armature control is more common because the response is faster. The transfer function of an armature-controlled DC motor is a second-order lag, often simplified to a first-order lag when armature inductance is small. A stepper motor moves by fixed angular steps for each input pulse and is used in open-loop positioning. Hydraulic and pneumatic actuators give high force for large loads. Dead-zone, backlash and saturation are common non-linearities; describing-function analysis is used to study limit cycles in non-linear systems.

Sampled-data and discrete systems

Sampled-data systems use the Z-transform. The sampling theorem requires the sampling frequency to be at least twice the highest signal frequency. A discrete system is stable if all poles of its transfer function lie inside the unit circle |z| = 1 in the z-plane. A zero-order hold keeps each sample constant until the next sample. Digital controllers are flexible and cheap but add delay, which reduces phase margin.

Worked examples

• 1. For G(s) = 25/(s² + 6s + 25): ωn = 5, 2ζωn = 6, so ζ = 0.6. Mp = 100 e^(−π×0.6/0.8) ≈ 9.5 %. ts = 4/(0.6 × 5) = 1.33 s.
• 2. Unity feedback with G(s) = 10/(s(s + 2)): type 1, Kv = 10/2 = 5, ramp error = 1/5 = 0.2.
• 3. Closed-loop gain of G = 10 with H = 0.1: T = 10/(1 + 1) = 5.

Exam traps

• Percentage overshoot depends on ζ only, not on ωn.
• Type of system counts poles at the origin of the open loop, not the closed loop.
• Phase margin is measured at gain crossover; gain margin at phase crossover.
• Lead compensation improves transient response; lag improves steady-state accuracy.
• Left-half-plane poles are stable; poles on the jω axis are marginally stable.
• Positive feedback is not the usual control configuration; the denominator is 1 − GH.
• Feedback reduces gain by the factor (1 + GH).
• Force-voltage analogy maps mass to inductance; force-current maps mass to capacitance.

One-liners

• 1. Characteristic equation: 1 + GH = 0.
• 2. ζ = 1 is critical damping.
• 3. Settling time (2 %) is about 4/(ζωn).
• 4. First-order step response is 63.2 % at one time constant.
• 5. A type-1 system has zero steady-state error for a step input.
• 6. Routh: sign changes in first column equal right-half-plane roots.
• 7. Root locus starts at poles and ends at zeros.
• 8. A simple pole gives −20 dB/decade.
• 9. Stable system has positive gain and phase margins.
• 10. Lead network adds positive phase.
• 11. Integral action removes steady-state error.
• 12. Tachogenerator measures speed.

Practice questions

1. A system that has no feedback from output to input is called

   1. open-loop system
   2. servo system
   3. closed-loop system
   4. regulator system
   Answer

   A. open-loop system

   Open-loop systems do not use output information.

2. The characteristic equation of a negative-feedback system is

   1. 1 − G(s)H(s) = 0
   2. G(s)H(s) = 0
   3. G(s) + H(s) = 0
   4. 1 + G(s)H(s) = 0
   Answer

   D. 1 + G(s)H(s) = 0

   Poles of the closed loop are roots of 1 + GH = 0.

3. A closed-loop system with forward gain G = 20 and feedback H = 0.2 has closed-loop gain

   1. 20
   2. 4
   3. 5
   4. 100
   Answer

   B. 4

   20/(1 + 4) = 4.

4. Negative feedback in a control system generally

   1. reduces sensitivity to parameter variation
   2. removes all poles
   3. increases overall gain
   4. always improves stability
   Answer

   A. reduces sensitivity to parameter variation

   Feedback makes the system less sensitive but reduces gain.

5. In the force-voltage analogy, mass corresponds to

   1. resistance
   2. capacitance
   3. conductance
   4. inductance
   Answer

   D. inductance

   Mass ↔ L, damper ↔ R, spring ↔ 1/C.

6. In the force-current analogy, mass corresponds to

   1. resistance
   2. capacitance
   3. inductance
   4. conductance
   Answer

   B. capacitance

   Mass ↔ C, damper ↔ 1/R, spring ↔ 1/L.

7. A second-order system with damping ratio ζ = 1 is

   1. critically damped
   2. overdamped
   3. underdamped
   4. undamped
   Answer

   A. critically damped

   ζ = 1 gives the fastest response without overshoot.

8. For G(s) = 25/(s² + 6s + 25), the damping ratio is

   1. 0.8
   2. 1.2
   3. 0.6
   4. 0.5
   Answer

   C. 0.6

   ωn = 5; 2ζωn = 6, so ζ = 0.6.

9. The percentage overshoot of an underdamped second-order system depends on

   1. natural frequency only
   2. both ζ and ωn equally
   3. damping ratio only
   4. the input amplitude
   Answer

   C. damping ratio only

   Mp = 100 e^(−πζ/√(1−ζ²)).

10. The 2 % settling time of a second-order system with ζ = 0.5 and ωn = 4 rad/s is

    1. 1 s
    2. 0.5 s
    3. 4 s
    4. 2 s
    Answer

    D. 2 s

    ts = 4/(ζωn) = 4/2 = 2 s.

11. The undamped natural frequency of s² + 4s + 16 = 0 is

    1. 8 rad/s
    2. 16 rad/s
    3. 2 rad/s
    4. 4 rad/s
    Answer

    D. 4 rad/s

    ωn² = 16, so ωn = 4.

12. A first-order system with time constant T = 2 s has a 2 % settling time of about

    1. 16 s
    2. 8 s
    3. 2 s
    4. 4 s
    Answer

    B. 8 s

    ts ≈ 4T = 8 s.

13. At t = T, a first-order step response reaches about

    1. 63.2 % of final value
    2. 86.5 % of final value
    3. 95 % of final value
    4. 36.8 % of final value
    Answer

    A. 63.2 % of final value

    1 − e⁻¹ = 0.632.

14. The type of a system is decided by the number of

    1. closed-loop poles in the right half plane
    2. complex poles
    3. poles at the origin in the open-loop transfer function
    4. zeros at the origin
    Answer

    C. poles at the origin in the open-loop transfer function

    Type = number of integrators (poles at s = 0).

15. A type-1 system has a steady-state error for a unit step input equal to

    1. 1/Kv
    2. zero
    3. infinity
    4. 1/(1 + Kp) with Kp finite
    Answer

    B. zero

    Type 1 has Kp = ∞, so ess = 0.

16. A unity-feedback system with G(s) = 10/(s(s + 2)) has a steady-state error for unit ramp input of

    1. 5
    2. 0
    3. 0.5
    4. 0.2
    Answer

    D. 0.2

    Kv = 10/2 = 5; ess = 1/Kv = 0.2.

17. The steady-state error of a type-0 system to a unit ramp input is

    1. infinite
    2. 1/Kv
    3. 1/(1 + Kp)
    4. zero
    Answer

    A. infinite

    Type 0 cannot follow a ramp.

18. In a Routh array, the number of sign changes in the first column gives the number of

    1. roots in the left half of the s-plane
    2. zeros of the system
    3. roots on the imaginary axis
    4. roots in the right half of the s-plane
    Answer

    D. roots in the right half of the s-plane

    Standard rule.

19. For s³ + 2s² + 3s + K = 0 to be stable, K must satisfy

    1. K < 0
    2. K > 6
    3. 0 < K < 6
    4. 0 < K < 2
    Answer

    C. 0 < K < 6

    Stability requires 2 × 3 > K and K > 0.

20. A system with simple poles on the imaginary axis and none in the right half plane is

    1. marginally stable
    2. unstable
    3. overdamped
    4. asymptotically stable
    Answer

    A. marginally stable

    Sustained oscillation: marginal stability.

21. A root locus branch begins at

    1. open-loop zeros
    2. origin always
    3. open-loop poles
    4. closed-loop zeros only
    Answer

    C. open-loop poles

    K = 0 gives closed-loop poles equal to open-loop poles.

22. The number of asymptotes of a root locus with 4 open-loop poles and 1 zero is

    1. 3
    2. 4
    3. 5
    4. 1
    Answer

    A. 3

    P − Z = 3.

23. Adding an open-loop pole to the transfer function tends to

    1. not change stability
    2. push the root locus toward the right half plane
    3. pull the root locus to the left
    4. make the system type lower
    Answer

    B. push the root locus toward the right half plane

    Poles reduce relative stability; zeros improve it.

24. The slope of the Bode magnitude plot of a simple pole beyond the corner frequency is

    1. 0 dB/decade
    2. −20 dB/decade
    3. +20 dB/decade
    4. −40 dB/decade
    Answer

    B. −20 dB/decade

    A first-order lag falls at 20 dB/decade.

25. At the corner frequency of a simple pole, the phase of 1/(1 + jωT) is

    1. −180°
    2. 0°
    3. −90°
    4. −45°
    Answer

    D. −45°

    tan⁻¹(1) = 45°.

26. Phase margin is measured at the

    1. resonant frequency
    2. gain crossover frequency
    3. phase crossover frequency
    4. corner frequency
    Answer

    B. gain crossover frequency

    Phase margin = 180° + phase at |GH| = 1.

27. A system has phase of −180° at the frequency where |GH| = 0.5. Its gain margin is

    1. −6 dB
    2. 3 dB
    3. 20 dB
    4. 6 dB
    Answer

    D. 6 dB

    GM = −20 log 0.5 ≈ 6 dB.

28. A lead compensator mainly

    1. improves steady-state accuracy only
    2. reduces bandwidth
    3. removes the integrator
    4. improves transient response by adding phase lead
    Answer

    D. improves transient response by adding phase lead

    Lead gives phase advance and more bandwidth.

29. Which compensator improves steady-state accuracy but reduces bandwidth?

    1. Lead compensator
    2. Lag compensator
    3. Derivative controller
    4. Feed-forward path only
    Answer

    B. Lag compensator

    Lag boosts low-frequency gain.

30. In a PID controller, the integral term mainly

    1. reduces steady-state error
    2. increases overshoot only
    3. adds phase lead
    4. speeds up noise rejection
    Answer

    A. reduces steady-state error

    Integral action drives the error to zero.

31. The derivative action in a PID controller mainly

    1. increases type of the system
    2. reduces bandwidth
    3. improves damping and reduces overshoot
    4. removes steady-state error
    Answer

    C. improves damping and reduces overshoot

    Derivative gives an anticipatory action.

32. In state-space form x' = Ax + Bu, the poles of the system are the

    1. eigenvalues of B
    2. rank of C
    3. eigenvalues of A
    4. elements of D
    Answer

    C. eigenvalues of A

    Poles are roots of det(sI − A) = 0.

33. A tachogenerator is used to measure

    1. speed
    2. torque
    3. temperature
    4. position
    Answer

    A. speed

    Output voltage is proportional to speed.

34. Consider the statements. 1. Poles in the left half of the s-plane give a stable system. 2. Poles in the right half of the s-plane give a stable system. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Only left-half-plane poles give stability.

35. Consider the statements. 1. Percentage overshoot depends on damping ratio. 2. Larger damping ratio gives smaller overshoot. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are true for an underdamped second-order system.

36. Consider the statements. 1. Gain margin is measured at the phase crossover frequency. 2. Phase margin is measured at the gain crossover frequency. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard definitions.

37. Consider the statements. 1. A lag compensator improves transient response. 2. A lead compensator increases the bandwidth. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Lag improves steady-state accuracy; statement 1 is false.

38. Consider the statements. 1. Feedback always increases system gain. 2. Feedback reduces sensitivity to parameter changes. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Negative feedback reduces gain by 1 + GH.

39. Consider the statements. 1. A type-2 system has zero steady-state error for a ramp input. 2. A type-0 system has zero steady-state error for a step input. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Type 0 has finite error 1/(1 + Kp) for a step.

40. Match the system type with the steady-state error to a unit ramp: P. Type 0 Q. Type 1 R. Type 2 1. Zero 2. Infinite 3. 1/Kv

    1. P-1, Q-2, R-3
    2. P-3, Q-1, R-2
    3. P-2, Q-1, R-3
    4. P-2, Q-3, R-1
    Answer

    D. P-2, Q-3, R-1

    Type 0 infinite, type 1 equals 1/Kv, type 2 zero.

41. Match the damping ratio with the response: P. ζ = 0 Q. 0 < ζ < 1 R. ζ > 1 1. Underdamped 2. Overdamped 3. Undamped

    1. P-1, Q-3, R-2
    2. P-2, Q-1, R-3
    3. P-3, Q-2, R-1
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    Standard classification.

42. Match the controller with its effect: P. Proportional Q. Integral R. Derivative 1. Removes steady-state error 2. Improves damping 3. Speeds up response

    1. P-2, Q-3, R-1
    2. P-1, Q-2, R-3
    3. P-3, Q-2, R-1
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    P speeds response, I removes error, D adds damping.

43. Consider the statements about the root locus. 1. It is symmetric about the real axis. 2. It ends at open-loop zeros or at infinity. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard properties.

44. Consider the statements about the first-order system 1/(Ts + 1). 1. Its step response has overshoot. 2. Its pole is at s = −1/T. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    A first-order system never overshoots.