Control Systems

What to remember

• Transfer function G(s) = L[output] / L[input] with zero initial conditions. Closed-loop with negative feedback: T = G / (1 + GH). The poles of T decide stability and response.
• Second-order system: ζ and ωn set the overshoot, peak time and settling time. Stable if all poles lie in the left half of the s-plane (Routh test).
• System type (number of poles at the origin) fixes steady-state error: Kp, Kv and Ka are the constants for step, ramp and parabolic input.

Basic ideas

A control system makes the output follow a desired input.

• Open-loop system: output is not fed back. Simple, cheap, stable, but not accurate; it cannot correct for disturbance. Example: a washing machine timer, a toaster.
• Closed-loop system: output is measured and compared with the reference. The error drives the controller. It is more accurate and reduces the effect of disturbance and parameter change, but it can become unstable and costs more.
• Negative feedback reduces gain, but improves accuracy, bandwidth and sensitivity.

Transfer function. The ratio of the Laplace transform of output to that of input, with zero initial conditions. It is for linear time-invariant (LTI) systems. Roots of the numerator are zeros; roots of the denominator are poles. The denominator set to zero is the characteristic equation.

Closed loop with feedback: T(s) = G(s) / (1 + G(s)H(s)). For G = 10, H = 0.4, T = 10 / (1 + 4) = 2.

Block diagram rules

• Series blocks multiply.
• Parallel blocks add.
• Moving a take-off point or summing point past a block needs compensating blocks.
• Mason's gain formula for a signal flow graph: T = Σ Pk Δk / Δ, where Pk are forward path gains and Δ is the graph determinant (1 − sum of loop gains + sum of products of non-touching loops − …).

Standard components

• Error detectors: potentiometer, synchro pair.
• Tachogenerator: gives speed feedback; the output voltage is proportional to speed.
• Servo motors: DC servo, AC two-phase servo (high resistance rotor for a near-linear torque-speed curve).
• Gear trains, amplifiers.

Time response

Standard test inputs: step, ramp, parabolic and impulse. In Laplace form: step 1/s, ramp 1/s², parabola 1/s³, impulse 1.

First-order system: G(s) = 1 / (τs + 1). Time constant τ; the step response rises to 63.2% of final value at t = τ. Settling time (2%) = 4τ. If τ = 0.5 s, settling time = 2 s.

Second-order system: T(s) = ωn² / (s² + 2ζωn s + ωn²)

Specifications (underdamped)

• Damped frequency ωd = ωn √(1 − ζ²).
• Rise time: time to go from 10% to 90% (or 0 to 100%) of the final value.
• Peak time tp = π / ωd.
• Peak overshoot Mp = exp(−πζ / √(1 − ζ²)) × 100%.
• Settling time ts = 4 / (ζωn) for 2% band; 3 / (ζωn) for 5% band.

Worked example. ωn = 10 rad/s, ζ = 0.5.

• ωd = 10 × 0.866 = 8.66 rad/s.
• tp = 3.14 / 8.66 = 0.363 s.
• Mp = exp(−3.14 × 0.5 / 0.866) = about 16.3%.
• ts (2%) = 4 / 5 = 0.8 s.

For ζ = 0.707, overshoot is about 4.3%.

Poles further to the left mean faster settling. Poles with larger imaginary parts mean higher oscillation frequency. The closed-loop system of unity feedback G = K / (s(s + 4)) gives s² + 4s + K = 0, so ωn = √K, ζ = 2/√K. For K = 16: ωn = 4, ζ = 0.5.

Steady-state error and system type

Type is the number of integrators (poles at origin) in the open-loop transfer function.

Here Kp = lim G(s)H(s) as s→0; Kv = lim s G(s)H(s); Ka = lim s² G(s)H(s).

Example: G = 10 / (s(s + 2)) with unity feedback is type 1; Kv = 10/2 = 5; ramp error = 0.2. For a type 0 system with Kp = 4, step error = 1/5 = 0.2.

Adding an integrator improves steady-state accuracy but reduces stability.

Stability

A linear system is stable if all poles of the closed-loop transfer function lie in the left half of the s-plane. Poles on the imaginary axis (simple) mean marginal stability. Poles in the right half mean instability.

Routh-Hurwitz criterion. Form the Routh array from the characteristic polynomial coefficients. The number of sign changes in the first column equals the number of poles in the right half plane. A necessary (not sufficient) condition: all coefficients present and of the same sign. A row of zeros indicates symmetric roots (imaginary axis or opposite pairs); the auxiliary equation is used.

Example: s³ + 6s² + 11s + 6 is stable (roots −1, −2, −3). For s³ + 5s² + 6s + K: stable for 0 < K < 30, since (5 × 6 − K)/5 > 0. s³ + 2s² + 3s + 10 is unstable because 2 × 3 < 10.

Root locus. Path of closed-loop poles as gain K changes from 0 to infinity.

• Number of branches = number of open-loop poles (or zeros, whichever is greater).
• Locus starts at open-loop poles (K = 0) and ends at zeros (K = ∞), or at infinity.
• Number of asymptotes = P − Z; angle = (2q + 1) × 180° / (P − Z).
• Centroid = (sum of poles − sum of zeros) / (P − Z).
• For poles at 0, −2, −4 and no zeros: three asymptotes at 60°, 180°, 300°; centroid = −6/3 = −2.
• Locus on the real axis lies to the left of an odd number of poles plus zeros.
• 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 and phase against log frequency. Each pole gives −20 dB/decade beyond its corner frequency; each zero gives +20 dB/decade. A simple pole 1/(1 + s/10) has a corner frequency at 10 rad/s. A gain of 100 equals 40 dB.
• Gain margin: extra gain that would bring the system to the edge of instability, measured at the phase crossover frequency (phase = −180°). Phase margin: extra phase lag that would make the system unstable, measured at the gain crossover frequency (|GH| = 1). Both must be positive for a stable closed loop (minimum phase systems).
• Nyquist criterion: Z = N + P, where Z is the number of closed-loop poles in the right half plane, P the open-loop poles in the right half plane, and N the number of encirclements of −1 + j0 (clockwise counted positive). For stable closed loop Z = 0.
• Resonant peak Mr relates to damping; bandwidth relates to speed of response.

Compensators and controllers

State space. x' = Ax + Bu; y = Cx + Du. Poles of the system are the eigenvalues of A. The transfer function is G(s) = C(sI − A)⁻¹B + D. Controllability and observability are properties of the pair (A, B) and (A, C).

Exam traps

• Open-loop systems have no feedback; closed-loop ones do.
• Negative feedback is the usual form; positive feedback tends to instability.
• ζ = 1 is critical damping; ζ > 1 is overdamped, with no overshoot.
• Settling time is 4/(ζωn) for 2% and 3/(ζωn) for 5%.
• Peak time uses ωd, not ωn.
• A step error exists only for type 0. A ramp error is finite only for type 1.
• Routh: sign changes in the first column count right-half-plane poles.
• Root locus starts at poles and ends at zeros.
• Lead compensator improves transient response; lag compensator improves steady-state accuracy.
• Phase margin is read at gain crossover; gain margin at phase crossover.
• Nyquist: Z = N + P, not Z = N − P with anticlockwise conventions mixed.
• Each pole gives −20 dB/decade, not −40.

One-liners

• 1. Transfer function is defined with zero initial conditions.
• 2. Closed-loop gain = G / (1 + GH) for negative feedback.
• 3. Time constant τ: first-order response reaches 63.2% at t = τ.
• 4. Characteristic equation is 1 + G(s)H(s) = 0.
• 5. ωd = ωn √(1 − ζ²).
• 6. Peak overshoot depends only on ζ.
• 7. ts (2%) = 4 / (ζωn).
• 8. Kv = lim s G(s)H(s) as s→0.
• 9. Type 1 system has zero error for a step input.
• 10. Stable system: all poles in the left half s-plane.
• 11. Root locus has as many branches as open-loop poles (when P ≥ Z).
• 12. PI controller removes steady-state error; PD controller adds damping.

Practice questions

1. A negative feedback system has forward gain G = 10 and feedback H = 0.4. The closed-loop gain is:

   1. 4
   2. 10
   3. 2.5
   4. 2
   Answer

   D. 2

   T = G/(1 + GH) = 10/(1 + 4) = 2.

2. A first-order system has transfer function 1/(0.5s + 1). The 2% settling time is about:

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

   C. 2 s

   ts = 4τ = 4 × 0.5 = 2 s.

3. A second-order system has ωn = 10 rad/s and ζ = 0.5. The damped natural frequency is about:

   1. 8.66 rad/s
   2. 10 rad/s
   3. 7.07 rad/s
   4. 5 rad/s
   Answer

   A. 8.66 rad/s

   ωd = ωn√(1 − ζ²) = 10 × 0.866 = 8.66 rad/s.

4. For ωn = 10 rad/s and ζ = 0.5, the peak overshoot is about:

   1. 16.3%
   2. 4.3%
   3. 50%
   4. 0.5%
   Answer

   A. 16.3%

   Mp = exp(−πζ/√(1 − ζ²)) = exp(−1.814) = 16.3%.

5. For ωn = 10 rad/s and ζ = 0.5, the 2% settling time is:

   1. 1.2 s
   2. 2 s
   3. 0.8 s
   4. 0.4 s
   Answer

   C. 0.8 s

   ts = 4/(ζωn) = 4/5 = 0.8 s.

6. For ωn = 10 rad/s and ζ = 0.5, the peak time is about:

   1. 0.314 s
   2. 0.363 s
   3. 1 s
   4. 0.8 s
   Answer

   B. 0.363 s

   tp = π/ωd = 3.14/8.66 = 0.363 s.

7. A unity feedback system has G(s) = K/(s(s + 4)). For K = 16, the undamped natural frequency is:

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

   D. 4 rad/s

   The characteristic equation s² + 4s + 16 = 0 gives ωn² = 16, so ωn = 4.

8. For the same system with K = 16, the damping ratio is:

   1. 0.5
   2. 1
   3. 0.25
   4. 2
   Answer

   A. 0.5

   2ζωn = 4, so ζ = 4/(2 × 4) = 0.5.

9. A unity feedback system has G(s) = 10/(s(s + 2)). The velocity error constant Kv is:

   1. 10
   2. 2
   3. 20
   4. 5
   Answer

   D. 5

   Kv = lim s·G(s) = 10/2 = 5.

10. For the same system, the steady-state error for a unit ramp input is:

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

    C. 0.2

    ess = 1/Kv = 1/5 = 0.2.

11. A type 0 unity feedback system has Kp = 4. The steady-state error for a unit step is:

    1. 0.25
    2. 0.2
    3. 4
    4. 0
    Answer

    B. 0.2

    ess = 1/(1 + Kp) = 1/5 = 0.2.

12. The characteristic equation s³ + 5s² + 6s + K = 0 gives a stable system for:

    1. 0 < K < 30
    2. K < 0
    3. K > 30
    4. 0 < K < 5
    Answer

    A. 0 < K < 30

    Routh: first-column term (30 − K)/5 must be positive, and K > 0.

13. A gain of 100 expressed in decibels is:

    1. 100 dB
    2. 10 dB
    3. 40 dB
    4. 20 dB
    Answer

    C. 40 dB

    dB = 20 log10(100) = 40.

14. For open-loop poles at 0, −2 and −4 and no zeros, the centroid of the root locus asymptotes is:

    1. 0
    2. −2
    3. −3
    4. −6
    Answer

    B. −2

    Centroid = (0 − 2 − 4)/3 = −2.

15. For three open-loop poles and no zeros, the root locus asymptote angles are:

    1. 45°, 135°, 225°
    2. 0°, 120°, 240°
    3. 60°, 180°, 300°
    4. 90°, 180°, 270°
    Answer

    C. 60°, 180°, 300°

    Angles = (2q + 1) × 180°/3 = 60°, 180°, 300°.

16. A factor 1/(1 + s/10) has a Bode corner frequency of:

    1. 0.1 rad/s
    2. 10 rad/s
    3. 100 rad/s
    4. 1 rad/s
    Answer

    B. 10 rad/s

    The corner frequency equals the pole value, 10 rad/s.

17. The polynomial s³ + 2s² + 3s + 10 has how many roots in the right half of the s-plane?

    1. 0
    2. 1
    3. 3
    4. 2
    Answer

    D. 2

    Routh first column: 1, 2, −2, 10 gives two sign changes.

18. A first-order system with time constant 2 s reaches 63.2% of its final value at:

    1. 2 s
    2. 4 s
    3. 1 s
    4. 0.632 s
    Answer

    A. 2 s

    At t = τ the step response is 63.2% of the final value.

19. The main advantage of a closed-loop control system over an open-loop system is:

    1. Lower cost
    2. Reduced effect of disturbances and parameter changes
    3. Simpler construction
    4. Always unconditionally stable
    Answer

    B. Reduced effect of disturbances and parameter changes

    Feedback corrects errors, but adds cost and a risk of instability.

20. Which of the following is an example of an open-loop system?

    1. An automatic washing machine timer sequence
    2. A thermostat-controlled room heater
    3. An aircraft autopilot
    4. A speed-controlled motor with tachogenerator
    Answer

    A. An automatic washing machine timer sequence

    It has no feedback of the output.

21. A transfer function is defined:

    1. With unit initial conditions
    2. Only for non-linear systems
    3. In the time domain
    4. With zero initial conditions
    Answer

    D. With zero initial conditions

    G(s) = L[output]/L[input] with zero initial conditions, for linear time-invariant systems.

22. The roots of the characteristic equation are:

    1. The closed-loop zeros
    2. The open-loop zeros
    3. The closed-loop poles
    4. The gain margin
    Answer

    C. The closed-loop poles

    They decide the stability and the transient response.

23. A second-order system with ζ = 1 is called:

    1. Critically damped
    2. Underdamped
    3. Undamped
    4. Overdamped
    Answer

    A. Critically damped

    Its poles are two equal real poles; it is the fastest response without overshoot.

24. A second-order system with ζ = 0 has a step response that is:

    1. Zero
    2. A sustained oscillation
    3. Overdamped
    4. Critically damped
    Answer

    B. A sustained oscillation

    The poles lie on the imaginary axis.

25. The peak overshoot of a standard second-order system depends only on:

    1. The gain margin
    2. The time constant
    3. The natural frequency
    4. The damping ratio
    Answer

    D. The damping ratio

    Mp = exp(−πζ/√(1 − ζ²)).

26. The type of a system is the number of:

    1. Closed-loop poles
    2. Zeros at the origin
    3. Poles of the open-loop transfer function at the origin
    4. Poles in the right half plane
    Answer

    C. Poles of the open-loop transfer function at the origin

    It decides the steady-state error constants.

27. A type 1 system has a steady-state error of zero for:

    1. A parabolic input
    2. A step input
    3. All inputs
    4. A ramp input
    Answer

    B. A step input

    Type 1 has one integrator: zero step error, finite ramp error.

28. A linear system is stable if all closed-loop poles lie in:

    1. The left half of the s-plane
    2. The right half of the s-plane
    3. The imaginary axis
    4. The origin only
    Answer

    A. The left half of the s-plane

    Poles with negative real parts give decaying responses.

29. A root locus branch starts at:

    1. Open-loop zeros (K = 0)
    2. The origin always
    3. Infinity
    4. Open-loop poles (K = 0)
    Answer

    D. Open-loop poles (K = 0)

    The branches end at the open-loop zeros or at infinity.

30. The gain margin is measured at the:

    1. Corner frequency
    2. Resonant frequency
    3. Phase crossover frequency
    4. Gain crossover frequency
    Answer

    C. Phase crossover frequency

    It is evaluated where the phase is −180°.

31. A lead compensator mainly:

    1. Adds an integrator
    2. Improves transient response and phase margin
    3. Reduces bandwidth
    4. Reduces the steady-state error to zero
    Answer

    B. Improves transient response and phase margin

    It adds phase lead near the crossover frequency.

32. A lag compensator mainly:

    1. Improves steady-state accuracy
    2. Increases the bandwidth
    3. Improves the speed of response
    4. Adds phase lead
    Answer

    A. Improves steady-state accuracy

    It raises the low-frequency gain, which reduces steady-state error.

33. The PI controller is used to:

    1. Reduce the rise time to zero
    2. Increase the noise
    3. Remove the integrator
    4. Eliminate steady-state error
    Answer

    D. Eliminate steady-state error

    The integral term drives the steady-state error to zero.

34. A PD controller mainly adds:

    1. Integral action
    2. Steady-state gain
    3. Damping
    4. Phase lag
    Answer

    C. Damping

    The derivative term anticipates error and reduces overshoot.

35. A tachogenerator in a servo system is used to:

    1. Give position command
    2. Provide speed feedback
    3. Supply power
    4. Drive the load
    Answer

    B. Provide speed feedback

    Its output voltage is proportional to speed.

36. In the Nyquist criterion, the number of closed-loop poles in the right half plane is given by:

    1. Z = N × P
    2. Z = P − 2N
    3. Z = N − P
    4. Z = N + P
    Answer

    D. Z = N + P

    With clockwise encirclements counted positive, Z = N + P; stability needs Z = 0.

37. Each simple pole in a Bode magnitude plot adds a slope of:

    1. −20 dB/decade beyond its corner frequency
    2. −40 dB/decade
    3. +20 dB/decade
    4. −6 dB/decade at all frequencies
    Answer

    A. −20 dB/decade beyond its corner frequency

    A simple zero adds +20 dB/decade.

38. Statements: 1. Closed-loop systems reduce the effect of disturbances. 2. Open-loop systems are generally more accurate than closed-loop systems.

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

    C. 1 only

    Feedback gives better accuracy; open-loop systems cannot correct errors.

39. Statements on damping: 1. A critically damped system has ζ = 1. 2. An underdamped system shows overshoot.

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

    B. Both 1 and 2

    Both are standard facts.

40. Statements on a type 1 system: 1. Its steady-state error for a ramp is finite. 2. Its steady-state error for a step is non-zero.

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

    A. 1 only

    A type 1 system has zero error for a step input.

41. Which pair is correctly matched?

    1. PI controller – more damping
    2. P controller – zero steady-state error
    3. Lag compensator – larger bandwidth
    4. Lead compensator – improved transient response
    Answer

    D. Lead compensator – improved transient response

    A lag compensator reduces bandwidth, PI gives zero steady-state error, and P control leaves an offset.

42. Statements on stability: 1. Poles in the right half plane make the system unstable. 2. Root locus branches end at open-loop poles.

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

    C. 1 only

    Branches start at poles and end at zeros or infinity.

43. Statements on time response: 1. Settling time (2%) is 4/(ζωn). 2. Peak time is π/ωn.

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

    B. 1 only

    Peak time is π/ωd, using the damped frequency.

44. Statements on Bode analysis: 1. Phase margin is measured at the gain crossover frequency. 2. Gain margin is measured at the phase crossover frequency.

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

    A. Both 1 and 2

    Both are correct.

45. Statements on the characteristic equation: 1. It is obtained from 1 + G(s)H(s) = 0. 2. Its roots are the open-loop zeros.

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

    D. 1 only

    Its roots are the closed-loop poles.