Chapter 2

Second-order systems

2.1 Time response of continuous time second-order linear systems

Concepts analyzed in the chapter

  • Modeling of linear time invariant dynamic system using a second-order linear differential equation.
  • Obtaining the transfer function of a second-order system from a linear differential equation.
  • Analysis of the time response of a second-order linear dynamic system for a step input.
  • Concept of static gain and its effect on the system time response for a step input.
  • Concept of relative damping factor and its effect on the system response for a step input.
  • Concept of undamped natural frequency and its effect on the system response for a step input.
  • Types of dynamic behavior in second-order systems: overdamped, critically damped, underdamped, critically stable and unstable.
  • Stability analysis in second-order linear systems.

    • Theory

    The second-order systems, as its name suggests, can be described by a standard second-order differential equation such as:

    d 2 y t d t 2 + 2 ζ ω n d y t d t + ω n 2 y t = k ω n 2 u t
    (2.1)

    where y t and u t are the system output and input respectively.

    There are systems with “pure” second-order dynamics or formed by the combination of two first-order systems in series (product of two transfer functions of first order). The standard transfer function of a second-order system is given by:

    G s = k ω n 2 s 2 + 2 ζ ω n s + ω n 2
    (2.2)

    wherein the denominator polynomial is called characteristic polynomial J s , which roots (solution of the characteristic equation J s = 0 ) are the poles of the transfer function, which in this case can be real or complex conjugates. The parameters that define the transfer function are:

  • k is the static gain or canonical gain of the system. It can be obtained by doing s = 0 in the transfer function, G 0 = k . In stable systems it represents the quotient of the amplitude of the steady-state response of the system and the amplitude of the input step.
  • ζ is the relative coefficient, ratio or damping factor of the system (dimensionless), which determine the shape of the transient response. Depending on its value, it can be deduced if the system is unstable ( ζ < 0 ), critically stable or not damped ( ζ = 0 ), underdamped ( 0 < ζ < 1 ), critically damped ( ζ = 1 ), or overdamped ( ζ > 1 ).
  • ω n natural undamped frequency (rad/s), which corresponds to the frequency of oscillation of the system if there were no damping ( ζ = 0 , sinusoidal response).

    • Obviously, to obtain a bounded response when the input signal has a step shape (step input and bounded signal), the poles of the system must be on the left side of the s -plane. If any of the roots is in the right-half s -plane, the system will be unstable. If the system characteristic equation J s = s 2 + 2 ζ ω n s + ω n 2 = 0 has its roots on the imaginary axis ( j ω axis), the output in steady-state when the input is a step signal will be of sinusoidal type (sustained oscillations), unless the input is a sine wave whose frequency is equal to the magnitude of the roots of the j ω axis. In this case, the output will be unbounded. Such a system is called marginally stable, because only some bounded inputs (sinusoids with the same frequency of the poles) will produce unbounded outputs.

    The time response when the input has a step shape of amplitude U e ( U s = U e / s ) can be obtained from:

    Y s = G s U s = k ω n 2 s 2 + 2 ζ ω n s + ω n 2 U e s

    by applying the inverse Laplace transform, y t = L − 1 Y s , or by solving the differential equation with:

    u t = 0 , t < 0 U e , t ≥ 0

    making it necessary to distinguish between cases: 0 ≤ ζ < 1 , ζ ≥ 1 and ζ < 0 . The time response of second-order systems without zeros starts at t = 0 with zero slope, as indicated by the initial conditions of integration.

  • Underdamped system: In the case 0 ≤ ζ < 1 the two poles of the system (roots of the characteristic equation) are complex conjugate, s 1 = − ζ ω n + j ω n 1 − ζ 2 and s 1 * = − ζ ω n − j ω n 1 − ζ 2 and the time response for a step input is:
    • y t = k U e 1 − e − ζ ω n t cos ω d t + ζ 1 − ζ 2 sen ω d t , t ≥ 0
    (2.3)

    where ω d = ω n 1 − ζ 2 is the natural damped frequency. It can be observed how the complex component of the poles produces a time response with presence of sines and cosines that gives rise to oscillations which are damped by the exponential envelope. The product σ = ζ ω n is called damping factor, which is a constant that determines the damping properties of a system. It determines the rate of growth or decay of the unit step response of an underdamped second-order system.

  • Overdamped system: When the relative damping factor ζ ≥ 1 , the poles of the second-order system are real s 1 = − 1 / τ 1 = − ζ ω n − ω n ζ 2 − 1 and s 2 = − 1 / τ 2 = − ζ ω n + ω n ζ 2 − 1 . The transfer function in this case is given by:
    • G s = k τ 1 s + 1 τ 2 s + 1
    (2.4)

    where τ 2 is the time constant associated with the pole closest to the imaginary axis, which causes the slower exponential response ( τ 2 > τ 1 ) and the time response can be obtained as a superposition of that given by two first-order systems:

    y t = k U e 1 − τ 2 τ 2 − τ 1 e − t τ 2 + τ 1 τ 2 − τ 1 e − t τ 1 , t ≥ 0
    (2.5)
  • When ζ = 1 , both roots are equal s 1 = s 2 ( τ 1 = τ 2 ) and the system is called critically damped. Its transfer function is given by:
    • G s = k τ s + 1 2
    (2.6)

    and the step response has the following analytical expression:

    y t = k U e 1 − e − t τ − t τ e − t τ , t ≥ 0
    (2.7)
  • Critically stable system: As can be seen in equation (2.3), when ζ = 0 the two complex conjugate poles are located on the imaginary axis (with zero real part) and the response shows a maintained oscillation given by:
    • y t = k U e 1 − cos ω n t , t ≥ 0
    (2.8)
  • Unstable system: If ζ < 0 the system will be unstable, with two complex conjugate poles with real part within the right-half s -plane if − 1 < ζ < 0 (unstable oscillatory response) or either two real poles in the right-half s -plane if ζ ≤ − 1 (unstable response of exponential type).

    • For both the overdamped and underdamped case, an interesting analysis is to study the location of the poles of a second-order system as a function of the characteristic parameters of the transfer function. From these parameters, several relationships with certain specific characteristics of the time response of the system can be found , can also be used as closed-loop performance specifications in control system design).

    The best known features of the time response are (see Figure 2.12):

  • Overshoot ( O S []): Represents the difference between the maximum peak value of the response and its steady-state value, relative to that steady-state value (in ). For the underdamped case, applying the time derivative to equation (2.3) and equating to zero, the maximum value of y t which defines the overshoot can be obtained, as well as the the time at which this maximum is reached (peak time). The value of the overshoot is given by:
    • S O % = 100 exp − ζ π 1 − ζ 2
    (2.9)

    This definition makes sense only in the underdamped case.

  • Peak time ( t p [s]): Time the system response takes to reach its maximum (peak) value measured from the moment a step input is applied. By applying the time derivative to equation (2.3) and equating to zero, it is possible to obtain the peak time as:
    • t p = π ω d = π ω n 1 − ζ 2
    (2.10)
  • Rise time ( t r [s]): In the underdamped case, it is the time elapsed since the output of the system begins to evolve until it reaches the steady-state value for the first time:
    • t r = π − φ ω d
    (2.11)

    with ζ = cos φ , φ being the angle formed by the complex conjugate poles with the x -axis. In overdamped systems it is defined as the time the system response takes to evolve from 10 to 90 of its steady-state value.

  • Settling time ( t s [s]): Time elapsed from the moment the output of the system begins to evolve until it lies stable at around 2 of the steady-state value. An upper bound calculated from the envelope is given by:
    • t s ≈ 4 ζ ω n
    (2.12)

    In the overdamped case ( ζ ≥ 1 ), its value is obtained as t s ≈ 4 τ 1 + τ 2 .

    It is also possible to analyze the effect of varying the location of the poles of an underdamped second-order system using an expression in terms of the real and imaginary parts of the roots (see Figure 2.12), where σ = ζ ω n and ω d = ω n 1 − ζ 2 . The time constant of the exponential envelope of the time response of an underdamped second-order system is τ = 1 / σ .

    G s = ω n 2 s 2 + 2 ζ ω n s + ω n 2 = ω d 2 + σ 2 s 2 + 2 σ s + ω d 2 + σ 2

    (interactive fig fig-3.2)

    Fig. 2.12 Parameters characterizing the time response and the complex plane representation of an underdamped second-order system

    Case I: Effects of increasing σ (with constant ω d )

  • The imaginary part of the poles remains constant and the real part (in absolute value) increases.
  • The settling time decreases.
  • The rise time decreases because the distance from the poles to the origin increases.
  • The overshoot is reduced because ζ increases.
  • The peak time remains constant because ω d has been fixed.
  • The bandwidth increases because it is proportional to ω n .

    • xs Case II: Effects of increasing ω d (with constant σ )

  • The real part of the poles remains constant while the imaginary part increases.
  • The settling time remains constant.
  • The overshoot and bandwidth increase.
  • The peak time and rise time decrease.

    • Case III: Effects of increasing ω n (with constant ζ )

  • The poles are moved radially away from the origin.
  • The overshoot remains constant.
  • The rise, peak and settling times decrease.
  • The bandwidth increases.

    • Case IV: Effects of increasing ζ (with constant ω n )

  • The rise time increases.
  • The overshoot and settling time decrease.
  • The peak time increases.

    • References

  • Bolzern, P., R. Scattolini and N. Schiavoni. Fundamentos de control automático (Fundamentals of automatic control). Mc Graw Hill, ISBN: 978-84-481-6640-3. Chapter 4, section 4, paragraph 4, pages 105-111, 2009.
  • Dorf, R. C. and R. H. Bishop. Modern control systems. Pearson - Prentice Hall, ISBN: 978-0-13-602458-3. Chapter 5, section 3, pages 308-314, 2011.
  • Franklin, G. F., J. D. Powell and A. Emani-Naeni. Feedback control of dynamic systems. 6th edition. Pearson. ISBN: 978-0-13-500150-9. Chapter 3, section 4, pages 134-137, 315-316, 2010.
  • Golnaraghi, F. and B.C. Kuo. Automatic control systems. 9th edition. John Wiley Sons, Inc. ISBN: 978-0470-04896-2. Chapter 5, section 6, pages 275-288, 2010.
  • Ogata, K. Modern control engineering. 5th edition. Pearson (International Edition), ISBN: 978-0-13-713337-6. Chapter 5, section 3, pages 174-189, 2010.
  • Shahian, B. and M. Hassul. Control system design using Matlab. 1st edition. Prentice Hall, ISBN: 0-13-174061-X. Chapter 1, section 5, paragraph 2, pages 11-16, 1993.
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