Chapter 1

First-order systems

1.1 Time response of continuous time first-order linear systems

Concepts analyzed in the chapter

  • Modeling of linear time invariant dynamic systems from first-order linear differential equation.
  • Obtaining the transfer function of a first-order system from a linear differential equation.
  • Time response of a first-order linear system for a step input (step response).
  • Concept of static gain and its effect on the system time response for a step input.
  • Concept of time constant and its effect on the system time response for a step input.
  • Stability analysis of first-order linear systems.

    • Theory

    This card analyzes the continuous time response of linear time invariant first-order systems without zeros. These systems represent the simplest dynamics, as derived from first-order differential equations of the form:

    τ d y t d t + y t = k u t
    (1.1)

    In expression (1.1), y t and u t represent the system output and input respectively. The representative transfer function can be written as:

    G s = k τ s + 1
    (1.2)

    wherein the denominator polynomial is called characteristic polynomial J s , whose only root (solution of the characteristic equation J s = 0 ) is called pole of the transfer function, placed on s = − 1 / τ . The two parameters that characterize the transfer function of a first-order system are:

  • k : Static gain or canonical gain of the system.
  • τ : Time constant.

    • The time response is linked to a certain excitation signal, traditionally the step ( U s = U e / s , with U e = 1 in the case of unit step). Therefore, using the inverse Laplace transform ( L − 1 ), one can obtain the time response of a first-order system when the input signal is a step. Using the expression of the output in the Laplace domain:

    Y s = G s U s = k τ s + 1 U e s

    if a partial-fraction expansion is used:

    Y s = k U e s − τ k U e τ s + 1

    the inverse Laplace transform of each one of the terms can be used, providing the time response when the input signal is a step:

    y t = k U e L − 1 1 s − L − 1 τ τ s + 1 = k U e 1 − e − 1 τ t , t ≥ 0
    (1.3)

    This time response can also be obtained by solving the differential equation with a step shape input:

    u t = 0 , t < 0 U e , t ≥ 0

    The behavior of the output of the system depends on the sign of τ :

  • If τ > 0 , the output is bounded ( ∀ u t ) and therefore it is said that the system is stable.
  • If τ < 0 , the output is unbounded for nonzero values of u t and therefore it is said that the system is unstable.

    • As a generalization of this result, it is stated that a necessary and sufficient condition for a system to be stable is that all poles of its transfer function have negative real parts. In this case we also speak of asymptotic stability, since the output tends to a steady-state value asymptotically.

    Note that if τ = 0 the system is not a dynamic one and the relationship between the output and the input is determined by the static gain of the system, k , which corresponds to the ratio between the value taken by the system output in steady-state ( t ∞ ) and the value of the input also in steady-state, as zero initial conditions are assumed. Applying the final value theorem of the Laplace transform to equation (1.3), we have:

    lim t ∞ y t = k U e = lim s 0 s Y s = lim s 0 s k τ s + 1 U e s = k U e

    As shown in the above equation, the static gain can be obtained by selecting s = 0 in the transfer function, G 0 = k .

    The designation of time constant, τ , stems from the fact that this parameter is indicative of the speed of the transient response of the system. The higher τ is, the slower the transient response of the system, that will take more time to reach its final value. Thus, according to equation (1.3):

  • For t = τ , the output of the system corresponds to k U e 1 − e − 1 , which is approximately 63 of the final value, y τ ≈ 0.63 k U e .
  • For t = 3 τ , the system output corresponding to 95 of the final value, y 3 τ ≈ 0.95 k U e .
  • For t = 4 τ , the output of the system is around 98 of the final value ( y 4 τ ≈ 0.98 k U e ), which is considered the settling time for first-order systems (the response remains within 2 of the final value). According to equation (1.3), the steady state is achieved mathematically in infinite time, so that in practice it is considered that the system output is in steady-state when it reaches 98 of the final value, equal to four time constants.

    • Another important feature is that the slope of the time response of the system in t = t 0 is k / τ (where t 0 is the instant the step input is applied to the system); see Figure 1.3.

    (interactive fig fig-3.1)

    Fig. 1.3 Parameters characterizing the time response and the complex plane representation of a first-order system

    A particular case study of a first-order system called integrator is that having its pole at the origin of the s -plane ( s = 0 ) and described by:

    d y t d t = k u t G s = k s
    (1.4)

    This system would be equivalent to a first-order one with a very large time constant ( τ ∞ ).

    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 3, pages 103-104, 2009.
  • 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 3, pages 126-128, 2010.
  • Golnaraghi, F. and B.C. Kuo. Automatic control systems. 9th edition. John Wiley Sons, Inc. ISBN: 978-0470-04896-2. Chapter 5, section 5, pages 274-275, 2010.
  • Ogata, K. Modern control engineering. 5th edition. Pearson (International Edition), ISBN: 978-0-13-713337-6. Chapter 5, section 2, pages 171-174, 2010.
  • Shahian, B. and M. Hassul. Control system design using Matlab. First edition. Prentice Hall, ISBN: 0-13-174061-X. Chapter 1, section 5, paragraph 1, pages 10-11, 1993.
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