Step Response

3 Key excerpts on "Step Response"

• 

  eBook - ePub

  System and Measurements

  • Yong Sang(Author)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  It is a useful idealization, although it is impossible in any real system. However, the step signal is easier to generate. Step Response recognition is one of the most important subjects in process control. Step input may be the most commonly used excitation signal in process industry. To deal with different practical problems in the recognition based on Step Response, many new methods have been proposed. If the input x (t) is the unit step signal, (3.40) x (t) = 1 The input and output signals are shown in Figure 3.33. Figure 3.33: The input and output signals. Then (3.41) X (s) = 1 s The dynamic characteristic Y (s) : (3.42) H (s) = Y (s) X (s) = Y (s) ⋅ s Let s = j ω,. then (3.43) H (ω) = Y (j ω) ⋅ j ω H (ω) consists of a real part and an imaginary part: (3.44) H (ω) = R (ω) + j I (ω) R (ω) and I (ω) are real functions over ω. You can plot R (ω) - ω or I (ω) - ω and get real or imaginary frequency characteristics. In Cartesian coordinates, the real part R (ω) is plotted on the X -axis and the imaginary part I (ω) is plotted on the Y -axis. This plot is called the Nyquist plot. Another useful plot is the Bode plot, which is a plot of either the magnitude or the phase of a transfer function as a function of ω. The magnitude plots are more common because they represent the gain of the system. Until now, dynamic characteristics can be described as h (t) in the time domain, H (ω) in the frequency domain and H (s) in the plural domain. The relationship between h (t), H (s) and H (ω) can be written as (3.45) h (t) ↔ H (s) L a p l a c e t r a n s f o r m p a i r s (3.46) h (t) ↔ H (ω) F o u r i e r t r a n s f o r m p a i r s A typical application of a Step Response method is. used to find the damping ratio and natural frequency of a second-order measurement system by experiment. Typical application The 1940 Tacoma Strait Suspension Bridge was built in Washington, DC. The new bridge collapsed 4 months after its completion

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  eBook - ePub

  Process Control Engineering

  • A. Ramachandro. Rao(Author)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  CHAPTER 3

  TIME DOMAIN ANALYSIS AND DESIGN

  3.1.  TEST SIGNALS

  When designing process control systems, the engineer requires an objective means of evaluating the performance of the control schemes under consideration. Performance criteria are quantitative measures of the system response to a specific input or test signal. The present section establishes the mathematical properties of some common test signals that may be used in the definition of various performance criteria.

  Step input

  As discussed in the previous chapter, a step input is a sudden and sustained change in input defined mathematically as

  u ( t ) = { M t ≥ 0 0 t < 0

  (3.1.1)

  (See Fig. 3.1.1a ). The constant M is known as the magnitude or size of the step function. u(t) is called a unit step function if M = 1. The Laplace transform of a step function was derived in Example 2.2.1:

  ℒ [ u ( t ) ] = M / s

  (3.1.2)

  The response of a system to a step input is referred to as its Step Response, and conveys information regarding the dynamic and steady state behavior of the system. The step function has been used extensively in evaluating the performance of control systems for the following reasons:

  i)  It is simple and easy to produce.

  ii)  It is considered to be a serious disturbance, since it has a very fast change at the initial moment. Many other kinds of bounded disturbances can be overcome if a step disturbance can be overcome.

  iii)  The response of the system to other types of disturbances can be inferred from the process Step Response. iv)  The Step Response is easy to measure, and thus get an approximated transfer function of the system.

  Ramp input

  A ramp input is a function that varies linearly with time in the following manner:

  r ( t ) = { R t t ≥ 0 0 t < 0

  (3.1.3)

  where R

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• 

  eBook - ePub

  Physiological Control Systems

  Analysis, Simulation, and Estimation

  • Michael C. K. Khoo(Author)
  • 2018(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  c will be the location (on the time axis) at which the center of mass acts.

  (4.68)

  Figure 4.7

  Descriptors of the (a) impulse and (b) Step Responses.

  4.4.2.2 Step Response Descriptors

  The most commonly used descriptors of the Step Response are shown in Figure 4.7b . As mentioned previously, the final value of the response is the steady-state level achieved by the system in question. If the input is a unit step, this final value will yield the steady-state gain GSS . If the peak value of the Step Response is larger than the final value, the overshoot will be the difference between this peak value and the final value. Frequently, this overshoot is expressed in percentage terms:

  (4.69)

  The time taken for the Step Response to achieve its peak value is known as the peak time or Tp , as illustrated in Figure 4.7b . Aside from peak time, there are two other measures of speed of response. One is the rise time Tr defined as

  (4.70)

  where t90% and t10% are the times at which the response first attains 90 and 10% of its final value, respectively.

  The other measure of speed of response is the settling time Ts , defined as the time taken for the Step Response to settle within ±δ% of the final value (Figure 4.7b ). The upper and lower levels of this band of values, that is, 100 + δ% and 100 − δ% of the final value, define the tolerance limits within which the Step Response will remain at all times greater than Ts . The values of δ generally employed range from 1 to 5%.

  4.5 Open-Loop versus Closed-Loop Dynamics: Other Considerations

  4.5.1 Reduction of the Effects of External Disturbances

  In our previous discussions of the first- and second-order models of lung mechanics, we showed that one clear consequence of introducing negative feedback into the control scheme is an increase in speed of system response. A second major effect of closing the loop is the reduction in overall system gain. For both first- and second-order models, closing the loop led to a significant reduction of the final values in the unit Step Responses (see Figures 4.3b and 4.5

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