Unit step

Often an unit impulse is not available as a signal to get the impulse response function g(x). Therefore an other characteristic signal, the unit step, is be used. 

The step response function h(x) is the response of a system to an unit step s(x) at the input. 

The unit step function s(x) is defined as a step from 0 to /. The function s(x) is shown in fig. 1 (centre) together with an example of a step response function h(x). 

Unit impulse fucnction Unit step function Example for an input signal 5(x) f s(x)f u( ),... 

Unit impulse response Unit step response responses for input example... 

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

FIG. 8-43 Step response for u, a unit step change in the input. 

The specific conductivities of molten salts are frequently represented, as a function of temperature by an AtTlrenius equation, but it is unlikely that the unit step in diffusion has a constant magnitude, as in the coiTesponding solids and the results for NaCl may be expressed, within experimental eiTor, by the alternative equations... 

Table 3.2 Unit step response of a first-order system...

Generalized second-order system response to a unit step input... 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

Fig. 3.19 Unit step response of a second-order system.

The generalized seeond-order system response to a unit step input is shown in Figure 3.19 for the eondition K = 1 (see also Appendix 1, sec ord.m). 

Figure 3.22 shows, in bloek diagram form, the transfer funetions for a resistanee thermometer and a valve eonneeted together. The input X[ t) is temperature and the output Xo t) is valve position. Find an expression for the unit step response funetion when there are zero initial eonditions. 

Determine the values of Wn and ( and also expressions for the unit step response for the systems represented by the following second-order transfer functions... 

When a unity gain seeond-order system is subjeet to a unit step input, its transient response eontains a first overshoot of 77%, oeeurring after 32.5 ms has elapsed. Find... 

When r t) is a unit step, and riit) is zero, the final value theorem (equation (3.10)) gives the steady-state response... 

Figure 4.34 shows the response to a unit step, or the process reaction curve. 

Fig. 4.38 Unit step response of ship autopilot control system. RiseTime (to 95%) = 23 seconds ...

From equations (3.58) and (3.59) the unit step response for the ship autopilot eontrol system is given by the expression... 

Figure 4.38 shows the system unit step response. From Figure 4.38... 

A sampler (i.e. an A/D eonverter) is represented by a switeh symbol as shown in Figure 7.5. It is possible to reeonstruet f t) approximately from f t) by the use of a hold deviee, the most eommon of whieh is the zero-order hold (D/A eonverter) as shown in Figure 7.6. From Figure 7.6 it ean be seen that a zero-order hold eonverts a series of impulses into a series of pulses of width T. Flenee a unit impulse at time t is eonverted into a pulse of width T, whieh may be ereated by a positive unit step at time t, followed by a negative unit step at time (t — T), i.e. delayed by T. 

If the sampling time is one seeond and the system is subjeet to a unit step input funetion, determine the diserete time response. (N.B. normally, a zero-order hold would be ineluded, but, in the interest of simplieity, has been omitted.) Now... 

Find the pulse transfer function and hence calculate the response to a unit step and unit ramp. T = 0.5 seconds. Compare the results with the continuous system response Xo t). The system is of the type shown in Figure 7.9(b) and therefore... 

Table 7.2 shows the discrete response x ikT) to a unit step function and is compared with the continuous response (equation 3.29) where... 

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