Unit step response

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

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

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

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... 

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... 

Example 3.2 Using the first order Pade approximation, plot the unit step response of the first order with dead time function ... 

Example 3.3. Make use of Eq. (3-37), show how the unit step response Cn(t) becomes more sluggish as n increases from 1 to 5. 

First order function unit step response... 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

With MATLAB, try do a unit step response of a lead-lag element in as in Eq. (3-49). 

The MATLAB statements to do the unit step response are already in Example 3.4. You may repeat the computation with a different time constant. The statements to attempt fitting the five tanks-in-series response are ... 

We can compare the unit step responses of the two transfer functions with... 

Now, go to the LTI Viewer window and select Import under the File pull-down menu. A dialog box will pop out to help import the transfer function objects. By default, a unit step response will be generated. Click on the axis with the right mouse button to retrieve a popup menu that will provide options for other plot types, for toggling the object to be plotted, and other features. With a step response plot, the Characteristics feature of the pop-up menu can identify the peak time, rise time, and settling time of an underdamped response. 

The transfer functions G and H will be imported automatically when the LTI Viewer is launched, and the unit step response plots of the two functions will be generated. 

In this case, the LTI Viewer will display both the unit step response plot and the Bode plot for the transfer function G. We will learn Bode plot in Chapter 8, so don t panic yet. Just keep this possibility in mind until we get there. 

The step () function also accepts state space representation, and to generate the unit step response is no more difficult than using a transfer function ... 

If this is not enough to convince you that everything is consistent, try step o on the transfer function and different forms of the state space model. You should see the same unit step response. 

As another example, consider the problem of estimating the gain K and time constants t/ for first-order and second-order dynamic models based on a measured unit step response of the process y(t). The models for the step response of these two processes are, respectively (Seborg et al., 1989),... 

Note that this is Jiot a hrsl-order lag because of the negative sign in the denominator. The system has an openloop pole in the RHP at s = -I- 1/tj,. The unit step response of this system is an exponential that goes olT to inhnity as time increases. 

Figure 3.7. Unit-step responses of an instrument (a) and of a high-frequency filter (b) in the same frequency range 1 - no filter 2 - with a filter 3 - ideal filter characteristic 4 - approximation by a second order Butterworth filter.

The modulus of the gain factor of such a filter (Butterworth) is shown in Fig. 3.7 b (curve 4). The total unit-step response of a measuring instrument equipped with a filter determined by the second-order expression (3.12) is presented in Fig. 3.7a (curve 2). Verification tests of the method have given good agreement between theory and experiment. 

Figure 4 Typical open-loop unit step responses, (a) First order, (b) Second (or higher)...

It can be shown that the unit step response of n tanks in series with mixing lag and no delays is given by... 

The feedforward controller contains a steady-state gain and dynamic terms. For this system the dynamic element is a first-order lead-lag. The unit step response of this lead-lag is an initial change to a value that is (- KJKm) tmItl), followed by an exponential rise or decay to the final steady-state value — KJKm- ... 

The MASIAB program below produces the unit-step response curve for the reference input eus shoun below. 

SiiKse the two closed-loop transfer fimctions are identical e get the unit-step response curves for the two systems. The following MATLAB program produces the response to tbs unlt-st disturbance iiput. The reeultiag response curve is shown belcm. 

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