Ramp input

Fig. 2.31 depicts the responses of the various control modes and their combinations to step and ramp inputs. 

Example 13. The overdamped system of Example 6.8 is now forced with a ramp input ... 

If the steadystate error is to go to zero, the term l/s(l + Gm, <,)) must go to zero as s goes to zero. This requires that B(,)Gj f(,) must contain a 1/s term. Double integration is needed to drive the steadystate error to zero for a ramp input (to make the output track the changing setpoint). 

To determine the speed of response of the instrument, a step change in the pH of the fluid around the electrode would have been ideal. It was not possible to do this experimentally in less than 0.030 sec, owing to the time required to wash out the electrode chamber. However, it was possible to estimate this speed using a ramp input, namely, the linear increase in pH produced by the dehydration of carbonic acid (I, 2, 5, 9) ... 

Ramp input. Gradually open the steam valve such that Q increases linearly. Ramp ends when the steam valve is fully open. 

The testing of polymers requires unique understanding of the viscoelastic nature of polymers. For example in a creep test it is required to suddenly apply a constant tensile, compression, or torsion stress to a bar of material. The most common description of a uniaxial tensile creep test is shown in Fig. 5.5(a). Several questions may arise one of which is How is the load to be applied suddenly without causing dynamic effects. One answer is for the load to be applied as ramp input as shown in Fig. 5.5(b). Obviously, the latter case is not a correct creep test. How big an error is involved A solution of the differential equation representative of the material for the ramp input of Fig 5.5(b) can be obtained and it can be shown that the error in the strain output is negligible if the loading time, to, is small compared to the retardation time of the material, x. 

Fig. 5.5c. Again, however, to avoid dynamic effects it is usual to use a ramp input of strain as shown in Fig. 5.5d and it can be shown that the error is negligible if the ramp time, to, is small compared to the relaxation time of the material, x (see homework problem 5.5).

The three important inputs discussed above—step, ramp, rectangular pulse—are depicted in Fig. 5.2. Note that many types of inputs can be represented as combinations of step and ramp inputs. For example, a unit height (isosceles) triangular pulse of width tyy can be constructed from three ramp inputs, as shown in Fig. 5.3. In this case, we write a single expression for the triangular pulse function... 

Figure 5.3 How three ramp inputs can be combined to form a triangular pulse.

We now evaluate how a first-order system responds to the ramp input, U(s) = a/ of Eq. 5-8. Performing a partial fraction expansion yields... 

Equation 5-22 implies that after an initial transient period, the ramp input yields a ramp output with slope equal to Ka, but shifted in time by the process time constant t (see Fig. 5.5). An unbounded ramp input will ultimately cause some process component to saturate, so the duration of the ramp input ordinarily is limited. A process input frequently will be ramped from one value to another in a fixed amount of time so as to avoid the sudden change associated with a step change. Ramp inputs of this type are particularly useful during the start-up of a continuous process or in operating a batch process. 

Any measurement transducer output contains some dynamic error an estimate of the error can be calculated if transducer time constant t and the maximum expected rate of change of the measured variable are known. For a ramp input, x(t) = at, and a first-order dynamic model (see Eq. 9-15), the transducer output y is related to x by ... 

For a ramp input it takes a period of time for the proportional action to reach the same level as the derivative action. This period of time is called the derivative time and is measured in minutes. Increasing the derivative time 7d increases MVd, or the contribution of the derivative action to the movement of the final control element. 

Figure 3.7. Calculated Permanent Memory Stress-Strain Response to an Interrupted Ramp Input...

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