Page 127 - Modern Control Systems

Section 2.8 Design Examples 101

where </> = tan (<w/ft). So, as t —> oo, we have

AQ 2 (0 ^ / ! 2 a 2 sm(<ot - ¢).
V f i + w
The maximum change in output flow rate is

lA^Olmax = - 7 = = 5 . (2.124)
V f i 2 + a) 2
The above analytic analysis of the linear system model to step and sinusoidal
inputs is a valuable way to gain insight into the system response to test signals. An-
alytic analysis is limited, however, in the sense that a more complete representa-
tion can be obtained with carefully constructed numerical investigations using
computer simulations of both the linear and nonlinear mathematical models. A
computer simulation uses a model and the actual conditions of the system being
modeled, as well as actual input commands to which the system will be subjected.
Various levels of simulation fidelity (that is, accuracy) are available to the con-
trol engineer. In the early stages of the design process, highly interactive design soft-
ware packages are effective. At this stage, computer speed is not as important as the
time it takes to obtain an initial valid solution and to iterate and fine tune that solu-
tion. Good graphics output capability is crucial. The analysis simulations are gener-
ally low fidelity in the sense that many of the simplifications (such as linearization)
made in the design process are retained in the simulation.
As the design matures usually it is necessary to conduct numerical experiments
in a more realistic simulation environment. At this point in the design process, the
computer processing speed becomes more important, since long simulation times
necessarily reduce the number of computer experiments that can be obtained and
correspondingly raise costs. Usually these high-fidelity simulations are programmed
in FORTRAN, C, C++, Matlab, Lab VIEW or similar languages.
Assuming that a model and the simulation are reliably accurate, computer sim-
ulation has the following advantages [13]:

1. System performance can be observed under all conceivable conditions.
2. Results of field-system performance can be extrapolated with a simulation model for
prediction purposes.
3. Decisions concerning future systems presently in a conceptual stage can be examined.
4. Trials of systems under test can be accomplished in a much-reduced period of time.
5. Simulation results can be obtained at lower cost than real experimentation.
6. Study of hypothetical situations can be achieved even when the hypothetical situation
would be unrealizable at present.
7. Computer modeling and simulation is often the only feasible or safe technique to
analyze and evaluate a system.
The nonlinear model describing the water level flow rate is as follows (using the
constants given in Table 2.7):
3
H = -0.0443 V / 7 + 1.2732 X 10" Q b (2.125)
= 34.77Vtf.
Q 2