Page 140 - Modern Control Systems

Chapter 2 Mathematical Models of Systems

We begin this section by analyzing a typical spring-mass-damper mathematical
model of a mechanical system. Using an m-file script, we will develop an interac-
tive analysis capability to analyze the effects of natural frequency and damping
on the unforced response of the mass displacement. This analysis will use the fact
that we have an analytic solution that describes the unforced time response of the
mass displacement.
Later, we will discuss transfer functions and block diagrams. In particular, we
are interested in manipulating polynomials, computing poles and zeros of transfer
functions, computing closed-loop transfer functions, computing block diagram re-
ductions, and computing the response of a system to a unit step input. The section
concludes with the electric traction motor control design of Example 2.14.
The functions covered in this section are roots, poly, conv, polyval, tf, pzmap,
pole, zero, series, parallel, feedback, minreal, and step.
Spring-Mass-Damper System. A spring-mass-damper mechanical system is
shown in Figure 2.2. The motion of the mass, denoted by y(t), is described by the dif-
ferential equation

My{t) + by{t) + ky{t) = r(t).
The unforced dynamic response y{t) of the spring-mass-damper mechanical
system is
2
y(t) = J^L_e-^' sin^Vl - £ t + Q\

-1
where o) n - vk/M, £ = b/(2vkM), and 6 = cos £. The initial displacement is
y(0). The transient system response is underdamped when £ < 1, overdamped
when £ > 1, and critically damped when £ = 1. We can visualize the unforced time
response of the mass displacement following an initial displacement of y(0). Consider
the underdamped case:

^
• 7 y(0) = 0.15 m, a> n = V 2 — , £ = -^= ( r = 2,-£-= 1 ).
' sec' 2V2 \M M J
The commands to generate the plot of the unforced response are shown in Figure 2.50.
In the setup, the variables y(0), w„, f, and £ are input at the command level. Then the
script unforced.m is executed to generate the desired plots. This creates an interac-
tive analysis capability to analyze the effects of natural frequency and damping on
the unforced response of the mass displacement. One can investigate the effects of
the natural frequency and the damping on the time response by simply entering new
values of <a n and £ at the command prompt and running the script unforced.m again.
The time-response plot is shown in Figure 2.51. Notice that the script automatically
labels the plot with the values of the damping coefficient and natural frequency. This
avoids confusion when making many interactive simulations. Using scripts is an im-
portant aspect of developing an effective interactive design and analysis capability.
For the spring-mass-damper problem, the unforced solution to the differential
equation was readily available. In general, when simulating closed-loop feedback