曝光台 注意防骗
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C.:\;B = -3.12 x lO-s, CtB = -4.9913 x 10-7, and Cnti = 1.6387 x 10-5. All
values are per radian.l
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Linear Systems, Theory, and Design:
A Brief Review
5.1 Introduction
Generally, a dynamical system is characterized by a differential equation that
gives a relation between the input and the output of that system. A dynamical
system may be linear'or nonlinear. It is said to be linear if the differential equation
that characterizes the system is linezu:
A differential equation is linear if the coefficients are constants or functions
of only the independent variable and not that of the dependent variable. The
most important property of the linear systems is the applicability of the prin-
ciple of superposition, i.e, if yi(t) and Y2(t) are two solutions to inputs ri(t)
and T2(t), then the solution to the new input r(t) = ciri(t) + c2r2(t) is given
by y(t) = ciyi(t) + c2Y2(t). This feature enables us to build system response to
any complex input function by expressing it as a sum of several simple input
functions.
A system is said to be nonlinear if the differential equation that characterizes it
is nonlinear. A differential equation is nonlinear ifit contains products or powers
of the dependent variable or its derivatives. Nonlinear differential equations, in
general, are quite difficult to solve. Furthermore, the property of superposition
does not hold for nonlinear systems.
A control system may be an open- or closed-loop system. An open loop system
is onein which the output has no 9ffect on theinput.ln other words,in an open-loop
system the output is n9t fed back for comparison with the input for regulation. An
open-loop control can be used in practice if the relation between the output and
the input is precisely known andl ~e system is not subject to internal parameter
variations or external disturbances.A closed-loop systemis one in which the output
is measured and is fed back to the input for comparison and system regulation. An
advantage ofthe closed-loop or feedback systemis that the system response will be
relatively insensitive to internal parameter variations or external disturbances. For
open-loop systems, stabilit)r is not a major concem. However, it is ofmaj or concern
for a closed-loop system because a closed-loop system may tend to overcorrect
itself and in that process develop instability.
In this chapter, we will re'view the basic principles oflinear time-invariant sys-
tems and the7representationin the transfer function form using Laplace transform.
We will also discuss system response to standard inputs such as unit-step function
and derive expressions for steady-state errors. Furthermore, we will briefly discuss
the frequency response and stability of closed- loop systems and the design of com-
pensators. Finally, we will grve a brief exposure to modern state-space analyses
and design methods.
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440 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
5.2 LaplaceTransform
For linear systems, the application of Laplace transform enables us to express
the given differential equation in an algebraic form that gready simplifies the
analyses of control systems. Obviously this type of simplification is not possible
for nonlinear systems. In view of this, one often introduces what is called an
equivalent linear system. Such a linearized system is valid for only a limited range
of parameter 'values. The linearization process may have to be repeated several
l .
times to cover the entire range of parameter values ofinterest.
In this section, we will briefiy review the main results on Laplace transform
that are useful in the analyses and design of aircraft control systems. For more
information, the-reader may refer to the standard texts on linear systems.l-3
The Laplace transform of a funcr"ion f (t) is defined as
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