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Cnr at M - 0.3 and a -. 8 deg using strip theory and compare your results with
those obtained using Datcom methods. Assume that the center ofgravity (moment
reference point) is located 0.2G ahead of the aerodynarnic center and the wing
root chord is located I m below the center of gravity (zw - 1.0 m).
4.16     Estimate the vertical tailcontribution to C,P, Cnp, C,r, Cnr, Cyr,, Ci/.i dand Clt,
for an aircraft with following data: wing sweep = 30 deg, wing mean aerodynamic
chord c = 2.5 m, wing dihedral = 3 deg, wing aspect ratio = 4, wing taper ratio =
0.5, wing span :  12 m,vertical taillength - 2.5 c,ay  = 0.08/deg, zv  =  0.85 m, and
ratio of vertical tail area to wing area = 0.25. Assume vertical tail parameter k =
0.84 and that-the aircraft is operating at an angle of attack of 6 deg and M N O.l0.
[Answer: Clp = -0.001908, Cnp ~ -0.06191, CLr - 0.0185, Cnr - -0.09464,
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
　
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