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                                    G(s) = s(s + 3) s + 6~
so that the closed-loop characteristic equation of the observer system is given by
                                                S3 + 90S2 + 2000s +10,000 - O
[;]
[;]
                                          6
Airplane Response and Closed-Loop Control
6.1 Introduction
    In Chapter 4, we studied various coordinate systems used in airplane dynamics
and derived equations of motion applicable for small disturbances. We also dis~
cussed the methods ofevaluating various stability and control derivatives appearing
in those equations. Under the assumption of small disturbance, the equations of
motion could be grouped into two sets of three equations each: one set for longitu-
dinal motion and another set for Iateral-directional motion of the aircraft. This kind
ofdecoupling enables us to study separately thelongitudinal and lateral-directional
response and closed~loop control.ln Chapter 5, we studied the basic principles of
linear system theory and design of closed-loop control systems.
    In this chapter we will discuss the solution of the small-disturbance equations
to determine the airplane fesponse. The airplane response depends on the initial
conditions and the input time history. The response to a given set of initial con-
ditions with zero input is called the natural or free response. The free response
is indicative of the transient behavior or the dynamic stability of the system. The
initial conditions are equivalent to suddenly i~posed disturbances. For example,
the response with u(0) - 0, Aa(0) : 5 deg, and q - AO -. 0 is equivalent to
the response for a suddenly imposed vertical gust that momentarily increases the
airplane angle of attack by 5 deg.
   The forced response is the solution of equations of motion with zero initial
conditions and a given input time history. The forced response is indicative of an
airplane's steady-state behavior. The common input test functions used to obtain
the forced response are the unit-step and impulse functions. For example, airplane
longitudinal response to a unit-step function describes the motion of the airplane to
a sudden unit d2fiection of the elevator. The steady-state solution gives the cor-
responding steady-state values of forward velocity, angle of attack, attitude, and
pitch rate.
       Finally, we will discuss the closed- loop control of the airplane to obtain desired
 level of handling qualities. The closed-loop control systems used for obtaining the
 specified Ievel of free response are called st~bility augjmentation systems, and those
used to establish and hold the desired flight conditions are called autopilots.In this
 chapter, we will discuss some of the important stability augmentation systems and
autopilots.
6.2 LongitudinaIResponse
     In this section we will discuss the longitudinal response of the airplane. We will
discuss two types of responses: 1) the free response and 2) the forced response.
The free response corresponds to the solution with a given set of initial conditions
and zero input and is indicative of the dynamic stability of the system. The forced
537
538          PERFORMANCE, STABILITY,.DYNAMICS, AND CONTROL
response corresponds to the solution with zero initial conditions and a given input
time history. For free response, we assume that the elevator is held fixed (stick-
fixed), and, for forced response, we assume that the elevatoris moved in a specified
manner to a new position and subsequently held fixed at that position.
   The longitudinal equations of motion for elevator control are given by Eqs.
(4.417-4.419),
 (m,dd _ Cx,.)u - (Cx,., ft + Cx.)A   - (CxqC,dd + C.,)A0 : CxteA8e
                     (6.1)
-Cz"u + [(m, dd  _ CzdCI dd )  - Cz  ] A,, - (m, dd + CzqC, dd  + Cz0) AO
= Cz3e A8e
(6.2)
　
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