# PART 4: Closed-Loop Control Model

An autopilot mechanism and the airplane it controls constitute a closed-loop (feedback) control system. The purpose of the autopilot is to maintain a specified heading, altitude, and speed. It accomplishes this objective by continuously measuring the aircrafts actual heading, altitude, and speed, and then bringing the airplane to the desired state by automatically adjusting control surfaces (e.g., rudder, ailerons, elevators) and throttle. The signals delivered to the control surface actuators are designed to effect a reduction in the computed differences between the desired and the measured value of some state variable (e.g., heading).

## The Control Laws

Equations 3-33, 3-34, and 3-35, the equations of motion, describe the flight dynamics of the aircraft state variables, . The form of the compensation applied to the flight system derives from this state-variable description. The compensation ensures that the state of the aircraft smoothly follows the reference inputs (desired heading, flight path angle, airspeed). The control laws, from which compensations are derived, are linear functions of these same three state variables.

Solving the equations of motion to obtain aircraft state requires knowledge of bank angle, and of the forces acting on the aircraft; namely, thrust, lift, and drag. Conventional models, empirically derived, which reasonably portray the aircrafts time response to applied controls (e.g., bank angle response to movements generated by ailerons) have been adopted for use here. The simple first-order differential equations which are used to model the dynamic behavior of thrust, lift, and bank angle are,

 (4-1) (4-2) (4-3)

where is the throttle command, is the lift command, and is the bank angle command. It is the throttle, lift, and bank angle commands that force the aircraft to follow the desired heading, flight path angle and airspeed. The gains are adjusted to give realistic response. Values for these gains are determined by aircraft characteristics, but nominal values of 2.0, 0.75, and 1.0, respectively, are reasonable values from which to begin the search for optimum responses.

Values of aerodynamic drag are also obtained from an empirical model. The equation relating drag to airspeed is,

 (4-4)

with nominal values for the coefficients and selected to be 3.8173x10-2 and 2.4789x10-2, respectively.

The control laws provide for feedback of the state variables (i.e., aircraft heading, flight path angle and airspeed) to the equations of motion through the throttle, bank angle, and lift commands. The control laws can be expressed as a linear combination of the state variables as,

 (4-5) (4-6) (4-7) (4-8) (4-9)

where , , and are commanded airspeed, flight path angle, and aircraft heading, respectively. It is natural and convenient to express the reference or desired values of airspeed, heading and flight path angle relative to the locally-level frame of reference without regard to a consideration of prevailing winds. The values of , , and are related to their associated desired or reference values, , , and according to the equations.

 (4-10) (4-11) (4-12)

where,

 (4-13)

## Computational Processes

The essential elements of the closed-loop control process is illustrated in the diagram of Figure 4.1. Comparison of an intended flight plan with the aircraft state yields state deviations. These deviations serve as input to control laws which relate them to state variable corrections. State variable corrections are used to drive control surface actuators and to compute the throttle commands used to control engine thrust.