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Fig. 18-1. Block Diagram of a General Spacecraft Attitude Feedback Control System. The system output is a measured angle 9M which is to be controlled near a reference or desired value 0REF. The controller issues a torque based on the error signal, 9 = 9M- 9REr, to control the effect of the disturbance torques on the spacecraft dynamics.

If the switch in Fig. 18-1 is open, we have an open-loop system in which the controller response is independent of the actual output. For example, the issuance of magnetic control commands from the ground is an open-loop procedure. Conversely, if the switch is closed, we have a closed-loop or feedback system in which the input to the controller is modified based on information available from the actual output. For example, for the attitude control of a three-axis stabilized, Earth-oriented spacecraft, we may continuously monitor pitch and roll angles (and often rates)-by attitude sensors (and gyros) and provide this information to the controller, which computes commands according to a control law and issues these commands to a torquing device or actuator. A control law is a principle on which the controller is designed to achieve the desired overall system performance.

The input-output relation of each element of the control system (i.e., the controller, plant, or feedback) is generally defined in terms of a transfer function (see Section 7.4). This idea of representing a physical system is a natural outgrowth of Laplace transform operational methods to solve linear differential equations (see Appendix F). The transfer function of each system element is defined as the ratio of the Laplace transform of its output to the Laplace transform of the input, assuming that all initial conditions are zero. Generally, the transfer function is represented as the ratio of two polynomials in s, as

The m values of s, for which n(s) is zero, are known as the zeros of G(i) and the n values of s, for which d(s) is zero, are known as the poles of G(s). The transfer function, G(s), thus has m zeros and n poles.

The transfer function of the plant element may be obtained by taking the Laplace transform of the equation which describes the system dynamics. For example, if the equation describing the plant is

10= N (18-1) where I is a constant, it may be transformed to obtain

/r2 £(*)=£( AT) (18-2) Thus, the transfer function, G(s), for the plant described by Eq. (18-1) is e (output) em ,

The transfer function of the feedback element commonly describes a filtering, smoothing, or calibration of the sensed output signal; however, in this section we will assume that the measured and reference angles are compared directly and thus the feedback transfer function is unity.

The transfer function of the controller is obtained by first relating the control torque to the error signal in terms of a control law. The simplest of the control laws is proportional control, for which

where Nc is the control torque and K is the system gain. Proportional control is rarely used because it results in large oscillations in 0.

A common method for spacecraft attitude control is a position-plus-rate control law for which

Here, the control torque, Nc, is directly proportional to the error signal and its time derivative. The KXB term provides damping. However, more sophisticated instruments, such as rate gyros, are needed to implement this control law. The transfer function for this controller is

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