The block diagram for a bang-bang control system is shown in Fig. 19-11. We will consider the basic bang-bang control law and several variations of it following the development of Hsu and Meyer [1968], who give a more extensive discussion of the subject with additional cases. .

Position-Only Control. As a simple case, consider position-only feedback, where the rate gain or amplification, a, is zero. The sampled output angle, 9, is fed back as input control signal by the function u(f)= - l/signfl(i) (19-65)

Because usually both position and rate are of interest, the attitude behavior can be conveniently visualized in state-space, or phase space, as it is more commonly called in physics. For this single-axis control system, state-space becomes the state plane of position versus rate. The time history of the state parameters 0 and 9 determine a trajectory in the state plane. The equation of motion is obtained by integrating Eq. (19-64) under the condition of Eq. (19-65). The state-plane trajectories of the system shown in Fig. 19-12(a) will consist of a set of parabolas given by fj±W=c, for u= ± U (19-66)

where c is a constant. They are connected about the 9 axis at the switching line as shown in Fig. 19-12(a). The 6 axis is called the switching line because u= + U to the left of it and u<= — U to the right of it. Thus, the wheel torque motor control signal will change sign as the state trajectory (9,9) crosses the switching line. Each state-plane trajectory is closed, with its size depending on the initial condition. Physically, the spacecraft will oscillate indefinitely about the equilibrium condition, without achieving the target attitude at the desired zero rate condition.

Position-PliB-Rate Control Adding a term to the control error signal which is proportional to the attitude rate provides damping and has the effect of a lead network in electrical systems in predicting the state at a future time. For positionplus-rate control the switching function becomes u{t)=-Usiga(0+a0) (19-67)

and the switching line, instead of being the 9 axis, becomes the straight line 0+a0 = 0, with slope -1 /a.

The state trajectories are found from integrating Eq. (19-64) with o>= — U for (0+a0)> 0 and with u= + U for (0+a0)<0. The trajectories will again be families of parabolas whose curvature, or acceleration, changes sign at the switching lines as shown in Fig. 19-12(b). A system originally at A will follow the trajectory shown, reversing control at B and again at C, spiraling in toward the center. As it approaches the origin from C, however, the system trajectory crosses the switching line after shorter and shorter time intervals, causing the control relay to rapidly switch states. This condition is called chattering, and although the system will continue to move toward the origin in a damped fashion, it could lead to actuator wear. In the discrete version of this system, the relay remains on for a

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