F IGURE 1 is a block diagram of a single axis attitude control system in which an attitude signal, 6*, is derived by integrating the signal from a precision rate gyro. A stabilizing signal e* is also derived in the computer e*=K R a*+K P 0* where e* is used to determine the times at which the thruster is fired, the direction of the thrust, and the duration of the thrust pulse. These three items of intelligence are represented as the valve command, r c , in the diagram. The thruster is characterized by the strength of the thrust, A (assumed constant during the pulse), a time delay r D between the instant a turnon command is received and the instant thrust actually occurs, and a minimum time, r 0 , which the pulse width modulator may command as thruster on-time, o> is the angular rate of the vehicle about its controlled axis and is related to the thrust M in the usual way: M=/d>. The gyro output signal m^ is dynamically related to the vehicle rate by where f is the damping ratio and u n the undamped natural frequency of the output axis of the gyro. Tis the cycle time of the computer, and the analog to digital conversion is represented by a zero order hold, the output of which is co* g , a piecewise constant representation of co^.A conventional method of control for this system is to program the computer to issue a thruster command at each sample instant. The pulse width command at instant t k is determined from e*(t k ) according to the curve in Fig. 2. A dead zone of total width 2Wis employed to prevent the valve from responding to noise which might exist on the e* signal. The parameters W, S, K R , K P , 7*,f, and u n may then all be adjusted to provide acceptable performance of the system where T D , T O > A, and / are considered fixed.Insofar as limit cycle performance is concerned these seven parameters are adjusted to minimize control fuel expenditure (total valve on-time) while maintaining acceptable limits on the excursions of co and 6 during the limit cycle oscillation. The conventional control scheme described above leads to a certain optimum limit cycle performance represented qualitatively by the e* (t) signal in Fig. 3. Here that portion of the limit cycle during which e*(0 reaches its peak is shown. Prior to t 0 the vehicle is drifting at a constant rate. t 0 is the sample instant at which e* (t) first exceeds the dead-zone value W. A pulse width is calculated from Fig. 2 and a thruster command is issued as soon as this calculation is completed. There is a delay, r D , between the initiation of the thruster command and the actual occurance of thrust. Also, the gyro does not DIGITAL COMPUTER 1
