## Steady State

Fig. 18-3. Time Response of a Simple Second-Order System With N0/K= 1. See text for explanation increases, the overshoot and the number of oscillations decrease, and the system eventually attains a steady-state value equal to N0/K. The gain K is chosen to achieve a specified steady-state error for an assumed magnitude of the disturbance torque, N0. When i/2 /2<p< 1, there is only one overshoot and no undershoots. When p> 1, the system is overdamped and acts as a simple first-order system. If p= 1, the system is critically damped. In many applications, overshoots are undesirable. However, if we choose a value of p> 1, die response of the system is slow; therefore, we will consider the value of p= 1 (critically damped).* For this case, Eq. (18-13) reduces to em i

The performance of a control system is generally expressed in terms of acceptable steady-state, error for a specified disturbance. The steady-state error is defined as the difference between the desired output, 9REF, and the actual output. The maximum steady-state error is determined using the final value theorem (see Appendix F) as

As an example of pitch control design, we will consider a solar radiation pressure torque of the order of 10~8 N-m (typical for MMS satellites). The steady-state error may be calculated using Eqs. (18-13) and (18-18) with t(ND) = 10"8j as

where K is in N • m. Using this expressiQn, the value of the pitch gain is chosen so that the steady-state error is within the given constraints. Having chosen the pitch gain, we can then calculate the lead time constant, r, of the pitch control system to achieve a desired damping rado from Eq. (18-12). The value of r so determined should be significantly smaller than the orbital period of the satellite.

A third common control law is bang-bang control defined by

where N^ is the maximum control torque and 9 is the angular error. An example of this law is the atdtude control of a spacecraft using jets to apply a constant torque in a direction to null the attitude error. The block diagram for a bang-bang control system is shown in Fig. 18-4. The control torque depends only on the sign of the difference between the desired and the actual output.

A block diagram for a bang-bang-plus-dead zone controller is shown in Fig. 18-5. Here the control torque is characterized by a dead zone followed by a

•The damping ratio p= 41/2 is frequently chosen because of its desirable frequency response characteristics. See Section 18.4 and DiStefano, el al^ [1967].

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