Question 13.4: Figure 13.17 shows a satellite with a reaction wheel mounted...
Figure 13.17 shows a satellite with a reaction wheel mounted along its 3 axis. The satellite’s moment of inertia about the 3 axis is I_{\text {sat }}=500 \mathrm{~kg}-\mathrm{m}^{2} (not including the inertia of the reaction wheel), and the wheel’s inertia about its spin axis is I_{w}=0.06 \mathrm{~kg}-\mathrm{m}^{2}. The DC motor that drives the reaction wheel has the following parameters: motor torque constant K_{m}=0.04 \mathrm{~N}-m / \mathrm{A}, back-emf constant K_{b}=0.04 \mathrm{~V}-s / \mathrm{rad}, resistance R=3 \Omega, and friction coefficient b=6\left(10^{-5}\right) \mathrm{N}-\mathrm{m}-s/rad. The DC motor has voltage limits of \pm 28 \mathrm{~V}. Design a PD controller for the reaction wheel that can provide a single-axis attitude maneuver with good damping and fast response speed. Demonstrate the closed-loop control system with an attitude maneuver starting from \phi_{0}=0, \dot{\phi}_{0}=0 with a target attitude \phi_{\text {ref }}=\pi / 3 (i.e., 60^{\circ} ). Because the satellite begins and ends with zero angular velocity, this maneuver is known as a “rest-to-rest maneuver.”
Figure 13.20 shows the closed-loop attitude control system using the reaction wheel. The PD controller transfer function is
G_{C}(s)=K_{P}+K_{D} s
The attitude error \phi_{e} (in rad) is the input to the controller, and its output is the voltage signal to the DC motor:
e_{\text {in }}=K_{P} \phi_{e}+K_{D} \dot{\phi}_{e}
Clearly, the PD controller gains have units of V/rad \left(K_{P}\right) and \mathrm{V}-\mathrm{s} / \mathrm{rad}\left(K_{D}\right). Equations (13.45) and (13.46)
Proportional gain: K_P=\frac{\omega_n^2}{a}=\frac{-R I_{\text {sat }} \omega_n^2}{K_m} (13.45)
Derivative gain: K_D=\frac{2 \zeta \omega_n}{a}=\frac{-2 R I_{\mathrm{sat}} \zeta \omega_n}{K_m} (13.46)
show that the gains can be obtained by specifying the desired undamped natural frequency \omega_{n} and damping ratio \zeta. However, we cannot “over gain” the controller because the motor’s input limit is \pm 28 \mathrm{~V}. Let us select a “good” damping ratio \zeta=0.7 and a settling time of t_{S}=250 \mathrm{~s}. Using the definition of settling time, Eq. (13.18), we can compute the undamped natural frequency:
\omega_{n}=\frac{4}{t_{S} \zeta}=4 /(250 \mathrm{~s})(0.7)=0.0229 \mathrm{rad} / \mathrm{s}
We use Eqs. (13.45) and (13.46) to obtain the PD controller gains:
\begin{aligned} & \text { Proportional gain: } K_{P}=\frac{-R I_{\mathrm{sat}} \omega_{n}^{2}}{K_{m}}=-19.5918 \mathrm{~V} / \mathrm{rad} \\ & \text { Derivative gain: } K_{D}=\frac{-2 R I_{\mathrm{sat}} \zeta \omega_{n}}{K_{m}}=-1,200 \mathrm{~V}-\mathrm{s} / \mathrm{rad} \end{aligned}
Recall that Eqs. (13.45) and (13.46) were derived using a simplified model of the reactionwheel system that ignored the friction torque and back-emf. At time t=0, the attitude error is \phi_{e}(0)=\phi_{\mathrm{ref}}-\phi_{0}=\pi / 3 \mathrm{rad}, and hence the motor’s initial voltage input is e_{\text {in }}(0)=K_{P} \phi_{e}(0)+K_{D} \dot{\phi}_{e}(0)=-20.52 \mathrm{~V} which is within the \pm 28 \mathrm{~V} limits (recall that the satellite’s initial angular velocity is zero).
We can use MATLAB’s Simulink to numerically simulate the closed-loop attitude control system presented in Figure 13.20 (we should again emphasize that although the PD controller was designed using approximate models, its performance is tested using the accurate modeling equations for the DC motor and reaction wheel). Figure 13.23 shows the satellite’s attitude response. Note that the satellite achieves the desired 60^{\circ} reference angle in about 230 \mathrm{~s} without any discernable overshoot. Figure 13.24 shows the angular momentum contributions due to the angular velocity of the complete satellite \left(I_{3} \omega_{3}\right) and spinning reaction wheel \left(I_{w} \omega_{w}\right). Note that both angular momentum components start at zero (the satellite and wheel are initially at rest), and that they are “mirror images” of each other because the total angular momentum is conserved and therefore remains zero, that is, H=I_{3} \omega_{3}+I_{w} \omega_{w}=0. The wheel spins with negative angular velocity (i.e., opposite the 3 axis) so that the satellite spins with positive angular velocity. Because the reaction wheel’s moment of inertia is very small \left(I_{w}=0.06 \mathrm{~kg}-\mathrm{m}^{2}\right) relative to the moment of inertia of the complete satellite and wheel \left(I_{3}=500.06 \mathrm{~kg}-\mathrm{m}^{2}\right), it must spin at a much higher rate in order to counteract the satellite’s angular momentum. For example, the wheel’s peak angular velocity is approximately -742 \mathrm{rpm} at about t=45 \mathrm{~s}, whereas the satellite’s peak angular velocity is about 0.09 \mathrm{rpm}(\approx 0.5 \mathrm{deg} / \mathrm{s}). Finally, Figure 13.25 shows the input voltage to the DC motor during the closed-loop response. The initial voltage of -20.5 \mathrm{~V} (due to the initial attitude error) is observed, and the motor voltage (and consequently the motor torque) go to zero as the satellite comes to rest at the desired 60^{\circ} attitude.
We can speed up the closed-loop attitude response by increasing the PD controller gains. However, we must keep the motor’s input voltage within the \pm 28 \mathrm{~V} limits. We do this by inserting a saturation block (or “limiter block”) immediately after the PD controller block as shown in Figure 13.26. The saturation block will allow a voltage command \left(e_{\mathrm{CMD}}\right) that is between \pm 28 \mathrm{~V} to reach the DC motor unaltered. However, the saturation block will limit the motor voltage e_{\text {in }} to +28 \mathrm{~V} or -28 \mathrm{~V} if the controller commands an excessive voltage (i.e., \left|e_{\mathrm{CMD}}\right|>28 \mathrm{~V} ). We can repeat this example with a settling time of 60 \mathrm{~s} and damping ratio \zeta=0.7; the resulting PD gains are K_{P}=-340.1361 \mathrm{V} / \mathrm{rad} and K_{D}=-5,000 \mathrm{~V}-s / \mathrm{rad}. Figure 13.27 shows that the satellite’s attitude reaches its 60^{\circ} reference in about 100 \mathrm{~s} (less than half the time of the previous case) and with a slight overshoot. Figure 13.28 shows that the satellite and reaction wheel both store more than twice as much angular momentum for this high-gain case when compared with the initial low-gain case. For example, the peak angular velocity of the reaction wheel is -2,090 \mathrm{rpm} and the satellite’s peak spin rate is 0.25 \mathrm{rpm}(1.5 \mathrm{deg} / \mathrm{s}). While the wheel’s peak spin rate is quite large, many reaction wheels have maximum angular velocities as high as 5,000 rpm. Figure 13.29 presents the input voltage to the DC motor. Here we see that the high controller gains and saturation block initially produce the largest possible (negative) input voltage (-28 \mathrm{~V}) to the motor in order to provide the greatest (negative) torque and angular acceleration for the wheel. At t=42 \mathrm{~s}, the controller’s damping term K_{D} \dot{\phi}_{e} causes the input voltage to switch toward +28 \mathrm{~V} so that the motor torque slows down the wheel’s spin rate as seen in Figure 13.28. For t>72 \mathrm{~s}, the motor’s input voltage is no longer saturated at its maximum value.
This example has demonstrated that we can adjust the PD controller gains to speed up the closed-loop attitude response. Consequently, many closed-loop controllers have “low-gain” and “high-gain” settings for added flexibility. Furthermore, the presence of a saturation block for limiting the control signal is a common feature in most operational systems. However, an active saturation block results in a nonlinear system, and therefore its closed-loop response can no longer be predicted by analytical formulas.
