State Feedback Decoupling Control of a Control Moment Gyroscope

Abstract

Control moment gyroscope (CMG) is an actuator commonly used in attitude control of satellites and spacecrafts, as well as in stabilization of marine vessels and unmanned vehicles. It is a nonlinear multivariable system and presents considerable coupling depending on the chosen operating point, i.e., the gyroscope gimbals angles. First, the complete modeling of a CMG with four degrees of freedom is shown using the Lagrangian dynamic formulation, resulting in a set of four nonlinear equations. Further, the system is linearized around an equilibrium point, resulting in a coupled two-input two-output system. Next, the application of a linear state feedback decoupling control to this linearized system is developed based on the classical Falb–Wolovich method. Aiming to deal with stationary errors, simple decentralized proportional-integral controllers are designed for each channel. Simulation and practical results with a didactic control moment gyroscope are presented in order to validate the methodology. The resulting system has good decoupling characteristics and presents satisfactory responses in terms of setpoint tracking and disturbance rejection.

Appendix

The full set of nonlinear equations resulting from the modeling is presented in Sect. 2 and related to Eqs. (3, 4, 5, 6). The inertia parameters are presented in Table 1.

$$\begin{aligned}&\tau _{4_{ext}} - J_{D} \left[ \dot{\omega }_{4}+\cos \left( \theta _{3}\right) \dot{\omega }_{2}+\cos \left( \theta _{2}\right) \sin \left( \theta _{3}\right) \dot{\omega }_{1}\right. \nonumber \\&\quad - \sin \left( \theta _{3}\right) \omega _{3}\omega _{2} + \cos \left( \theta _{2}\right) \cos \left( \theta _{3}\right) \omega _{3}\omega _{1}\nonumber \\&\quad - \left. \sin \left( \theta _{2}\right) \sin \left( \theta _{3}\right) \omega _{2}\omega _{1}\right] =0 \end{aligned}$$

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$$\begin{aligned}&\tau _{3_{ext}} - \left( I_{C} + I_{D} \right) \dot{\omega }_{3} + \left( I_{C} + I_{D} \right) \sin \left( \theta _{2}\right) \dot{\omega }_{1} \nonumber \\&\quad + \dfrac{\left( I_{D} + K_{C} - J_{C} - J_{D} \right) }{2} \sin \left( 2\theta _{3}\right) \omega _{2}^2 \nonumber \\&\quad + \left( J_{C} + J_{D} - I_{D} - K_{C} \right) \cos ^2\left( \theta _{2}\right) \cos \left( \theta _{3}\right) \sin \left( \theta _{3}\right) \omega _{1}^2 \nonumber \\&\quad + \left( I_{C} + J_{C} + J_{D} - K_{C} \right) \cos \left( \theta _{2}\right) \omega _{1}\omega _{2} \nonumber \\&\quad + 2 \left( I_{D} + K_{C} - J_{C} - J_{D} \right) \cos \left( \theta _{2}\right) \sin ^2\left( \theta _{3}\right) \omega _{1}\omega _{2}\nonumber \\&\quad +J_{D}\cos \left( \theta _{2}\right) \cos \left( \theta _{3}\right) \omega _{1}\omega _{4} - J_{D}\sin \left( \theta _{3}\right) \omega _{2}\omega _{4}=0 \end{aligned}$$

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$$\begin{aligned}&\tau _{2_{ext}} - {J_D}\cos \left( {{\theta _3}} \right) {{\dot{\omega }}_4} - \left( J_B + J_C + J_D \right) {{\dot{\omega }}_2} \nonumber \\&\quad + \left( J_C + J_D - K_C - I_D \right) \sin ^2 \left( {{\theta _3}} \right) {{\dot{\omega }}_2} + \nonumber \\&\quad + \left( K_C + I_D - J_C - J_D \right) \cos \left( {{\theta _2}} \right) \cos \left( {{\theta _3}} \right) \sin \left( {{\theta _3}} \right) {{\dot{\omega }}_1} \nonumber \\&\quad + \dfrac{ \left( I_B + I_C - K_B - K_C \right) }{2} \sin \left( {2{\theta _2}} \right) \omega _1^2 \nonumber \\&\quad + \left( K_C + I_D - J_C - J_D \right) \cos \left( {{\theta _2}} \right) \sin \left( {{\theta _2}} \right) \sin ^2 \left( {{\theta _3}} \right) \omega _1^2 \nonumber \\&\quad + \left( J_C + J_D - K_C - I_D \right) \sin \left( {2{\theta _3}} \right) {\omega _2}{\omega _3} \nonumber \\&\quad + \left( K_C - I_C - J_C - J_D \right) \cos \left( {{\theta _2}} \right) {\omega _1}{\omega _3} \nonumber \\&\quad + 2 \left( J_C + J_D - K_C - I_D \right) \cos \left( {{\theta _2}} \right) \sin ^2 \left( {{\theta _3}} \right) {\omega _1}{\omega _3} \nonumber \\&\quad + {J_D}\sin \left( {{\theta _3}} \right) {\omega _3}{\omega _4} - {J_D}\sin \left( {{\theta _2}} \right) \sin \left( {{\theta _3}} \right) {\omega _1}{\omega _4} =0 \end{aligned}$$

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$$\begin{aligned}&\tau _{1_{ext}} - {J_D}\cos \left( {{\theta _2}} \right) \sin \left( {{\theta _3}} \right) {{\dot{\omega }}_4} + \left( I_C + I_D \right) \sin \left( {{\theta _2}} \right) {{\dot{\omega }}_3} \nonumber \\&\quad + \left( K_C + I_D - J_C - J_D \right) \cos \left( {{\theta _2}} \right) \cos \left( {{\theta _3}} \right) \sin \left( {{\theta _3}} \right) {{\dot{\omega }}_2} \nonumber \\&\quad - \left( K_A + K_B + K_C + I_D \right) {{\dot{\omega }}_1}\nonumber \\&\quad + \left( K_B + K_C - I_B - I_C \right) \sin ^2 \left( {{\theta _2}} \right) {{\dot{\omega }}_1} \nonumber \\&\quad + \left( K_C + I_D - J_C - J_D \right) \cos ^2 \left( {{\theta _2}} \right) \sin ^2 \left( {{\theta _3}} \right) {{\dot{\omega }}_1}\nonumber \\&\quad + \left( J_C + J_D - K_C - I_D \right) \cos \left( {{\theta _3}} \right) \sin \left( {{\theta _2}} \right) \sin \left( {{\theta _3}} \right) \omega _2^2 \nonumber \\&\quad + 2 ( K_C + I_D - J_C - J_D ) \cos ^2 \left( {{\theta _2}} \right) \cos \left( {{\theta _3}} \right) \sin \left( {{\theta _3}} \right) {\omega _1}{\omega _3} \nonumber \\&\quad + \left( K_C + I_C + 2 I_D - J_C - J_D \right) \cos \left( {{\theta _2}} \right) {\omega _2}{\omega _3} \nonumber \\&\quad + 2 \left( J_C + J_D - K_C - I_D \right) \cos \left( {{\theta _2}} \right) {\sin ^2}\left( {{\theta _3}} \right) {\omega _2}{\omega _3}\nonumber \\&\quad + \left( K_B + K_C - I_B - I_C \right) \sin \left( {2{\theta _2}} \right) {\omega _1}{\omega _2} \nonumber \\&\quad + 2 ( J_C + J_D - K_C - I_D ) \cos \left( {{\theta _2}} \right) \sin \left( {{\theta _2}} \right) {\sin ^2}\left( {{\theta _3}} \right) {\omega _1}{\omega _2} \nonumber \\&\quad + {J_D}\sin \left( {{\theta _2}} \right) \sin \left( {{\theta _3}} \right) {\omega _2}{\omega _4}\nonumber \\&\quad - {J_D}\cos \left( {{\theta _2}} \right) \cos \left( {{\theta _3}} \right) {\omega _3}{\omega _4} =0 \end{aligned}$$

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