STATE-DEPENDENT RICCATI EQUATION CONTROL OF A SMALL UNMANNED HELICOPTER
Alexander Bogdanov, Magnus Carlsson, Geoff Harvey , John Hunt, Dick Kieburtz, Rudolph van der Merwe, Eric Wan OGI School of Science and Engineering, OHSU 20000 NW Walker Rd, Beaverton, Oregon 97006 This paper is an initial report on flight experiments with a small, unmanned helicopter using a state dependent Riccati Equation (SDRE) controller for autonomous, agile maneuvering. The control design is based upon a full, 6-DoF, analytic nonlinear dynamic model, which is manipulated into a pseudo-linear form in which system matrices are given explicitly as a function of the current state. A standard Riccati equation is then solved numerically in each frame of a 50 Hz. control loop to design the state feedback control law on-line. Several flights have been flown with the helicopter to evaluate the accuracy of tracking under SDRE control in comparison with simulation results.
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y r s })
y r s } y r s }-
vdy r s })r s
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where y r s } is a steady state solution of the difference Riccati equation, obtained by solving the discrete-time algebraic Riccati equation (DARE) ~
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using state-dependent matrices ~y rs.} and y rsz} , which f are treated as being constant. For tracking problems with V the desired state rs , the SDRE control can be implemented as {|s u
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The SDRE control generally exhibits greater stability and better performance than linear control laws, and thus is attractive in nonlinear control problems. To maintain a steady flight along some trim trajectory, a trim control corresponding to the trim trajectory is usually introduced: { s uvdy rs.} s {
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This is a typical solution for error-linear control laws. $; The trim control, { , is a function of the trim trajectory, which is a steady state condition, thus partitioning vehicle dynamics into linear+angular velocity (fuselage frame) and attitude+position (inertial frame) components we can write:
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(7)
For a feasible arbitrary desired trajectory, the reference control can be approximated by scheduling between a number of trim trajectories. Another approach is to treat the current commanded trajectory (e.g. commanded velocities) as if it were a trim trajectory and recompute the $; trim control for this trajectory, { s , by solving Eqn. (7) at each time step, . With such a solution, the steadystate error of the vehicle can be made as small as desired, given a gain matrix dy r s } that stabilizes wy r s { s } . For helicopter control, we define observable states to > correspond to the standard 12 states of a 6-DoF rigid y g W } > , body model: rs u and vector of controls {s u y #"%$ &( ) $+&( } corresponding to rotor blade pitch angles. The helicopter model can be described as a generic 6DOF rigid body model with external forces and moments originating from the main and tail rotors, vertical
3 A MERICAN I NSTITUTE OF A ERONAUTICS AND A STRONAUTICS
and horizontal fins and fuselage drag. The model we used to design the SDRE control law is discussed in detail by Gavrilets, et al.5 ¡
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then one can derive thrust derivative with respect to the blade collective pitch (i.e.collective control input) as 9 :
