Limit Cycle Oscillation Control of Aeroelastic System Using Model-Error Control Synthesis

Model-error control synthesis is a nonlinear robust control approach that uses an optimal solution to cancel the effects of modelling errors and external disturbances on a system. In this paper the optimal solution for a modified approximate receding-horizon problem is used to determine the model error in an aeroelastic system, which has several uncertain parameters, including a nonlinear spring constant. To verify the control performance the approach is applied to compensate the limit cycle oscillation of the system. Simulation results show the performance of the model-error control synthesis approach. INTRODUCTION Model-Error Control Synthesis (MECS) is a signal synthesis adaptive control method. Robustness is achieved by applying a correction control, which is determined during the estimation process, to the nominal control vector thereby eliminating the effects of modelling errors at the system output. The model-error vector is estimated by using either a onestep ahead prediction approach, an Approximate Receding-Horizon (ARH) approach, or a Modified Approximate Receding-Horizon (MARH) approach. Choosing among the one-step ahead prediction approach, the ARH approach, or the MARH approach to determine the model error depends on the particular properties and required robustness in the system to be controlled. In Ref. [1] MECS with the one-step ahead prediction approach is first applied to suppress the wing rock motion of a slender delta wing, which is described by a highly nonlinear differential equation. Results indicated that this approach provides adequate robustness Post-Doctoral Fellow, Member AIAA, jongrae@ece.ucsb.edu Associate Professor, Associate Fellow AIAA, johnc@eng. buffalo.edu for this particular system. In Ref. [3] a simple study to test the stability of the closed-loop system is presented using a Pade approximation for the time delay, which showed the relation between the system zeros and the weighting in the cost function. The analysis proved that some systems may not be stabilized using the original model-error estimation algorithm, which lead to the ARH approach in the MECS design to determine the model-error vector in the system. The closed-form solution of the ARH approach using Quadratic Programming (QP) is first presented by Lu. The model-error vector is determined by the ARH optimal solution. Using the ARH approach, the capability of MECS is expanded so that unstable nonminimum phase systems can be stabilized. Furthermore, Ref. [4] shows a method to calculate the stable regions with respect to the weighting and the length of receding-horizon step-time using the Hermite-Biehler theorem. After the stable region is found, the weighting and the length of receding-horizon step-time are chosen to minimize the∞-norm of the sensitivity function. The ARH solution for an r-order relative degree system shows that the model-error solution is zero before the end of receding-horizon step-time is reached. Some parts of the model-error vector are separated completely from the constraints, so that the optimal solution for those parts are automatically zero. To avoid this situation for all model-error elements of each constraint at the time before the end of recedinghorizon step-time, the state prediction is substituted by an r-order Taylor series expansion instead of a repeated first-order expansion in the ARH approach. We call this the Modified Approximate Receding-Horizon (MARH) approach, which leads to an even more robust MECS law than with the ARH solution. In Ref. [5] the MARH approach is used to the spacecraft attitude control problem for the case where the only available information is attitude-angle measurements, i.e., with no angular-velocity measurements. 1 American Institute of Aeronautics and Astronautics _ + + ) (t u Plant ) (t u ) (t r