Yes, complexity pseudoinverse methods Ola Hrkegrd Active Set - PDF document

Efficient Active Set Algorithms for Solving Constrained LS Problems in Aircraft Control Allocation Ola Härkegård Linköpings universitet, Sweden Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation Main Message Today: Efficient but approximate methods Can standard QP methods be used efficiently? � Yes, complexity ≈ pseudoinverse methods � Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 1

What is Control Allocation? M ( u ) Bu v u � Control design → = = � Actuator constraints u ≤ u ≤ u � Position: min max r u r ≤ � ≤ � Rate: min max M ( u ) v = Bu ( t ) = v ( t ) u ( t ) u ( t T ) − − u ( t ) ≈ → � u ( t ) u ( t ) u ( t ) ≤ ≤ T Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation Least Squares Formulation � Sequential least squares: Bu = v Ω ( ) arg min W Bu v Ω = − u v 2 u ≤ u ≤ u ( ) u u = arg min W u − u p u p 2 u ∈ Ω Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 2

Numerical Methods � Active set methods (this paper) � Pseudoinverse methods (dominate) � ... Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation Active Set min Bu v min u − � 2 � 2 u u u Bu v ≤ ≤ = u u u ≤ ≤ Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 3

Pseudoinverse � Successful case � Failing case Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation Why Active Set? � Always finds optimal solution � Can reuse previous solution � All iterates are feasible Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 4

Example (Durham and Bordignon, 1996) � Aerodynamic coefficients pitch roll yaw � 8 actuators, 3 moments � Position and rate limits Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation Simulations Active set Active set Pseudoinverse Pseudoinverse Mean: 0.9 ms Max: 2.5 ms Mean: 0.9 ms Max: 1.5 ms Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 5

Conclusions Active set methods are well Active set methods are well suited for control allocation suited for control allocation � Find optimal control input � Complexity ≈ pseudoinverse methods Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation 6