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

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

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

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

Control design Actuator constraints

Position: Rate:

v Bu ) u ( M = = →

max min

u u u ≤ ≤

max min

r u r ≤ ≤ v ) u ( M = u ) t ( u ) t ( u ) t ( u ) t ( v ) t ( Bu ≤ ≤ = → − − ≈ T ) T t ( u ) t ( u ) t ( u

• What is Control Allocation?

Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

Least Squares Formulation

Sequential least squares:

( )

u u u v Bu W min arg

2 v

≤ ≤ − = Ω

( )

Ω ∈ − = u u u W min arg u

2 p u

v Bu = Ω u

p

u

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

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

u u u v Bu min

2

≤ ≤ −

• u

u u v Bu u min

2

≤ ≤ =

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

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

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

Example (Durham and Bordignon, 1996)

8 actuators, 3 moments Position and rate limits Aerodynamic coefficients

pitch roll yaw

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

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Ola Härkegård Active Set Algorithms for Solving LS Problems in Aircraft Control Allocation

Conclusions

Find optimal control input Complexity ≈ pseudoinverse methods

Active set methods are well suited for control allocation Active set methods are well suited for control allocation