Presentation at 2011 American Control Conference, San Francisco, CA.

1. Generalized Dynamic Inversion for Multiaxial
Nonlinear Flight Control
Ismail Hameduddin
Research Engineer
King Abdulaziz University
Jeddah, Saudi Arabia
29th June 2011
American Control Conference, San Francisco

2. Content
● Goals/summary.
● Outline of generalized dynamic inversion.
● Aircraft mathematical model.
● Brief introduction to aircraft states.
● Nonlinear model.
● Controller design.
● Generalized dynamic inversion control via Greville formula.
● Generalized inverse singularity robustness strategy.
● Null-control vector design.
● Results/simulation.
● Conclusions.

3. Goals
● Demonstrate effectiveness of generalized dynamic
inversion (GDI) for control of large order, nonlinear,
MIMO systems.
● Aircraft good example of such a system.
● Framework for future work in GDI – tools and
strategies for large order, nonlinear, MIMO systems

4. Outline of GDI
1. Form expression to measure error of state variable from desired trajectory – so-
called “deviation function.”
2. Differentiate deviation function along trajectories of system until explicit
appearance of control terms.
3. Use derivatives from step 2 to construct stable dynamic system representing the
error response of the closed-loop system – so called “servo-constraint.”
4. Invert system using the Moore-Penrose generalized inverse and Greville
formula to obtain desired control vector.
5. Exploit redundancy (null-control vector) in Greville formula to ensure stability
of closed-loop system.

5. Aircraft Mathematical Model
● Rigid six degree-of-freedom nonlinear aircraft
model with 9 states.
Euler angles Tangential Angular
Aerodynamic angles velocity body rates
● Aircraft model affine in control terms.
● Dogan & Venkataramanan in AIAA Journal of
Guidance, Control & Dynamics.

6. Euler Angles
● Defined with respect to
the inertial frame.
● φ – Roll angle.
● θ – Pitch angle.
● ψ – Heading angle.

7. Aerodynamic Angles:
Angle-of-attack, α
● Angle between aircraft centerline and relative wind
(or velocity vector).

8. Aerodynamic Angles: Sideslip, β
● Angle between relative
wind aircraft centerline.
● Positive when “wind in
pilot's right ear.”

9. Other States
● Tangential velocity: magnitude of total velocity
vector.
● Velocity vector (magnitude & direction) is completely
described with tangential velocity + aerodynamic angles.
● Body angular rates:
● p – body roll rate.
● q – body pitch rate.
● r – body yaw rate.

10. Controls
● Four controls, typical of aircraft:
Elevator Aileron Rudder Throttle
● In general:
● Elevator controls body pitch rate.
● Ailerons control body roll rate.
● Rudder controls body yaw rate.
● Throttle controls tangential velocity.

11. Kinematic Equations
● Coordinate transformation of angular rates from
body to inertial frame.

12. Dynamics: Aerodynamic Angles
● L – lift force, T – thrust force, S – side force.

13. Dynamics: Tangential Velocity
● δ is a constant representing the offset angle of the
thrust vector from the aircraft centerline.

14. Forces
● L – lift force, T – thrust force, S – side force.
● In terms of dimensionless coefficients:
Dynamic
Planform Dimensionless
pressure
area coefficients
● Thrust:
Maximum
thrust available

15. Forces: Dimensionless Coefficients

16. Dynamics: Angular Rates
● Define the vector of body angular rates
● Then dynamics of body angular rates given by
Inertia
matrix
External
Cross-product
moment vector
matrix of angular
velocities

17. External Moments
● Rolling and yawing moments, respectively:
Wingspan Dimensionless
coefficients
● Pitching moment:
Mean Offset distance
aerodynamic of thrust vector
chord from aircraft
centerline

18. External Moments: Dimensionless
Coefficients

19. State Decomposition
● Define unactuated state vector
● Define outer state vector, “slow dynamics”
● Define inner state vector, “fast dynamics”
● Hence, entire state vector given by

20. Deviation Functions
● Let error of states from their desired values be given
by
Subscript d implies
desired trajectory
● Then a choice for the deviation functions is
Go to zero if system
at desired trajectories

21. Servoconstraints
● Define servoconstraints based on deviation functions as
● Differential order of servoconstraints related to relative
degree.
● Constants chosen to ensure stability and good response.
● Time-varying constraints incorporated to reduce
peaking (at t = 0).

22. Generalized Dynamic Inversion Control
Law
● Servoconstraints may be expressed in linear form
● Invert using Greville formula to obtain
Projection matrix “Null-control vector”
(free)
● Two controllers acting on two orthogonal subspaces
(inherently noninterfering).

23. Dynamically Scaled Generalized
Inverse
● Moore-Penrose generalized inverse has singularity when matrix
changes rank.
● New development: dynamically scaled generalized inverse
(DSGI)
where
● Asymptotic convergence to true MPGI without singularity
(proof available in paper).

24. Null-control Vector
● Stability guaranteed via null-control vector; validity
of entire architecture (including singularity avoidance)
depends on proper selection of null-control vector.
● Null-control vector designed to ensure asymptotic
stability of inner states.
● Choose null-control vector
where K is a gain to be determined.

25. Design of Stabilizing Gain K
● K maybe designed any number of ways; we use the null-
projected control Lyapunov function
● Defined along the closed-loop system, the following null-
control vector guarantees stability
where Q is an arbitrary positive definite matrix.
● Proof is elementary and is available in paper.

26. Schematic of Controller

27. Simulation Parameters
● Euler angles
● φ – sinusoidal signal with 30° angle.
● θ – 4°, set to ensure 0° flight-path angle.
● ψ – +180° heading change (exponential growth to limit).
● Aerodynamic angles
● α – angle-of-attack left uncontrolled.
● β – 0° sideslip angle to ensure coordinated flight.
● Body angular rates
● All set to stability; p = q = r = 0.
● Tangential velocity: increase up to maximum throttle (approx. 230
m/s).

28. Results: Euler Angles

29. Roll Tracking (close-up)

30. Heading Tracking (close-up)

31. Results: Aerodynamic Angles

32. Results: Inner States

33. Results: Control Surface Deflections

34. Results: Throttle

35. Conclusions & Future Work
● New nonlinear flight control methodology derived and
validated via nonlinear UAV simulation.
● Methodology allows use of linear system tools on
nonlinear systems.
● Provides a framework for noninterfering controllers.
● Future work:
● Robustness/disturbance rejection.
● Output feedback.
● Adaptive control/nonaffine in control systems, etc.

36. Thank you for listening!
● Questions?

37. Appendix A: GDI Control Matrices

38. Appendix B: Degree of Interference