Generalized Dynamic Inversion for Multiaxial Nonlinear Flight Control
Ismail HameduddinResearch Engineer
King Abdulaziz UniversityJeddah, Saudi Arabia
29th June 2011American Control Conference, San Francisco
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.
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
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.
Aircraft Mathematical Model
● Rigid six degree-of-freedom nonlinear aircraft model with 9 states.
● Aircraft model affine in control terms.● Dogan & Venkataramanan in AIAA Journal of
Guidance, Control & Dynamics.
Euler anglesAerodynamic angles
Angularbody rates
Tangentialvelocity
Euler Angles
● Defined with respect to the inertial frame.
● φ – Roll angle.● θ – Pitch angle.● ψ – Heading angle.
Aerodynamic Angles: Angle-of-attack, α
● Angle between aircraft centerline and relative wind (or velocity vector).
Aerodynamic Angles: Sideslip, β
● Angle between relative wind aircraft centerline.
● Positive when “wind in pilot's right ear.”
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.
Controls
● Four controls, typical of aircraft:
● In general:● Elevator controls body pitch rate.
● Ailerons control body roll rate.
● Rudder controls body yaw rate.
● Throttle controls tangential velocity.
ThrottleElevator Aileron Rudder
Kinematic Equations
● Coordinate transformation of angular rates from body to inertial frame.
Dynamics: Aerodynamic Angles● L – lift force, T – thrust force, S – side force.
Dynamics: Tangential Velocity
● δ is a constant representing the offset angle of the thrust vector from the aircraft centerline.
Forces
● L – lift force, T – thrust force, S – side force.● In terms of dimensionless coefficients:
● Thrust:
Dimensionless coefficients
Dynamicpressure Planform
area
Maximumthrust available
Forces: Dimensionless Coefficients
Dynamics: Angular Rates
● Define the vector of body angular rates
● Then dynamics of body angular rates given by
Inertiamatrix
Cross-productmatrix of angular
velocities
Externalmoment vector
External Moments
● Rolling and yawing moments, respectively:
● Pitching moment:
Dimensionless coefficients
Wingspan
Meanaerodynamic
chord
Offset distanceof thrust vector
from aircraftcenterline
External Moments: Dimensionless Coefficients
State Decomposition
● Define unactuated state vector
● Define outer state vector, “slow dynamics”
● Define inner state vector, “fast dynamics”
● Hence, entire state vector given by
Deviation Functions
● Let error of states from their desired values be given by
● Then a choice for the deviation functions is
Subscript d impliesdesired trajectory
Go to zero if systemat desired trajectories
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).
Generalized Dynamic Inversion Control Law
● Servoconstraints may be expressed in linear form
● Invert using Greville formula to obtain
● Two controllers acting on two orthogonal subspaces (inherently noninterfering).
Projection matrix “Null-control vector”(free)
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).
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.
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.
Schematic of Controller
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).
Results: Euler Angles
Roll Tracking (close-up)
Heading Tracking (close-up)
Results: Aerodynamic Angles
Results: Inner States
Results: Control Surface Deflections
Results: Throttle
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.
Thank you for listening!
● Questions?
Appendix A: GDI Control Matrices
Appendix B: Degree of Interference