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