Generalized Dynamic Inversion for Multiaxial Nonlinear Flight Control
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
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 GDI1. 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. Euler angles Tangential AngularAerodynamic angles velocity body rates ● Aircraft model affine in control terms. ● Dogan & Venkataramanan in AIAA Journal of Guidance, Control & Dynamics.
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 pilots 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: 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.
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:Dynamic Planform Dimensionlesspressure area coefficients ● Thrust: Maximum thrust available
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 Subscript d implies desired trajectory ● Then a choice for the deviation functions isGo 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 Projection matrix “Null-control vector” (free)● Two controllers acting on two orthogonal subspaces (inherently noninterfering).
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.
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).
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.