1. February 24, 20051 Final Presentation AAE 666 - Final Presentation Backstepping Based Flight Control Asif Hossain

2. February 24, 2005 2 Overview Modern Aircraft Configuration Aircraft Dynamics Force, Moment and Attitude Equations Current Approaches to Flight Control Design Backstepping Approaches to Flight Control Design Backstepping Backstepping Design for Flight Control Flight Control Laws Simulation

3. February 24, 2005 3 Modern Aircraft Configuration

4. February 24, 2005 4 Aircraft Dynamics, Aircraft position expressed in an Earth-fixed coordinate system;, The velocity vector expressed in the body-axis coordinate system;, The Euler angles describing the orientation of the aircraft relative to the Earth-fixed coordinate system;, The angular velocity of the aircraft expressed in the body axes coordinate system;

5. February 24, 2005 5 Aerodynamics Forces and Moments Body axis coordinate system

6. February 24, 2005 6 Force Equations (Body-axes) Rewrite the force equations in terms of

7. February 24, 2005 7 Force Equations (wind-axes) Where the contributions due to gravity are given by,

8. February 24, 2005 8 Moment and Attitude Equations Moment Equations: Attitude Equations:

9. February 24, 2005 9 Current Approaches to Flight Control Design Gain Scheduling  Divide and conquer approach is tedious since controller must be designed for each flight envelope.  Stability is guaranteed only for low angles of attack and low angular rates. Dynamic Inversion (feedback linearization)  Cancels valuable nonlinear dynamics.  Relies on precise knowledge of the aerodynamic coefficients

10. February 24, 2005 10 Backstepping based flight control design Constructive (systematic) control design for nonlinear systems. Lyapunov based control design method Avoid cancellation of “useful nonlinearities” (unlike feedback linearization). Stability is guaranteed for all angles of attack (unlike gain scheduling). Different flavors: Adaptive, robust and observer backstepping.

11. February 24, 2005 11 LaSalle-Yoshizawa Theory The time-invariant system, Let be a scalar continuously differentiable function of the state such that is positive definite is radially unbounded a Then, all solutions satisfy In addition, if is positive definite, then the equilibrium is Globally Asymptotically Stable (GAS).

12. February 24, 2005 12 Control Lyapunov Function (clf) The time-invariant system, A smooth, positive definite, radially unbounded function is called a control Lyapunov function (clf) for the system if for all, Given a clf for the system, we can thus find a globally stabilizing control law. In fact, the existence of a globally stabilizing control law is equivalent to the existence of a clf, and vice versa.

13. February 24, 2005 13 Backstepping Consider the system Where are state variables and is the control input. Assume a virtual control law is known such that 0 is GAS equilibrium of the system.

14. February 24, 2005 14 Backstepping Let, be a clf for the subsystem such that Then, the clf for the augmented system is given by Moreover, a globally stabilizing control law, satisfying is given by

15. February 24, 2005 15 Strict Feedback System By recursive applying backstepping, globally stabilizing control laws can be constructed for systems of the following lower triangular form:

16. February 24, 2005 16 Backstepping design for flight control Controlled variables: General maneuvering

17. February 24, 2005 17 Control Objectives

18. February 24, 2005 18 Assumptions: Control surface deflections only produce aerodynamic moments, and not forces. The speed, altitude and orientation of the aircraft vary slowly compared to the controlled variables. Therefore, their time derivatives can be neglected. Longitudinal and lateral commands are assumed not to be applied simultaneously. The control surface actuator dynamics are assumed to be fast enough to be disregarded.

19. February 24, 2005 19 Backstepping design for flight control The roll rate to be controlled,, is expressed in the stability axes coordinate system. The stability axes angular velocity,,is related to the body axes angular velocity,, through the transformation: where Note that the transformation matrix Introducing:

20. February 24, 2005 20 Aircraft Dynamics Revisited 1 2 3 4 5 Roll rate dynamics: Equation 1 Angle of attack dynamics: Equation 2-3 Sideslip dynamics: Equation 4-5

21. February 24, 2005 21 The nonlinear control problem The angle of attack dynamics and the sideslip dynamics can be written as For notational convenience it is favorable to make the origin the desired equilibrium. Let, is the desired equilibrium.

22. February 24, 2005 22 The nonlinear control problem The dynamics become We will use backstepping to construct a globally stabilizing feedback control laws for the system assuming general nonlinearity.

23. February 24, 2005 23 The nonlinear control problem Assume there exists a constant,, such that Then a globally stabilizing control law can be given by where, and

24. February 24, 2005 24 Block Diagram The nonlinear system is globally stabilized through a cascaded control structure

25. February 24, 2005 25 Aircraft Application

26. February 24, 2005 26 Flight Control Laws Angle of attack control Sideslip regulation Stability axis roll control

27. February 24, 2005 27 Gain Selection, How should the control law parameters be selected? For control, linearize the angle of attack dynamics around a suitable operating point and then select to achieve some desired linear closed behavior locally around the operating point. For regulation, can be selected by choosing some desired closed loop behavior using linearization of the sideslip dynamics.

28. February 24, 2005 28 Roll rate demand,

29. February 24, 2005 29 Angle of attack demand,

30. February 24, 2005 30 Questions Polygonia interrogationis known as Question Mark