Master Thesis Preparations: Modelling and Identification of the - PowerPoint PPT Presentation

Master Thesis Preparations: Modelling and Identification of the HalfWing Jonas Schlagenhauf September 5, 2016 1 / 1

Objective of the master thesis ”NMPC of a constrained model airplane” ✓ Develop system and tools ✓ Record data → Choose model and identify ✗ Develop controller(s) ✗ Evaluate 2 / 1

The HalfWing Model airplane, fixed via two joints to the carousel 3 DOF: carousel rotation ψ , elevation φ , pitch θ Input: elevator angle Output: pitch / elevation angle ψ φ θ elevator 3 / 1

Choosing a model 1 Model type pros cons White-box explicitly modeled phys- extendable, values cor- gets complex quickly, ical properties respond to real-world easy to neglect impor- properties tant factors Grey-box only prescribe general remove complexity changing requirements characteristics while keeping structure may need larger modi- fications Black-box no explicit knowledge choose best result re- same as grey-box, also about the system struc- gardless of the means no direct correspon- ture dence to real-world values 1 L.Ljung, ’Approaches to Identification of Nonlinear Systems’ 4 / 1

Black Box Model I Apply multisine exitation to system: 10 8 2 elevator angle 10 1 magnitude 0 10 − 6 10 − 13 − 2 10 − 20 0 5 10 15 20 25 30 35 40 45 50 10 − 1 10 0 10 1 time [s] frequency [rad/s] Record response: 4 φ 10 5 θ 3 . 5 10 3 angle [rad] Magnitude 3 10 1 10 − 1 2 . 5 measured input φ 10 − 3 θ 2 10 12 14 16 18 20 22 24 26 28 30 10 − 1 10 0 10 1 time [s] frequency [rad/s] 5 / 1

Black Box Model II Filter out chosen frequencies: 10 3 u φ θ 10 2 Magnitude 10 1 10 0 10 − 1 10 − 1 10 0 10 1 frequency [rad/s] 6 / 1

Black Box Model III Estimate transfer function with varying order (n poles, n-1 zeros): 10 1 10 0 Magnitude φ /u Magnitude θ /u 10 − 1 10 − 2 measured measured order 1 order 1 10 − 3 order 2 order 2 order 3 order 3 order 4 order 4 10 − 4 10 − 1 10 0 10 1 10 − 1 10 0 10 1 frequency [rad/s] frequency [rad/s] 7 / 1

Black Box Model IV However... 0 . 6 measured simulated 0 . 4 0 . 2 φ [rad] 0 − 0 . 2 − 0 . 4 − 0 . 6 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 0 . 4 measured simulated 0 . 2 θ [rad] 0 − 0 . 2 − 0 . 4 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 time [s] 8 / 1

Grey Box Model - Structure Linear state space model with assumptions about the dynamics:  ˙        θ 0 1 0 0 θ 0 ¨ ˙ θ a 21 a 22 a 23 a 24 θ b 2          =  +  u         ˙ 0 0 0 1 0 φ φ      ¨ ˙ a 41 a 42 a 43 a 44 b 4 φ φ rearrange for parameters, set up least squares scheme: � ¨ � � θ i � ˙ ˙ θ i θ i φ i φ i 0 0 0 0 u i 0 � � ⊤ = a 21 . . . b 4 ¨ ˙ ˙ 0 0 0 0 0 φ i θ i θ i φ i φ i u i � �� � ���� � �� � p y i Ψ i p ∗ = Ψ + y = (Ψ ⊤ Ψ) − 1 Ψ ⊤ y 9 / 1

Grey Box Model - Identification First results: 0 . 5 measured simulated 0 φ [rad] − 0 . 5 − 1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0 . 4 measured simulated 0 . 2 θ [rad] 0 − 0 . 2 − 0 . 4 − 0 . 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time [s] To do: Record more data, see if model still works 10 / 1

White Box Model - Structure Determine all relevant forces: F g , F Lift , Wing , F Lift , Elev , F Drag , Wing , F Drag , Elev Calculate resulting torque using M = r × F : M = M g + . . . + M Drag , Elev + M Constr + M Friction Obtain angular accelerations via the inertia tensor: M = I · α Set up state space model:       θ θ x 2 ˙ ˙ θ d θ α 1       x =  ,  =       φ φ x 4  dt    ˙ ˙ α 2 φ φ 11 / 1

White Box Model - Identification Combination of measuring and estimating: Measure as much as possible: geometric features, weight, COM, ... Estimate intricate parameters: lift / drag coefficients, inertia tensor, ... A lot of parameters to estimate from two angle measurements: 4x aerodynamic coefficients, 3x moment of inertia, friction, actuator delay, ... Good chance of being nonlinear in the parameters Solution: Identify parts separately (e.g. by fixing one axis) WIP 12 / 1

Black Box Model (Again) Using Matlab’s prepacked modelling tools: Measured and simulated model output 0 . 6 Grey Box n4s3 0 . 5 measured 0 . 4 0 . 3 0 . 2 φ [rad] 0 . 1 0 − 0 . 1 − 0 . 2 − 0 . 3 − 0 . 4 − 0 . 5 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 Time [s] Drawback: non-trivial estimator needed 13 / 1

What’s next? Implement LQR and compare to PID ACADO + NMPC 14 / 1