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

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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

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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

φ θ ψ elevator

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Choosing a model1

Model type pros cons White-box

explicitly modeled phys- ical properties extendable, values cor- respond to real-world properties gets complex quickly, easy to neglect impor- tant factors

Grey-box

• nly prescribe general

characteristics remove complexity while keeping structure changing requirements may need larger modi- fications

Black-box

no explicit knowledge about the system struc- ture choose best result re- gardless of the means same as grey-box, also no direct correspon- dence to real-world values

1L.Ljung, ’Approaches to Identification of Nonlinear Systems’ 4 / 1

Black Box Model I

Apply multisine exitation to system:

5 10 15 20 25 30 35 40 45 50 −2 2 time [s] elevator angle 10−1 100 101 10−20 10−13 10−6 101 108 frequency [rad/s] magnitude

Record response:

10 12 14 16 18 20 22 24 26 28 30 2 2.5 3 3.5 4 time [s] angle [rad] φ θ 10−1 100 101 10−3 10−1 101 103 105 frequency [rad/s] Magnitude measured input φ θ 5 / 1

Black Box Model II

Filter out chosen frequencies:

10−1 100 101 10−1 100 101 102 103 frequency [rad/s] Magnitude u φ θ

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Black Box Model III

Estimate transfer function with varying order (n poles, n-1 zeros):

10−1 100 101 10−4 10−3 10−2 10−1 100 101 frequency [rad/s] Magnitude φ/u measured

• rder 1
• rder 2
• rder 3
• rder 4

10−1 100 101 frequency [rad/s] Magnitude θ/u measured

• rder 1
• rder 2
• rder 3
• rder 4

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Black Box Model IV

However...

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 −0.6 −0.4 −0.2 0.2 0.4 0.6 φ [rad] measured simulated 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 −0.4 −0.2 0.2 0.4 time [s] θ [rad] measured simulated

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Grey Box Model - Structure

Linear state space model with assumptions about the dynamics:     ˙ θ ¨ θ ˙ φ ¨ φ     =     1 a21 a22 a23 a24 1 a41 a42 a43 a44         θ ˙ θ φ ˙ φ     +     b2 b4     u rearrange for parameters, set up least squares scheme: ¨ θi ¨ φi

• yi

= θi ˙ θi φi ˙ φi ui θi ˙ θi φi ˙ φi ui

• Ψi
• a21

. . . b4 ⊤

• p

p∗ = Ψ+y = (Ψ⊤Ψ)−1Ψ⊤y

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Grey Box Model - Identification

First results:

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 −1 −0.5 0.5 φ [rad] measured simulated 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 −0.6 −0.4 −0.2 0.2 0.4 time [s] θ [rad] measured simulated

To do: Record more data, see if model still works

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White Box Model - Structure

Determine all relevant forces: Fg, FLift,Wing, FLift,Elev, FDrag,Wing, FDrag,Elev Calculate resulting torque using M = r × F: M = Mg + . . . + MDrag,Elev + MConstr + MFriction Obtain angular accelerations via the inertia tensor: M = I · α Set up state space model: x =     θ ˙ θ φ ˙ φ     , d dt     θ ˙ θ φ ˙ φ     =     x2 α1 x4 α2    

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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

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Black Box Model (Again)

Using Matlab’s prepacked modelling tools:

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 Time [s] φ [rad] Measured and simulated model output Grey Box n4s3 measured

Drawback: non-trivial estimator needed

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What’s next?

Implement LQR and compare to PID ACADO + NMPC

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