Cessna 182 model

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Cessna 182 model
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System Identification Modeling
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1 Introduction to System Identification of Cessna 182 Model UAV

Typically system identification is employed in order to determine model parameters for a system whose parameters are unknown or whose system model is complex. Complex models can often be approximated accurately using test data.

This particular article is on the use of system identification with test data to create a system model for a quarter scale model of a Cessna 182 UAV. The article will briefly discuss system identification via the following machine learning algorithms

  1. General Least Squares
  2. Coordinate Descent
  3. Recursive Estimation with Gaussian Noise
  4. Learn-Lagged-Linear

2 Vehicle Dynamical Model - the basis for System Identification

The determination of system identification parameters requires a model form. In other words, the control designer or modeler needs to have a basic idea of the form the model will take. In the case of a SISO system this a matter of deciding how many poles and zeros the system model will have.

The student authors of the project this is based on an assumption of straight and steady flight. This allows the lateral and longitudinal dynamics to be decoupled with each having 4 states. After linearizing based on the small angle approximation ad zero cross product term LaTeX: I_{xz}=0, the lateral equations of motion become

LaTeX: \frac{d}{dt}\begin{bmatrix}\Delta \nu \\ \Delta p \\ \Delta r \\ \Delta \phi\end{bmatrix}=\begin{bmatrix}Y_{\nu} & Y_{p} & -\left(u_{0}-Y_{r}\right) & g\mbox{cos}\theta_{0} \\ L_{\nu} & L_{p} & L_{r} & 0 \\ N_{\nu} & N_{p} & N_{r} & 0 \\ 0 & 1 & 0 & 0\end{bmatrix}\begin{bmatrix}\Delta \nu \\ \Delta p \\ \Delta r \\ \Delta \phi\end{bmatrix} + \begin{bmatrix}0 & Y_{\delta r} \\ L_{\delta a} & L_{\delta r} \\ N_{\delta a} & N_{\delta r} \\ 0 & 0\end{bmatrix}\begin{bmatrix}\Delta \delta_{a} \\ \Delta \delta_{r}\end{bmatrix}

and the longitudinal dynamics are

LaTeX: \frac{d}{dt}\begin{bmatrix}\Delta u \\ \Delta w \\ \Delta q \\ \Delta \theta\end{bmatrix}=\begin{bmatrix}X_{u} & X_{w} & 0 & -g \\ Z_{u} & Z_{x} & u_{0} & 0 \\ M_{u}+M_{wZ}Z_{u} & M_{w}+M_{wZ}Z_{w} & M_{q}+M_{wZ}u_{0} & 0 \\ 0 & 1 & 0 & 0\end{bmatrix}\begin{bmatrix}\Delta u \\ \Delta w \\ \Delta q \\ \Delta \theta\end{bmatrix} + \begin{bmatrix}X_{\delta e} & Y_{\delta T} \\ Z_{\delta e} & L_{\delta T} \\ M_{\delta e}+M_{wZ}Z_{\delta e} & M_{\delta T}+M_{wZ}Z_{\delta T} \\ 0 & 0\end{bmatrix}\begin{bmatrix}\Delta \delta_{e} \\ \Delta \delta_{T}\end{bmatrix}


LaTeX: \left [ \phi, \theta, \gamma \right ] is the roll, pitch, and yaw Euler angles,
LaTeX: \left [ p, q, r \right ] is the angular velocity of body-fixed axis frame relative to earth-fixed frame,
LaTeX: \left [ u, v, w \right ] is the linear velocity of the body-fixed axis frame, and
LaTeX: \left [ \delta a, \delta r, \delta e, \delta t \right ] is the aileron, rudder, elevator and throttle inputs.

Remember that these equations can be written in the form of

LaTeX: \dot{x}=Ax+Bu

3 Machine Learning for System Identification

3.1 General Least Squares

For system identification, least squares regression can be used to find the system parameters. This is accomplished by minimizing the following objective function

LaTeX: \begin{Vmatrix}\dot{x}-\left(Ax+Bu\right)\end{Vmatrix}_{2}^{2}

The data can be rearranged into the conventional form

LaTeX: \begin{Vmatrix}\hat{y}-\hat{A}\hat{s}\end{Vmatrix}_{2}^{2}

wher the state data and unknown parameters are reordered as LaTeX: \hat{A}<\math> and <math>\hat{s}. The pseudo-inverse is used to find LaTeX: \hat{s}.

3.2 Coordinate Descent

The performance of the dynamic model is based on its simulated trajectory relative to the actual trajectory. Minimizing the mean squared error between simulated states and actual states at each time step. However, this appears to be farr too computationally expensive.

3.3 Recursive Estimation with Gaussian Noise

Recursion methods are ideal for in-flight applications since they are formulated to use only present data thus minimizing calculation time at each step. For this formulation all past data is contained in the current state and covariance estimates.

LaTeX: Q_{k+1}=Q_{k}-Q_{k}A_{k+1}^{T}\left(A_{k+1}Q_{k}A_{k+1}^{T}+\Sum_{k+1}\rihgt)^{-1}A_{k+1}Q_{k}

LaTeX: \hat{s}_{k+1}=\hat{s}_{k}+Q_{k}A_{k+1}^{T}\left(A_{k+1}Q_{k}A_{k+1}^{T}+\Sum_{k+1}\right)^{-1}\left(y_{k+1}-A_{k+1}\hat{s}_{k}\right)


LaTeX: \hat{s}_{k} is the optimal state and
LaTeX: Q_{k} is the covariance estimate given at k measurements LaTeX: y_{1} to LaTeX: y_{k}.

Application of this method to system identification of the model requires an estimate of the prior covariances and parameters. The initial estimates on the parameters were chosen to be 0. The covariances are large and decoupled giving more weight to current data than prior data. This leads to the observation that these initial values are most likely not optimal.

3.4 Learn-Lagged-Linear

LaTeX: \Sum_{t=1}^{T-H} \Sum_{h=1}^{H}\begin{Vmatrix}\hat{s}_{t+h|t}-s_{t+h}\end{Vmatrix}_{2}^{2}

3.5 Results

General Least Squares: Fair results
Coordinate Descent: Initial trials suggested that it was too numerically intensive for in-flight use
Recursive Estimation with Gaussian Noise: Performs well enough

4 References

Student Project this is based on