LMIs in Control/Click here to continue/Applications of Linear systems/LMI-based State Feedback Design for mass varying Quadcopter Optimal path control and Tracking

Introduction
An LMI-based  state  feedback  approach  that  ensures optimum  path  tracking  and  improved  steady  state performance  of  a  quadrotor  in  both  translational  and rotational movements.

Quadcopter Dynamics
The motion of Quad Copter in 6DOF is controlled by varying the rpm of four rotors individually, thereby changing the vertical, horizontal and rotational forces.


 * ASSUMPTIONS:
 * 1) The structure is symmetric, thus the inertia matrices are diagonal.
 * 2) The center of mass corresponds to the origin of the physical coordinate system.
 * 3) A quadcopter is a rigid body.

State Space Representation


\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$

where x(t) is state vector, y(t) is output vector and u(t) is Input or control vector.

$$ \dot x(t)=\frac{d}{dt} x(t) $$
 * A is the system matrix
 * B is the input matrix
 * C is the output matrix
 * D is the feed forward matrix

Quadcopter modelling with 6 degree of freedom

 * REQUIRED 12 STATES:

The state vector x is $$ x^T= $$ $$ \begin{bmatrix} x & y  & z  & x' & y' & z' & \phi & \theta & \psi & \phi' & \theta' & \psi' \end{bmatrix} $$

The Input matrix u is, $$ u^T=$$$$ \begin{bmatrix} U1 & U2 & U3  & U4 \end{bmatrix} $$, where


 * U1 is the Total Upward Force on the quadrotor along z-axis ( T-mg)
 * U2 is the Pitch Torque (about x-axis)
 * U3 is the Roll Torque (about y-axis)
 * U4 is the Yaw Torque (about z-axis)

The Output matrix y is $$ y^T=$$$$ \begin{bmatrix} x & y & z & \phi & \theta & \psi \end{bmatrix} $$

The State differential equations written in matrix form as,

$$ \begin{bmatrix} x'\\ y'\\ z' \\ x''\\ y''\\ z''\\ \phi'\\ \theta'\\ \psi'\\ \phi''\\ \theta''\\ \psi''\\

\end{bmatrix}$$     = $$ \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\               0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & -g & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & g & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\               0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{bmatrix} $$ $$ \begin{bmatrix} x\\ y\\ z \\ x'\\ y'\\ z'\\ \phi\\ \theta\\ \psi\\ \phi'\\ \theta'\\ \psi'\\

\end{bmatrix}$$ + $$ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{m} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \frac{1}{l_{x}} & 0 & 0 \\ 0 & 0 & \frac{1}{l_{y}} & 0 \\ 0 & 0 & 0 & \frac{1}{l_{z}} \\ \end{bmatrix} $$$$ \begin{bmatrix} U1\\ U2 \\ U3 \\ U4 \end{bmatrix} $$

The above martices represents the equation $$ x'=Ax+Bu$$

$$ \begin{bmatrix} x\\ y\\ z \\ \phi\\ \theta\\ \psi\\\end{bmatrix} $$=$$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\               0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \end{bmatrix} $$$$ \begin{bmatrix} x\\ y\\ z \\ x'\\ y'\\ z'\\ \phi\\ \theta\\ \psi\\ \phi'\\ \theta'\\ \psi'\\

\end{bmatrix}$$+$$ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$$$ \begin{bmatrix} U1\\ U2 \\ U3 \\ U4 \end{bmatrix} $$

The above martices represents the equation $$ y=Cx+Du$$

LMI for stability Analysis
$$ \begin{bmatrix} PA^T-W^TB^T+AP-BW+\mu^2\delta & -PA^T+W^TB^T-DP & I \\ PD^T-AP+BW & D^TP+PD & -P \\ I & -P & -\delta I \\ \end{bmatrix} <0 $$


 * $$K=WP^-1$$

Solving the above LMI yields the unknown coefficients of the feedback control. The system will be then asymptotically stable and path track will be achieved.

LPV Attitude State Feedback Controller Design
To design a $$ H\infty $$ LPV feedback control scheme for the altitude/attitude stabilization of the quadrotor aircraft,


 * Output $$y=\begin{bmatrix} \phi & \theta& \psi&  \Z \end{bmatrix}^T$$ of the system must track r =$$\begin{bmatrix} \phi_{ref} & \theta_{ref}&  \psi_{ref}&  \Z_{ref} \end{bmatrix}^T$$, a reference trajectory.


 * To achieve the objective outputs of the integrator are considered as extra state variables $$x_{e}=\begin{bmatrix}x_{\phi} & x_{\theta}& x_{\psi}&  x_{Z} \end{bmatrix}^T$$ as

$$x_{\phi}=\int_{0}^{t} e_{\phi} (\delta) d\delta \,dx ,e_{\phi}=\phi_{ref}-\phi$$

$$x_{\theta}=\int_{0}^{t} e_{\theta} (\delta) d\delta \,dx ,e_{\theta}=\theta_{ref}-\theta$$

$$x_{\psi}=\int_{0}^{t} e_{\psi} (\delta) d\delta \,dx ,e_{\psi}=\psi_{ref}-\psi$$

$$x_{z}=\int_{0}^{t} e_{z} (\delta) d\delta \,dx ,e_{z}=z_{ref}-z$$

In this case the error signal e=y-r ,for the outputs $$ U_{1},U_{2},U_{3},U_{4}$$,the weight functions $$ W_{ui},i=1,2,3,4 $$are added to the system. The system matrices of weight functions are $$A_{ui},B_{ui},C_{ui}and D_{ui}.$$

The dynamic of all the weight functions $$ W_{u1},W_{u2},W_{u3},W_{u4}$$ can be constituted as,

\begin{align} \dot{x}_{u}=A_{u}x_{u}+B_{u}u\\ y_{u}=C_{u}x_{u}+D_{u}u \end{align}$$

where $$x_{u}=\begin{bmatrix} x_{u1}&x_{u2}&x_{u3}&x_{u4}\end{bmatrix}^T$$ is the state,$$u=\begin{bmatrix}U_{1} & U_{2}&U_{3}&U_{4}\end{bmatrix}^T$$ represents the input $$y_{u}=\begin{bmatrix} Z_{1}&Z_{2}&Z_{3}&Z_{4}\end{bmatrix}^T$$

$$\Delta_{u}=\begin{bmatrix} \Delta_{u1}&0&0&0\\0&\Delta_{u2}&0&0\\0&0&\Delta_{u3}\\0&0&0&\Delta_{u4}\end{bmatrix}$$,$$\Delta\in{A,B,C,D}$$
 * The system matrices of the weight function can be deducted as,

$$w=\begin{bmatrix} r&d\end{bmatrix}^T$$,$$z=\begin{bmatrix} y_{u}&e\end{bmatrix}^T$$,and $$\bar{x}=\begin{bmatrix} x&x_{e}&x_{u}\end{bmatrix}^T$$ are the exogenous input and exogenous output respectively.


 * The system differential equations with augumented ststes and weight functions are,

\begin{align} \dot{\bar x}=\sum_{i=1}^{16} \mu_{i} (\bar A_{i}\bar x+\bar B1_{i}w+\bar B_{2}u)\\ z=C_{\bar x}+D11 w+D12 u \end{align}$$

where,

$$\bar A_{i}=\begin{bmatrix} \bar A_{i} &0&0&0\\ -C&0 &0\\0 &0&A_{u} \end{bmatrix}$$;

$$ \bar B_{1}=\begin{bmatrix} 0 & \bar E_{i}\\ -I_{4} & 0\\0 &0 \end{bmatrix}$$

$$ \bar B_{2}=\begin{bmatrix} \bar B_{i}\\ 0\\ B_{u}\end{bmatrix}$$

$$ C_{1}=\begin{bmatrix} 0& 0& C_{u}\\ C &0&0\end{bmatrix}$$

$$ D_{11}=\begin{bmatrix} 0 &0\\-I_{4}&0\end{bmatrix}$$

$$ D_{12}=\begin{bmatrix} D_{u}\\0\end{bmatrix}$$

Making the closed loop system,

$$dot{\bar x}=\sum_{i=1}^{16} \mu_{i} ((\bar A_{i}+\bar B_{2}ki)\bar x(t)+\bar B_{1}w)$$

LMI for H∞ optimal state-feedback
$$ X=X^T>0$$

$$\begin{bmatrix} He(\bar A_{i}X+\bar B_{2}Y_{i} &(*)^T &(*)^T\\ \bar B_{1i} ^T & -\gamma I&(*)^T \\ \bar C_{1i}X+D_{12}Y_{i}& \bar D_{11} &-\gamma I \end{bmatrix}<0 $$

$$He(\bar A_{i}X+\bar B_{2}Y_{i}+2\alpha X <0$$ ; i=1 to 16

By solving the LMI,the optimal H\inf state feed-back controller with the smallest attenuation level \gamma >0 for the attitude/altitude subsystem of the mass-varying quadcopter is $$K(\rho)=\sum_{i=1}^{16} \mu_{i} Y_{i} X^-1$$

Conclusion
This LMI can be used to analyze the state feedback control and path tracking of a quadcopter.

Implementation
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,.