LMIs in Control/Click here to continue/Applications of Linear systems/H∞-based Synthesis of Quadrotor Guidance and Control System

System Dynamics
Consider the motion of a quadrotor along a horizontal plane alone. The control law is synthesised for roll motion(rotation wrt. X-axis) coupled with linear motion along Y-axis. The equations of motion governing the 2-D motion are given by(taken from the paper provided below):

$$ \begin{align} &\quad \quad \begin{cases} \frac{d^2x}{dt^2}=-\frac{\mu}{m_{Q}}\frac{dx}{dt}-g\frac{tan\theta}{cos\phi};\\ \frac{d^2y}{dt^2}=-\frac{\mu}{m_{Q}}\frac{dy}{dt}-gtan\phi,\\ \end{cases} \end{align}$$

$$ where $$


 * $$x$$ and $$y$$ are the displacements along X & Y axis, respectively.
 * $$\phi$$ and $$\theta$$ are the angular displacement about X & Y axis, respectively.
 * $$m_{Q}$$ and $$\mu$$ are constants that depend on the intrinsic properties of the quadrotor.

The state space representation of The $$H_\infty$$ Quadrotor Guidance is given below,

$$ \begin{align} &\quad \quad \begin{cases} \frac{dX(t)}{dt}=AX(t)+B_{u}u(t)+B_{d}(t)d(t); \\ Y=CX(t), \\ \end{cases}\end{align} $$

$$ where: $$

$$ X = [X_{1},X_{2},X_{3},X_{4},X_{5}]^T = [x,\frac{dx}{dt},y,\frac{dy}{dt},\Delta \Omega]^T,$$ (where $$\Delta \Omega$$ is the increment in motor's rotation rate)

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & -\frac{\mu}{m_{Q}} & g & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\               0 & 0 & 0 & 0 & K_{1}\\ 0 & 0 & 0 & 0 & K_{2}\\ \end{bmatrix} $$

$$ B_{u} = \begin{bmatrix} 0\\ 1\\                           K_{3} \\ 0 \\                           K_{4}\\ \end{bmatrix}$$

$$ B_{d} = \begin{bmatrix} 0\\ 0\\                           0 \\                            0 \\                            K_{3}\\ \end{bmatrix}$$

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

$$C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} $$

$$ D_1 = 10^{-3} \times L_1, D_2 = 10^{-3} \times L_2 $$

The Optimization Problem
For the above-linearized model of the quadrotor, the linear control would be as below:



u = -K*V(t) = -K*C*X(t), $$

where V is the voltage input, $$ K \in R^{1x4}$$ and $$ C \in R^{4x5}.$$

The cost function of this controller could be defined as:



j=\int_0^\infty \! ||z(t)||^2 \, \mathrm{d}t=\int_0^\infty \! ||x^TQx+u^T Ru||^2 \, \mathrm{d}t $$ ,

where,

$$ z=

\begin{bmatrix} \sqrt{Q} & 0\\ 0 & \sqrt{R} \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix} .$$

The LMI
$$\begin{bmatrix} PA+A^TP+Q& PB_{u}& PB_{d}&L^T \\B^T_{u}P &-R& 0& 0\\B^T_{d}P&0&-\gamma^2I&0\\L&0&0&-R \end{bmatrix}$$ < 0.

Solving the above LMI gives the value of $$\gamma$$ for X>0, $$\gamma$$ >0 and Z.

Conclusion
This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,. This can be used to analyze the state feedback control and path tracking of a quadcopter.