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):

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

$where$

• $x$ 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,

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

$where:$

$X=[X_{1},X_{2},X_{3},X_{4},X_{5}]^{T}=[x,{\frac {dx}{dt}},y,{\frac {dy}{dt}},\Delta \Omega ]^{T},$ (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:

$j=\int _{0}^{\infty }\!||z(t)||^{2}\,\mathrm {d} t=\int _{0}^{\infty }\!||x^{T}Qx+u^{T}Ru||^{2}\,\mathrm {d} t$ ,

where,

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

The LMI

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