# A Comparative Study for Control of Quadrotor UAVs
## 2 Materials and Methods
### System Model
\begin{pmatrix}\mathrm{U_{1}}\\ \mathrm{U_{2}}\\ \mathrm{U_{3}}\\ \mathrm{U_{4}}\end{pmatrix} =\mathrm{M_{bd}}\mathrm{\mathbf{w}}^{2}= \begin{pmatrix} \mathrm{b}&\mathrm{b}&\mathrm{b}&\mathrm{b}\\ 0&\mathrm{b}&0&-\mathrm{b}\\ \mathrm{b}&0&-\mathrm{b}&0\\ \mathrm{d}&-\mathrm{d}&\mathrm{d}&-\mathrm{d}\end{pmatrix}\begin{pmatrix} \mathrm{w_{1}^{2}}\\ \mathrm{w_{2}^{2}}\\ \mathrm{w_{3}^{2}}\\ \mathrm{w_{4}^{2}}\end{pmatrix}  \tag{18}\]  
It is worth noting that Equation (18) stands for an artificial input transformation made for ease of representation and control law definition. U\({}_{1}\) denotes the total lift force produced by the rotors, U\({}_{2}\) and U\({}_{3}\) are the resultant forces generating pitch and roll movement respectively and \(\mathrm{U}_{4}\) is the resultant moment acting on the yaw axis. The presented quadrotor dynamical model has 12 state variables \(\mathbf{x}=(\mathrm{x}_{1},\ \mathrm{x}_{2},\ \mathrm{x}_{3},\mathrm{x}_{4}, \mathrm{x}_{5},\mathrm{x}_{6},\mathrm{x}_{7},\mathrm{x}_{8},\mathrm{x}_{9}, \mathrm{x}_{10},\mathrm{x}_{11},\ \mathrm{x}_{12})=(\mathrm{X},\mathrm{Y}, \mathrm{Z},\ \mathrm{X}^{\prime},\ \mathrm{Y}^{\prime},\ \mathrm{Z}^{\prime}, \mathrm{\Phi},\mathrm{\Phi},\mathrm{\Phi},\mathrm{p},\mathrm{q},\mathrm{r})\) and 4 input variables \(\mathbf{u}=(\mathrm{u}_{1},\mathrm{u}_{2},\mathrm{u}_{3},\mathrm{u}_{4})=( \mathrm{U}_{1},\mathrm{U}_{2},\mathrm{U}_{3},\mathrm{U}_{4})\). Note that the rotors' velocities giving the gyroscopic effect contribution still appear in Equations (12) and (13) but can be rewritten as a nonlinear function of the artificial input variables \(\mathrm{f}(\mathbf{u})\) by inversion and manipulation of Equation (18). \[\mathrm{f}(\mathbf{u})=\sum_{i=1}^{4}(-1)^{i+1}\sqrt{\mid(\mathrm{M}_{\mathrm{ bd}}{}^{-1}\mathbf{u})_{i}\mid}=\mathrm{w}_{1}+\mathrm{w}_{3}-\mathrm{w}_{2}- \mathrm{w}_{4} \tag{19}\]  
The system, besides presenting under-actuation characteristics that always yield to challenging control design aspects, is highly nonlinear and coupled. When dealing with a dynamical system with such properties, it is convenient to derive a linearized model in order to obtain a clearer understanding of the relation between control and state variables and the system's stability. Furthermore, linearized models are often exploited for the design of the control law itself. \[\mathrm{X}^{\prime\prime} = -\mathrm{\theta g} \tag{20}\] \[\mathrm{Y}^{\prime\prime} = -\mathrm{\phi g}\] (21) \[\mathrm{Z}^{\prime\prime} = \mathrm{g}-\frac{1}{\mathrm{m}}\mathrm{U}_{1}\] (22) \[\mathrm{\Phi}^{\prime\prime} = \frac{1}{\mathrm{I}_{x}}\mathrm{LU}_{2}\] (23) \[\mathrm{\Theta}^{\prime\prime} = \frac{1}{\mathrm{I}_{y}}\mathrm{LU}_{3}\] (24) \[\mathrm{\Psi}^{\prime\prime} = \frac{1}{\mathrm{I}_{z}}\mathrm{U}_{4} \tag{25}\]  
The Equations (20)-(25) refer to the linearization of the model represented by the Equations (9)-(17) by means of a Taylor expansion of the first order and adopting approximations for small angles, under the following assumptions:  
* The quadrotor is hovering: external forces and moments are neglectable, linear and rotational velocities are small (the Euler rates coincide with the rotational velocities in the body frame). * The equilibrium control input and state vectors are \(\left(\mathrm{u}_{1}^{*},\mathrm{u}_{2}^{*},\mathrm{u}_{3}^{*},\mathrm{u}_{4}^ {*}\right)=(\mathrm{mg},0,0,0)\) and \(\left(\mathrm{x}_{1},\ \mathrm{x}_{2},\ \mathrm{x}_{3},\mathrm{x}_{4}, \mathrm{x}_{5},\mathrm{x}_{6},\mathrm{x}_{7},\mathrm{x}_{8},\mathrm{x}_{9}, \mathrm{x}_{10},\mathrm{x}_{11},\ \mathrm{x}_{12}\right)=(0,0,0,0,0,0,0,0,0,0,0)\), respectively. * All terms of order higher than one are neglected.