# Critically Leveraging Theory for Optimal Control of Quadrotor Unmanned Aircraft Systems
## 2 The Structure and Mathematical Model of the Quadrotor
### Nonlinear Control \(H_{\infty}\) for Rotational Subsystem  
The H\({}_{\infty}\) controller [29] for the rotational subsystem is studied to achieve stability under the influence of external noises and uncertain parameters (Figures 5 and 6). According to the theory of nonlinear control H\({}_{\infty}\), and assuming the system \(\sum\) is nonlinear:  
\[ \begin{array}{l}\dot{x}=f(x,u,d)\\ \sum:y=g(x,u,d)\\ z=h(x,u,d)\end{array}  \tag{34}\]  
has two inputs, u and d, two outputs, y and z, and state variable x:  
where u is the control vector, d is the input noise (noise to be suppressed or the signal to be tracked), y is the metering output, and z is the control lever output (tracking error, cost function). The optimal control problem H\({}_{\infty}\), roughly speaking, is to find a controller C that handles the output y and modulates the input u so that it is a closed loop:  
Having the L\({}_{2}\)-gain from [30], the input to output noise z is minimized; moreover, this closed loop must be stable in some sense (H\({}_{\infty}\) comes from the linear system case, L\({}_{2}\)-gain of the stable system is the H\({}_{\infty}\) norm of the transfer function matrix). Normally, optimizing H\({}_{\infty}\) is a difficult problem; instead, it can approach a suboptimal controller H\({}_{\infty}\), with a given noise reduction factor \(\gamma\) such that the closed system has L\({}_{2}\)-gain \(\leq\gamma\) and stable. The solution for H-optimal control can be approximated by iteration of the suboptimal H controller. Going back to the Quadrotor problem, the kinematics of a smooth nonlinear system of order n, affected by unknown perturbations, can be expressed as follows:  
\[\dot{x}=f(x,t)+g(x,t)u+k(x,t)d \tag{35}\]  
Figure 5: Model of the H-inf nonlinear system. Figure 6: Optimized control structure H-inf. where:  
\( \begin{tabular}{l} u \(\epsilon\) RP is the vector of control input signals. \\ d \(\epsilon\) RP is the vector of external noise variables. \\ x \(\epsilon\) RP is a vector of state variables. \\ Cost variable definition \(\xi\), \(\epsilon\) R\({}^{(m+p)}\): \\ \(\xi\) = W\({}^{[}\)h\((x)\)\(u\)\(]\) \\ \end{tabular}  (36)  
where \(\mathrm{h}(x)\)\(\epsilon\) R\({}^{m}\) is a function of the state vector x and W \(\epsilon\) R\({}^{(m+p)\times(m+p)}\) is the weight matrix. Assuming the state variable x can be measured, the optimization problem H\({}_{\infty}\) is posed as follows [31, 32]:  
Find the minimum value of \(\gamma^{*}\geq 0\) such that for any \(\gamma^{*}\geq\gamma\), there exists a state response \(\mathrm{u}=\mathrm{u}(x,t)\), such that the L\({}_{2}\)-gain from d to \(\xi\), is less than or equal to \(\gamma\):  
\[\int_{0}^{T}\|\xi\|_{2}^{2}\mathrm{dt}\leq\gamma^{2}\int_{0}^{T}\|\mathrm{d} \|_{2}^{2}\mathrm{dt} \tag{37}\]  
The function enclosed within the integral sign on the left-hand side of the inequality is expressed as:  
\[\|\xi\|_{2}^{2}=\xi^{\prime}\xi= \begin{bmatrix}\mathrm{h}^{\prime}(x)&\mathrm{ u}^{\prime}\end{bmatrix} \mathrm{W}^{\prime}\mathrm{W} \begin{bmatrix}\mathrm{h}(x)\\ \mathrm{u}\end{bmatrix}  \tag{38}\]  
And the positive definite symmetry matrix W'W has the form:  
\[\mathrm{W}^{\prime}\mathrm{W}= \begin{bmatrix}\mathrm{Q}&\mathrm{S}\\ \mathrm{S}^{\prime}&\mathrm{R}\end{bmatrix}  \tag{39}\]  
where Q and R are positively definite symmetric matrices, and for W\({}^{\prime}\mathrm{W}>0\), then Q - SR\({}^{-1}>\) O (O is a 0-order matrix of n). With those assumptions, there is an optimal control signal \(\mathrm{u}^{*}(x,t)\)[33] as follows:  
\[\mathrm{u}^{*}=-\mathrm{R}^{-1}\bigg{(}\mathrm{S}^{\prime}\mathrm{h}(x)+ \mathrm{g}^{\prime}(x,t)\frac{\partial\mathrm{V}(x,t)}{\partial x}\bigg{)} \tag{40}\]  
Let the system (20), where V(x,t) (provided that V\((x_{0},t)\equiv 0\) when \(t\geq 0\)), be the solution of the Hamilton-Jacobi equation [34]:  
\[