# Bounded Attitude Control with Active Disturbance Rejection Capabilities for Multirotor UAVs
## 4 ADRC Design for Attitude Tracking
### Active Disturbance Rejection Bounded Attitude Control  
In this subsection a control law is proposed that stabilizes the system; this law is described by (8). The objective is to design a control torque that is restricted. **Definition 2**. : _Given a positive constant \(M\), a continuous, non-decreasing function \(\sigma_{M}:\mathbb{R}\rightarrow\mathbb{R}\) is defined by_  
\[ \begin{array}{l}\sigma_{M}=s\text{ if }|s|<M;\\ \sigma_{M}=\text{sign}(s)M\text{ elsewhere};\end{array}  \tag{15}\]Let us remember that the disturbance vector \(\xi\) is assumed to be bounded, i.e., \(\sup_{t}\ |\xi_{i}(\cdot)|\leq K_{0_{i}}\). Furthermore, let \(K_{0_{i}}\) denote the bound for the disturbance function about the \(i^{th}\) axis. Assuming that after a sufficiently long time \(\xi\) is estimated via the observer (13), that is, \(\xi=\mathbf{J}\eta_{1}\), then one has the following result. Consider the attitude dynamics error described by (11) with the following bounded control inputs \(\Gamma=(\Gamma_{1}\ \Gamma_{2}\ \Gamma_{3})^{T}\) such that  
\[\Gamma_{i}=-\sigma_{M_{i2}}\big{(}\xi_{i}+\sigma_{M_{i1}}(\bar{\lambda}_{i}[ \omega_{e_{i}}+\rho_{i}\bar{q}_{i}])\big{)} \tag{16}\]  
with \(i\in\{1,2,3\}\) and where \(\sigma_{M_{i1}}\) and \(\sigma_{M_{i2}}\) are saturation functions such that \(K_{0_{i}}<M_{i2}-M_{i1}\) and \(M_{1i}\geq 3\bar{\lambda}_{i}\rho_{i}\). \(\bar{\lambda}_{i}\) and \(\rho_{i}\) are positive parameters. \(\xi=\mathbf{J}\eta_{1}\) with \(\eta_{1}\) the estimation of the unknown disturbance \(\mathbf{J}^{-1}\xi\). Then the inputs (16) stabilize robustly (11) to the origin of the error space \((1\ 0^{7}\ 0^{7})^{T}\) (i.e., \(\bar{q}_{0}=1\), \(\bar{\mathbf{q}}_{v}=0\) and \(\mathbf{\omega}_{e}=0\)) with a domain of attraction equal to \(\mathbb{S}^{3}\times\mathbb{R}^{3}\setminus(-1\ 0^{7}\ 0^{7})^{T}\). The proof is given in Appendix B. However, we would like to present the idea behind the construction of feedback (16). The proposed control law is composed of a feedforward term represented by \(\xi_{i}\) and the feedback term \(\sigma_{M_{i1}}(\cdot)\). Furthermore, both terms represent the argument for the function \(\sigma_{M_{i2}}\), which at the same time represents the bound of the control torque \(\Gamma_{i}\). The objective is to design a control law that assures that physically, the system has the capability to track a desired attitude while it rejects the total disturbance \(\xi\). Since disturbance \(\xi_{i}\) about the \(i^{th}\) axis is assumed uniformly bounded by \(K_{0_{i}}\), the closed-loop stability will be guaranteed if the following constraint is satisfied \(K_{0_{i}}+M_{i1}<M_{i2}\), i.e., \(\Gamma_{i}\) must be authoritative enough to reject the disturbance. Actually, \(M_{i2}\) plays the role of an explicit reference governor [57], which ensures constraint satisfaction by suitably manipulating the feedforward and the feedback terms. Consider the active disturbance rejection controller (ADRC) composed of the extended state observer (ESO) (13) and the trajectory-tracking bounded control (16) working together to drive the attitude dynamics (10) in order to follow a desired angular trajectory (see Figure 1). In view of the Propositions 1 and 2, the observer dynamics error and the tracking dynamics error can be viewed as two ISS systems in cascade (see Figure 2). Then, the following and main result for the ADRC-based attitude tracking stands. Figure 1: ADRC: ESO and bounded attitude tracking control. **Proposition 3**. : _The active disturbance rejection controller (ADRC) composed of the extended state observer (ESO) (13) and the trajectory tracking controller (16) is ISS when the unknown total disturbance \(\xi\in\mathbb{R}^{3}\) is viewed as the input and the tracking error \(z=(\omega_{e}^{T}\ \tilde{\mathbf{q}}_{v}^{T})^{T}\) as the output._  
Proof. : The claim follows from the properties for the interconnection of nonlinear systems that are input-to-state stable (ISS) [52; 54]. Consider the estimation error dynamics (A2) and the tracking error dynamics (10).