# RCS and Aero Surfaces Control Allocation Research on RLV's Re-Entry Phase
## 3 Control Allocation Scheme Design
### Multi-Object-Based Optimal Allocation  
In order to solve the allocation problem with regards to allocation error minimization, fuel consumption minimization, control efficiency maximization, and other objective functions [20], an optimization allocation method is presented. The optimized objective function is weighted, and then combined with all the necessary objective functions, through which the multi-object problem is transformed into a single-object optimization problem. Moreover, the weight values are designed to be time-varying, to adapt for the flight requirements and the control allocation feedback. Therefore, the multi-objective optimization problem is given as follows:  
\[\begin{split}\min_{\mathbf{v}\in\mathbb{U}}& J=w_{e}\|\mathbf{e}\|_{1}+w_{-1}\|\mathbf{e}+\mathbf{e}_{-1}\|_{ 1}+\sum\limits_{i=1}^{N}w_{i}v_{i}+\sum\limits_{i=1}^{m-N}w_{\delta i}\| \delta_{i}-\delta_{pi}\|_{1}+w_{p}\|q_{p}/q_{i}\|+\cdots\\ s.t.&\left\{\begin{array}{ll}\underline{T}_{t}\leq T_{i} \leq\overline{T}&\underline{\Delta T}_{t}\leq\Delta\overline{T}\\ \overline{\int}_{RCS}\left(\sum\limits_{i=1}^{N}v_{i}\right)\!m_{rRS}(t)dt\leq M _{RCS}&\sum\limits_{i=1}^{N}(v_{i})_{on/off}\leq N_{on/off}&i=1,2,\cdots,N\\ \underline{\delta}_{i}\leq\delta_{i}\leq\overline{\delta}_{i}&\underline{ \delta}_{i}\leq\overline{\delta}_{i}&i=N+1,N+2,\cdots,m\end{array}\right.\end{split} \tag{11}\]  
where \(\mathbf{e}=\mathbf{M}_{R}+\mathbf{M}_{RCS}-\mathbf{M}_{c}\) is the control allocation error, \(\mathbf{e}_{-1}\) is the allocation error from the last simulation step. The variable \(\delta_{pi}(i=1,\cdots,m-N)\) denotes the baseline of the deflection angle of the corresponding aero surface, and \(w_{e},w_{-1},w_{i}(i=1,2,\cdots,N),w_{\delta i}(i=1,\cdots,m-N),w_{p}\) stand for the weight coefficient of the corresponding single-object optimization function, which can improve its priority by increasing its weight coefficient. \(T_{i}\) is the current working duration of the \(i^{th}\) thruster, and  
Figure 2: The structure diagram of the daisy-chaining allocation scheme.  
\(\underline{T}_{I},\overline{T}_{I}\) are the lower and upper limits of the \(i^{th}\) thruster's working duration; \(\Delta T_{i}\) is the current 'ON/OFF' switching interval of the \(i^{th}\) thruster, and \(\Delta\underline{T}_{I},\Delta\overline{T}_{I}\) are the lower and upper limits of the \(i^{th}\) thruster's switching interval; \(T_{RCS}\) is the total working time of RCS, \(\dot{m}_{rcs}\) is the fuel consumption per second of the single thruster, and \(\mathbf{M}_{RCS}\) is the total fuel mass of RCS; \((v_{i})_{\text{on/off}}\) is the switching times of the \(i^{th}\) thruster, and \(N_{\text{on/off}}\) is the maximum switching times of RCS. \(\underline{\delta}_{i},\overline{\delta}_{i}\) are the lower and upper limits of the corresponding aero surface's deflection angle; \(\underline{\delta}_{i},\overline{\delta}_{i}\) are the lower and upper limits of the corresponding aero surface's deflection angular rate. Figure 3 presents the structure diagram of this allocation scheme.  
Obviously, this control allocation scheme takes the dynamic characteristics and necessary constraints into account, and considers the allocation error as the feedback to the multi-object based algorithm, which can effectively improve the accuracy and adaptivity of the control allocation.