## 4 Optimization Design of Acceleration Control Plan  
### Optimization Scheme of Control Plan Based on Target Shooting Method  
In the 1980s, [PERSON] proposed the idea of a direct multiple target shooting method, whose essence is to transform the optimal control problem into a parameter optimization problem [16,17]. Consider the following optimal control problem:  
\[\dot{x}(t)=f(x(t),u(t),t). \tag{16}\]  
The variation range of control variables and state variables is constrained as:  
\[\left\{ \begin{array}{l}l_{1}\leq u(t)\leq h_{1}\\ l_{2}\leq x(t)\leq h_{2}\end{array} \right.. \tag{17}\]  
For initial state and terminal state, there are the following boundary conditions:  
\[\left\{ \begin{array}{l}x(0)=x_{0}\\ x(T)=x_{T}\end{array} \right.. \tag{18}\]  
The optimal performance index is:  
\[J=\int_{0}^{T}L(x(t),u(t),t)dt. \tag{19}\]  
Figure 14: Group A of dual-sensors fault grouping diagnosis—fault in the same group. Figure 15: Group B of dual-sensor faults grouping diagnosis—fault in the same group. Terminal time \(T\) can be fixed or free. For the case where \(T\) is free, a new time variable \(\tau\in[0,1]\) can be introduced. Let \(T\) be a variable parameter in a range and define \(t=T\tau\), \(\frac{dt}{d\tau}=T\) and \(\frac{dT}{d\tau}=0\). At the same time, the augmented state variable \(x_{l+1}\) and state equation are introduced:  
\[\dot{x}_{l+1}=T\cdot L(x(\tau),u(\tau),T\tau),x_{l+1}(0)=0. \tag{20}\]  
Thus, the above optimal trajectory problem is equivalent to the following Mayer problem [18] with fixed terminal time. \[\text{min}f_{1}(x_{l+1}(1),T). \tag{21}\]  
The Equation of state of the system is as follows:  
\[\dot{x}=Tf(x(\tau),u(\tau),T\tau)=F(x(\tau),u(\tau),T,\tau). \tag{22}\]  
The starting and ending constraints become:  
\[\left\{ \begin{array}{l}x(0)=x^{0}\\ x(1)=x^{f}\end{array} \right.. \tag{23}\]  
Control variables, state variables, and process time constraints are shown in the following Equation:  
\[\left\{ \begin{array}{l}L_{1}\leq u(\tau)\leq H_{1}\\ L_{2}\leq x(\tau)\leq H_{2}.\\ L_{3}\leq T\leq H_{3}\end{array} \right. \tag{24}\]  
Next, the Mayer problem can be solved by using a direct multiple target shooting method. The basic idea of the target shooting method is to transform the infinite dimensional problem into a finite dimension problem. The optimization process is divided into several intervals, and the design values of control variables and state variables are given in the interval nodes. In this way, the constraints on the whole process state and control variables can be transformed into constraints on these design values, and the performance indicators can also be transformed into functions of the design values. The following matching conditions should be considered: Set the change rule of the control quantity in each interval. It can be in the form of a constant value or polynomial. Then, the state variables at the end of each interval can be obtained from the guess value of the state variables at the beginning of the interval and the system state equation as shown in Equation (22). This value needs to be equal to the estimated value of the state here, which will be met by continuously revising the design value through the optimization algorithm. The specific steps of the target shooting method are as follows:  
1. Divide the fixed time interval [0, 1] into m equal parts to get the node: \(\tau_{i}=\frac{i}{m}\), where \(i=0,1,\ldots,m\). 2. Parameterize the control quantity. Introduce a set of vectors \(P_{i}\in R^{q}\) where \(i=0,1,\ldots,m\) and define: \[u(\tau)=u_{i}(P_{0},P_{i},\ldots,P_{m}),\ \tau\in[\tau_{i},\tau_{i+1}],\] (25) where \(u_{i}(\tau)\) can be a constant value or some interpolation function. 3. Set up the initial value problem. A set of vectors \(S_{i}\in R^{n}\) is selected as an estimate of the state variables \(x_{i}(\tau_{i})\) at the node \(\tau_{i}\).