# A Predefined-Time Control for the Laser Acquisition in Space Gravitational Wave Detection Mission
## 4 Controller Design  
### Predefined-Time Stable Theory  
Consider the nonlinear autonomous dynamical system:  
\[\dot{\mathbf{x}}=f(\mathbf{x}) \tag{12}\]where \(\mathbf{x}\in\mathbb{R}^{n}\) denotes the state vector and \(f\): \(\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a nonlinear and locally Lebesgue-integrable function, with \(f(0)=0\). The initial conditions are denoted by \(\mathbf{x}_{0}=\mathbf{x}(0)\in\mathbb{R}^{n}\). **Definition 1**. : _(Globally finite-time stable): System (12) is globally finite-time stable if it is globally asymptotically stable and any solution \(\mathbf{x}(t,\mathbf{x}_{0})\) of (12) reaches the equilibrium point at some finite time moment, that is \(\forall\mathbf{x}\geq T(\mathbf{x}_{0})\): \(\mathbf{x}(t,\mathbf{x}_{0})=0\), where \(T\): \(\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}\cup\{0\}\) is called the settling-time function._  
**Definition 2**. : _(Fixed-time stable): System (12) is fixed-time stable if it is globally finite-time stable and the settling-time function is bounded, \(\exists T_{\max}>0\): \(\forall\mathbf{x}_{0}\in\mathbb{R}^{n}\): \(T(\mathbf{x}_{0})\leq T_{\max}\)._  
**Definition 3**. : _(Predefined-time stable): System (12) is predefined-time stable if it is fixed-time stable and exists a predefined constant \(T_{\mathrm{C}}>0\), the settling-time function \(T\) satisfies_  
\[T(\mathbf{x}_{0})\leq T_{\mathrm{C}},\forall\mathbf{x}_{0}\in\mathbb{R}^{n} \tag{13}\]  
\(T_{\mathrm{C}}\) _is called a predefined time._  
The predefined-time stability can be attained by a method similar to the Lyapunov method, which can be summarized as the following theorems:  
**Theorem 1**. : _(Lyapunov function-like method of predefined-time stabilization): Consider the uncertain dynamical system:_  
\[\dot{\mathbf{x}}=f(t,\mathbf{x},\mathbf{d}) \tag{14}\]  
_where \(\mathbf{d}\in\mathbb{R}^{n}\) is a vector of unknown disturbances and dynamic uncertainties. If there exists a radially unbounded Lyapunov function \(V(\mathbf{x})\): \(\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}\cup\{0\}\), and it satisfies the following properties [39]:_  
1. \(V(\mathbf{x})=0\) _if and only if_ \(\mathbf{x}=0\)_._
2. \(V(\mathbf{x})\geq 0\) _for any_ \(\mathbf{x}\)_._
3. _For_ \(\forall\mathbf{x}\)_, the derivative of_ \(V(\mathbf{x})\) _satisfies:_ \[\dot{V}\leq-\frac{\pi}{\eta\gamma\beta T_{\mathrm{C}}}\Big{(}\gamma^{2}V^{1- \eta/2}+\beta^{2}V^{1+\eta/2}\Big{)}\] (15)  
_where_ \(T_{\mathrm{C}}>0\) _is a predefined-time constant and_ \(\eta\in(0,1)\)_,_ \(\gamma>0\)_,_ \(\beta>0\) _are parameters. Then, system (_14_) is globally predefined-time stable with predefined time_ \(T_{\mathrm{C}}\)_._  
**Proof :**  
According to (15), we have:  
\[\frac{dV}{dt}\leq-\frac{\pi}{\eta\gamma\beta T_{\mathrm{C}}}\Big{(}\gamma^{2} V^{1-\eta/2}+\beta^{2}V^{1+\eta/2}\Big{)} \tag{16}\]  
If \(V(\mathbf{x})=0\), Equation (15) is obvious. Consider that \(V(\mathbf{x})>0\), we have:  
\[\frac{\pi}{\eta\gamma\beta T_{\mathrm{C}}}\Big{(}\gamma^{2}V^{1-\eta/2}+\beta^ {2}V^{1+\eta/2}\Big{)}>0 \tag{17}\]  
Then Equation (16) will be:  
\[\frac{dV}{-\frac{\pi}{\eta\gamma\beta T_{\mathrm{C}}}\big{(}\gamma^{2}V^{1- \eta/2}+\beta^{2}V^{1+\eta/2}\big{)}}\geq dt \tag{18}\]By integrating the Equation (21), we can obtain:  
\[