## 3 Control System Design
### Upper Controller
#### 3.2.2 Definition of the High-Order Sliding Mode Control Algorithm  
For discontinuous differential equations:  
\[\dot{x}=v(x) \tag{16}\]  
In the formula, \(x\in R^{n}\), \(v\) is a locally bounded and measurable (Lebesgue) vector function, such that Formula (16) can be expressed by an equivalent differential set:  
\[\dot{x}\in V(x) \tag{17}\]  
Assuming that the range of \(v\) is continuous everywhere, when \(y\to x\), the function \(V(x)\) is the convex closed set of all of the possible limits of \(V(y)\) in the domain, and \(\{y\}\) is called the continuous point of \(v\). The solution set of this equation can be expressed as an absolutely continuous function \(x(t)\), which is differentiable everywhere. **Definition 1**. : _Let \(L\) be a sliding mode manifold, which is defined as a first-order sliding. The point set \(x\in L\) is defined as a second-order sliding, where \(V(x)\) is completely in the tangential space \(T_{L}\) at the point \(x\) relative to the manifold \(L\)._  
**Definition 2**. : _Suppose that there is a first-order (or second-order) sliding mode near the first-order (second-order) sliding point \(x\) on the manifold \(L\) if the first-order (second-order) sliding set near the point \(x\) is an integrable set composed of trajectories in the meaning of Filippov (that is, a combination of Filippov trajectories of discontinuous dynamic systems). Let \(L_{1}=L\), and let \(L_{2}\) represent the second-order sliding point set of the relative manifold \(L\). Assuming that \(L_{2}\) is a sufficiently smooth manifold, similarly, \(L_{3}\) represents a second-order sliding set relative to \(L_{2}\), \(L_{3}\) represents a third-order sliding set relative to \(L\), and so on._  
A constraint condition is represented by the equation \(\sigma(x)=0\), where \(\sigma\): \(R^{n}\to R\) is a sufficiently smooth constraint function. Assuming that all of the trajectories \(\sigma,\ddot{\sigma},\ddot{\sigma},\ldots,\sigma^{(r-1)}\) of the derivatives with respect to time exist and are univariate functions with respect to \(x\), this means that discontinuities will not appear in the first \(r-1\) order derivative of the constraint function \(\sigma\), and the \(r\)-th order sliding set is given by the following equation:  
\[\sigma=\dot{\sigma}=\ddot{\sigma}=\cdots=\sigma^{(r-1)}=0 \tag{18}\]  
**Definition 3**. : _Let the \(r\)-th order sliding set in Definition 1 be non-empty, and assume that it is a locally integrable set in the sense of Filippov; then, the corresponding motion satisfies Definition 1, which is called the \(r\)-th order sliding mode of the relative constraint function \(\sigma\)._Set \(\sigma,\dot{\sigma},\ddot{\sigma},\ldots,\sigma^{(r-2)}\) as the derivative function with respect to \(x\), and have  
\[\text{rank}\Big{\{}\
abla\sigma,\
abla\ddot{\sigma},\ldots,\
abla\sigma^{(r-2)} \Big{\}}=r-1 \tag{19}\]  
\(L_{r}\) can be obtained by Formula (19), and \(L_{i}\), \(I=1,...,r-1\) are all smooth manifolds. If the differential manifold of \(L_{r}\) is required, then Equation (20) is extended to:  
\[\text{rank}\Big{\{}\
abla\sigma,\
abla\ddot{\sigma},\ldots,\
abla\sigma^{(r-1 )}\Big{\}}=r \tag{20}\]  
The condition under which the derivative of Formula (20) and its corresponding \(\sigma\) becomes the differentiable function of \(x\) is called the sliding regularity condition, and Formula (21) is called the weak sliding regularity condition. If Formula (20) holds, then a new local coordinate system can be established, and the system can be re-expressed as  
\[y_{1}=\sigma,\dot{y}_{1}=y_{2},\cdots,\dot{y}_{r-1}=y_{r} \tag{21}\]  
\[\sigma^{(r)}=\dot{y}_{r}=\Phi(y,\dot{\xi}) \tag{22}\]  
\[\dot{\xi}=\Psi(y,\xi),\dot{\xi}\in R^{n-r} \tag{23}\]  
**Theorem 1**.