## 3 Control System Design
### Upper Controller
#### 3.2.2 Definition of the High-Order Sliding Mode Control Algorithm
: _If Equation (20) is full rank and Equation (18), the r-th order sliding manifold, is not empty, then if and only if the intersection of the Filippov vector set domain and the tangential space of the Equation (18) manifold for any r when the first-order sliding points are not empty, the r-th-order sliding mode of the constraint function \(\sigma\) exists._  
Now suppose that \(\sigma\) is a smooth vector function, \(\sigma\): \(R^{n}\!\rightarrow\!\!R^{m}\), \(\sigma\) = (\(\sigma_{1}\),..., \(\sigma_{m}\)), \(r\) = (\(r_{1}\),..., \(r_{m}\)), where \(r_{i}\) is a natural number. **Definition 4**. : _Assuming that all of the derivatives of the first \(r_{i}\) continuous \(\sigma_{i}\) with respect to time are smooth functions, the set given by the following formula is a locally integrable set in the sense of Filippov._  
\[\sigma_{i}=\dot{\sigma_{i}}=\ddot{\sigma_{i}}=\cdots=\sigma_{i}^{r_{i}-1}=0,i= 1,\ldots,m \tag{24}\]  
Then, the motion mode existing in the set is called the sliding mode of the vector sliding order \(r\) with respect to the vector constraint function \(\sigma\), and the corresponding sliding regularity condition is in the form:  
\[\text{rank}\Big{\{}\
abla\sigma_{i},\ldots,\
abla\sigma_{i}^{(r_{i}-1)} \Big{|}i=1,\ldots,m\Big{\}}=r_{1}+\cdots+r_{m} \tag{25}\]  
\[\dot{x}\in V(x) \tag{17}\]0  
In the formula, \(t\) is the time, \(u\) is the control quantity, and \(f\) and \(\sigma\) are smooth functions. \(u\) is determined by the feedback \(u=U(t,x)\), where \(U\) is a discrete function. For the sake of simplicity, \(\sigma\) and \(u\) are considered scalars, but the following descriptions are all applicable to the vector sliding order. The regular sliding mode satisfies the following conditions: the velocity \(V\) is not in the tangential space \(T_{L}\), but intersects with the manifold \(\sigma=0\), and there is a trajectory in the manifold of the velocity vector at \(T_{L}\). This mode is the main working mode of the variable structure system, and is called the first-order mode. When there is a switching error, the trajectory will leave the manifold at a certain angle. On the other hand, in the second-order sliding mode, even if there is a switching error, all of the possible velocities exist in the tangential space of the manifold, and the state trajectory is tangent to the manifold when it leaves. Consider the following system:  
\[\dot{x}\in V(x) \tag{17}\]1  
Among them, \(a,b,\sigma\) are smooth vector functions; let the system have a degree of freedom \(r\) relative to the output variable \(\sigma\), which means that the derivative \(L_{b}\sigma\), \(L_{b}L_{a}\sigma\),..., \(L_{b}L_{a}^{r-2}\sigma\) near a given point is always equal to 0, and \(L_{b}L_{a}^{r-1}\sigma\) is not equal to 0. Consider the following dynamic system:  
\[\dot{x}\in V(x) \tag{17}\]2  
**Theorem 2**. : _Assume that the degree of freedom of the relative output function \(\sigma\) of the system at a certain \(r\)-order sliding point (\(t_{0}\), \(x_{0}\)) is \(r\), and the range of the discontinuous function is [\(K\), \(\infty\)), and (\(-\infty\), \(-K\)] is the upper point (for some non-zero sets on any region near any order sliding point near \(t_{0}\), \(x_{0}\)), then as long as \(K\) is large enough, there must be an \(r\)-order sliding mode near the point (\(t_{0}\), \(x_{0}\))._