# Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System
## 2 Classical Sliding Mode Control
Strategies aiming at a reduction of the chattering phenomenon by a stability preserving interval based gain adaptation strategy were investigated in [20; 21; 22; 23; 24; 25]. To ensure that the system will reach the sliding surface in finite time, the condition:  
\[\frac{1}{2}\frac{\mathrm{d}s^{2}(\mathbf{x})}{\mathrm{d}t}=s(\mathbf{x})\cdot\dot{s}( \mathbf{x})\leq-\eta|s(\mathbf{x})|,\text{ where }\eta>0, \tag{5}\]  
is typically introduced in a classical sliding mode control synthesis. The inequality (5) means that the squared distance to the sliding surface has to decrease along all possible trajectories with a constant velocity \(\eta\). This condition makes the set \(s=0\) an invariant set due to the invariance theorem of [PERSON]. If this condition is satisfied for the complete state space, the system will always converge to the sliding surface. To reach the sliding mode, the state dependent controller has to be designed. There are various ways to do that, where we restrict ourselves to one of the available options. To deal with possible singularities in the control law and to avoid them during perfect trajectory tracking in the sliding mode, the input \(u\) is chosen according to the concept of equivalent control. To find the equivalent control \(u_{\mathrm{eq}}\) that ensures perfect tracking, \(\dot{s}=0\) should be valid on the sliding surface in order for \(s\) not to change and thus the system to remain in sliding mode. For a system like (1) with a (nonlinear) feedback controller \(u(\mathbf{x})\), the time derivative of \(s(\mathbf{x})\) is calculated by the following Lie derivative of \(s(\mathbf{x})\) in the direction of the vector field \(\mathbf{f}(\mathbf{x},u)\), assuming that \(s(\mathbf{x})\) does not explicitly depend on the time variable \(t\):  
\[\dot{s}(\mathbf{x})=\left(\frac{\partial s(\mathbf{x})}{\partial\bm  
With the help of the relation (7), the control signal can be found in symbolic form. As an example, consider a two-dimensional input affine system of the form:  
\[\dot{x}_{1} =x_{2} \tag{9}\] \[\dot{x}_{2} =c(\mathbf{x})+u,\]  
for which Equation (7) becomes equal to:  
\[\dot{s}(\mathbf{x})= \frac{\partial s(\mathbf{x})}{\partial x_{1}}\dot{x}_{1}+\frac{ \partial s(\mathbf{x})}{\partial x_{2}}\dot{x}_{2}=-s(\mathbf{x})\cdot p(\mathbf{x}) \tag{10}\]  
as well as equal to:  
\[\dot{s}(\mathbf{x})= \frac{\partial s(\mathbf{x})}{\partial x_{1}}x_{2}+\frac{\partial s( \mathbf{x})}{\partial x_{2}}(c(\mathbf{x})+u)=-s(\mathbf{x})\cdot p(\mathbf{x}). \tag{11}\]  
The latter expression can now be solved for the control input according to:  
\[u=u(\mathbf{x})=\left(-s(\mathbf{x})\cdot p(\mathbf{x})-\frac{\partial s(\mathbf{x})}{ \partial x_{1}}x_{2}\right)\cdot\frac{1}{\frac{\partial s(\mathbf{x})}{\partial x _{2}}}-c(\mathbf{x}). \tag{12}\]  
To avoid singularities in the controller and in this way in the closed-loop system dynamics, the following condition has to be satisfied:  
\[-s(\mathbf{x})\cdot p(\mathbf{x})-\frac{\partial s(\mathbf{x})}{\partial x_{1}}x_{2}=k( \mathbf{x})\frac{\partial s(\mathbf{x})}{\partial x_{2}}, \tag{13}\]  
so that all fractions in the expression for the input \(u\) cancel out analytically. After simple rearrangement, this expression becomes:  
\[\frac{\partial s(\mathbf{x})}{\partial x_{2}}k(\mathbf{x})+s(\mathbf{x})\cdot p(\mathbf{x})=- \frac{\partial s(\mathbf{x})}{\partial x_{1}}x_{2}, \tag{14}\]  
which is [PERSON]'s identity, where \(p(\mathbf{x})\) and \(k(\mathbf{x})\) have to be found in terms of non-singular analytic (polynomial) expressions. **Remark 1**.