# Systematic Analysis and Design of Control Systems Based on [PERSON]'s Direct Method
## 3 Stability Analysis  
### Lyapunov's Direct Method  
In this section, we present how the previously introduced methods can be used to test nonlinear systems concerning different stability properties. For the following, autonomous time-invariant systems of the form  
\[\dot{x}=f(x) \tag{23}\]  
are considered. Here, \(x(t)\in\mathbb{R}^{n}\) is the state of the system (23) and \(f:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is a (locally) _Lipschitz_ vector field. This is true for polynomial systems as a result of the mean value theorem of differential calculus. According to the _theorem of Picard-Lindelof_, the Lipschitz condition guarantees the (local) existence and uniqueness of the solution of the system of differential Equation (23) [4; 55]. To formulate the stability conditions, functions of class \(\mathcal{K}\) and class \(\mathcal{K}_{\infty}\) are introduced. A function \(\alpha:[0,a)\to[0,\infty)\) with \(a>0\) belongs to the class \(\mathcal{K}\) if it is strictly increasing and \(\alpha(0)=0\) holds. A function \(\alpha\) with \(a=\infty\) belongs to the class \(\mathcal{K}_{\infty}\) if in addition \(\alpha(r)\to\infty\) holds for \(r\to\infty\)[2; 56; 3]. For systems of the form (23), the stability of the equilibrium \(x=0\) can be analyzed using the _direct method of [PERSON] [3; 4]_. **Theorem 1** ([PERSON]'s direct method). : _Let \(x=0\) be a equilibrium of (23) with a locally Lipschitz vector field \(f\) and \(\mathbb{U}=\{x\in\mathbb{R}^{n},|x|<a\}\). If there exists a continuously differentiable function \(V:\mathbb{U}\to[0,\infty)\) and functions \(\bar{\alpha},\bar{\alpha}\in\mathcal{K}\), defined on \([0,a)\), such that_  
\[\bar{\alpha}(|x|) \leq V(x)\leq\bar{\alpha}(|x|) \tag{24}\] \[\dot{V}(x) \leq 0 \tag{25}\]  
_for all \(x\in\mathbb{U}\) holds, then the equilibrium is (locally) stable in the sense of Lyapunov. If in addition there exists a function \(\alpha\) of class \(\mathcal{K}\) defined on \([0,a)\) such that_  
\[\forall x\in\mathbb{U}:\quad\dot{V}(x)\leq-\alpha(|x|) \tag{26}\]  
_is satisfied, then \(x=0\) is a (locally) asymptotically stable equilibrium. If \(\alpha,\bar{\alpha},\bar{\alpha}\in K_{\infty}\) also holds, then the equilibrium is globally asymptotically stable._  
In Equation (27), the (total) time derivative \(\dot{V}\) of \(V\) along the solutions of the system (23) is described by the Lie derivative  
\[\dot{V}(x)=L_{f}V(x)=\frac{\partial V}{\partial x}(x)\,f(x) \tag{27}\]  
of the scalar field \(V\) in the direction of the vector field [4]. A function \(V\) satisfying the conditions of Theorem 1 is called a Lyapunov function. Given a polynomial system (23), the procedures described in Sections 2.2 and 2.3 can be used directly to verify the conditions of Theorem 1. For a system (2)-(6) polynomialized according to Section 2.1, some adjustments are necessary ([30] Proposition 4):  
**Theorem 2** (Lyapunov conditions for a recasted system). : _Let a system (2)-(6) with the main denominator \(N(z)\) be given. Moreover, let \(z_{2,0}=T(0)\) hold. Given a polynomial function \(\dot{V}(z)\), column vectors of polynomial functions \(\mu_{1}(z)\), \(\mu_{2}(z)\); column vectors of SOS polynomials \(\sigma_{1}(z)\)\(\sigma_{2}(z)\) and a scalar polynomial function \(\phi(z),\;\phi(z_{1},T(z_{1}))>0\), \(\forall z_{1}\in\mathbb{D}_{1}\setminus\{0\}\) with suitable dimensions exist such that_  
\[\tilde{V}(0,z_{2,0}) =0 \tag{28}\] \[\tilde{V}(z)-\mu_{1}^{T}(z)G_{1}(z)-\sigma_{1}^{T}(z)G_{2}(z)-\phi (z) \in\mathcal{S}\] (29) \[-N(z)\bigg{(}\frac{\partial\tilde{V}}{\partial z_{1}}(z)f_{1}(z) +\frac{\partial\tilde{V}}{\partial z_{2}}(z)f_{2}(z)\bigg{)}-\mu_{2}^{T}(z)G_{ 1}(z)-\sigma_{2}^{T}(z)G_{2}(z) \in\mathcal{S} \tag{30}\]  
_holds, then \(x=0\) is a stable equilibrium of the original system (1)._  
The proof of this theorem is given in [30]. The conditions presented in Theorem 2 suggest only stability and not asymptotic stability. This stems from the fact that the condition (30) requires only positive semidefiniteness.