## 3 Robust Control of Matched Uncertain Nonlinear Systems  
This section considers the problem of robust control when the system (1) meets the matched conditions (2) and (3). By constructing appropriate performance indexes, the problem of robust control is transformed into calculating the optimal control law of a corresponding nominal system. Based on the optimal control of the nominal system, a PI algorithm is proposed to obtain robust feedback controller.  
For the nominal system  
\[\dot{x}=f(x)+g(x)u, \tag{4}\]  
find the controller \(u=u(x)\) to minimize performance index  
\[J(x_{0},u)=\int_{0}^{\infty}[f_{max}^{2}(x)+x^{T}x+u^{T}u]dt \tag{5}\]  
where \(f_{max}(x)\) is the supremum function of uncertainty \(h(x)\), that is \(\|h(x)\|\leq f_{max}(x)\).  
The definition of admissible control in optimal control problem is given below [26].  
**Definition 5**.: _The control policy \(u(x)\) is called an admissible control of the system (4) with regard to the performance function (5) on compact set \(\Omega\subseteq\mathbb{R}^{n}\) if \(u(x)\) is continuous on \(\Omega\), \(u(0)=0\), it can stabilize the system (4) on \(\Omega\), and the performance function (5) is limited for any \(x\in\Omega\)._  
According to the performance index (5), the cost function corresponding to the admissible control \(u(x)\) is given by  
\[V(x,u)=\int_{t}^{\infty}[f_{max}^{2}(x)+x^{T}x+u^{T}u]dt \tag{6}\]  
Taking time derivative on both side of (6), it follows the Bellman equation  
\[f_{max}^{2}(x)+x^{T}x+u^{T}u+\
abla V^{T}[f(x)+g(x)u]=0, \tag{7}\]  
where \(\
abla V\) is the gradient vector of the cost function \(V(x,u)\) with respect to \(x\).  
Definite Hamiltonian function  
\[H(x,u,\
abla V)=f_{max}^{2}(x)+x^{T}x+u^{T}u+\
abla V^{T}[f(x)+g(x)u] \tag{8}\]  
\[\frac{dV^{*}}{dt}\leq -f_{max}^{2}(x)-x^{T}x-\frac{1}{4}\
abla{V^{*}}^{T}g(x)g^{T}(x)\
abla V ^{*}+\
abla{V^{*}}^{T}g(x)h(x) \tag{15}\] \[= -f_{max}^{2}(x)-x^{T}x-\frac{1}{4}[\
abla{V^{*}}^{T}g(x)g^{T}(x) \
abla V^{*}-4\
abla{V^{*}}^{T}g(x)h(x)+4h(x)^{T}h(x)]+h(x)^{T}h(x)\] \[= -x^{T}x+h(x)^{T}h(x)-f_{max}^{2}(x)-\frac{1}{4}[g^{T}(x)\
abla V^{ *}-2h(x)]^{T}[g^{T}(x)\
abla V^{*}-2h(x)]\] \[\leq -x^{T}x\]  
Therefore, by [PERSON] stability theory [30], the optimal control \(u^{*}(x)=-\frac{1}{2}g^{T}(x)\
abla V^{*}\) can make the matched uncertain system (1) asymptotically stable. Thus, for a constant \(c>0\), there is a neighborhood \(N=\{x:\|x\|<c\}\) such that if the state \(x(t)\) enters the neighborhood \(N\), then \(x\to 0\) when \(t\to\infty\). However, \(x(t)\) cannot stay out of the domain \(N\) forever; otherwise, for all \(t>0\), there is \(\|x(t)\|\geq c\). This implies that  
\[V^{*}[x(t)]-V^{*}[x(0)] =\int_{0}^{t}\dot{V}^{*}(x(\tau))dt\] \[\leq\int_{0}^{t}(-x^{T}x)dt\] \[\leq-\int_{0}^{t}c^{2}dt\] \[=-c^{2}t\]  
Therefore, when \(t\to\infty\), \(V^{*}[x(t)]\leq V^{*}[x(0)]-c^{2}t\to-\infty\). This contradicts that \(V^{*}[x(t)]\) is positive definite. Consequently, the system (1) is globally asymptotically stable.  
**Remark 1**.: _For matched nonlinear systems, the robust controller can be obtained by solving the optimal cost function \(V^{*}[x(t)]\) from HJB Equation (10). In Section 4, we will use the PI algorithm to solve the HJB equation, which is a difficult partial differential equation._