## 4 Robust Control of Nonlinear Systems with Mismatched Uncertainties  
In this section, we consider the robust control problem when the system (1) does not satisfy the matched condition (2). At this time, the system is a mismatched nonlinear uncertain system. By constructing the appropriate auxiliary system and performance index, the robust control for the mismatched uncertain system is transformed into solving optimal control law of an auxiliary system. Firstly, the following assumptions are given. **Assumption 3**. : _Suppose that the uncertainty of system (1) satisfies \(\Delta f(x)=c(x)h(x)\), \(\Delta g(x)=g(x)m(x)\), where \(c(x)\) is a known function matrix of appropriate dimensions, \(h(x)\) and \(m(x)\) are uncertain functions, and \(m(x)\geq 0\)._  
The goal of robust control is to find a control function \(u(x)\), which makes the closed-loop system  
\[\dot{x}=f(x)+c(x)h(x)+[g(x)+g(x)m(x)]u(x) \tag{16}\]  
globally asymptotically stable for all uncertainties \(h(x)\) and \(m(x)\). In order to obtain the robust controller, an optimal control problem is constructed as follow. For the following auxiliary systems  
\[\dot{x}=f(x)+g(x)u+[I-g(x)g(x)^{+}]c(x)v \tag{17}\]find the controller \(u=u(x)\), \(v=v(x)\), such that the performance index  
\[J(x_{0},u)=\int_{0}^{\infty}[f_{max}^{2}(x)+g_{max}^{2}(x)+\beta^{2}x^{T}x+u^{T}u +v^{T}v]dt \tag{18}\]  
is minimized, where \(\beta\) is the design parameter, \(g(x)^{+}=[g^{T}(x)g(x)]^{-1}g^{T}(x)\) is a pseudo inverse of the matrix function \(g(x)\). Moreover, \(f_{max}(x)\) and \(g_{max}(x)\) are nonnegative functions and satisfy the conditions  
\[\|g(x)^{+}c(x)h(x)\|\leq f_{max}(x),\|h(x)\|\leq g_{max}(x) \tag{19}\]  
According to the performance index (18), the cost function corresponding to the admissible control \((u(x),v(x))\) is  
\[V(x)=\int_{t}^{\infty}[f_{max}^{2}(x)+g_{max}^{2}(x)+\beta^{2}x^{T}x+u^{T}u+v^ {T}v]dt \tag{20}\]  
The following Bellman equation is obtained by taking the time derivation on both sides of (20)  
\[f_{max}^{2}(x)+g_{max}^{2}(x)+\beta^{2}x^{T}x+\bar{u}^{T}\bar{u}+\
abla V^{T}[ f(x)+\bar{g}(x)\bar{u}]=0, \tag{21}\]  
where \(\
abla V\) is the gradient vector of \(V(x)\) with respect to \(x\), \(\bar{g}(x)=[g(x),(I-g(x)g(x)^{+})c(x)]\), \(\bar{u}=[u^{T},v^{T}]^{T}\). Defining Hamiltonian functions as  
\[H(x,u,\
abla V)=f_{max}^{2}(x)+g_{max}^{2}(x)+\beta^{2}x^{T}x+\bar{u}^{T}\bar{ u}+\
abla V^{T}[f(x)+\bar{g}(x)\bar{u}] \tag{22}\]  
Assuming that the minimum value exists and is unique in (22), the optimal control law is given by  
\[\bar{u}^{*}(x) =\left[ \begin{array}{c}u^{*}(x)\\ v^{*}(x)\end{array} \right]=-\frac{1}{2}\bar{g}^{T}(x)\
abla V^{*} \tag{23}\] \[=-\frac{1}{2}\left[ \begin{array}{c}g^{T}(x)\
abla V^{*}\\ c^{T}(x)[I-g(x)g(x)^{+}]^{T}\
abla V^{*}\end{array} \right]\]  
By substituting (23) into (21), the HJB equation is given by  
\[f_{max}^{2}(x)+g_{max}^{2}(x)+\beta^{2}x^{T}x+\
abla{V^{*}}^{T}f(x)-\frac{1}{4 }\
abla{V^{*}}^{T}\bar{g}(x)g^{T}(x)\
abla V^{*}=0 \tag{24}\]  
and the initial value \(V^{*}(0)=0\). **Remark 2**. : _Generally, the pseudo-inverse of \(g(x)\), \(g(x)^{+}\) will exist if its columns are linearly independent when Assumptions 1 and 2 are true [31]. In practical control systems, the function, \(g(x)\), is usually column full-rank. Therefore, the pseudo-inverse of the function \(g(x)\) is generally satisfied. Furthermore, the pseudo-inverse \(g(x)^{+}\) satisfies \(g(x)^{+}g(x)=1\). However, it does not satisfy \(g(x)g(x)^{+}=1\). In addition, the auxiliary system constructed above is not a nominal system, but a compensation control term \(v(x)\) is added to the nominal system._  
If we can choose an appropriate parameter \(\beta\), the optimal cost \(V^{*}(x)\) can be computed from HJB Equation (24). Then, we can get the optimal control law of system (17) with performance index (18). The following theorem shows that optimal control \(u^{*}(x)=-\frac{1}{2}\bar{g}^{T}(x)\
abla V^{*}\) is a robust controller for uncertain systems. **Theorem 2**.