## 3 Control Law and Stability
### Description of the System  
According to the partial differential equations established above, the flexible manipulator is a nonlinear multiple-input multiple-output system. For this system, the flexible manipulator can be written as a nonlinear state space:  
\[\left\{\begin{array}{l}\dot{x}_{i}(t)=b_{i}x_{i+1}(t)+f_{i}(x_{1}(t),x_{2}(t), \ldots,x_{i}(t))+w_{i}(t),\;i\leq i\leq n-1\\ \dot{x}_{n}(t)=b_{n}u(t)+f_{n}(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))+w_{n}(t)\\ y(t)=x_{1}(t)\end{array}\right. \tag{10}\]  
where \(x=[x_{1},x_{2},\;\ldots,x_{n}]^{T}\in R^{n}\), \(u\in R\), and \(y\in R\) are the state variables of the system, control input and object output; \(w_{i}(t)\) is external interference; \(f_{i}\) is unknown nonlinear function; and \(b_{i}\) is an unknown constant.  
**Hypothesis 1**.: _The positive constants \(b_{im}\) and \(b_{iM}\) satisfy the inequality: 0 < \(b_{im}\leq\;\mid b_{i}\mid\leq b_{iM}\), \(i\) = 1, 2, \(\ldots\), \(n\)._  
**Hypothesis 2**.: _For smooth functions \(f(x)\) and fuzzy systems, there is an optimal constant parameter \(\stackrel{{\rightarrow*}}{{\theta}}\) that minimizes the approximation error. Here the optimal parameter constant is defined as \(\stackrel{{\rightarrow*}}{{\theta}}=\underset{\theta\in\Omega_{0}}{ \operatorname{argmin}}\left[\underset{x\in\Omega}{\operatorname{sup}}\left|f( x)-\stackrel{{\rightarrow T}}{{\theta}}\xi(x)\right|\right],\,\Omega_{0}\) and \(\Omega\) are the bounded sets of \(\stackrel{{\rightarrow}}{{\theta}}\) and \(x\)._  
The control goal is to make the output \(y(t)\) of the system well clear of vibration errors, and all signals are bounded. For simplicity, the symbols \(\overline{x}_{i}\) and \(y_{di}\) are introduced. Where \(\overline{x}_{i}=[x_{1},x_{2},\;\ldots,\;x_{i}]^{T}\in R^{i}\) (\(i\) = 1,2, \(\ldots\), \(n\)) is the input of the system, \(y_{di}=[y_{d},\dot{y}_{d^{\prime}},\;\ldots,y_{d}^{(i)}]^{T}\in R^{i}\) (\(i\) = 1,2, \(\ldots\), \(n\)\(-\)1) is the expected trajectory value of the system.  
Using single-value fuzzers, product reasoners and barycentric average defuzzers, the fuzzy rules can be expressed as follows: IF \(x_{1}\) is F\({}_{1}^{j}\) and \(\ldots\) and \(x_{n}\) is F\({}_{n}^{j}\) then \(y\) is \(B^{j}\) (\(j\) = 1,2, \(\ldots\), \(N\)). The fuzzy output is:  
\[y(x)=\frac{\sum\limits_{j=1}^{N}\theta_{j}\Pi_{i=1}^{n}\mu_{i}^{j}(x_{i})}{ \sum\limits_{j=1}^{N}\Pi_{i=1}^{n}\mu_{i}^{j}(x_{i})} \tag{11}\]  
where \(x=[x_{1},x_{2},\;\ldots,x_{n}]^{T}\in R^{n}\) and \(\mu_{i}^{j}(x_{i})\) are the membership function \(\mu_{i}^{j}\), \(\theta_{j}=\underset{y\in R}{\operatorname{max}}B^{j}(y)\). Defining \(\xi(x)=[\xi_{1}(x),\;\xi_{2}(x),\;\ldots,\xi_{N}(x)]^{T}\), \(\stackrel{{\rightarrow}}{{\theta}}=[\theta_{1},\;\theta_{2},\; \ldots,\theta_{N}]^{T}\), then the equation can be obtained: \(y(x)=\xi^{T}(x)\stackrel{{\rightarrow}}{{\theta}}\).  
According to the fuzzy universal approximation theorem, if \(f(x)\) is a continuous function defined on a compact set \(\Omega\), then for any constant \(\varepsilon>0\), there exists a fuzzy system, as shown above, that satisfies \(\underset{y\in\Omega}{\operatorname{sup}}|f(x)-y(x)|\leq\varepsilon\).