Distributed control of networked uncertain Euler–Lagrange systems in the presence of stochastic disturbances: a prescribed performance approach

Abstract

In order to synchronize a network of Euler–Lagrangian (EL) systems, a distributed output-feedback control is proposed in this paper. The adopted mathematical model of EL systems possesses nonlinear uncertainties, unmeasured states, and stochastic disturbances. By introducing prescribed performance control (PPC) in a dynamic surface control approach, a novel distributed neural adaptive PPC design is presented. Simultaneously, one outstanding characteristic is that the minimal learning parameter approach is employed in the dynamic surface control methodology. It is proved that the norm of distributed tracking errors is confined to compact set with predefined convergence rates and maximum overshoots by using stochastic Lyapunov analysis and graph theory. Finally, simulation examples validate the effectiveness of the proposed approach.

Appendix 1

Because in the proposed distributed controller design two RBFNNs are employed to approximate uncertainties for each agent, it can be proved that there exist two compact and invariant sets such that over them the approximation property holds. However, since all the signals in the systems are bounded, there exists a compact set such that the approximation property holds throughout [26].

For state-feedback case, from (61), one can obtain that

$$\begin{aligned} ||\bar{z}_{N,1}||\le & {} \sqrt{2} \sum _{i=1}^{N} ||\bar{\eta }_{N,1}||~||\bar{1}^*-\bar{\zeta }_{N,1}|| \nonumber \\&\times \,\sqrt{\left( 1-e^{-\rho t}\right) \frac{\lambda }{\rho }+V(0)}. \end{aligned}$$

(82)

where $\bar{z}_{N,1}=[z_{1,1},z_{2,1},\ldots ,z_{N,1}]^T,\bar{\eta }_{N,1}=[\eta _{1,1},\eta _{2,1},\ldots ,\eta _{N,1}]^T,$ $\bar{\zeta }_{N,1}=[\zeta _{1,1},\zeta _{2,1},\ldots ,\zeta _{N,1}]^T$, and $\bar{1}^*=[1,1,\ldots ,1]{^T}\in \mathfrak {R}^{N.n}$.

Note the fact that [21]

$$\begin{aligned} \bar{z}_{N,1}=H\left( \bar{y}-\bar{1}y_L\right) \end{aligned}$$

(83)

where $\bar{y}=[y_1,y_2,\ldots ,y_N]^T$ and $y_i$ is the output of i-th agent for $i=1,\ldots ,N$.

Therefore, we have

$$\begin{aligned} ||\bar{T}||\le \frac{\sqrt{2} \sum _{i=1}^{N} ||\bar{\eta }_{i,1}||~||\bar{1}^*-\bar{\zeta }_{i,1}||\sqrt{\left( 1-e^{-\rho t}\right) \frac{\lambda }{\rho }+V(0)}}{\sigma (H)}, \end{aligned}$$

(84)

where $\sigma (H)$ denotes minimum singular value of H, $\bar{T}=\bar{y}-\bar{1}y_L$, $\bar{T}=[T_1,T_2,\ldots ,T_N]^T$, and $T_i=y_i-y_L$ is the consensus error for $(i=1,2,\ldots ,N)$.

Via Assumption 4, we have

$$\begin{aligned} ||\bar{y}||&\le \frac{\sqrt{2} \sum _{i=1}^{N} ||\bar{\eta }_{i,1}||~||\bar{1}^*-\bar{\zeta }_{i,1}||\sqrt{\left( 1-e^{-\rho t}\right) \frac{\lambda }{\rho }+V(0)}}{\sigma (H)}\nonumber \\&\quad +\sqrt{N}\mu , \end{aligned}$$

(85)

where $||y_L||\le \mu$.

Therefore, we have

$$\begin{aligned} \Omega _4:=\Bigg \{y_i\in \mathfrak {R}^n: ||y_i||\le \frac{AB}{\sigma (H)}+\sqrt{N}\mu \Bigg \}, \end{aligned}$$

(86)

where  $A=\sqrt{2} \sum _{i=1}^{N} ||\bar{\eta }_{i,1}||~||\bar{1}^*-\bar{\zeta }_{i,1}||$           and

$B=\sqrt{\left( 1-e^{-\rho t}\right) \frac{\lambda }{\rho }+V(0)}.$

It is obvious that $\Omega _4$ is a closed set, because of it is the union of points form a boundary set $\{||y_i||=\frac{AB}{\sigma (H)}+\sqrt{N}\mu \}$ and the points of an open set $\{||y_i||<\frac{AB}{\sigma (H)}+\sqrt{N}\mu \}$. Moreover, due to $y_i$ in $\Omega _4$ have upper and lower bounds; therefore, it is bounded. Hence, it can be concluded that $\Omega _4$ is a compact set. On the other hand, from (86), it is obvious that $\Omega _4$ is an invariant set, because of $y_i$ for all times stay in $\Omega _4$. Therefore, it can be inferred that universal approximation ability for

$$\begin{aligned} \bar{f}_{i,2}(\bar{Z}_{i,2})=\Theta _{i,2}^{*T}\varphi _{i,2}{(\bar{Z}_{i,2})}+\delta _{i,2}^*{(\bar{Z}_{i,2})}, \end{aligned}$$

(87)

holds over a compact and invariant set $\Omega _4$.

On the other hand, from (20), (21) and (22), we have

$$\begin{aligned} x_{i,2}=&~S_{i,2}+\pi _i \nonumber \\ ||x_{i,2}||\le&~||S_{i,2}||+||\pi _i|| \nonumber \\ \le&~||S_{i,2}||+||\chi _i||+||\alpha _i||\nonumber \\ \le&~||S_{i,2}||+||\chi _i||+||c_{i,1}||~||h_{i,1}||~||S_{i,1}|| \nonumber \\&+\frac{1}{2}||r_{i,1}^{-1}||~||\hat{W}_{i,1}||~||h_{i,1}||~||S_{i,1}||+||S_{i,2}|| \nonumber \\&+||\dot{\eta }_{i,1}||~||\zeta _{i,1}||. \end{aligned}$$

(88)

From (11)−(13), one obtains $\dot{\eta }_{i,1_k}=a_{i,1_k}\rho _{i,1_k\infty }e^{-a_{i,1_k}t}\le a_{i,1_k}\rho _{i,1_k\infty }$. Therefore, $||\dot{\eta }_{i,1}||\le \upsilon$, where $\upsilon$ is a unknown positive constant. Due to the fact $\rho _{i,1_k\infty }\le \eta _{i,1_k} \le 2\rho _{i,1_k\infty }+\rho _{i,1_k0}$, by noting (14) and definition $h_{i,1_k}$, one gets $||h_{i,1}||\le \beta$, where $\beta$ is a positive constant. Furthermore, for simplicity, we assume that $||c_{i,1}||\le c$ and $||r_{i,1}^{-1}||\le r$, where c and r are known positive parameters.

Hence, one obtains

$$\begin{aligned} ||x_{i,2}||&\le 2\sqrt{\frac{ 2(V(0)+\frac{\lambda }{\rho })}{\lambda _{\text {min}}\left( M_i(x_{i,1}\right) )}} +\sqrt{2{\left( V(0)+\frac{\lambda }{\rho }\right) }}\nonumber \\&\quad +c\beta \sqrt{\acute{S}_{i,1}^T \acute{S}_{i,1}+2\left( \acute{V}_W+\frac{\lambda }{\rho }\right) }\nonumber \\&\quad +\frac{1}{2}r\Bigg (||W_{i,1}^*||+\sqrt{2\lambda _{\text {min}}(\gamma _{i,1})\left( \acute{V}+\frac{\lambda }{\rho }\right) } \nonumber \\&\quad \times \beta \sqrt{\acute{S}_{i,1}^T \acute{S}_{i,1}+2\left( \acute{V}_W+\frac{\lambda }{\rho }\right) }\Bigg ) + \upsilon \sqrt{n}\overset{\Delta }{=} \Upsilon . \end{aligned}$$

(89)

Moreover, due to Assumption 4, it can be observed that $\dot{y}_L$ is bounded, i.e., $||\dot{y}_L|| \le m$, where m is an unknown positive constant.

Therefore, we have

$$\begin{aligned} \Omega _5= \Big \{[x_{i,2},\dot{y}_L]^T:||x_{i,2}||\le \Upsilon , ~~||\dot{y}_L||\le m\Big \} \end{aligned}$$

(90)

Similar to previous discussion, from (90) it can be obtained that $\Omega _5$ is a compact and an invariant set. Hence, it can be concluded that universal approximation ability for

$$\begin{aligned} \bar{f}_{i,1}(\bar{Z}_{i,1})=\Theta _{i,1}^{*T}\varphi _{i,1}{(\bar{Z}_{i,1})}+\delta _{i,1}^*{(\bar{Z}_{i,1})}, \end{aligned}$$

(91)

holds over compact and invariant set $\Omega _5$.

Similar to (82)−(91), it can be proved that universal approximation ability for output-feedback case holds over compact and invariant sets and, hence, is omitted here.