Multilayer neural networks-based control of underwater vehicles with uncertain dynamics and disturbances

Abstract

In the presence of uncertain dynamic terms and external disturbances, the problem of trajectory tracking with application to an underactuated underwater vehicle is addressed in this paper. Based on Lyapunov theory and properties of neural networks, a nonlinear neural controller is designed, where multilayer neural networks are adopted to approximate the unmodeled dynamic terms and external disturbances. In order to confine the values of estimated weights within predefined bounds, smooth projection functions are employed. Moreover, measurement noises are considered so as to simulate realistic operation scenario, while filters are designed to get cleaner states. From the stability analysis, it is proven that the tracking errors are globally uniformly ultimately bounded. Numerical examples are provided to demonstrate the robustness of the controller in the presence of unmodeled terms, disturbances and measurement noises.

Appendix

Proof of Theorem 1

Rewrite $\dot{V}_\mathrm{d}$ as

$$\begin{aligned} \dot{V}_\mathrm{d}&\le -\lambda _{1}\mathbf {e}_{1}^{\top }\mathbf {M}^{-1}\mathbf {e}_{1}-\lambda _{2}\mathbf {e}_{2}^{\top }\mathbf {e}_{2}-\lambda _{3}\mathbf {e}_{3}^{\top }\mathbf {e}_{3}\nonumber \\&\quad +\mathbf {e}_{1}^{\top }\varvec{\rho }+\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top } \Big (\mathbf {M}\varvec{\epsilon }_{1}(\mathbf {x}_{1})-\mathbf {M}\varvec{\mu }_1\Big )\nonumber \\&\quad +\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }(\varvec{\epsilon }_{2}(\mathbf {x}_{2})-\varvec{\mu }_2)+a \end{aligned}$$

(56)

where $a=2\varsigma _{n}$ is an arbitrarily small positive constant, and

$$\begin{aligned} \varvec{\mu }_1&=\widetilde{\mathbf {W}}_{1}^{\top }\widehat{\sigma }_1^{'}\mathbf {V}_{1}^{\top }\mathbf {x}_{1} - \mathbf {W}_{1}^{\top } \mathbf {o}(\widetilde{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1})^{2},\\ \varvec{\mu }_2&=\widetilde{\mathbf {W}}_{2}^{\top }\widehat{\sigma }_{2}^{'}\mathbf {V}_{2}^{\top }\mathbf {x}_{2} - \mathbf {W}_{2}^{\top } \mathbf {o}(\widetilde{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2})^{2}, \end{aligned}$$

and

$$\begin{aligned} \mathbf {x}_1&=[1,\,\varvec{\nu }_{1}^{\top }]^{\top }, \\ \mathbf {x}_2&=[1,\,\hat{w}_{11},\,\ldots ,\hat{w}_{43},\,(\partial \sigma (\varvec{\nu }_{1})/\partial \varvec{\nu }_{1})^{\top }, \,\varvec{\nu }_{1}^{\top },\,\varvec{\nu }_{2}^{\top }, \, (\mathbf {J}(\varvec{\eta }_{2})^{\top }\dot{\varvec{\eta }}_\mathrm{d})^{\top }]^{\top }. \end{aligned}$$

It is noted that $\varvec{\nu }_1$ and $\varvec{\nu }_2$ can be written as

$$\begin{aligned} \varvec{\nu }_1&=\mathbf {e}_{2}+\mathbf {J}(\varvec{\eta }_{2})^{\top }\dot{\varvec{\eta }}_\mathrm{d}-\lambda _1\mathbf {M}^{-1}\mathbf {e}_{1}+\varvec{\rho }, \\ \varvec{\nu }_2&=\mathbf {e}_3+\mathbf {h}_2{\varvec{\varpi }}_\mathrm{d} \end{aligned}$$

where ${\varvec{\varpi }}_\mathrm{d}$ is defined in (29). Inspired by [57], the following facts are introduced, given as, $\square$

Fact 1

For each time t, $\mathbf {x}_1$ is bounded by

$$\begin{aligned} ||\mathbf {x}_1|| \le (a_0+a_1 \xi _1)+a_2||\mathbf {e}_1||+a_3||\mathbf {e}_2|| \end{aligned}$$

where $a_{i},\,i=0,\,1,\,2,\,3$, are computable positive constants.

Fact 2

For each time t, $\mathbf {x}_2$ is bounded by

$$\begin{aligned} ||\mathbf {x}_2|| \le (b_0+b_1\xi _1+b_2\xi _2+b_3W_{\max }+b_4\sigma '_{\max })+b_4||\mathbf {e}_1||+b_5||\mathbf {e}_2||+b_6||\mathbf {e}_3|| \end{aligned}$$

where $b_{i},\,i=0,\,1,\,\ldots ,\,6$, are computable positive constants.

Fact 3

For radial basis functions activation functions, the higher-order terms $\mathbf {o}(\widetilde{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1})^{2}$ and $\mathbf {o}(\widetilde{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2})^{2}$ satisfy the following inequalities:

$$\begin{aligned} \mathbf {o}(\widetilde{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1})^{2}&\le c_{10}+||\widetilde{\mathbf {V}}_{1}||_\mathrm{{F}}(c_{11}+c_{12}||\mathbf {e}_1||+c_{13}||\mathbf {e}_2||) \\ \mathbf {o}(\widetilde{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2})^{2}&\le c_{20}+||\widetilde{\mathbf {V}}_{2}||_\mathrm{{F}}(c_{21}+c_{22}||\mathbf {e}_1||+c_{23}||\mathbf {e}_2||+c_{24}||\mathbf {e}_3||) \end{aligned}$$

where $c_{1i},\,c_{2j},\,i=0,\,1,\,2,\,3,\,\,j=0,\,1,\ldots ,4$ are computable positive constants.

Fact 4

For $\varvec{\mu }_1$ and $\varvec{\mu }_2$, they satisfy

$$\begin{aligned} ||\varvec{\mu }_1||&\le d_{10}+||\widetilde{\mathbf {Z}}_1||_\mathrm{{F}}\Big (d_{11}+d_{12}||\mathbf {e}_1||+d_{13}||\mathbf {e}_2||\Big ) \\ ||\varvec{\mu }_2||&\le d_{20}+||\widetilde{\mathbf {Z}}_2||_\mathrm{{F}}(d_{21}+d_{22}||\mathbf {e}_1||+d_{23}||\mathbf {e}_2||+d_{24}||\mathbf {e}_3||) \end{aligned}$$

where $d_{1i},\,d_{2j},\,i=0,\,1,\,2,\,3,\,\,j=0,\,1,\ldots ,4$ are computable positive constants, and $\widetilde{\mathbf {Z}}_1={\mathrm {diag}}(\widetilde{\mathbf {W}}_1,\, \widetilde{\mathbf {V}}_1)$, $\widetilde{\mathbf {Z}}_2={\mathrm {diag}}(\widetilde{\mathbf {W}}_2,\, \widetilde{\mathbf {V}}_2)$.

Fact 5

As we have assumed $||\mathbf {W}||_\mathrm{{F}}\le W_{\max }$, $||\mathbf {V}||_\mathrm{{F}}\le V_{\max }$ and guaranteed $||\hat{\mathbf {W}}_1||^2_\mathrm{{F}}\le W_{m1}$, $||\hat{\mathbf {V}}_{1}||^{2}_\mathrm{{F}}\le V_{m1}$, $||\hat{\mathbf {W}}_2||^2_\mathrm{{F}}\le W_{m2}$, $||\hat{\mathbf {V}}_{2}||^{2}_\mathrm{{F}}\le V_{m2}$, we thereby get the result that

$$\begin{aligned} ||\widetilde{\mathbf {Z}}_1||_\mathrm{{F}}\le z_{m1},\, ||\widetilde{\mathbf {Z}}_2||_\mathrm{{F}}\le z_{m2} \end{aligned}$$

where $z_{m1},\,z_{m2}$ are computable positive constants.

Use the following inequalities:

• 1. $-\lambda _{1}\mathbf {e}_{1}^{\top }\mathbf {M}^{-1}\mathbf {e}_{1} \le -\lambda _{1}\gamma _{\min }(\mathbf {M}^{-1})||\mathbf {e}_1||^2$

• 2. $\mathbf {e}_{2}^{\top }\mathbf {M}\varvec{\epsilon }_{1}(\mathbf {x}_1)\le \gamma _{\max }(\mathbf {M})\Big ( \dfrac{||\mathbf {e}_2||^2}{2\varepsilon }+\dfrac{\varepsilon \epsilon _{\max }^2}{2}\Big )$

• 3. $\mathbf {e}_{1}^{\top }\varvec{\rho }\le \dfrac{||\mathbf {e}_1||^2}{2\varepsilon }+\dfrac{\varepsilon ||\varvec{\rho }||^2}{2}$, $\mathbf {e}_3^{\top }\varvec{\epsilon }_{2}(\mathbf {x}_2) \le \dfrac{||\mathbf {e}_3||^2}{2\varepsilon }+\dfrac{\varepsilon \epsilon _{\max }^2}{2}$

• 4. $-\mathbf {e}_2^{\top }\mathbf {M}\varvec{\mu }_1\le \gamma _{\max }(\mathbf {M})\Big ( \dfrac{f_{10}^2}{2\varepsilon }+\dfrac{\varepsilon ||\mathbf {e}_2||^2}{2}+\dfrac{f_{11}||\mathbf {e}_1||^2}{2\varepsilon }+\dfrac{f_{11}\varepsilon ||\mathbf {e}_2||^2}{2}+f_{12}||\mathbf {e}_2||^2\Big )$

• 5. $-\mathbf {e}_3^{\top }\varvec{\mu }_2 \le \dfrac{f_{20}^2}{2\varepsilon }+\dfrac{\varepsilon ||\mathbf {e}_3||^2}{2}+\dfrac{f_{21}||\mathbf {e}_1||^2}{2\varepsilon }+\dfrac{f_{21}\varepsilon ||\mathbf {e}_3||^2}{2}+\dfrac{f_{22}||\mathbf {e}_2||^2}{2\varepsilon }+\dfrac{f_{22}\varepsilon ||\mathbf {e}_3||^2}{2}+f_{23}||\mathbf {e}_3||^2$

where $\gamma _{\min }(\mathbf {M}^{-1})$ and $\gamma _{\max }(\mathbf {M})$ are minimum and maximum eigenvalues of $\mathbf {M}^{-1}$ and $\mathbf {M}$, respectively, $f_{10},\,f_{11},\, f_{12},\,f_{20},\, f_{21},\, f_{22},\, f_{23}$ are computable positive constants, and $\varepsilon$ is an arbitrarily small positive constant. Based on these above inequalities, and the facts $0\le \vartheta (||\mathbf {x}_1||) \le 1,\, 0\le \vartheta (||\mathbf {x}_1||) \le 1$, (56) is rewritten as

$$\begin{aligned} \dot{V}_\mathrm{d}&\le -\bigg (\lambda _{1}\gamma _{\min }(\mathbf {M}^{-1})-\dfrac{1}{2\varepsilon }-\dfrac{\gamma _{\max }(\mathbf {M})f_{11}}{2\varepsilon }-\dfrac{f_{11}}{2\varepsilon }\bigg )||\mathbf {e}_1||^2 \nonumber \\&\qquad -\bigg (\lambda _{2}-\dfrac{\gamma _{\max }(\mathbf {M})}{2\varepsilon }-\dfrac{\gamma _{\max }(\mathbf {M})\varepsilon }{2}\nonumber \\&\qquad -\dfrac{\gamma _{\max }(\mathbf {M})f_{11}\varepsilon }{2}-\gamma _{\max }(\mathbf {M})f_{12} -\dfrac{f_{22}}{2}\bigg )\nonumber \\&\qquad ||\mathbf {e}_2||^2-\bigg (\lambda _{3}-\dfrac{\varepsilon }{2}-\dfrac{f_{21}\varepsilon }{2}-\dfrac{f_{22}\varepsilon }{2}-f_{23}\bigg ) ||\mathbf {e}_3||^2 \\&\qquad +\dfrac{\varepsilon ||\varvec{\rho }||^2}{2} +\dfrac{\gamma _{\max }(\mathbf {M})\epsilon _{\max }^2}{2}+\dfrac{\varepsilon \epsilon _{\max }^2}{2}+\dfrac{\gamma _{\max }(\mathbf {M})}{2\varepsilon }f_{10}^2+\dfrac{f_{20}^2}{2\varepsilon }\nonumber \\&\quad =-\kappa _1\Vert \mathbf {e}_{1} \Vert ^2-\kappa _2\Vert \mathbf {e}_{2}\Vert ^2 -\kappa _3\Vert \mathbf {e}_{3}\Vert ^2+\varsigma , \end{aligned}$$

(57)

with

$$\begin{aligned} \kappa _1&= \lambda _{1}\gamma _{\min }(\mathbf {M}^{-1})-\dfrac{1}{2\varepsilon }-\dfrac{\gamma _{\max }(\mathbf {M})f_{11}}{2\varepsilon }-\dfrac{f_{11}}{2\varepsilon },\\ \kappa _2&= \lambda _{2}-\dfrac{\gamma _{\max }(\mathbf {M})}{2\varepsilon }-\dfrac{\gamma _{\max }(\mathbf {M})\varepsilon }{2}-\dfrac{\gamma _{\max }(\mathbf {M})f_{11}\varepsilon }{2}\\&\quad -\gamma _{\max }(\mathbf {M})f_{12}-\dfrac{f_{22}}{2},\\ \kappa _3&=\lambda _{3}-\dfrac{\varepsilon }{2}-\dfrac{f_{21}\varepsilon }{2}-\dfrac{f_{22}\varepsilon }{2}-f_{23}, \\ \varsigma&=\dfrac{\varepsilon ||\varvec{\rho }||^2}{2}+\dfrac{\gamma _{\max }(\mathbf {M})\epsilon _{\max }^2}{2}+\dfrac{\varepsilon \epsilon _{\max }^2}{2}\\&\quad +\dfrac{\gamma _{\max }(\mathbf {M})}{2\varepsilon }f_{10}^2+\dfrac{f_{20}^2}{2\varepsilon }. \end{aligned}$$

By choosing parameters properly, we can guarantee that $\kappa _1,\,\kappa _2,\,\kappa _3$ are positive definite. Define $\mathbf {a}=[\Vert \mathbf {e}_{1}\Vert ,\,\Vert \mathbf {e}_{2}\Vert ,\,\Vert \mathbf {e}_{3}\Vert ]^{\top }$ and $\kappa _{\min }=\min \{\kappa _{1},\kappa _2,\kappa _3\}$; one further can rewrite (57) as

$$\begin{aligned} \dot{V}_\mathrm{d}\le -\kappa _{\min }\mathbf {a}^{\top }\mathbf {a}+\varsigma \end{aligned}$$

(58)

which is negative definite for $\Vert \mathbf {a}\Vert > \sqrt{\varsigma /\kappa _{\min }}$ which can be made arbitrarily small by adjusting $\varsigma$ and $\kappa _{\min }$. As a result, global uniformly ultimately bounded is achieved.

At last, inspired by [60], we can prove that the estimated weights $\widehat{\mathbf {W}}_{1},\, \widehat{\mathbf {W}}_{2},\, \widehat{\mathbf {V}}_{1},\, \widehat{\mathbf {V}}_{2}$ are bounded, and the following inequalities are satisfied:

• 1. $-\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top }\mathbf {M}\widetilde{\mathbf {W}}_{1}^{\top }\big (\widehat{\sigma }_1- \widehat{\sigma }_{1}^{'} \widehat{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1} \big )+{\mathrm {tr}}(\widetilde{\mathbf {W}}_{1}^{\top }\varGamma _{W_1}^{-1}\dot{\widehat{\mathbf {W}}}_{1}) \le 0$

• 2. $-\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top }\mathbf {M}\widehat{\mathbf {W}}_{1}^{\top }\widehat{\sigma }_1^{'}\widetilde{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1} +{\mathrm {tr}}(\widetilde{\mathbf {V}}_{1}^{\top }\varGamma _{V_1}^{-1}\dot{\widehat{\mathbf {V}}}_{1}) \le 0$

• 3. $-\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }\widetilde{\mathbf {W}}^{\top }_{2}(\widehat{\sigma }_{2}- \widehat{\sigma }_{2}^{'} \widehat{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2} )+{\mathrm {tr}}(\widetilde{\mathbf {W}}_{2}^{\top }\varGamma _{W_2}^{-1}\dot{\widehat{\mathbf {W}}}_{2}) \le 0$

• 4. $-\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }\widehat{\mathbf {W}}_{2}^{\top }\widehat{\sigma }_{2}^{'}\widetilde{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2}+{\mathrm {tr}}(\widetilde{\mathbf {V}}_{2}^{\top }\varGamma _{V_2}^{-1}\dot{\widehat{\mathbf {V}}}_{2}) \le 0$.

It is noted that due to the above inequalities, the original $\dot{V}_\mathrm{d}$

$$\begin{aligned} \dot{V}_\mathrm{d}&\le -\lambda _{1}\mathbf {e}_{1}^{\top }\mathbf {M}^{-1}\mathbf {e}_{1}-\lambda _{2}\mathbf {e}_{2}^{\top }\mathbf {e}_{2}-\lambda _{3}\mathbf {e}_{3}^{\top }\mathbf {e}_{3}+\mathbf {e}_{1}^{\top }\varvec{\rho }+\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top }\nonumber \\&\qquad \Big (\mathbf {M}\varvec{\epsilon }_{1}(\mathbf {x}_{1})-\mathbf {M}\varvec{\mu }_1\Big )+\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }(\varvec{\epsilon }_{2}(\mathbf {x}_{2})-\varvec{\mu }_2) +a-\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top }\mathbf {M}\widetilde{\mathbf {W}}_{1}^{\top }\big (\widehat{\sigma }_1\nonumber \\&\quad - \widehat{\sigma }_{1}^{'} \widehat{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1} \big )+{\mathrm {tr}}(\widetilde{\mathbf {W}}_{1}^{\top }\varGamma _{W_1}^{-1}\dot{\widehat{\mathbf {W}}}_{1}) -\vartheta (||\mathbf {x}_1||)\mathbf {e}_{2}^{\top }\mathbf {M}\widehat{\mathbf {W}}_{1}^{\top }\widehat{\sigma }_1^{'}\widetilde{\mathbf {V}}_{1}^{\top }\mathbf {x}_{1}\nonumber \\&\quad +{\mathrm {tr}}(\widetilde{\mathbf {V}}_{1}^{\top }\varGamma _{V_1}^{-1}\dot{\widehat{\mathbf {V}}}_{1})-\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }\widetilde{\mathbf {W}}^{\top }_{2}(\widehat{\sigma }_{2}- \widehat{\sigma }_{2}^{'} \widehat{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2} )\nonumber \\&\quad +{\mathrm {tr}}(\widetilde{\mathbf {W}}_{2}^{\top }\varGamma _{W_2}^{-1}\dot{\widehat{\mathbf {W}}}_{2}) -\vartheta (||\mathbf {x}_2||)\mathbf {e}_{3}^{\top }\widehat{\mathbf {W}}_{2}^{\top }\widehat{\sigma }_{2}^{'}\widetilde{\mathbf {V}}_{2}^{\top }\mathbf {x}_{2}+{\mathrm {tr}}(\widetilde{\mathbf {V}}_{2}^{\top }\varGamma _{V_2}^{-1}\dot{\widehat{\mathbf {V}}}_{2}) \end{aligned}$$

(59)

can be rewritten as (56).