Experimental Study of a Modified Command Governor Adaptive Controller for Depth Control of an Unmanned Underwater Vehicle

Abstract

Command governor–based adaptive control (CGAC) is a recent control strategy that has been explored as a possible candidate for the challenging task of precise maneuvering of unmanned underwater vehicles (UUVs) with parameter variations. CGAC is derived from standard model reference adaptive control (MRAC) by adding a command governor that guarantees acceptable transient performance without compromising stability and a command filter that improves the robustness against noise and time delay. Although simulation and experimental studies have shown substantial overall performance improvements of CGAC over MRAC for UUVs, it has also shown that the command filter leads to a marked reduction in initial tracking performance of CGAC. As a solution, this paper proposes the replacement of the command filter by a weight filter to improve the initial tracking performance without compromising robustness and the addition of a closed-loop state predictor to further improve the overall tracking performance. The new modified CGAC (M-CGAC) has been experimentally validated and the results indicate that it successfully mitigates the initial tracking performance reduction, significantly improves the overall tracking performance, uses less control force, and increases the robustness to noise and time delay. Thus, M-CGAC is a viable adaptive control algorithm for current and future UUV applications.

Appendix: Lyapunov stability proof of M-CGAC

In order to derive the stable adaptive laws, consider the following Lyapunov function candidate

$$ {\displaystyle \begin{array}{l}V\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{un f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)={\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}{\boldsymbol{e}}_m+{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\hat{\boldsymbol{e}}+\mathrm{trace}\left[\begin{array}{l}{\left({\overset{\sim }{\boldsymbol{W}}}_{un}{\Lambda}^{\frac{1}{2}}\right)}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}\left({\overset{\sim }{\boldsymbol{W}}}_{un}{\Lambda}^{\frac{1}{2}}\right)+\\ {}{\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma }{\Lambda}^{\frac{1}{2}}\right)}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma }{\Lambda}^{\frac{1}{2}}\right)\end{array}\right]\\ {}\kern0.5em \\ {}\kern26.25em +\alpha \mathrm{trace}\left[\begin{array}{l}{\left({\overset{\sim }{\boldsymbol{W}}}_{un f}{\Lambda}^{\frac{1}{2}}\right)}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un f}^{-1}\left({\overset{\sim }{\boldsymbol{W}}}_{un f}{\Lambda}^{\frac{1}{2}}\right)+\\ {}{\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma f}{\Lambda}^{\frac{1}{2}}\right)}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}{\Lambda}^{\frac{1}{2}}\right)\end{array}\right]\end{array}}\kern9em $$

(49)

where ${\overset{\sim }{\boldsymbol{W}}}_{un}(t)\triangleq {\hat{\boldsymbol{W}}}_{un}(t)-{\boldsymbol{W}}_{un}\in {\mathbb{R}}^{q\times q}$ and ${\overset{\sim }{\boldsymbol{W}}}_{\sigma }(t)\triangleq {\hat{\boldsymbol{W}}}_{\sigma }(t)-{\boldsymbol{W}}_{\sigma}\in {\mathbb{R}}^{s\times q}$ are weight estimation errors, ${\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}(t)\triangleq {\hat{\boldsymbol{W}}}_{u{n}_f}(t)-{\boldsymbol{W}}_{un}\in {\mathbb{R}}^{q\times q}$and ${\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}(t)\triangleq {\hat{\boldsymbol{W}}}_{\sigma }(t)-{\boldsymbol{W}}_{\sigma}\in {\mathbb{R}}^{s\times q}$are low-pass-filtered weights estimation errors.

Note that V(0,  0,  0,  0, 0,  0) = 0 and $V\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)>0$ for all $\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\ne \left(0,0,0,0,0,0\right)$. In addition, $V\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)$ is radially unbounded.

Now, differentiating Eq. (49) yields

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)={\dot{\boldsymbol{e}}}_m^{\mathrm{T}}\boldsymbol{P}{\boldsymbol{e}}_m+{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}{\dot{\boldsymbol{e}}}_m+{\dot{\hat{\boldsymbol{e}}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\hat{\boldsymbol{e}}+{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\dot{\hat{\boldsymbol{e}}}\\ {}+2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\kern5em \end{array}} $$

(50)

Substituting from error dynamics of Eqs. (11) and (31) in Eq. (50) yields

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)={\left({\boldsymbol{A}}_m{\boldsymbol{e}}_m-\boldsymbol{H}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\right)}^{\mathrm{T}}\boldsymbol{P}{\boldsymbol{e}}_m\\ {}+{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}\left({\boldsymbol{A}}_m{\boldsymbol{e}}_m-\boldsymbol{H}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}(t){\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\right)+{\left({\boldsymbol{A}}_m\hat{\boldsymbol{e}}+\boldsymbol{H}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\right)}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\hat{\boldsymbol{e}}\ \\ {}\kern1.75em +{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\left({\boldsymbol{A}}_m\hat{\boldsymbol{e}}+\boldsymbol{H}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\right)+2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]\kern0.5em \\ {}\kern1.75em +2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\end{array}} $$

(51)

Simplyfing Eq. (51) yields

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)={\boldsymbol{e}}_m^T\left[{\boldsymbol{A}}_m^T\boldsymbol{P}+\boldsymbol{P}{\boldsymbol{A}}_m\right]{\boldsymbol{e}}_m+{\hat{\boldsymbol{e}}}^{\mathrm{T}}\left[{\boldsymbol{A}}_{prd}^{\mathrm{T}}{\boldsymbol{P}}_{prd}+{\boldsymbol{A}}_{prd}{\boldsymbol{P}}_{prd}\right]\boldsymbol{e}\\ {}\kern1.75em -\mathbf{2}{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{PH}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]+\mathbf{2}{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}\boldsymbol{H}\boldsymbol{\Lambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\\ {}\kern2em +2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\kern3em \end{array}}\kern4.5em $$

(52)

Substituting in Eq. (52) from the Lyapunov equations $\mathbf{0}={\boldsymbol{A}}_m^{\mathrm{T}}\boldsymbol{P}+\boldsymbol{P}{\boldsymbol{A}}_m+\boldsymbol{Q}$ for some Q = Q^T > 0 and$\mathbf{0}={\boldsymbol{A}}_{prd}^{\mathrm{T}}{\boldsymbol{P}}_{prd}+{\boldsymbol{P}}_{prd}{\boldsymbol{A}}_{prd}+{\boldsymbol{Q}}_{prd}$ for some ${\boldsymbol{Q}}_{prd}={\boldsymbol{Q}}_{prd}^{\mathrm{T}}>0$ yields

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}+\mathbf{2}\left({\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}\right)\boldsymbol{H}\boldsymbol{\Lambda } \left[{{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right]\\ {}\kern8em +2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\kern4.75em \end{array}} $$

(53)

Defining $\overline{\boldsymbol{e}}=\left({\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}\right)\boldsymbol{H}$ Eq. (53) simplify to

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}-\mathbf{2}{\overline{\boldsymbol{e}}}^T\boldsymbol{\Lambda} {{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{u}}_n(t)-2{\overline{\boldsymbol{e}}}^{\mathrm{T}}\boldsymbol{\Lambda} {{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\\ {}\kern7.5em +2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\kern3em \end{array}} $$

(54)

Using the trace identity a^Tb = trace(ba^T) in Eq. (54) gives

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}-2\mathrm{trace}\left[{{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{u}}_n(t){\overline{\boldsymbol{e}}}^{\mathrm{T}}\boldsymbol{\Lambda} \right]-2\mathrm{trace}\left[{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right){\overline{\boldsymbol{e}}}^{\mathrm{T}}\boldsymbol{\Lambda} \right]\\ {}+2\mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{un}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{un}+{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}{\boldsymbol{\Gamma}}_{\sigma}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}{\boldsymbol{\Gamma}}_{u{n}_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{u{n}_f}+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}{\boldsymbol{\Gamma}}_{\sigma_f}^{-1}{\dot{\hat{\boldsymbol{W}}}}_{\sigma_f}\right)\Lambda \right]\\ {}\kern3em \end{array}} $$

(55)

Substituting from the filtered weights Eq. (22) and proposed M-CGAC weight update laws given by (32) and (33) in Eq. (55) yields

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}-2\mathrm{trace}\left[{{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}{\boldsymbol{u}}_n(t){\overline{\boldsymbol{e}}}^{\mathrm{T}}\boldsymbol{\Lambda} \right]-2\mathrm{trace}\left[{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right){\overline{\boldsymbol{e}}}^{\mathrm{T}}\boldsymbol{\Lambda} \right]+\\ {}2\mathrm{trace}\left[\left(\begin{array}{l}{{\overset{\sim }{\boldsymbol{W}}}_{un}}^{\mathrm{T}}\left({\boldsymbol{u}}_n(t){\overline{\boldsymbol{e}}}^{\mathrm{T}}-\alpha \left({\hat{\boldsymbol{W}}}_{un}-{\hat{\boldsymbol{W}}}_{u{n}_f}\right)\right)+\\ {}{{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}\left(\boldsymbol{\sigma} \left(\boldsymbol{x}(t)\right){\overline{\boldsymbol{e}}}^{\mathrm{T}}-\alpha \left({\hat{\boldsymbol{W}}}_{\sigma }-{\hat{\boldsymbol{W}}}_{\sigma_f}\right)\right)\end{array}\right)\Lambda \right]+2\alpha \mathrm{trace}\left[\left({{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{u{n}_f}\left({\hat{\boldsymbol{W}}}_{un}-{\hat{\boldsymbol{W}}}_{u{n}_f}\right)+{{\overset{\sim }{\boldsymbol{W}}}^{\mathrm{T}}}_{\sigma_f}\left({\hat{\boldsymbol{W}}}_{\sigma }-{\hat{\boldsymbol{W}}}_{\sigma f}\right)\right)\Lambda \right]\\ {}\kern0.75em \\ {}\end{array}} $$

(56)

Further simplifying Eq. (56)

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}\\ {}\kern1em -2\alpha \mathrm{trace}\left[\left({\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)\right)\Lambda \right]-2\alpha \mathrm{trace}\left[\left({\left({{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}-{{\overset{\sim }{\boldsymbol{W}}}^T}_{\sigma_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma }-{\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\right)\Lambda \right]\\ {}\kern1em \end{array}} $$

(57)

By definition

${\displaystyle \begin{array}{l}\mathrm{trace}\left[\left({\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)\right)\Lambda \right]=\kern0.5em \sum \limits_{i=1}^N\sum \limits_{j=1}^M{\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}_{ij}^2{\Lambda}_{ii}\ge {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right\Vert}_F^2{\Lambda}_{\mathrm{min}}\\ {}\kern0.5em \end{array}}$ $\mathrm{where}\ {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right\Vert}_F^2=\sum \limits_{i=1}^N\sum \limits_{j=1}^M{\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}_{ij}^2$ is the Forbenius norm of $\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)$ and Λ_min is the minimum diagonal element of Λ. A similar definition is applicable to $\mathrm{trace}\left[\left({\left({{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}-{{\overset{\sim }{\boldsymbol{W}}}^T}_{\sigma_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma }-{\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\right)\Lambda \right]$.

Thus, Eq. (57) reduces to

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\le -{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}-2\alpha {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right\Vert}_F^2{\Lambda}_{\mathrm{min}}-2\alpha {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{\sigma }-{\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right\Vert}_F^2{\Lambda}_{\mathrm{min}}\\ {}\kern11em \end{array}} $$

(58)

Therefore, as Λ_min is positive and the Forbenius norm is positive

$$ {\displaystyle \begin{array}{l}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\le -{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{Q}{\boldsymbol{e}}_m-{\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{Q}}_{prd}\boldsymbol{e}\le 0\\ {}\kern11em \end{array}} $$

Hence, this proves that the closed-loop system is Lyapunov stable and that ${\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em$ and ${\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}$ are uniformly ultimately bounded. Since c(t)is bounded and A_mis Hurwitz, then x_m(t) and ${\dot{\boldsymbol{x}}}_m(t)$ are bounded. Hence, the system state x(t) is bounded. This implies $\hat{\boldsymbol{x}}(t)$, u_n(t), and σ(x) are bounded. Since the weight estimation errors ${\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma }$ are bounded and the ideal weights W_un, W_σ are constant, the weight estimations ${\hat{\boldsymbol{W}}}_{un},{\hat{\boldsymbol{W}}}_{\sigma }$ are also bounded. Thus, it follows from Eqs. (11) and (31) that ${\dot{\boldsymbol{e}}}_m,\kern0.5em \dot{\hat{\boldsymbol{e}}}$ are also bounded. Therefore, $\ddot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)$ is bounded. Now, it follows from Barbalat’s lemma (Ioannou and Fidan 2006) that $\underset{t\to \infty }{\lim}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=0$, which consequently shows that e_m(t) and $\hat{\boldsymbol{e}}(t)$ asymptotically converge to zero as t → ∞. Moreover, since the Lyapunov function V is radially unbounded, this convergence is global.

Thus, the system error and prediction error are globally, uniformly asymptotically stable, i.e., $\underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0$ and $\underset{t\to \infty }{\lim}\left\Vert \hat{\boldsymbol{e}}\right\Vert =0$. This completes the proof.