Multilayer neural network based asymptotic motion control of saturated uncertain robotic manipulators

Abstract

Composite influences coming from signal measurement noises, unknown nonlinear dynamics, external disturbances and input saturation nonlinearity make it challenging to synthesize high-performance closed-loop control algorithms for uncertain robotic manipulators. In the face of these challenges, we employ the nonlinear multilayer neural networks to approach uncertain nonlinear dynamics and exploit the robust adaptive control to deal with external disturbances without knowing their bounds in advance. More importantly, robust adaptive based auxiliary functions are creatively introduced to offset the possible input saturation nonlinearity. Furthermore, the desired trajectory based model compensation technology is integrated into the control scheme to reduce measurement noises as much as possible. In theory, the global closed-loop stability of the dynamical uncertain system is testified and significant asymptotic tracking result can be acquired. The application verification under different working conditions including severe high-frequency working conditions is implemented to indicate the high-performance effect of the synthesized intelligent controller.

Appendix 1

Proof of Theorem 1

A continuously differentiable Lyapunov function candidate V_L can be chosen as

$$ {\displaystyle \begin{array}{c}{V}_L=\frac{1}{2}{e}_1^T{e}_1+\frac{1}{2}{e}_2^T{e}_2+\frac{1}{2}{\eta}_1^T{\eta}_1+\frac{1}{2}{\eta}_2^T{\eta}_2+\frac{1}{2} tr\left({\overset{\sim }{W}}^T{\Upsilon}^{-1}\overset{\sim }{W}\right)\\ {}+\frac{1}{2} tr\left({\overset{\sim }{V}}^T{\Gamma}^{-1}\overset{\sim }{V}\right)+\frac{1}{2r}{\overset{\sim }{\phi}}^T\overset{\sim }{\phi }+\frac{1}{2s}{\overset{\sim }{\omega}}^T\overset{\sim }{\omega}\end{array}} $$

(26)

where tr(•) represents the trace of the matrix •.

Take the time derivative of the Eq. (26) yields

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L={e}_1^T{\dot{e}}_1+{e}_2^T{\dot{e}}_2+{\eta}_1^T{\dot{\eta}}_1+{\eta}_2^T{\dot{\eta}}_2+ tr\left({\overset{\sim }{W}}^T{\Upsilon}^{-1}\dot{\overset{\sim }{W}}\right)\\ {}+ tr\left({\overset{\sim }{V}}^T{\Gamma}^{-1}\dot{\overset{\sim }{V}}\right)-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}-\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}} $$

(27)

Substituting (9), (15) and (19) into (27) yields

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L={e}_2^T\left[-{k}_2{e}_2+{P}_1+{P}_2+\overset{\sim }{\varphi}\left({\zeta}_1,{\dot{\zeta}}_1,{\zeta}_{1r},{\dot{\zeta}}_{1r}\right)+\Omega (t)+{\upsilon}_s\right]\\ {}+{e}_1^T\left({e}_2-{k}_1{e}_1\right)+{\eta}_1^T\left({\eta}_2-{k}_1{\eta}_1\right)+{\eta}_2^T\left(\Delta u-{k}_2{\eta}_2+{\eta}_s\right)\\ {}+ tr\left({\overset{\sim }{W}}^T{\mathrm{Y}}^{-1}\dot{\overset{\sim }{W}}\right)+ tr\left({\overset{\sim }{V}}^T{\Gamma}^{-1}\dot{\overset{\sim }{V}}\right)-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}-\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}} $$

(28)

Afterwards, (28) can be reorganized as

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L=-{k}_1{\left\Vert {e}_1\right\Vert}^2-{k}_2{\left\Vert {e}_2\right\Vert}^2-{k}_1{\left\Vert {\eta}_1\right\Vert}^2-{k}_2{\left\Vert {\eta}_2\right\Vert}^2\\ {}+{e}_1^T{e}_2+{\eta}_1^T{\eta}_2+{e}_2^T{P}_1+{e}_2^T\overset{\sim }{\varphi}\left({\zeta}_1,{\dot{\zeta}}_1,{\zeta}_{1r},{\dot{\zeta}}_{1r}\right)\\ {}\begin{array}{c}+{e}_2^T\left[{P}_2+\Omega (t)\right]+{e}_2^T{\upsilon}_s+{\eta}_2^T\Delta u+{\eta}_2^T{\eta}_s\\ {}+ tr\left({\overset{\sim }{W}}^T{\mathrm{Y}}^{-1}\dot{\overset{\sim }{W}}\right)+ tr\left({\overset{\sim }{V}}^T{\Gamma}^{-1}\dot{\overset{\sim }{V}}\right)-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}-\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}\end{array}} $$

(29)

The upper bound of (29) can be achieved as

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L\le -{k}_1{\left\Vert {e}_1\right\Vert}^2-{k}_2{\left\Vert {e}_2\right\Vert}^2-{k}_1{\left\Vert {\eta}_1\right\Vert}^2-{k}_2{\left\Vert {\eta}_2\right\Vert}^2+{e}_1^T{e}_2+{\eta}_1^T{\eta}_2\\ {}+\left\Vert {e}_2\right\Vert \left\Vert \overset{\sim }{\varphi}\left({\zeta}_1,\dot{\zeta_1},{\zeta}_{1r},{\dot{\zeta}}_{1r}\right)\right\Vert +\left\Vert {e}_2\right\Vert \left\Vert {P}_2+\Omega (t)\right\Vert +{e}_2^T{\upsilon}_s\\ {}+\left\Vert {\eta}_2\right\Vert \left\Vert \Delta u\right\Vert +{\eta}_2^T{\eta}_s-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}-\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}} $$

(30)

Applying on Young’s inequality, we have

$$ {e}_1^T{e}_2\le \frac{1}{2}{\left\Vert {e}_1\right\Vert}^2+\frac{1}{2}{\left\Vert {e}_2\right\Vert}^2,{\eta}_1^T{\eta}_2\le \frac{1}{2}{\left\Vert {\eta}_1\right\Vert}^2+\frac{1}{2}{\left\Vert {\eta}_2\right\Vert}^2 $$

(31)

Noting the expressions of $\overset{\sim }{\varphi}\left({\zeta}_1,{\dot{\zeta}}_1,{\zeta}_{1r},{\dot{\zeta}}_{1r}\right)$and Δu, and (21), one yields

$$ \left\Vert \overset{\sim }{\varphi}\left({\zeta}_1,{\dot{\zeta}}_1,{\zeta}_{1r},{\dot{\zeta}}_{1r}\right)\right\Vert \le \rho \left(\left\Vert e\right\Vert \right)\left\Vert e\right\Vert, \left\Vert {P}_2+\Omega (t)\right\Vert \le \omega, \left\Vert \Delta u\right\Vert \le \phi $$

(32)

where ρ(•)∈ℝ⁺indicates any globally invertible non-decreasing function and e = [e₁, e₂]^T.

Noting (24), (31) and (32), (29) can be rearranged as

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L\le -\left({k}_1-\frac{1}{2}\right){\left\Vert {e}_1\right\Vert}^2-\frac{1}{2}{\left\Vert {e}_2\right\Vert}^2-\left({k}_1-\frac{1}{2}\right){\left\Vert {\eta}_1\right\Vert}^2\\ {}-\left({k}_2-\frac{1}{2}\right){\left\Vert {\eta}_2\right\Vert}^2-\left({k}_1-1\right){\left\Vert {e}_2\right\Vert}^2+\rho \left(\left\Vert e\right\Vert \right)\left\Vert {e}_2\right\Vert \left\Vert e\right\Vert \\ {}+\omega \left\Vert {e}_2\right\Vert +{e}_2^T{\upsilon}_s+\phi \left\Vert {\eta}_2\right\Vert +{\eta}_2^T{\eta}_s-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}-\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}} $$

(33)

Depending on the expressions of η_s and υ_s, one has

$$ {\displaystyle \begin{array}{c}{\dot{V}}_L\le -{\gamma}_1{\left\Vert e\right\Vert}^2+\frac{\rho^2\left(\left\Vert e\right\Vert \right){\left\Vert e\right\Vert}^2}{4\left({k}_2-1\right)}-{\gamma}_2{\left\Vert \eta \right\Vert}^2+\overset{\sim }{\phi}\left\Vert {\eta}_2\right\Vert \\ {}-\frac{1}{r}{\overset{\sim }{\phi}}^T\dot{\hat{\phi}}+\overset{\sim }{\omega}\left\Vert {e}_2\right\Vert -\frac{1}{s}{\overset{\sim }{\omega}}^T\dot{\hat{\omega}}\end{array}} $$

(34)

where γ₁ = min{k₁–1/2, 1/2} and γ₂ = min{k₁–1/2, k₂–1/2}.

Furthermore, we have

$$ {\dot{V}}_L\le -{\gamma}_3{\left\Vert e\right\Vert}^2-{\gamma}_2{\left\Vert \eta \right\Vert}^2+\beta (t)+\delta (t) $$

(35)

where γ₃ = γ₁–ρ²(||e||)/[4(k₂–1)] and η = [η₁, η₂]^T.

After integrating both sides of (35), one has

$$ {V}_L(t)+{\gamma}_2{\int}_0^t{\left\Vert \eta (v)\right\Vert}^2 dv+{\gamma}_3{\int}_0^t{\left\Vert e(v)\right\Vert}^2 dv\le {V}_L(0)+{\beta}_m+{\delta}_m $$

(36)

Consequently, it follows from (36) that V_L(t),$\hat{\omega}$,$\hat{\phi}$, η and e are all bounded. Moreover, it is also easy to infer that ζ is bounded. Based on the Lemma 1, we can get that η_s is bounded. Moreover, we can also prove that ||υ_s||≤$\hat{\omega}+\varepsilon {\hat{\omega}}^2$or ||υ_s||≤$\hat{\omega}$, which means that υ_s is always bounded. Thus, all system signals can be guaranteed bounded. Furthermore, it can be concluded that ${\int}_0^t{\left\Vert \eta (v)\right\Vert}^2 dv$≤[V_L(0) + β_m + δ_m]/γ₂ and ${\int}_0^t{\left\Vert e(v)\right\Vert}^2 dv$≤ [V_L(0) + β_m + δ_m]/γ₃, which means η∈L₂ and e∈L₂ [37,38,39]. Recalling the boundedness of all system signals under the closed-loop operation and observing the right hand side of the expressions (9), (15) and (19), we can conclude that $\dot{\eta}$ and $\dot{e}$ are bounded. In accordance with Barbalat’s lemma [40], we can achieve e₁ → 0 as t → ∞. Therefore, all results in Theorem 1 are proved.