Intelligent wavelet fuzzy brain emotional controller using dual function-link network for uncertain nonlinear control systems

Abstract

This study aims to propose a more efficient hybrid algorithm to achieve favorable control performance for uncertain nonlinear systems. The proposed algorithm comprises a dual function-link network-based multilayer wavelet fuzzy brain emotional controller and a sign(.) functional compensator. The proposed algorithm estimates the judgment and emotion of a brain that includes two fuzzy inference systems for the amygdala network and the prefrontal cortex network via using a dual-function-link network and three sub-structures. Three sub-structures are a dual-function-link network, an amygdala network, and a prefrontal cortex network. Particularly, the dual-function-link network is used to adjust the amygdala and orbitofrontal weights separately so that the proposed algorithm can efficiently reduce the tracking error, follow the reference signal well, and achieve good performance. A Lyapunov stability function is used to determine the adaptive laws, which are used to efficiently tune the system parameters online. Simulation and experimental studies for an antilock braking system and a magnetic levitation system are presented to verify the effectiveness and advantage of the proposed algorithm.

Appendix

Proof for Theorem 1

Proof: A Lyapunov function is determined as:

$$ {\displaystyle \begin{array}{c}V(t)=\frac{1}{2}{s}^T(t){g}_0^{-1}s(t)+\frac{1}{2{\lambda}_p} tr\left({\overset{\sim }{p}}^T\overset{\sim }{p}\right)+\frac{1}{2{\lambda}_q} tr\left({\overset{\sim }{q}}^T\overset{\sim }{q}\right)+\frac{1}{2{\lambda}_{z_k^a}} tr\left(\tilde{z}_{k}^{aT}\tilde{z}_{k}^a\right)+\frac{1}{2{\lambda}_{z_k^o}} tr\left(\tilde{z}_{k}^{oT}\tilde{z}_{k}^o\right)+\\ {}\frac{1}{2{\lambda}_m}{\overset{\sim }{m}}^T\overset{\sim }{m}+\frac{1}{2{\lambda}_{\sigma }}{\overset{\sim }{\sigma}}^T\overset{\sim }{\sigma }+\frac{{\overset{\sim }{\varDelta}}^T\overset{\sim }{\varDelta }}{2{\lambda}_{\varDelta }}\end{array}} $$

(65)

Next, taking the derivative of (65), then using (49), gives:

$$ {\displaystyle \begin{array}{c}\dot{V}(t)={s}^T(t){g}_0^{-1}\dot{s}(t)+\frac{1}{\lambda_p} tr\left({\overset{\sim }{p}}^T\dot{\overset{\sim }{p}}\right)+\frac{1}{\lambda_q} tr\left({\overset{\sim }{q}}^T\dot{\overset{\sim }{q}}\right)+\frac{1}{\lambda_{z_k^a}} tr\left(\tilde{z}_{k}^{aT}{\dot{\overset{\sim }{z}}}_k^a\right)+\frac{1}{\lambda_{z_k^o}} tr\left(\tilde{z}_{k}^{oT}{\dot{\overset{\sim }{z}}}_k^o\right)+\frac{1}{\lambda_m}{\overset{\sim }{m}}^T\dot{\overset{\sim }{m}}+\frac{1}{\lambda_{\sigma }}{\overset{\sim }{\sigma}}^T\dot{\overset{\sim }{\sigma }}+\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }}\\ {}={s}^T(t)\Big[\tilde{z}_{k}^{aT}\hat{a}+{\hat{z}}_k^{aT}\left({\overline{a}}_m^T\overset{\sim }{m}+{\overline{a}}_{\sigma}^T\overset{\sim }{\sigma}\right)+{\hat{z}}_k^{aT}\left({\overline{a}}_{p_k}^T\tilde{p}_{k}+{\overline{a}}_{q_k}^T\tilde{q}_{k}\right)-\\ {}\tilde{z}_{k}^{oT}\hat{o}-{\hat{z}}_k^{oT}\left({\overline{o}}_m^T\overset{\sim }{m}+{\overline{o}}_{\sigma}^T\overset{\sim }{\sigma}\right)-{\hat{z}}_k^{oT}\left({\overline{a}}_{p_k}^T\tilde{p}_{k}+{\overline{a}}_{q_k}^T\tilde{q}_{k}\right)+\varTheta -{u}_{rb}\Big]+\\ {}\frac{1}{\lambda_p} tr\left({\overset{\sim }{p}}^T\dot{\overset{\sim }{p}}\right)+\frac{1}{\lambda_q} tr\left({\overset{\sim }{q}}^T\dot{\overset{\sim }{q}}\right)+\frac{1}{\lambda_{z_k^a}} tr\left(\tilde{z}_{k}^{aT}{\dot{\overset{\sim }{z}}}_k^a\right)+\frac{1}{\lambda_{z_k^o}} tr\left(\tilde{z}_{k}^{oT}{\dot{\overset{\sim }{z}}}_k^o\right)+\\ {}\frac{1}{\lambda_m}{\overset{\sim }{m}}^T\dot{\overset{\sim }{m}}+\frac{1}{\lambda_{\sigma }}{\overset{\sim }{\sigma}}^T\dot{\overset{\sim }{\sigma }}+\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }}\end{array}} $$

(66)

• Remark 1

${\boldsymbol{p}}^{\ast },{\boldsymbol{q}}^{\ast },{{\boldsymbol{z}}_k^a}^{\ast },{{\boldsymbol{z}}_k^o}^{\ast },{\boldsymbol{m}}^{\ast },\mathrm{and}\ {\boldsymbol{\sigma}}^{\ast }$are constants; hence,

$$ \dot{\tilde{\boldsymbol{q}}}=-\dot{\hat{\boldsymbol{q}}},\dot{\tilde{\boldsymbol{p}}}=-\dot{\hat{\boldsymbol{p}}},{\dot{\tilde{\boldsymbol{z}}}}_k^a=-{\dot{\hat{\boldsymbol{z}}}}_k^a,{\dot{\tilde{\boldsymbol{z}}}}_k^o=-{\dot{\hat{\boldsymbol{z}}}}_k^o,\dot{\tilde{\boldsymbol{\sigma}}}=-\dot{\hat{\boldsymbol{\sigma}}},\mathrm{and}\ \dot{\tilde{\boldsymbol{m}}}=-\dot{\hat{\boldsymbol{m}}} $$

(67)

• Remark 2

$$ \left\{{\displaystyle \begin{array}{c}{\hat{z}}_k^{aT}{\overline{a}}_m^T\overset{\sim }{m}={\overset{\sim }{m}}^T{\overline{a}}_m{\hat{z}}_k^a\\ {}{\hat{z}}_k^{aT}{\overline{a}}_{\sigma}^T\overset{\sim }{\sigma }={\overset{\sim }{\sigma}}^T{\overline{a}}_{\sigma }{\hat{z}}_k^a\\ {}{\hat{z}}_k^{aT}{\overline{a}}_{p_k}^T\tilde{p}_{k}=\tilde{p}_{k}^T{\overline{a}}_{p_k}{\hat{z}}_k^a\\ {}{\hat{z}}_k^{aT}{\overline{a}}_{q_k}^T\tilde{q}_{k}=\tilde{q}_{k}^T{\overline{a}}_{q_k}{\hat{z}}_k^a\end{array}},\right\{{\displaystyle \begin{array}{c}{\hat{z}}_k^{oT}{\overline{o}}_m^T\overset{\sim }{m}={\overset{\sim }{m}}^T{\overline{o}}_m{\hat{z}}_k^o\\ {}{\hat{z}}_k^{oT}{\overline{o}}_{\sigma}^T\overset{\sim }{\sigma }={\overset{\sim }{\sigma}}^T{\overline{o}}_{\sigma }{\hat{z}}_k^o\\ {}{\hat{z}}_k^{oT}{\overline{a}}_{p_k}^T\tilde{p}_{k}=\tilde{p}_{k}^T{\overline{a}}_{p_k}{\hat{z}}_k^o\\ {}{\hat{z}}_k^{oT}{\overline{a}}_{q_k}^T\tilde{q}_{k}=\tilde{q}_{k}^T{\overline{a}}_{q_k}{\hat{z}}_k^o\end{array}}\ and\Big\{{\displaystyle \begin{array}{c} tr\left(\tilde{z}_{k}^{aT}{\dot{\overset{\sim }{z}}}_k^a\right)=\sum \limits_{k=1}^L\tilde{z}_{k}^{aT}{\dot{\hat{z}}}_k^a\\ {} tr\left(\tilde{z}_{k}^{oT}{\dot{\overset{\sim }{z}}}_k^o\right)=\sum \limits_{k=1}^L\tilde{z}_{k}^{oT}{\dot{\hat{z}}}_k^o\\ {} tr\left({\overset{\sim }{p}}^T\dot{\hat{p}}\right)=\sum \limits_{k=1}^L{{\overset{\sim }{p}}^T}_k{\dot{\hat{p}}}_k\\ {} tr\left({\overset{\sim }{q}}^T\dot{\hat{q}}\right)=\sum \limits_{k=1}^L{{\overset{\sim }{q}}^T}_k{\dot{\hat{q}}}_k\end{array}} $$

(68)

Using (66)–(68), (65) becomes

$$ {\displaystyle \begin{array}{c}\dot{V}(t)=\left[\sum \limits_{k=1}^L\tilde{p}_{k}^T\left({s}_k(t)\left({\overline{a}}_{p_k}{\hat{z}}_k^a-{\overline{o}}_{p_k}{\hat{z}}_k^o\right)-\frac{{\dot{\hat{p}}}_k}{\lambda_p}\right)\right]+\left[\sum \limits_{k=1}^L\tilde{q}_{k}^T\left(\sum \limits_{k=1}^L{s}_k(t)\left({\overline{a}}_{q_k}{\hat{z}}_k^a-{\overline{o}}_{q_k}{\hat{z}}_k^o\right)-\frac{{\dot{\hat{q}}}_k}{\lambda_q}\right)\right]+\\ {}\left[\sum \limits_{k=1}^L\tilde{z}_{k}^{aT}\left({s}_k(t)\hat{a}-\frac{{\dot{\hat{z}}}_k^a}{\lambda_{z^a}}\right)\right]+\left[\sum \limits_{k=1}^L\tilde{z}_{k}^{oT}\left(-{s}_k(t)\hat{o}-\frac{{\dot{\hat{z}}}_k^o}{\lambda_{z^o}}\right)\right]+{\overset{\sim }{m}}^T\left[\sum \limits_{k=1}^L{s}_k(t)\left[{\overline{a}}_m{\hat{z}}_k^a-{\overline{o}}_m{\hat{z}}_k^o\right]-\frac{\dot{\hat{m}}}{\lambda_m}\right]+\\ {}{\overset{\sim }{\sigma}}^T\left[\sum \limits_{k=1}^L{s}_k(t)\left[{\overline{a}}_{\sigma }{\hat{z}}_k^a-{\overline{o}}_{\sigma }{\hat{z}}_k^o\right]-\frac{\dot{\hat{\sigma}}}{\lambda_{\sigma }}\right]+{s}^T(t)\left[\varTheta -{u}_{rb}\right]+\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }}\end{array}} $$

(69)

Via the adaptive laws in (50)–(53) and the robust controller in (54), then (69) becomes

$$ \dot{V}(t)={s}^T(t)\left(\varTheta -{u}_{rb}\right)+\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }}={s}^T(t)\left(\varTheta -\hat{\varDelta}\operatorname{sgn}\left(s(t)\right)\right)+\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }}={s}^T(t)\varTheta -\hat{\varDelta}\mid s(t)\mid +\frac{{\overset{\sim }{\varDelta}}^T\dot{\overset{\sim }{\varDelta }}}{\lambda_{\varDelta }} $$

(70)

If the error bound for $\dot{\hat{\Delta}}$ is chosen as

$$ \dot{\tilde{\Delta}}=-\dot{\hat{\Delta}}=-{\lambda}_{\Delta}\left|\boldsymbol{s}(t)\right|, $$

(71)

then (60) is rewritten as

$$ {\displaystyle \begin{array}{c}\dot{V}(t)={s}^T(t)\varTheta -\hat{\varDelta}\mid s(t)\mid -\left(\varDelta -\hat{\varDelta}\right)\mid s(t)\mid \\ {}=\left({s}^T(t)\varTheta -\varDelta |s(t)|\right)\le \left(|\varTheta \Big\Vert s(t)|-\varDelta |s(t)|\right)\\ {}=-\left(\varDelta -|\varTheta |\right)\mid s(t)\mid \le 0\end{array}} $$

(72)

As $\dot{V}(t)\le 0$, it indicates that $\tilde{\Delta}$ and s(t) are bounded. Set $\Psi (t)\equiv \left(\Delta -\left|\boldsymbol{\varTheta} \right|\right)\boldsymbol{s}(t)\le \left(\Delta -\left|\boldsymbol{\varTheta} \right|\right)\left|\boldsymbol{s}(t)\right|\le -\dot{V}(t)$, then integrate Ψ(t) with respect to time, gives

$$ {\int}_0^t\Psi \left(\tau \right) d\tau \le \dot{V}(0)-\dot{V}(t) $$

(73)

$\dot{V}(0)$ is bounded and $\dot{V}(t)$ does not increase, obtains:

$$ \underset{t\to \infty }{\lim}\underset{0}{\overset{t}{\int }}\Psi \left(\tau \right) d\tau <\infty $$

(74)

Furthermore, $\dot{\Psi}(t)$ is bounded via Barbalat’s Lemma, then $\underset{t\to \infty }{\lim}\Psi (t)\to 0$. That means s → 0 when t→∞ [50]. Consequently, the WDFLFBEC is asymptotically stable. Hence, the control system is stable.