Hermite broad-learning recurrent neural control with adaptive learning rate for nonlinear systems

Abstract

Although conventional control systems are simple and widely used, they may not be effective for complex and uncertain systems. This study proposes a Hermite broad-learning recurrent neural network (HBRNN) with a wide network structure and an internal feedback loop that enables good approximation of system dynamics without the need for a large number of training parameters. Furthermore, a Hermite broad-learning recurrent neural control (HBRNC) with HBRNN as the main controller is proposed, which requires no prior knowledge about the system dynamics and has no off-line learning phase. All the HBRNN network parameters are updated according to parameter learning laws through the gradient descent approach. To prevent network parameter overtraining of the HBRNN, a discrete-type Lyapunov function is used to determine the least upper bound for the learning rates. Additionally, an adaptive learning rate (ALR) approach is designed to dynamically fine-tune the learning rates within these specified limits, thereby achieving an optimal convergence speed between network parameters and tracking error. Finally, the HBRNC system with ALR is applied to a chaotic circuit and a reaction wheel pendulum, and its effectiveness is validated through simulation and experimentation.

Appendix

Theorem 1

Let $\eta_\alpha$, $\eta_\beta$, $\eta_w$ and $\eta_v$ be the learning rates for the parameters learning laws of HBRNN. Define $P_{\alpha \max }$ as $P_{\alpha \max } = {\mathop {\max }\limits_N} \left\| {P_\alpha (N)} \right\|$, where $P_\alpha (N) = \frac{{\partial \tau_{nc} }}{\partial \alpha_j }$; define $P_{\beta \max }$ as $P_{\beta \max } = {\mathop {\max }\limits_N} \left\| {P_\beta (N)} \right\|$, where $P_\beta (N) = \frac{{\partial u_{nc} }}{\partial \beta_k }$; define $P_{w\max }$ as $P_{w\max } = {\mathop {\max }\limits_N} \left\| {P_w (N)} \right\|$, where $P_w (N) = \frac{{\partial \tau_{nc} }}{{\partial w_{jk} }}$; define $P_{v\max }$ as $P_{v\max } = {\mathop {\max }\limits_N} \left\| {P_v (N)} \right\|$, where $P_v (N) = \frac{{\partial u_{nc} }}{{\partial v_{ji} }}$. Thus, the system stability can be guaranteed if $\eta_\alpha$ and $\eta_\beta$ are chosen as $\eta_\alpha^* = \frac{1}{{m(V_{\max } )^2 }}$ and $\eta_\beta^* = \frac{2}{n}$, respectively, in which $V_{\max } = {\mathop {\max }\limits_j} \left| {V_j } \right|$; $\eta_w$ and $\eta_v$ are chosen as $\eta_w^* = \frac{2}{{mn(\beta_{\max } V_{\max } )^2 }}$ and $\eta_v^* = \frac{2}{{n(\alpha_{\max } + \beta_{\max } w_{\max } )^2 }}$, respectively, in which $\alpha_{\max } = {\mathop {\max }\limits_j} \left| {\alpha_j } \right|$, $\beta_{\max } = {\mathop {\max }\limits_k} \left| {\beta_k } \right|$ and $w_{\max } = {\mathop {\max }\limits_{j,k}} \left| {w_{jk} } \right|$.

Proof

Since $P_\alpha (N) = \frac{{\partial \tau_{nc} }}{\partial \alpha_j } = V_j$ and $P_\beta (N) = \frac{{\partial u_{nc} }}{\partial \beta_k } = W_k$, the following result can be concluded:

$$ \left\| {P_\alpha (N)} \right\| < \sqrt {m} V_{\max } $$

(A1)

$$ \left\| {P_\beta (N)} \right\| < \sqrt {n} $$

(A2)

The upper bounds of $P_w (N)$ and $P_v (N)$ can be derived as follows:

$$ \begin{aligned}P_w (N) &= \frac{{\partial u_{nc} }}{{\partial w_{jk} }} = \frac{{\partial u_{nc} }}{\partial W_k }\frac{\partial W_k }{{\partial w_{jk} }} \\ &= \beta_k \left( {1 - W_k^2 } \right)V_j \\ &\le \beta_k V_j\end{aligned}$$

(A3)

$$ \begin{aligned}P_v (N) &= \frac{{\partial u_{nc} }}{{\partial v_{ji} }} = \left( {\frac{{\partial \,u_{nc} }}{\partial \,V_j } + \sum_{k = 1}^n {\frac{{\partial \,u_{nc} }}{\partial \,W_k }\frac{\partial \,W_k }{{\partial \,V_j }}} } \right)\frac{\partial \,V_j }{{\partial \,v_{ji} }} \\ &= \left( {\alpha_j + \sum_{k = 1}^n {\beta_k w_{jk} } \left( {1 - W_k^2 } \right)} \right)h_{ji}\\ &\le \alpha_j + \sum_{k = 1}^n {\beta_k w_{jk} } \left( {1 - W_k^2 } \right)\\ &\le \alpha_j + \sum_{k = 1}^n {\left| {\beta_k } \right|\left| {w_{jk} } \right|} \end{aligned}$$

(A4)

From (A3) and (A4), the inequalities can be obtained as:

$$ \left\| {P_w (N)} \right\| \le \left\| {\beta_k V_j } \right\| \le \left\| {\beta_k } \right\|\left\| {V_j } \right\| \le \sqrt {mn} \beta_{\max } V_{\max } $$

(A5)

$$ \begin{aligned}\left\| {P_v (N)} \right\| &\le \left\| {\alpha_j + \sum_{k = 1}^n {\left| {\beta_k } \right|\left| {w_{jk} } \right|} } \right\| \\ & \le \left\| {\alpha_j } \right\| + \left\| {\sum_{k = 1}^n {\left| {\beta_k } \right|\left| {w_{jk} } \right|} } \right\| \\ &\le \alpha_{\max } + \sqrt {n} \beta_{\max } w_{\max } \end{aligned}$$

(A6)

To ensure the system stability, consider a discrete-type Lyapunov function as follows:

$$ V_2 (N) = \frac{1}{2}e^2 (N) $$

(A7)

where N denotes the number of iteration. The change of discrete-type Lyapunov function can be expressed as:

$$ \Delta V_2 (N) = V_2 (N + 1) - V_2 (N) = \frac{1}{2}\left[ {e^2 (N + 1) - e^2 (N)} \right] $$

(A8)

The error difference can be represented by:

$$ \begin{aligned}e(N + 1) &= e(N) + \Delta e(N) \\ & = e(N) + \left[ {\frac{\partial e(N)}{{\partial \alpha_j }}} \right]^T \Delta \alpha_j + \left[ {\frac{\partial e(N)}{{\partial \beta_k }}} \right]^T \Delta \beta_k + \left[ {\frac{\partial e(N)}{{\partial w_{jk} }}} \right]^T \Delta w_{jk} + \left[ {\frac{\partial e(N)}{{\partial v_{ji} }}} \right]^T \Delta v_{ji} \end{aligned}$$

(A9)

where $\Delta e(N)$ respects a change in system output, and $\Delta \alpha_j$, $\Delta \beta_k$, $\Delta w_{jk}$, and $\Delta v_{ji}$ respect a parameter change in output layer, enhancement layer and recurrent feature layer, respectively. Define $\xi = \frac{\partial e}{{\partial u_{nc} }}$ as a positive constant designed by the user, (A9) using (33)–(36), (A1), (A2), (A5) and (A6) can be obtained as:

$$ \begin{aligned}\left\| {e(N + 1)} \right\| &= \left\| {e(N)\left( {1 - \eta_\alpha \xi^2 P_\alpha^T (N)P_\alpha (N)} \right)} \right. + e(N)\left( {1 - \eta_\beta \xi^2 P_\beta^T (N)P_\beta (N)} \right) \\ &\quad + e(N)\left( {1 - \eta_w \xi^2 P_w^T (N)P_w (N)} \right) \left. + e(N)\left( {1 - \eta_v \xi^2 P_v^T (N)P_v (N)} \right) \right\| \\ &\le \left\| {e(N)} \right\|\left\| {1 - \eta_\alpha \xi^2 P_\alpha^T (N)P_\alpha (N)} \right\| + \left\| {e(N)} \right\|\left\| {1 - \eta_\beta \xi^2 P_\beta^T (N)P_\beta (N)} \right\| \\ &\quad + \left\| {e(N)} \right\|\left\| {1 - \eta_w \xi^2 P_w^T (N)P_w (N)} \right\| +\left\| {e(N)} \right\|\left\| {1 - \eta_v \xi^2 P_v^T (N)P_v (N)} \right\| \end{aligned}$$

(A10)

If the learning rates for the parameters learning laws of HBRNN are selected as follows:

$$ \eta_\alpha = \frac{1}{{(\xi P_{\alpha \max } )^2 }} = \frac{1}{{m(\xi V_{\max } )^2 }} $$

(A11)

$$ \eta_\beta = \frac{1}{{(\xi P_{\beta \max } )^2 }} = \frac{1}{n\xi^2 } $$

(A12)

$$ \eta_w = \frac{1}{{(\xi P_{w\max } )^2 }} = \frac{1}{{mn(\xi \beta_{\max } V_{\max } )^2 }} $$

(A13)

$$ \eta_v = \frac{1}{{(\xi P_{v\max } )^2 }} = \frac{1}{{(\xi (\alpha_{\max } + \sqrt {n} \beta_{\max } w_{\max } ))^2 }} $$

(A14)

the term $\left\| {1 - \eta_\alpha \delta^2 P_\alpha^T (N)P_\alpha (N)} \right\|$, $\left\| {1 - \eta_\beta \delta^2 P_\beta^T (N)P_\beta (N)} \right\|$, $\left\| {1 - \eta_w \delta^2 P_w^T (N)P_w (N)} \right\|$ and $\left\| {1 - \eta_v \delta^2 P_v^T (N)P_v (N)} \right\|$ are less than 1. According to $\left\| {e(N + 1)} \right\| < \left\| {e(N)} \right\|$, the Lyapunov stability of $V_2 (N) > 0$ and $\Delta V_2 (N) < 0$ can be guaranteed. Thus, the discrete-type Lyapunov approach can find the learning rate ranges that allows the ALR can efficiently train HBRNN, where the learning rates are designed as $\eta_\alpha^* = \frac{\eta_\alpha }{2}$, $\eta_\beta^* = \frac{\eta_\beta }{2}$, $\eta_w^* = \frac{\eta_w }{2}$ and $\eta_v^* = \frac{\eta_v }{2}$ for the parameters learning laws of HBRNN (33)–(36), respectively.□