Self-tuning output recurrent cerebellar model articulation controller for a wheeled inverted pendulum control

Abstract

In this study, a model-free self-tuning output recurrent cerebellar model articulation controller (SORCMAC) is investigated to control a wheeled inverted pendulum (WIP). Since the proposed SORCMAC captures the system dynamics, it has superior capability compared to the conventional cerebellar model articulation controller in terms of an efficient learning mechanism and dynamic response. The dynamic gradient descent method is also adopted to adjust the SORCMAC parameters online. Moreover, an analytical method based on a Lyapunov function is proposed to determine the learning rates of the SORCMAC so that the convergence of the system can be guaranteed. Finally, the effectiveness of the proposed control system is verified by simulations of the WIP control. Simulation results show that the WIP can move forward and backward stably with uncertainty disturbance by using the proposed SORCMAC.

Appendix

1.1 Proof of Theorem 1

Since

$$ {\mathbf{P}}_{s} (N) = {\frac{{{\kern 1pt} \partial u}}{{\partial \,{\mathbf{s}}}}} \quad \hbox{for} \, s = w, n, v \; \hbox{and}\; r$$

it follows that

$$ {\mathbf{P}}_{w} (N) = {\frac{\partial u}{{\partial {\mathbf{w}}}}} = \left[ {{\frac{\partial u}{{\partial w_{1} }}}, \ldots ,{\frac{\partial u}{{\partial w_{k} }}}, \ldots ,{\frac{\partial u}{{\partial w_{{n_{R} }} }}}} \right]^{T} $$

(A1)

$$ {\mathbf{P}}_{m} (N) = {\frac{\partial u}{{\partial {\mathbf{m}}}}} = \left[ {{\frac{\partial u}{{\partial m_{11} }}}, \ldots ,{\frac{\partial u}{{\partial m_{n1} }}}, \ldots ,{\frac{\partial u}{{\partial m_{1k} }}}, \ldots ,{\frac{\partial u}{{\partial m_{ik} }}}, \ldots ,{\frac{\partial u}{{\partial m_{nk} }}}, \ldots, {\frac{\partial u}{{\partial m_{{1n_{R} }} }}}, \ldots ,{\frac{\partial u}{{\partial m_{{nn_{R} }} }}}} \right]^{T} $$

(A2)

$$ {\mathbf{P}}_{v} (N) = {\frac{\partial u}{{\partial {\mathbf{v}}}}} = \left[ {{\frac{\partial u}{{\partial v_{11} }}}, \ldots ,{\frac{\partial u}{{\partial v_{n1} }}}, \ldots ,{\frac{\partial u}{{\partial v_{1k} }}}, \ldots ,{\frac{\partial u}{{\partial v_{ik} }}}, \ldots ,{\frac{\partial u}{{\partial v_{nk} }}}, \ldots ,{\frac{\partial u}{{\partial v_{{1n_{R} }} }}}, \ldots ,{\frac{\partial u}{{\partial v_{{nn_{R} }} }}}} \right]^{T} $$

(A3)

$$ {\mathbf{P}}_{r} (N) = {\frac{\partial u}{{\partial {\mathbf{r}}}}} = \left[ {{\frac{\partial u}{{\partial r_{1} }}},{\frac{\partial u}{{\partial r_{2} }}}, \ldots ,{\frac{\partial u}{{\partial r_{n} }}}} \right]^{T} $$

(A4)

where ${\frac{\partial u}{{\partial w_{k} }}} = b_{k}$, ${\frac{\partial u}{{\partial m_{ik} }}} = w_{k} b_{k} {\frac{{2(x_{ri} - m_{ik} )}}{{v_{ik}^{2} }}}$, ${\frac{\partial u}{{\partial v_{ik} }}} = w_{k} b_{k} {\frac{{2(x_{ri} - m_{ik} )^{2} }}{{v_{ik}^{3} }}}$ and ${\frac{\partial u}{{\partial r_{i} }}} = {\frac{1}{{r_{i} }}}y(N - 1).$ Define a Lyapunov function as

$$ V(N) = {\frac{1}{2}}\,e_{m}^{2} (N) $$

(A5)

Then, the change of the Lyapunov function is obtained as

$$ \Updelta V(N) = V(N + 1) - V(N) = {\frac{1}{2}}\left[ {e_{m}^{2} (N + 1) - e_{m}^{2} (N)} \right] $$

(A6)

where $e_{m} (N + 1)$ is represented by

$$ e_{m} (N + 1) = e_{m} (N) + \Updelta e_{m} (N) = e_{m} (N) + \left[ {{\frac{{\partial e_{m} (N)}}{{\partial {\mathbf{s}}}}}} \right]^{T} \Updelta {\mathbf{s}} $$

(A7)

Using (17), it is clear that

$$ {\frac{{\partial e_{m} }}{{\partial {\mathbf{s}}}}} = {\frac{{\partial e_{m} }}{{\partial \theta_{p} }}}\,{\frac{{\partial \theta_{p} }}{\partial u}}\,{\frac{\partial u}{{\partial {\mathbf{s}}}}} = - {\frac{{\rho_{p} }}{{e_{m} (N)}}}\,{\mathbf{P}}_{s} (N) $$

(A8)

Thus,

$$ \begin{aligned} e_{m} (N + 1) = &\; e_{m} (N) - \left[ {{\frac{{\rho_{p} }}{{e_{m} (N)}}}\,{\mathbf{P}}_{s} (N)} \right]^{T} \beta_{s} \rho_{p} {\mathbf{P}}_{s} (N) \\ = &\; e_{m} (N)\left[ {1 - \beta_{s} \left( {{\frac{{\rho_{p} }}{{e_{m} (N)}}}} \right)^{2} {\mathbf{P}}_{s}^{T} (N){\mathbf{P}}_{s} (N)} \right] \\ \end{aligned} $$

(A9)

From (A6) and (A9), $\Updelta V(N)$ can be represented as

$$ \Updelta V(N) = {\frac{1}{2}}\,\beta_{s} \rho_{p}^{2} ||{\mathbf{P}}_{s} (N)||^{2} \left[ {\beta_{s} \left( {{\frac{{\rho_{p} }}{{e_{m} (N)}}}} \right)^{2} ||{\mathbf{P}}_{s} (N)||^{2} - 2} \right] $$

(A10)

If β _s is chosen as $0 < \beta_{s} < {\frac{2}{{||{\mathbf{P}}_{s} (N)||^{2} \left[ {{{\rho_{p} } \mathord{\left/ {\vphantom {{\rho_{p} } {e_{m} (N)}}} \right. \kern-\nulldelimiterspace} {e_{m} (N)}}} \right]^{2} }}}$, $\Updelta V(N)$ in (A10) is less than 0. Therefore, the Lyapunov stability of $V > 0$ and $\Updelta V < 0$ is guaranteed. Moreover, the optimal learning rates are chosen as $\beta_{s}^{*} = {\frac{1}{{||{\mathbf{P}}_{s} (N)||^{2} \left[ {{{\rho_{p} } \mathord{\left/ {\vphantom {{\rho_{p} } {e_{m} (N)}}} \right. \kern-\nulldelimiterspace} {e_{m} (N)}}} \right]^{2} }}}$. This shows an interesting result of the variable optimal learning rates which can be adjusted online at each instant.