Friction coefficient estimation in servo systems using neural dynamic programming inspired particle swarm search

Abstract

Parameter estimation of static friction torques in servo control systems is of great significance to their robust control. Many researchers are devoted to pursuing the solutions to estimating the coefficients of the static friction torques. In order to tackle the troublesome matter more effectively, in this paper, we address a neural dynamic programming inspired particle swarm search algorithm. We call the algorithm direct BP neural dynamic programming inspired PSO (NDPSO) since we incorporate direct back propagation (BP) and neural dynamic programming (NDP) into particle swarm optimization (PSO). In NDPSO, critic BP neural network is trained to balance the Bellman equation while action BP neural network is used to train the inertia weight, the cognitive coefficient, and the social coefficient of the PSO algorithm. The training target is to enable the critic BP neural network output to approach the ultimately successful objective. Successively, NDPSO, together with standard PSO (SPSO) and genetic algorithm (GA), is applied to the parameter identification of the static friction torque in a servo control system with single input and single output (SISO). The experimental results clearly demonstrate that NDPSO is effective and outperforms SPSO and GA in identifying the parameters of the static friction torque in the servo control system.

Appendices

Appendix A: Derivation of the adaptive rule of the critic neural network

$$\begin{array}{@{}rcl@{}} {\Delta} w_{c_{i}}^{(2)}&=&l_{c}(t)\cdot [-\frac{\partial E_{c}(t)}{\partial w_{c_{i}}^{(2)}}]\\ &=&-l_{c}(t)\cdot \frac{\partial E_{c}(t)}{\partial J(t)}\cdot \frac{\partial J(t)}{\partial w_{c_{i}}^{(2)}}\\ &=&-l_{c}(t)\cdot \beta\cdot e_{c}(t)\cdot p_{i}(t) \end{array} $$

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$$ \begin{array}{lll} {\Delta} w_{c_{ij}}^{(1)}\!\!\!&=&\!l_{c}(t)\cdot [-\frac{\partial E_{c}(t)}{\partial w_{c_{ij}}^{(1)}}] \\ &=&\!-l_{c}(t)\cdot \frac{\partial E_{c}(t)}{\partial J(t)}\cdot \frac{\partial J(t)}{\partial p_{i}(t)}\cdot \frac{\partial p_{i}(t)}{\partial q_{i}(t)}\cdot \frac{\partial q_{i}(t)}{\partial w_{c_{ij}}^{(1)}} \\ &=&\!-l_{c}(t)\!\cdot\!\beta\!\cdot\! e_{c}(t)\!\cdot\! {w_{c_{i}}^{(2)}\!\cdot\! [\frac{1}{2}\!\cdot\! (1\,-\,{p_{i}^{2}}(t))]}\!\cdot\! x_{j}(t) \end{array} $$

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Appendix B: Derivation of the updated rule of the action neural network

$$\begin{array}{@{}rcl@{}} {\Delta} w_{a_{i}}^{(2)}\!\!&=\!&l_{a}(t)\cdot [-\frac{\partial E_{a}(t)}{\partial w_{a_{i}}^{(2)}}]\\ &=\!&-l_{a}(t)\cdot \frac{\partial E_{a}(t)}{\partial J(t)}\cdot \frac{\partial J(t)}{\partial X_{k}(t)}\cdot \frac{\partial X_{k}(t)}{\partial w_{a_{i}}^{(2)}}\\ &=\!&-l_{a}(t)\!\cdot\! e_{a}(t)\!\cdot\!\! \sum\limits_{j=1}^{N_{ch}}\{{w_{c_{i}}^{(2)}(t)\!\cdot\! [\frac{1}{2}\!\cdot\! (1\,-\,\!{p_{i}^{2}}(t))]\!\cdot\! w_{c_{ij}}^{(1)}(t)}\}\\ &&\cdot [\frac{1}{2}\cdot (1-x_{k}(t))] \cdot g_{i}(t) \end{array} $$

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$$ \begin{array}{ll} {\Delta} w_{a_{ij}}^{(1)}\!\!&=l_{a}(t)\cdot [-\frac{\partial E_{a}(t)}{\partial w_{a_{ij}}^{(1)}}] \\ &=-l_{a}(t)\!\cdot\! \frac{\partial E_{a}(t)}{\partial J(t)}\!\cdot\! \frac{\partial J(t)}{\partial X_{k}(t)}\!\cdot\! \frac{\partial X_{k}(t)}{\partial g_{i}(t)}\!\cdot\! \frac{\partial g_{i}(t)}{\partial h_{i}(t)}\!\cdot\! \frac{\partial h_{i}(t)}{\partial w_{a_{ij}}^{(1)}} \\ &=-l_{a}(t)\!\cdot\! e_{a}(t)\!\cdot\!\! {\sum}_{j=1}^{N_{ch}}\{{w_{c_{i}}^{(2)}(t)\!\cdot\! [\frac{1}{2}\!\cdot\! (1\!\,-\,\!{p_{i}^{2}}(t))]\!\cdot\! w_{c_{ij}}^{(1)}\!(t)\!}\}\\ &{\kern7pt} \cdot [\frac{1}{2}\cdot (1\,-\,x_{k}(t))]\cdot w_{a_{ki}}^{(2)}(t)\cdot [\frac{1}{2}\cdot (1-{g_{i}^{2}}(t))]\cdot x_{j}(t) \end{array} $$

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