Neural-adaptive control using alternate weights

Abstract

This paper proposes a novel robust neural-adaptive control method for controlling underdamped non-minimum phase system. Without robust modifications to the training rule, adaptive approximators experience weight weight drift which typically causes control chatter and excitation of the natural frequency. Popular robust modifications, like e-modification and deadzone, significantly reduce performance. In the proposed method, an alternate neural network, providing approximately the same output, guides the training. The proposed algorithm trains the alternate weights in a manner so as to avoid the weight drift caused by underdamped vibrations. Experimental results show dramatic improvement in performance over e-modification when controlling a flexible-joint robot.

Appendix

An n-link flexible joint robot with large gear ratios has model

$$ {\bf M} ({\bf x}_1) \dot{{\bf x}}_2 = {\bf F}_1({\bf x}_1,{\bf x}_2) + {\bf K}({\bf x}_3-{\bf x}_1), $$

(39)

$$ {\bf J} \dot{{\bf x}}_4 = {\bf F}_2({\bf x}_1,{\bf x}_2,{\bf x}_3,{\bf x}_4) + {\bf K} ({\bf x}_1 - {\bf x}_3) + {\bf u} , $$

(40)

where

${\bf x}_1$ :

are link angles

${\bf x}_2$ :

are link angular velocities

${\bf x}_3$ :

are rotor angles

${\bf x}_4$ :

are rotor angular velocities

${\bf M}({\bf x}_1) \in {\mathcal{R}}^{n \times n}$ :

is the link inertia matrix

${\bf K} \in {\mathcal{R}}^{n \times n}$ :

is a diagonal matrix of (positive) spring constants

${\bf J} \in {\mathcal{R}}^{n \times n}$ :

is the rotor inertia matrix (after gear reduction)

${\bf u} \in {\mathcal{R}}^{n}$ :

is control torque

${\bf F}_1, {\bf F}_2 \in {\mathcal{R}}^n$ :

contain linear and non-linear terms.

Given a desired trajectory $\varvec{\theta}_d,\dot{\varvec{\theta}}_d,\ddot{\varvec{\theta}}_d$ for the link coordinates, then the filtered error tracking error is defined

$$ {\bf z}_1 = \Lambda ({\bf x}_1-\varvec{\theta}_d) + ({\bf x}_2-\dot{\varvec{\theta}}_d) =\Lambda {\bf e}_1 + {\bf e}_2. $$

(41)

To use backstepping, the desired rotor states are considered to be virtual controls. The first virtual control ${\bf v}_1$ contains the desired rotor positions, and the second, ${\bf v}_2$, contains the desired rotor velocities. The virtual control errors are then defined as

$$ {\bf z}_2 = {\bf x}_3 -{\bf v}_1, \qquad {\bf z}_3 = {\bf x}_4 -{\bf v}_2. $$

(42)

The (virtual) controls using CMACs are chosen

$$ {\bf v}_1 =-\varvec{\Gamma}_1 {\hat{\bf w}}_1 +{\bf x}_1 -{\bf G}_1 {\bf z}_1, $$

(43)

$$ {\bf v}_2 =-\varvec{\Gamma}_2 {\hat{\bf w}}_2 +{\bf x}_2 - {\bf z}_1 -{\bf G}_2 {\bf z}_2, $$

(44)

$$ {\bf u} =-\varvec{\Gamma}_3 {\hat{\bf w}}_3 -{\bf z}_2 -{\bf G}_3 {\bf z}_3, $$

(45)

where each ${\bf G}_i$ is a positive-definite matrix of control gains. The closed-loop error dynamics can be written

$$ {\mathcal{I}} \dot{{\bf z}} = -{\mathcal{G}} {\bf z} + {\mathcal{F}} - \varvec{\Gamma} {\hat{\bf w}}, $$

(46)

where

$$ \begin{aligned} {\mathcal{I}} &= \left[ \begin{array}{ccc} {\bf K}^{-1} {\bf M} & {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf I} & {\bf 0} \\ {\bf 0} & {\bf 0} & {\bf J} \end{array} \right] \in {\mathcal{R}}^{6 \times 6}, \qquad {\bf z} = \left[ \begin{array}{c} {\bf z}_1 \\ {\bf z}_2 \\ {\bf z}_3 \end{array} \right]\in {\mathcal{R}}^{6}, \qquad {\mathcal{G}}= \left[ \begin{array}{ccc} {\bf G}_1 & -{\bf 1} & {\bf 0} \\ {\bf 1} & {\bf G}_2 & -{\bf 1} \\ {\bf 0} & {\bf 1} & {\bf G}_3 \end{array} \right],\\ \varvec{\Gamma}&= \left[ \begin{array}{c} \varvec{\Gamma}_1 \\ \varvec{\Gamma}_2 \\ \varvec{\Gamma}_3 \end{array} \right], \qquad {\hat{\bf w}} = \left[ \begin{array}{c} {\hat{\bf w}}_1 \\ {\hat{\bf w}}_2 \\ {\hat{\bf w}}_3 \end{array} \right], \qquad {\mathcal{F}} = \left[ \begin{array}{c} {\mathcal{F}}_1({\bf x}_1,{\bf x}_2, \varvec{\theta}_d, \dot{\varvec{\theta}}_d,\ddot{\varvec{\theta}}_d) +{\bf x}_1\\ {\mathcal{F}}_2({\bf x}_1,{\bf x}_2,{\bf x}_3, \varvec{\theta}_d, \dot{\varvec{\theta}}_d,\ddot{\varvec{\theta}}_d,\varvec{\theta}^{(3)}_d) +{\bf x}_2 \\ {\mathcal{F}}_3 ({\bf x}_1,{\bf x}_2,{\bf x}_3,{\bf x}_4, \varvec{\theta}_d, \dot{\varvec{\theta}}_d,\ddot{\varvec{\theta}}_d,\varvec{\theta}^{(3)}_d,\varvec{\theta}^{(4)}_d) \end{array} \right]. \end{aligned} $$

Since (46) has the same basic form as (5) the stability proof of the new weight update method is still valid.