Robust control based on adaptive neural network for Rotary inverted pendulum with oscillation compensation

Abstract

A new stable adaptive neural network (ANN) control scheme for the Furuta pendulum, as a two-degree-of-freedom underactuated nonlinear system, is proposed in this paper. This approach aims to address the control problem of the Furuta pendulum in the steady state and also in the presence of external disturbances. The adaptive classical control laws such as e-modification present some limitations in particular when oscillations are presented in the input. To avoid this problem, two ANNs are implemented using filtered tracking error in the control loop. The first one is a single hidden layer network, used to approximate the equivalent control online, and the second is the feed-forward network, used to minimize the oscillations. The goal of the control is to bring the pendulum close to the upright position in the presence of the various uncertainties and being able to compensate oscillations and external disturbances. The main purpose of the second ANN is to minimize the chattering phenomenon and response time by finding the optimal control input signal, which also leads to the reduction of energy consumption. The learning algorithms of the two ANNs are obtained using the direct Lyapunov stability method. The simulation results are given to highlight the performances of the proposed control scheme.

Appendix

1.1 Proof of the closed-loop system

Define the Lyapunov function candidate

$$\begin{aligned} L & = \frac{1}{2}\frac{{r^{2} }}{G} + \frac{1}{2}{\text{tr}} \left\{ {\tilde{W}^{\text{T}} F_{w}^{ - 1} \tilde{W}} \right\} + \frac{1}{2}{\text{tr}}\left\{ {\tilde{V}^{\text{T}} F_{v}^{ - 1} \tilde{V}} \right\} + \frac{1}{2}{\text{tr}}\left\{ {\tilde{B}^{\text{T}} F_{B}^{ - 1} \tilde{B}} \right\} \\ & \quad + \frac{1}{2}{\text{tr}}\left\{ {\tilde{ \propto }^{\text{T}} F_{ \propto }^{ - 1} \tilde{ \propto }} \right\} \\ \end{aligned}$$

(49)

where L is the proposed Lyapunov function. Differentiating yields

$$\begin{aligned} \dot{L} & = r\frac{{\dot{r}}}{G} - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{tr}} \left\{ {\tilde{W}^{\text{T}} F_{w}^{ - 1} \dot{\tilde{W}}} \right\} + tr\left\{ {\tilde{V}^{\text{T}} F_{v}^{ - 1} \dot{\tilde{V}}} \right\} + tr\left\{ {\tilde{B}^{\text{T}} F_{B}^{ - 1} \dot{\tilde{B}}} \right\} \\ & \quad + {\text{tr}}\left\{ {\tilde{ \propto }^{\text{T}} F_{ \propto }^{ - 1} \dot{\tilde{ \propto }}} \right\} \\ \end{aligned}$$

(50)

Whence substitution from (38) (with $w_{1} = 0)$ yields

$$\begin{aligned} \dot{L} & = - K_{v} r^{2} - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{tr}}\left\{ {\tilde{W}^{\text{T}} \left( {F_{w}^{ - 1} \dot{\tilde{W}} + \hat{\sigma }r^{\text{T}} } \right)} \right\} + {\text{tr}} \left\{ {\tilde{V}^{\text{T}} \left( {F_{v}^{ - 1} \dot{\tilde{V}} + xr^{\text{T}} \hat{W}^{T} \hat{\sigma }^{'} } \right)} \right\} \\ & \quad + {\text{tr}} \left\{ {\tilde{B}^{\text{T}} \left( {F_{B}^{ - 1} \dot{\tilde{B}} - r^{\text{T}} \hat{Z}} \right)} \right\} + tr \left\{ {\tilde{ \propto }^{\text{T}} \left( {F_{ \propto }^{ - 1} \dot{\tilde{ \propto }} + x^{e} r^{\text{T}} \hat{B}\hat{Z}^{'} } \right)} \right\} \\ \end{aligned}$$

(51)

Since $\hat{W} = W - \hat{W}$, the W is constant, so ${\text{d}}\frac{{\hat{W}}}{{{\text{d}}t}} = - {\text{d}}\hat{W}/{\text{d}}t$, as for V, B and ∝, the tuning rules from (32) and (25) yield.

$$\begin{aligned} \dot{L} & = - K_{v} r^{2} - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{tr}}\left\{ {\tilde{W}^{\text{T}} \left( {{\text{kr}}\tilde{W}} \right)} \right\} + {\text{tr}} \left\{ {\tilde{V}^{\text{T}} \left( {{\text{kr}}\hat{V}} \right)} \right\} \\ & \quad + {\text{tr}} \left\{ {\tilde{B}^{\text{T}} \hat{Z} \left( {r - r_{e} } \right)} \right\} + {\text{tr}} \left\{ {\tilde{ \propto }^{\text{T}} \left( {x^{e} \hat{B}\hat{Z}^{'} \left( {r - r_{e} } \right) - k_{\alpha } \left| {r_{e} } \right|\left( {\bar{\alpha } - \hat{\alpha }} \right)} \right)} \right\} \\ \end{aligned}$$

(52)

$$\begin{aligned} \dot{L} & = - K_{v} r^{2} - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{krtr}}\left\{ {\tilde{W}^{\text{T}} \left( {W - \tilde{W}} \right)} \right\} + {\text{krtr}} \left\{ {\tilde{V}^{\text{T}} \left( {V - \tilde{V}} \right)} \right\} \\ & \quad + {\text{tr}} \left\{ {\tilde{B}^{\text{T}} \hat{Z} \left( {r - r_{e} } \right)} \right\} + {\text{tr}} \left\{ {\tilde{ \propto }^{\text{T}} \left( {x^{e} \hat{B}\hat{Z}^{'} \left( {r - r_{e} } \right) - k_{\alpha } \left| {r_{e} } \right|\left( {\bar{\alpha } - \hat{\alpha }} \right)} \right)} \right\} \\ \end{aligned}$$

(53)

Define the matrix of all the NN weights as

$$T \equiv \left[ {\begin{array}{*{20}c} W & 0 \\ 0 & V \\ \end{array} } \right]$$

(54)

Assumption: On any compact subset of $\Re^{n}$, the ideal NN weights are bounded so that

$$\left\| T \right\|_{F} \le T_{B}$$

with $T_{B}$ known, and $\left\| \cdot \right\|_{\text{F}}$ is the Frobenius norm. Then

$$\begin{aligned} \dot{L} & = - K_{v} r^{2} - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{krtr}}\left\{ {\tilde{T}^{\text{T}} \left( {T - \tilde{T}} \right)} \right\} \\ & \quad + \left( {r - r_{e} } \right){\text{tr}} \left\{ {\tilde{B}^{\text{T}} \hat{Z} + \tilde{ \propto }^{\text{T}} x^{e} \hat{B}\hat{Z}^{'} } \right\} - k_{\alpha } \left| {r_{e} } \right|{\text{tr}} \left\{ {\left( {\bar{\alpha } - \hat{\alpha }} \right)} \right\} \\ \end{aligned}$$

(55)

Since ${\text{tr}}\left\{ {\tilde{T}^{\text{T}} (T - \,\tilde{T})} \right\} \le \left\| {\tilde{T}} \right\|_{F} \left\| T \right\|_{F} - \left\| {\tilde{T}} \right\|_{F}^{2}$, the results are as follows[28]:

$$\begin{aligned} \dot{L} & \le - rK_{v} r - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} + {\text{kr}} \cdot \tilde{T}_{F} \left( {T_{B} - \tilde{T}_{F} } \right) \\ & \quad + \left( {r - r_{e} } \right){\text{tr}} \left\{ {\tilde{B}^{\text{T}} \hat{Z} + \tilde{ \propto }^{\text{T}} x^{e} \hat{B}\hat{Z}^{'} } \right\} - k_{\alpha } \left| {r_{e} } \right|{\text{tr}} \left\{ {\left( {\bar{\alpha } - \hat{\alpha }} \right)} \right\} \\ \dot{L} & \le - rK_{v} r - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} - {\text{kr}} \cdot \left( {\tilde{T}_{F} - D} \right)^{2} + {\text{kr}}D^{2} \\ & \quad + \left( {r - r_{e} } \right)tr \left\{ {\tilde{B}^{\text{T}} \hat{Z} + \tilde{ \propto }^{\text{T}} x^{e} \hat{B}\hat{Z}^{'} } \right\} - k_{\alpha } \left| {r_{e} } \right|{\text{tr}} \left\{ {\left( {\bar{\alpha } - \hat{\alpha }} \right)} \right\} \\ \dot{L} & \le - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} - {\text{kr}} \cdot \left( {\tilde{Z}_{F} - D} \right)^{2} - r(K_{v} r - kD^{2} ) \\ & \quad - \left( {\bar{w}_{g} + k_{ \propto } \frac{{ \propto_{m}^{2} }}{4} + k_{ \propto } \tilde{ \propto }\bar{ \propto }} \right)\left| {r_{e} } \right| \\ \dot{L} & \le - \frac{1}{2}\frac{{\dot{G}}}{{G^{2} }}r^{2} - {\text{kr}} \cdot \left( {\tilde{Z}_{F} - D} \right)^{2} - r(K_{v} r - kD^{2} ) \\ & \quad - \left( {\bar{w}_{g} + k_{ \propto } \frac{{ \propto_{m}^{2} }}{4}} \right)\left| {r_{e} } \right| - k_{ \propto } \left| {r_{e} } \right| \propto_{n}^{2} \\ \end{aligned}$$

(56)

where $D = \frac{{T_{B} }}{2}$. Suppose that k > 0, using the inequality in (15), the first term is negative; then, we prove that the Lyapunov first-time derivative becomes negative if:

$$r > \frac{{kD^{2} }}{{K_{v} }}$$

Then, we prove that the Lyapunov first-time derivative will be:

$$\dot{L} \le 0$$

which guarantees the stability of closed-loop system [3, 27, 28].