A design of active disturbance rejection control with higher convergence rate and its application in inertia wheel pendulum stabilization

Abstract

In this paper, active disturbance rejection control (ADRC) with higher convergence rate is designed by using linear quadratic (LQ) method and Lyapunov stability theory. Firstly, the state equation of the system is transformed into error-based form so as to simplify the ADRC structure, and it is linearized around its equilibrium point. Then the state feedback controller of ADRC is designed by using LQ method. The non-linearized part, model uncertainty, and external disturbances are regarded as ‘total disturbance’ and estimated by generalized extended state observer (ESO). In the conventional ADRC design, the disturbance estimated by ESO was fully cancelled by disturbance compensator. In this paper, the disturbance compensator is designed not to simply cancel the estimated disturbance, but to make the derivative of Lyapunov function smaller so as to increase the convergence rate. Proposed controller is adopted to stabilize inertia wheel pendulum (IWP) to verify its effectiveness.

Appendix: Modelling of IWP

Moment equilibrium equation of the inertia wheel pendulum is given as [44]:

$$ m_{\text{b}} gl_{\text{b}} \sin \theta_{b} + m_{\omega } gl\sin \theta_{b} - T_{\text{m}} + M_{\omega } - M_{\text{b}} = I_{{{\text{total}}}} \ddot{\theta }_{b} $$

(32)

where $m_{\text{b}}$ is the mass of the body, $m_{\omega }$ is the mass of the wheel, $T_{\text{m}}$ is the angular moment of the motor, $M_{\omega }$ is the friction moment of the wheel, $M_{\text{b}}$ is the friction moment of the body, $I_{{{\text{total}}}}$ is the inertia moment of the whole pendulum. $M_{\omega } ,M_{\text{b}} ,I_{\text{total}}$ are given as follow:

$$ M_{\omega } = C_{\omega } \dot{\theta }_{\omega } , M_{\text{b}} = C_{\text{b}} \dot{\theta }_{b} , I_{\text{total}} = I_{\text{b}} + m_{\omega } l^{2} $$

(33)

where $C_{\omega } , C_{\text{b}}$ are friction coefficient of the wheel and the body, respectively and $I_{\text{b}}$ is inertia moment of the body.

Driving torque occurred by the rotation of DC motor is given as:

$$ T_{\text{m}} = I_{\omega } \ddot{\theta }_{\omega } + M_{\omega } + I_{\omega } \ddot{\theta }_{\omega } = K_{\text{m}} i $$

(34)

where $i$ is the current of DC motor, $K_{\text{m}}$ is moment constant, and $I_{\omega }$ is inertia moment of the wheel. Current feedback or speed feedback can be used as the internal loop of stabilization. In our case, angular speed feedback is used, because angular speed is measurable but current of DC motor is not available.

Substituting Eq. (34) into Eq. (32) gives:

$$ m_{\text{b}} gl_{\text{b}} \sin \theta_{b} + m_{\omega } gl\sin \theta_{b} - I_{\omega } \ddot{\theta }_{\omega } - I_{\omega } \ddot{\theta }_{b} - M_{\text{b}} = I_{{{\text{total}}}} \ddot{\theta }_{b} $$

(35)

Taking $\ddot{\theta }_{\omega }$ as an input and $\theta_{b}$ as an output, one can get:

$$ \ddot{\theta }_{b} = \frac{{\left( {m_{\text{b}} l_{\text{b}} + m_{\omega } l} \right)g\sin \theta_{b} - C_{\text{b}} \dot{\theta }_{b} - I_{\omega } \ddot{\theta }_{\omega } }}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }} $$

(36)

Taking $x_{1} = \theta_{b}$, $x_{2} = \dot{\theta }_{b}$, $u = \ddot{\theta }_{\omega }$, the state space model is given as:

$$ \left\{ {\begin{array}{*{20}l} {\dot{x}_{1} = x_{2} } \hfill \\ {\dot{x}_{2} = \frac{{\left( {m_{\text{b}} l_{\text{b}} + m_{\omega } l} \right)g\sin x_{1} - C_{\text{b}} x_{2} }}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }} - \frac{{I_{\omega } }}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }}u} \hfill \\ \end{array} } \right. $$

(37)

Equation (37) can be rewritten by using matrix form as follows:

$$ \left\{ {\begin{array}{*{20}l} {\dot{x}(t) = Ax(t) + B_{u} u(t) + B_{f} f(t)} \hfill \\ {y(t) = Cx(t)} \hfill \\ \end{array} } \right. $$

(38)

$$ \begin{aligned} A & = \left[ {\begin{array}{*{20}c} 0 & 1 \\ {\frac{{(m_{\text{b}} l_{\text{b}} + m_{\omega } l)g}}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }}} & {\frac{{C_{\text{b}} }}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }}} \\ \end{array} } \right], \;B_{u} = \left[ {\begin{array}{*{20}c} 0 \\ { - \frac{{I_{\omega } }}{{I_{\text{b}} + I_{\omega } + m_{\omega } l^{2} }}} \\ \end{array} } \right],\;B_{f} = I_{2 \times 2} \\ C & = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right],\;f(t) = \left[ {\begin{array}{*{20}c} 0 \\ {w(t)} \\ \end{array} } \right] \\ \end{aligned} $$

where ${\varvec{w}}({\varvec{t}})$ is an extended disturbance containing nonlinear-part, model uncertainties, measurement noise, and external disturbance.