# Simplified Schmid pendulum

## Model description:

Simplified Schmid pendulum:

\begin{align*} \ddot{\psi} + a_{21}\omega - a_{11}\sin{\psi} &= -b_1u, \\ \dot{\omega} + a_{22}\omega + a_{12}\sin{\psi} &= b_2u, \end{align*}

where $\psi$ is the pendulum angle; $\omega$ is the wheel angular rate; $u$ is the controlling voltage, applied to the motor; $a_{11},a_{21},a_{12},b_1,b_2$ are positive constants, depending on the design parameter of the pendulum. It is assumed that the upper (unstable) equilibrium point corresponds to $\psi=0$.

## Publication details:

 Title Hybrid quantised observer for multi-input-multi-output nonlinear systems Publication Type Conference Paper Year of Publication 2008 Authors Fradkov, Alexander L., Andrievsky Boris, and Evans Robin J. Conference Name IEEE International Conference on Control Applications, 2008. CCA 2008. Date Published 09/2008 Publisher IEEE Conference Location San Antonio, Texas, USA ISBN Number 978-1-4244-2222-7 Accession Number 10235153 Keywords IMO systems, nonlinear control systems, observers, oscillators, pendulums Abstract Limit possibilities of state estimation under information constraints (limited information capacity of the coupling channel) for multi-input-multi-output (MIMO) nonlinear systems are evaluated. We give theoretical analysis for state estimation of nonlinear systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) with a first-order coder-decoder. It is shown that the upper bound of the limit estimation error is proportional to the upper bound of the transmission error. As a consequence, the upper bound of limit estimation error is proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to state estimation of a nonlinear self-excited mechanical oscillator and a reaction-wheel pendulum. DOI 10.1109/CCA.2008.4629572