Control Stability of TP Model Transformation Design via Probabilistic Error Bound of Plant Model

Recently, an emerging numerical controller design based on Tensor-Product model transformation has proven to work on a number of nonlinear systems by both simulations and practical implementations. In this paper, aiming at fully utilizing the power of approximation of TP model transformation, a new control stability analysis is proposed in a probabilistic framework. We utilize the matrix 2-norm to characterize the maximum deviation of the approximated system from the actual one. However, the theoretical boundary of the maximum deviation is hard to determine. Therefore, probabilistic error bound estimation is introduced with the aid of Chebyshev inequality to define a practical bound of deviation for the approximated system. This new analysis technique allows practical controller design for approximated TP system model by solving a set of Linear Matrix Inequalities (LMI). Examples using the benchmark Translational Oscillator with Rotational Actuator (TORA) system are presented to illustrate our formulation, showing that the designed controller can guarantee asymptotic stability with sufficiently high probability.