Mixed \(\mathcal {H}_{2}\)/\(\mathcal {H}_{\infty }\) fault detection and control for uncertain delta operator systems with mixed random delays and multiple data packet dropouts

This paper investigates the fault detection (FD) and control problem in delta operator networked control systems (NCSs) with mixed random delays and multiple data packet dropouts. A novel approach utilizing a unified \(\varvec{\mathcal {H}_{2}}\) ${\mathcal{H}}_{\mathbf{2}}$ / \(\varvec{\mathcal {H}_{\infty }}\) ${\mathcal{H}}_{\mathit{\infty }}$ optimization framework is developed for the system. The \(\varvec{\mathcal {H}_{2}}\) ${\mathcal{H}}_{\mathbf{2}}$ and \(\varvec{\mathcal {H}_{\infty }}\) ${\mathcal{H}}_{\mathit{\infty }}$ indexes are employed to measure the detection and control performances, respectively. Probability density functions (PDFs) are used to accurately analyze packet dropouts, overcoming limitations of the Bernoulli distribution. The random delays are modeled as stochastic Markov processes and transformed into mixed stochastic delays with linearly dependent and randomly generated multiplicative variables. These delay-dependent coefficients are uniformly modeled by multiple mutually correlated Bernoulli processes, whose relevance are formulated by a priori Pearson correlation coefficient matrix using statistical tests. The detector and control parameters are then derived in terms of solving a set of linear matrix inequalities (LMIs). Simulation results demonstrate the effectiveness of the proposed method.