$\mathscr{H}_{2}$ and $\mathscr{H}_{\infty}$ mode-independent state-feedback control of generalized Bernoulli jump systems with uncertain probabilities

Many practical problems arising in networked control systems can be suitably modeled by linear stochastic systems described in terms of discrete-time generalized Bernoulli models, that are a particular case of the so called Markov jump linear systems. Motivated by real world applications where the transition probability matrix is uncertain, this paper proposes a general framework to deal with the problem of control design for Bernoulli systems, providing synthesis conditions for
$\mathscr{H}_{2}$
and
$\mathscr{H}_{\infty}$
state-feedback controllers that are sufficient in the uncertain case and also necessary (optimal) for precisely known models. The conditions can be solved in terms of LMI relaxations of increasing accuracy, allowing the user to tradeoff precision and computational cost in the search for better solutions. The networked control of a mechanical plant is presented to illustrate the applicability of the method.