Positive $L_1$L1 filter design for positive piecewise homogeneous Markovian jump T–S fuzzy system

This study introduces a problem of positive $L_{1}$L1 filter design for positive piecewise homogeneous Markovian jump Takagi–Sugeno (T–S) fuzzy systems. The difference between the existing achievements is that the considered transition rate of the positive Markovian jump system is time-varying. This time-varying nature is finite piecewise homogeneous. The time variation of the transition rate is characterised by two cases: stochastic variation and arbitrary variation. First of all, sufficient conditions are obtained for stochastic stability and $L_{1}$L1 performance under the transition rate in a manner of stochastic variation and arbitrary variation by means of choosing a linear co-positive Lyapunov function. Then, based on the obtained achievements, the problem of positive $L_{1}$L1 filter design for the positive piecewise homogeneous Markovian jump T–S fuzzy system is studied. All the mentioned problems in this study can be solved by linear programming. Finally, a biological model is proposed to illustrate the effectiveness of theoretical results.