H-infinity stability analysis and output feedback control for fuzzy stochastic networked control systems with time-varying communication delays and multipath packet dropouts

Abstract

The H-infinity stability analysis and delay-dependent Takagi–Sugeno (T–S) fuzzy dynamic output feedback control are proposed for the T–S fuzzy discrete networked control systems with time-varying communication delay and multipath packet dropouts. T–S fuzzy model is employed to approximate the discrete networked control system with time-varying state delay and external disturbance. Stochastic system theory and Bernoulli probability distribution are employed to describe the time-varying communication delay and multipath packet dropouts. Delay-dependent T–S fuzzy dynamic output feedback controller is designed. The delay-dependent T–S fuzzy dynamic output feedback controller is employed to relax the design conditions and enhance the design flexibility. The delay-dependent Lyapunov–Krasovskii functional, stochastic system theory and Bernoulli probability distribution are introduced to guarantee the stochastic mean-square stability and prescribed H-infinity performance. Some slack matrices are introduced to reduce the computation complexity. Finally, simulation examples are presented to show the effectiveness and advantages of the proposed methods.

Appendix

Definition 1

[65, 66] For any initial condition and $w\left( k \right) = 0$, if there exists the matrix H with appropriate dimension for system (19) such that

$$ E\left\{ {\left. {\mathop \sum \limits_{k = 0}^{ + \infty } \left\| {\bar{x}\left( k \right)} \right\|^{2} \,} \right|\,\bar{x}\left( 0 \right)} \right\} < \bar{x}^{\rm T} \left( 0 \right)H\bar{x}\left( 0 \right),\quad w\left( k \right) = 0 $$

then system (19) is said to be stochastically mean-square stable.

Definition 2

[67] For the zero initial condition and $w\left( k \right) \ne 0$, if system (19) is stochastically mean-square stable and the control output $z\left( k \right)$ satisfies

$$ \mathop \sum \limits_{k = 0}^{ + \infty } E\left\{ {z^{\rm T} \left( k \right)z\left( k \right)} \right\} \le \gamma^{2} \mathop \sum \limits_{k = 0}^{ + \infty } E\left\{ {\bar{w}^{\rm T} \left( k \right)\bar{w}\left( k \right)} \right\},\quad w\left( k \right) \ne 0 $$

then system (19) is said to be stochastically mean-square stable and the prescribed H-infinite performance is guaranteed.

Assumption 1

The matrix $B_{i} \in R^{n \times m}$ is a column full rank matrix, i.e., ${\text{Rank}}\left( {B_{i} } \right) = m$.

Lemma 1

[68] For the given matrix $B_{i} \in R^{n \times m}$ satisfying Assumption 1, if there exist the matrices $\Omega_{1} \in R^{m \times m}$, $\Omega_{2} \in R^{{\left( {n - m} \right) \times \left( {n - m} \right)}}$ and $\Omega$ with appropriate dimension such that

$$ \Omega = U \cdot {\text{diag}}\left\{ {\Omega_{1} ,\Omega_{2} } \right\}U^{\rm T} $$

then

$$ \Omega B_{i} = B_{i} \bar{\Omega } $$

where $\bar{\Omega }$ is a nonsingular matrix.

Lemma 2

[69] For the given matrix $S$, if there exist the symmetric matrices $Q$, $R$ and $T$ with appropriate dimensions such that

$$ \left\{ {\begin{array}{*{20}l} {T - Q \le 0} \hfill \\ {\left[ {\begin{array}{*{20}c} Q & S \\ {S^{\rm T} } & R \\ \end{array} } \right] \le 0} \hfill \\ \end{array} } \right. $$

then

$$ \left[ {\begin{array}{*{20}c} T & S \\ {S^{\rm T} } & R \\ \end{array} } \right] \le 0 . $$