Three-phase two-level inverters are the most widely used power converters in power electronics and motor drives. The main components of the inverters are controllable power switches such as IGBTs, MOSFETs, GTOs, etc. [1]. By implementing appropriate control strategies, they can output active and reactive power as required, meanwhile operate with desirable performance in terms of good current quality, high power efficiency, and strong robustness against disturbances. In real applications, the inverter faces uncertain conditions including time-varying grid voltage, filter capacitance and inductance changes, measurement delays, and environment noises. To effectively control the inverter, all these uncertainties should be taken into account [2,3].

Linear parameter-varying (LPV) systems are used to describe systems with time-varying parameters. Since the seminal work of Shamma in 1991 [4], the LPV control and filtering approach has attracted a lot of attention from the control field. This is because $1)$ it is effective to cope with nonlinearities and time-varying dynamics of the system, and $2)$ the maturely developed LTI control methodologies, such as sensitivity shaping and modeling tools, can be extended to LPV systems. During the past decades, several advances in theoretical and practical aspects have been done. Among others, theoretical works are related to analysis and synthesis issues [5,6], robust filtering problems [7,8], fault detection and isolation [9], etc. Practical engineering works are also popular, for instance, an LPV static output feedback control method is proposed to improve lateral stability and driving comfort of narrow tilting vehicles [10], LPV modeling method is adopted to ensure large signal stability for multi-mode buck-boost converters [11], the uncertain robot system is controlled by an LPV controller with sliding mode optimization [12]. Time delays exist in many practical systems, such as network controlled systems, electronic networks, hydraulic systems, and chemical processes, which are mostly nonlinear systems. Time delays may cause instability and oscillation in these systems and severely degrade performance [13,14]. To target this problem, research on time-delayed LPV system starts coming on stage in recent years, see [15,16].

Stochastic systems are used to model systems with stochastic perturbations. This approach is used in several fields and engineering applications, such as mechanical systems, economic systems, etc. Over the past decades, the study of stabilization and filtering problems for stochastic systems has received much attention, and many valuable results have been obtained. For stabilization problems, the asymptotic stability of semiswitched stochastic systems has been investigated in [17], the problem of input-to-state stability for nonlinear systems with stochastic impulses is systematically studied [18]. For filtering problems, the recursive filtering problem for a class of uncertain stochastic systems with amplify and forward relays has been investigated in [19], the problem of distributed robust filtering for switched stochastic time-delay systems with fading measurements over sensor networks is addressed in [20].

This paper aims to propose a stochastic LPV model approach to deal with the uncertain conditions and time delay faced by a three-phase two-level inverter, where the grid voltage, filter capacitance, and inductance are treated as time-varying parameters, and the environment noises as stochastic disturbances. Sufficient conditions for stability analysis and controller synthesis are obtained for the stochastic LPV system. The theoretical results obtained are applied to the inverter, whose currents are treated as state variables and controlled to reach the equilibrium point. The contributions of this paper are of three folds: 1) the LPV approach is used to model the three-phase two-level inverter, which is a new practical method; 2) comparing with previous theoretical works, this paper further considers the stochastic perturbation and parameter-varying time delay, based on which the stochastic LPV model is established and the stability conditions are obtained; 3) the slack matrix technique is adopted to improve the feasibility of the stability conditions, which facilitates the controller design process. The remaining of this work includes system description and inverter modeling in Section II. Main theoretical analysis is presented in Section III. Simulations to validate the proposal are provided in Secition IV. Finally, conclusioms are addressed in Section V.

where $V_{in}$ is the input dc power source, $\omega$ is the frequency of grid voltage, L is the filtering inductor and r is its equivalent series resistance, and the dq variables are transformed from the three-phase abc variables, i.e., $v_d,v_q$ are grid voltages transformed from $v_a$, $v_b$ and $v_c$, $i_d, i_q$ are grid currents transformed from $i_a$, $i_b$ and $i_c$, and $u_d, u_q$ are switching functions generated by controller, which is to be used to generate the switching signals $S_a$, $S_b$ and $S_c$, though PWM modulator. To provide required power to grid, the dq currents must be controlled to desired value. Taking into account the inverter parameter variations, stochastic grid perturbations, and control/communication delays, the inverter model can be formulated as a general time-delayed stochastically perturbed LPV system: where $x\left(t\right)=[i_d;i_q]$ is the state vector, $A\left(\rho \left(t\right)\right)$ is the system matrix related to L, r, $\omega$; $u\left(t\right)=[u_d{V}_{in}-v_d;u_q{V}_{in}-v_q]$ is the control input, $B\left(\rho \left(t\right)\right)$ is the control matrix related to L; $v\left(t\right)\in {\mathcal{R}}^q$ belonging to ${\mathcal{L}}_2[0,\infty )$, is either a disturbance input or a reference signal; $\omega \left(t\right)$ is a one-dimensional (1-D) Brownian motion satisfying $E\left\{d\omega \right(t\left)\right\}=0$ and $E\left\{d{\omega }^2\left(t\right)\right\}=dt$; $\rho \left(t\right)\in {\mathcal{R}}^s$ is vector-valued parameter evolving continuously over time and its range is limited to a compact subset; $h\left(\rho \left(t\right)\right)$ is the parameter varying delay and satisfies: where H and $\sigma$ are constant scalars; $x\left(\theta \right)$ is the initial data function given in the time interval $[-H,0]$. The value of $\rho \left(t\right)$ is unknown, but can be measured in real time.

Disregarding the control input $u\left(t\right)$, the autonomous system of (2) is formulated as:

On the other hand, to carry out stability analysis and synthesis, the exogenous disturbance “$v\left(t\right)$” in (2) is considered to be zero, yielding following system: Therefore the autonomous system of (4) can be formulated as

For simplicity, $\rho$ will be used instead of $\rho \left(t\right)$ in the rest of this paper. Before going further, some definitions and lemmas should be given, which are important for deriving main results.

Definition 1: For any initial state $x\left(0\right)\in R^n$, the time-delayed LPV stochastic system (3) is said to be robustly stable with disturbance attenuation index $\gamma$ if for all $v\left(t\right)\in {\mathcal{L}}_2[0,\infty )]$ and all parameter trajectories, it holds that

Also, the system is asymptotically mean square stable according to Definition 1 in [22].