# Optimal Robust LQI Controller Design for Z-Source Inverters
## 1 Introduction  
The substantial progress of modern power inverters is well underway for the penetration of renewable energy sources into power systems and many other industrial applications [1]. The Impedance Source Inverters (also called Z-Source Inverters or ZSIs) form a class of high-performance, highly reliable, and highly efficient yet low-cost inverters equipped with an X-shaped impedance circuit that enables shoot-through state [2, 3]. Hence, they can overcome the limitations of traditional inverters and fulfill both buck and boost capability in a single-stage topology without the necessity for any kind of delays [4, 5]. Therefore, the output voltage can be adjusted freely to provide a wider voltage gain range without any additional cost or distortion [6]. Additionally, besides the advent and advance of new pulse width modulation (PWM) methods, all the traditional PWM schemes can be used to drive the inverter bridge and control the ZSIs [7]. These interesting features have paved the ZSI's way for penetrating into different industrial applications such as photovoltaic power generation [8], wind energy [9], electric vehicles (EV) [10], motor drive [11], and battery energy storage [12]. Multiple new versions of ZSIs are proposed in the literature to improve their performance and consolidate their position in the industry, which can be classified into four general categories: constantboost ratio, improved boost ratio, multilevel and multiplex, and parameter optimization topologies. The first category can be further subdivided into six classes:  
* Basic topology
* Improved topologies
* Neutral point topologies
* Reduced leakage current topologies
* Quasi topologies
* Other topologies  
The second category can be further subcategorized into four classes:  
* Switched components topologies
* Tapped inductor topologies
* Cascaded quasi topologies
* Coupled inductor topologies (such as the T-source inverter, gamma-source inverter, Y-source inverter (YSI), improved YSI, \(\Delta\)-source inverter, trans-quasi ZSI, improved trans-quasi ZSI, and transformer ZSI)  
Multilevel topologies consist of three, four, or more-level cascaded and dual-input or dual-output topologies. Finally, parameter optimization topologies involve high-frequency transformer-isolated ZSI, inductor ZSI, extended quasi YSI, low dc-link voltage spikes YSI, and other optimized topologies. The ZSI possesses a non-minimum phase characteristic with a real right-half plane zero, which imposes special constraints on the performance of the control system [13]. It is well known that real non-minimum phase zeros lead to an undershoot and also overshoot in the transient response, increase the harmonic distortions, and jeopardize system stability, thereby making the control problem complicated. Besides, parameter scanning of the pole-zero map reveals that altering system parameters cannot completely eliminate the non-minimum phase zeros [14]. Hence, in contrast with two-stage inverters, the ZSI closed-loop control is not an easy task. Over recent years, a variety of single-loop, dual-loop, and nonlinear control actions have been successfully applied generally using state-space averaging or small-signal modeling techniques [15]. The single-loop methods commonly employ the feedback by the input voltage [15], capacitor voltage [16], or dc-link voltage [17] to the closed-loop control of ZSIs through traditional PI controllers. A PID controller is proposed in [18] to provide a constant capacitor voltage with an excellent transient performance besides enhanced disturbance rejection. Variations of the dc-link voltage caused can be further suppressed by using dual-loop methods with an additional PI control loop [19]. Finally, nonlinear control approaches, for instance, the neural network control [20], sliding mode control [21; 22], fuzzy control [23], model predictive control [24], and adaptive control [25] have also been presented.