## 1 Introduction  
In recent years, clean and renewable energy, such as solar power and wind power, has been vigorously promoted and applied [1; 2; 3]. However, the photovoltaic (PV) systems [1] and typical wind power systems [2] require inverters to operate. They cause power quality issues such as a low power factor and current harmonic pollution, which result in the failure of electrical equipment, the interference of communication circuits, etc., bringing economic losses to societies [4; 5]. In order to solve the power quality problem, capacitor bank (CB) was proposed and used for reactive power compensation in the 1900s. In the 1940s, passive power filter (PPF) was proposed to achieve both reactive power and harmonic compensation [6]. However, it can only compensate fixed harmonic frequency and suffers from resonance problems. Therefore, the active power filter (APF), which combines the voltage source inverter (VSI) and inductor, was proposed in 1976 [7]. It overcomes the shortcomings of CB and PPF, and can achieve dynamic compensation. However, APF has the problem of high cost due to the high DC-link voltage. The hybrid active power filter (HAPF) was proposed to overcome the disadvantages of PPF and APF. It can reduce the operation cost by reducing the DC-link voltage [8; 9; 10]. There are different topologies of HAPF. Among them, the \(LC\)-coupling hybrid active power filter (\(LC\)-HAPF), which has an \(LC\) branch and a voltage source inverter (VSI) in series, was proposed in 2003. It can achieve harmonic current and reactive power compensation [8; 9; 10; 11]. The thyristor-controlled _LC_-coupling hybrid active power filter (TCLC-HAPF) was proposed in 2014 [12], and can provide a wide range of reactive power compensation for both capacitive and inductive loads in the medium-voltage-level power system. However, it also increases the control complexity and power consumption. Next, _LCLC_-coupling hybrid active power filter (_LCLC_-HAPF) was proposed in 2020 [13]. It has a good high-order harmonic attenuation characteristic and lower inductor cost. However, it has stability problems and needs the assistance of active damping control. Therefore, considering the complexity and the system stability, the _LC_-HAPF is mainly focused on in this paper. In addition to the topology of the _LC_-HAPF, the controller design also plays an important role in the compensation performance. Hysteresis current control (HCC) is the most widely used control method and provides fast transient response and acceptable steady-state performance [14]. However, due to its varied switching frequency, it generates harmonic current in a wide frequency range, resulting in a large output current ripple. Thus, the classical fixed switching frequency control, proportional current control (PCC), is proposed [15] to mitigate the current ripple for a better current tracking ability comparing with the HCC. However, the control gain optimization of PCC requires a trial-and-error process, therefore retaining a certain degree of steady-state error. To obtain a lower steady-state error, a linear quadratic regulator control (LQRC) [16; 17; 18; 19; 20; 21; 22; 23] was proposed for APF, HAPF and other applications due to the good steady-state performance. Moreover, some research suggests that LQRC with integral action (LQRIC) [24; 25; 26; 27] is able to achieve a better steady-state performance because of the integral term. Therefore, this paper proposes the LQRIC for the _LC_-HAPF to achieve superior steady-state performance, and the LQRC for the _LC_-HAPF for the comparative study. Moreover, the gain matrix K of LQRC and LQRIC is obtained from the cost function of state errors, weighting matrices and system state space model. It results in a better steady-state performance than the PCC, whose gain optimization is just the trial-and-error process. The objective of this paper is to design a LQRIC for the _LC_-HAPF to improve the steady-state performance under low DC-link voltage. Figure 1 is the graphical abstract of this paper.