The basic idea behind EID can be illustrated by a conceptual diagram as Figure 4. If for all t ≥ 0, the output of Figure 4A,B is the same that always stands up, then the disturbance d _e (t) is called EID of d(t); as shown in Figure 4C, the disturbance can be obtained by EID estimator and then inverse feedback it to input channel to eliminate the influence of d _e (t) on the output y(t). Generally, the grid voltage feedforward method is used to decouple in the dq reference (Eq. 5). However, its decoupling performance may be reduced because of the communication time delay from grid voltage sensor to controller. In this paper, the model is modified by regarding the coupling terms and grid voltage disturbances as system disturbances together. Transform Eq. 5 into Eq. 6 that d i d d t = − R L i d + 1 L u d + ω L i q − e d d i q d t = − R L i q + 1 L u q + − ω L i d − e q (6) considering the existence of disturbance signals in the disturbance input channel, then state space (Eq. 7) is obtained x ̇ t = A x t + B u t + d c p t + B d 1 d 1 t + B d 2 d 2 t y t = C x t + D u t (7) where d _cp (t) is the disturbance caused by the coupling terms, d ₁(t) is the disturbance caused by the fluctuation of the grid and the dead time effect, and d ₂(t) is the disturbance caused by the uncertainties of internal parameters.

All the disturbances in Eq. 7are lumped, and the modified model Eq. 8 is obtained: x ̇ t = A x t + B u t + d e t y t = C x t + D u t (8) where x = i d , i q ⊤ is the state variable, u = u d , u q ⊤ is the control signal, y = i d , i q ⊤ is the output signal, d c p = ω L i q − e d , − ω L i d − e q ⊤ is the crosscoupling terms, and d _e (t) = d _cp (t) + B ⁻¹ B _d1 d ₁(t) + B ⁻¹ B _d2 d ₂(t) is the equivalent lumped disturbances, where A = − R / L 0 0 − R / L , B = − 1 / L 0 0 − 1 / L , C = 1 0 0 1 , and D = 0, respectively.

Obviously, d _e (t) is composed of coupling terms and comprehensive disturbances. The purpose of model decoupling and comprehensive disturbance suppression can be achieved by eliminating d _e (t). Figure 5 shows the configuration of the EID-based d-axis current control system for system Eq. 8. It is composed of five parts: gain module, internal model, state feedback controller, state observer, and EID estimator. The separation theory (Anderson and Moore, 2007) holds for the system, so each part can be designed separately. The proposed EID-based controller directly estimates the lumped disturbance d _e (t) and adds it to the control signal u ( t ) = u f ( t ) − d ̃ e ( t ) , without considering the prior information of the disturbances.

The internal model is used to accurately track the reference input; its form is shown in Eq. 9: x ̇ R t = A R x R t + B R i d * t − i d t y R t = x R t (9)

However, the inertial link and integral action will lead to response delay. To improve the dynamic performance of the internal model, a gain module K _s is proposed to parallel connect with the internal model in this paper. The state equation obtained by parallel connection gain K _s with the internal model is shown in Eq. 10: x ̇ R t = A R x R t + B R i d * t − i d t y R t = K R i d * t − i d + x R t (10)

Assuming there is no disturbance, the augmented state system is as follows, which consists of the internal model and the plant: x ̄ ̇ t = A ̄ x ̄ t + B ̄ u ̄ t + B ̄ R i d * t u ̄ t = K x ̄ t (11) where gain K = [α,β] ^⊤ , u ̄ ( t ) = u f ( t ) , x ̄ ( t ) = x ( t ) , x R ( t ) ⊤ , A ̄ = A 0 − B R C A R , B ̄ = [ B , 0 ] ⊤ , and B ̄ R = [ 0 , B R ] ⊤ , respectively.

To design the state-feedback gain, we build the quadratic performance index (Eq. 12), which is relative to the state and control: J K = ∫ 0 ∞ x ̄ ⊤ t Q K x ̄ t + R K u 2 t d t (12)

Select an appropriate diagonal matrix Q _K > 0 and a real scalar R _K > 0 to minimize Eq. 12, and then, [ α , β ] = − R K − 1 B ̄ ⊤ P , where P is the solution of R i c c a t i e q u a t i o n A ̄ ⊤ P + P A ̄ + Q K − P B ̄ R K − 1 B ̄ ⊤ P = 0 .

The state observer of EID controller is constructed as follows: d x ̂ t d t = A x ̂ t + B u f t + Γ i d t − C d x t d t (13)

Defining Δ x ( t ) = x ̂ ( t ) − x ( t ) and substituting it into Eq. 8, we obtain Eq. 14: d x ̂ t d t = A d x t d t + B u t + B d e t + d Δ x t d t − A Δ x t (14)

If there exists a control signal Δd(t), which satisfies Eq. 15: d Δ x t d t − A Δ x t = B Δ d t (15) define d ̂ e ( t ) = d ε ( t ) + Δ d ( t ) , and then, Eqs. 14 and 15 give Eq. 16: d x ̂ t d t = A x ̂ t + B u t − d ̂ e t (16)

From Eqs. 8, 13, and 16, the estimated value d ̂ e ( t ) of EID can be solved: d ̂ e t = Ξ Γ C x t − x ̂ t + u f t − u t (17) where Ξ = B ⊤ B − 1 B ⊤ .

To ensure the causality of the system and suppress the high-frequency measurement noise, d ̃ e ( t ) is obtained by d ̂ e ( t ) through a low-pass filter F(s): d ̃ e t = F s d ̂ e t (18) and the filter satisfies the following: | F j ω | ≈ 1 , ∀ ω ∈ 0 , ω r (19) where ω _r is the highest angular frequency estimated by the selected disturbance; usually, the cutoff frequency of the filter is 5 ∼ 10ω _r . The first-order filter F ( s ) = 1 T s + 1 can be satisfied well. The control law of EID controller is as follows: u t = u f t − d ̃ e t (20)

The constructed state observer is Eq. 13 without the need of inverter dynamic of the plant. Consider that the system in Eq. 10 has no disturbance, let the reference input and disturbance be zero, and then, the transfer function from d ̃ e ( t ) to d ̂ e ( t ) is Eq. 21: G s = Ξ s I − A s I − A − Γ C − 1 B (21)

Theorem I: The control row in Eq. 21 guarantees the stability of the system in Eq. 5 if the following conditions hold:

1) A − ΓC is Hurwitz;

The dual model of plant is as follows: x ̇ o t = A T x o t + C T u o t y o t = B T x o t (22) and construct a state feedback controller as follows: u o t = Γ x o t (23)

Minimize Eq. 24 by selecting the appropriate diagonal matrix Q _o > 0, the real scalar R _o > 0 and ρ > 0: J o = ∫ 0 ∞ ρ x 0 T t Q o x o t + u o T t R o u o t d t (24) meanwhile, ensuring Eq. 25: lim ρ → ∞ s I − A − Γ C − 1 B = 0 (25) then the observer gain Γ can be solved, and Theorem I is guaranteed.

By employing EID-based control, Figure 6 shows that the system stabilizes quickly in normal condition, and the PV output power, output, and modulated dc voltage of the BOOST circuit basically stay the same, respectively. Figure 7 shows the active power and reactive power of grid-tied inverter output; the power factor in steady state is 1. Figure 8 presents the d- and q-axis disturbances estimation. It can be seen from Figure 6 that the PV output power is stable at about 0.8 s; so before 0.8 s, the disturbance caused by the voltage fluctuation of the dc side of the inverter is also estimated by EID, as shown in Figure 8. After 0.8 s, the dc side voltage is stable, and there is no other external disturbance. At this time, the EID estimation in Figure 8 is mainly caused by the non-linear switch, unmodeled dynamic, coupling terms, and so on. Figure 9 shows the high sinusoidal degree grid-tied current waveform, whose THD value is only 0.31%, and its harmonic spectrum is shown in Figure 10.

Figure 11 summarizes the THD values of the grid-tied current under the different dead time, which compares with the grid voltage feedforward PI controller (Zammit et al., 2021) and the PI-RC controller (Li et al., 2020). The dead time of SPWM signal is set from 2 to 6 µs. In real situations, the dead time of SPWM signal is usually 20%∼50% of the switching cycle (Lai et al., 2021); so in this paper, the dead time of SPWM signal is set from 2 to 6 µs. As the dead time increases, the THD value greatly increases using the PI control and increases slightly using the PI-RC. However, by the proposed EID-based controller, the THD value is still less than 2% and tends to be flat. Clearly, the EID-based controller is much better to inhibit the dead time effect.

In addition, the inverter system under the different control strategies is tested when the grid voltage distortion is occurred at 0.8 s. The grid voltage contains 0.05 pu 3rd and 0.03 pu 5th harmonics (Lai et al., 2021), and the dead time is set as 5 µs.

Figure 12A shows the compressive grid-tied current harmonic spectrum among the PI, PI-RC, and proposed EID-based controllers, and Figure 12B shows the disturbance estimation of the proposed controller. The detailed data of Figure 12A are listed in Table 2. Figure 13 shows the three-phase grid-tied current waveforms by the different controller, respectively.

When non-ideal grid voltage and dead time effect occur simultaneously in system, the current distorts seriously by PI controller, and the THD value reaches 6.25%. It mainly contains low-order harmonics as shown in Table 2. Clearly, the steady-state performance is much better by PI-RC controller, its THD value is 4.5%. Figure 12B shows that the periodic disturbance is estimated quickly by proposed controller, all order current harmonics almost have been reduced as shown in Figure 12A, and the THD value is only 1.4%. Moreover, compared with the results in Figure 13A and Figure 13B, the three phases have the same amplitude in Figure 13C. Obviously, because the coupling term of the model will lead to cross coupling of dq axis current, the three-phase current under PI and PI-RC controller is unbalanced, but the three-phase current amplitude of the controller proposed in this paper is still the same, which implies that the proposed controller has an excellent model decouple capability.

Overall, the inverter system with the EID-based controller suppresses both dead time effect and grid voltage distortion satisfactorily. It is clear that the proposed current controller is better than others.