Design of composite control scheme for single-phase inverter based on PID

As a classic control algorithm , PID control has the characteristics of simple structure, easy debugging, fast dynamic response characteristics, and strong robustness. However, for medium and low frequency periodic signals, this algorithm still cannot achieve zero static error control; and its ability to adjust the output waveform distortion caused by nonlinear load is also poor.

This paper introduces a scheme of using a series topology structure for PID controller and repetitive controller, taking the stable PID+ controlled object closed-loop system as the controlled object of the repetitive controller, while ensuring the steady-state error and dynamic performance of the system, simplifying the design of the repetitive controller.

1 Inverter model

Where u0 is the output voltage; i is the inductor current; is the load resistance; C is the filter capacitor; ￡ is the capacitor equivalent series resistance:

Taking the sampling frequency and the switching frequency equal, and considering the inverter bridge as a zero-order holder, the pulse transfer function of the object obtained by discretizing equation (2) is:

2 PID controller design

Figure 2 shows the open-loop frequency characteristic diagram (Bode diagram) of the PID control system. Where G0 is the controlled object; G is the PID controller; G=Gp×G0

According to traditional PID design theory, the open-loop coefficient is first set to K=200, in order to increase the low-frequency gain of the system and reduce the steady-state error. However, if the K value is too large, the stability of the system will be reduced, so a zero point is added in the low-frequency band to form a lag correction with the integral link. The lag link has two main functions: one is to increase the gain of the system's low-frequency response and reduce the steady-state error of the system while ensuring that the transient performance of the system remains basically unchanged; the other is to use its low-pass filtering characteristics to attenuate the high-frequency response gain of the system and increase the phase margin of the system to improve the stability of the system.

A zero point is added at 60° in the intermediate frequency band, and a pole is added at the high frequency band, thus forming a lead correction. It has two main functions: first, it uses the phase lead characteristic to increase the phase margin of the system, increase the cutoff frequency of the system, and ensure the fast dynamic response of the system; second, it attenuates the high-frequency response gain of the system, suppresses high-frequency noise, and improves the robustness of the system.

In Figure 2, G is the open-loop transfer function of the PID control system. From its frequency characteristic curve, we can see that the low-frequency open-loop gain of the system is very large; the phase margin in the frequency band near the cutoff frequency increases; the open-loop gain in the high-frequency band is very small, which suppresses high-frequency interference signals. The PID controller is designed by the lag-lead phase correction method to achieve the desired purpose. After the above analysis, the PID transfer function is:

Taking the sampling frequency and the switching frequency equal, using the zero-order holder to discretize equation (4) to obtain the pulse transfer function of the object:

3. Repeated Controller Design

According to the internal model principle, by adding the dynamic model of the external periodic signal to the closed-loop system, the system can achieve the purpose of asymptotic tracking of the external periodic signal. The repetitive control strategy is based on this principle. Figure 3 is the basic repetitive controller system structure diagram, where the repetitive controller discrete expression is:

Where IV is the number of sampling times of the output voltage per fundamental wave cycle.

From equation (6), it can be seen that when the frequency is ∞=2~k/T (K=0, 1, 2, ..., T is the fundamental period), since z=1, if a repetitive controller is embedded in the closed-loop system, the open-loop gain will tend to infinity. In this case, the non-harmonic input signal will be strongly attenuated, achieving the purpose of accurately tracking the input signal. However, since the dynamic characteristics of the controlled object cannot be accurately known, the open-loop gain tending to infinity will deteriorate the stability of the closed-loop system. In order to ensure the stability of the system, the basic repetitive control system needs to be improved.

The structure diagram of the composite repetitive control system proposed in this scheme is shown in Figure 4.

Q(z), G, (.) are low-pass filter compensators, which are the focus of repetitive controller design. The role of P () is to adjust the open-loop gain to a large finite value, ensuring system stability without affecting the steady-state accuracy; the role of G, () is to improve the robustness of the system by limiting the frequency band of the repetitive controller. From the figure, the error transfer function of the system can be obtained as:

In the formula,

According to the small gain theorem, the stability condition of the above system is:

① The closed-loop system G ( ) is stable.

②

From the error transfer function (7), we can see that if:

Then formula (7) can be rearranged as:

If we construct Q(z) so that at frequency ∞=2,rrk/T (k=0, 1, 2, ...):

Then we can get E(z): 0. Therefore, when the system satisfies equations (10) and (12), the steady-state error of each order harmonic will theoretically approach zero. However, since the actual system is a non-ideal system, the above design requirements cannot meet the harmonics of all frequency bands. Usually, within a certain frequency range, a repetitive controller is designed according to the stability conditions (8), (9) and the controller conditions (10), (12) to meet the system's steady-state and dynamic requirements.

According to equations (8) and (10), the compensator G, ( ) can be directly designed as the inverse function of G ( ). However, if G (z) is a non-minimum phase system, although equation (10) still holds and the external performance is stable, due to the unstable zero-pole cancellation, this will cause internal instability in the system. In this case, other types of compensators must be used to design G, ( ).

The solution proposed in this paper controls a stable closed-loop system stabilized by a PID controller, which is itself a minimum phase system, so the compensator can be designed directly using the inverse function, namely:

Formula (12) theoretically requires p(z) = 1; however, Formula (9) shows that since G(z) tends to 0 in the high frequency band, Q(z) should be less than 1 in the high frequency band, so Q(z) should be a low-pass filter with zero phase shift, and its expression is:

In practical applications, the use of a first-order low-pass filter can fully meet the system requirements:

Through the above analysis, the two filters of the repetitive controller can now be designed according to equations (13) and (15).

To further understand the role of the repetitive controller in the system, the open-loop frequency characteristics of the system with and without the repetitive controller can be compared, as shown in Figure 5.

In the high frequency band, the open-loop gain becomes very small, which is very helpful for suppressing high-frequency noise and improving system stability and robustness. However, at non-harmonic frequencies, the open-loop gain of the system without embedded repetitive controller is larger, which means that the repetitive controller has poor control effect on the signal at this frequency. Therefore, in addition to improving the dynamic response speed of the system, the PID controller in the system also needs to adjust the non-harmonic signal to make up for the shortcomings of the repetitive controller.

4 Simulation Experiment Analysis

Based on the above analysis, the author conducted simulation experiments on digital PID control, repetitive control and the proposed composite control. The system parameters are as follows:

The input DC voltage is 270 V, the output AC voltage is 110 V/50 Hz, the switching frequency is 10 kHz, the output filter inductor is 1.5 mH, the output filter capacitor is 20 F, and the load resistance is 10 n.

In Figure 6 and Figure 7, is the given voltage; Uo is the output voltage.

PID control cannot achieve error-free tracking of periodic signals, and there are periodic steady-state errors. For a system with a repetitive controller embedded, the output can track the input signal well, the system quickly enters a steady state, and exhibits good dynamic performance.

Figure 8 and Figure 9 are spectrum analysis of the system output voltage waveform. The figures intuitively reflect that repetitive control can effectively suppress harmonics and reduce the distortion rate of the output waveform.

5 Conclusion

The above process analyzes the working principle of the repetitive controller in detail, combines the advantages and disadvantages of PID control and repetitive control, and designs a composite controller with a series topology structure, while giving full play to the repetitive controller's ability to track periodic signals without error and the PID controller's ability to respond quickly to sudden disturbances. The simulation results show that repetitive control is effective in reducing output waveform distortion, and the composite control strategy based on PID control and repetitive control is a practical sinusoidal inverter control solution.