Fight or Flee? Active damping control strategies in resonating system

What is your best move being asked to transport a glass of water without spilling? Perhaps walks in a conservative manner that does not excite the content too much, or adaptively adjust your pace that matches the motion of the content and moves aggressively? This specific task, controlling a system with a resonating mode that can’t be directly accessed, turns out to be a common one in numerous domains and will be discussed in this article. For instance, the structural resonance in the positioning control of a mechanical system and the instability caused by grid impedance mismatch in power electronics depicts in the following figure.

Control problem with resonating mode that was not directly accessible

Modeling and problem formulation

Roughly speaking, the control problem contains two major objectives:

1. Tracking current reference while avoiding unwanted resonating mode
2. Suppressing the oscillation caused by initial-condition or internal and external disturbances

LCL circuit, which is common in the grid-connected power system will be used as an example in the following sections. With “Vp” (from power stages) and “Vg” (from AC grid) defined as input and disturbance correspondingly, the derivation of states equations of “Ip”, “Ig”, and “Vc” is straightforward. Readers may refer to the following block diagram and the state equations that encodes the resonating nature.

Block diagram of LCL circuit with “Vp” (power stages) defined as input and “Vg” (grid) as disturbances, the inductance “Lg” serves as a lumped model of the grid impedance

Alternatively, the Bode-plot provides an intuitive perspective of a system with undamped resonate mode. The frequency responses of each transfer from “Vp” (input) to three of the states were depicted in the following figure with a clear resonating frequency of around 4.8kHz. Resonating mode creates multiple 0dB cross-overs and poses a challenge in designing an adequate controller (responsive and stable). Methods that handle such an issue were usually referred to as an “Active Damping Controller” and two of the popular types will be discussed.

Bode-plots of LCL circuit with input from power stage “Vp”, the resonation could be observed from all transfer functions since they all share the same poles

Numerical results throughout this article were carried with parameters: “Lp=100uH”, “rp=5mΩ”, “Lg=100uH”, “rg=5mΩ”, and “C=22uH”.

Active damping based on cascaded filter

Filtering the input signal around resonating frequency, depicted in the block diagram below was possibly the most instinctive and common solution [1,2]. The central concept of this approach is to avoid any excitation caused by the controller, therefore referred to as a “Flee” type of design.

Typical feedback regulating system with a cascaded “Active damping” filter “F”

For instance, a Notch filter matching the resonant frequency helps smooth out the resonance peak with a price of decreased phase margin shown in the following Bode-plot. You may already notice a mismatched Notch frequency was deployed at 5.0kHz instead of the actual 4.8kHz, this was purposely done to emphasize the robustness issue of this type of approach and was basically unacceptable in real applications.

Bode-plot of the loop transfer with and without slightly mismatched Notch filter, notice the drastically decreased phase margin which limits the resulting bandwidth, not to mention if the resonance frequency was varied due to any reason, this approach was likely to fail

Another critical downside of a “Flee” type of approach can be revealed by viewing the system block diagram shown as follows. That is, the actual damping ratio (or the quality of resonance) was not improved at all. In other words, with the disturbance type of excitation either from the AC grid or noises from the input port, the resonance peak was still solid as before. This dilemma is a common symptom of most pole-zero cancellation motivated technics and again, was not favorable in applications such as close-loop control.

The system was unaffected with controller signal “u” attenuated around the resonance frequency by “F”, which also implies the damping factor of the system was not improving with such a strategy :(

Active damping based on virtual damping

A more “initiative” approach will be generating control signals that attenuate the resonance, essentially a “Fight back” type. Such behavior can be accomplished by imitating or approximating a system with damped LCL parameters. For example, two properly designed filters “G” shown as the feedback controller in the figure below will be discussed in the following sections. As a result, the fundamental difference between a “Fight” and “Flee” approach is that the former actually generates a control signal containing the resonance frequency whereas the latter tends to avoid any.

Active damping based on virtual damping controller “G”, assuming the state “Vc” was measured

Ideal decoupling case with G=-1,

With an ideal filter “G=-1”, the signal from “Vc” to “Ip” was effectively canceled and left with two uni-directionally decoupling systems shown in the following block diagram.

Equivalent block diagram with “G=-1”, frequently and abusively termed as a feed-forward control from “Vc”, such technic was in fact feedback seeing from the big-picture

The stability of such a control loop could be verified in the Bode-plot shown in the bottom left and the closed-loop transfer from “Vp” (input) to three of the states were shown in the right. Compared to the original plant without active damping “G=-1”, the resonate peaks from “Vp” to “I_p” vanishes while the other two outputs “I_g and Vc” still suffer from the resonance peaks at a lower frequency of 3.4kHz, both were straightforward results from the equivalent block diagram.

LHS: Loop gain with “G=-1”, a stable one despite a small phase margin, RHS: Bode-plots with closed active damping control loop “G=-1”, peak vanishes in the perspective of “i_p”

Long story short, this method hides resonate from the perspective of “i_p” and help the design of the consecutive current regulator. But on the other hand, the resonating mode still exists and can still be excited by the input “Vc” or the grid disturbances “Vg”, a piece of bad news if “i_g” is what we care about.

Ideal damping case with G=s*Kd

Alternatively, feeding signal proportion to the capacitor current to each of the voltage inputs of both the inductors “Lp and Lg” creates a virtual resistor cascaded with the capacitor. Despite the limitation that only “Vp” was accessible as input, this technic still works without having a realizable equivalent circuit. With the capacitor current approximated from a highpass filtered voltage signal “Vc”, the block diagram is shown as follows.

Equivalent block diagram with “G=LPF(s*Kd)” assuming the state “Vc” was measured

Similar to the previous case, the stability is verified in the Bode-plot shown in the bottom left and the closed-loop transfer in the right. With properly selected gain “Kd=2*sqrt(C*(Lg+Lp)*Lp/Lg)”, the resulting transfer of all the outputs is free from resonating mode compared to the case with “G=-1”. Although not directly shown in this article, such a damped characteristic is also held from the perspective of the grid disturbances “Vg”.

LHS: A stable loop gain with “G=LPF(s*Kd)”, RHS: Bode-plots with closed active damping control loop “G=LPF(s*Kd)”, peak vanishes in all of the outputs

A state-space point of view

Apparently, the last approach outperforms others in the sense of damping resonance mode. This is reasonable since the proportional feedback of capacitor current was closely related to a state feedback gain from both “I_p and I_g”. In other words, (almost) a full-state feedback controller design such that all three system poles were assignable.

Conclusion

The cascaded (Flee) type of active damping controller could provide a baseline performance under a situation with a limited sensor, analogously, this is the strategy one will choose if asked to carry a glass of water without spilling while blindfolded. With additional information, more aggressive approaches such as ideal decoupling, ideal damping, and the general full-state feedback can be performed. Furthermore, only the case with ideal damping (or the general full-state feedback) actually improves the damping of the transfer function from the perspective of external disturbances such as grid voltages “Vg”.

Limitations with digitally realized active damping

As one may concern, ideal damping requires stability by considering the loop transfer. Under the phase dropping caused by digital controlled output delay (Zero Order Hold) and the sample-calculate-execute delay, the phase margin of the ideal damping case (~60deg) was very likely to be violated if care was not taken.

References

[1] M. Liserre, A. Dell’Aquila and F. Blaabjerg, “Genetic algorithm based design of the active damping for a LCL-filter three-phase active rectifier,” Eighteenth Annual IEEE Applied Power Electronics Conference and Exposition, 2003. APEC ‘03., 2003, pp. 234–240 vol.1, doi: 10.1109/APEC.2003.1179221.

[2] M. Liserre, R. Teodorescu and F. Blaabjerg, “Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values,” in IEEE Transactions on Power Electronics, vol. 21, no. 1, pp. 263–272, Jan. 2006, doi: 10.1109/TPEL.2005.861185.