A Nonlinear Control Strategy for DC-DC Converter with Unknown Constant Power Load Using Damping and Interconnection Injecting

DC-DC converters with constant power loads are mostly used in DC microgrids. Negative impedance and large disturbances of constant power loads may lead to the instability of DC-DC converters. To address this issue, a nonlinear control strategy consisting of an improved passivity-based controller and nonlinear power observer is proposed in this paper. First, an improved passivity-based controller is designed based on the port-controlled Hamiltonian with dissipation model. By proper damping and interconnection injecting, the fast dynamic response of output voltage and stability of the DC-DC converter is achieved. Second, the constant power load is observed by a nonlinear power observer, which is adopted to estimate the power variation of the constant power load within a small settling time and improve the adaptability of the DC-DC converter under power disturbance. Finally, the simulation and experimental results are presented, which illustrate the proposed control strategy not only ensures the stability of the DC-DC converter under large disturbances, but also can track the desired operating point with low voltage overshoot in no more than 10 milliseconds.


Introduction
With the development of the DC Microgrid (DM), the DM is becoming an effective way of distributing power supplies [1][2][3]. Meanwhile, the DC-DC converters (DDCs) are becoming increasingly common and play an important role in the DM due to their advantages in efficiency, flexibility, isolation, controllability, etc. [4].
The typical diagram of DM is shown in Figure 1; the DC sources and loads are commonly connected to DC BUS through the DDCs. The DDC is acting as a load from the input side and exhibits the constant power load (CPL) behavior. The negative impedance characteristic of CPL behavior [5,6] will lead to the instability of the DDC [7,8]. Therefore, it is necessary to develop some advanced control methods which can eliminate the influence of negative impedance on the DDC, achieve the fast dynamic response, and guarantee the DDC stability.
To solve the instability problem of the CPL, linear control methods have been introduced in the literature. In [9][10][11], the small signal model is established in order to analyze the influence of CPL on the stability of the DDC, and the virtual impedance method is proposed to realize the stability of the DDC with CPL. However, as the small signal model is only applicable to linear systems or linearized systems near the desired equilibrium point, the accuracy of the model will decrease in the case of large disturbances [12]. Furthermore, the traditional linear control strategy only provides an accurate control performance in close proximity to the desired equilibrium point and has a slow dynamic response when the CPL changes significantly [13,14]. In summary, research on the stability and control of the DDC using the small signal model has some limitations. Therefore, the numerous nonlinear control strategies are introduced based on large signal. Based on the affine nonlinear model and feedback linearization control theory [15], the state feedback controller and input/output feedback linearization controller of the DDC with CPL are designed in [16,17], which improve the dynamic response of the converter and enhance the stability of DC voltage. However, they need full state measurement, and have complex controllers and a large dependence on parameters. In order to improve the disturbance rejection ability of the DDC, the active disturbance rejection controller (ADRC) [18] with CPL is designed in [19]; this improves the performance of the converter and has the robustness to face various disturbances and uncertain factors. Although satisfactory disturbance rejection performance of the DDC can be obtained by ADRC, there are many parameters of the controller which need to be determined, and the optimization of these parameters is difficult. The sliding-mode duty-ratio controller is proposed in [20], which can maintain the stability of the DDC in the operating domain under the significant changes in load power and input voltage, and it improves the dynamic response and steady performance of the DDC. However, the controller is very complex, and the output voltage fluctuation of the DDC is large. Moreover, the switching frequency of a sliding-mode controller is not fixed, and the filter design is difficult. In [21], the backstepping controller is designed in order to achieve better dynamic performance and robustness of the DDC. However, the virtual controller needs to be differentiated repeatedly, and becomes complex with the increase in system order.
The passivity-based controller (PBC) [22] has been widely employed in many physical systems [23,24] based on the energy function, and is an effective practical nonlinear control technique due to its simplicity, efficiency, and ease of implementation in comparison to the other nonlinear control techniques. The various types of PBC are designed in [25][26][27] based on the Euler-Lagrange (EL) model and the port-controlled Hamiltonian with dissipation (PCHD) model, which can ensure the large-range stability and reliability of the DDC. However, the output voltage has the steady-state error, and the transient stability of the DDC will be reduced under the variations and uncertainties of system parameters and operating conditions. In [28], the interconnection and damping assignment passivity-based controller (IDA-PBC) is designed based on the PCHD model with PI controller. It can guarantee the stability of the DDC under disturbance. However, the global stability of the DDC cannot be guaranteed, and its dynamic response is slow. In [29], the stability of the cascaded LC filter DDC is realized by the modified IDA-PBC based on the port-controlled Hamiltonian (PCH). However, the control law of this strategy is complex, and does not allow for the principle of parameters design.
To obtain excellent dynamic and steady characteristics of the DDC with CPL, this paper will utilize the DC-DC boost converter (DDBC) to study the stability and control based on the passivity. An improved passivity-based controller (IPBC) is proposed to realize the dynamic response and stability of DDBC based on the PCHD model, and the damping and interconnection injecting in IPBC is determined according to the dynamic energy storage function. Meanwhile, the nonlinear power observer (NPO) is used with IPBC, which can improve the adaptability of DDBC, and ensure the desired operating point tracking when CPL and input voltage change. In summary, the nonlinear control strategy which consists of the IPBC and NPO is proposed in this paper. Finally, the comparative study of the proposed control strategy with IDA-PBC and PBC is verified by detailed simulation and experimental results in this paper, suggesting that the DDBC has a better dynamic response characteristic while ensuring stability by using the proposed control strategy.
This paper is organized as follows. The stability analysis of DDBC with CPL based on passivity is analyzed in Section 2. The IPBC based on PCHD model is proposed in Section 3. The NPO of unknown CPL is designed and the selection principle of IPBC parameters is analyzed in Section 4. Subsequently, some realistic simulation and experimental results are given in Section 5. The conclusion of this paper appears in Section 6.

Constant Power Load Characteristic
The U-I characteristic of a CPL is illustrated in Figure 2. The voltage-current characteristic of a CPL is given by: where P CPL is the power of CPL, i and u are the instantaneous values of input current and voltage of the CPL. For a given operating point M (U I), the rate of change in current can be obtained from (1) as follows: Therefore, the curve representing the current versus voltage for a CPL can be approximated by a straight line tangent to the curve at the operating point. The equation for this line is as follows: According to (3), at a given operating point, a CPL can be equivalent to a controlled current source I CPL in parallel with a negative resistance. R CPL , I CPL and R CPL are given by the following: From (4), the CPL has a negative impedance characteristic and this will pose negative impact on system performance [30].

Effect of CPL on Stability of DDBC
The power circuit of DDBC with CPL is shown in Figure 3, where u S is the input voltage; L and C are the inductance and capacitance in DDBC, respectively; i L and i C are the inductor and capacitor current, respectively; u C is the output voltage; i CPL is the current of CPL; T is insulated gate bipolar transistor; s g is the gate drive signal of T; and D is the diode. To facilitate the analysis, the components are considered as ideal devices. According to the principle of volt-second balancing, the average switch model of DDBC [31] can be given as follows: where σ is the duty ratio of s g . According to (5), the phase portrait and simulation results of DDBC can be concluded when the desired average output voltage U CD = 60 V, desired average inductor current I LD = 2 A, switching frequency f = 10 kHz, P CPL = 60 W, σ = 0.5, L = 2 mH, C = 940 µF, and u S = 30 V.
As shown in Figure 4, the phase portrait displays clearly the characteristics of DDBC with a CPL. The blue line is the trajectory of the state variable (i L , u C ); the blue arrow indicates the evolving direction of the trajectory. The state plane is divided into two regions by the separatrix: S. The region above the separatrix is a stable region, and the region below is an unstable region; the DDBC is unstable when the state variable is in the unstable region. Moreover, the trajectory will converge to a limit cycle when the state variable is in the stable region. Due to the existence of the limit cycle, the DDBC cannot operate stably at the desired voltage and current. As can be seen in Figure 5, the state variables oscillate in a periodic steady state. When the oscillation is significant, it is easy to damage the switching devices and capacitors, and result in the damage of the DDBC.

Passivity Analysis of DDBC with CPL
Consider the system with input u and output y as follows: .
The system depicted in (6) exists a nonnegative storage function H(x) and a positive definite function Q(x) such that: If .
H(x) < 0 when Q(x) > 0, the system is said to be strictly passive; it must be internally stable [32].
From (5), it is further changed to: According to the characteristic of CPL, the P CPL = i CPL 2 R CPL and R CPL = -u C 2 /P CPL in DDBC with CPL, the following equation can be obtained by combining the two equations in (9).
The energy storage function can be expressed as: From (10) and (11), .
H O (t) can be written as: .
Let the u S = u, i L = y, according to (8) and (12), Q(x) < 0, the DDBC with CPL is nonpassive and is the reason why the DDBC cannot work at a constant stable state. Therefore, in order to eliminate the limit cycle in Figure 4 and realize the stable work of the DDBC, the DDBC needs to be passivated.

PCHD Model of DDBC
In order to establish the PCHD model of DDBC, let inductive magnetic flux x 1p = ϕ L = Li L , capacitive charge x 2p = q C = Cu C , then take the vector as T , i l and u c are the error of the inductor current and output voltage, respectively. Therefore, (5) can be changed to: According to (13), the PCHD model of DDBC is: where J is a skew symmetric matrix and satisfies J = −J T , and J = 0 −1 1 0 , x P T Jx P = 0,

IPBC Design of DDBC
In order to overcome the non-passivity and realize the stable work of the DDBC, interconnection and damping injecting are added in (16). If there is a dynamic energy storage function H p (x P ) and a skew symmetric matrix j a > 0, and a positive definite matrix R P = R + R a , where R a = r a 0 0 g a , r a > 0, g a > 0, r a , g a are positive damping injecting and conductance injecting, the controller of DDBC is taken as From (15), the dynamic energy storage function H p (x P ) can be written as where M P = diag(1/L 1/C). According (15) and (16), According to (17), M P is a positive-definite matrix and H p (x P ) is qualified as a Lyapunov function [33], and the passivation of the DDBC with CPL is realized by (15). Therefore, (15) is the IPBC.
From (15), the duty cycle corresponding to the IPBC is represented by According to the direction of energy flow, the i L should be established first, and the u C should be established later. It can be obtained from the first equation of Equation (19): where I LD = P CPL /U SD . From (19), duty ratio σ is associated with u C and i L , and a correlation degree is decided by j a and r a . Therefore, the good dynamic and steady state performance of DDBC is obtained by selecting proper j a and r a .

Nonlinear Power Observer of Unknown CPL
In practice, the power of CPL may be changed under the influences of various uncertain factors, and the operating point of DDBC may be changed. Therefore, the nonlinear power observer of CPL is introduced in the proposed control strategy to adjust the I LD in real time to ensure the operating point tracking. Learning from [34,35], the NPO of CPL for DDBC can be expressed as whereP CPL is the observed value of P CPL , γ is a NPO gain. Let power observed error be: According to (21) and (5), . P CPL is transformed as: .
Equation (22) can be written compactly as: Equation (23) shows that P CPL goes to zero exponentially. How fast P CPL goes to zero depends on the magnitude of γ. The larger γ is, the faster P CPL goes to zero, and P CPL = P CPL . Furthermore, the dynamic response of the NPO needs to be fast enough to adjust the I LD in real time, and ensure the CPL tracking, and the large NPO gain, is selected in general [36]. Therefore, an appropriate γ can be selected to realize effective real-time observation ofP CPL on unknown P CPL , and can ensure the operating point tracking when CPL changes.

Design of Damping Injecting and Conductance Injecting for IPBC
The convergence rate of the output voltage and inductor current is affected by the values of damping injecting r a and conductance injecting g a . Therefore, the parameters are designed from the perspective of energy conservation.
H p (x) can be expressed as: .
where β > 0. Equation (25) can be transformed as: . and Equation (27) shows that H P (x) can approach zero exponentially, and the convergence rate of H P (x) depends on β. The larger β is, the faster H P (x) will approach zero. As the H P (x) goes to zero, desired output voltage and inductor current will be realized.
By substituting (16) into (25), it can be expressed as: Originating from (28), r a and g a satisfy: Further, β and r a are selected by: According to (30), α and r a depend on g a , P CPL and u C . In order to improve the robustness to the load and let H P (x E ) rapidly approach zero, a large g a can be selected. g a is determined by: Therefore, r a is mainly determined by g a as g a > P CPL /(u C 2 ) according to (30) and (31). This means the tuning of r a is not significantly affected by operating points to ensure the robustness to the load.

Design of Interconnection Injecting for IPBC
In order to analyze the ability of interconnection injecting j a to track U CD and I LD , substituting (19) into (5) results in a voltage equation of It can be seen from (32) that j a is mainly related to the voltage and current. According to (32), when u c > 0 and u C > U CD , i l will become a negative incremental trend, which makes the i L and u C decrease; when u c < 0 and u C < U CD , i l will become a positive incremental trend, which makes the i L and u C increase. From the dynamic perspective, the regulation speed depends on j a . The larger j a is, the faster the regulation is, which means smaller dynamic error and shorter regulation time. On the contrary, the smaller j a will have the slower adjustment speed and lead to greater dynamic error. Thus, j a can be selected according to the maximum change in CPL to obtain the desired transient response of DDBC.
In summary, the structure of the proposed control strategy is shown in Figure 6. The implementation procedures for the proposed control strategy are as follows: (1) First, the parameters γ, r a , and j a are designed according to (23), (30) and (31).
(2) Second, the power of CPL is estimated according to (20), and I LD is calculated with the measured input voltage. (3) Finally, the duty cycle is calculated according to (19) with determined parameters and measured inductor current, output voltage, and input voltage.

Simulation Results
The proposed control strategy in Figure 6 is simulated in MATLAB/Simulink to validate its effectiveness. The system parameters are listed in Table 1. First, the procedure for tuning design parameters is illustrated, then the simulation results under different operating conditions will be presented. It can be observed from Figures 7 and 8 that the NPO and u C can track the CPL variation and U CD with fast dynamic response, and the larger γ will result in a faster convergence rate for output voltage and operating point tracking when the CPL changes. As the CPL steps, the power of the CPL and U CD need to be tracked quickly to maintain DDBC stability, and the large power observer gain is typically selected [37]. Therefore, γ can be selected as 2000 to ensure the faster dynamic response can be obtained. This set of the parameters will be used in the simulations and experiments below to verify the feasibility of the proposed control strategy, and show its advantage in terms of dynamic characteristics by comparing with other strategies.

Positive Damping Injecting and Conductance Injecting
From (25), (30) and (31), the convergence rate of H P (x E ) depends on g a and r a , and impacts on the dynamic response of the u C . The dynamic responses of the u C when the CPL steps from 60 W to 90 W and 90 W to 60 W at 0.5 s under different values of g a and r a (g a = 1, r a = 2.12; g a = 2, r a = 4.24; g a = 3, r a = 6.36; g a = 4, r a = 8.48; g a = 5, r a = 10.6) are shown in Figure 9. It can be observed from Figure 9 that the larger g a and r a will lead to a faster convergence rate for H P (x E ), and the u C can track the U CD within a short transition while the voltage fluctuation peak is lower. Thus, the proper values of g a and r a should be selected to obtain a fast dynamic response of u C while avoiding a large voltage overshoot.

3.
Interconnection Injecting Figure 10 shows the simulation results of the output voltage dynamic response when the CPL steps from 60 W to 90 W and 90 W to 60 W at 0.5 s with different values of j a (j a = 3, 5, 7, 9 and 11). It is clearly observed in Figure 10 that a larger j a will lead to a shorter settling time and a lower voltage fluctuation peak for the U CD tracking. Thus, j a can be selected at a larger value in order to obtain a better dynamic response of u C .
In order to further verify the influence of control parameters on DDBC stability, and guide the selection of parameters, the phase portraits of DDBC using IPBC under different value of parameters are shown in Figures A1 and A2 (Appendix A). It can be observed that larger g a , r a , and j a will lead to a larger unstable region. According to the influence of control parameters on dynamic response and stability, the g a , r a and j a can each be made a compromise choice. Therefore, g a and r a can be selected as 3 and 6.36, respectively, and j a can be selected as 7; these parameters will be used in the following simulations and experiments.
Based on the selection of the above parameters, the phase portrait of DDBC by using IPBC as shown in Figure 11 is compared with Figure 4; the output voltage and inductor current can converge to the desired equilibrium point P and realize the stable work of the DDBC. Figure 11. Phase portrait of DDBC by using IPBC when feeding CPL.

Simulation Verification
Simulations are carried out based on MATLAB/Simulink to validate the proposed control strategy. The simulation parameters are shown in Table 1 of DDBC with CPL, and used in the simulations for comparative purpose. To verify the advantages of the proposed control strategy, the IDA-PBC [28] and PBC [22] are implemented for the DDBC with the same simulation parameters. The u C is considered to operate into a steady state when it fluctuates at U CD ± 0.2 V.

1.
Step Change in u S Figure 12 shows the simulation results of the output voltage when u S steps from 30 V to 40 V at 0.5 s with different control strategies. The u C has approximately 21 V deviation from the U CD when the PBC is used, while the steady-state output voltage error is eliminated by using the proposed control strategy and IDA-PBC. Furthermore, the voltage overshoot of the proposed control strategy is less than 0.2 V, significantly smaller than the IDA-PBC under the u S steps, and it has the faster convergence rate for U CD .

2.
Step Change in P CPL Figure 13 gives the comparison results of the output voltage when the CPL steps from 60 W to 90 W at 0.5 s and 90 W to 60 W at 1 s with different control strategies.
As shown in Figure 13, the output voltage performances of the proposed control strategy, IDA-PBC and PBC, can be obtained. Comparing the simulation results, the u C has approximately 3.5 V deviation from the U CD when the PBC is used, and the PBC cannot realize the operating point tracking when the CPL steps. Meanwhile, the steady-state output voltage error is eliminated by using the proposed control strategy and IDA-PBC, and the voltage overshoot of proposed control strategy is significantly smaller than the IDA-PBC. Moreover, the u C has the faster dynamic performance by using the proposed control strategy, and it can track the U CD accurately within a very short transition, which is much better than the IDA-PBC. Therefore, the proposed control strategy can realize the faster convergence rate of operating point tracking when the CPL steps.   PBC [22] 3.5 -3.5 IDA-PBC [28] 0.8 16 ≈0

Experimental Results
The experimental system setup is shown in Figure 14, and the experimental parameters are the same as Table 1. It consists of a boost converter, a DC source, an oscilloscope, and a DC electronic load which is used to emulate the CPL. The u C of the DDBC is considered to enter a steady state when the output voltage fluctuates at U CD ± 1 V.

1.
Step Change in u S The impact of the u S steps is shown in Figure 15. In Figure 15a, the u C has little or no overshoot, and quickly tracks the U CD when the u S steps from 30 V to 40 V by using the proposed strategy. In Figure 15b,c, the output voltage overshoots are much larger than the proposed control strategy, and the u C of PBC has approximately 23 V deviation from the U CD when the PBC is used. Therefore, compared to other control strategies, the proposed control strategy can track the U CD within a very short transition, and has better dynamic response when the u S changes.

2.
Step Change in P CPL As shown in Figure 16a, when the CPL steps from 60 W to 90 W and 90 W back to 60 W, the output voltage has the overshoot below 1 V and then drops back to U CD by using the proposed strategy. However, in Figure 16b, the PBC cannot track the desired value and has approximately 8 V deviation from the U CD under the CPL steps. In Figure 16c, the overshoot of output voltage is much larger than the proposed control strategy, and the settling time is more than 350 milliseconds using IDA-PBC. Thus, the proposed control strategy can realize the faster convergence rate of u C and ensure the operating point tracking when the CPL changes significantly.  Based on the experiment results from Tables 4 and 5, the proposed control strategy can ensure DDBC stability with a fast dynamic performance, and realize the accurate output voltage and operating point tracking under various disturbances. The dynamic response and overshoot for experimental cases are slightly slower and larger than the corresponding simulation results. This is acceptable, as in the simulation studies the ideal CPL with the step change is used, whereas in the experimental tests, the u S and CPL steps are restricted by their controller bandwidth. Moreover, the experimental results suffer from real-life system conditions such as Electro-Magnetic Interference (EMI), noise, delays, and various uncertainties disturbances which are not considered in the simulations.

Conclusions
The instability problem caused by the CPL for DDBC is addressed in this paper. To solve this problem, the stability of DDBC with CPL is studied based on passivity, and the nonlinear control strategy which consists of the IPBC and NPO is proposed. First, the IPBC is studied using damping and interconnection injecting based on the PCHD model and the parameters of IPBC are designed from the perspective of energy conservation. From the phase portraits of DDBC, the stability of DDBC is realized by IPBC. Second, the adaptability of the DDBC is improved as the CPL and input voltage change through the design of NPO. The NPO with large γ is employed to estimate load power with a fast dynamic response to ensure the accurate desired operating point tracking. Meanwhile, extensive numerical simulation and experimental results are given to verify the validity of the proposed control strategy. Under different large disturbances, both results indicate that the output voltage has the voltage overshoot below 1 V and can accurately track the U CD within 10 milliseconds by using the proposed strategy, which are significantly better than PBC and IDA-PBC. Furthermore, the consistency of simulation and experimental results is validated. Finally, the proposed control algorithm is also applicable to other DDC topologies.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available as the data also forms part of an ongoing study.