Tradeoff between the Output Voltage Deviation and Recovery Time of Boost Converters

Abstract

The time-optimal control for boost converters can achieve the minimum recovery time. However, their output voltage deviation is quite large. Since the minimum output voltage deviation and minimum recovery time cannot be obtained at the same time, a novel energy control is proposed to achieve a superior tradeoff between them in this paper. The peak value of the inductor current can be decreased as well. Its control parameter is easy to choose. When compared with the conventional control methods, the proposed control shows a better dynamic performance. Experimental results, which are in agreement with the theoretical analysis, are provided to verify the proposed control method.

I. INTRODUCTION

Various control schemes have been developed to enhance the regulation performance of DC-DC converters over the past few decades. Since good dynamic performance has been emphasized, some time-optimal control methods have been proposed. Boundary control, which is a geometric based control method [1], [2], is a typical example of these control methods. There are a variety of studies on different switching surfaces, e.g., first-order and second-order switching surfaces, which can improve the dynamic performance [3]-[5]. Furthermore, minimum-time transient recovery during sudden load changes can be achieved by using natural switching surfaces [6], [7]. Some other time optimal controls can obtain the same effect [8]-[10]. Most of them can aslo achieve the minimum recovery time.

For buck converters, both the minimum output voltage deviation and the minimum recovery time can be achieved at the same time. Unfortunately, this is impossible with boost converters. The smaller the output voltage deviation, the smaller the capacitance can be. Therefore, a small voltage deviation is important in reducing the cost and volume of electrolytic capacitors, which is the weakest link in power electronic circuits. Augmented DC-DC converters can substantially reduce or eliminate output voltage deviations [11], [12]. However, additional circuits, which increase the cost and complexity of the control, are needed. [13] and [14] propose constrained control which can limit the inductor peak current. However, this is only for buck converters and the voltage deviation during sudden load changes has not been analyzed.

In addition, boost converters are non-minimum phase systems. As a result, conventional linear control methods are not able to achieve very good performance. Energy-based controls are nonlinear control techniques that are based on the concept of energy [15]. They use the measured or estimated energy in the inductors or capacitors of converters to obtain simple control structures. Lyapunov-based control employs the Lyapunov energy function to derive a linear control law with asymptotic stability [16]-[18]. Passivity-based control relies on the assumption that the system is made up of energy-transforming blocks. By adding damping and modifying the dissipation structure, which greatly affects the dynamics and stability, it can modify the system energy to achieve desired behavior [19]-[23]. Similarly, Hamiltonian control and dissipativity-based control rely on the energy-balancing principle or energy storage functions as well [24], [25].

Since the minimum output voltage deviation and the minimum recovery time cannot be achieved at the same time for boost converters, a good tradeoff between them is valuable. A novel high-order energy control, which considers the total energy in the inductor and capacitor, is proposed in this paper. Unlike the conventional energy-based controls mentioned above, the proposed control method directly modifies the energy storage in the circuit. The order of its corresponding Lyapunov energy function is higher as well. Actually, the realization of the proposed control looks similar to the sliding-mode control and the synergetic control, which force the system to operate on a desired control manifold [26]-[30]. From the viewpoint of the sliding-mode control, it has an energy-based nonlinear sliding trajectory. Since its switching frequency is constant, it is more like the synergetic control. However, from the viewpoint of the synergetic control, its macro-variable is a high-order nonlinear function as well. Therefore, the proposed control can achieve a better dynamic performance, i.e., reduce the output voltage deviation in less time. Moreover, its control parameters are easier to choose. Experimental results are provided to verify the theoretical analysis.

 

II. OUTPUT VOLTAGE DEVIATION AND RECOVERY TIME OF BOOST CONVERTERS

The classic configuration of boost converters is shown in Fig. 1, where L is the inductor, C is the capacitor, S is the switch, D is the diode, R is the load resistance, Uin is the input supply voltage, iL is the inductor current, iC is the capacitor current, uC is the capacitor voltage, i.e., output voltage uO, and iO is the output current.

Fig. 1.Configuration of boost converters.

For buck converters, both the minimum output voltage deviation and the minimum recovery time can be achieved at the same time by employing the time-optimal control [10]. However, this is impossible for the boost converters in Fig. 1.

The waveforms of the boost converter with the time-optimal control are shown in Fig. 2. When the output power pout suddenly increases at t0, iL will be made to increase. The input power pin is equal to pout at t1. However, uC will keep decreasing until t2 since an increase of iL requires more energy. Then the minimum recovery time t3-t0 can be achieved. Nevertheless, the output voltage deviation is not the minimum. If iL stops increasing before t2, uC will stop decreasing. The output voltage deviation can be smaller, but the recovery time will be longer.

Fig. 2.Waveforms of time-optimal control.

As shown in Fig. 3, the output power is larger than the input power during t0~t1, and the input power is larger than the output power during t1~t3. It is worth pointing out that the energy changes E1 and E2 in these two periods have constant differences. The smaller the maximum input power, the longer the recovery time and the smaller the output voltage deviation.

Fig. 3.Energy changes.

 

III. NOVEL ENERGY CONTROL FOR TRADEOFF

Since the minimum output voltage deviation and the minimum recovery time cannot be achieved at the same time for a boost converter, a good tradeoff between them is valuable. A novel high-order energy control is proposed to achieve a better tradeoff.

The energy changes of the boost converter are the energy changes of the inductor and capacitor in the circuit. The total energy in the inductor and the capacitor is:

When the target output voltage and inductor current are uO_r and iL_r, the target energy is:

The energy in the circuit can be modified to approach the target energy step by step if:

where k is a control parameter and 0

When uO < uO_r, from (3), the energy is:

which indicates that the energy in the circuit is always smaller than the target energy until the output voltage reaches the target.

When uO > uO_r, the energy is:

which indicates that the energy in the circuit is always larger than the target energy until the output voltage reaches the target.

Therefore, from (3), the energy control low can be:

The control diagram is shown in Fig. 4, where the inside of the dashed box is the current loop and outside is the energy loop with the control law (6). Considering the control delay: is added.

Fig. 4.Control diagram.

In order to study the stability of the proposed control, the small-signal analysis method is employed as follows:

where donate large signals, and donate their corresponding small signals. The large signals constitute a stable operation point and the high order items of the small signals can be ignored in the small-signal analysis method. Thus, from (6) and (7), the small signal mode of the control law can be derived as:

According to Fig. 5 which shows a control diagram of the small-signal mode based on (8), the closed-loop transfer function of the output voltage is:

Fig. 5.Control diagram of small-signal mode.

From (9), the locations of the poles of the system can be obtained as shown in Fig. 6. As can be seen, all of the poles are on the left side of the imaginary axis when k changes from 0.1 to 1, the inductor changes from 0.5 L to 1.5 L, or the capacitor changes from 0.5 C to 1.5 C. Therefore, the system is stable and has a good robustness.

Fig. 6.Location of poles.

 

IV. PERFORMANCE OF THE ENERGY CONTROL

According to (6), the maximum inductor current of the energy control is related to parameter k. The larger k is, the larger the maximum inductor current becomes. Since the input power is proportional to the inductor current, the maximum input power depends on the maximum inductor current. The larger k is, the larger the input power becomes, as shown in Fig. 7. As the energy changes E1_k1 and E2_k2 of different values of k should be the same, the transition time t4-t1 of k2 should be longer than the transition time t3-t1 of k1. Moreover, if the maximum inductor current is smaller, the voltage deviation which represents the energy loss of the capacitor can be smaller by energy balancing. Therefore, the control parameter k is chosen to decide the trade off, while 0

Fig. 7.The influence of k.

The performance of the proposed control with different values os k will be investigated in the following. The parameters of the circuit are listed in Table I.

TALBE IPARAMETERS OF CIRCUIT

When the load resistance suddenly changes from 41.6 Ω to 10.2 Ω, the trajectory of the time-optimal (TO) control and the different trajectories in the proposed control with different values of tr are shown in Fig. 8.

Fig. 8.Different trajectories and tr.

With an increase of k, the trajectory in the proposed control becomes closer to TO and tr becomes smaller. Meanwhile the peak current of the inductor iLmax and the output voltage deviation ΔuO both increase. The relationship between ΔuO, tr and k is shown in Fig. 9. The relationship between ΔuO and iLmax is shown in Fig. 10.

Fig. 9.The relation of ΔuO, tr and k.

Fig. 10.The relation of ΔuO and iLmax.

ΔuO and tr are the both of interest. A comparison of PI control, synergetic control and the proposed control is shown in Fig. 11. Obviously, the proposed control can achieve a better dynamic performance, because the curve of the proposed control is under the curves of both the PI control and the synergetic control. With the same tr, ΔuO of the proposed control is smaller, especially when tr is small.

Fig. 11.The performances of PI control, synergetic control and the proposed control.

Furthermore, the control parameter k of the proposed control is easier to choose since it should always be larger than 0 and smaller than 1 for different systems. In order to reduce the output voltage deviation with less time, the points before the inflection point where tr is about 2.5 ms, are better choices for the system in Table I. Thus, k should be larger than 0.45 and smaller than 1 according to Fig. 11.

When the value of the inductor or the capacitor has an error of ±10%, the proposed control still can achieve nearly the same results with the exact parameters, as shown in Fig. 12. Thus, the proposed control is insensitive to the system parameters.

Fig. 12.Tolerance of L and C. (a) ±10% error of L. (b) ±10% error of C.

 

V. EXPERIMENTAL RESULTS

In order to confirm the correctness of the theoretical analysis and to verify the validity of the proposed energy control, a boost converter prototype of 500 W is established. A TMS320F28335 digital signal processor (DSP) which samples at 20 kHz is utilized for the experiment. The parameters of the experimental circuit are the same as those in Table I.

An experimental comparison of the PI control, the synergetic control and the proposed control is shown in Fig. 13, which is in agreement with the simulation in Fig. 11. The proposed control shows the best performance.

Fig. 13.Experimental results of PI control, synergetic control and the proposed control.

The experimental waveforms of the time-optimal control are shown in Fig. 14. The load changes from 41.6 Ω to 10.2 Ω as well. In Fig. 14(a), the peak value of the inductor current is 37 A. The voltage deviation is 8.4 V. The recovery time is 1.6 ms, which is the shortest time. Fig. 14(b) shows the phase plane curve, which indicates the relationship between the output voltage and the inductor current.

Fig. 14.Experimental results of time-optimal control. (a) Output voltage, output current and inductor current. (b) Phase plane.

The experimental waveforms of the proposed control are shown in Fig. 15. The recovery time is 2.2 ms, which is longer than the 1.6 ms in Fig. 14(a). However, the voltage deviation is 6.4 V, which is much smaller than 8.4 V. Moreover, its peak inductor current is smaller as well. When compared to the experimental waveforms of the synergetic control in Fig. 16, the recovery time and voltage deviation of the proposed control are both smaller than those of the synergetic control, i.e., 2.8 ms and 6.6 V. However, both the proposed control and the synergetic control have much better performances than the PI control in Fig. 17, which agrees with the theoretical analysis. Therefore, a better tradeoff, i.e., a reducing output voltage deviation with less time, can be achieved by the proposed control. As can be seen, the phase plane curves of the different controls are quite different from each other. This implies that their energy changing processes are different.

Fig. 15.Experimental results of proposed control. (a) Output voltage, output current and inductor current. (b) Phase plane.

Fig. 16.Experimental results of synergetic control. (a) Output voltage, output current and inductor current. (b) Phase plane.

Fig. 17.Experimental results of PI control. (a) Output voltage, output current and inductor current. (b) Phase plane.

 

VI. CONCLUSION

The minimum output voltage deviation and the minimum recovery time cannot both be achieved at the same time for boost converters, so a good tradeoff between them is valuable. This paper proposes a novel energy control to modify the energy storage in the circuit. The performance of the proposed control whose control parameters are easy to choose is analyzed. The experimental results obtained from a 500 W prototype are in agreement with the theoretical analysis, and they show that the proposed control achieves a better tradeoff between the minimum output voltage deviation and the minimum recovery time.