Combined feedback–feedforward control of Ćuk CCM converter for achieving fast transient response

This research was supported by Kyungpook National University Research Fund, 2020 Abstract The Ćuk converters operating in continuous conduction mode (CCM) can be preferred in applications such as microprocessor power delivery and pulsed load because these circuits have advantages of being able to step up/down, a small number of power components, and low input/output current ripples. However, they show poor transient performance due to right‐half‐plane‐zeros (RHPZs) in the closed‐loop transfer function of the Ćuk CCM converter. To enhance the transient response, a combined feedback–feedforward control for the Ćuk CCM converter is proposed. The proposed control scheme comprises a feedback control signal based on a Lyapunov function and a duty‐ratio feedforward control signal. A Lyapunov‐function‐based controller (LBC) achieves fast dynamic response even under large‐signal variations from the operating point. The duty ratio feedforward controller (DFFC) is developed to predict the effect of the disturbances and compensate it, while alleviating the burden of LBC. The proposed control logic makes the closed‐loop system of the Ćuk CCM converter globally exponentially stable and thus provides a fast transient response even under large‐signal variations. To construct the proposed controller, the authors make use of the large‐signal averaged model of the Ćuk CCM converter, and consider the parasitic elements. To verify the proposed control scheme, numerical simulations and experimental tests are conducted.


| INTRODUCTION
A Ćuk converter, which is a widely used dc/dc converter, provides an output voltage less than or greater than the input voltage depending on the duty ratio. Also, it has low input and output current ripples due to the inherited inductors at the input and output sides. Furthermore, the Ć uk converter is more efficient with lower current ripple when it operates in (CCM) than when it operates in discontinuous conduction mode (DCM). Thus, the Ć uk CCM converter can be preferred in applications such as microprocessor power delivery and pulsed load. However, the Ć uk CCM converter shows poor transient response due to RHPZs in its transfer function [1,2]. Therefore, the output regulation of the Ć uk CCM converter is a difficult challenge.
In control design of the Ć uk converter operating in CCM, the use of its small signal model around a fixed operating point has been popular due to its simplicity [3,4]. Using this linearised model, the control that has been applied to Ć uk converters includes proportional-integral (PI) control [5], PI fuzzy control [6], PI and sliding mode control (SMC) [7,8], cascaded PI-SMC [9], and optimal control via a jump parameter technique [10]. However, the linearised model cannot reflect the Ć uk CCM converter completely, especially when large variations from the operating point occur. For this reason, these control schemes show poor transient performance caused by large signal variations. Variable structure control (VSC) is a robust control technique, which has low sensitivity to parameter variations and unmodelled dynamics [11]. Nevertheless, in applying VSC to switching power converter, the output voltage error in the steady state, variation of switching frequency, and chattering are still problems that need to be solved. One-cycle control is a non-linear control method, which has the ability to follow the control reference instantly [12]. However, its input stage may exhibit an oscillation as a result of Neimark-Sacker bifurcation. The oscillation increases the voltage stress on the input stage switch. To reduce the oscillation, [13] proposed to use a timedelay feedback controller and correctly chose the delayed time and feedback gain.
A Lyapunov function-based control (LBC) scheme generally guarantees globally asymptotically stability of the closedloop system while achieving excellent dynamic response [14][15][16]. For this control scheme, a control signal is constructed to ensure that the total system energy is constantly dissipated and thus the tracking error of the control system asymptotically converges to zero. However, no published research has considered compensating for the use of the LBC for Ć uk CCM converters. Furthermore, the use of the LBC only would not achieve a fast transient response because it adopts feedback signal to implement its control input, and this feedback signal would generate the inevitable delay in control. Therefore, the LBC should be combined with other types of control techniques.
Herein, the authors propose a combined feedbackfeedforward control for a Ć uk CCM converter under large variations of operating points. The proposed control scheme is composed of a feedback control signal based on a Lyapunov function and a duty-ratio feedforward control signal, which is constructed based on the large signal dynamic model. The Lyapunov function-based controller (LBC) guarantees global exponential stability and exhibits a fast dynamic response even under large-signal variations from the operating point. This controller is supplemented with duty-ratio in the feedforward loop that helps the regulation of the output voltage. Moreover, the authors deal with parasitic components to build the dynamic model and proposed control scheme of the Ć uk CCM converter. They performed experimental tests to demonstrate the remarkable tracking precision of the proposed control scheme.
Section 2 introduces the modelling of the Ć uk CCM converter with parasitic components. Section 3 proposes a controller scheme for the Ć uk CCM converter and shows corresponding stability analysis. Section 4 presents the simulation and experimental results, and Section 5 provides the conclusion.

| MODELLING OF THE Ć UK CCM CONVERTER WITH PARASITIC COMPONENTS
The circuit of a Ć uk CCM converter ( Figure 1) consists of input voltage source V i , inductor L 1 , controllable switch S 1 , capacitor C 1 , transformer T with turns ratio n = N p /N s , capacitor C 2 , diode D 1 , inductor L 2 , capacitor C 3 , and output load resistance R o . The parasitic components are presented as the direct current resistance r 1 of L 1 and r 2 of L 2 [17]. The equivalent series resistances of used capacitors are very small, so they are neglected here.
The circuit is designed to operate in CCM, and experiences two phases in each switching period: (1) switch turned on and diode turned off; (2) switch turned off and diode turned on (Figures 2 and 3). When S 1 is turned on and D 1 is turned off, the current through L 1 increases and L 1 stores energy; C 1 is discharged and T transmits the energy to the secondary part of the circuit; C 2 is also discharged and the energy is transmitted to the output stage formed by L 2 , C 3 , and R o . When S 1 is turned off and D 1 is turned on, the currents of both L 1 and L 2 decrease, and C 1 and C 2 are charged by the energy stored in L 1 .
By applying Kirchhoff's voltage law and Kirchhoff's current law, the state-space equations are obtained as follows.
Phase 1 (switch turned-on): F I G U R E 1 Circuit diagram of the Ćuk continuous conduction mode (CCM) converter with parasitic components. i 1 is the current through L 1 , i 2 is the current through L 2 ; v 1 is the voltage across C 1 , v 2 is the voltages across C 2 ; v o is the output voltage F I G U R E 2 Current and voltage waveforms of the Ćuk continuous conduction mode (CCM) converter Phase 2 (switch turned-off): where i 1 is the current through L 1 , i 2 is the current through L 2 ; Using the average modelling technique, the averaged model of the Ć uk CCM converter can be obtained as where 0 ≤ u ≤ 1 is the control duty and � i 1 , � i 2 , � v 12 , and � v o are averaged values of i 1 , i 2 , v 12 , and v o , respectively, for one switching period.

| COMBINED FEEDBACK-FEEDFORWARD CONTROLLER DESIGN
The purpose of controlling the Ć uk CCM converter is to To enhance the dynamic response even under large-signal variations from the operating point, the authors propose an LBC. To reduce the burden that this controller imposes, a duty-ratio feedforward controller (DFFC) is also proposed and supplemented in the feedforward loop. Moreover, the global exponential stability of the Ć uk CCM converter system can be ensured. In this section, the feedback plus feedforward control scheme is first derived, then, the stability of this closed-loop control system is analysed.

| Duty-ratio feedforward controller
The desired current of L 1 and L 2 as I 1 and I 2 , the desired current of C 12 as V 12 , and the desired output voltage as V o are first defined. Assuming that the Ć uk CCM converter operates in the steady state, substituting I 1 , ð1 − u f f Þ I 1 n ¼ u f f I 2 ; ð15Þ where u ff is duty-ratio feedforward control input. Substituting Equation (16) into Equations (14) and (15) yields Then substituting Equations (17) and (18) into Equation (13) yields which in turn can be arranged as ð20Þ u ff can then be calculated from Equation (20) as where ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Between the two values of u ff , the one with the plus sign makes the inductor current become too large (u ff > 1). Thus, the following feedforward signal is used u ff predicts the effect of the disturbances and compensates it, thus it significantly improves the transient response. In practice, u ff itself cannot very accurately determine v o of the Ć uk CCM converter because of converter parameters deviation, but it alleviates the burden of the LBC. Therefore, a high gain is not required in the feedback loop.

| Lyapunov function-based feedback controller
In The error dynamics in Equations (26)-(29) can be compactly represented as  : ð34Þ According to Lyapunov's direct method, if a radially unbounded scalar function V(x) is positive definite and V ⋅ ðxÞ is negative definite, then the system is globally asymptotically stable. Moreover, the system becomes exponentially stable if V(x) satisfies a 1 ‖x‖ 2 ≤ V(x) ≤ a 2 ‖x‖ 2 and V ⋅ ðxÞ ≤ −a 3 kxk 2 where c 1 , c 2 and c 3 are strictly positive constants.
If a Lyapunov function is chosen as the energy stored in the Ć uk circuit, the following is obtained where P ¼ diag L 1 = ½ n L 2 C 12 C 3 �, its time derivative can be computed: When the LBC is chosen as where α > 0, the derivative of the Lyapunov function becomes where ðb T PÞ T ðb T PÞ ¼ Thus, we can conclude that the closed-loop system of the Ć uk CCM converter becomes globally exponentially stable.
The complete control input is obtained from Equations (25) and (39) as HAN ET AL. -5 The proposed control scheme consists of two control components: the DFFC term that predicts the effect of the disturbances and compensates it while alleviating the burden imposed by the LBC, and the LBC term that drives the closedloop system to become exponentially stable.
Remark 1 The proposed control strategy requires more sensors to measure the current flowing through the inductor and voltage across the capacitor. However, its control accuracy is very high compared to the control accuracy when conventional controllers are used. In the data centre application, the backup battery unit should supply the constant voltage to the load under the load variation at the steady state. Thus, in the data centre application, the proposed control strategy is needed for the converters, which determine the output voltage of the backup battery unit. Also, the pulsed voltage should be developed to generate the electric field in the electroporation of the skin. In that case, the converter should generate the accurately controlled pulsed voltage for the skin electroporation device, because a very small voltage surge is able to present an electric shock to the patient. Therefore, the proposed control strategy is also required for the converter used in the skin electroporation application.

Parameters
Symbols Value performed using a prototype of the Ć uk CCM converter (Figure 4) implemented on a TMS320F28377D microcontroller. The major parameters and components of the Ć uk CCM converter are listed in Tables 1 and 2. The total system configuration is shown in Figure 5.  To evaluate the performance of the proposed controller, the rising time, settling time, and percentage overshoot (PO) of the transient responses were examined under output voltage variation conditions. Also, the PO and settling time were also measured under output load variation conditions. The proposed controller exhibits the faster response compared with other controllers (Table 3).

| Experiment
Experiments were also carried out to evaluate the transient performance of the Ć uk CCM converter under variations of V o and R o . V o alternated between 15 V and 24 V and R o alternated between 10 Ω to 20 Ω; both patterns ran at 2.5 Hz.
The observed tracking results were similar to the simulated results. Under desired output voltage variations, when VC was used, v o tracked V o rather slowly (Figure 8a). When CC was used, v o tracked V o faster than VC, but the settling time was too long (Figure 8b). Even when the controller gain was increased to improve tracking during transient periods, the transient response was not improved noticeably. When SC was used, v o tracked V o with an acceptably short time (Figure 8c). When the LBC was applied, its global exponential stability allowed v o to track V o well even during the transient period  Experimental results were also analysed in terms of the rising time, settling time, and PO of the transient responses under output voltage and load variation conditions ( Table 4). The proposed controller exhibits a faster response compared with other controllers.

| CONCLUSION
A combined feedback-feedforward control strategy for the Ć uk CCM converter is presented herein. The LBC ensures global exponential stability of the Ć uk CCM converter and thus provides superior transient performance even under large-signal variations; the DFFC is supplemented to predict the effect of the disturbances and compensate it while reducing the burden of LBC, thereby illustrating that the proposed control strategy is suitable for applications such as fast processors and pulsed loads. In the proposed control scheme, the authors used the large-signal averaged model of the Ć uk CCM converter while considering the parasitic components. They conducted numerical simulations and experimental tests to validate the superior performance of the proposed control scheme.  -9