Comparison of first- and second-order sliding-mode controllers for a dc-dc dual active bridge

This paper compares four sliding-mode control strategies for a dual active bridge. The proposed control algorithms define different switching functions resulting in first- and second-order controllers, and using both discontinuous and continuous sliding-mode algorithms. The paper details the design stage and includes simulation and experimental tests to better compare the performance for each control scheme.


I. INTRODUCTION
T HE dual active bridge (DAB) is a power converter widely used in several applications such as microgrids [1], [2], electric vehicles [3], aeronautic field (the so-called More Electric Aircraft, MEA) [4], energy storage systems [5], solid-state transformer in medium-voltage and lowvoltage distribution networks [6], among others. The main features of DAB converters are high power density, bidirectional power flow, galvanic isolation, and the possibility of soft switching [7].
The DAB has two active bridges interconnected with a high-frequency transformer. The model of the DAB converter results in a nonlinear switched dynamical system that mixes two dc stages (input and output) with an ac stage in between due to the magnetic transformer. Usually, the switched model is averaged based on the power flowing between the two ports resulting in first-order nonlinear dynamics for the output voltage. See [8] for a detailed discussion on this behavioural modelling.
From the averaged model, most of the approaches devoted to the control of DAB propose a linearization around a working point and use standard linear techniques such as PI controllers [9], phase compensators [10], or linear observers [11]. Alternatively, other works use nonlinear control strategies, including passivity-based techniques [12], [13], discrete-time linear controllers [14], or the feedback linearization approach [15], [16]. Sliding-mode controllers (SMC) have also been proposed for the control of DAB, including a first-order design [17], the double integral slidingmode control [18], and the super-twisting algorithm [4]. The main properties of SMC are finite-time stabilization, the robustness in front of parametric uncertainties, and the rejection of disturbances. The SMC designs mentioned above, and most of the control designs available in the literature, propose a change of variables increasing the algorithm complexity, or approximations to overcome the strong nonlinearity in the control input function. This paper offers a different alternative: to convert the system into an affine form with the control input via a dynamic extension. From the extended model, four different sliding-mode strategies are compared; the classical First-Order (FO) sliding-mode controller, the Twisting-Algorithm (TA), the Super-Twisting Algorithm (STA), and the Discontinuous Integral Controller (DIC). These strategies include VOLUME X, 20XX both Discontinuous Sliding-Mode Controllers (DSMCs) and Continuous Sliding-Mode Controllers (CSMCs). In DSMCs, such as FO and TA, the control action is switched, implying undesired oscillations (known as the chattering phenomena). The STA and DIC strategies are CSMCs developed to reduce the chattering through the replacement of switching functions by continuous ones but retaining some robustness properties and finite-time convergence.
The remainder of the paper is organized as follows. In Section II, the DAB converter is presented, and the dynamical model is obtained. The sliding-mode controllers are defined in Section III, including a description of the design and tuning. Preliminary simulation results to assess the proposed controllers are shown in Section IV and, Section V include the experimental setup description and tests. The overall comparison among the four controllers can be found in Section VI, and, finally, the conclusions are stated in Section VII. Figure 1 shows a simplified scheme of the DAB converter. It consists of a two-port high frequency transformer with a two full-bridge switches connected to each transformer winding, a dc-voltage source, E, in the primary and a capacitor in the secondary side, ports A and B, respectively. The control goal is regulating the dc-voltage capacitor. Neglecting the magnetizing current of the transformer and considering unity turns ratio, the DAB dynamics can be written as [23]

A. THE DUAL ACTIVE BRIDGE CONVERTER
where i is the transformer current in the secondary, v is the dc input voltage, i L is the load current, L is the equivalent inductance, C is the capacitance of the output capacitor, and r represents the transformer losses. All the parameters are referred to the secondary of the transformer. The gate signals β A , β B are square wave signals with switching frequency, f s , and a certain phase-shift, δ. See Figure 2, where T s = 1 fs . A mathematical description of the gate signals is given by where ω s = 2πf s . The phase shift, δ, between signals β A and β B is used as control input and is updated at every sampled time, T s .

B. AVERAGED MODEL AND DYNAMIC EXTENSION
The system (1) is a switched nonlinear system. The switching frequency, f s , is selected high enough such the effect of the discontinuous signals β A and β B in the capacitor dynamics can be neglected. Thanks to this assumption, averaged models are usually adopted for the control design. Among others, a standard model based on averaging the power flowing through the DAB converter is [8] C dv dt Assuming an static load composed by a resistor and a constant power load (CPL), the load current can be expressed as where R L is the resistance value and P L is the power consumed by the CPL. The system (3) is still nonlinear and non-affine with the control input, δ. In the literature two different alternatives are found to attain the control design: linear approximations, or the inversion of the function δ 1 − |δ| π , see [24] or [16], respectively. In this work, a different approach is proposed consisting in a dynamic extension defined by where u is the new control input. Combining (3), (4) and (5), the overall dynamics results in The control objective is to regulate the average of the capacitor voltage, v, to a desired value, v d , by acting on the control input u in (6b). Therefore, the system has relative degree two.
Proposed control methods depending on the relative degree of the sliding manifold (σ1 has relative degree two; σ2 has relative degree one) and the discontinuous (DSMC) or continuous (CSMC) control actions.

III. SLIDING-MODE CONTROLLERS
When applying SMC for systems of relative degree two, different alternatives exist. A first approach is to define the voltage error as the switching function, i.e., and use a Second-Order Sliding-Mode Controller. Alternatively, the time derivative of the output can be included in the switching function which has relative degree one. For example where the parameter τ > 0 corresponds to the time constant of the error dynamics once the sliding motion is reached. Notice that assuming, σ 2 = 0, the voltage dynamics is easily identified as the first order dynamics The sliding motion in (7) can be enforced applying the Twisting Algorithm (TA) [20] u = −k TA,1 sign σ 1 − k TA,2 signσ 1 .
On the other hand, the classical First-Order Sliding-Mode Control (FO) [19] achieves sliding motion on (8). Both TA and FO result in discontinuous control actions that imply in oscillations related to the chattering phenomena. Alternatively, CSMCs are proposed in the literature to reduce the chattering still preserving robustness properties and finite-time response. For systems of relative degree two, the Discontinuous Integral Controller (DIC) is proposed in [22], and is defined as For systems of relative degree one, the Super-Twisting Algorithm (STA) [21] is a well-known alternative of CSMCs, and takes the form (13b) In this paper the four SMC mentioned above are proposed for the voltage regulation of a DAB. Figure 3 shows how the control methods are classified depending on the relative degree and the continuous or discontinuous properties. The corresponding control schemes of the proposed controllers are depicted in Figure 4.

A. THE TWISTING ALGORITHM
The TA uses the switching function in (7). Using (6), the second time derivative of σ 1 can be written as where Functions φ and γ can be bounded as where R L,min , v min are the minimum values of the load resistance and output voltage, and P L,max is the maximum expected values of the CPL, respectively. Note that δ ∈ − π 2 , π 2 is assumed. From [20], the control parameters must be tuned according to Using the values of the prototype described in Table 1, with R L,min = 9 Ω, P L,max = 108 W, v min = 25 V, and max(|δ|) = 85 • (a value close to π 2 ), the conditions (17a) and (17c) result in

B. THE FIRST-ORDER SLIDING-MODE CONTROLLER
In this section a FO sliding-mode controller is designed. Differentiating (8) with respect to time and using (6) one gets where, γ is given in (15b) and The equivalent control, u eq , is defined as the control input guaranteeing σ 2 = 0 andσ 2 = 0. Hence, from (19), The sliding motion is ensured if the reachability condition, σ 2σ2 < 0, is fulfilled. Using (19) and (21) one can write and replacing the control law (11) hence the sliding motion is guaranteed if The control tuning is based on the first condition. The second condition is ensured since δ ∈ (− π 2 , π 2 ). With the same conditions described at the end of Section III-A with max(|δ|) = 85 • , the sliding motion is guaranteed with k FO > 36.3654.
Notice that large values of k FO imply fastest convergence to σ 2 but higher chattering.

C. THE DISCONTINUOUS INTEGRAL CONTROLLER
Combining (6a)-(6b), with (7) one gets where γ is given in (15b). Assuming small variations of γ, the system (26)-(27) can be seen as a double integrator with a disturbance as requested in [22]. Hence, the DIC with the form (12a)-(12b) stabilize at σ 1 = 0 in finite time. The selection of the DIC gains follows the procedure given in [22].

D. THE SUPER-TWISTING ALGORITHM
Using the time derivative of σ 2 in (19) together with (13a)-(13b) results in Again, assuming small variations of γ, the system (28a)-(28b) matches with the perturbed STA. See the gain selection in [25].

IV. SIMULATION RESULTS
Numerical simulations in the Matlab-Simulink environment have been carried out to test the proposed controllers. The performance of the controllers has been compared under the averaged model (3) and the switched model (1a)-(1b) that contains the sampling period, T s , associated to the switching signals β A , β B . The parameters of the DAB converter correspond to the ones used in Section V and are described in Table 1. The gains of the controllers, shown in Table 2, have
With the initial values of v(0) = 25 V, R L = 18 Ω and P L = 0 W, three tests have been simulated: a reference change to a desired value v d = 30 V, a resistive load change to 9 Ω, and finally, a 108 W CPL is connected (disconnecting the resistive load). The simulation has been run at a fixed step size of 1 · 10 −8 s with the ode4 (Runge-Kutta) solver.
Figures from 5 to 8 show the simulation results. The simulations of the TA exhibits the expected performance when using the averaged model, with finite-time convergence and robustness against load changes, see Figure 5. When the algorithm is implemented with the switched model the voltage oscillates around the regulation point with low frequency. This phenomena appears when the sampling of the gate signals is included in the switched model. Low frequency oscillations have been studied when implementing SMC with the use of the Describing Function method [26]. However, in this case the analysis turns to be more complicated because the nonlinearities of the switched system (1a)-(1b) with (2a)-(2b).
The use of the FO sliding-mode controller is shown in Figure 6. As expected, the output voltage behaves as a first order system, with the desired time constant of τ , see (8). The asymptotic convergence is the main difference between FO and TA when applied to the averaged model. When the FO controller is tested with the switched model, the behaviour remains close to the one exhibit with the averaged model. The small differences and the overshoot in the response are due to the sample in time of β A , β B . The chattering phenomena when using both TA and FO schemes becomes masked by the intrinsic switching of signals β A , β B .
The third test evaluates the DIC. The tuning procedure for this algorithm is complicated and the best performance obtained is shown in Figure 7, with a good rejection of load changes but a settling time of 20 ms. Moreover, the algorithm does not stabilize when is tested with the switched model.
Finally, the behaviour of the STA is shown in Figure 8.
Similarly to the FO strategy, also based on σ 2 , the voltage is regulated with a first-order response with the desired time constant. The controller provides robustness against changes of resistive loads and CPLs. The behaviour is similar when it is applied to the averaged model and the switched model.

V. EXPERIMENTAL RESULTS
The experimental prototype consists of an isolated dc-dc DAB converter with a high-frequency transformer having a unity turns ratio. The parameters of the DAB converter are shown in Table 1. The control strategies were implemented in a TMS320F28377D floating point DSP of Texas Instrument. The power switches used in the converter are IRFP260 MOSFETs. The CPL consists of a dc-dc buck converter with a resistive load of 4.5 Ω. The output voltage is controlled with 22V achieving a constant power of 108 W. Figure 9 shows a picture of the experimental prototype. The same tests performed for the numerical simulations have been carried out experimentally, for the TA, FO and STA control approaches. As predicted in the simulation stage, the DIC scheme does not stabilize the voltage and it is not included in this section. The gains for all control strategies are the same as the simulation tests given by Table 2. Figure 10 shows the performance of the output voltage, v, and the phase-shift control signal, δ, for the TA scheme. Similarly to the simulation tests with the switched model, the sampling related to the gate signals result in low-frequency oscillations. The amplitude of the oscillations is smaller than the one obtained in the simulations because of unmodelled losses in the DAB components.
The experimental results with the FO controller are presented in Figure 11. In the first plot (left) can be observed how the output voltage, starting at 25 V, reaches the new reference value of 30 V in the desired time (2 ms) following a first-order response, as designed in (9). The second plot (centre) shows the output voltage when the resistive load changes from 18 Ω to 9 Ω. After a short transient, the voltage recovers the desired value. Finally, the third plot (right) confirms the robustness of the FO algorithm in face of CPL connection. Notice that the simulations performed   with the switched model in Section IV are close to the results with the prototype. The overshoot observed in the numerical simulations is unnoticed in the experimental tests because of the unmodelled losses. Figure 12 shows the results with the STA. The obtained performance is similar to the one obtained for the FO scheme, and also matches with the simulation tests. Is worth noticing that the STA rejects better the load changes than the FO algorithm, see centre and left plots. Similar waveforms for the current and voltage are obtained using TA and STA, for this reason they are not included in this section. Figure 13 shows a detail of the high-frequency transformer current and output voltage for the FO sliding-mode controller. The reference voltage is 30 V and the system is feeding the CPL. It can be observed that in steady-state the output voltage is regulated on the reference value and the high-frequency transformer current has a mean value equal to zero which prevents the magnetic saturation of the transformer.
Finally, the computational burden of each control scheme is compared. Table 3 shows the execution time of each control routine. Thanks to the simplicity, FO spends less time than TA and STA, using less CPU resources.

VI. EVALUATION DISCUSSION
The comparison of the controllers is performed according to the following indicators: • Stability accuracy: indicates how accurate is the stability proof concerning the assumptions made along the VOLUME X, 20XX   in the spider chart in Figure 14. The conclusion is that the FO and STA are suitable for the control of a DAB, with similar performance. The FO turns to be easiest in the implementation and the gain tuning stage.

VII. CONCLUSIONS
Four control schemes based on sliding-modes have been proposed for a DAB. A significant difference compared to other control algorithms available in the literature is the use of a dynamic extension to overcome the structural problem of the nonlinear function of the input and redefine the control system affine with the control input. In contrast with other approaches, this alternative does not approximate (or linearize) the function nor use a change of variables requiring more computational resources. The four alternatives arise from the definition of two different switching functions (resulting in first-and secondorder sliding-mode strategies) combined with discontinuous and continuous approaches. All the algorithms have been tested through numerical simulations, and three of them have been experimentally validated.
From a practical point of view, the FO and STA schemes offer excellent regulation performance and robustness properties. In opposite, the second-order approaches are more sensitive to the sampling and discretization stages necessary for the implementation, resulting in undesired oscillations (TA) and instability (DIC).