Design of a Nonlinear Integral Terminal Sliding Mode Controller for a PEM Fuel Cell Based on a DC-DC Boost Converter

This paper proposes a robust nonlinear controller for a proton exchange membrane fuel cell coupled with a DC-DC boost converter. The key feature is to maintain the desired reference output voltage despite significant disturbances while improving transient stability. In order to do this, a nonlinear robust integral terminal sliding mode controller (ITSMC) is proposed, along with the derivation of the control law, explanation of the reachability analysis, and stability condition. An adapted integral sliding surface is used to capture the dynamics caused by variations in the input voltage and loads. In addition, a modified reaching law is proposed to guarantee the finite time convergence while reducing the chattering. Afterward, the Lyapunov control theory is employed to evaluate the DC-DC boost converter’s large-signal stability while guaranteeing the resilience of the proposed controller. Finally, the applicability of the proposed controller is evaluated through comprehensive analyses on both the simulation platform and the real-time processor-in-loop (PIL) platform under various operating conditions, such as the variation in load resistance, reference output voltage, etc. All of the results, when compared to an existing integral terminal sliding mode controller, indicate quick reference tracking capability with reduced overshoots and robustness against disturbances.

challenging task [10], [11]. To address such a difficulty while 56 improving the performance of the DC-DC boost converter in 57 FC applications, it is required to include a suitable controller 58 with it [12]. 59 The output voltage obtained from a DC-DC boost con-60 verter is commonly controlled using proportional-integral-61 derivative (PID) and a few additional linear controllers that 62 use other approaches [1]. Linear controllers, on the other 63 hand, have a relatively limited operating band since they often 64 employ linearized converter models with a single working 65 point. Furthermore, some of these strategies do not neces-66 sitate the use of a model in which the gains are manually 67 or automatically tweaked. As a result, even with a slight 68 step change in the load, these controllers do not achieve 69 the necessary performance [13]. Basically, the switching 70 states of the boost converter give it a nonlinear shape, which 71 makes it difficult to implement in linear controllers [14]. 72 Therefore, nonlinear controllers are essential as these con-73 trollers can provide satisfactory results by overcoming all 74 of the constraints of linear controllers. Fuzzy logic con-75 trollers [15], feedback linearization controllers [5], backstep-76 ping controllers [9], and sliding mode controllers [13] are 77 some examples of nonlinear controllers for FC applications. 78 For its simplicity and robustness, the data-driven fuzzy-79 logic controller (FLC) is widely used [15]. In [16], a FLC 80 based on adaptive law is used for a DC-DC step up con-81 verter in applications of PEMFCs. However, a significant 82 drawback of FLCs is that the computational requirements 83 have to be incremented for the fuzzy set growth when com-84 plex strategies are used during their implementation [5]. 85 Again, the accuracy of the fuzzy system is compromised  ric uncertainties is proposed in [10]. However, the main 96 limitation of this adaptive approach is that it cannot ensure 97 robustness against state-dependent uncertainties [11]. 98 As a result, the nonlinear backstepping controller (NBC) 99 might be utilized to overcome the disadvantages of FBLCs 100 since it can provide satisfactory performance without requir-101 ing precise knowledge of the system's parameters, as men-102 tioned in [17]. Ref. [18] discusses a NBC strategy for ensuring 103 the system's dynamic stability. Though this approach can 104 be used to achieve the desired performance, the derivative 105 of a stabilizing function creates some complexities, espe-106 cially for higher order systems, while overcoming the FBLC's 107 limitations. Furthermore, if the user-defined constants are 108 not appropriately selected, the system's response could be 109 delayed [19].

110
For mitigating the aforementioned limitations, the slid-111 ing mode controller (SMC) is the most operative approach 112 due to its stability, robustness, better energetic response, 113 and high compatibility with the innate switching nature of 114 power converters [20], [21]. Furthermore, as discussed in [22] 115 and [23], an SMC scheme can provide a promising solution 116 for rejecting disturbances and compensating for uncertainty. 117 An SMC approach of dual loop is presented in [24] where 118 the output side voltage is controlled for a fixed input side 119 voltage to establish the asymptotic stability of closed-loop 120 systems. But the controller design is not easy for any dual 121 loop control structure. In association with the FC, a similar 122 type of controller is proposed in [25], but chattering prob-123 lems are limiting its application practically. To reduce the 124 chattering problem, higher order SMCs are initiated in [26] 125 and [27]. Though these controllers can enhance the output 126 voltage regulation of DC-DC boost converters in the case of 127 FC applications, the tracking error of steady-state cannot be 128 fully diminished. For this reason, a robust SMC scheme is 129 required that not only improves the transient performance of 130 the system but also eliminates the steady-state tracking error. 131 It is well-known that in the SMC technique, it is required to 132 find a control law conditional on a particular sliding surface 133 that changes the dynamics of the system and ensures the 134 asymptotic convergence. But the equilibrium point cannot be 135 gained in a finite time if a conventional reaching law is used, 136 which is challenging for this strategy, Consequently, to solve 137 this challenge and to obtain a finite-time response, a terminal 138 SMC (TSMC) was introduced in [28]. But still, this controller 139 gives a slower response at a distant place from the origin and 140 it terminates with an unbounded control signal due to singu-141 larities found. An integral TSMC (ITSMC) can improve the 142 convergence time as well as enhance the chattering reduction 143 stuff and the dynamics of the overall system as discussed in 144 [29] and [30].

145
Being motivated from the above discussion, the main 146 objective of this paper is to design an ITSMC that is based 147 on a new reaching law to regulate the output voltage of 148 the boost converter. The overall stability and the robust-149 ness of the system are proved by the help of control Lya-150 punov functions (CLFs) at the end of the design procedure. 151 Finally, the proposed controller has been implemented on a 152 MATLAB/SIMULINK platform to assess its performance in 153 the effectiveness of the proposed control strategy in practice, 155 the controller performance is also tested on a processor-in-156 loop (PIL) platform. Basing on the aforementioned analysis, 157 the key offerings of this paper can be described as follows:  and activation polarization. Moreover, before developing the 196 PEMFC model, a few essential presumptions would be taken 197 into account, which are discussed in [32]. According to the 198 assumptions as discussed in [32], the PEMFC equivalent elec-199 trical circuit can be represented by Fig. 1 (a). From Fig. 1 (a), 200 the cell voltage of the PEMFC might be written as follows [1], 201 [33]: where T is the temperature, P H 2 is the potential pressure of 213 hydrogen, and P O 2 is the potential pressure of oxygen.

214
The concentration voltage drop, V con can be expressed as 215 follows: where F, n, R u , i L , I FC and A are used to represent Fara-218 day's constant, number of electrons, constant for the universal 219 gas, limiting current, cell current, and area of the active cell, 220 respectively.

221
The ohmic voltage drop, V ohmic can be written as follows: 222 where R ohmic is the ohmic resistance. Finally, the activation 224 voltage drop, V act can be expressed as follows [35]: where i with i=1, 2, 3, 4 represents the cell's parametric 227 coefficient and C o 2 is the dissolved oxygen concentration.

228
At this stage, the FCs output voltage can be calculated 229 by combining all of the preceding equations. However, the 230 single cell's output voltage of a PEMFC is extremely low. So, 231 to boost the output voltage, numerous FCs must be linked to 232 a bipolar plate. Therefore, the stack voltage of a PEMFC can 233 be expressed as follows: where n FC represents the total number of series connected 236 single FCs.

237
As discussed in [36], if the operating frequency of the 238 stack reaches to 10 kHz, then R act and R conc can be omit-239 ted. Consequently, it could be viewed as a pure and constant 240 resistance, which would then simplify the proposed model to 241 a DC voltage source and an impedance of resistive nature, 242 as illustrated in Fig. 1 (b). For the convenience expression, 243 a voltage source, as indicated in Fig. 1 (c), might be utilized 244 VOLUME 10,2022 to replace the structure in Fig. 1 (b). Based on this description, it is clear that the input voltage of a DC-DC boost converter 246 follows the same dynamics as the PEMFC. 247 An appropriate controller with that of a DC-DC converter 248 might be installed in the real application to increase the 249 PEMFC voltage from lower to a higher and stable DC-bus 250 voltage while staying within the conversion capacity limit.

251
The performance of the current of PEMFC and the DC bus   When the switch is OFF: When the switch is ON: where V o is the average voltage obtained at the output, I L is 284 the current flowing through the inductor and u {0, 1} is the 285 control input of the converter. Therefore, the entire dynamical 286 average model can be represented as follows: Eqs. (12), (13) reflect the complete propellant paradigm of a 290 DC-DC boost converter, and the suggested controller will be 291 designed conditioned on this model in the next section.

293
The control law (u) must be designed to gain quicker transient 294 response, tighter output voltage regulation, and resilience 295 with fewer steady state tracking errors, which is the main goal 296 of this section. The output voltage of the boost converter must 297 be controlled by the tracking of the inductor current. Before 298 continuing further with the design method, the tracking error 299 must be identified. The duty cycle is adjusted to make the 300 output voltage of the converter equal to the reference voltage. 301 To achieve the final control law (u), two steps must be fol-302 lowed according to the controller design technique, which is 303 detailed in the following.

305
The initial step of this sub-section is to select a stable sliding 306 surface that is aimed to meet the system's requirements. Fol-307 lowing the objectives of the design procedure, the tracking 308 error of the inductor current can be expressed as follows: where I L(ref ) is the value representing the reference of the 311 inductor current which is calculated using the following 312 formula: Now, along with the tracking error, the proposed nonlinear 319 integral terminal sliding surface might be defined as follows: 320 where α, β > 0, and 0 < γ < 1 are the tuning parameters 322 that are required to control the speed of convergence of the 323 controller. It is worth mentioning that when the frequency 324 is extremely high or infinite, it is feasible to achieve the 325 required control target by simply considering the first term. 326 For a constant frequency, however, this is not practicable 327 since steady-state tracking errors in the inductor current and 328 output voltage will remain. As a result, the integral terminal 329 term is examined in this study to enhance the steady-state 330 tracking error. Using Eq. (16), the derivative is expressed as 331 follows: In the second step, a control law (u) is required to be found to 335 bring the sliding surface into finite time convergence. It is 336 well-known that in the SMC approach, the overall control 337 consists of two parts, which are written as follows: where u eq is the equivalent part of control law and u rl is the 340 reaching law. To obtain u eq , it is obvious to setṠ = 0, i.e., From Eq. (20), the equivalent control law can be written 343 as: 345 However, to ensure the overall system's stability an improved 346 reaching law u rl is incorporated with the original control law 347 which is written as follows: , and (22), the overall control law is Once the system state trajectory reaches the sliding surface, 373 the sliding surface design is such that the system state trajec-374 tory will be reaching the equilibrium point in a finite time. 375 Now, the finite time of convergence, t will be determined in 376 this section by considering the nonzero initial state S(0).

377
By considering the first terms of Eq. (24), it can be detailed 378 as After doing the mathematical manipulation, the required time 381 t 1 can be detailed as follows: Similarly, for the second term of Eq. (24), the required time 384 t 2 can be detailed as follows: Now, by combining Eqs. (29) and (30), the total finite time of 387 convergence (t) is expressed as follows: Thus, from Eq. (31), it is confirmed that both the tracking 391 error and the derivative of it will converge to zero in a finite 392 time. In the next section, the efficacy of the designed con-393 troller is analyzed.

396
This section includes both simulation and experimental 397 processor-in-loop (PIL) studies to evaluate the efficacy of the 398 designed controller. The simulation investigation has been 399 carried out on the MATLAB/Simulink platform. The imple-400 mentation block diagram in Fig. 3 depicts the overall simula-401 tion structure in which the FC output is provided as input to 402 the DC-DC boost converter. Afterward, the sliding surface is 403 determined using the system's output. Finally, the actual con-404 trol signal is computed and supplied back via the PWM to the 405 DC-DC boost converter switch. The simulation settings for 406 the designed nonlinear controller are selected by prioritizing 407 future prototype development while keeping practical situa-408 tions in mind. As a result, the system and controller param-409 eters utilized in this simulation are provided in Table 1 and 410 Table 2 respectively. For convenience, the controller param-411 eters are chosen basing on the trial and error methodology to 412 fulfil the cherished control objectives.

413
By using these parameters, the final control law (u) is eval-414 uated that is to be applied to the converter's switch through a 415 pulse width modulator (PWM) where 100 kHz is chosen as 416 the switching frequency. To test the ability of output voltage 417 tracking and to assess the proposed controller's performance, 418   it is compared with an existing integral terminal sliding mode 419 controller (ITSMC) as presented in [29] by considering three 420 cases. These cases are listed as follows:   The cases are elaborately discussed in the following. In the study of this case, the performance of the designed 434 controller is validated by varying the load resistance. The load 435 resistance is varied according to the following mathematical 436 equation: The corresponding response is presented in Fig. 4. From 439 the above shown diagrams, it can be stated that the first 440 transient response occurs at a time period of 0.2 s and the 441 following occurs at a time period of 0.6 s. The red-colored 442 curve indicates the controller performance of the existing 443 one, while the black-colored curve indicates the performance 444 measure of the proposed controller. From Fig. 4, it can be 445 seen that the initial evaluation has started from a time period 446 of 0.05 s as priority is given to the two transient conditions 447 and has ended after 1 s. From Fig. 4 (a), it is also observed 448 that the proposed controller exhibits an overshoot of 1.57%, 449 an undershoot of 15.75%, and a settling time of 0.041s, while 450 the existing one shows an overshoot of 2.33%, an undershoot 451 of 16.67%, and a settling time of 0.043 s at an initial transient 452 time of 0.2 s. Similarly, an overshoot, undershoot, and settling 453 time of 18.17%, 7.13%, and 0.096s, respectively, is observed 454 for the proposed controller at a second transient time of 0.6 s, 455 while 19.25%, 9.13%, and 0.122 s of overshoot, undershoot, 456 and settling time are observed, respectively for the existing 457 controller. Fig. 4 (b) represents the inductor current response 458 versus time (s). This curve also represents the same type of 459 result at transient conditions as that of the response curve of 460 output voltage. So, the above mentioned observations clearly 461 shows the supremacy of the proposed controller design with 462 respect to the existing one. Table 3 is representing the quan-463 titative analysis of the voltage for Case I.            (as illustrated in Fig. 7). (Rasberry Pi 3B Quad-Core 64-bit 548 Microprocessor Development Board). The control signal gen-549 erated from the processor is feedback to the Simulink plat-550 form, which indicates that the control signal is generated on 551 a real-time platform where an analog signal is received by 552 the microprocessor board. Thus, a suitable control signal is 553 generated on the development board by a combination of ref-554 erence values created in real time on the platform and actual 555 board values. This analog control signal is then transferred as 556 the input of the switching pulse generator. Fig. 7 depicts an 557 Ethernet cable that sends and receives data between the simu-558 lator and the processor. It is noteworthy that a similar system 559 as that of the simulation study is used here with the only 560 difference being the implementation of the controller in the 561 real-time environment. Here, analysis is performed to show 562 the robustness of both the controllers under the variations of 563 reference output voltage as described in case-III of the earlier 564 discussed subsection.

565
The output voltage response curve in Fig. 8 indicates that in 566 the PIL platform, the proposed ITSMC controller has neither 567 overshoot or undershoot and has a faster settling time than 568 the existing controller when both transient times of 0.35 s 569 and 0.7 s are taken into account. This experimental setup and 570 approach can demonstrate the developed controller's real-571 time performance improvement in all circumstances. Finally, 572 the proposed controller provides a better dynamic response 573 than the existing one.

575
In this paper, a nonlinear robust integral terminal sliding 576 mode controller (ITSMC) for a DC-DC boost converter 577