Active disturbance rejection temperature control of open-cathode proton exchange membrane fuel cell

Applied Energy 261 (2020) 114381

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Active disturbance rejection temperature control of open-cathode proton exchange membrane fuel cell Li Suna, Yuhui Jina, Fengqi Youb, a b

T

⁎

Key Lab of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, United States

H I GH L IG H T S

uncertainties are lumped and compensated by real-time estimation. • Various proof is given to guarantee the control performance against uncertainties. • Rigorous • Experiments are given to show nonlinearity and validate the controller efficiency.

A R T I C LE I N FO

A B S T R A C T

Keywords: PEMFC Nonlinearity analysis Extended state observer ADRC

Open-cathode proton exchange membrane fuel cell (PEMFC) is promising in small-scale power generation due to its compact channel that integrates air supply and coolant flow. The operational temperature is significant for the safety and efficiency, but the temperature control is challenging due to the various uncertainties resulted from model inaccuracy and unexpected disturbances. To this end, the uncertainties and disturbances are lumped as a unified item, which is then augmented as an extended state to the original system. The extended uncertain state is estimated via the real-time input-output data and then compensated by active disturbance rejection control (ADRC). A series of linear models are identified via step response tests, showing the strong nonlinearity. Besides, PI and ADRC controllers are respectively designed based on the nominal linear model. The performance guarantee of ADRC is theoretically proved under uncertainty. Extensive simulations of the proposed models demonstrate the uncertainty compensation ability of ADRC. Finally, practical tests on a 300 W PEMFC experimental bench show that the proposed ADRC method has the obvious advantage over the conventional PI controller in both tracking and regulation performances.

1. Introduction With the ever-increasing concern on the environmental and climate issues, past years witnessed the booming development of the renewable energy power generation around the world, such as solar, wind, geothermal, and so on [1]. However, the increasing integration of intermittent renewable resources in the energy system introduces inevitable uncertainties that may result in significant challenges to the stable operations of the energy system, especially for the small-scale residential applications [2,3]. Hydrogen, emerging as a secondary energy sources, is considered as potential candidate medium for energy storage [4,5], thanks to its diverse production sources such as biomass direct gasification [6] and water electrolysis from redundant renewable energy [7]. Fuel cell is considered as a disruptive technology in directly converting the chemical energy stored in hydrogen to electrical energy

⁎

via electrochemical reaction, in which no moving part is necessary. Compared to the conventional fossil fuel power generation, fuel cell is superior in terms of high efficiency, low noise, zero emission [8,9] and controllable power output in compensating the renewable intermittency [10]. Among various fuel cell types, proton exchange membrane fuel cell (PEMFC), featured by the low-temperature operation [11,12], has many successful applications in both mobile vehicles [13] and stationary systems [14], demonstrating its technology reliability and maturity [15]. For some small-scale application scenarios (less than 5 kW), open-cathode PEMFC is extremely appealing because of its compact structure by integrating the fan channel to supply the cathode oxygen and to cool down the stack simultaneously [16]. Although promising, some obstacles still impede the further commercialization of PEMFC, among which the stability of stack temperature is a critical concern [17]. A frequent fluctuation of temperature

Corresponding author. E-mail address: [email protected] (F. You).

https://doi.org/10.1016/j.apenergy.2019.114381 Received 7 September 2019; Received in revised form 24 November 2019; Accepted 14 December 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Recently, ADRC was also successfully used in both voltage control [37] and oxygen excess ratio control [38] of fuel cell, showing superiority over conventional methods via simulation. A drawback of the existing literature on ADRC is that the expected closed-loop performance is derived in an approximate manner based on the ideal compensation assumption. This paper presents a rigorous theoretical analysis by studying the positive invariant set of the closed-loop system under nonlinear unknown dynamics. It is shown that ADRC can ensure both tracking error and estimation error that are bounded under any given size of nonlinear uncertainties. Furthermore, the tacking error during both transient and steady state can be governed by ADRC’s parameters. The motivations of using the data-driven ADRC for PEMFC temperature control rather than the advanced model-based control systems are listed as follows,

would give rise to thermal fatigue to the membrane layers and thus shorten the cell-life expectancy [18]. Moreover, in the case of combined heat and power (CHP) applications, a rapid temperature tracking ability is necessary in order to optimize the economic performance and maintain the highest efficiency [19]. Specifically, for an open cathode fuel cell stack, whose fan plays dual roles of oxygen supply and heat dissipation, the temperature set point indicates the tradeoffs between the operational efficiency and heat quality [20]. Several significant steps were carried out towards efficient PEMFC temperature regulation. Based on the coolant circuit physical modelling, a fuzzy control strategy [21] was introduced to adaptively handle the load current disturbance. A first-principle based semi-physical model [22] was developed based on which three comparative control strategies are designed to regulate the operating temperature within a certain range robustly against various operation conditions. A Lyapunov based adaptive control law [23] was developed to overcome the system nonlinearity and achieve a guaranteed performance with prescribed target. Model predictive control [24] was also utilized to address the current constraint with consideration of limited cooling capacity. Although promising, these model-based methods are, however, computationally sophisticated and difficult to be programmed and realized via short and simple codes of the microcontroller or other tiny chips [25]. Thus, experimental verification is mostly absent in the PEMFC control studies because of the implementation difficulty. On the other hand, PEMFC is usually utilized in a compact environment, such as the mobile vehicle and residential building, which might not have sufficient and cost-effective computational power for the model-based sophisticated computation. PEMFC temperature control is challenging primarily because of the inevitable uncertainties, such as hypothesis simplification, model linearization, time-varying parameter and external disturbances. The conventional model-based controller is usually designed based on a nominal linear model, which is approximately linearized from an essential nonlinear description under a certain operating condition. Thus, the control performance may deteriorate when the condition changes. The nonlinearity and unmodeled dynamics will make the control performance perturb away from the desired set point, and potentially lose stability [26]. Besides, the dynamics of fuel cell stack would vary over time, which results in additional uncertainties. Moreover, the frequent external disturbances such as ambient temperature change and load demand shifting may inevitably cause temperature fluctuation. Additional difficulty comes from the fact that the response from the cooling fan (control input) to the controlled temperature is much slower than that from the load disturbance to temperature [27]. Therefore, a modelbased controller perturbing from the designed condition may not be able to deal with the above uncertainties, resulting in large overshoot, sluggish response or oscillation. To sum up, it necessitates an alternative approach from the data-driven perspective to handle the uncertainties with acceptable online computation. Active disturbance rejection control (ADRC), inherited from the conventional error-driven proportional-integral-differential (PID) controller [28], is emerging as a promising data-driven method without requiring a priori accurate model. Under the ADRC framework, all the uncertainties, including model uncertainties and unmeasurable disturbances, are lumped as a unified term, which is expected to be estimated and compensated by analyzing the real-time input-output data [29]. Being able to compensate the uncertainties promptly, the resulting tracking performance can be guaranteed as expected in spite of the perturbations of the operating conditions. Besides, the operational fluctuation can be greatly reduced by timely responding to the external disturbance. It is revealed that ADRC is actually of two-degrees-offreedom structure that is able to achieve satisfactory set-point tracking and disturbance rejection performances simultaneously [30]. The growing applications of ADRC in both academia and industry, for example, in motion control [31–33] and process control [34–36], can be attributed to its simple structure, ease of tuning and strong robustness.

• It could be expensive and time-consuming to obtain an accurate • •

model that is able to describe the global dynamics of the PEMFC temperature, because of the sophisticated electrochemical mechanisms. Both PEMFC applications in mobile vehicle and home generation require the microcontroller hardware to be installed in a compact environment. On this regard, the light computational demand and ease of coding makes ADRC more appealing. ADRC only has two parameters, and its parameter tuning/optimization is simple thanks to their clear physical meanings related to the frequency bandwidth.

The above features make ADRC an engineering-friendly solution that is particularly compatible with the practical requirements of the PEMFC applications. It is able to give a considerable performance improvement by introducing little complexity to the software and hardware implementation. The novel contributions of this work include:

• Various uncertainties, such as nonlinearity, parametric perturbation • •

and external load disturbance, are treated as a lumped term and compensated via real-time estimation. A theoretic analysis gives a performance guarantee condition that is able to make the temperature response sufficiently insensitive to the uncertainties. Experimental results are given to identify the models for simulation and to verify the superiority of the proposed method.

2. System description and data-driven identification 2.1. Fuel cell system description A fuel cell consists of an electrolyte sandwiched between two electrodes. A unique feature of PEMFC is the special proton-conducting polymer electrolyte, which only allows positive ions (protons) to pass but block the electrons. The overall reaction to convert hydrogen chemical energy into electricity is given by, (1)

2H2 + O2 → 2H2 O

Several layers of cells piled up into a fuel cell stack. Under the ideal condition, all reversible work to move the hydrogen ions around the circuit can be converted to electricity. The reversible voltage of each cell can be derived from Nernst formulation [39]:

E=

−Δgf 2F

=

−Δgf0 2F

1

⎡ p H pO2 ⎤ ln ⎢ 2 2 ⎥ + 2F ⎢ p H2 O ⎥ ⎣ ⎦ ¯ fc RT

(2)

where Δgf is the change of Gibbs free energy at operational pressure while Δgf0 represents the one at standard pressure (1 bar); F is the Faraday constant (=96485 Coulombs) which represents electric charge 2

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̇ is the heat produced by heat flows with the gas flow, respectively; Qrea the electrochemical reaction, due to various irreversibility in Eq. (3), which can be determined as the difference between the higher heating value (HHV) of hydrogen and the usable electrical power output. The ̇ natural heat loss Qamb is caused by the heat convection and radiation to ̇ by the the ambient environment. The enforced heat removal Qcool cooling fan can be determined by the air speed vair and the temperature difference between the stack temperature Ts and the ambient temperature Tamb ,

of one mole of electrons; R¯ is the universal gas constant, 8.31451 J/(kg·K) ; Tfc is the fuel cell temperature in Kelvin; and p H2 , pO2 and p H2 O respectively expressed the partial pressure of the hydrogen, oxygen and vapor in bar. However, multiple irreversible factors inevitably bring losses and theoretical voltage drop. The ultimate stack voltage is given as

vst = n (E − vact − vohm − vconc )

(3)

where n is the number of layers, vact , vohm, vconc are respectively the activation loss, ohmic loss and concentration loss, which vary under different operating conditions. It follows Eq. (2) that the stack voltage is closely related to the operating temperature, which normally ranges from 40 °C to 100 °C [39]. If the temperature is too low, it could decrease the voltage as well as the electrical efficiency because of the slow electrochemical reaction rate. On the other hand, a high temperature could have serious consequences, such as the degradation of transport effect [40] or irreversible membrane damage [41]. In addition, a frequent temperature fluctuation will cause severe thermal stress, thus impairing the membrane permanently. Therefore, stable temperature with a suitable set point is a prerequisite for efficient and safe operations. A typical open-cathode PEMFC experimental platform is built, and its schematic diagram is shown in Fig. 1. Besides the fuel cell stack working as the power source, some other auxiliary components are indispensable, such as air compressor, cooling module and DC/AC inverter. An electronic load is used to give the power changing disturbance. Anode hydrogen gas is supplied from the hydrogen tank while the cathode oxygen is sucked from the ambient air by the axial-flow fans. The air flow is usually sufficient for electrochemical breathing [42]. Thus, the flow rate is primarily regulated to maintain the stack temperature operating at a desired set-point. The thermal model is governed by the energy balance between the heat sources and sinks [43],

̇ + Qrea ̇ − Qcool ̇ − Qamb ̇ dTS Q̇ − Qout −P = in dt ms cp, s

̇ = ρait Ainlet vair Cp, air (Ts − Tamb) Qcool

(5)

where ρair is the ambient air density, Ainlet is the cross-sectional area of the inlet manifold and Cp, air is the heat capacity. Details of the mechanistic model are available in the literature [44]. The air speed vair is determined by the duty-cycle of pulse-width-modulation (PWM) signal given by the controller, which is the original manipulated input (u) to regulate the stack temperature output (y), in the presence of load disturbance (d). 2.2. Data-driven identification As discussed above, the temperature of PEMFC is influenced by many factors, such as the operating condition, ambient environment and the power output. It is analyzed [44] that the accurate mechanistic model Eq. (4) is challenging to obtain, because it needs to accurately estimate many geometric and physical property parameters. Moreover, the temperature is highly intertwined with the liquid formation and transportation, leading to a strong nonlinearity of PEMFC stack temperature in terms of the operating conditions. To avoid the complicated first-principle modelling, this paper utilizes the data-driven identification to obtain a series of typical transfer function models in terms of the high (270 W), medium (240 W) and low (194 W) power conditions, as well as a consecutive change of PWM duty cycle. Three sets of data are modest to describe the feature dynamics of PEMFC within wide-range of operations. For higher accuracy and integrity, more power condition nodes are necessary for experimentation. In a practical application, linear model of nominal operating condition will be used for controller design, and the mitigation of the model variations against different operating conditions is left to the

(4)

where Ts denotes the temperature of fuel cell stack. The variable ms is the stack mass and cp, s is the specific heat capacity, which usually needs ̇ , Qout ̇ represent the inlet and outlet to be determined experimentally. Qin

Fig. 1. Schematic diagram of a PEMFC experimental platform. 3

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36

38

34

36

Temperature (°C)

Temperature (°C)

L. Sun, et al.

32 30 28 26

75

Duty Cycle (%)

Duty Cycle (%)

32 30 28 26 90

90

60 45 30 15

34

0

100

200

300

400

80 70 60 50 40 30

500

0

Time (s)

100

200

300

400

500

Time (s)

Fig. 2. Open-loop step response at 270 W.

Fig. 4. Open-loop step response at 194 W.

duty of the data-driven feedback control. The open-loop temperature step responses are shown in Figs. 2–4, where the PWM duty cycle changes along the path from 30–50% to 70–90% position. The transfer functions under nine operation conditions are identified in Table 1. Fig. 5 takes the operating condition (2) as an example, showing that the model output agrees well with the experimental validation results. These experiments demonstrate that the temperature response changes significantly with the operating power conditions. Moreover, even under the same operating power condition, the step responses may vary under different fan speeds. These differences can be attributed to the inherent nonlinearity. To compromise, the medium transfer function G5 is used as the nominal model for controller design.

Table 1 Identified transfer functions at different stack power rates and manipulating duty cycles. Duty Cycle (%)

270 W

30–50

G1 (s ) =

50–70

G2 (s ) =

70–90

G3 (s ) =

204 W −0.1949 1 + 43.38s −0.1388 1 + 38.47s −0.0339 1 + 43.51s

194 W

G4 (s ) = G5 (s ) =

G6 (s ) =

−0.2334 1 + 80.66s −0.1639 1 + 123.10s −0.0700 1 + 76.21s

G 7 (s ) =

G8 (s ) =

G9 (s ) =

−0.1924 1 + 47.08s −0.1428 1 + 38.50s −0.03690 1 + 40.05s

0.0 -0.5

2.3. Problem statement

Temperature (°C)

-1.0

As mentioned above, the difficulties of PEMFC stack temperature control can be attributed to the model simplification, systems nonlinearity, time-varying parameters and frequent external disturbances. The stack temperature model is usually described by a group of multivariable nonlinear differential equations [45]. Thus, reduced order modelling can be used for controller design. However, over-simplification missing key information will reduce the model accuracy and efficacy, while a conservative one will perplex the controller design and

Temperature (°C)

-1.5 -2.0 -2.5 -3.0

40

-3.5

38

0

25

50

75

100

125

150

175

200

Time (s)

36 34

Fig. 5. Comparison between the change of the experimental and model outputs when increasing duty cycle from 50% to 70% under 270 W.

32 30 28

parameter tuning. In this paper, a lumped-parameter first-order temperature model described by Eq. (4) is used by concentrating on the temperature dynamics and ignoring the liquid coupling and the uneven temperature distribution, for controller deign simplicity. Although such non-dominant details omitted, the model is still challenging for controller design because of the unmeasurable time-varying heat flows. Besides, the time-varying characteristics of fuel cell will also lead to model perturbation. Even under the same ambient temperature with the same operating condition, the former temperature experimental results may not be duplicated. The difficulty in modelling leads to challenges in the design of model-based controllers. The frequent and undetectable external disturbances, such as ambient temperature, load demand and inlet humidity, may cause

26 90

Duty Cycle (%)

Experimental Result Identified Result

80 70 60 50 40 30 0

100

200

300

400

500

600

700

800

Time (s) Fig. 3. Open-loop step response at 240 W. 4

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0.00

uncertain environment requires a new paradigm. 3. Data-driven controller design

Temperature (°C)

-0.05

G1 G2 G3

-0.10

G4 G5 G6

Based on the model set identified in Table 1, there are two mainstream approaches to efficient control. On the one hand, model-based advanced control methods, such as model predictive control (MPC) [47], received widespread attention because of its superiority in achieving multivariable constrained dynamic optimization. Although powerful, the computational complexity usually requires an independent high-performance computer to calculate the real-time optimal control decisions. Thus, it usually serves in the supervisory level [48] to achieve plant-wide optimization. For some single-loop applications, the benefits brought by MPC should be carefully justified for the additional hardware investment and implementation complexity [49]. On the other hand, PI/PID controller and their variations still dominate the practical applications. A recent survey shows that even the simplest PI controller accounts for almost 95% of the controllers used in coal-fired power plant [50]. This paper will use a modified PI controller based on data-driven uncertainty compensation to achieve a satisfactory performance with negligible computation and ease of use.

G7 G8 G9

-0.15

-0.20

-0.25

0

100

200

300

400

500

600

700

800

Time (s) Fig. 6. Step response of models G1-G9 identified in Table 1.

temperature fluctuation. The change of ambient temperature with time and weather will prompt the change in stack heat dissipating, which is reflected in the open-loop gain of the model transfer function. Stack power supply is user defined and related to heat generation. Moreover, the power fluctuation happens rapidly, and the slow response of cooling system may cause large overshoot and integral saturation. To avoid the influence of disturbances, the data-driven rather than error-driven control method should be applied to impair the overshoot and expedite the sluggish response to multiple external disturbances [46]. It is further proven in [46] that ADRC is able to achieve desired transient performance of the tracking error despite uncertainties, which is quite important for process control. Step responses of the nine transfer functions are depicted together in Fig. 6. It is shown that both the static gain and lag time distinguishes from each other significantly, depending on the power and temperature conditions. Larger temperature difference enhances the convection and conduction, thus magnifying the gain under the low duty-cycle conditions. Besides, the change of ambient environment may also contribute to the variation of the lag time. To sum up, the parameters of PEMFC stack temperature model depend on many unknown and inaccessible factors, such as ambient environment, the internal humidity and reaction conditions. The identified model can be approximately described by a first-order form with indeterminate gain values Kn and time constants Tn

Gn (s ) =

3.1. PI controller design It is widely acknowledged that PI controller is adequate to well control a first-order process [51]. Therefore, we adopt a single-loop PI controller to control the exhaust temperature. The mathematical form of PI controller is denoted as

Gc (s ) = kp +

(7)

where kp is the proportional gain and ki is the integral parameter. They are tuned based on the identified nominal model G5 , by balancing the computational efficiency and robustness measures. The resulting parameters are kp = −0.2035 and ki = −0.01029 which are able to produce a reasonable set-point tracking with phase margin being around 45°. 3.2. ADRC controller design The primary deficiency of PI controller lies in its poor robustness against model uncertainties because of its dumpish reaction in integrating the operational error. ADRC, using a different manner to mine the operational error, aims to estimate and compensate the modelling uncertainties by analyzing the real-time input-output data. A first-order ADRC structure is used in this paper, and the corresponding structure is depicted in Fig. 7. It can be deemed as a slight modification of PI controller by replacing the integrator with an extended state observer (ESO) [30]. Considering the parametric perturbation in Table 1, system nonlinearity and disturbances, the uncertain first-order system can be represented in the following form,

Kn 1 + Tn s

⎧ Pn = fP (Tab, Pd, t , ···) ⎨ ⎩Tn = fT (Tab, Pd, t , ···)

ki s

(6)

where n denotes a particular working condition, fP and fT are unknown functions in terms of the ambient temperature Tab , the external load demand Pd and the running time t. Controller design under such an

Fig. 7. The control structure of ADRC. 5

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30

x ̇ = f (x ) + d (t ) + bu y=x

where x is state variable which equals to the system output y, i.e., the temperature. u is the control signal, i.e., the PWM duty cycle. d represents the external disturbance, and f is the unknown function, which contains the internal uncertainties of the system and external disturbances d. Denoting b0 as an estimation of the parameter b, eq. (8) can be rewritten as,

where (10)

is the total disturbance, which needs to be estimated. If we denote δ as an extended state x2 and assume δ is differentiable[28], an augmented model of Eq. (9) can be expressed as,

⎧⎡ x1̇ ⎤ = ⎪⎢ ⎣ x2̇ ⎥ ⎦ ⎨ y = [1 ⎪ ⎩

x b ⎡ 0 1 ⎤ ⎡ 1⎤ + ⎡ 0 ⎤ u + ⎡ 0 ⎤ δ ⎣1⎦ ⎣ 0 0 ⎦ ⎣ x2 ⎦ ⎣ 0 ⎦ x1 0] ⎡ x ⎤ ⎣ 2⎦

24

PI

29

G1 G4 G2 G5 G3 G6 Set point

28 27

G7 G8 G9

26 25 24 0

250

500

750

1000

1250

1500

1750

2000

2250

2500

Fig. 8. Temperature tracking results while set point changes.

3.3. Theoretic analysis of performance guarantee

(11)

Since the expected closed-loop transfer function is an approximation result assuming an absolutely accurate estimation of δ . However, it is challenging in practice because of the limited bandwidth and the time discretization. This section aims to rigorously prove the performance guarantee that the actual output can be controlled as close to the expectation as possible, in spite of the uncertainties raised from different operating conditions. Consider the uncertain plant given in Eq. (9), the ADRC controller law in Eqs. (11)–(16) can be summarized as follows,

(12)

(13)

⎧⎧ x ̇ ̂ = −2ω0 (x ̂ − x ) + b0 u + δ ̂ ⎪ ̇ 2 ̂ ⎪⎨ ⎩ δ = ω0 (x ̂ − x ) ⎨u = −k (x − r ) − δ ̂ ⎪ b0 ⎪ ̇ ̂ ( ) (t0), δ (̂ t0) = 0 x t x = 0 ⎩

where the bandwidth value ω0 is tuned as 0.5 for sufficiently fast realtime estimation in consideration of the time scale of the temperature dynamics. A significant theoretic step was made in [52] to prove that z1 and z2 can track x1 and x2 well provided that δ ̇ is bounded. By compensating the estimated δ , i.e., z2 , the inner-loop control law in Fig. 7 becomes

u − z2 u= 0 b0

25

Time (s)

where β1 and β2 are observer parameters. For observation accuracy, the values of β1 and β2 can be uniformly tuned such that the observer eigenvalues are located at − ω0 , leading to [36],

β1 = 2ω0 , β2 = ω02

26

23

In order to estimate the value of total disturbance on time, an extended state observer (ESO) is designed as,

b0 β1 ⎤ u − β1 1 ⎤ z1 ⎡ z1̇ ⎤ = ⎡ ⎡z ⎤ + ⎡ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ y 2⎦ ̇ z 0 β − ⎣ 2 2 ⎣ ⎦ ⎣ ⎦ ⎣ 0 β2 ⎦ ⎣ ⎦

27

23

(9)

δ (x , t , u) = f (x ) + d (t ) + (b − b0) u

28

Temperature (°C)

y ̇ = δ (x , t , u) + b0 u

ADRC

29

Temperature (°C)

(8)

(18)

The ideal trajectory (expectation) x ∗ can be derived from eq. (17), ∗ ∗ ⎧ x ̇ = −k (x − r ) ∗ (t ) = x (t ) x ⎨ 0 0 ⎩

(14)

the equivalent system of the inner loop can thus be approximately compensated as follows,

(19)

Theorem 1. Assume that

⌢

y ̇ = δ + b0 ⎛ ⎝ ⎜

⎛u − δ ⎞ u 0 − z2 ⎞ = δ + b0 ⎜ 0 ≈ u0 b0 ⎠ b0 ⎟ ⎝ ⎠

α1 ⩽

⎟

(15)

(16)

α 1 5 ⎞ ω0 > 2η + 2kη2 + 2k ⎛ + 2 + α1 ⎠ 2 4 4 ⎝ −1 ⎤ 2 η ⎡ ⎢ 12 + 25α ⎥ (g (ρ1 + |r|) + (1 + α 2 )k ) 1⎦ ⎣ ⎜

where as shown in Fig. 8, r is input and k is proportional parameter. Combining Eqs. (15) and (16), the closed-loop transfer function will behave similarly as follows: 1

Gcl (s ) =

(20)

where α1, α2 , ρ0 and h are any given positives, and g (ρ) is any given increasing function. Then, the design law

It can be easily controlled by proportional control,

u 0 = k (r − y )

∂f (x ) b ⩽ g (ρ) ⩽ α2, |x (t0)| ⩽ ρ0 , |d ̇ (t )| ⩽ h, sup ∂x b0 |x| ⩽ ρ

ks y (s ) k = = 1 r (s ) s+k 1 + ks

⎟

(21)

ensures that

(17)

|x (t ) − r| ⩽ ρ1 , ∀ t ⩾ t0

if the estimation of δ is accurate enough. Thus, the time constant of the closed-loop tracking response is 1/ k . To get a tracking performance with 5 s inertia lag, the proportional gain is tuned as 0.2. For implementation issue, note that the ADRC controller contains the anti-windup scheme by [36] simply using the real manipulated variable as the input of ESO (12). Note that the algorithm introduced in this section has limitations and needs modifications when used in the processes with time-delay [53] or non-minimum phase characteristics [54].

(22)

where

⎧ ⎨ ⎩ Moreover, there ρ1 ≜

2 max

1 1 ⎡ 1 − 15 ⎤ h2 , 2 e (t0)2 + 4 + ⎢ 2 + 2α ⎥ 2kω0 1⎦ ⎣ exists a positive constant γ such that

sup |x (t ) − x * (t )| ⩽ t

6

(

γ ω0

5 4α1

) δ (t ) ⎫⎬. 0

2

⎭

(23)

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34

⎧ ex = x − x ̂ ⎨ eδ = δ − δ ̂ = f (x ) + d (t ) + (b − b0 ) u − δ ̂ ⎩

Temperature (°C)

Proof. Firstly, we define

(24)

Thus, the closed-loop system can be rewritten as

⎧ x ̇ = −k (x − r ) + eδ ⎪ eẋ = −2ω0 ex + eδ ⎪ ∂f (x ) 2 ̇ ⎨ eδ̇ = −ω0 ex + ∂x (−k (x − r ) + eδ ) + d (t ) ⎪ b−b + b 0 (−k (−k (x − r ) + eδ ) − ω02 ex ) ⎪ 0 ⎩

b

0

)

(

Temperature (°C)

(25)

∂f (x ) ∂x

b − b0 k b0

−

)e

δ

−k

(

∂f (x ) ∂x

−

b − b0 k b0

G1 G2 G3

22 20

G4 G5 G6

G7 G8 G9

PI

30 28 26 24 22 20 0

100

200

300

400

500

600

700

800

Fig. 9. Simulation of disturbance rejection. Step disturbances occur at 0 s, 300 s and 600 s.

) (x − r ) + d ̇ (t )

perturbation, while the performance of the PI controller changes significantly from overshoot to sluggish response. Fig. 9 shows the disturbance rejection performance. The disturbance enters into the system at 0 s, 300 s and 600 s with amplitude of +2, −5 and +4, respectively. Compared with the PI controller, the ADRC controller yields shorter recovery time, smaller overshoot and better robustness. The disturbance rejection was significantly accelerated because of the two degrees-of-freedom nature of ADRC. It can be found from Figs. 8 and 9 that the deviation of operating condition will degrade the control performances of both controllers, but ADRC performs much better than the competitor. Most obviously, the extreme slow response of conditions (3) and (9) is due to the excessive operating conditional deviation. Even so, ADRC still rapidly settles down to the set point. From this point of view, ADRC shows less model dependency compared with the PI controller because of the real-time data-driven compensation.

(27)

e = x − r , e¯x = ω0 ex Consequently, Eqs. (26) and (27) lead to,

⎧ e ̇ = −ke + eδ ⎪ e¯ ̇ = −2ω e¯ + ω e 0 x 0 δ x ⎨ ∂f (x ) b ̇ e ω e ¯ = − + − ⎪ δ b0 0 x ∂x ⎩

(

b − b0 k b0

)e

δ

−k

(

∂f (x ) ∂x

−

b − b0 k b0

) e + d ̇ (t )

(28)

Define 1

b

1 2 e¯ T ⎡ 4 + 4b0 e + ⎡ x⎤ ⎢ 2 ⎣ eδ ⎦ ⎢ − 0.5 ⎣

1 4

− 0.5 ⎤ e¯ ⎥⎡ x⎤ e 5b + 4b0 ⎥ ⎣ δ ⎦ ⎦

(29) 1

b

⎡ 4 + 4b0 − 0.5 ⎤ ⎥ ⩽ α2 , it can be verified that ⎢ ⎢ − 0.5 1 + 5b0 ⎥ 4 4b ⎦ ⎣ is a positive definite matrix. Next, it is proved in Appendix (A1) that under the condition of (21), V (e (t ), e¯x (t ), eδ (t )) ⩽

24

Time (s)

Introduce the following definitions

Firstly, since α1 ⩽

26

18

(26)

V≜

28

32

⎧ ⎪ ⎨ ⎪ ⎩ x ̇ = −k (x − r ) + eδ eẋ = −2ω0 ex + eδ

(

30

18

Equivalently, there is

eδ̇ = − b ω02 ex +

ADRC

32

b b0

ρ12 2

for ∀ t ⩾ t0

4. Experimental results In this section, the experimental setup is introduced in terms of the hardware integration, sensor placement and controller communication. Disturbance rejection performance is tested against the sudden change of the load requirement. Set-point tracking is also carried out to examine the overshoot and integrated error. Finally, the comparison between the conventional PI and the proposed ADRC is discussed.

(30)

Therefore, It follows from (29) and (30) that

ρ2 1 2 e ⩽V⩽ 1 2 2

(31)

Thus, eq. (22) holds. Finally, eq. (23) is proved in Appendix (A2).□

4.1. Experimental setup The experimental platform is shown in Fig. 10. To avoid distraction, some irrelevant components to the experiment, such as gas flowmeter, switches, relays, and valves, are not detailed in the paper, but they are indispensable for safe and normal operation. The core component is the fan-cooling PEMFC stack that is shown in the photo. The rated power of the experimental PEMFC is 300 W, with a rated voltage of 25 V and a rated current of 12A. The stack size is 267 × 136 × 132 mm and weighs 3.4 kg . The PEMFC stack usually operates around 30 °C because of the large excessive cathode air supply in the open-cathode structure [42]. Note that this temperature setting is remarkably different from the popular closed-cathode PEMFC where the temperature is usually set around 70 °C by using circulating water for cooling [43]. Fig. 1 also concisely illustrates the main components related to the communication of each components. The system is controlled by the host computer. The communication between heat management

Remark. Theorem 1 states that the difference between the actual output x (t ) and the ideal output x ∗ (t ) can be governed by the observer parameter ω0 and can be arbitrarily small with a sufficiently large ω0 . 3.4. Simulation comparison To evaluate and compare the robustness against parameter perturbation under the overall operating region, the above nominal PI and ADRC controllers are simulated based on the whole model set. Both disturbance rejection and temperature tracking are examined to give a comprehensive evaluation. Fig. 8 shows temperature tracking performances of the fixed ADRC and PI controllers against the varying transfer functions. Obviously, the tracking performance of ADRC is almost insensitive to the model 7

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L. Sun, et al.

response of temperature. However, with the proposed ADRC, the temperature was kept nearly invariant regardless of the high-frequency noise, and only fluctuated within ± 0.3 °C from the beginning to the end. This attributes to the estimation of total disturbance and fast compensation to it. The sensitivity of ADRC controller to noise can be reduced by incorporating the nominal model into ESO [37]. 4.3. Set-point tracking Sometimes the temperature set-points need to be changed, depending upon the waste heat demand, ambient temperature and system efficiency. This section will examine the set-point tracking performance, in which the same controllers designed above are used for comparison. Fig. 14 compares the temperature tracking performance of PI and ADRC controllers. The IAE of ADRC is much smaller than that of PI controller, indicating an about 30% performance improvement. The experimental results of both controllers are consistent with the simulation results. The tracking performance of PI controller varies significantly with the operating conditions. In comparison, the tracking performances of ADRC are almost consistent in spite of the condition change, indicating a stronger robustness than PI. Besides, the temperature outputs of PI controller exhibit obvious overshoot while the overshoot of ADRC is negligible, which is realized the ADRC closed-loop transfer function derived in (17). In the PI controller design, the closed-loop transfer function should contain zeroes and cannot be maintained invariant to the process uncertainty. Thus the PI control performance will contain overshoot and vary with the operating conditions.

Fig. 10. Photo of PEMFC experimental platform.

subsystem and the upper computer is based on the NI (National Instruments) USB-6002 communication card. Stack current and voltage signals are collected via MODBUS protocol. The load change is realized by an electricity load working in piece-constant current mode. The experimental PEMFC system is a self-powered system. The system can operate independently without external electric power supply. All the components, including cooling fans, relays, switches and other loads, are of high couplings and powered by the fuel cell itself while operating. The operating voltage of the cooling fans is 12 V and the rotational speed of fans changes with the duty cycle of controllable PWM wave. Thus, a Voltage/PWM module is used to convert the analog output into PWM wave. The host computer calculates the control input and sends it to the PWM/Voltage module via an NI card analog output paths. The correspondence of the duty cycle to the rotational speed is nearly linear, while n% duty cycle corresponds to n% of the maximum rotating speed. The controlled variable is the operating temperature.

4.4. Discussion Simulation and experimental results verified the promising perspectives of ADRC application in PEMFC temperature control. The merits include reducing tracking error, quick recovery to steady state, and strong robustness to parameter perturbations. The successful implementation of ADRC illustrates the superiority of data-driven methodologies in tackling nonlinearity and disturbances. By compensating the nonlinear model into a pure integral process using a real-time datadriven ESO, the data-driven ADRC greatly simplifies the controller design and parameter tuning. The stronger control ability inevitably brings higher sensitivity from measurement noise to control action due to the rotational speed of cooling fans. A traditional way to restrain this is to introduce filters at the cost of sacrificing control performance. However, under this case, the fan rotating speed is determined by the duty cycle of PWM wave delivered by Voltage/PWM conversion module. As the PWM wave resembles the square wave, the duty cycle actually corresponds to the length of on-off period of the control signal. For example, 30% duty cycle means the voltage is low during 30% of the time, and high during the rest of the time. The frequency of PWM voltage wave change is much higher than the fluctuation of control signal, so it will do no harm to the actuator fans.

4.2. Disturbance rejection The controller designed by simulation is used to control the stack temperature. In order to prevent the oxygen starvation, the lowest analog output is 2 V, corresponding to the 20% of maximum rotational speed of the fan. Under the application scenario [42] given in the open-cathode fuel cell cogeneration the set-point of the exhaust air is determined as 26 °C for comfortable indoor environment. The ambient temperature during the experiment is around 0 °C. It is analyzed in literature [42] that the reuse of the waste heat can significantly improve the overall efficiency. The electric power during the experiment fluctuated between 160 W and 200 W. For the sake of comparison, integral absolute error (IAE) is used to evaluate the control performance, which is defined as

IAE =

∫0

t

|y − r| dt

(32)

5. Conclusion

The disturbance rejection results of PI and ADRC are given in Fig. 11 and Fig. 12, respectively. As expected from simulation, ADRC gives a much better regulation performance than PI controller, both in perturbation magnitude and settling time. By comparing the IAE value denoted on the upper left corner of the figures, ADRC shows overwhelming superiority than the PI controller. Fig. 13 is a detailed comparison of PI and ADRC when electricity power stepped downward from 200 W to 175 W. With PI controller, temperature gradually dropped to tmin = 25.25 °C and took 149.2 s for recovery. Afterwards, the temperature slightly increased to tmax = 26.47 °C and returned to the set-point at 344.9 s, reaching to the stable response at the end. The overshoot is caused by the accumulation of integral and the sluggish

This paper presented an integrated modelling and experimental approach for the temperature control of open-cathode proton exchange membrane fuel cell (PEMFC) in a data-driven manner, from both modelling and control design perspectives. First, the model was identified based on a series of step responses. It was shown that the models differ from each other under various operation conditions, indicating strong nonlinearity of PEMFC system. Moreover, the model nonlinearity, disturbances and time-varying dynamics were lumped as a unified uncertainty term, which was then estimated and compensated by active disturbance rejection controller (ADRC). The experimental results showed that the method developed in this paper was able to 8

Applied Energy 261 (2020) 114381

L. Sun, et al.

30

IAE=433.49

Temperature (°C)

29 28 27 26 25 24 23 22 65

Duty Cycle (%)

60 55 50 45 40 35 30 25 20

Power (W)

220 200 180 160 140

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (s) Fig. 11. PI regulation performance during power disturbance.

Declaration of Competing Interest

improve the tracking and regulation performances simultaneously for fuel cell temperature control, compared with the conventional PI controller. Besides, it was shown that ADRC had stronger capability in dealing with uncertainties and disturbances, keeping the temperature stable in spite of the operating condition change and power shift.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement Acknowledgements Li Sun: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Supervision, Funding acquisition, Writing - original draft, Visualization. Yuhui Jin: Funding acquisition, Writing - original draft, Visualization. Fengqi You: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Supervision, Writing - review & editing.

Li Sun would like to give his thanks to the support from National Natural Science Foundation of China, China under Grant 51806034 & 51936003, the Natural Science Foundation of Jiangsu Province, China under Grant BK20170686 and the open funding of the state key lab of power systems, Tsinghua University.

Appendix (A1). The proof of V (e (t ), e¯x (t ), eδ (t )) ⩽

ρ12 2

for ∀ t ⩾ t0 consists of next two steps:

Step 1. By Eq. (29), x (̂ t0) = x (t0) , δ ̂(t0) = 0 and the definitions of e¯x and eδ , there is 9

Applied Energy 261 (2020) 114381

L. Sun, et al.

30

IAE=124.53

Temperature (°C)

29 28 27 26 25 24 23 22 65

Duty Cycle (%)

60 55 50 45 40 35 30 25 20

Power (W)

220 200 180 160 140

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (s) Fig. 12. ADRC regulation performance during power disturbance.

V (e (t0), e¯x (t0), eδ (t0)) ⩽ According

to

1 1 5 ⎞ e (t0)2 + ⎛ + δ (t0)2 2 4 4 α1 ⎠ ⎝ ⎜

the

V (e (t0), e¯x (t0), eδ (t0)) ⩽

ρ12 2

definition

⎟

of

ρ1,

(33) we

have

that

.

Step 2. By using (28) and (29), there is

e¯ T e¯ V̇ (e (t ), e¯x (t ), eδ (t )) = −ke 2 + eeδ − ω0 ⎡ x ⎤ ⎡ x ⎤ ⎣ eδ ⎦ ⎣ eδ ⎦ T −0.5 ⎤ ∂f (x ) e¯ b−b + 2⎡ x ⎤ ⎡ − b 0 k eδ − k 1 5b0 ∂x ⎥ 0 + ⎣ eδ ⎦ ⎢ 4b ⎦ ⎣4

((

)

(

∂f (x ) ∂x

−

b − b0 k b0

+ d ̇ (t )) ) e(34)

According to Cauchy-Schwarz inequality and the square of the vector difference is non-negative,

Fig. 13. Temperature regulation comparison at power stepping downward from 200 W to 175 W.

10

Applied Energy 261 (2020) 114381 30

Temperature (°C)

30

28 26 24

IAE=1031.01

22

28 26 24

80

70

70

60 50 40 30 20

0

500

1000

1500

2000

2500

60 50 40 30 20

3000

IAE=714.29

22

80

Duty Cycle (%)

Duty Cycle (%)

Temperature (°C)

L. Sun, et al.

0

500

1000

1500

2000

2500

3000

Time (s)

Time (s)

(a)

(b)

Fig. 14. Composition between temperature tracking experiments. (a): Conventional PI; (b) ADRC.

−1 ⎤ ⎛ k 1 V̇ (e (t ), e¯x (t ), eδ (t )) ⩽ −ke 2 + 4 e 2 + k eδ2 − ω0 − ⎡ ⎢ 1 + 5b0 ⎥ ⎜ 2b ⎦ ⎣2 ⎝ 2 2 − 1 2 ∂f (x ) k b−b ω ⎤ ⎡ e¯x ⎤ − b 0k + 20 + 4 e2 + k ⎡ ∂x ⎢ 1 + 5b0 ⎥ eδ ⎦ 0 ⎣ b 2 2 ⎦ ⎣ ⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

⎛ω k 1 ⩽ − 2 e 2 − ⎜ 20 − k − ⎝ 2 − 1 h2 ⎤ + ⎡ 1 5b0 2ω0 ⎥ ⎢ + 2b ⎦ ⎣2

∂f (x ) ∂x

b − b0 k b0

−

∂f (x ) ∂x

−

⎡ e¯x ⎤ ⎣ eδ ⎦

x = x (t *)

⎞ ⎟ ⎠

⎡ e¯x ⎤ ⎣ eδ ⎦

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

2

+ 2

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

−k

∂f (x ) ∂x

−

2

2 h2 2ω0

b − b0 2 ⎞ k ⎟ b0

⎠

⎡ e¯x ⎤ ⎣ eδ ⎦

2

(35)

Assume that there exists a time t ∗ such that V (e (t *), e¯x (t *), eδ (t *)) =

∂f (x ) ∂x

b − b0 k b0

ρ12 2

, then |e (t *)| ⩽ ρ1. Thus, |x (t *)| ⩽ ρ1 + |r| which means

⩽ g (ρ1 + |r|)

(36)

Combination of (21) and (34)–(36) leads to

V̇ (e (t *), e¯x (t *), eδ (t *)) ⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

⎛ω k 1 ⩽ − 2 e 2 (t *) − ⎜ 20 − k − ⎝ 2 − 1 h2 ⎤ + ⎡ 1 5b0 2ω0 ⎥ ⎢ + 2b ⎦ ⎣2 k

⩽ − 2 e 2 (t *) − k

(

1 2

+

α2 4

+

5 4α1

∂f (x ) ∂x x = x (t *)

2

2 * ⎡ e¯x (t ) ⎤ ⎢ e 2 (t *) ⎥ ⎣ δ ⎦

)

−

b − b0 k b0

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

+

−k

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

2 ∂f (x ) ∂x x = x (t *)

−

b − b0 k b0

2⎞

⎟ ⎠

* ⎡ e¯x (t ) ⎤ ⎢ eδ (t *) ⎥ ⎣ ⎦

2

2 h2 2ω0

(37)

On the other hand, it can be verified that 1

b

T⎡ + 4b0 ⎡ e¯x ⎤ ⎢ 4 e δ ⎣ ⎦ ⎢ − 0.5 ⎣

− 0.5 ⎤ e¯ 1 α 5 ⎞ ⎥⎡ x⎤ ⩽ ⎛ + 2 + e 5b 4 4α1 ⎠ + 4b0 ⎥ ⎣ δ ⎦ ⎝ 2 ⎦ ⎜

1 4

⎡ e¯x ⎤ ⎣ eδ ⎦

⎟

2

(38)

From (37) and (38), there is k

V̇ (e (t *), e¯x (t *), eδ (t *)) ⩽ − 2 e 2 (t *) − k ⩽ − kV (e (t *), e¯x (t *), eδ (t *)) +

⩽ −k

ρ12 2

+

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

(

1 2

+

α2 4

+

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

5 4α1

2 * ⎡ e¯x (t ) ⎤ ⎢ e 2 (t *) ⎥ ⎣ δ ⎦

)

2

+

⎡ −1 ⎤ ⎢ 1 + 5b0 ⎥ 2b ⎦ ⎣2

2 h2 2ω0

2 h2 2ω0

2 h2 2ω0

⩽0 (39)

11

Applied Energy 261 (2020) 114381

L. Sun, et al. ρ12

Note that V (e (t *), e¯x (t *), eδ (t *)) = ρ12

V (e (t ), e¯x (t ), eδ (t )) ⩽

and (39), then V (e (t0), e¯x (t0), eδ (t0)) ⩽

2

ρ12 2

implies that (e (t ), e¯x (t ), eδ (t )) will not escape from

forever.

2

ρ2

Thus, V (e (t ), e¯x (t ), eδ (t )) ⩽ 21 for ∀ t ⩾ t0 . (A2) The proof of Eq. (23) Based on (20) and (22), there exists a positive constant α3 > 0 such that

⎡ − 0.5 ⎤ ⎢ 1 + 5b0 ⎥ 4b ⎦ ⎣4

⎛ ∂f (x ) − b − b0 k ⎞ eδ − k ⎛ ∂f (x ) − b − b0 k ⎞ e + d ̇ (t ) ⩽ α3 b0 b0 ⎠ ⎠ ⎝ ∂x ⎝ ∂x ⎜

⎟

⎜

⎟

(40)

Define 1

b

e¯ T ⎡ 4 + 4b0 V2 = ⎡ x ⎤ ⎢ ⎣ eδ ⎦ ⎢ − 0.5 ⎣

1 4

− 0.5 ⎤ e¯ ⎥⎡ x⎤ e 5b + 4b0 ⎥ ⎣ δ ⎦ ⎦

(41)

It can be verified that

1 4

2

⎡ e¯x ⎤ ⎣ eδ ⎦

1 α 5 ⎞ ⩽ V2 ⩽ ⎛ + 2 + 4 4α1 ⎠ ⎝2 ⎜

2

⎡ e¯x ⎤ ⎣ eδ ⎦

⎟

(42)

Regarding to system (26), we have that

⩽ −

ω0 ⎛ 1 + α2 + 5 ⎞ 4 4α1 ⎠ ⎝2

2

⎡ e¯x ⎤ ⎣ eδ ⎦

V2̇ (¯ex (t ), eδ (t )) ⩽ −ω0

⎡ e¯x ⎤ ⎣ eδ ⎦

+2

α3

V2 (¯ex (t ), eδ (t )) + 4 V2 (¯ex (t ), eδ (t )) α3 (43)

Thus,

d V2 (¯ex (t ), eδ (t )) ⩽ − dt 2

ω0

(

1 2

+

α2 4

+

5 4α1

)

V2 (¯ex (t ), eδ (t )) + 2α3 (44)

Consequently, −

V2 (¯ex (t ), eδ (t )) ⩽ e

ω0 α 2 12 + 42 + 4α5

(

1

)

(t − t0 )

4α3 V2 (¯ex (t0), eδ (t0)) +

(

1 2

+

α2 4

+

5 4α1

) (45)

ω0

According to Eqs. (42) and (45), we have, −

|eδ (t )| ⩽ e

ω0 α 2 12 + 42 + 4α5

(

1)

(t − t0 )

8α3 2 V2 (¯ex (t0), eδ (t0)) +

On the other hand, since

x * (t

(

1 2

+

α2 4

+

5 4α1

) (46)

ω0

= x (t0) , we have,

0)

t

|x (t ) − x * (t )| ⩽ ∫t e−k (t − τ ) |eδ (t )| dτ 0 t

⩽ ∫t 0

⩽

e−k (t − τ )

⎛ − 2⎛ 1 ⎜⎜e ⎝ 2 ⎝

+

ω0 (τ − t0 ) α2 + 5 ⎞ 4 4α1 ⎠ 2

V2 (¯ex (t0), eδ (t0)) +

1 5 ⎞ α 8α3 ⎛ + 2 + 4 4α1 ⎠ ⎞ ⎝2 ⎟⎟ dτ ω0

⎠

1 5 ⎞ α 4⎛ + 2 + V2 (¯ex (t0), eδ (t0)) 4 4α1 ⎠ 1 ⎝2 ω0 1 α 5 ⎞ 1 2 1 − 2k ⎛ + + 4 4α1 ⎠ ω0 ⎝2

+

1 α 5 ⎞ 8α3 ⎛ + 2 + 4 4α1 ⎠ ⎝2 kω0

(47)

Combining Eq. (47) with (21) leads to

⎛ |x (t ) − x * (t )| ⩽ ⎜ ⎜ 1 − 2k ⎝

4

(

1 2

+

(

5 4α1

)

1 2

+

α2 4

+

α2 4

+

5 4α1

)/(2η + 2kη

V2 (¯ex (t0), eδ (t0)) 2

+ 2k

(

1 2

+

α2 4

+

5 4α1

))

⎞ 1 α 5 ⎞⎟ 1 + 8α3 ⎛ + 2 + 4 4α1 ⎠ ⎟ ω0 ⎝2 ⎠ ⎜

⎟

(48)

Thus, Eq. (23) holds.

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