Proton Exchange Membrane Fuel Cell (PEMFC) is considered a type of fuel cell in industry and is significantly welcome since it is an environmentally friendly technology that eliminates the pollution caused by burning fossil fuels. Minimizing the consumption power internally through the air-feed systems of PEMFCs along with maximizing the net power output lead to the most optimal control results. Hence, the present paper is conducted to control these parameters optimally. Most previous works used linear models of PEMFC to deal with control problems. Linear models cannot exactly model the nonlinear behavior of the PEMFC stacks. Accordingly, the open-loop nonlinear system is presented as state-dependent polynomial equations, and the state feedback method is employed to design the controller. Furthermore, semi-definite programming (SDP) based on the sum of square (SOS) relaxation is used to solve the state-dependent polynomial nonlinear model numerically. On the other hand, due to the high response time of SOS-based control design methods, the feasibility problem of SOS-based method is conceived as an optimization problem to decrease the response time. The most important advantage of the proposed method is that this method guarantees the stability. This method is applied to PEMFC stack. A comparison is also made between the results obtained in this study and the results of one of the most valuable previous studies. Accordingly, the proposed method could outperform its counterpart method regarding the accuracy and efficiency.
Research Article
Design a novel nonlinear controller based on sum of square relaxation to obtain an optimal controller with stability guarantee applied on PEMFC nonlinear model
https://doi.org/10.21203/rs.3.rs-1598214/v1
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Proton Exchange Membrane Fuel Cell
Control method
SOS Relaxation
Stability Guarantee
With the rapid development of society, the use of traditional fossil fuels has increased dramatically, resulting in a 14% increase in global greenhouse gas emissions [1]. Therefore, it is urgent to find alternative energy sources and promote the development of new energy sources [2]. A fuel cell system is a clean and environmentally friendly technology that can eliminate the pollution caused by burning fossil fuels [3]. There are different types of fuel cells; however, the main idea behind them is the same. PEMFC is an electrochemical device that converts chemical energy into electrical energy with the advantages of high power density, low for zero emission, rapid response and cold start [4], which can be widely used in automotive, military and power plants in the future [5]. However, there are still many problems about the control of PEMFC, one of the significant control problems of PEMFCs is to reduce the consumption of power internally through their air-feed systems with the aim of increasing the net power output [6]. A PEMFC requires an adequate quantity of air (excess oxygen) in its cathode for responding the load variations/transitions with no damage. Interestingly, consumed power in the air-feed compressor has the highest value compared to the whole fuel cell's auxiliary systems and includes the total PEMFC power output [7]. Thus, an accurate control needs to be designed to run the compressor at its optimum operational point for the air-feed system. Therefore, the net power output can be significantly increased, and the high amount of the oxygen excess ratio is maintained for efficient operation. Most previous works used linear models of PEMFC to deal with control problems. Linear models cannot exactly model the nonlinear behavior of the PEMFC stacks. So, it’s required to design the controller for PEMFC nonlinear model.
Nonlinear control is one of the essential branches of control engineering that deals with nonlinear dynamics, time variables, or both [8, 9]. Stability study and design of stabilizer controller for nonlinear systems is a significant issue in control systems. Nonlinear controllers are beneficial in terms of speed, power consumption, and easy implementation compared to the linear controller [10]. However, the major drawback of nonlinear controllers is that they require more complicated mathematical analyses. Considering Lyapunov's theory is the only way to evaluate the stability of linear systems and nonlinear ones [11–13]. One of the remarkable problems that scientists mainly face in this regard is the lack of a systematic and specific method for finding a function with the characteristics of Lyapunov's function. To tackle this problem, complex design methods like nonlinear controller design based on regression, sliding-mode method, and neural network control are considered [14]. Besides, the numerical stability analysis methods have come to the Lyapunov's controller’s aid and made them more practical. Yang et al. [15] developed a new controlling approach based on Takagi-Sugeno fuzzy theory and predictive control for an efficient and precise air supply control. According to the results, their proposed approach could efficiently control the oxygen excess ratio, leading to enhanced power output and improved energy efficiency. Deng et al. [16] designed a robust nonlinear controller regarding cascade adaptive sliding mode control for air supply control in PEMFC. They applied adaptive laws for online regulation of controller parameters. According to their obtained results, their proposed controller could outperform the conventional PID controller and constant sliding mode controller.
Recently, the SOS method has been suggested as an advantageous method for the formation of Lyapunov functions and closed-loop stability [17, 18]. Since this method has the unique capability for solving nonlinear control problems, it has been proposed in many studies for various applications. This method uses algebraic calculations and convex optimization to prepare efficient convex relaxation for complicated computational problems. The SOS method is actually a set of optimization problems in which a linear cost function must be solved under the constraints of a polynomial matrix. This method is considered for the analysis and design of control systems and is very suitable for many systems, especially hybrid systems, nonlinear systems, time-delay systems, and indeterminate systems [19, 20]. Overall, the polynomial of a sum of squares (SOS) is written as the sum of several monomial squares. Hence the positive-definite of polynomial functions is proved, and this method is beneficial for the applications such as Lyapunov stability. Many nonlinear problems with polynomial constraints can be converted to equivalent convex SOS optimizations using the SOS method. In this case, the typical computational issues of the traditional nonlinear control methods are tackled. This method is utilized for solving the H∞ output feedback control problem, and the results are obtained by solving convex state-dependent LMI inequalities [21]. Notably, in this method, the precise rate of state changes is proposed to form the Lyapunov function, reducing system conservatism [22].
Many studies have been proposed so far concerning the SOS method and its practicalities [23–25]. For instance, Jiafeng et al. employed a technique based on the robust SOS to design a nonlinear controller and solve the uncertainty in the model of an ultrasonic aircraft [26]. The results obtained in this study represent the capability of the designed controller to ensure aircraft stability even when the rate of change in the aerodynamic coefficients is above 50%. In fact, a systematic computational approach using SOS methods was presented to design a controller for polynomial systems with indeterminate parameters in this research. Also, the performance of the controller under nonlinear vibrations and indeterminate parameters was examined. In a major advance, a robust nonlinear controller using the SOS method along with Lyapunov's theory of stability is proposed to stabilize a ship with operator dynamics and uncertain parameters resulting from variable loading conditions and ship speed [27]. Also, the LMI technique has been widely used as a numerical method based on convex optimization [28]. In order to design the controllers for nonlinear models in terms of LMIs, Takagi Sugeno fuzzy or Linear Parameter Variant (LPV) models have been significantly utilized. The nonlinear base model is modeled in these methods using a polyhedron region. In these regions, vertices are the local presentation of the base model. Due to the convexity of the achieved polyhedron, the control design conditions are presented based on vertices [29].
Further, squared decomposition methods were proposed to design a controller with less conservatism and complexity through semi-finite programming and the definition of polynomial matrix inequalities (PMI) [30]. Compared to LMI conditions, PMIs can be applied to a broader class of dynamic and static systems, especially polynomial systems. In addition, Takagi Sugeno LPV or fuzzy models are derived using some approximations, such as Taylor’s theorem, that maintain the system in only one local area.
For linear systems, there are different methods in terms of Riccati equations [31] or LMI [32]. Hence, the researchers are motivated to provide linear state equations for nonlinear systems, and the state-dependent Riccati equations are presented for nonlinear optimal control [33]. However, this method does not guarantee asymptotic stability. So, it’s required to analyze their stability. In addition, the approaches based on nonlinear matrix inequalities [34] that are indeed LMI equations, have also been developed. For example, the quasi-LPV method [35] has been presented based on LPV control [36]. The nonlinear systems are considered LPV systems in this method, and their parameters are state-dependent.
In addition to the methods mentioned above, the iterative methods can be employed for determining Lyapunov functions or controllers [37–39]. The main problem associated with these methods is that they typically tend to local optimal points. On the other hand, various techniques such as feedback linearization have been proposed so far to tackle the limitations of the previous methods [40]. The method considered in this study is based on the methods proposed in [35, 36]. Accordingly, the open-loop nonlinear system is presented in state-dependent polynomial equations, and the state feedback method is used to obtain appropriate control responses. Furthermore, SDP based on SOS method is utilized to numerically solve the state-dependent LMIs. This computational method was presented in the reference [37] to solve the parameter-dependent LMI equations in LPV control. Based on the reviewed literature, no study to date has yielded the results presented in this study. The nonlinear model shown in the study of Laghrouche et al. [7] is converted into a nonlinear polynomial model, and the response speed considered in this study is improved. The obtained method is employed on the nonlinear polynomial model. The considerable benefit of the designed controller is its constant stability which has been rarely observed in the literature. Then, this method is applied to PEMFC, and the obtained results are reported. A careful comparison is also made between the achieved results of this paper and the results of the study conducted by Laghrouche et al. in 2014 [7]. The comparison results represent the unique capability of the proposed method regarding the accuracy, speed, and authenticity.
Main contributions and novelties
As stated before, PEMFC is an environmentally friendly technology that eliminates the pollution caused by burning fossil fuels. Thus, it got very popular to generate the electricity among renewable energy technologies. Minimizing the consumption power internally through the air-feed systems of PEMFCs along with maximizing the net power output lead to the most optimal control results. Hence, the present paper is conducted to control these parameters optimally. The open-loop nonlinear system is presented as state-dependent polynomial equations in the present work, and the state feedback method is employed to design the controller. Furthermore, semi-definite programming (SDP) based on the sum of square (SOS) relaxation is used to solve the state-dependent polynomial nonlinear model numerically.
The main contributions of this work are given as follows:
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The nonlinear model of PEMFC system is used in this paper instead of the linear model, which simulates the system behavior more exactly;
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The nonlinear model is converted to polynomial model in order to apply the SOS-based method to design the controller for nonlinear system;
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The basic SOS-based method is applied to the obtained polynomial nonlinear system to design a controller with stability guarantee;
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The SOS-based feasibility problem is converted to an optimization problem to decrease the response time keeping the stability guarantee;
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Comparing the obtained results with one of most valuable previous work to show advantages of the proposed method;
The methods considered in this paper for solving the problem are explained in this part. Also, the information regarding the SOS method, decomposition of sum squares, and state-dependent LMIs are given.
The SOS method
In recent years, the SOS method and semidefinite programming (SDP) have been proposed to form the Lyapunov functions for closed-loop stability. The squared sum method uses algebraic calculations and convex optimization to create efficient convex relaxation for complex computational problems. This method is a set of optimization problems in which a linear cost function needs to be solved under the constraints of a polynomial matrix. The SOS programming is considered to analyze and design the control systems and is significantly advantageous for many systems, especially hybrid systems, nonlinear systems, time-delay systems, and uncertain systems [19, 20]. Considering the theory of ordinary differential equations (ODEs), Lyapunov functions are known as the scalar functions which can be employed for proving the stability of an equilibrium of an ODE. In order to guarantee the stability for specific classes of ODEs, the Lyapunov functions need to be considered. A model with a polynomial vector such as \(\dot{x}=f\left(x\right)\) where f (0) = 0 is considered according to which the main idea is to find a polynomial V (x) function that satisfies the following conditions:
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\(V\left(x\right)-\varphi \left(x\right) is SOS\) |
(1) |
|---|---|
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\(-\frac{\partial V}{\partial x}f\left(x\right) is SOS\) |
(2) |
where ϕ (x) denotes a definite positive polynomial, the condition of the sum of the squares ensures that the above functions are non-negative, and V(x) is a Lyapunov function that proves that the system is stable.
For example, \(\dot{x}=f\left(x\right)+g\left(x\right)u\) can be considered as a system in which f(x) and g(x) are polynomial functions. The method mentioned above can be employed to design state feedback controllers and specify the polynomial state feedback u = k(x) rule. Through the use of the Lyapunov function, the following condition is satisfied:
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\(-\frac{\partial V}{\partial x}\left(f\left(x\right)+g\left(x\right)k\left(x\right)\right) is SOS\) |
(3) |
However, the set of V(x) and k(x) that meet these conditions isn’t a common convex problem yet. According to this condition, the simultaneous search is difficult for V(x) and k(x) and is equivalent to solving several Bilinear Matrix Inequality (BMI). Therefore, a dual method for designing state feedback based on the density function was proposed with a better convexity [30, 41]. However, the question that needs to be raised here is how a target function can be considered in this method based on density.
The decomposition of SOS and state-dependent LMIs
The computational method used here is based on analyzing SOS equations of multivariate polynomials. A multivariate functional polynomial is indicated below in which \(x\in {R}^{n}\).
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\(f\left(x\right)=\sum _{i=1}^{m}{f}_{i}^{2}\left(x\right)\) |
(4) |
where, the real constants ci, i = 1,…, m are employed to parameterize a series of candidate Lyapunov functions in an affine form. For example:
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\(\text{Ѵ}=\left\{V\left(x\right): V\left(x\right)={v}_{0}\left(x\right)+{\sum }_{i=1}^{m}{c}_{i}{v}_{i}\left(x\right)\right\}\) |
(5) |
where \(V\left(x\right)\) represents some monomials in terms of x, is considered. Then, \(\text{u}\text{s}\text{i}\text{n}\text{g} \text{s}\text{e}\text{m}\text{i}-\text{f}\text{i}\text{n}\text{i}\text{t}\text{e} \text{p}\text{r}\text{o}\text{g}\text{r}\text{a}\text{m}\text{m}\text{i}\text{n}\text{g} \left(\text{S}\text{I}\text{P}\right), V\left(x\right)\in \text{Ѵ}\), or equivalently the ci coefficients are specified to meet the constraints (Equations 1 and 2). Notably, SIP has a finite number of variables and infinite number of constraints and is practical in an optimization problem. The lower-level variables of SIP are not considered in the objective function. In this paper, an object optimization in which a quasi-linear function is utilized to determine the coefficients can be considered if necessary [42]. This method can be employed for a set of state-dependent LMIs, which are as follows:
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\({min}{\sum }_{i=1}^{m}{a}_{i}{c}_{i}\) |
(6) |
|---|---|
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\(s.t. {F}_{0}\left(x\right)+{\sum }_{i=1}^{m}{c}_{i}{F}_{i}\left(x\right)\ge 0\) |
(7) |
According to Equations 6 and 7, ai is the real constant coefficients, and ci is the decision variable. Also, \({F}_{i}\left(x\right)\) indicates the symmetric matrix functions of indefinite \(x\in {R}^{n}\). The left side of the inequality is semidefinite for all positive values of \(x\in {R}^{n}\). Solving the high-optimization problem requires solving many LMIs and is computationally challenging. On the other hand, when \({F}_{i}\left(x\right)\) is the symmetric polynomial matrices in terms of x, the analysis of the sum of the squares can provide computational programming for Equations 3–7, which can be represented as a theorem.
Theorem 1[30]: \(F\left(x\right)\) is assumed to be a symmetric polynomial matrix N*N with the degree of 2d in terms of \(x\in {R}^{n}\). Also, Z(x) is a column vector whose objects are monomials in terms of x with degrees less than d. Thus, there are:
- \(F\left(x\right)\ge 0 \forall x\in {R}^{n}\)
- \({v}^{T}F\left(x\right)v is SOS, v\in {R}^{N}\)
- There is a positive semidefinite matrix in such that \({v}^{T}F\left(x\right)v={\left(v\otimes Z\left(x\right)\right)}^{T}Q\left(v\otimes Z\left(x\right)\right)\) in which \(\otimes\) represents the multiple of Kronecker. It should be noted that the Kronecker is considered as an operation on two matrices with any size that results in a block matrix.
According to this theorem, the solutions of the SOS optimization problem are considered for state-dependent (6) and (7) LMIs:
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\({min}{\sum }_{i=1}^{m}{a}_{i}{c}_{i}\) \(s.t. {v}^{T}\left({F}_{0}\left(x\right)+{\sum }_{i=1}^{m}{c}_{i}{F}_{i}\left(x\right)\right)v is SOS\) |
(8) |
The state feedback controller design
The state feedback control system is designed for a set of nonlinear polynomial systems by the SOS decomposition. In fact, \(\dot{x}=f\left(x\right)+g\left(x\right)u\)is a quasi-linear system that depends on the following state:
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\(\dot{x}=A\left(x\right)Z\left(x\right)+B\left(x\right)u\) |
(9) |
In which, A(x) and B(x) indicate that the polynomial matrices are in terms of x accordingly, Z(x) is a vector N*1 of monomials in terms of x that satisfies the following hypothesis.
Hypothesis 1
, so that:
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\({M}_{ij}\left(x\right)=\frac{\partial {Z}_{i}}{\partial {x}_{j}}\left(x\right), i=1, 2, \dots , N and j=1, 2, \dots , n , \tilde{x}=\{{x}_{{j}_{1}}, \dots , {x}_{{j}_{m}}\}\) |
(10) |
The aim is to meet the condition of the equilibrium point (x = 0) for the state feedback controller \(u=k\left(x\right)=F\left(x\right)Z\left(x\right)\).
Lemma
[30]: For a symmetric polynomial matrix P(x) that is non-singular based on the x values, (11) is considered:
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\(\frac{\partial P}{\partial {x}_{i}}\left(x\right)=-P\left(x\right)\left(\frac{\partial {P}^{-1}}{\partial {x}_{i}}\right)P\left(x\right)\) |
(11) |
Theorem
[30]: For the system given in (9), it is assumed that a symmetric polynomial matrix is a polynomial matrix (K (x)) n * N. There are a constant and a set of squares exists in such a way that below two conditions are met:
In these equations, the feedback stabilization problem can be solved easily, and the system stabilization controller is presented as follows:
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\(u\left(x\right)=K\left(x\right){P}^{-1}\left(\tilde{x}\right)Z\left(x\right)\) |
(15) |
In addition, (13) is satisfied based on \(x\ne 0\) and \({ϵ}_{2}\left(x\right)>0\), and Lyapunov system stability is obtained. If \(P\left(\tilde{x}\right)\) is a constant matrix, the total stability for the system is obtained.
In the next sections, the nonlinear optimal controller is designed, which is, actually, the extension of the quadratic linear regulator (LQR) problem to a nonlinear regulator.
The optimal nonlinear controller design
The nonlinear optimal control problem is solved accurately using a Hamilton-Jacobi partial differential relation. Hence, a solution considering the state-dependent LMIs can be obtained through the non-optimal state. The Lyapunov function obtained here provides a high limit for the optimal cost function. Notably, this control design is known as guaranteed cost control. For the sake of simplicity, the following structure is considered:
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\(\left[\begin{array}{c}\dot{X}\\ {Z}_{1}\\ {Z}_{2}\end{array}\right]=\left[\begin{array}{cc}A\left(x\right)& x\\ {C}_{1}\left(x\right)& 0\\ {C}_{2}\left(x\right)& I\end{array}\right]\left[\begin{array}{c}Z\left(x\right)\\ u\end{array}\right]\), \(X\left(0\right)={x}_{0}\) |
(16) |
where \(Z\left(x\right)\) is the monomial vector that satisfies Hypothesis 1 and \({z}_{1}\in {\text{R}}^{\text{M}}\). The major aim here is to design the state feedback rule \(\text{u}=\text{K}\left(\text{x}\right)\text{Z}\left(\text{x}\right)\) such that the closed-loop system has asymptotic stability, and the cost function is obtained as follows:
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\({\left|\left|z\right|\right|}_{2}^{2}={\int }_{0}^{{\infty }}\left[{z}_{1}^{2}\left(t\right)+{z}_{2}^{2}\left(t\right)\right]dt\) |
(17) |
Theorem 3
[30]: A symmetric polynomial matrix is assumed for (16), and the constant of and a sum of squares are considered for the following equations.
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\({v}_{1}^{T}\left(P\left(\stackrel{-}{x}\right)-{\in }_{1}I\right){v}_{1}= {\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right]}^{T}\left[\begin{array}{cc}M\widehat{A}P+P{\widehat{A}}^{T}{M}^{T}-MB{B}^{T}{M}^{T}-{\varSigma }_{j\in J}\frac{\partial P}{\partial {x}_{j} }\left({A}_{j}Z\right)+{ϵ}_{2}I& P{C}_{1}^{T}\\ {C}_{1}P& -(1-{ϵ}_{2})I\end{array}\right]{\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right]}^{ }\) |
(18) |
and
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\(\widehat{A}\left(x\right)=A\left(x\right)-B\left(x\right){C}_{2}\left(x\right)\) |
(19) |
|---|---|
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\(u\left(x\right)=-[{B}^{T}\left(x\right){M}^{T}\left(x\right){P}^{-1}\left(\stackrel{-}{x}\right)+{C}_{2}(x\left)\right]Z\left(x\right)\) |
(20) |
The zero equilibrium point of a closed-loop system is asymptotic, and if \(P\left(\tilde{x}\right)\) is constant, the stability will be universal. Also, the following equation is considered for each initial condition inside the zero equilibrium absorption region.
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\({‖z‖}_{2}^{2}\le {z}^{T}\left({x}_{0}\right){P}^{-1}\left({\tilde{x}}_{0}\right)Z\left({x}_{0}\right)\) |
(21) |
The control method proposed in this study is applied to the PEMFC which is explained in the following section.
The Equations Considered For The Control Method
The equations used in this study are outlined in this section based on which the control method will be applied to PEMFC.
The feedback controller design with transient improvement
According to the explanations and equations illustrated above, Theorem 4 is presented for the control system considered in this study. The exponential stability of the system under the desired controller is achieved through the obtained equation in this theorem.
Theorem 4
According to the system defined in (9), the matrices and K(x), the number γ, a constant, and a set of squares are considered to satisfy the following equations:
$$minimize \gamma$$
Subject to
$${v}_{1}^{T}\left[P\left(\tilde{x}\right)-{\text{ϵ}}_{1}I\right]{v}_{1} is SOS$$
$$-{v}_{2}^{T}\left[\begin{array}{ccc}{{\Psi }}_{1}\left(x\right)-{\text{ϵ}}_{2}I& \text{*}& \text{*}\\ I& 0& \text{*}\\ 0& \gamma I& P\left(\tilde{x}\right)\end{array}\right]{v}_{2} is SOS$$
Then, the system under the following controller is exponentially stable:
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\(u\left(x\right)=K\left(x\right){P}^{-1}\left(\tilde{x}\right)Z\left(x\right)\) |
(22) |
Proof: To prove Theorem 4, the condition below is considered in the beginning:
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\(\frac{dV}{dt}\left(x\left(t\right)\right)\le -\lambda V\left(x\right)\) |
(23) |
where \(\lambda\) is a positive value, and also a positive constant, namely L can be defined in such a way that the system operates considering the below equation in terms of \(x\left(0\right)\).
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\(‖x\left(t\right)‖=L‖x\left(0\right)‖\text{*}\text{exp}\left(-\lambda t\right)\) |
(24) |
On the other hand, considering Lyapunov function:
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\(V\left(x\right)={Z}^{T}\left(x\right){P}^{-1}\left(\tilde{x}\right)Z\left(x\right)\) |
(25) |
The closed-loop system can be presented by:
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\(\dot{x}=\left(A\left(x\right)+B\left(x\right)K\left(x\right){P}^{-1}\left(\tilde{x}\right)\right)Z\left(x\right)\) |
(26) |
Since the matrices \(P\left(\tilde{x}\right)\) and \({P}^{-1}\left(\tilde{x}\right)\) are definite positive in terms of the whole values of x, the Lyapunov function is positive based on the whole values of x.
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\(\frac{dV}{dt}\left(x\left(t\right)\right)={Z}^{T}\left(x\right)\left({\left(A\left(x\right)+B\left(x\right)K\left(x\right){P}^{-1}\left(\tilde{x}\right)\right)}^{T}{M}^{T}{P}^{-1}\left(\tilde{x}\right)+{P}^{-1}\left(\tilde{x}\right)M\left(x\right)\left(A\left(x\right)+B\left(x\right)K\left(x\right){P}^{-1}\left(\tilde{x}\right)\right)+\sum _{j\in J}\frac{\partial {P}^{-1}}{\partial {x}_{j}}\left(\tilde{x}\right)\left({A}_{j}\left(x\right)Z\left(x\right)\right)\right)Z\left(x\right)\) |
(27) |
Moreover, the whole exponentially stable systems are considered asymptotically stable. Since exponential stability is assumed here, the following matrix will be definite for all negative x(t) values.
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\({\left(A\left(x\right)+B\left(x\right)K\left(x\right){P}^{-1}\left(\tilde{x}\right)\right)}^{T}{M}^{T}{P}^{-1}\left(\tilde{x}\right)+{P}^{-1}\left(\tilde{x}\right)M\left(x\right)\left(A\left(x\right)+B\left(x\right)K\left(x\right){P}^{-1}\left(\tilde{x}\right)\right)+\sum _{j\in J}\frac{\partial {P}^{-1}}{\partial {x}_{j}}\left(\tilde{x}\right)\left({A}_{j}\left(x\right)Z\left(x\right)\right)+\lambda {P}^{-1}\left(\tilde{x}\right)\le 0\) |
(28) |
Multiplying the matrix \(P\left(\tilde{x}\right)\) on both sides, it can be concluded that the following matrix must be negative:
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\(-\sum _{j\in J}\frac{\partial {P}^{-1}}{\partial {x}_{j}}\left(\tilde{x}\right)\left({A}_{j}\left(x\right)Z\left(x\right)\right)+P\left(\tilde{x}\right){A}^{T}\left(x\right){M}^{T}\left(x\right)+M\left(x\right)A\left(x\right)P\left(\tilde{x}\right)+{K}^{T}\left(x\right){B}^{T}\left(x\right){M}^{T}\left(x\right)+M\left(x\right)B\left(x\right)K\left(x\right)+2\lambda P\left(\tilde{x}\right)+\lambda {P}^{-1}\left(\tilde{x}\right)\le 0\) |
(29) |
The optimization problem is considered to improve the transient state by maximizing the value of λ. In the above equation, \(\lambda {P}^{-1}\left(\tilde{x}\right)\) is used to convert an inequality to a BMI problem that is non-convex. To convert the above inequality to an LMI problem, the Schur complement is employed as follows:
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\(\left[\begin{array}{cc}{\varPsi }_{1}\left(x\right)& I\\ I& -\frac{{P}^{-1}\left(\tilde{x}\right)}{\lambda }\end{array}\right]=\left[\begin{array}{cc}{\varPsi }_{1}\left(x\right)& I\\ I& 0\end{array}\right]-\left[\begin{array}{c}0\\ \frac{1}{\sqrt{\lambda }}I\end{array}\right]{P}^{-1}\left(\tilde{x}\right)\left[\begin{array}{cc}0& \frac{1}{\sqrt{\lambda }}I\end{array}\right]\le 0\) |
(30) |
The matrix \(M=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\) can be considered a (p + q)*(p + q) matrix according to which p and q are positive integers. Notably, A, B, C, D are p*p, p*q, q*p, and q*q matrices of complex numbers. Now, when A is invertible, M/A:=D-CA-1B defines the q*q matrix for the Schur complement of the block A of the matrix M. Accordingly, M/D:=A-BD-1C defines the p*p matrix for the Schur complement of the block D of the matrix M when D is invertible.
Using the Schur complement, the equation below is obtained:
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\(\left[\begin{array}{ccc}{{\Psi }}_{1}\left(x\right)& \text{*}& \text{*}\\ I& 0& \text{*}\\ 0& \gamma I& P\left(\tilde{x}\right)\end{array}\right]\le 0, \gamma =\frac{1}{\sqrt{\lambda }}\) |
(31) |
Now, a linear matrix that needs to be definite for all negative x(t) values is achieved. It should be noted that the minimum value of the variable γ is equal to the maximum value of λ.
The Pemfc Control
The structure of PEMFC and its unique features are illustrated in detail. Besides, the process of applying the control methods proposed in this research to PEMFC is explained.
The structure of PEMFC
Figure 1 illustrates the PEMFC structure, which is strikingly popular due to its high efficiency and density, low operating temperature, quick implementation, long lifetime of the stack and cell, and usability in hybrid vehicles. Also, PEMFC has low fatigue and can efficiently operate at low temperatures of 40–80°C, which leads to a relative increase in its implementation speed [43]. According to Fig. 1, the electrolyte material used is a polymer membrane known as a PEM and is broadly utilized in transportation applications. Notably, this material varies regarding the FC system type according to which fuel (usually hydrogen) is injected into the anode and air (or pure oxygen) enters towards the cathode. On the anode side, electrons are released when gas tries to pass through the electrolyte membrane, whose task is to filter the electrons and allows only 2H + ions to pass through. These ions are combined with oxygen atoms to produce water (H2O) on the cathode side as a by-product. Heat is also a by-product of other reactions that occur. It is noteworthy that unlike the internal combustion engines, in which fuel is mixed with air and ignited, there is a barrier between the fuel and the oxidizer in a fuel cell that prevents combustion. Thus, the pollution caused by the fuel cell and internal combustion engine is not the same. In other words, water and heat are the only by-products of the fuel cell [44].
Applying the control method to PEMFC
In this system, Oxygen partial pressure and Nitrogen partial pressure are obtained by the law of conservation of mass and the rule of the ideal gas.
|
\(\frac{d{p}_{{O}_{2}}}{dt}=\frac{R{T}_{fc}}{{V}_{ca}{M}_{{O}_{2}}}\text{*}\left[{W}_{{O}_{2}, in}-{W}_{{O}_{2}, out}-{W}_{{O}_{2}, react}\right]\) and \(\frac{d{p}_{{N}_{2}}}{dt}=\frac{R{T}_{fc}}{{V}_{ca}{M}_{{N}_{2}}}\text{*}\left[{W}_{{N}_{2}, in}-{W}_{{N}_{2}, out}\right]\) | (32) |
\({M}_{{O}_{2}}\) and \({M}_{{N}_{2}}\) denote molar masses of Oxygen and Nitrogen; Vca and R represent the cathode volume and the universal gas constant, respectively; Tfc denote FC temperature; \({W}_{{O}_{2}, in}\) and \({\text{W}}_{{\text{O}}_{2}, \text{o}\text{u}\text{t}}\)represent the mass flow rate of oxygen inlet and outlet. Whereas \({W}_{{N}_{2}, in}\) and \({W}_{{N}_{2}, out}\) determine the mass flow rate of Nitrogen inlet and outlet flow. In addition, \({\text{W}}_{{\text{O}}_{2}}\) indicates the mass flow rate of the oxygen reaction flow. Here, the mass flow of the input flow is obtained through the following equations:
|
\({W}_{{O}_{2}, in}={x}_{{O}_{2}}{W}_{ca, in}\), \({W}_{{N}_{2}, in}=\left(1-{x}_{{O}_{2}}\right)\text{*}{W}_{ca, in}\) |
(33) |
|---|---|
|
\({W}_{ca, in}=\frac{1}{1+{\omega }_{atm}}\text{*}{k}_{ca, in}\text{*}\left[{p}_{sm}-{p}_{ca}\right]\), \({\omega }_{atm}=\frac{{M}_{v}}{{M}_{a}}\text{*}\left[\frac{{\varphi }_{atm}\text{*}{p}_{sat}\left({T}_{atm}\right)}{{p}_{atm}-\left({\varphi }_{atm}\text{*}{p}_{sat}\left({T}_{atm}\right)\right)}\right]\) | (34) |
\({x}_{{O}_{2}}\) represents the mass of oxygen, and \({\varphi }_{atm}\) indicates the relative humidity (which is 0.5 on average); \({p}_{atm}\) and \({p}_{atm}\left({T}_{atm}\right)\) show the atmospheric and saturation pressures; \({k}_{ca}\) is the constant of the cathode inlet. Also, \({p}_{ca}\)is the cathode pressure calculated by the following equation:
|
\({p}_{ca}={p}_{{O}_{2}}+{p}_{{N}_{2}}+{p}_{sat}\) |
(35) |
The mass flow of outlet flow is calculated through the following equations:
|
\({W}_{{O}_{2}, out}={W}_{ca, out}\text{*}\left[\frac{{M}_{{O}_{2}}{p}_{{O}_{2}}}{{M}_{{O}_{2}}{p}_{{O}_{2}}+{M}_{{N}_{2}}{p}_{{N}_{2}}+{M}_{v}{p}_{sat}}\right]\) \({W}_{{N}_{2}, out}={W}_{ca, out}\text{*}\left[\frac{{M}_{{N}_{2}}{p}_{{N}_{2}}}{{M}_{{O}_{2}}{p}_{{O}_{2}}+{M}_{{N}_{2}}{p}_{{N}_{2}}+{M}_{v}{p}_{sat}}\right]\) |
(36) |
where, \({W}_{ca, out}\) denotes the mass flow for the outlet flow of cathode and is obtained by (37), according to which \({k}_{ca, out}\) is the constant of the outlet flow of cathode.
|
\({W}_{ca, out}={k}_{ca, out}\text{*}{\left({p}_{ca}-{p}_{atm}\right)}^{0.5}\) |
(37) |
The consumption rate of oxygen that reacts at the cathode is represented by the basic electrochemical principles below:
|
\({W}_{{O}_{2}, react}={M}_{{O}_{2}}\text{*}\left[\frac{n{I}_{st}}{4F}\right]\) |
(38) |
In this formula, F indicates the Faraday number and n is the number of cell stacks. In addition, \({I}_{st}\) shows the stack flow.
The following mechanical relation presents angular velocity dynamics (angular acceleration):
|
\(\frac{d{\omega }_{cp}}{dt}=\frac{\left[{\tau }_{cm}-{\tau }_{cp}-{\tau }_{f}\right]}{{J}_{cp}}\) |
(39) |
\({J}_{cp}\) denotes the compressor motor's inertia; \({\tau }_{cm}\), \({\tau }_{cm}\), and \({\tau }_{f}\) represent the compressor torques, the compressor flow, and orifice.
|
\({\tau }_{cm}={\eta }_{cm}{k}_{t}{I}_{q}\) \({\tau }_{f}=f{\omega }_{cp}\) \({\tau }_{cp}=\left[\frac{{C}_{p}{T}_{atm}}{{\eta }_{cp}{\omega }_{cp}}\right]\text{*}\left[{\left(\frac{{p}_{sm}}{{p}_{atm}}\right)}^{\left(1-\frac{1}{\gamma }\right)}-1\right]\text{*}{W}_{cp}\) |
(40) |
where \({k}_{t}\) denotes the motor constant, \({\eta }_{cp}\) is the compressor performance, and f indicates the engine friction. Also, \({{\eta }}_{\text{c}\text{m}}\) represents the engine's mechanical efficiency; γ indicates the air-specific heat ratio, and \({C}_{p}\) denotes the specific heat capacity of the air. Finally, the mass flow of the compressor flow is represented by \({W}_{cp}\) [45].
It is noteworthy that the following differential equation represents the air pressure dynamics of the power port:
|
\(\frac{d{p}_{sm}}{dt}=\left[\frac{R{T}_{cp}}{{M}_{a}{V}_{sm}}\right]\text{*}\left[{W}_{cp}-{k}_{ca, in}\text{*}\left({p}_{sm}-{p}_{ca}\right)\right]\) |
(41) |
where \({V}_{sm}\) and \({T}_{cp}\)indicate the power port volume and the temperature of the compressor outlet gas.
|
\({T}_{cp}=\frac{{T}_{atm}}{{\eta }_{cp}}\left({\left[\frac{{p}_{sm}}{{p}_{atm}}\right]}^{1-\frac{1}{\gamma }}-1\right)+{T}_{atm}\) |
(42) |
Clearly, the state equations of the system contain many parameters and are fundamentally complex. The fixed system parameters have been aggregated for simplification, and the complete model has been simplified as follows:
|
\(x={\left[{p}_{{O}_{2}} {p}_{{N}_{2}} {\omega }_{cp} {p}_{sm}\right]}^{T}={\left[{x}_{1} {x}_{2} {x}_{3} {x}_{4}\right]}^{T}\) |
(43) |
In general, the complete system is defined as follows:
|
\({\dot{x}}_{1}={c}_{1}\left[{x}_{4}-{x}_{1}-{x}_{2}-{c}_{2}\right]-{c}_{17}\left[\frac{{c}_{3}{x}_{1}}{{c}_{4}{x}_{1}+{c}_{5}{x}_{2}+{c}_{6}}\right]{\left[{x}_{1}+{x}_{2}+{c}_{2}-{c}_{11}\right]}^{0.5}-{c}_{7}\zeta\) \({\dot{x}}_{2}={c}_{8}\left[{x}_{4}-{x}_{1}-{x}_{2}-{c}_{2}\right]-{c}_{17}\left[\frac{{c}_{3}{x}_{2}}{{c}_{4}{x}_{1}+{c}_{5}{x}_{2}+{c}_{6}}\right]{\left[{x}_{1}+{x}_{2}+{c}_{2}-{c}_{11}\right]}^{0.5}\) \({\dot{x}}_{3}={c}_{13}u-{c}_{9}{x}_{3}-{c}_{10}\left({\left[\frac{{x}_{4}}{{c}_{11}}\right]}^{{c}_{12}}-1\right)\) \({\dot{x}}_{4}={c}_{14}\left({c}_{15}\left({\left[\frac{{x}_{4}}{{c}_{11}}\right]}^{{c}_{12}}-1\right)+1\right)\left({W}_{cp}-{c}_{16}\left[{x}_{4}-{x}_{1}-{x}_{2}-{c}_{2}\right]\right)\) |
(44) |
with respect to (34), \(u={I}_{q}, and \zeta ={I}_{st}\).
In this study, the quadratic current of motor is considered as the control input (u), and the stack current (ζ) is regarded as a measurable turbulence input. The pressures x1, x2, and x4 are positive, and since the air supply is unidirectional, the compressor speed x3 is always positive.
The next sections presented the equations that need to be considered for the proposed control method. Also, the control purposes are outlined in this section.
The control outputs
In the commercial PEM fuel cells, below set of variables are considered as measureable parameters, and they can be introduced as outputs:
|
\(y={\left[{y}_{1} {y}_{2} {y}_{3}\right]}^{T}={\left[{V}_{st} {p}_{sm} {W}_{cp}\right]}^{T}\) |
(45) |
where \({V}_{st}\) and \({p}_{sm}\) are the stack voltage and the pressure of the supply port, respectively, and \({W}_{cp}\) indicates the compressor airflow. The airflow at the compressor outlet is a function of the angular velocity of the compressor and the pressure of the supply port, which is defined as follows [6]. Accordingly, \({\eta }_{v-c}\) is the volume efficiency.
|
\({y}_{3}\left({x}_{3}\right)=\frac{1}{2\pi }\left[{V}_{\frac{cpr}{tr}}{\eta }_{v-c}{\rho }_{a}{x}_{3}\right]={c}_{21}{x}_{3}\) |
(46) |
The controller’s aims
In fact, the controllers that bring the most accurate results and work with the highest efficiency are significantly welcome. This aim is obtained when most of the uncertainties can be overcome through these controllers. Eq. (47) outlines the system efficiency that is the central attention of this work.
|
\(z={\left[{z}_{1} {z}_{2}\right]}^{T}={\left[{P}_{net} {\lambda }_{{O}_{2}}\right]}^{T}\) |
(47) |
where \({P}_{net}\) denotes the FC net power (the difference between the whole power generated of the stack and the power used of the compressor), also, \({\lambda }_{{O}_{2}}\) the oxygen excess ratio (excess air in the stack). This parameter is the rate of fed and used oxygen in the reaction period at the cathode. These variables can be presented mathematically by:
|
\({P}_{net}={P}_{st}-{P}_{cm}\), \({\lambda }_{{O}_{2}}=\frac{{W}_{{O}_{2}, in}}{{W}_{{O}_{2}, react}}\), \({P}_{st}={V}_{st}{I}_{st}\), \({P}_{cm}={\tau }_{cm}{\omega }_{cp}\) |
(48) |
The above equation can be written as follows:
|
\({z}_{1}={y}_{1}\zeta -{c}_{18}{x}_{3}u\), \({z}_{2}=\frac{{c}_{19}}{{c}_{20}\zeta }\left[{x}_{3}-{x}_{1}\right]\) | (49) |
The control purpose of the fuel cell is considered for the optimized \({P}_{net}\), that is measured at the stack output. As it’s shown, \({P}_{net}\) and \({\lambda }_{{O}_{2}}\) are interconnected through ζ. As illustrated in the literature, the maximum net power for various currents is between 2 and 2.5 [46]. Therefore, according to the load current, the control purpose is met by regulating \({\lambda }_{{O}_{2}}\) at the operation points.
The optimal operation points are already known concerning the obtained results, and the optimization algorithm does not need to be used. Also, the net power is easily maximized through regulating the OER for the pre-defined operation point. The excess rate of the reference oxygen is extracted from the experimental results and is presented as a net power-dependent polynomial and stack current. The characteristic curve is presented as a third-degree polynomial which is shown below:
|
\({z}_{2, ref}=5e-8{\zeta }^{3}-2.87e-5{\zeta }^{2}+2.23e-3\zeta +2.5\) |
(50) |
The criterion below is considered for the optimization process:
|
\(PI=1-\frac{{P}_{net}}{{P}_{net, ref}}\) |
(51) |
and
|
\({P}_{net, ref}=1.562e-2{\zeta }^{2}+5.671e+1\zeta +6.063e+2\) |
(52) |
The main results obtained in this study are illustrated in this section, and the outputs are analyzed further. In this analysis, \(\dot{x}=f\left(x\right)\) is considered as the nonlinear system in which \(f\left(x\right)\) is the analytic nonlinear functions’ vector \({f}_{i}\left(x\right)\), and \(x\in {R}^{n}\) is the system state vector. Also, \(\stackrel{-}{x}\) is considered to be an equilibrium point and \(f\left(\stackrel{-}{x}\right)-\stackrel{-}{x}=0\).
Applying the Taylor series for the system
Each function \({\text{f}}_{i}\left(x\right)\) can be approximated around the point \(\stackrel{-}{x}\in {\text{R}}^{\text{n}}\) in the Taylor series.
|
\({f}_{i}\left(\stackrel{-}{X}+W\right)={f}_{i}\left(\stackrel{-}{x}\right)+\sum _{i=1}^{l}\frac{1}{j!}{\left(w\nabla \right)}^{i}f\left(\stackrel{-}{x}\right)+{R}^{(l+1)}≔f\left(\stackrel{-}{x}\right)+{g}_{i}^{\left(l\right)}\left(w\right)+{R}^{(l+1)}\) |
(53) |
where, \(w=x-\stackrel{-}{x}\) and\(\nabla =(\frac{\partial }{\partial {x}_{1}},\frac{\partial }{\partial {x}_{2}},\dots ,\frac{\partial }{\partial {x}_{n}})\)
Applying the Taylor expansion and removing the higher-order expression, the nonlinear system \(\dot{x}=f\left(x\right)\) can be approximated by the h-order polynomial system below.
|
\({f}_{i}\left(x\right)=\sum _{{j}_{i}=0}^{{h}_{1}}\dots \sum _{{j}_{i}=0}^{{h}_{1}}{{a}_{i{j}_{1}\dots {j}_{n}}}_{ }{x}_{1}^{{j}_{1}}\dots {x}_{n}^{{j}_{n}}, i=\text{1,2},\dots ,n\) |
(54) |
where the expression \(h={h}_{1}+\dots +{h}_{n}\) specifies the polynomial system’s order based on \({a}_{i{j}_{1}\dots {j}_{n}}\ne 0\). In the Taylor series, the expressions with orders higher than h are removable for \({f}_{i}\left(x\right)\). Also, the expansion of the Taylor series around the equilibrium point \(\stackrel{-}{x}=0\) is equal to the original system.
Applying the above approximation to the dynamics of the PEMFC system presented in (34), its polynomial model is extracted. For evaluation of the accuracy of the approximated system, the open-loop PEMFC nonlinear system is simulated assuming that the control input is zero, and the trajectory and time series of the open-loop system states are compared with the polynomial model states. The simulation and comparison results of the two systems are highlighted in Figs. 2–7. To be more detailed, the difference between the nonlinear PEMFC system and approximated systems based on the examined parameters is indicated in these figures.
According to Fig. 2, there is a high agreement between the PEMFC nonlinear system and approximated system except when the oxygen partial pressure varies from 3*105 to 5*105 and the compressor angular speed varies from zero to 800. The same result is observed in Fig. 3 where the compressive angular speed varies from zero to 800 and the nitrogen partial pressure 12*105 to 16*105.
It should be noted that there is a minor difference between the results of the nonlinear PEMFC system and the approximated system in Figs. 4 and 5. However, when the values reach their peak, a slight gap is created between the results. This observation is related to both Figs. 4 and 5 where the oxygen and the nitrogen partial pressures variations over time in the PEMFC system are considered. As depicted in Fig. 6, the two diagrams thoroughly match each other and the difference between them is about zero. The supply manifold pressure variations over time are highlighted in Fig. 7 whose results are similar to the ones shown in Figs. 4 and 5. Accordingly, the nonlinear PEMFC system results and the approximated system results find a minor difference when the parameters' values reach their highest level.
According to the results illustrated in Figs. 2–7, it is clear that the approximated polynomial system accurately highlights the nonlinear system's behavior. Therefore, the optimal controller can be designed for a polynomial system and applied to a nonlinear system, which is discussed in the following.
Applying the proposed method to the PEMFC system
The results obtained through the use of the proposed controller irrespective of the operator saturation and uncertainty in the dynamic system of PEMFC are outlined here. The parameters’ values used in this study are indicated in Table 1. A permanent magnet synchronous motor (PMSM) is used to feed the compressor. The 3-phase PMSM current is computed in DQ coordinates and it’s fed by an inverter. Also, the control input is produced using the suggested controller that is implemented in a real-time control system and applied to the inverter. Notably, the measured airflow of the compressor is given to the FC simulation model in real-time. The output load is considered the variable steps of the stack current between 150 and 400 A, shown in Fig. 9. Considering the quasi-paradigm model obtained by Taylor expansion and the excess rate of reference oxygen given in (40), the results shown in Figs. 8–11are obtained.
|
Symbol |
Parameter |
Value |
Symbol |
Parameter |
Value |
|---|---|---|---|---|---|
|
n |
Number of cells in the stack |
90 |
\({A}_{T}\) |
The opening area of the nozzle |
0.00138 m2 |
|
R |
Universal gas constant |
8.314 J/(mol K) |
\(\gamma\) |
The ratio of specific heats of air |
1.4 |
|
\({R}_{a}\) |
Air-gas constant |
286.9 J/(kg K) |
\({J}_{cp}\) |
Compressor inertia |
671.9 \(\times\)10− 5 |
|
\({P}_{atm}\) |
Atmospheric pressure |
1.01325 \(\times\)105 Pa |
\(f\) |
Motor friction |
0.00136 v/(rad/s) |
|
\({T}_{atm}\) |
Atmospheric temperature |
298.15 K |
\({k}_{t}\) |
Motor constant |
0.31 N m.A |
|
\({T}_{fc}\) |
Universal gas constant |
353.15 K |
\({C}_{p}\) |
Constant pressure specific heat of air |
1004 J/(kg K) |
|
\(F\) |
Faraday constant |
96485 C/mol |
\({\eta }_{cp}\) |
Compressor efficiency |
80% |
|
\({M}_{a}\) |
Air molar mass |
28.9644 \(\times\) 10− 3 kg/mol |
\({\eta }_{cm}\) |
Motor mechanical efficiency |
98% |
|
\({{M}_{o}}_{2}\) |
Oxygen molar mass |
32 \(\times\)10− 3 kg/mol |
\({V}_{ca}\) |
Cathode volume |
0.0015 m3 |
|
\({{M}_{N}}_{2}\) |
Nitrogen molar mass |
28 \(\times\) 10− 3 kg/mol |
\({V}_{sm}\) |
Supply manifold volume |
0.003 m3 |
|
\({M}_{v}\) |
Vapor molar mass |
18.02 \(\times\) 10− 3 kg/mol |
\({V}_{cpr/tr}\) |
Compressed volume per turn |
5\(\times\) 10− 4 |
|
\({C}_{D}\) |
Discharge of the nozzle |
0.0083 |
\({K}_{ca, in}\) |
Cathode inlet orifice constant |
0.3629 \(\times\)10 − 5 kg/(Pa s) |
|
\({\rho }_{a}\) |
Air density |
1.23 kg/m3 |
\({X}_{{O}_{2}, C{O}_{2}}\) |
Oxygen mass fraction |
0.23 |
|
\(\mu\) |
Smoothing filter time constant |
0.01 s |
|||
It is worth mentioning that the stack current variations over time are indicated in Fig. 8. As it’s shown, a set of variable stack current steps is presented as the output load that varies between 150 and 400 A. As shown in Fig. 9, the excess rate of oxygen follows its reference precisely without tracing error. The speed changes’ diagram of the compressor over time is highlighted in Fig. 10. Also, the control input (Iq) is shown in Fig. 11, which varies between 0 and 3 amps.
For indicating the reliability of the proposed and its remarkable benefits over the previous ones, the results of this study are compared to the results of the study conducted by Laghrouche et al. [7]. The robust and adaptive controller results considered in the study of Laghrouche et al. [7] are outlined in Figs. 12 and 13. Clearly, Figs. 12-a and 13-a indicate the oxygen excess ratio over time which are compared with Fig. 10. Similarly, Figs. 12-b and 13-b show the current value over time which are compared with Fig. 11.
According to Fig. 9 and Figs. 12-a and 13-a, by the suggested control method in the present work result in less fluctuation, and the control purpose is achieved more satisfactorily. Unlike the adaptive controller used in [7], the fluctuation is not observed in this study's control method, which is considered a unique benefit.
With respect to Figs. 12-b, 13-b, and 8, the rise time in the proposed method is more quickly and reaches its peak earlier than the robust and adaptive controller presented in [7]. In contrast to the outputs of this reference, the control method proposed in this research has less overshoot and smoothly reaches the desired point. However, in the beginning, the proposed method has an undershoot and then reaches the final point, and this is considered the drawback of this method compared to the robust and adaptive controller. Nevertheless, the overshoot of this method in each point is generally less than the methods indicated in Figs. 12-b and 13-b. Hence, the remarkable advantages of the proposed method outweigh its pitfalls since it has a better response speed, less settling time, and fluctuation compared to the results of reference [7] (Figs. 12 and 13).
As regards the explanations above, the proposed method generally outperforms the control method of reference [7] in terms of settling time, response speed, and accuracy. Regardless of being more accurate and applicable (The comparison between Figs. 9 and 11 and Figs. 12 and 13), the proposed controller also ensures stability and efficiency under the system's parametric uncertainties, and this is the main benefit of the proposed method against the shortcomings of the previous methods.
In summary, the current paper aims to control the PEMFC’s parameters with the highest accuracy and stability. For this purpose, the open-loop nonlinear system is given in state-dependent polynomial equations, and the state feedback method is considered to provide the control responses. Furthermore, SDP based on the SOS method is utilized to numerically solve the state-dependent LMIs. The nonlinear model presented in the study of Laghrouche et al. was converted into a nonlinear polynomial model, and the response speed of the controller was enhanced. The saturation constraint is considered for the controller, and by expressing a theorem, the optimal-state feedback controller is employed for PEMFC. Due to the high parametric and nonlinear uncertainties of the studied system, the linear model cannot meet the control needs of this system well, and a nonlinear model is proposed to tackle this problem. According to this system's literature and performance criteria, the significant control criteria were introduced that can be solved using the control method proposed in this research. Notably, modeling and how to convert a nonlinear model to a polynomial model were presented. Finally, the improved controller designed in this study was applied to the polynomial system to regulate the rate of excess oxygen. The results of this method were compared to the result of a valid study and turned out to be more advantageous in terms of accuracy and speed. In fact, using the proposed method, the rate of the excess oxygen is regulated and is maintained optimally to optimize the output power. Unlike most of the related studies proposed up to now, stability and efficiency under the parametric uncertainties are ensured in this method. Besides, this method presents a good starting point for discussion and further investigations. In the future, further investigations could prove quite beneficial to the literature.
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript
Competing Interests
The authors have no relevant financial interests to disclose.
Author Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by behnam sobhani and Zahra Rahmani. The first draft of the manuscript was written by Behnam Sobhani and Zahra Rahmani commented on the manuscript. All authors read and approved the final manuscript.
Data Availability
The datasets generated during the current study are not publicly available but are available from the corresponding author.
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