Advanced Control Strategies for the Oxygen in the Cathode

Abstract

This chapter presents two advanced control strategies based on model predictive control to control the oxygen level in the cathode of a proton exchange membrane (PEM) fuel cell system. The objectives are to achieve a better efficiency and to maintain the necessary level of the oxygen in the cathode to prevent short circuit and membrane damage. First, a methodology of control based on dynamic matrix control (DMC) is proposed. This strategy includes a stationary and dynamic study of the advantages of using a regulating valve for the cathode outlet flow in combination with the compressor motor voltage as manipulated variables in a PEM fuel cell system. The influence of this input variable is exploited by implementing a predictive control strategy based on DMC, using these manipulated variables. The objectives of this control strategy are to regulate both the fuel cell voltage and oxygen excess ratio in the cathode, and thus, to improve the system performance. Second, a methodology of control based on adaptive predictive control with robust filter (APCWRF) is proposed. The APCWRF is designed for controlling the compressor motor voltage. Because of the wide working range the algorithm is improved with three different zones supported by three nominal linear models.

Appendix

1.1 Recursive Identification Algorithms

In this section a brief description about recursive system identification algorithms is presented.

Remembering the linear regression form in Eq. 43

$$ y(k)=\psi(k)^{T}\theta $$

(63)

and supposing the parameters vector $\theta$ unknown but the input–output data were collected suitably from an identification experiment, so

$$ Z^N=\{u(1),y(1),\ldots,u(N),y(N)\} $$

(64)

are known and N is the data base dimension. Then, the predictions in Eq. 63 can be used to fit the data base observations in a least squares sense by mean of cost criterion presented in Eq.  65

$$ \min_{\theta} V_N\left(\hat{\theta},Z^N\right), $$

(65)

where $\hat{\theta}$ is the estimated parameters vector and

$$ V_N\left(\hat{\theta},Z^N\right)={\frac{1}{N}}\sum_{k=1}^{N}\left(y(k)-\hat{y}(k)\right)^2= {\frac{1}{N}}\sum_{k=1}^{N}\left(y(k)-\psi(k)^T\hat{\theta}\right)^2. $$

(66)

In this case, the least square estimate can be represented as

$$ \hat{\theta}_{N}=\arg\min_{\hat{\theta}} V_N\left(\hat{\theta},Z^N\right), $$

(67)

where $\hat{\theta}_{N}$ indicates that N input–output samples were used.

Being $V_N$ a quadratic function in $\hat{\theta}$ the optimal solution can be easily found by

$$ 0={\frac{\hbox{d}V_N\left(\hat{\theta},Z^N\right)}{\hbox{d}\hat{\theta}}}= {\frac{2}{N}}\sum_{k=1}^{N}\psi(k)\left(y(k)-\psi(k)^T\hat{\theta}\right). $$

(68)

Reordering:

$$ \sum_{k=1}^{N}\psi(k)y(k)=\sum_{k=1}^{N}\psi(k)\psi(k)^T\hat{\theta}, $$

(69)

or similarly:

$$ \hat{\theta}_N=\left[\sum_{k=1}^{N}\psi(k) \psi(k)^T\right]^{-1}\sum_{k=1}^{N}\psi(k)y(k). $$

(70)

1.1.1 Recursive Least Squares

The least squares methodology presented in the previous section can be modified to update the parameters vector recursively as new input–output data are acquired from the process [8, 25].

Rewriting the covariance matrix in Eq. 70 as

$$ P_{N}=\left[\sum_{k=1}^{N}\psi(k)\psi(k)^T\right]^{-1}. $$

(71)

Then,

$$ P^{-1}_{N}=\sum_{k=1}^{N-1}\psi(k)\psi(k)^T+\psi(N)\psi(N)^T= P^{-1}_{N-1}+\psi(N)\psi(N)^T $$

(72)

and the estimated parameters vector

$$ \begin{aligned}[b] \hat{\theta}_{N}&=P_{N}\left[\sum_{k=1}^{N-1}\psi(k)y(k)+\psi(N)y(N)\right]\\ &=P_{N}\left[P_{N-1}^{-1}\hat{\theta}_{N-1}+\psi(N)y(N)\right]\\ &=\hat{\theta}_{N-1}+P_{N}\psi(N)\left[y(N)-\psi(N)^T\hat{\theta}_{N-1}\right]. \end{aligned} $$

(73)

Thus, the following recursive algorithm can be obtained:

$$ \begin{aligned} \hat{\theta}_{N}&= \hat{\theta}_{N-1}+K(N)\left[y(N)-\psi(N)^T\hat{\theta}_{N-1}\right]\\ K(N)&= P_{N}\psi(N)\\ \varepsilon(N) &= y(N)-\psi(N)^T\hat{\theta}_{N-1}. \end{aligned} $$

(74)

From Eq. 74, $\varepsilon(N)$ represents the output prediction error in $k=N$ computed by using the parameters vector estimated in the previous sample time $k=N-1.$ The first equation shows that the parameters vector estimate is computed by mean of the previous one and a correction factor that depends on the prediction error. The $K(N)$ term indicates how the previous estimate must be modified in the updating procedure.

The main problem with the recursive algorithm shown in Eq. 74 is the matrix inversion in $P_{N}$ for each sample instant. This problem can be avoided by using the matrix inversion lemma, which defines that

$$ \left[A+BCD\right]^{-1}=A^{-1}-A^{-1}B\left[DA^{-1}B+C^{-1}\right]^{-1}DA^{-1}. $$

(75)

Rewriting Eq. 71:

$$ P_{N}=\left[P^{-1}_{N-1}+\psi(N)\psi(N)^T\right]^{-1} $$

(76)

and applying the previous lemma:

$$ P_{N}=P_{N-1}-{\frac{P_{N-1}\psi(N)\psi(N)^TP_{N-1}}{1+\psi(N)^TP_{N-1}\psi(N)}}, $$

(77)

that is a recursive expression for $P_{N}.$ In this context, the recursive least squares algorithm is

$$ \begin{aligned} [b] \hat{\theta}_{N} &= \hat{\theta}_{N-1}+K(N)\left[y(N)-\psi(N)^T\hat{\theta}_{N-1}\right]\\ K(N)&= P_{N}\psi(N)\\ \varepsilon(N) &= y(N)-\psi(N)^T\hat{\theta}_{N-1}\\ P_{N} &= P_{N-1}-{\frac{P_{N-1}\psi(N) \psi(N)^TP_{N-1}}{1+\psi(N)^TP_{N-1}\psi(N)}} \end{aligned} $$

(78)

An interesting variant of this algorithms is the called RLS with forgetting factor. In this strategy the functional cost stated in Eq. 66 is augmented by introducing a parameter $\lambda$ as is shown in Eq. 79:

$$ V_N\left(\hat{\theta},Z^N\right) ={\frac{1}{N}}\sum_{k=1}^{N}\lambda^{N-k}\left(y(k)-\psi(k)^T\hat{\theta}\right)^2 $$

(79)

with $0<\lambda\leq1.$ Thus, an exponential weighting is made on the prediction error giving more importance to the new data ($\lambda^{N-k}$ near to 1) and less emphasis to the old ones ($\lambda^{N-k}$ near to 0). Applying the same procedure as for the RLS case, the following algorithm can be obtained,

$$ \begin{aligned} \hat{\theta}_{N} &= \hat{\theta}_{N-1}+K(N)\left[y(N)-\psi(N)^T\hat{\theta}_{N-1}\right]\\ K(N)&= P_{N}\psi(N)\\ \varepsilon(N) &= y(N)-\psi(N)^T\hat{\theta}_{N-1}\\ P_{N}&={\frac{1}{\lambda}}\left[P_{N-1}-{\frac{P_{N-1}\psi(N) \psi(N)^TP_{N-1}}{\lambda+\psi(N)^TP_{N-1}\psi(N)}}\right].\\ \end{aligned} $$

(80)

This approach is called RLS with forgetting factor. Note the only difference with the classical RLS in Eq. 78.

1.1.2 Recursive Least Squares with UD-Factorization

It is very helpful, from a numerical point of view, to represent the matrices by factorization. This avoids problems with large ill-conditioned matrices. A classical method is the UD-factorization. The basic idea is to represent the covariance matrix error in the RLS algorithms as is shown in Eq. 81:

$$ {\mathbf{P}}(k)={\mathbf{U}}_*(k){\mathbf{D}}_*(k){\mathbf{U}}_*(k), $$

(81)

where ${\mathbf U}_*(k)$ is a triangular superior matrix with unitary principal diagonal and y ${\mathbf D}_*(k)$ a diagonal matrix [8, 9, 25, 26].

The RLS with forgetting factor and UD-factorization is

A Initialization, $k=0$

$$ {\mathbf D}_*(0)=\delta{\mathbf I},\,\hbox{con} \,\delta=10^2-10^4 $$

B For$k>0$

$$ \begin{aligned} {\mathbf f}(k)&={\mathbf U}_*^T(k-1)\psi(k)\\ {\mathbf v}(k)&={\mathbf D}_*(k-1)f(k)\\ \alpha_0(k)&=\lambda \end{aligned} $$

• 1. For $j=1,2,\ldots,N$

  $$ \begin{aligned} \alpha_j(k)&=\alpha_{j-1}(k)+f_{j}(k)v_{j}(k)\\ D_{jj}(k)&=\alpha_{j-1}(k)D_{jj}(k-1)/\alpha_{j}(k)\lambda\\ \tilde{\gamma}_{j}(k)&=v_j(k)\\ u_j(k)&=-f_j(k)/\alpha_{j-1}(k)\\ \end{aligned} $$

• 2. For $i=1,\ldots,j-1$

  $$ \begin{aligned} U_{ij}(k)&=U_{ij}(k-1)+\tilde{\gamma}_{i}(k)u_j(k)\\ \tilde{\gamma}_{i}(k) &\leftarrow \tilde{\gamma}_{i}(k)+U_{ij}(k-1)\tilde{\gamma}_{j}(k)\\ \end{aligned} $$

• 3. Compute

  $$ \gamma(k)={\frac{1}{\alpha_N(k)}}\left[\tilde{\gamma}_{1}(k), \tilde{\gamma}_{2}(k),\ldots,\tilde{\gamma}_{N}(k)\right]^T $$

• 4. Update

  $$ \hat{\theta}(k)=\hat{\theta}(k-1)+\gamma(k)\left[y(k)-\psi^T(k)\hat{\theta}(k-1)\right] $$

• 5. Store${\mathbf U}_*(k),$${\mathbf D}_*(k)$ and $\hat{\theta}(k)$

where $y(k)$ is the process output, $\psi(k)$ the regressor vector (inputs for FIR model) and $\lambda$ the forgetting factor. With this algorithm the covariance matrix, ${\mathbf P}(k),$ is updated recursively by actualizing its factorization ${\mathbf U}_*(k)$ and ${\mathbf D}_*(k).$ Moreover, the parameters vector is estimated recursively also.