Adaptive neural output feedback control of automobile PEM fuel cell air-supply system with prescribed performance

Abstract

Oxygen excess ratio (OER) is a key specification of fuel cells, which influences the net power and health state. To reconstitute the unmeasurable variable and achieve precise tracking accuracy, the observer-based adaptive neural network control using a prescribed performance function is proposed for the polymer electrolyte membrane (PEM) fuel cell air-supply system. Firstly, an observer is designed to recover the unmeasurable variable based on the transformed canonical system. Secondly, a finite-time prescribed performance function is constructed to guarantee the maximal overshoot and steady-state tracking error within the quantitative boundary. The contribution of the proposed control scheme can be concluded that: 1) the different errors are simultaneously used to update the neural network weights for the improvement of the observer performance; 2) the restriction that the initial error is required to be within the performance function bound is relaxed by proposing a tuning function and 3) the convergence time and residual set of OER tracking error can be determined qualitatively. The signals included in the air-supply system are proved to be uniformly ultimately bounded. Different numerical simulations and hardware-in-loop (HIL) experiments show that the more accurate estimation is provided by the proposed observer. Meanwhile, the tracking errors are restricted within the predefined bounds. From the experimental results, the proposed observer and controller show the best performance indexes including the root mean square error (RMSE), the mean absolute error (MAE) and the standard deviation (SD) in different conditions.

Appendix

The proof of smoothness of function ς(t) is explained as follows:

1) When 0 < t < T_f or t ≥ T_f, it is obvious that ς(t) is continuous. Meanwhile, because of $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {\varsigma }(t)=\underset {t\to T_{f}^{+}}{\mathop {\lim }} {\varsigma }(t)={\varsigma }_{T_{f}}$, the function ς(t) is continuous when t > 0.

2) When t ≥ T_f, the derived function ς⁽ⁿ⁾(t) = 0, n ≥ 1 is continuous and it is viewed that $\underset {t\to T_{f}^{+}}{\mathop {\lim }} {\varsigma }^{(n)}(t)=0$.

3) When 0 < t < T_f, the smoothness of function ς(t) can be proven step by step.

Step 1. The first-order derivative of ς(t) is given as:

$$ \begin{array}{@{}rcl@{}} \frac{{\partial{\varsigma}(t)}}{\partial t}\!\!\!&&=\frac{\partial \ell e^{\hbar(1-\frac{T_{f}}{T_{f}-t})}}{\partial t}{\varUpsilon}(t)+\frac{\partial {\varUpsilon}(t)}{\partial t}\ell e^{\hbar (1-\frac{T_{f}}{T_{f}-t})}\\ &&=\ell \frac{dq(t)}{dt}e^{q(t)}{\varUpsilon}(t)+\ell\frac{d{\varUpsilon}(t)}{dt}e^{q(t)} \end{array} $$

(70)

where $q(t)=\frac {\hbar t}{t-T_{f}}$. It follows from (70), one can obtain that $\frac {dq(t)}{dt}=-\frac {\hbar T_{f}}{(t-T_{f})^{2}}$, $\frac {d{\varUpsilon }(t)}{dt}=-\frac {\pi }{2T_{f}}\sin \limits (\frac {\pi t}{T_{f}})$, $\underset {t\to T_{f}^{-}}{\mathop {\lim }} e^{q(t)}=0$, $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {\varUpsilon }(t)=0$. Thus, $\underset {t\to T_{f}^{-}}{\mathop {\lim }} \frac {d{\varUpsilon }(t)}{dt}e^{q(t)}=0$. Based on the L’Hospital’ rule, the following one can be obtained through the double derivations of numerator and denominator:

$$ \begin{array}{@{}rcl@{}} &\underset{t\to T_{f}^{-}}{\mathop{\lim}} \frac{dq(t)}{dt}e^{q(t)}=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{\frac{\hbar T_{f}}{(T_{f}-t)^{2}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{\frac{2}{T_{f}-t}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{2e^{\frac{\hbar t}{t-T_{f}}}}{\hbar T_{f}}\right]=0 \end{array} $$

(71)

Based on the above results, one can get $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {\varsigma }^{(1)}(t)=\underset {t\to T_{f}^{+}}{\mathop {\lim }} {\varsigma }^{(1)}(t)=0$ and it proves that ς⁽¹⁾(t) is continuous.

Step 2. The second-order derivative of ς(t) is given:

$$ \begin{array}{@{}rcl@{}} \frac{{\partial^{2}{\varsigma}(t)}}{\partial t^{2}}\!\!\!&&=\ell\frac{d^{2}{\varUpsilon}(t)}{dt^{2}}e^{q(t)}+2\ell\frac{dq(t)}{dt}\frac{d{\varUpsilon}(t)}{dt}e^{q(t)}+\\ &&~~~~\ell\frac{\partial^{2} e^{q(t)}}{\partial t^{2}}{\varUpsilon}(t)\\ &&=\ell\frac{d^{2}{\varUpsilon}(t)}{dt^{2}}e^{q(t)}+2\ell\frac{dq(t)}{dt}\frac{d{\varUpsilon}(t)}{dt}e^{q(t)}+\\ &&~~~~\ell{\varUpsilon}(t)\frac{d^{2}q(t)}{dt^{2}}e^{q(t)}+\ell{\varUpsilon}(t)\left( \frac{dq(t)}{dt}\right)^{2}e^{q(t)} \end{array} $$

(72)

It follows from (72) that $\frac {d^{2}{\varUpsilon }(t)}{dt^{2}}=-\frac {\pi ^{2}}{2{T_{f}^{2}}}\cos \limits (\frac {\pi t}{T_{f}})$, $\frac {d^{2}q(t)}{dt^{2}}=\frac {2\hbar T_{f}}{(t-T_{f})^{3}}$. Furthermore, $\underset {t\to T_{f}^{-}}{\mathop {\lim }} \ell \frac {d^{2}{\varUpsilon }(t)}{dt^{2}}e^{q(t)}=0$ and $\underset {t\to T_{f}^{-}}{\mathop {\lim }} 2\ell \frac {dq(t)}{dt}\frac {d{\varUpsilon }(t)}{dt}e^{q(t)}=0$.

Similarly, use the L’Hospital’ rule and one can get:

$$ \begin{array}{@{}rcl@{}} &&\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{d^{2}q(t)}{dt^{2}}e^{q(t)}=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left[-\frac{\frac{2\hbar T_{f}}{(T_{f}-t)^{3}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{\frac{6}{(T_{f}-t)^{2}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{\frac{12}{T_{f}-t}}{\hbar T_{f}e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}} \left[-\frac{12e^{\frac{\hbar t}{t-T_{f}}}}{(\hbar T_{f})^{2}}\right]=0 \end{array} $$

(73)

and

$$ \begin{array}{@{}rcl@{}} &&\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left( \frac{dq(t)}{dt}\right)^{2}e^{q(t)}=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left[\frac{\frac{(\hbar T_{f})^{2}}{(t-T_{f})^{4}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left[\frac{\frac{4\hbar T_{f}}{(T_{f}-t)^{3}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=0 \end{array} $$

(74)

Therefore, one has $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {\varsigma }^{(2)}(t)=\underset {t\to T_{f}^{+}}{\mathop {\lim }} {\varsigma }^{(2)}(t)=0$ and it proves that ς⁽²⁾(t) is continuous.

Step n. The n th-order (n ≥ 3) derivative of ς(t) is given as:

$$ \begin{array}{@{}rcl@{}} \frac{{\partial^{n}{\varsigma}(t)}}{\partial t^{n}}&={C_{n}^{0}}\ell\frac{d^{n}{\varUpsilon}(t)}{dt^{n}}e^{q(t)}+\ell\sum\limits_{i=1}^{n}{C_{n}^{i}}\frac{d^{n-i}{\varUpsilon}(t)}{dt^{n-i}}\frac{\partial^{i}e^{q(t)}}{\partial t^{i}} \end{array} $$

(75)

where $\frac {d^{n}{\varUpsilon }(t)}{dt^{n}}=\frac {1}{2}\left (\frac {\pi }{T_{f}}\right )^{n}\cos \limits \left (\frac {\pi t}{T_{f}}+\frac {n\pi }{2}\right )$ is a bounded value. Thus, one can obtain $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {C_{n}^{0}}\ell \frac {d^{n}{\varUpsilon }(t)}{dt^{n}}e^{q(t)}=0$. The term $\ell \sum \limits _{i=1}^{n}{C_{n}^{i}}\frac {d^{n-i}{\varUpsilon }(t)}{dt^{n-i}}\frac {d^{i}e^{q(t)}}{dt^{i}}$ includes the polynomials of $\frac {d^{i}q(t)}{dt^{i}}=(-1)^{i}\prod \limits _{j=1}^{i}(j)\frac {\hbar T_{f}}{(t-T_{f})^{i+1}}$ and $\left (\frac {dq(t)}{dt}\right )^{i}=\frac {(-\hbar T_{f})^{i}}{(t-T_{f})^{2i}}$.

As a result, $\frac {\partial ^{n}{\varsigma }(t)}{\partial t^{n}}$ can be expressed as polynomials of e^q(t) and $\frac {\zeta _{i}}{(T_{f}-t)^{r}}e^{q(t)}$ where ζ_i is a bounded constant and r ≥ 2. Therefore, use the L’Hospital’ rule successively:

$$ \begin{array}{@{}rcl@{}} &&\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left[\frac{\zeta_{i}}{(T_{f}-t)^{r}}e^{q(t)}\right]=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\left[\frac{\frac{\zeta_{i}}{(T_{f}-t)^{r}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{\zeta_{i}r}{\hbar T_{f}}\left[\frac{\frac{1}{(T_{f}-t)^{(r-1)}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{\zeta_{i}r(r-1)}{(\hbar T_{f})^{2}}\left[\frac{\frac{1}{(T_{f}-t)^{(r-2)}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{\zeta_{i}r(r-1)(r-2)}{(\hbar T_{f})^{3}}\left[\frac{\frac{1}{(T_{f}-t)^{(r-3)}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]=...\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{\zeta_{i}\prod\limits_{j=0}^{r-1}(r-j)}{(\hbar T_{f})^{(r-1)}}\left[\frac{\frac{1}{(T_{f}-t)^{}}}{e^{\frac{\hbar t}{T_{f}-t}}}\right]\\ &&=\underset{t\to T_{f}^{-}}{\mathop{\lim}}\frac{\zeta_{i}\prod\limits_{j=0}^{r-1}(r-j)}{(\hbar T_{f})^{r}}e^{q(t)}\\ &&=0 \end{array} $$

(76)

Thus, one can obtain $\underset {t\to T_{f}^{-}}{\mathop {\lim }} {\varsigma }^{(n)}(t)=\underset {t\to T_{f}^{+}}{\mathop {\lim }} {\varsigma }^{(n)}(t)=0$ and it proves that ς⁽ⁿ⁾(t) is continuous and ς(t) is n th-order differentiable. As a result, the proof of smoothness of ς(t) is completed.