General regression neural network-based data-driven model-free predictive functional control for a class of discrete-time nonlinear systems

Abstract

In this paper, a novel general regression neural network-based data-driven model-free nonlinear predictive functional control approach is proposed for a class of unknown single-input/single-output discrete-time nonlinear systems. The historical measurable data are used to construct an explicit model for control design and estimate system time-varying parameters online, making the approach model free. By using basis functions to structuralize system input, online computational load (OCL) of the controller can be reduced dramatically compared with existing model-free predictive control approaches. Moreover, the prediction function is realized by a general regression neural network-based system parameter predictive algorithm, bringing the controller good robustness and strong capability against sudden changes from prescribed references and external circumstances simultaneously. Simulation results show that, by using much less OCL, the proposed approach-based closed-loop systems can achieve even smaller overshoot and shorter response time than the existing model-free predictive control approaches.

Appendices

Appendix A: Proof of Theorem 1

From formula (12), it is easy to obtain that:

$$\begin{aligned} \frac{\partial {y_m(k+n_i)}}{\partial {\varDelta u(k+n_i-1)}}={\hat{\phi }}(k+n_i-1) \end{aligned}$$

(26)

where $i=1,2$. Let’s consider the RHPI (14). Considering the necessary condition $\frac{\partial {J(k)}}{\partial {\varDelta u(k+n_i-1)}}=0$ and formula (13) yields:

$$\begin{aligned} \begin{aligned} 0&=-{\hat{\phi }}(k+n_i-1)[y_r(k+n_i)-y_p(k+n_i)]\\&\quad +\lambda _i \varDelta u(k+n_i-1) \\&=-{\hat{\phi }}(k+n_i-1)[y_r(k+n_i)-y_m(k+n_i)\\&\quad -e(k+n_i)]+\lambda _i \varDelta u(k+n_i-1) \\&=-{\hat{\phi }}(k+n_i-1)[y_r(k+n_i)\\&\quad -y_m(k+n_i)-y_p(k)+y_m(k)]\\&\quad +\lambda _i \varDelta u(k+n_i-1) \end{aligned} \end{aligned}$$

(27)

Then bringing formula (11) and the internal predictive model (12) into (27) yields:

$$\begin{aligned}&-{\hat{\phi }}(k+n_i-1)[y_r(k+n_i)-{\hat{\phi }}(k) \cdot \varDelta u(k)\nonumber \\&\quad - ({\hat{\phi }}(k+1)+ \cdots +{\hat{\phi }}(k+n_i-1)) \cdot \mu _2-y_p(k)]\nonumber \\&\quad +\lambda _i \mu _2=0 \end{aligned}$$

(28)

Then by considering $i=1,2$, formula (28) can be written by:

$$\begin{aligned} \left\{ \begin{array}{ll} {\hat{\phi }}(k) {\hat{\phi }}(k+n_1-1) \varDelta u(k)+[{\hat{\phi }}(k+n_1-1)\\ \qquad \times \sum \limits _{i=1}^{n_1-1}{\hat{\phi }}(k+i)+\lambda _1] \mu _2\\ ={\hat{\phi }}(k+n_1-1)[y_r(k+n_1)-y_p(k)]\\ ~\\ {\hat{\phi }}(k) {\hat{\phi }}(k+n_2-1) \varDelta u(k)+[{\hat{\phi }}(k+n_2-1)\\ \qquad \times \sum \limits _{i=1}^{n_2-1}{\hat{\phi }}(k+i)+\lambda _2] \mu _2\\ ={\hat{\phi }}(k+n_2-1)[y_r(k+n_2)-y_p(k)]\\ \end{array} \right. \end{aligned}$$

(29)

Treat $\varDelta u(k)$ and $\mu _2$ as variables, then, the incremental control law $\varDelta u(k)$ can be derived by solving formula (29):

$$\begin{aligned} \left\{ \begin{array}{ll} D={\hat{\phi }}(k+n_1-1) \cdot [y_r(k+n_1)-y_p(k)][{\hat{\phi }}(k+n_2-1) \\ \qquad \times \sum \limits _{i=1}^{n_2-1} {\hat{\phi }}(k+i)+\lambda _2]\\ \qquad -{\hat{\phi }}(k+n_2-1) \cdot [y_r(k+n_2)-y_p(k)][{\hat{\phi }}(k+n_1-1)\\ \qquad \times \sum \limits _{i=1}^{n_1-1}{\hat{\phi }}(k+i)+\lambda _1]\\ D_0={\hat{\phi }}(k) \cdot [\lambda _2 {\hat{\phi }}(k+n_1-1) - \lambda _1 {\hat{\phi }}(k+n_2-1) \\ \qquad -{\hat{\phi }}(k+n_1-1) {\hat{\phi }}(k+n_2-1) \cdot sign(n_2-n_1) \cdot \\ \qquad \times \sum \limits _{i=min \{ n_1,n_2 \}}^{max \{ n_1,n_2 \}-1} {\hat{\phi }}(k+i)]\\ \varDelta u(k)=\frac{D}{D_0}\\ u(k)= \varDelta u(k) + u(k-1)\\ \end{array}\right. \nonumber \\ \end{aligned}$$

(30)

Thus, Theorem 1 is proved.

Appendix B: Proof of Theorem 2

Denote $\phi _e(k)={\hat{\phi }}(k)-\phi (k)$. Subtracting $\phi (k)$ from both sides of the first equation of formula (20) yields:

$$\begin{aligned} \begin{aligned} \phi _e(k)&=\phi _e(k-1)-\phi (k)+\phi (k-1)\\&\quad +\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2} \Bigg [\varDelta y_p(k)+ \varDelta w(k) \\&\quad -{\hat{\phi }}(k-1) \varDelta u(k-1\Bigg ] \\&=[1-\frac{\eta |\varDelta u(k-1)|^2}{\mu +|\varDelta u(k-1)|^2}]\phi _e(k-1)\\&\quad +\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2} \varDelta w(k)-\phi (k)\\&\quad +\phi (k-1) \end{aligned} \end{aligned}$$

(31)

Considering that function $f(x)=1-\frac{\eta x}{\mu +x}$ is a decreasing function with respect to $x \ge 0$ and using the conditions $\mu \ge 0$ and $\eta \in (0,1]$ yields:

$$\begin{aligned}&0<\frac{\eta \varepsilon ^2}{\mu +\varepsilon ^2} \le \frac{\eta |\varDelta u(k-1)|^2}{\mu +|\varDelta u(k-1)|^2} \le \eta \le 1 \end{aligned}$$

(32)

$$\begin{aligned}&\frac{\eta |\varDelta u(k-1)|}{\mu +|\varDelta u(k-1)|^2}=\frac{\eta }{\frac{\mu }{|\varDelta u(k-1)|}+|\varDelta u(k-1)|}\nonumber \\&\quad \le \frac{\eta }{2\sqrt{\mu }} \end{aligned}$$

(33)

Recall in Remark 1 that $|\phi (k)| \le {\bar{b}}$. Thus, by using (33), it has:

$$\begin{aligned}&|\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2}\varDelta w(k)-\phi (k)+\phi (k-1)| \nonumber \\&\quad \le |\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2}\varDelta w(k)|+|\phi (k)+\phi (k-1)| \end{aligned}$$

(34)

Recall that $|w(k)| \le {\bar{W}}$. Thus, one can derive $|\varDelta w(k)|=|w(k)-w(k-1)| \le |w(k)|+|w(k-1)| \le 2{\bar{W}}$. Hence, one can infer that $0 \le |\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2}\varDelta w(k)| \le c$, c is a bounded number. Let $\delta =1-\frac{\eta \varepsilon ^2}{\mu +\varepsilon ^2}$. Taking absolute value of both sides of formula (31) and considering formulas (32)−(34) yield:

$$\begin{aligned} \begin{aligned} |\phi _e(k)|&=|1-\frac{\eta |\varDelta u(k-1)|^2}{\mu +|\varDelta u(k-1)|^2}| |\phi _e(k-1)|\\&\quad +|\frac{\eta \varDelta u(k-1)}{\mu +|\varDelta u(k-1)|^2}\varDelta w(k)-\phi (k)\\&\quad +\phi (k-1)| \\&\le (1-\delta )|\phi _e(k-1)|+c\\&\le (1-\delta )^2|\phi _e(k-2)|+c(1-\delta )+c\\&\le \cdots \le (1-\delta )^k |\phi _e(0)|+\frac{c}{\delta }\\&\le M_1 \end{aligned} \end{aligned}$$

(35)

Formula (35) means that $\phi _e(k)$ is bounded. By recalling the boundedness of the PPD $\phi (k)$, it can be inferred that ${\hat{\phi }}(k)$ is bounded. Then Theorem 2 is proved.

Appendix C: Proof of Theorem 3

Denote:

$$\begin{aligned} \begin{aligned} \varepsilon (k+1)&=y_r(k+1)-y_p(k+1)-w(k+1)\\&=y^*-y_p(k+1)-w(k+1) \end{aligned} \end{aligned}$$

(36)

Then the incremental input $\varDelta u(k)$ in formula (21) can be re-written into:

$$\begin{aligned} \varDelta u(k)=\frac{\varepsilon (k)}{{\hat{\phi }}(k)}-\frac{w(k)}{{\hat{\phi }}(k)} \end{aligned}$$

(37)

On one hand, bringing formula (4) into (36) and considering formulas (21) and (37) yield:

$$\begin{aligned} \begin{aligned} \varDelta \varepsilon (k+1)&=\varepsilon (k+1)-\varepsilon (k)\\&=y^*-y_p(k+1)-w(k+1)\\&\quad -[y^*-y_p(k)-w(k)] \\&=-\phi (k) \cdot \varDelta u(k)- \varDelta w(k+1) \\&=-\frac{{\phi }(k)}{{\hat{\phi }}(k)}\varepsilon (k)+\frac{{\phi }(k)}{{\hat{\phi }}(k)} w(k)-\varDelta w(k+1) \end{aligned} \end{aligned}$$

(38)

Thus, it has:

$$\begin{aligned} \varepsilon (k{+}1){=}[1{-}\frac{{\phi }(k)}{{\hat{\phi }}(k)}]\varepsilon (k){+}\frac{{\phi }(k)}{{\hat{\phi }}(k)} w(k)-\varDelta w(k+1) \end{aligned}$$

(39)

On the other hand, it also has:

$$\begin{aligned} |{\hat{\phi }}(k)|= & {} |\phi (k)+\phi _e(k)| \le |\phi (k)|\nonumber \\&+|\phi _e(k)| \le |\phi (k)|+M_1 \end{aligned}$$

(40)

Then, by recalling assumptions A1 and A2, it can be inferred that:

$$\begin{aligned} \frac{{\phi }(k)}{{\hat{\phi }}(k)}= & {} |\frac{\phi (k)}{\phi (k)+\phi _e(k)}| \ge \frac{\phi (k)}{|\phi (k)|+|\phi _e(k)|}\nonumber \\\ge & {} \frac{\phi (k)}{|\phi (k)|+M_1} \end{aligned}$$

(41)

Notice that $M_1>0$. Hence, there exists a positive number $d_2 \in (0,1)$ such that:

$$\begin{aligned} 0< & {} |1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}|=1-\frac{\phi (k)}{{\hat{\phi }}(k)}\nonumber \\= & {} d_2 \le 1- \frac{\phi (k)}{|\phi (k)|+M_1} \le 1 \end{aligned}$$

(42)

It also has:

$$\begin{aligned} 0<\frac{{\phi }(k)}{{\hat{\phi }}(k)} = d_1 < 1 \end{aligned}$$

(43)

Thus, it has:

$$\begin{aligned} |\frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)- \varDelta w(k+1)| \le d_1 {\bar{W}}+2 {\bar{W}}=d_3 \end{aligned}$$

(44)

Hence, taking absolute value of both sides of formula (39) yields:

$$\begin{aligned} \begin{aligned} |\varepsilon (k+1)|&=|[1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}]\varepsilon (k)+\frac{{\phi }(k)}{{\hat{\phi }}(k)} w(k)\\&\quad -\varDelta w(k+1)|\\&\le d_2|\varepsilon (k)|+d_3\\&\le d^2_2|\varepsilon (k-1)|+d_3 d_2+d_3\\&\le \cdots \le d^{k+1}_2|\varepsilon (0)|+\frac{1-d^{k-1}_2}{d_1}d_3 \end{aligned} \end{aligned}$$

(45)

Therefore, by considering formula (44), it has:

$$\begin{aligned}&\lim _{k \rightarrow \infty }|y^*-y_p(k+1)-w(k+1)|\nonumber \\&\quad =\lim _{k \rightarrow \infty }|\varepsilon (k+1)|=\frac{2+d_1}{d_1}=\lambda {\bar{W}} \end{aligned}$$

(46)

Particularly, $w(k+1)=0 \Rightarrow {\bar{W}}=0 \Rightarrow \lim \limits _{k \rightarrow \infty }|y^*-y_p(k+1)-w(k+1)|=0$.

To prove the UUB, select following Lyapunov function:

$$\begin{aligned} V(k+1)=\varepsilon ^2(k+1) \end{aligned}$$

(47)

Bringing formula (38) into (47) yields:

$$\begin{aligned} \begin{aligned} \varDelta V(k+1)&=V(k+1)-V(k)\\&=[\varDelta \varepsilon (k+1)+\varepsilon (k)]^2-\varepsilon ^2(k)\\&=\left\{ \left[ 1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}\right] ^2-1 \right\} \varepsilon ^2(k)\\&\quad +2\left[ 1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}\right] \\&\quad \times \left[ \frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)-\varDelta w(k+1)\right] \varepsilon (k)\\&\quad +\left[ \frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)-\varDelta w(k+1)\right] ^2 \end{aligned} \end{aligned}$$

(48)

The rest task of the proof is to show $\varDelta V(k+1) \le 0$.

In formula (48), denote $a=[1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}]^2-1$, $b=2[1-\frac{{\phi }(k)}{{\hat{\phi }}(k)}][\frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)-\varDelta w(k+1)]$ and $c=[\frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)-\varDelta w(k+1)]^2$. It is clear that $a<0$ and $0 \le \sqrt{\varDelta }=\sqrt{b^2-4ac}=2|\frac{{\phi }(k)}{{\hat{\phi }}(k)}w(k)-\varDelta w(k+1)| \le 2(d_1+1) {\bar{W}}$. This means that the function $a \cdot \varepsilon ^2(k)+b \cdot \varepsilon (k)+c=0$ has solutions $\varepsilon _1$ and $\varepsilon _2$ (there might be $\varepsilon _1=\varepsilon _2$). Let $\varepsilon _0=max\{ \varepsilon _1, \varepsilon _2 \}$. Thus, there must exist a STP $k_0$ such that when $k \ge k_0$, it has $\varepsilon (k) \ge \varepsilon _0$, making $\varDelta V(k+1) \le 0$.

Then UUB of the system is proved.