Output Feedback Controller for a Class of Unknown Nonlinear Discrete Time Systems Using Fuzzy Rules Emulated Networks and Reinforcement Learning

Figures & data

Figure 1. Closed-loop system architecture.

Table

IF $e(k)$ is NL	THEN $u(k)$ is NL,
IF $e(k)$ is NM	THEN $u(k)$ is NM,
IF $e(k)$ is NS	THEN $u(k)$ is NS,
IF $e(k)$ is Z	THEN $u(k)$ is Z,
IF $e(k)$ is PS	THEN $u(k)$ is PS,
IF $e(k)$ is PM	THEN $u(k)$ is PM,
IF $e(k)$ is PL	THEN $u(k)$ is PL,

Figure 2. Action network or controller based on FREN.

Figure 3. Estimated cost function or critic network.

Table 1. MiFRENc: IF–THEN rules.

	$e^2(k)$
$\hat{L}(k)$		Z	S	L
$u^2(k)$	Z	${\varphi }_1;$ Z${}_{\mathrm{Z}}$	${\varphi }_4;$ S${}_{\mathrm{Z}}$	${\varphi }_7;$L${}_{\mathrm{Z}}$
	S	${\varphi }_2;$ Z${}_{\mathrm{S}}$	${\varphi }_5;$ S${}_{\mathrm{S}}$	${\varphi }_8;$L${}_{\mathrm{S}}$
	L	${\varphi }_3;$ Z${}_{\mathrm{L}}$	${\varphi }_6;$ S${}_{\mathrm{L}}$	${\varphi }_9;$L${}_{\mathrm{L}}$

Figure 4. FRENa membership functions: simulation case.

Figure 5. MiFRENc membership functions: simulation case.

Table 2. Initial setting ${\beta }_{◻}(1)$: simulation system.

FRENa		MiFRENc
Parameter	Value	Parameter	Value
${\beta }_{\mathrm{NL}}(1)$	−0.25	${\beta }_{\mathrm{ZZ}}(1)$	0
${\beta }_{\mathrm{NM}}(1)$	−0.15	${\beta }_{\mathrm{ZS}}(1)$	0.1
${\beta }_{\mathrm{NS}}(1)$	−0.05	${\beta }_{\mathrm{ZL}}(1)$	0.2
${\beta }_{\mathrm{Z}}(1)$	0	${\beta }_{\mathrm{SZ}}(1)$	0.3
${\beta }_{\mathrm{PS}}(1)$	0.05	${\beta }_{\mathrm{SS}}(1)$	0.4
${\beta }_{\mathrm{PM}}(1)$	0.15	${\beta }_{\mathrm{SL}}(1)$	0.5
${\beta }_{\mathrm{PL}}(1)$	0.25	${\beta }_{\mathrm{LZ}}(1)$	0.6
		${\beta }_{\mathrm{LS}}(1)$	0.7
		${\beta }_{\mathrm{LL}}(1)$	0.8

Figure 6. Tracking performance $y(k)$ and $e(k)$: simulation system.

Figure 7. Control effort $u(k)$: simulation system.

Figure 8. Estimated cost function $\hat{L}(k)$: simulation system.

Figure 9. $u(k)$ and $e(k)$: simulation system.

Figure 10. FRENa membership functions: experimental system.

Figure 11. MiFRENc membership functions: experimental system.

Table 3. Initial setting ${\beta }_{◻}(1)$: experimental system.

FRENa		MiFRENc
Parameter	Value	Parameter	Value
${\beta }_{\mathrm{NL}}(1)$	−3.25	${\beta }_{\mathrm{ZZ}}(1)$	0
${\beta }_{\mathrm{NM}}(1)$	−2.5	${\beta }_{\mathrm{ZS}}(1)$	0.1
${\beta }_{\mathrm{NS}}(1)$	−1.05	${\beta }_{\mathrm{ZL}}(1)$	0.2
${\beta }_{\mathrm{Z}}(1)$	0	${\beta }_{\mathrm{SZ}}(1)$	0.3
${\beta }_{\mathrm{PS}}(1)$	1.05	${\beta }_{\mathrm{SS}}(1)$	0.4
${\beta }_{\mathrm{PM}}(1)$	2.5	${\beta }_{\mathrm{SL}}(1)$	0.5
${\beta }_{\mathrm{PL}}(1)$	3.25	${\beta }_{\mathrm{LZ}}(1)$	0.6
		${\beta }_{\mathrm{LS}}(1)$	0.7
		${\beta }_{\mathrm{LL}}(1)$	0.8

Figure 12. Tracking performance $y(k)$ and $e(k)$: experimental system.

Figure 13. Control effort $u(k)$: experimental system.

Figure 14. Estimated cost function $\hat{L}(k)$: experimental system.

Figure 15. $u(k)$ and $e(k)$: experimental system.

Figure 16. Pulse response: experimental system.