Feedback control of the COVID-19 outbreak based on active disturbance rejection control

Figures & data

Table 1. Discrete-time ADRC scheme parameters.

ADRC part	Parameter	Value	Parameter	Value
NLESO	$p_1$	−7.64578	A	21.57116
	${\text{β}}_1$	493.89822	${\text{β}}_2$	−78.32702
	$b_0$	1578.40920
NLESF	$k_1$	1568.86329	$a_1$	0.17138
	$\mathrm{delt}{a}_1$	11,165.80953

Figure 1. The comparison of the simulation of discrete-time ADRC and PID scheme. Top: the reproduction number $R_0$ input into the SIR-T model. Bottom: the output number of infected individuals $I\left(t\right)$.

Table 2. Performance indexes for the discrete-time ADRC scheme and the discrete PID scheme.

Tuning method	ITAE	$\sigma \mathrm{\%}$	$T_r/s$	$T_s/s$
ADRC	$2.6852\times {10}^3$	0.4379	84.9922	84.4512
PID	$2.8105\times {10}^3$	1.1705	85.4126	84.7734

Figure 2. Simulation results for the two control schemes when the simulation time is 2000 days. (a) The evolution of reproduction number $R_0$; (b) the evolution of $\mathrm{rate}\left(t\right)$; (c) Susceptible individuals $S\left(t\right)$; (d) Infected individuals $I\left(t\right)$; (e) Removed individuals $R\left(t\right)$.

Figure 3. Simulation results for two control schemes when ${\gamma }_{\mathrm{real}}=1.05{\gamma }_{\mathrm{model}}$. (a) Input signal: the reproduction number; (b) Output signal: infected individuals $I\left(t\right)$.

Figure 4. Simulation results for two control schemes when ${\gamma }_{\mathrm{real}}=0.95{\gamma }_{\mathrm{model}}$. (a) Input signal: the reproduction number;(b) Output signal: infected individuals $I\left(t\right)$.

Table 3. Robustness comparison of two control schemes.

Parameter uncertainty	Tuning method	ITAE	$\sigma \mathrm{\%}$	$T_r/s$	$T_s/s$
$\begin{array}{l}{\gamma }_{\mathrm{real}}=\frac{1.05}{30}\\ {\gamma }_{\mathrm{model}}=\frac{1}{30}\end{array}$	ADRC	$3.3422\times {10}^3$	14.8087	81.9128	282.2972
	PID	$4.8817\times {10}^3$	19.1147	82.1806	298.8819
$\begin{array}{l}{\gamma }_{\mathrm{real}}=\frac{0.95}{30}\\ {\gamma }_{\mathrm{model}}=\frac{1}{30}\end{array}$	ADRC	$3.2768\times {10}^3$	6.7192	95.1379	155.7862
	PID	$4.3560\times {10}^3$	9.8862	94.6078	295.9852
$\begin{array}{l}{N}_{\mathrm{real}}=1.05\times {10}^7\\ N_{\mathrm{model}}=1\times {10}^7\end{array}$	ADRC	$2.6829\times {10}^3$	0.4335	84.9934	84.4526
	PID	$2.8089\times {10}^4$	1.1171	85.4123	84.7734
$\begin{array}{l}{N}_{\mathrm{real}}=0.95\times {10}^7\\ N_{\mathrm{model}}=1\times {10}^7\end{array}$	ADRC	$2.6879\times {10}^3$	0.4430	84.9908	84.4498
	PID	$2.8124\times {10}^4$	1.1695	85.4130	84.7735

Figure 5. Simulation results for two control schemes when $N_{\mathrm{real}}=1.05N_{\mathrm{model}}$. (a) Input signal: the reproduction number; (b) Output signal: infected individuals $I\left(t\right)$.

Figure 6. Simulation results for two control schemes when $N_{\mathrm{real}}=0.95N_{\mathrm{model}}$. (a) Input signal: the reproduction number; (b) Output signal: infected individuals $I\left(t\right)$.