Model predictive control of nonlinear system based on adaptive fuzzy neural network

doi:10.11949/0438-1157.20191531

Abstract

Abstract:

Aiming at the control problem of nonlinear dynamic systems, a model predictive control (MPC) method based on adaptive fuzzy neural network (AFNN) was proposed. First, in the offline modeling phase, a rule automatic splitting technique is used to generate initial fuzzy rules, and an improved adaptive LM learning algorithm is adopted to optimize network parameters. Second, in the real-time control process, the network parameters of the AFNN are adjusted according to the error between the system output and the predicted output, so that providing an accurate prediction model for the MPC. Furthermore, the gradient descent optimization algorithm with adaptive learning rate is used to solve optimization problem and obtain the nonlinear control law online, which is applied to control the dynamic system. In addition, the convergence and stability analysis of the proposed AFNN-MPC are given to ensure its successful application in practical engineering. Finally, numerical simulation and two-CSTR process experiments are used to verify the effectiveness of AFNN-MPC algorithm. The results show that the proposed AFNN-MPC has superior control performance.

Key words: nonlinear systems, dynamic modeling, model predictive control, process control, fuzzy neural network, adaptive learning rate

摘要：

针对非线性动态系统的控制问题，提出了一种基于自适应模糊神经网络(adaptive fuzzy neural network, AFNN)的模型预测控制(model predictive control, MPC)方法。首先，在离线建模阶段，AFNN采用规则自分裂技术产生初始模糊规则，采用改进的自适应LM学习算法优化网络参数；然后，在实时控制过程，AFNN根据系统输出和预测输出之间的误差调整网络参数，从而为MPC提供一个精确的预测模型；进一步，AFNN-MPC利用带有自适应学习率的梯度下降寻优算法求解优化问题，在线获取非线性控制量，并将其作用到动态系统实施控制。此外，给出了AFNN-MPC的收敛性和稳定性证明，以保证其在实际工程中的成功应用。最后，利用数值仿真和双CSTR过程进行实验验证。结果表明，AFNN-MPC能够取得优越的控制性能。

关键词: 非线性系统, 动态建模, 模型预测控制, 过程控制, 模糊神经网络, 自适应学习率

CLC Number:

TP 273

Hongbiao ZHOU, Yu ZHANG, Xiaoying BAI, Baolian LIU, Huanyu ZHAO. Model predictive control of nonlinear system based on adaptive fuzzy neural network[J]. CIESC Journal, 2020, 71(7): 3201-3212.

周红标, 张钰, 柏小颖, 刘保连, 赵环宇. 基于自适应模糊神经网络的非线性系统模型预测控制[J]. 化工学报, 2020, 71(7): 3201-3212.

Figures/Tables 17

Fig.1 Architecture of MIMO-type FNN

Table 1 Algorithm flow of RAS

Algorithm 1: RAS
Set the cluster number r = 1
Calculate the center of the first cluster v₁;
Calculate the mean variance $g ˉ j$ according to Eq. (10) and Eq.(11);
while $g ˉ I$ >g_thdo
Split the Ith cluster into two new centers;
r = r + 1;
while current iteration times t₁<T₁ & \|J_m(t₁) - J_m(t₁-1)\| > εdo
forj=1:rdo
Find input training samples for cluster v_j ;
Calculate the center of cluster v_j according to Eq. (13);
end
t₁++;
end while
end while
return r clusters;

Table 1 Algorithm flow of RAS

Algorithm 1: RAS
Set the cluster number r = 1
Calculate the center of the first cluster v₁;
Calculate the mean variance $g ˉ j$ according to Eq. (10) and Eq.(11);
while $g ˉ I$ >g_thdo
Split the Ith cluster into two new centers;
r = r + 1;
while current iteration times t₁<T₁ & \|J_m(t₁) - J_m(t₁-1)\| > εdo
forj=1:rdo
Find input training samples for cluster v_j ;
Calculate the center of cluster v_j according to Eq. (13);
end
t₁++;
end while
end while
return r clusters;

Table 2 Algorithm flow of AFNN

Algorithm 2: AFNN
Set the maximum iteration times T_max;
Obtain r clusters using RAS algorithm;
Create an initial four-layer FNN;
while current iteration times t<T_max do
for p=1:Pdo
Calculate the outputs of FNN y_p(t) by Eq.(8);
Make the error of each sample e_p(t) by Eq. (15);
Make the submatrices ψ_p(t) by Eq. (21);
Make the subvectors ω_p(t) by Eq. (22);
end
Update the adaptive learning rate η(t) by Eq. (18);
Make the quasi-Hessian matrix Ψ(t) by Eq. (19);
Make the gradient vector Ω(t) by Eq. (20);
Update the parameter vector Θ(t) by Eq. (17);
t++;
end while

Fig.2 Control architecture of AFNN-MPC

Table 3 Algorithm flow of AFNN-MPC

Algorithm 3: AFNN-MPC
Generate offline data using GMN signal;
Develop AFNN predictive model offline;
Initialize the parameters of MPC;
fork←1 to ndo
Sample the plant output y(k);
Update the parameters of AFNN by Eq.(24)—Eq.(26);
Calculate the prediction outputs ?(k+h);
while current iteration times t_c<T_cdo
Compute the control increment Δu^k(t_c) by Eq. (39);
Compute the control signal u^k(t_c) by Eq. (31);
t_c++;
end while
Apply u(k+1) to the controlled process;
end

Fig.3 MMSE curve during AFNN structure identification

Fig.4 RMSE curve during AFNN parameter training

Fig.5 Prediction results of testing samples

Fig.6 Control performance of AFNN-MPC

Fig.7 Change of the control signal u

Fig.8 Schematic of two-CSTR process

Fig.9 GMN excitation signal and CSTR response signal

Fig.10 RMSE and modeling results of AFNN

Fig.11 Control results of AFNN-MPC

Fig.12 Changes of control signal Qcw1 and Qcw2

Table 4 Control results of different algorithms for To1

Controller	No.	IAE	ISE	Dev^max
PID	-	147.20	110.22	3.05
FLC	49	139.45	102.71	2.78
GPC	-	135.58	107.29	2.80
FNN-MPC	20	130.48	100.43	2.66
WFNN-MPC	20	132.66	101.10	2.68
AFNN-MPC	14	126.00	95.31	2.59

Table 5 Control results of different algorithms for To2

Controller	No.	IAE	ISE	Dev^max
PID	-	180.72	170.82	3.28
FLC	49	160.04	158.90	2.99
GPC	-	175.13	156.29	3.01
FNN-MPC	20	156.72	149.64	2.81
WFNN-MPC	20	152.54	151.08	2.82
AFNN-MPC	14	149.66	144.70	2.70

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