位置式PID算法
$\{\begin{array}{l}error(k)=rin(k)-yout(k)\\ x(1)=error(k)\\ x(2)=x(2)+Ts\ast error(k)\\ x(3)=error(k)-error(k-1)\\ pid=[K_p{K}_i{K}_d]\\ u(k)=pid\ast x\end{array}$
$u(k)$为控制量相当于状态量x与pid参数的线性组合，为了动态调整pid参数构建一个单神经元网络，性能指标函数取$J=\sum (rin(k)-yout(k){)}^2/2$;根据梯度下降法调整pid参数，即$\Delta K_p=-\eta \frac{\partial J}{\partial K_p}$,这个怎么求偏导？
$\Delta K_p=-\eta \frac{\partial J}{\partial K_p}=-\eta \frac{\partial J}{\partial yout}\frac{\partial yout}{\partial \Delta u}\frac{\partial \Delta u}{\partial K_p}=\eta .eorror(k).x(1).\frac{\partial yout}{\partial \Delta u}$,利用RBF网络的逼近能力，识别系统的Jacobian阵，即$\frac{\partial yout}{\partial \Delta u}$。

function [sys,x0,str,ts,simStateCompliance] = PID_RBF_s(t,x,u,flag)
% 网络结构3-6-1

% 采样时间
Ts = 0.001;
switch flag
    case 0
        [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes(Ts);
    case 2
        sys=mdlUpdate(x,u,Ts);
    case 3
        sys=mdlOutputs(t,x,u);
    case {1,4,9}
        sys=[];
    otherwise
        DAStudio.error('Simulink:blocks:unhandledFlag', num2str(flag));
end

function [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes(Ts)
sizes = simsizes;

sizes.NumContStates  = 0;
sizes.NumDiscStates  = 3;
sizes.NumOutputs     = 2;
sizes.NumInputs      = 4;
sizes.DirFeedthrough = 1;
sizes.NumSampleTimes = 1;   

sys = simsizes(sizes);
x0  = [0;0;0];
str = [];
ts  = [Ts 0];
simStateCompliance = 'UnknownSimState';

function sys=mdlUpdate(x,u,Ts)

sys=[u(1);
    x(2) + u(1)*Ts;
    (u(1) - u(2))];

function sys=mdlOutputs(t,x,u)
persistent w w_1 w_2 h c PID_parm
% PID三个三个参数学习效率
xitePID = [1 1 1];
% RBF网络学习效率
xite = 0.5;
% RBF网络动量因子
alfa = 0.5;
% 高斯基函数宽度
b = 3;
if t == 0
    % PID参数初值
    PID_parm = [0.7 0 1];
    % 高斯基函数中心矩阵，取值范围在输入信号范围内
    c = [linspace(-1,1,6);linspace(0,1,6);linspace(0,1,6)];
    h = zeros(6,1);
    w = zeros(6,1);
    w_1 = w;
    w_2 = w_1;
end
uu = PID_parm*x;
RBF_input = [uu u(3) u(4)]';
for j = 1:6
    h(j) = exp(-norm(RBF_input - c(:,j))^2/(2*b^2));
end
ymout =  w'*h;
d_w = xite*(u(3) - ymout)*h;
w = w_1 + d_w + alfa*(w_1 - w_2);
yu = w.*h.*(-RBF_input(1) + c(1,:))'/b^2;
% Jacobian阵
dyout = sum(yu);
PID_parm = PID_parm + u(1)*dyout*x'.*xitePID;
w_2 = w_1;
w_1 = w;
sys = [uu;ymout];