用s-function一个系统的控制器以及自适应律，其中自适应律为向量形式，写一个实例

以下是一个使用S-Function实现控制器和自适应律的示例：

function [sys,x0,str,ts] = adaptive_control_sfun(t,x,u,flag,params)
% ADAPTIVE_CONTROL_SFUN S-Function for a system with adaptive controller
% and adaptive law
%   The system is described by the following differential equation:
%
%       dx/dt = A*x + B*u
%
%   The adaptive controller is given by:
%
%       u = -K*xhat + r
%
%   where K is the feedback gain matrix, xhat is the estimated state,
%   and r is the reference input.
%
%   The adaptive law is given by:
%
%       dtheta/dt = gamma*P'*xtilde
%
%       where theta is the vector of adaptive parameters, P is a matrix of
%       regressor functions, xtilde is the tracking error, and gamma is a
%       positive constant.
%
%   The adaptive parameters are updated by:
%
%       theta = theta + dtheta*dt
%
%   The regressor functions are given by:
%
%       P = [phi1(x,u), phi2(x,u), ..., phim(x,u)]
%
%   where phi1, phi2, ..., phim are basis functions.
%
%   The S-Function takes the following inputs:
%
%       u(1) - reference input r
%       u(2) - measured output y
%
%   and generates the following outputs:
%
%       y(1) - control input u
%       y(2) - estimated state xhat
%
%   The S-Function has the following parameters:
%
%       params.A - state matrix A
%       params.B - input matrix B
%       params.K - feedback gain matrix K
%       params.gamma - adaptation rate gamma
%       params.phi - basis function handle
%       params.theta0 - initial value of adaptive parameters
%       params.m - number of basis functions
%       params.n - number of states
%
%   The S-Function has the following states:
%
%       x(1:n) - state vector x
%       x(n+1:end) - adaptive parameters theta
%
%   The S-Function is called with flag = 0 at initialization and with
%   flag = 2 at each simulation step.

switch flag
    case 0 % Initialization
        [sys,x0,str,ts] = init(params);
    case 2 % Simulation step
        sys = step(t,x,u,params);
    case 3 % Output sizing
        sys = sizes(params);
    case {1,4,9} % Unused flags
        sys = [];
    otherwise % Error handling
        error(['Unhandled flag = ',num2str(flag)]);
end

function [sys,x0,str,ts] = init(params)
% Initialization function
%
%   The initial state is set to zero, and the initial value of the
%   adaptive parameters is set to theta0.

n = params.n; % number of states
m = params.m; % number of basis functions

% Initialize state vector x to zero
x0 = zeros(n+m,1);

% Initialize adaptive parameters to theta0
x0(n+1:end) = params.theta0;

% Return system sizes
sizes(params); % call sizing function

% Set sample time
ts = [params.dt 0];

% Set state derivatives to zero
str = [];

% Set output vector to empty
sys = [];

function sys = step(t,x,u,params)
% Simulation step function
%
%   The state derivatives are computed using the system equations.
%   The control input is computed using the estimated state and the
%   feedback gain matrix. The tracking error is computed using the
%   reference input and the measured output. The regressor functions
%   are computed using the current state and input. The adaptive law
%   is computed using the tracking error and the regressor functions.
%   The adaptive parameters are updated using the adaptive law.
%
%   The estimated state and control input are returned as outputs.

n = params.n; % number of states
m = params.m; % number of basis functions

% Extract input values
r = u(1); % reference input
y = u(2); % measured output

% Extract state and adaptive parameter vectors
xhat = x(1:n); % estimated state
theta = x(n+1:end); % adaptive parameters

% Extract system matrices and vectors
A = params.A;
B = params.B;
K = params.K;
gamma = params.gamma;
phi = params.phi;

% Compute state derivatives
xdot = A*xhat + B*(-K*xhat + r);

% Compute control input
u = -K*xhat + r;

% Compute tracking error
xtilde = xhat - y;

% Compute regressor functions
P = zeros(m,1);
for i = 1:m
    P(i) = phi{i}(xhat,u);
end

% Compute adaptive law
dtheta = gamma*P'*xtilde;

% Update adaptive parameters
theta = theta + dtheta*params.dt;

% Update estimated state
xhat = xhat + xdot*params.dt;

% Update state vector
x = [xhat;theta];

% Set output vector
sys = [u;xhat];

function sys = sizes(params)
% Output sizing function
%
%   The output vector has two elements: the control input and the
%   estimated state.

n = params.n; % number of states

% Set output vector sizes
sys = [2,1,n];

function phi = basis_functions()
% Basis function definition
%
%   This function defines the basis functions used in the adaptive law.
%   In this example, we use a set of radial basis functions.

phi = {...
    @(x,u) exp(-norm(x)^2),...
    @(x,u) exp(-norm(x-u)^2),...
    @(x,u) exp(-norm(x+u)^2),...
    @(x,u) exp(-norm(x-2*u)^2),...
    @(x,u) exp(-norm(x+2*u)^2)...
    };

end

% Define system parameters
A = [0 1; -1 0];
B = [0; 1];
K = [1 0];
gamma = 0.1;
phi = basis_functions();
theta0 = [1; 1; 1; 1; 1];
m = length(phi);
n = size(A,1);
dt = 0.1;

% Define parameter struct
params.A = A;
params.B = B;
params.K = K;
params.gamma = gamma;
params.phi = phi;
params.theta0 = theta0;
params.m = m;
params.n = n;
params.dt = dt;

% Simulate system using S-Function
sim('adaptive_control_sfun',[0 10],[],params);

% Plot results
t = ans.tout;
u = ans.u;
xhat = ans.xhat;
subplot(2,1,1); plot(t,u); xlabel('Time (s)'); ylabel('Control input');
subplot(2,1,2); plot(t,xhat); xlabel('Time (s)'); ylabel('Estimated state');

此示例实现了一个二阶系统的自适应控制器和自适应律，其中自适应律为向量形式。该系统由以下微分方程描述：

$$\frac{d}{dt}\begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}0&1\\-1&0\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} + \begin{bmatrix}0\\1\end{bmatrix}u$$

自适应控制器由以下控制律描述：

$$u = -K\hat{x} + r$$

其中$K$是反馈增益矩阵，$\hat{x}$是估计的状态，$r$是参考输入。

自适应律由以下微分方程描述：

$$\frac{d}{dt}\theta = \gamma P^T\tilde{x}$$

其中$\theta$是自适应参数向量，$P$是回归函数矩阵，$\tilde{x}$是跟踪误差，$\gamma$是正常数。自适应参数通过以下更新方程进行更新：

$$\theta = \theta + \frac{d\theta}{dt}\Delta t$$

回归函数由以下基函数定义：

$$\begin{aligned}\phi_1(x,u) &= \exp(-||x||^2)\\\phi_2(x,u) &= \exp(-||x-u||^2)\\\phi_3(x,u) &= \exp(-||x+u||^2)\\\phi_4(x,u) &= \exp(-||x-2u||^2)\\\phi_5(x,u) &= \exp(-||x+2u||^2)\end{aligned}$$

在S-Function中，我们使用以下参数传递系统参数：

% Define system parameters
A = [0 1; -1 0];
B = [0; 1];
K = [1 0];
gamma = 0.1;
phi = basis_functions();
theta0 = [1; 1; 1; 1; 1];
m = length(phi);
n = size(A,1);
dt = 0.1;

% Define parameter struct
params.A = A;
params.B = B;
params.K = K;
params.gamma = gamma;
params.phi = phi;
params.theta0 = theta0;
params.m = m;
params.n = n;
params.dt = dt;

其中`basis_functions`函数定义了基函数集合。

在S-Function的`init`函数中，我们将状态向量`x`初始化为零，并将自适应参数向量初始化为给定的初始值。

function [sys,x0,str,ts] = init(params)
% Initialization function
%
%   The initial state is set to zero, and the initial value of the
%   adaptive parameters is set to theta0.

n = params.n; % number of states
m = params.m; % number of basis functions

% Initialize state vector x to zero
x0 = zeros(n+m,1);

% Initialize adaptive parameters to theta0
x0(n+1:end) = params.theta0;

% Return system sizes
sizes(params); % call sizing function

% Set sample time
ts = [params.dt 0];

% Set state derivatives to zero
str = [];

% Set output vector to empty
sys = [];

在S-Function的`step`函数中，我们使用系统方程计算状态导数，使用估计状态和反馈增益矩阵计算控制输入，使用参考输入和测量输出计算跟踪误差，使用当前状态和输入计算回归函数，使用跟踪误差和回归函数计算自适应律，使用自适应律更新自适应参数，使用状态导数更新估计状态。

function sys = step(t,x,u,params)
% Simulation step function
%
%   The state derivatives are computed using the system equations.
%   The control input is computed using the estimated state and the
%   feedback gain matrix. The tracking error is computed using the
%   reference input and the measured output. The regressor functions
%   are computed using the current state and input. The adaptive law
%   is computed using the tracking error and the regressor functions.
%   The adaptive parameters are updated using the adaptive law.
%
%   The estimated state and control input are returned as outputs.

n = params.n; % number of states
m = params.m; % number of basis functions

% Extract input values
r = u(1); % reference input
y = u(2); % measured output

% Extract state and adaptive parameter vectors
xhat = x(1:n); % estimated state
theta = x(n+1:end); % adaptive parameters

% Extract system matrices and vectors
A = params.A;
B = params.B;
K = params.K;
gamma = params.gamma;
phi = params.phi;

% Compute state derivatives
xdot = A*xhat + B*(-K*xhat + r);

% Compute control input
u = -K*xhat + r;

% Compute tracking error
xtilde = xhat - y;

% Compute regressor functions
P = zeros(m,1);
for i = 1:m
    P(i) = phi{i}(xhat,u);
end

% Compute adaptive law
dtheta = gamma*P'*xtilde;

% Update adaptive parameters
theta = theta + dtheta*params.dt;

% Update estimated state
xhat = xhat + xdot*params.dt;

% Update state vector
x = [xhat;theta];

% Set output vector
sys = [u;xhat];

最后，我们使用S-Function进行仿真，并绘制结果。

% Simulate system using S-Function
sim('adaptive_control_sfun',[0 10],[],params);

% Plot results
t = ans.tout;
u = ans.u;
xhat = ans.xhat;
subplot(2,1,1); plot(t,u); xlabel('Time (s)'); ylabel('Control input');
subplot(2,1,2); plot(t,xhat); xlabel('Time (s)'); ylabel('Estimated state');

此示例演示了如何使用S-Function实现具有向量形式自适应律的控制器。