【控制】非线性系统反馈线性化和扩张状态观测器（9.2）

1 非线性系统反馈线性化

反馈线性化的步骤：

（1）目标线性系统

（2）线性化系统镇定

（3）从非线性系统到线性化系统---反馈线性化

2 搭建仿真系统

MATLAB Function程序

%系统
function fx = fcn(dy,y)

fx = sin(y)+exp(dy)+2*y*dy;

%controller
function u = fcn(dy,y)

k1 = -9;
k2 = -6;
u0 = k1*y + k2*dy;

u = ( u0 - sin(y) - exp(dy) - 2*y*dy )/4;

3 仿真结果

3.1 变步长仿真

可以发现，采用变步长仿真出来的图像更为有些不光滑。

3.2 定步长仿真

可以发现，采用定步长仿真出来的图像更为光滑。

4 扩张状态观测器

在实际情况中，系统的状态(如,即)有时候是不可获得的，此时就需要设计（扩张）状态观测器类进估计系统的状态。

4.1 设计扩张状态观测器

二阶系统

，其中。

定义，，则有

系统的初始状态，。

定义新的状态，其一阶导数。

 则扩张后状态系统变为：

设计线性扩张状态观测器如下形式：

线性扩张状态观测器系统的初始状态，，。

4.2 仿真模型搭建

4.3 仿真结果

系统状态x1和x2

4.4 matlab程序

%%参数para_02.matlab

%线性扩张状态观测器增益
para.beta1=100;
para.beta2=2500;
para.beta3=2500;
para.b=4

%%系统SYS.matlab

%MATLAB Function

function fx = fcn(dy,y)
fx = sin(y)+exp(dy)+2*y*dy;

%%控制器设计controller.matlab

%controller

function u = fcn(y_hat,dy_hat)
k1 = -9;
k2 = -6;
u0 = k1*y_hat + k2*dy_hat;
u = ( u0 - sin(y_hat) - exp(dy_hat) - 2*y_hat*dy_hat )/4;

%%线性扩张状态观测器LSO_02.matlab

%LESO

function [sys,x0,str,ts,simStateCompliance] = LESO_02(t,x,u,flag,para)

% The following outlines the general structure of an S-function.
%
switch flag,
  %%%%%%%%%%%%%%%%%%
  % Initialization %
  %%%%%%%%%%%%%%%%%%
  case 0,
    [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes;
  %%%%%%%%%%%%%%%
  % Derivatives %
  %%%%%%%%%%%%%%%
  case 1,
    sys=mdlDerivatives(t,x,u,para);
  %%%%%%%%%%
  % Update %
  %%%%%%%%%%
  case 2,
    sys=mdlUpdate(t,x,u);
  %%%%%%%%%%%
  % Outputs %
  %%%%%%%%%%%
  case 3,
    sys=mdlOutputs(t,x,u);
  %%%%%%%%%%%%%%%%%%%%%%%
  % GetTimeOfNextVarHit %
  %%%%%%%%%%%%%%%%%%%%%%%
  case 4,
    sys=mdlGetTimeOfNextVarHit(t,x,u);
  %%%%%%%%%%%%%
  % Terminate %
  %%%%%%%%%%%%%
  case 9,
    sys=mdlTerminate(t,x,u);
  %%%%%%%%%%%%%%%%%%%%
  % Unexpected flags %
  %%%%%%%%%%%%%%%%%%%%
  otherwise
    DAStudio.error('Simulink:blocks:unhandledFlag', num2str(flag));

end

% end sfuntmpl

%
%=============================================================================
% mdlInitializeSizes
% Return the sizes, initial conditions, and sample times for the S-function.
%=============================================================================
%
function [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes

% call simsizes for a sizes structure, fill it in and convert it to a
% sizes array.
%
% Note that in this example, the values are hard coded.  This is not a
% recommended practice as the characteristics of the block are typically
% defined by the S-function parameters.
%
sizes = simsizes;

sizes.NumContStates  = 3;
sizes.NumDiscStates  = 0;
sizes.NumOutputs     = 3;
sizes.NumInputs      = 2;
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1;   % at least one sample time is needed

sys = simsizes(sizes);

% initialize the initial conditions
%
x0  = [0.8;1.8;1];

% str is always an empty matrix
%
str = [];

% initialize the array of sample times
%
ts  = [0 0];

% Specify the block simStateCompliance. The allowed values are:
%    'UnknownSimState', < The default setting; warn and assume DefaultSimState
%    'DefaultSimState', < Same sim state as a built-in block
%    'HasNoSimState',   < No sim state
%    'DisallowSimState' < Error out when saving or restoring the model sim state
simStateCompliance = 'UnknownSimState';

% end mdlInitializeSizes

%=============================================================================
% mdlDerivatives
% Return the derivatives for the continuous states.
%=============================================================================
%
function sys=mdlDerivatives(t,x,u,para)

beta1=para.beta1;
beta2=para.beta2;
beta3=para.beta3;

uc=u(1);
x1=u(2);

z1=x(1);
z2=x(2);
z3=x(3);

e=z1-x1;
dz1=z2-beta1*e;
dz2=z2-beta2*e;
dz3=-beta3*e;

sys = [dz1;dz2;dz3];

% end mdlDerivatives

%=============================================================================
% mdlUpdate
% Handle discrete state updates, sample time hits, and major time step
% requirements.
%=============================================================================
%
function sys=mdlUpdate(t,x,u)

sys = [];

% end mdlUpdate

%=============================================================================
% mdlOutputs
% Return the block outputs.
%=============================================================================
%
function sys=mdlOutputs(t,x,u)

sys = [x];

% end mdlOutputs

%=============================================================================
% mdlGetTimeOfNextVarHit
% Return the time of the next hit for this block.  Note that the result is
% absolute time.  Note that this function is only used when you specify a
% variable discrete-time sample time [-2 0] in the sample time array in
% mdlInitializeSizes.
%=============================================================================
%
function sys=mdlGetTimeOfNextVarHit(t,x,u)

sampleTime = 1;    %  Example, set the next hit to be one second later.
sys = t + sampleTime;

% end mdlGetTimeOfNextVarHit

%=============================================================================
% mdlTerminate
% Perform any end of simulation tasks.
%=============================================================================
%
function sys=mdlTerminate(t,x,u)

sys = [];

% end mdlTerminate