控制理论–观测器设计

一、弹簧-质量-阻尼系统

• 系统参数： $m=1,K=1,B=0.5$
• 状态变量: $z_1=x\text{(位置)};z_2=\dot{x}\text{ (速度，不可测)}$
• 输入: $u=F\text{（外力）}$

1. 系统状态空间方程

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\end{array}\right]& =\left[\begin{array}{cc}0& 1\\ -1& -0.5\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u\\ y=& \left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]=z_1\end{array}$$

2. 观测器状态空间方程

$$\begin{array}{rl}\left[\begin{array}{c}{\dot{\hat{z}}}_1\\ {\dot{\hat{z}}}_2\end{array}\right]& =\left[\begin{array}{cc}-2.5& 1\\ 0.25& -0.5\end{array}\right]\left[\begin{array}{c}{\hat{z}}_1\\ {\hat{z}}_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u+\left[\begin{array}{c}2.5\\ -1.25\end{array}\right]y\end{array}$$

二、仿真(Simulink s-function函数)

1. 搭建Simulink仿真模型

2. s-function函数代码

• Mode_system.m

function [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes
sizes = simsizes;
sizes.NumContStates  = 2;  %
sizes.NumDiscStates  = 0;
sizes.NumOutputs     = 3;  %z1,z2,y
sizes.NumInputs      = 1;  %u
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1;   % at least one sample time is needed
sys = simsizes(sizes);
x0  = [0;1];

function sys=mdlDerivatives(t,x,u)
M = 1;
K = 1;
B = 0.5;
uc = u(1);
z1 = x(1);
z2 = x(2);
dz1 = z2;
dz2 = (-K/M) * z1-(B/M) * z2 + (1/M)*uc;
sys = [dz1;dz2];

function sys=mdlOutputs(t,x,u)
z1 = x(1);
y = z1;
sys = [y;x]; %sys = [z1,z2,y]

• Observer.m

function [sys,x0,str,ts,simStateCompliance]=mdlInitializeSizes
sizes = simsizes;
sizes.NumContStates  = 2;
sizes.NumDiscStates  = 0;
sizes.NumOutputs     = 2; %z1_hat,z2_hat
sizes.NumInputs      = 2; %u和y
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1;   % at least one sample time is needed
sys = simsizes(sizes);
x0  = [0;0];

function sys=mdlDerivatives(t,x,u)
A = [-2.5,1;0.25,-0.5];
B = [0;1];
L = [2.5;-1.25];
z_hat = x; %z_hat = [z1_hat;z2_hat]; 
u_input = u(1); %输入1
y_meas = u(2);  %输入2
dz_hat = A * z_hat + B * u_input + L * y_meas;
sys = dz_hat;

function sys=mdlOutputs(t,x,u)
sys = x; %sys = [z1_hat; z2_hat];

3. 仿真效果

由图可知，$z1,z2$ 分别被${\hat{z}}_1,{\hat{z}}_2$所观测，且效果不错。