估计方法

扩展(Extension)

$\begin{aligned} y(t) &=x_{1}(t) \theta_{1}+x_{2}(t) \theta_{2}, \\ y(t-\tau) &=x_{1}(t-\tau) \theta_{1}+x_{2}(t-\tau) \theta_{2}, \end{aligned}$ ：

$Y(t)=\left[\begin{array}{c} y(t) \\ y(t-\tau) \end{array}\right] \quad X(t)=\left[\begin{array}{cc} x_{1}(t) & x_{2}(t) \\ x_{1}(t-\tau) & x_{2}(t-\tau) \end{array}\right]$

$\begin{aligned} Y(t) &=X(t) \theta \\ X^{-1}(t) Y(t) &=X^{-1}(t) X(t) \theta \\ X^{-1}(t) Y(t) &=\theta \\ \frac{\operatorname{adj} X(t)}{\operatorname{det} X(t)} Y(t) &=\theta \end{aligned}$

混合(Mixing)

$\begin{aligned} \operatorname{adj} X(t) Y(t) &=\operatorname{det} X(t) \theta \\ Z(t) &=\Delta \theta \end{aligned}$

其中 $Z(t)=\left[\begin{array}{l} z_{1}(t) \\ z_{2}(t) \end{array}\right]=\operatorname{adj} X(t) Y(t), \Delta(t)=\operatorname{det} X(t)$

评估策略：

$\begin{aligned} \dot{\hat{\theta}}_{1} &=\gamma \Delta\left(z_{1}-\Delta \hat{\theta}_{1}\right) \\ \dot{\hat{\theta}}_{2} &=\gamma \Delta\left(z_{2}-\Delta \hat{\theta}_{2}\right) \\ \dot{\hat{\theta}} &=\gamma \Delta(Z-\Delta \hat{\theta}) \end{aligned}$

仿真

待估计系统：

$y=\sin (t)+2 \cos (t)$

编写m代码确定模型参数

theta = [1;2];
tau=0.5*pi;
gamma = 100;

simulink仿真模型：

使用估计参数的系统与原系统输出比较

模型及m文件下载地址：drem方法对系统参数进行估计-智慧城市文档类资源-CSDN下载

参考：Parameters Estimation via Dynamic Regressor Extension and Mixing 2016 American Control Conference (ACC) Boston Marriott Copley Place
July 6-8, 2016. Boston, MA, USA