系列文章目录

【系统辨识】最小二乘估计
【系统辨识】递推最小二乘法的推导及matlab仿真
【系统辨识】最小二乘辨识模型的建立

前言

$y(t)={\phi }^T(t)\theta +v(t)$描述的辨识模型示意图如上图所示，它把输出表示为参数的线性组合形式，是辨识问题的一个基本表达形式，称为最小二乘格式。因为这个最小二乘格式辨识模型的参数向量易于用最小二乘方法辨识。本文主要说明如何把系统模型化为最小二乘格式。

一、线性系统辨识模型

1.线性系统脉冲响应模型

$$\begin{gathered} y(t)=b_{1}u(t-1)+b_{2}u(t-2)+...+b_{n}u(t-n)+v(t) \\ =\left[u(t-1),u(t-1),...,u(t-n)\right]\left[\begin{array}{c}{b}_1\\ b_2\\ ...\\ b_n\end{array}\right]+v(t) \end{gathered}$$对应的辨识模型为$$\{\begin{array}{l}y(t)={\phi }^T(t)\theta +v(t),\\ \phi (t)=[u(t-1),u(t-2),...,u(t-n){]}^T\in R^n,\\ \theta =[b_1,b_2,...,b_n{]}^T\in R^n.\end{array}$$

2.时变系统差分方程描述的线性系统

$$y(t)+a_1(t)y(t-1)+a_2(t)y(t-2)=b_{1}u(t-1)+b_{2}u(t-2)$$
对应的辨识模型为
$$\{\begin{array}{l}y(t)={\phi }^T(t)\theta ,\\ \phi (t)=[-y(t-1),-y(t-2),u(t-1),u(t-2){]}^T\in R^4,\\ \theta =[a_1(t),a_2,b_1,b_2{]}^T\in R^4.\end{array}$$

3.线性动态系统（CAR模型）

$$A(z)y(t)=B(z)u(t)+v(t),$$其中$A(z)$和$B(z)$是$z^{-1}$的常系数时不变多项式，定义如下，$$\begin{gathered} A(z)=1+a_1{z}^{-1}+a_2{z}^{-2}+...+a_{n_a}{z}^{-n_a}, \\ B(z)=b_1{z}^{-1}+b_2{z}^{-2}+...+b_{n_b}{z}^{-n_b}. \end{gathered}$$将多项式$A(z)$和$B(z)$带入CAR模型中可得
$$(1+a_1{z}^{-1}+a_2{z}^{-2}+...+a_{n_a}{z}^{-n_a})y(t)=(b_1{z}^{-1}+b_2{z}^{-2}+...+b_{n_b}{z}^{-n_b})u(t)+v(t)$$利用移位算子的性质，上式可以写为
$$y(t)+a_{1}y(t-1)+a_{2}y(t-2)+...+a_{n_a}y(t-n_a)=b_{1}u(t-1)+b_{2}u(t-1)+...+b_{n_b}u(t-n_b)+v(t)$$移项可得$$y(t)=-a_{1}y(t-1)-a_{2}y(t-2)-...-a_{n_a}y(t-n_a)+b_{1}u(t-1)+b_{2}u(t-1)+...+b_{n_b}u(t-n_b)+v(t)$$对应的辨识模型为
$$\{\begin{array}{l}y(t)={\phi }^T(t)\theta +v(t),\\ \phi (t)=[-y(t-1),-y(t-2),...,-y(t-n_a),u(t-1),u(t-2),...,u(t-n_b){]}^T\in R^{n_a+n_b},\\ \theta =[a_1,a_2,...,a_{n_a},b_1,b_2,...,b_{n_b}{]}^T\in R^{n_a+n_b}.\end{array}$$

二、非线性系统辨识模型

1.静态非线性系统

$$y=a{x}^2+bx+c$$
将该模型写为向量形式
$y=\left[\begin{array}{ccc}{x}^2& x& 1\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$
对应的辨识模型为
$$\{\begin{array}{l}y={\phi }^T\theta ,\\ \phi =[x^2,x,1{]}^T\in R^3,\\ \theta =[a,b,c{]}^T\in R^3.\end{array}$$

2.动态非线性系统

$$y(t)=a_{1}y(t-1)+a_{2}y(t-2)y(t-3)+b_{1}u(t)+b_2{u}^2(t-1)+c\cos (\sqrt{t})+v(t)$$
可以写为
$y(t)=\left[\begin{array}{ccccc}y(t-1)& y(t-2)y(t-3)& u(t)& u^2(t-1)& \cos (\sqrt{t})\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\\ b_1\\ b_2\\ c\end{array}\right]+v(t)$
对应的辨识模型为
$$\{\begin{array}{l}y(t)={\phi }^T(t)\theta +v(t),\\ \phi (t)=[y(t-1),y(t-2)y(t-3),u(t),u^2(t-1),\cos (\sqrt{t}){]}^T\in R^5,\\ \theta =[a_1,a_2,b_1,b_2,c{]}^T\in R^5.\end{array}$$

3.非线性系统

$$y(t)=\frac{b[e^t+u^2(t-3)]}{u^2(t)+a_{1}u(t-1)+a_{2}u(t-2)}$$
为了将参数分离出来，可以通过定义新的输出$y_1(t)=1/y(t)$,新的参数${\theta }_1=1/b,{\theta }_2=a_1/b,{\theta }_3=a_2/b$,该模型可以变化为线性参数系统：
$$\begin{gathered} y_1(t)=\frac{1}{y(t)}=\frac{u^2(t)+a_{1}u(t-1)+a_{2}u(t-2)}{b[e^t+u^2(t-3)]} \\ =\frac{1}{b}\frac{u^2(t)}{e^t+u^2(t-3)}+\frac{a_1}{b}\frac{u(t-1)}{e^t+u^2(t-3)}+\frac{a_2}{b}\frac{u(t-2)}{e^t+u^2(t-3)} \\ =\left[\begin{array}{ccc}\frac{u^2(t)}{e^t+u^2(t-3)}& \frac{u(t-1)}{e^t+u^2(t-3)}& \frac{u(t-2)}{e^t+u^2(t-3)}\end{array}\right]\left[\begin{array}{c}{\theta }_1\\ {\theta }_2\\ {\theta }_3\end{array}\right] \end{gathered}$$
对应的辨识模型为
$$\{\begin{array}{l}{y}_1(t)={\phi }^T(t)\theta ,\\ \phi (t)=[\frac{u^2(t)}{e^t+u^2(t-3)},\frac{u(t-1)}{e^t+u^2(t-3)},\frac{u(t-2)}{e^t+u^2(t-3)}{]}^T\in R^3,\\ \theta =[\frac{1}{b},\frac{a_1}{b},\frac{a_2}{b}{]}^T\in R^3.\end{array}$$

4.状态空间系统

$$\{\begin{array}{l}x(t+1)=Ax(t)+bu(t)+\omega (t),\\ y(t)=cx(t)+du(t)+v(t),\end{array}$$
其中$\omega (t)$为过程噪声向量，$v(t)$为观测噪声。
定义：
$$\begin{gathered} Y(t)=\left[\begin{array}{c}x(t+1)\\ y(t)\end{array}\right],\phi (t)=\left[\begin{array}{c}x(t)\\ u(t)\end{array}\right], \\ {\theta }^T=\left[\begin{array}{cc}A& b\\ c& d\end{array}\right],e(t)=\left[\begin{array}{c}\omega (t)\\ v(t)\end{array}\right]. \end{gathered}$$
则状态空间模型对应的辨识模型为
$Y(t)={\theta }^T\phi (t)+e(t).$

参考文献

丁锋.系统辨识新论[M].北京:科学出版社,2013:10.