【系统辨识】最小二乘辨识模型的建立
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
文章目录
前言
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y(t)= \pmb{\varphi^{T}}(t)\pmb{\theta}+v(t)
y(t)=φT(t)θ+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
y ( t ) = b 1 u ( t − 1 ) + b 2 u ( t − 2 ) + . . . + b n u ( t − n ) + v ( t ) = [ u ( t − 1 ) , u ( t − 1 ) , . . . , u ( t − n ) ] [ b 1 b 2 . . . b n ] + v ( t ) 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 ] \begin{bmatrix} b_{1}\\ b_{2}\\ ...\\ b_{n} \end{bmatrix}+v(t) y(t)=b1u(t−1)+b2u(t−2)+...+bnu(t−n)+v(t)=[u(t−1),u(t−1),...,u(t−n)] b1b2...bn +v(t)对应的辨识模型为 { y ( t ) = φ T ( t ) θ + v ( t ) , φ ( t ) = [ u ( t − 1 ) , u ( t − 2 ) , . . . , u ( t − n ) ] T ∈ R n , θ = [ b 1 , b 2 , . . . , b n ] T ∈ R n . \left\{ \begin{array}{l} y(t)= \pmb{\varphi^{T}}(t)\pmb{\theta}+v(t),\\ \pmb{\varphi}(t)=[u(t-1),u(t-2),...,u(t-n) ]^T\in R^{n},\\ \pmb{\theta}=[b_1,b_2,...,b_n]^T\in R^n. \end{array} \right. ⎩ ⎨ ⎧y(t)=φT(t)θ+v(t),φ(t)=[u(t−1),u(t−2),...,u(t−n)]T∈Rn,θ=[b1,b2,...,bn]T∈Rn.
2.时变系统差分方程描述的线性系统
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y(t)+a_1(t)y(t-1)+a_2(t)y(t-2)=b_1u(t-1)+b_2u(t-2)
y(t)+a1(t)y(t−1)+a2(t)y(t−2)=b1u(t−1)+b2u(t−2)
对应的辨识模型为
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\left\{ \begin{array}{l} y(t)= \pmb{\varphi^{T}}(t)\pmb{\theta},\\ \pmb{\varphi}(t)=[-y(t-1),-y(t-2),u(t-1),u(t-2) ]^T\in R^{4},\\ \pmb{\theta}=[a_1(t),a_2,b_1,b_2]^T\in R^4. \end{array} \right.
⎩
⎨
⎧y(t)=φT(t)θ,φ(t)=[−y(t−1),−y(t−2),u(t−1),u(t−2)]T∈R4,θ=[a1(t),a2,b1,b2]T∈R4.
3.线性动态系统（CAR模型）
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A(z)y(t)=B(z)u(t)+v(t),
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A(z)=1+a_1z^{-1}+a_2z^{-2}+...+a_{n_a}z^{-n_a}, \\B(z)=b_1z^{-1}+b_2z^{-2}+...+b_{n_b}z^{-n_b}.
A(z)=1+a1z−1+a2z−2+...+anaz−na,B(z)=b1z−1+b2z−2+...+bnbz−nb.将多项式
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(1+a_1z^{-1}+a_2z^{-2}+...+a_{n_a}z^{-n_a})y(t)=(b_1z^{-1}+b_2z^{-2}+...+b_{n_b}z^{-n_b})u(t)+v(t)
(1+a1z−1+a2z−2+...+anaz−na)y(t)=(b1z−1+b2z−2+...+bnbz−nb)u(t)+v(t)利用移位算子的性质，上式可以写为
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y(t)+a_1y(t-1)+a_2y(t-2)+...+a_{n_a}y(t-n_a)=b_1u(t-1)+b_2u(t-1)+...+b_{n_b}u(t-n_b)+v(t)
y(t)+a1y(t−1)+a2y(t−2)+...+anay(t−na)=b1u(t−1)+b2u(t−1)+...+bnbu(t−nb)+v(t)移项可得
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y(t)=-a_1y(t-1)-a_2y(t-2)-...-a_{n_a}y(t-n_a)+b_1u(t-1)+b_2u(t-1)+...+b_{n_b}u(t-n_b)+v(t)
y(t)=−a1y(t−1)−a2y(t−2)−...−anay(t−na)+b1u(t−1)+b2u(t−1)+...+bnbu(t−nb)+v(t)对应的辨识模型为
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\left\{ \begin{array}{l} y(t)= \pmb{\varphi^{T}}(t)\pmb{\theta}+v(t),\\ \pmb{\varphi}(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},\\ \pmb{\theta}=[a_1,a_2,...,a_{n_a},b_1,b_2,...,b_{n_b}]^T\in R^{n_a+n_b}. \end{array} \right.
⎩
⎨
⎧y(t)=φT(t)θ+v(t),φ(t)=[−y(t−1),−y(t−2),...,−y(t−na),u(t−1),u(t−2),...,u(t−nb)]T∈Rna+nb,θ=[a1,a2,...,ana,b1,b2,...,bnb]T∈Rna+nb.
二、非线性系统辨识模型
1.静态非线性系统
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y=ax2+bx+c
将该模型写为向量形式
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y=\begin{bmatrix}x^2 & x & 1 \end{bmatrix} \begin{bmatrix}a \\ b \\ c\end{bmatrix}
y=[x2x1]
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对应的辨识模型为
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\left\{ \begin{array}{l} y= \pmb{\varphi^{T}}\pmb{\theta},\\ \pmb{\varphi}=[x^2,x,1 ]^T\in R^{3},\\ \pmb{\theta}=[a,b,c]^T\in R^{3}. \end{array} \right.
⎩
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⎧y=φTθ,φ=[x2,x,1]T∈R3,θ=[a,b,c]T∈R3.
2.动态非线性系统
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y(t)=a_1y(t-1)+a_2y(t-2)y(t-3)+b_1u(t)+b_2u^2(t-1)+c\cos (\sqrt{t})+v(t)
y(t)=a1y(t−1)+a2y(t−2)y(t−3)+b1u(t)+b2u2(t−1)+ccos(t)+v(t)
可以写为
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y(t)=\begin{bmatrix}y(t-1) & y(t-2)y(t-3) & u(t) & u^2(t-1) & \cos(\sqrt t) \end{bmatrix} \begin{bmatrix}a_1 \\ a_2 \\ b_1 \\ b_2 \\ c \end{bmatrix} +v(t)
y(t)=[y(t−1)y(t−2)y(t−3)u(t)u2(t−1)cos(t)]
a1a2b1b2c
+v(t)
对应的辨识模型为
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\left\{ \begin{array}{l} y(t)= \pmb{\varphi^{T}(t)}\pmb{\theta}+v(t),\\ \pmb{\varphi}(t)=[y(t-1),y(t-2)y(t-3),u(t),u^2(t-1),\cos(\sqrt t) ]^T\in R^{5},\\ \pmb{\theta}=[a_1,a_2,b_1,b_2,c]^T\in R^{5}. \end{array} \right.
⎩
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⎧y(t)=φT(t)θ+v(t),φ(t)=[y(t−1),y(t−2)y(t−3),u(t),u2(t−1),cos(t)]T∈R5,θ=[a1,a2,b1,b2,c]T∈R5.
3.非线性系统
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y(t)=\frac{b[e^t+u^2(t-3)]}{u^2(t)+a_1u(t-1)+a_2u(t-2)}
y(t)=u2(t)+a1u(t−1)+a2u(t−2)b[et+u2(t−3)]
为了将参数分离出来，可以通过定义新的输出
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y_1(t)=1/y(t)
y1(t)=1/y(t),新的参数
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\theta_1=1/b,\theta_2=a_1/b,\theta_3=a_2/b
θ1=1/b,θ2=a1/b,θ3=a2/b,该模型可以变化为线性参数系统：
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y_1(t)=\frac{1}{y(t)}=\frac{u^2(t)+a_1u(t-1)+a_2u(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)} \\=\begin{bmatrix}\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{bmatrix} \begin{bmatrix}\theta_1 \\ \theta_2 \\ \theta_3 \end{bmatrix}
y1(t)=y(t)1=b[et+u2(t−3)]u2(t)+a1u(t−1)+a2u(t−2)=b1et+u2(t−3)u2(t)+ba1et+u2(t−3)u(t−1)+ba2et+u2(t−3)u(t−2)=[et+u2(t−3)u2(t)et+u2(t−3)u(t−1)et+u2(t−3)u(t−2)]
θ1θ2θ3
对应的辨识模型为
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\left\{ \begin{array}{l} y_1(t)= \pmb{\varphi^{T}(t)}\pmb{\theta},\\ \pmb{\varphi}(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},\\ \pmb{\theta}=[\frac{1}{b},\frac{a_1}{b},\frac{a_2}{b}]^T\in R^{3}. \end{array} \right.
⎩
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⎧y1(t)=φT(t)θ,φ(t)=[et+u2(t−3)u2(t),et+u2(t−3)u(t−1),et+u2(t−3)u(t−2)]T∈R3,θ=[b1,ba1,ba2]T∈R3.
4.状态空间系统
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\left\{ \begin{array}{l} \pmb x(t+1)=\pmb {Ax}(t)+\pmb bu(t)+\pmb \omega(t),\\ y(t)=\pmb{cx}(t)+du(t)+v(t), \end{array} \right.
{x(t+1)=Ax(t)+bu(t)+ω(t),y(t)=cx(t)+du(t)+v(t),
其中
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定义：
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Y(t)=\begin{bmatrix} \pmb x(t+1) \\ y(t) \end{bmatrix},\pmb \varphi(t)=\begin{bmatrix} \pmb x(t) \\ u(t) \end{bmatrix},\\ \pmb \theta^T=\begin{bmatrix} \pmb A & \pmb b\\ \pmb c & d\end{bmatrix},\pmb e(t)=\begin{bmatrix} \pmb \omega(t) \\ v(t) \end{bmatrix}.
Y(t)=[x(t+1)y(t)],φ(t)=[x(t)u(t)],θT=[Acbd],e(t)=[ω(t)v(t)].
则状态空间模型对应的辨识模型为
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\pmb Y(t)=\pmb {\theta^T \varphi}(t)+\pmb e(t).
Y(t)=θTφ(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.
AtomGit 是由开放原子开源基金会联合 CSDN 等生态伙伴共同推出的新一代开源与人工智能协作平台。平台坚持“开放、中立、公益”的理念，把代码托管、模型共享、数据集托管、智能体开发体验和算力服务整合在一起，为开发者提供从开发、训练到部署的一站式体验。
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