【系统辨识】递推最小二乘法的推导及matlab仿真
递推最小二乘算法(RLS)是一种用于在线计算线性回归的方法。该算法可以在不需要保存所有数据的情况下，使用最小二乘法递推地计算线性回归系数。具体地说，该算法在每次接收一个新的样本时，会根据已经处理过的样本和相应的预测值，递推地更新线性回归系数。这样，就可以利用新的样本来更新模型，而不需要重新计算所有样本。递推最小二乘算法的优点是可以在不需要保存全部数据的情况下，快速计算出线性回归系数。因此，它在处理
一、递推最小二乘
1.什么是递推最小二乘
递推最小二乘算法(RLS) 是一种用于在线计算线性回归的方法。该算法可以在不需要保存所有数据的情况下，使用最小二乘法递推地计算线性回归系数。
具体地说，该算法在每次接收一个新的样本时，会根据已经处理过的样本和相应的预测值，递推地更新线性回归系数。这样，就可以利用新的样本来更新模型，而不需要重新计算所有样本。
递推最小二乘算法的优点是可以在不需要保存全部数据的情况下，快速计算出线性回归系数。因此，它在处理大量数据时很有用。
2.最小二乘和递推最小二乘的区别
最小二乘是一种常见的回归分析方法，通过寻找最小化残差平方和的参数估计值来拟合数据。而递推最小二乘是对最小二乘方法的一种改进，它使用过去的数据估计未来的值，通过递推公式来计算给定时间点的预测值，同时不断更新参数估计值，以最小化当前误差的平方和。
两者区别在于，最小二乘是一次性拟合整个模型，属于离线辨识方法。而递推最小二乘则是通过递推过程逐步建立模型，并不断根据新的数据进行参数更新和模型修正，属于在线辨识方法。递推最小二乘方法更适用于时间序列数据的预测，能够更好地反映数据的动态变化和趋势。
3.递推最小二乘算法的推导
根据最小二乘算法可得参数向量
θ
\pmb \theta
θ的最小估计值：
θ
^
(
t
)
=
(
H
t
T
H
t
)
−
1
H
t
T
Y
t
(1-1)
\hat{\pmb{\theta}}(t)=(\pmb{H_t}^T\pmb{H_t})^{-1}\pmb{H_t}^T\pmb{Y_t}\tag{1-1}
θ^(t)=(HtTHt)−1HtTYt(1-1)上式的推导可以看【系统辨识】最小二乘估计。请务必明白式(1-1)的由来和含义，有利于明白后文整个过程的推导。下面讨论将公式(1-1)改写为递推计算式，即参数估计
θ
^
(
t
)
\hat{\pmb{\theta}}(t)
θ^(t)等于前一时刻的参数估计
θ
^
(
t
−
1
)
\hat{\pmb{\theta}}(t-1)
θ^(t−1)加上一校正项，构成一个递推计算式。
首先定义非降矩阵，
P
−
1
(
t
)
=
H
t
T
H
t
=
∑
j
=
1
t
φ
(
j
)
φ
T
(
j
)
=
P
−
1
(
t
−
1
)
+
φ
(
t
)
φ
T
(
t
)
.
(1-2)
\begin{aligned}\pmb{P}^{-1}(t)&=\pmb{H}^{T}_t\pmb{H}_t=\sum_{j=1}^{t}\pmb{\varphi(j)}\pmb{\varphi^T(j)}\\&=\pmb{P}^{-1}(t-1)+\pmb{\varphi(t)}\pmb{\varphi^T(t)}.\tag{1-2}\end{aligned}
P−1(t)=HtTHt=j=1∑tφ(j)φT(j)=P−1(t−1)+φ(t)φT(t).(1-2)由此递推计算式可得
P
−
1
(
t
)
=
P
−
1
(
t
−
1
)
+
φ
(
t
)
φ
T
(
t
)
=
P
−
1
(
t
−
2
)
+
∑
j
=
t
−
1
t
φ
(
j
)
φ
T
(
j
)
=
.
.
.
.
.
.
.
.
.
.
.
=
P
−
1
(
0
)
+
∑
j
=
1
t
φ
(
j
)
φ
T
(
j
)
=
P
−
1
(
0
)
+
H
t
T
H
t
.
\begin{aligned}\pmb{P}^{-1}(t)&=\pmb{P}^{-1}(t-1)+\pmb{\varphi(t)}\pmb{\varphi^T(t)} \\&=\pmb P^{-1}(t-2)+\sum_{j=t-1}^{t}\pmb{\varphi(j)}\pmb{\varphi^T(j)} \\&=........... \\&=\pmb P^{-1}(0)+\sum_{j=1}^{t}\pmb{\varphi(j)}\pmb{\varphi^T(j)} \\&=\pmb P^{-1}(0)+\pmb{H}^{T}_t\pmb{H}_t. \end{aligned}
P−1(t)=P−1(t−1)+φ(t)φT(t)=P−1(t−2)+j=t−1∑tφ(j)φT(j)=...........=P−1(0)+j=1∑tφ(j)φT(j)=P−1(0)+HtTHt.将上式与式(1-2)比较可以得到，
P
−
1
(
t
)
\pmb P^{-1}(t)
P−1(t)初值应该取
P
−
1
(
0
)
=
0
\pmb P^{-1}(0)=0
P−1(0)=0，但实际中取为一个非常小的正定矩阵，即
P
−
1
(
0
)
=
I
/
p
0
\pmb P^{-1}(0)=\pmb I/p_0
P−1(0)=I/p0或
P
(
0
)
=
p
0
I
\pmb P(0)=p_0\pmb I
P(0)=p0I,这里
p
0
p_0
p0为一个大的常数，如
p
0
=
1
0
6
.
p_0=10^6.
p0=106.
另外，根据
Y
t
\pmb{Y}_t
Yt和
H
t
\pmb{H}_t
Ht的定义可知(在【系统辨识】最小二乘估计中有相关定义)，
Y
t
=
[
y
(
1
)
y
(
2
)
.
.
.
y
(
t
−
1
)
y
(
t
)
]
=
[
Y
t
−
1
y
(
t
)
]
∈
R
t
,
H
t
=
[
φ
T
(
1
)
φ
T
(
2
)
.
.
.
φ
T
(
t
−
1
)
φ
T
(
t
)
]
=
[
H
t
−
1
φ
T
(
t
)
]
∈
R
t
×
n
.
\pmb{Y}_t=\begin{bmatrix} y(1)\\y(2)\\...\\y(t-1)\\y(t)\end{bmatrix}=\begin{bmatrix} \pmb{Y}_{t-1}\\y(t)\end{bmatrix}\in R^t,\\ \pmb{H_t}=\begin{bmatrix} \pmb{\varphi^T(1)}\\ \pmb{\varphi^T(2)}\\...\\\pmb{\varphi^T(t-1)}\\\pmb{\varphi^T(t)}\end{bmatrix}=\begin{bmatrix}\pmb{H_{t-1}}\\\pmb{\varphi^T(t)}\end{bmatrix}\in R^{t\times n}.
Yt=
y(1)y(2)...y(t−1)y(t)
=[Yt−1y(t)]∈Rt,Ht=
φT(1)φT(2)...φT(t−1)φT(t)
=[Ht−1φT(t)]∈Rt×n.把式(1-2)带入式(1-1)中可得
θ
^
(
t
)
=
(
H
t
T
H
t
)
−
1
H
t
T
Y
t
=
P
(
t
)
H
t
T
Y
t
=
P
(
t
)
[
H
t
−
1
T
φ
(
t
)
]
[
Y
t
−
1
y
(
t
)
]
=
P
(
t
)
[
H
t
−
1
T
Y
t
−
1
+
φ
(
t
)
y
(
t
)
]
=
P
(
t
)
[
P
−
1
(
t
−
1
)
P
(
t
−
1
)
H
t
−
1
T
Y
t
−
1
+
φ
(
t
)
y
(
t
)
]
=
P
(
t
)
[
P
−
1
(
t
−
1
)
θ
^
(
t
−
1
)
+
φ
(
t
)
y
(
t
)
]
=
P
(
t
)
[
[
P
−
1
(
t
)
−
φ
(
t
)
φ
T
(
t
)
]
θ
^
(
t
−
1
)
+
φ
(
t
)
y
(
t
)
]
=
θ
^
(
t
−
1
)
−
P
(
t
)
φ
(
t
)
φ
T
(
t
)
θ
^
(
t
−
1
)
+
P
(
t
)
φ
(
t
)
y
(
t
)
=
θ
^
(
t
−
1
)
+
P
(
t
)
φ
(
t
)
[
y
(
t
)
−
φ
T
(
t
)
θ
^
(
t
−
1
)
]
\begin{aligned}\hat{\pmb{\theta}}(t)&=(\pmb{H^T_t}\pmb{H_t})^{-1}\pmb{H^T_t}\pmb{Y_t}=\pmb{P}(t)\pmb{H^T_t}\pmb{Y_t}\\&=\pmb{P}(t)\begin{bmatrix}\pmb{H^T_{t-1}}&\pmb{\varphi}(t)\end{bmatrix}\begin{bmatrix} \pmb{Y}_{t-1}\\y(t)\end{bmatrix} \\&=\pmb P(t)[\pmb{H^T_{t-1}}\pmb{Y}_{t-1}+\pmb{\varphi}(t)y(t)] \\&=\pmb P(t)[\pmb P^{-1}(t-1)\pmb P(t-1)\pmb{H^T_{t-1}}\pmb{Y}_{t-1}+\pmb{\varphi}(t)y(t)] \\&=\pmb P(t)[\pmb P^{-1}(t-1)\pmb{\hat{\theta}}(t-1)+\pmb{\varphi}(t)y(t)] \\&=\pmb P(t)[[\pmb{P}^{-1}(t)-\pmb{\varphi(t)}\pmb{\varphi^T(t)}]\pmb{\hat{\theta}}(t-1)+\pmb{\varphi}(t)y(t)] \\&=\pmb{\hat{\theta}}(t-1)-\pmb{P}(t)\pmb{\varphi(t)}\pmb{\varphi^T(t)}\pmb{\hat{\theta}}(t-1)+\pmb P(t)\pmb{\varphi}(t)y(t) \\&=\pmb{\hat{\theta}}(t-1)+\pmb{P}(t)\pmb{\varphi(t)}[y(t)-\pmb{\varphi^T(t)}\pmb{\hat{\theta}}(t-1)] \end{aligned}
θ^(t)=(HtTHt)−1HtTYt=P(t)HtTYt=P(t)[Ht−1Tφ(t)][Yt−1y(t)]=P(t)[Ht−1TYt−1+φ(t)y(t)]=P(t)[P−1(t−1)P(t−1)Ht−1TYt−1+φ(t)y(t)]=P(t)[P−1(t−1)θ^(t−1)+φ(t)y(t)]=P(t)[[P−1(t)−φ(t)φT(t)]θ^(t−1)+φ(t)y(t)]=θ^(t−1)−P(t)φ(t)φT(t)θ^(t−1)+P(t)φ(t)y(t)=θ^(t−1)+P(t)φ(t)[y(t)−φT(t)θ^(t−1)]联立上式和式(1-2)可以得到参数向量
θ
\pmb \theta
θ的递推最小二乘算法(RLS):
θ
^
(
t
)
=
θ
^
(
t
−
1
)
+
P
(
t
)
φ
(
t
)
[
y
(
t
)
−
φ
T
(
t
)
θ
^
(
t
−
1
)
]
,
(1-3)
\hat{\pmb{\theta}}(t)=\pmb{\hat{\theta}}(t-1)+\pmb{P}(t)\pmb{\varphi(t)}[y(t)-\pmb{\varphi^T(t)}\pmb{\hat{\theta}}(t-1)],\tag{1-3}
θ^(t)=θ^(t−1)+P(t)φ(t)[y(t)−φT(t)θ^(t−1)],(1-3)
P
−
1
(
t
)
=
P
−
1
(
t
−
1
)
+
φ
(
t
)
φ
T
(
t
)
,
P
(
0
)
=
p
0
I
>
0.
(1-4)
\pmb{P}^{-1}(t)=\pmb{P}^{-1}(t-1)+\pmb{\varphi(t)}\pmb{\varphi^T(t)},\pmb P(0)=p_0\pmb I>0.\tag{1-4}
P−1(t)=P−1(t−1)+φ(t)φT(t),P(0)=p0I>0.(1-4)
4.递推最小二乘的等价形式
为了避免协方差矩阵
P
(
t
)
\pmb P(t)
P(t)的求逆运算，通常运用矩阵求逆公式和引入一个增益向量
L
(
t
)
=
P
(
t
)
φ
(
t
)
\pmb L(t)=\pmb P(t)\pmb \varphi(t)
L(t)=P(t)φ(t),将式(1-3)和(1-4)化为一种等价形式。
首先引入一个矩阵求逆引理：设
A
∈
R
n
×
n
,
B
∈
R
n
×
r
,
C
∈
R
r
×
n
\pmb A \in R^{n \times n},\pmb B\in R^{n \times r},\pmb C \in R^{r \times n}
A∈Rn×n,B∈Rn×r,C∈Rr×n,假设矩阵
A
\pmb A
A和
(
I
+
C
A
−
1
B
)
(\pmb I+\pmb{CA^{-1}B})
(I+CA−1B)可逆，则下列等式成立，
(
A
+
B
C
)
−
1
=
A
−
1
−
A
−
1
B
(
I
+
C
A
−
1
B
)
−
1
C
A
−
1
.
(1-5)
(\pmb A+\pmb {BC})^{-1}=\pmb A^{-1}-\pmb A^{-1}\pmb{B}(\pmb I+\pmb {CA^{-1}B})^{-1}\pmb{CA}^{-1}.\tag{1-5}
(A+BC)−1=A−1−A−1B(I+CA−1B)−1CA−1.(1-5)通过式(1-5)应用到式(1-2)中可得
P
(
t
)
=
P
(
t
−
1
)
−
P
(
t
−
1
)
φ
(
t
)
φ
T
(
t
)
P
(
t
−
1
)
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
(1-6)
\pmb P(t)=\pmb P(t-1)-\frac{\pmb P(t-1)\pmb{\varphi(t)}\pmb{\varphi^T(t)}\pmb P(t-1)}{1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}}\tag{1-6}
P(t)=P(t−1)−1+φT(t)P(t−1)φ(t)P(t−1)φ(t)φT(t)P(t−1)(1-6)将上式两边右乘向量
φ
(
t
)
\pmb \varphi(t)
φ(t)可得
P
(
t
)
φ
(
t
)
=
P
(
t
−
1
)
φ
(
t
)
−
P
(
t
−
1
)
φ
(
t
)
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
=
P
(
t
−
1
)
φ
(
t
)
[
1
−
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
]
=
P
(
t
−
1
)
φ
(
t
)
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
=
L
(
t
)
.
\begin{aligned}\pmb P(t)\pmb \varphi(t)&=\pmb P(t-1)\pmb \varphi(t)-\frac{\pmb P(t-1)\pmb{\varphi(t)}\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}}{1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}} \\&=\pmb P(t-1)\pmb \varphi(t)[1-\frac{\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}}{1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}}] \\&=\frac{\pmb P(t-1)\pmb{\varphi(t)}}{1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}}=\pmb L(t). \end{aligned}
P(t)φ(t)=P(t−1)φ(t)−1+φT(t)P(t−1)φ(t)P(t−1)φ(t)φT(t)P(t−1)φ(t)=P(t−1)φ(t)[1−1+φT(t)P(t−1)φ(t)φT(t)P(t−1)φ(t)]=1+φT(t)P(t−1)φ(t)P(t−1)φ(t)=L(t).将
L
(
t
)
\pmb L(t)
L(t)代入式(1-6)可得
P
(
t
)
=
P
(
t
−
1
)
−
P
(
t
−
1
)
φ
(
t
)
φ
T
(
t
)
P
(
t
−
1
)
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
=
P
(
t
−
1
)
−
L
(
t
)
φ
T
(
t
)
P
(
t
−
1
)
=
[
I
−
L
(
t
)
φ
T
(
t
)
]
P
(
t
−
1
)
,
P
(
0
)
=
p
0
I
.
\begin{aligned}\pmb P(t)&=\pmb P(t-1)-\frac{\pmb P(t-1)\pmb{\varphi(t)}\pmb{\varphi^T(t)}\pmb P(t-1)}{1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}} \\&=\pmb P(t-1)-\pmb L(t)\pmb{\varphi^T(t)}\pmb P(t-1) \\&=[\pmb I-\pmb L(t)\pmb{\varphi^T(t)}]\pmb P(t-1),\pmb P(0)=p_0\pmb I.\end{aligned}
P(t)=P(t−1)−1+φT(t)P(t−1)φ(t)P(t−1)φ(t)φT(t)P(t−1)=P(t−1)−L(t)φT(t)P(t−1)=[I−L(t)φT(t)]P(t−1),P(0)=p0I.
综上，递推最小二乘(RLS)算法还可以表达为
θ
^
(
t
)
=
θ
^
(
t
−
1
)
+
L
(
t
)
[
y
(
t
)
−
φ
T
(
t
)
θ
^
(
t
−
1
)
]
,
(1-7)
\hat{\pmb{\theta}}(t)=\pmb{\hat{\theta}}(t-1)+\pmb L(t)[y(t)-\pmb{\varphi^T(t)}\pmb{\hat{\theta}}(t-1)],\tag{1-7}
θ^(t)=θ^(t−1)+L(t)[y(t)−φT(t)θ^(t−1)],(1-7)
L
(
t
)
=
P
(
t
−
1
)
φ
(
t
)
[
1
+
φ
T
(
t
)
P
(
t
−
1
)
φ
(
t
)
]
−
1
,
(1-8)
\pmb L(t)=\pmb P(t-1)\pmb{\varphi(t)}[1+\pmb{\varphi^T(t)}\pmb P(t-1)\pmb{\varphi(t)}]^{-1},\tag{1-8}
L(t)=P(t−1)φ(t)[1+φT(t)P(t−1)φ(t)]−1,(1-8)
P
(
t
)
=
[
I
−
L
(
t
)
φ
T
(
t
)
]
P
(
t
−
1
)
,
P
(
0
)
=
p
0
I
.
(1-9)
\pmb P(t)=[\pmb I-\pmb L(t)\pmb{\varphi^T(t)}]\pmb P(t-1),\pmb P(0)=p_0\pmb I.\tag{1-9}
P(t)=[I−L(t)φT(t)]P(t−1),P(0)=p0I.(1-9)
注：在递推算法中，初值通常选择
P ( 0 ) = \pmb P(0)= P(0)=很大对称正定矩阵,如 P ( 0 ) = p 0 I , p 0 = 1 0 6 ≫ 1 , P(0)=p_0\pmb I,p_0=10^6\gg1, P(0)=p0I,p0=106≫1,
θ ^ ( t ) = \hat{\pmb{\theta}}(t)= θ^(t)=很小实向量，如 θ ^ ( t ) = 1 n / p 0 . \hat{\pmb{\theta}}(t)=1_n/p_0. θ^(t)=1n/p0.
二、MATLAB仿真
例 考虑CAR模型仿真对象,模型如下图所示。
A
(
z
)
y
(
t
)
=
B
(
z
)
u
(
t
)
+
v
(
t
)
,
A
(
z
)
=
1
+
a
1
z
−
1
+
a
2
z
−
2
+
a
3
z
−
3
=
1
−
1.40
z
−
1
+
0.50
z
−
2
+
0.10
z
−
3
,
B
(
z
)
=
b
1
z
−
1
+
b
2
z
−
2
+
b
3
z
−
3
=
0.50
z
−
1
−
0.60
z
−
2
−
0.70
z
−
3
,
θ
=
[
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
]
T
=
[
−
1.40
,
0.50
,
0.10
,
0.50
,
−
0.60
,
−
0.70
]
T
.
\begin{aligned}A(z)y(t)&=B(z)u(t)+v(t), \\A(z)&=1+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}=1-1.40z^{-1}+0.50z^{-2}+0.10z^{-3}, \\B(z)&=b_1z^{-1}+b_2z^{-2}+b_3z^{-3}=0.50z^{-1}-0.60z^{-2}-0.70z^{-3}, \\\pmb \theta&=[a_1,a_2,a_3,b_1,b_2,b_3]^T=[-1.40,0.50,0.10,0.50,-0.60,-0.70]^T.\end{aligned}
A(z)y(t)A(z)B(z)θ=B(z)u(t)+v(t),=1+a1z−1+a2z−2+a3z−3=1−1.40z−1+0.50z−2+0.10z−3,=b1z−1+b2z−2+b3z−3=0.50z−1−0.60z−2−0.70z−3,=[a1,a2,a3,b1,b2,b3]T=[−1.40,0.50,0.10,0.50,−0.60,−0.70]T.
参考程序：
%注：第一次运行sigma=0.1，程序将估计到的参数保存到data1的文件中。再令sigma=1，程序将会比较sigma=0.1和sigma=1时参数估计的误差。
clear all;
%初始化模型
FF =1;% The Forgettingfactor
sigma =0.1;% sigma =0.10 and 1.0 噪声方差
PlotLength =3000; length1 = PlotLength +100;
na =3; nb =3; n = na + nb ;
a =[1,-1.4,0.5,0.1]; b =[0,0.5,-0.6,-0.7]; d =[1 0 0 0];
par0=[a(2:na+1), b(2:nb+1)]';%参数theta真实值
p0=1e6; P=eye(n)*p0; r=1;%P(0)=p0*I，很大对称正定矩阵
par1 = ones(n,1)*1e-6;%参数估计初值theta=1/p0，很小实向量
%生成输入输出数据
rand ('state',40);% randn('state',1);
eta = randn(length1); %得到服从标准正态分布的随机数
u = eta(:,1); v = eta(:,2)*sigma ; %得到输入序列u和方差为sigma的干扰v
clear eta ;
Gz = tf(b ,a ,1), Gn=tf(d ,a,1) %建立传递Gz，Gn函数
y = lsim(Gz,u)+ lsim(Gn ,v); %根据输入u和干扰v得到系统输出y
% 递推最小二乘(RLS)算法
jj =0;j1=0;
for t =20:length1
jj = jj +1;
varphi =[-y(t-1:-1:t-na); u(t-1:-1:t-nb )];
L = P * varphi /( FF + varphi'* P * varphi);
P = P - L * varphi'* P ;
par1=par1+ L *( y ( t )- varphi'*par1);
delta1 = norm(par1 -par0)/norm(par0);%与真值的误差
ls(jj,:)=[jj,par1',delta1 ];%存储每一step的参数估计和误差
if (jj==100)|(jj==200)|(jj==500)|mod(jj,1000)==0
j1=j1+1;
ls_100(j1,:)=[jj,par1',delta1*100];%存储t=100，200,500,1000,2000,3000时刻的参数估计和误差
end
if jj == PlotLength %t=3000时停止计算
break
end
end
%作图程序
ls_100(j1+1,:)=[0,par0',0];
figure (1)
jk =(17:10:PlotLength-1)';
plot(ls(jk ,1),ls(jk,n+2));
if sigma ==0.1
data1 =[ls(:,1),ls(:,n +2)];
save data1 data1
else % sigma ==1.0
load data1
z0=[ data1 , ls(:, n +2)];
figure (3);
plot (z0(jk ,1),z0(jk ,2),' k ',z0(jk,1),z0( jk ,3),' b ')
axis ([0,3000,0,0.33])
text (600,0.058,'{\it\sigma^2}=1.00*2')
text (600,0.13,'{\it\sigma^2}=0.10*2')
line ([247,620],[0.024,0.119])
end
运行结果：
下表由数组ls_100得出
总结
（1）递推最小二次算法有很强的鲁棒性
（2）随着数据长度的增加，参数估计误差不断减小
（3）输入输出信号的幅值越大，参数估计精度越高
参考文献
丁锋.系统辨识新论[M].北京:科学出版社,2013:120.
开放原子开发者工作坊旨在鼓励更多人参与开源活动，与志同道合的开发者们相互交流开发经验、分享开发心得、获取前沿技术趋势。工作坊有多种形式的开发者活动，如meetup、训练营等，主打技术交流，干货满满，真诚地邀请各位开发者共同参与！
更多推荐
30
0- 0
已为社区贡献3条内容
所有评论(0)