经典控制理论
动态系统建模
通过配置系统输入u(t)，使u(s)G(s)的极点使系统满足一定特性
一阶系统特性
G ( s ) = a s + a G(s) = \frac{a}{s+a} G ( s ) = s + a a 1 a \frac{1}{a} a 1 τ \tau τ 4 τ 4\tau 4 τ
二阶系统特性
m x ¨ + B x ˙ + k x = F m\ddot x+B\dot x+kx=F m x ¨ + B x ˙ + k x = F x ¨ + 2 ω n ξ x ˙ + ω n 2 x = F m \ddot x+2\omega_n\xi \dot x+\omega_n^2x=\frac{F}{m} x ¨ + 2 ω n ξ x ˙ + ω n 2 x = m F
阻尼比固有频率:ω n 1 − ξ 2 \omega_n\sqrt{1-\xi^2} ω n 1 − ξ 2
单位化：u ( t ) = F ω n 2 u(t)=\frac{F}{\omega_n^2} u ( t ) = ω n 2 F H ( s ) = ω n 2 s 2 + 2 ξ ω n s + ω n 2 H(s) = \frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2} H ( s ) = s 2 + 2 ξ ω n s + ω n 2 ω n 2
零极点图：
极点全部在左，系统稳定
虚轴长度代表振荡周期
实轴长度代表衰减速度
cos θ \cos \theta cos θ 阻尼比
SISO system稳定性判据
特征多项式系数判断传递函数稳定性
Hurwitz霍尔维兹判据：构建霍尔维兹行列式，全部为正
D 1 = a 1 D1 = a_1 D 1 = a 1
D 2 = ( a 1 a 3 a 0 a 2 ) D2 = \begin{pmatrix}
a_1&a_3\\
a_0&a_2
\end{pmatrix} D 2 = ( a 1 a 0 a 3 a 2 )
D 3 = ( a 1 a 3 a 5 a 0 a 2 a 4 0 a 1 a 3 ) D3 = \begin{pmatrix}
a_{1}& a_{3}& a_{5}\\
a_{0}& a_{2}& a_{4}\\
0& a_{1}& a_{3}
\end{pmatrix} D 3 = a 1 a 0 0 a 3 a 2 a 1 a 5 a 4 a 3
Lienard-Chipard林纳德-齐帕特判据：系数都大于零，奇数或偶数阶次行列式
Routh劳斯判据：
求e s s e_{ss} e ss e s s = lim s → 0 s E ( s ) e_{ss} = \lim \limits_{s\rightarrow0}sE(s) e ss = s → 0 lim s E ( s )
频率稳定判据：
H. Nyquist奈奎斯特判据，开环频率特性，判断闭环稳定性
F ( s ) = 1 + G ( s ) H ( s ) F(s) = 1 +G(s)H(s) F ( s ) = 1 + G ( s ) H ( s ) R = P − Z R=P-Z R = P − Z
频率特性
只适用于线性定常模型，否则不能拉式变换
稳定条件下使用
bode图单位用dB：20log(Mo/Mi)，表征了能量
幅值相应:magnitude response
M o M i = ∣ G ( j ω ) ∣ \frac{M_o}{M_i} = \left | G(j\omega)\right | M i M o = ∣ G ( j ω ) ∣
幅角响应:Phase response
ϕ o − ϕ i = ∠ G ( j ω ) \phi_o-\phi_i = \angle G(j\omega) ϕ o − ϕ i = ∠ G ( j ω )
带阻尼比的共振频率:
ω = ω n 1 − 2 ζ 2 \omega = \omega_n \sqrt{1-2\zeta^2}\\ ω = ω n 1 − 2 ζ 2 1 2 ζ 1 − ζ 2 \frac{1}{2\zeta\sqrt{1-\zeta^2}} 2 ζ 1 − ζ 2 1
幅值裕度h：相位为-π时，幅值距0dB的差值
相位裕度γ \gamma γ
不同频段信息
低频段G ( j ω ) G(j\omega) G ( j ω )
中频段：穿越0dBω c \omega_c ω c
高频段反映了系统对高频干扰抑制能力
系统矫正
串联矫正
超前矫正
G c ( s ) = 1 + a T s 1 + T s , a > 1 G_c(s)=\frac{1+aTs}{1+Ts},a>1 G c ( s ) = 1 + T s 1 + a T s , a > 1
滞后矫正
G c ( s ) = 1 + b T s 1 + T s , b < 1 G_c(s)=\frac{1+bTs}{1+Ts},b<1 G c ( s ) = 1 + T s 1 + b T s , b < 1
滞后超前矫正
两个合起来
PID矫正器
复合矫正
前置矫正：指令->Gc(s)->误差，一般补偿分母s，开环前向增益1
干扰前置补偿：干扰测量->Gc(s)->误差，误差->干扰端传函G s − 1 Gs^{-1} G s − 1
根轨迹
（开环->闭环稳定性）:分析G(s)的N、P，看闭环系统稳定性
开环传递函数中开环增益K从0-无穷时，闭环特征根的移动轨迹
单位负反馈闭环传递函数
ϕ ( s ) = C ( s ) R ( s ) = G ( s ) 1 + G ( s ) \phi(s) = \frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)} ϕ ( s ) = R ( s ) C ( s ) = 1 + G ( s ) G ( s )
非线性系统
叠加原理不适用
常规分类：
死区
饱和
间隙-滞环
系统收敛：消耗系统能量
系统发散：从外界获取能量
#相关词汇
X s s ( t ) X_{ss}(t) X ss ( t ) T s T_s T s T r T_r T r M p M_p M p T s s T_{ss} T ss ∀ \forall ∀ ∃ \exists ∃ ∥ ⋅ ∥ \left \| \cdot \right \| ∥ ⋅ ∥
工程数学基础
1. 特征值,特征向量,过渡矩阵→ \rightarrow →
特征值λ \lambda λ λ v = A v \lambda v=Av λ v = A v ∣ λ I − A ∣ = 0 \ | \lambda I-A\ | = 0 ∣ λ I − A ∣ = 0 λ \lambda λ ( λ I − A ) ∗ v = 0 ( \lambda I - A)* v = 0 ( λ I − A ) ∗ v = 0 λ 1 、 λ 2 \lambda_1 、\lambda_2 λ 1 、 λ 2 v 1 、 v 2 v_1 、v_2 v 1 、 v 2 过渡矩阵 ：特征向量组成的矩阵
P = ( v 1 v 2 ) P =
\begin {pmatrix} v_1&v_2
\end {pmatrix} P = ( v 1 v 2 ) A P = A [ v 1 v 2 ] = [ A v 1 A v 2 ] = [ λ 1 v 1 λ 2 v 2 ] = [ λ 1 v 11 λ 2 v 21 λ 1 v 12 λ 2 v 22 ] = P Λ AP=A[v_1 v_2] = [Av_1 Av_2]=[\lambda_1v_1 \lambda_2 v_2]=
\begin{bmatrix}
\lambda_1v_{11} & \lambda_2v_{21}\\
\lambda_1v_{12} & \lambda_2v_{22}
\end{bmatrix}
=P\Lambda
A P = A [ v 1 v 2 ] = [ A v 1 A v 2 ] = [ λ 1 v 1 λ 2 v 2 ] = [ λ 1 v 11 λ 1 v 12 λ 2 v 21 λ 2 v 22 ] = P Λ P − 1 A P = Λ P^-1AP = \Lambda P − 1 A P = Λ
2. 线性化 Linearization
非线性：1 / x , x , x n 等 1/x,\sqrt{x},x^n等 1/ x , x , x n 等
用泰勒级数展开
在平衡点(Fixed point)x 0 x_0 x 0
令导数项为0，求得平衡点x的值x = x 0 x=x_0 x = x 0
把x σ = x 0 + x d x_\sigma = x_0 + x_d x σ = x 0 + x d f ( x σ ) = f ( x 0 ) + f ′ ( x 0 ) ( x σ − x 0 ) f(x_\sigma)=f(x_0)+f'(x_0)(x_\sigma-x_0) f ( x σ ) = f ( x 0 ) + f ′ ( x 0 ) ( x σ − x 0 )
把x = x σ x = x_\sigma x = x σ σ \sigma σ x ˙ = A x + b u \dot x = A x + b u x ˙ = A x + b u
3. 卷积与LTI冲激响应（LTI：linear time invariant system）
卷积：x ( t ) = f ( t ) ∗ h ( t ) = ∫ 0 t f ( τ ) h ( t − τ ) d τ x(t) = f(t)*h(t)=\int_0^t f(\tau)h(t-\tau)d\tau x ( t ) = f ( t ) ∗ h ( t ) = ∫ 0 t f ( τ ) h ( t − τ ) d τ f ( t ) f(t) f ( t ) h ( t ) h(t) h ( t ) L 卷积 L_{卷积} L 卷积
e i θ = cos ( θ ) + i sin ( θ ) e^{i\theta}=\cos(\theta)+i\sin(\theta) e i θ = cos ( θ ) + i sin ( θ )
5. 复数Complex Number
sin ( x ) = C → x = π / 2 + 2 k π + ln ( C ± C 2 − 1 ) i \sin(x) = C\rightarrow x = \pi/2+2k\pi + \ln(C\pm\sqrt{C^2-1})i sin ( x ) = C → x = π /2 + 2 k π + ln ( C ± C 2 − 1 ) i Z = a + b i Z = a + b i Z = a + bi R e ( Z ) = a Re(Z) =a R e ( Z ) = a I m ( Z ) = b Im(Z)=b I m ( Z ) = b ∣ Z ∣ = a 2 + b 2 \left | Z \right | = \sqrt{a^2+b^2} ∣ Z ∣ = a 2 + b 2 Z = ∣ Z ∣ ⋅ ( cos θ + i sin θ ) = ∣ Z ∣ ⋅ e i θ Z = \left | Z \right | \cdot (\cos\theta+i\sin\theta)= \left | Z \right | \cdot e^{i\theta} Z = ∣ Z ∣ ⋅ ( cos θ + i sin θ ) = ∣ Z ∣ ⋅ e i θ Z 1 ⋅ Z 2 = ∣ Z 1 ∣ ∣ Z 2 ∣ e θ 1 + θ 2 Z_1 \cdot Z_2 = \left | Z_1 \right | \left | Z_2 \right | e^{\theta_1+\theta_2} Z 1 ⋅ Z 2 = ∣ Z 1 ∣ ∣ Z 2 ∣ e θ 1 + θ 2 Z + Z ˉ = 2 a Z+\bar Z = 2a Z + Z ˉ = 2 a Z − Z ˉ = 2 b i Z- \bar Z = 2bi Z − Z ˉ = 2 bi
6. 阈值选取
Normal Distribution正态分布、高斯分布
X = ( μ , σ 2 ) X = (\mu,\sigma^2) X = ( μ , σ 2 )
Advanced控制理论
状态空间：State-Space，包含输入、输出、状态，写成一阶微分方程的形式
x ˙ = A x + B u \dot x = A x + B u x ˙ = A x + B u y = C x + D u y = Cx+Du y = C x + D u
稳定性
两种类型
Lyapunov稳定性：有界
∀ t 0 , ∀ ϵ > 0 , ∃ δ ( t 0 , ϵ ) : ∥ x ( t 0 ) ∥ < δ ( t 0 , ϵ ) ⇒ ∀ t ⩾ t 0 , ∥ x ( t ) ∥ < ϵ \forall t_0, \forall \epsilon >0, \exists \delta (t_0, \epsilon):\left \| x(t_0)\right \|<\delta(t_0,\epsilon)\Rightarrow \forall t \geqslant t_0, \left \| x(t) \right \| < \epsilon ∀ t 0 , ∀ ϵ > 0 , ∃ δ ( t 0 , ϵ ) : ∥ x ( t 0 ) ∥ < δ ( t 0 , ϵ ) ⇒ ∀ t ⩾ t 0 , ∥ x ( t ) ∥ < ϵ a o f λ i ⩽ 0 a \, of\, \lambda_i \leqslant 0 a o f λ i ⩽ 0
渐进稳定性：
∃ δ ( t 0 ) > 0 : ∥ x ( t 0 ) ∥ < δ ( t 0 ) ⇒ lim t → ∞ ∥ x ( t ) ∥ = 0 \exists \delta(t_0)>0: \left \|x(t_0)\right \|<\delta(t_0) \Rightarrow
\lim \limits_{t \rightarrow \infty }
\left \| x(t)\right \| = 0
∃ δ ( t 0 ) > 0 : ∥ x ( t 0 ) ∥ < δ ( t 0 ) ⇒ t → ∞ lim ∥ x ( t ) ∥ = 0 a o f λ i < 0 a \, of\, \lambda_i < 0 a o f λ i < 0
判别方法
直接方法：解微分方程(Direct method)
求解λ的值，判断正负
第二方法：(2nd method)
( i ) V ( 0 ) = 0 (i)V(0) = 0 ( i ) V ( 0 ) = 0 ( i i ) V ( x ) ⩾ 0 , i n D − 0 (ii) V(x) \geqslant 0 , in\, D-{0} ( ii ) V ( x ) ⩾ 0 , in D − 0 ( i i i ) V ˙ ( x ) ⩽ 0 , i n D − 0 (iii)\dot V(x) \leqslant 0 , in\, D-{0} ( iii ) V ˙ ( x ) ⩽ 0 , in D − 0 ⇒ x = 0 \Rightarrow x = 0 ⇒ x = 0
3. 不稳定
存在至少一个特征值实部大于零
相图分析-phase-portrait
plot(x,x ˙ \dot x x ˙
% 画解微分方程组的相图 clear;cla;clc; [x,y]=meshgrid(linspace(-5,5)); streamslice(x,y,0 * x + 2 * y,-3 * x + 0 * y ); xlabel('x');ylabel('y');
特征值和相图的关系
齐次状态方程解x ˙ = A x \dot x = A x x ˙ = A x
x ˙ = a x → x ( t ) = e a t x ( 0 ) \dot x = a x\rightarrow x(t) = e^{at}x(0) x ˙ = a x → x ( t ) = e a t x ( 0 ) x ˙ = a x → x ( t ) = e A t x ( 0 ) \dot x = a x\rightarrow x(t) = e^{At}x(0) x ˙ = a x → x ( t ) = e A t x ( 0 ) 状态转移矩阵Φ ( t ) \Phi(t) Φ ( t )
数值法：
Φ ( t ) = e A t = I + A t + 1 2 ! A 2 t 2 + . . . + 1 k ! A k t k \Phi(t) = e^{At}=I+At+\frac{1}{2!}A^2t^2+...+\frac{1}{k!}A^kt^k Φ ( t ) = e A t = I + A t + 2 ! 1 A 2 t 2 + ... + k ! 1 A k t k
解析法：
Φ ( t ) = L − 1 [ s I − A ] − 1 \Phi(t) = L^{-1}[sI-A]^{-1} Φ ( t ) = L − 1 [ s I − A ] − 1
性质：
Φ ( 0 ) = I \Phi(0) = I Φ ( 0 ) = I x ( t ) = Φ ( t − t 0 ) x ( t 0 ) x(t) = \Phi(t-t_0)x(t_0) x ( t ) = Φ ( t − t 0 ) x ( t 0 ) Φ − 1 ( t ) = Φ ( − t ) \Phi ^{-1}(t) = \Phi(-t) Φ − 1 ( t ) = Φ ( − t )
非齐次状态方程x ˙ = A x + B u \dot x = A x + B u x ˙ = A x + B u
x ( t ) = Φ ( t ) x ( 0 ) + ∫ 0 t Φ ( t − τ ) B u ( τ ) d τ x(t) = \Phi (t)x(0)+ \int_0^t\Phi(t-\tau)Bu(\tau)d\tau x ( t ) = Φ ( t ) x ( 0 ) + ∫ 0 t Φ ( t − τ ) B u ( τ ) d τ
线性系统可控性与可观测性
可控性：∀ x ( 0 ) , x ( t f ) , ∃ t f < + ∞ , u [ 0 , t f ] , s t . x ( 0 ) → x ( t f ) \forall x(0),x(t_f), \exists t_f < +\infty , u[0,t_f], st. x(0)\rightarrow x(t_f) ∀ x ( 0 ) , x ( t f ) , ∃ t f < + ∞ , u [ 0 , t f ] , s t . x ( 0 ) → x ( t f )
S = [ b A b A 2 . . . A n − 1 b ] S = [b\, Ab\, A^2...\, A^{n-1}b] S = [ b A b A 2 ... A n − 1 b ]
x 0 = 0 x 1 = A x 0 + B u 0 = B u 0 x 2 = A x 1 + B u 1 = A B u 0 + B u 1 x 3 = A x 2 + B u 2 = A 2 B u 0 + A B u 1 + B u 2 x_ 0 = 0\\
x_1 = Ax_0 + Bu_0 = Bu_0\\
x_2 = Ax_1 + Bu_1 = ABu_0 + B u_1\\
x_3 = Ax_2 + Bu_2 = A^2Bu_0 + AB u_1 + B u_2\\ x 0 = 0 x 1 = A x 0 + B u 0 = B u 0 x 2 = A x 1 + B u 1 = A B u 0 + B u 1 x 3 = A x 2 + B u 2 = A 2 B u 0 + A B u 1 + B u 2 Matlab 求解，Co矩阵 “ctrb(A,B)”
r a n k [ S ] = n , d e t S ≠ 0 rank[S] = n, det \, S \neq 0 r ank [ S ] = n , d e t S = 0
可观性：∀ t ∈ [ t 0 , t f ] , 已知 y ( t ) , u ( t ) , 可求 x ( t 0 ) \forall t \in [t_0,t_f],已知y(t),u(t),可求x(t_0) ∀ t ∈ [ t 0 , t f ] , 已知 y ( t ) , u ( t ) , 可求 x ( t 0 ) r a n k [ C C A C A 2 . . . C A n − 1 ] = n rank
\begin{bmatrix}
C\\
CA\\
CA^2\\
...\\
CA^{n-1}
\end{bmatrix}
=n
r ank C C A C A 2 ... C A n − 1 = n
引理
f ( λ ) = ∑ i = 0 n a i λ i f(\lambda) = \sum_{i=0}^{n}a_i\lambda ^i f ( λ ) = ∑ i = 0 n a i λ i f ( A ) = 0 → A n = ∑ i = 0 n − 1 a i A i f(A) = 0 \rightarrow A^n = \sum_{i=0}^{n-1}a_iA^i f ( A ) = 0 → A n = ∑ i = 0 n − 1 a i A i
求解∣ λ I − A ∣ \left | \lambda I - A\right | ∣ λ I − A ∣ λ = A \lambda = A λ = A A n A^n A n
状态反馈与状态观测器
取u = v − k x u=v-kx u = v − k x
不改变可控性，有可能改变可观性
闭环特征值
状态观测器
平凡观测器
对于系统
x ˙ = A x + B u y = C x + D u \dot x = Ax+Bu\\
y = Cx + Du x ˙ = A x + B u y = C x + D u 观测器形式（模拟器）：x ^ ˙ = A x ^ + B u \dot {\hat x }=A\hat x +Bu x ^ ˙ = A x ^ + B u e = x − x ^ e=x-\hat x e = x − x ^ e ˙ = x ˙ − x ^ ˙ = A ( x − x ^ ) = A e \dot e = \dot x - \dot {\hat x }=A(x-\hat x)=Ae e ˙ = x ˙ − x ^ ˙ = A ( x − x ^ ) = A e A A A det ( A ) = 0 \det(A)=0 det ( A ) = 0
完全Luenberger观测器
观测器分为，模拟器，修正器部分，通过输出y的信息来修正观测器的收敛性。
x ˙ = A x + B u y = C x + D u \dot x = Ax+Bu\\
y = Cx + Du x ˙ = A x + B u y = C x + D u x ^ ˙ = A x ^ + B u + L ( y − y ^ ) \dot {\hat x} = A\hat x + Bu + L (y - \hat y) x ^ ˙ = A x ^ + B u + L ( y − y ^ ) y ^ ˙ = C x ˙ + D u \dot {\hat y} = C \dot x+ Du y ^ ˙ = C x ˙ + D u 将( 3 ) (3) ( 3 ) ( 1 ) (1) ( 1 )
x ^ ˙ = ( A − L C ) x ^ + ( B − L D ) u + L y \dot {\hat x} = (A-LC)\hat x + (B-LD)u + Ly x ^ ˙ = ( A − L C ) x ^ + ( B − L D ) u + L y 定义e = x − x ^ e = x - \hat x e = x − x ^ e ˙ \dot e e ˙ ( 1 ) (1) ( 1 ) ( 4 ) (4) ( 4 )
e ˙ = A x + B u − ( A − L C ) x ^ − ( B − L D ) u − L y \dot e = Ax+Bu-(A-LC)\hat x -(B-LD)u-Ly e ˙ = A x + B u − ( A − L C ) x ^ − ( B − L D ) u − L y 将( 1 ) (1) ( 1 ) ( 5 ) (5) ( 5 )
e ˙ = ( A − L C ) e \dot e = (A-LC)e e ˙ = ( A − L C ) e 测量噪声的影响
考虑测量噪声
y ( t ) = C x ( t ) + r ( t ) y(t) = Cx(t) + r(t) y ( t ) = C x ( t ) + r ( t )
系统的观测误差动态为
e ˙ ( t ) = ( A − L C ) e ( t ) − L r ( t ) \dot e (t) = (A-LC)e(t)-Lr(t) e ˙ ( t ) = ( A − L C ) e ( t ) − L r ( t ) L L L
分离原理
分别对系统
设计观测器
x ^ ˙ = A x ^ + B u + L ( y − y ^ ) y ^ = C x ^ \dot {\hat x} = A\hat x + Bu + L (y - \hat y)\\\hat y = C\hat x x ^ ˙ = A x ^ + B u + L ( y − y ^ ) y ^ = C x ^
设计控制率
u = − K x ^ + S w u = -K\hat x + Sw u = − K x ^ + S w
对分别设计的观测器和控制器构建扩展状态方程
[ x ˙ e ˙ ] ⏟ x ˙ e = [ A − B K B K 0 A − L C ] ⏟ A e [ x e ] ⏟ x e + [ B S 0 ] ⏟ B e w \underbrace{\left[\begin{array}{c}\dot{\mathbf{x}} \\ \dot{\mathbf{e}}\end{array}\right]}_{\dot{\mathbf{x}}_{e}}=\underbrace{\left[\begin{array}{cc}\mathbf{A}-\mathbf{B K} & \mathbf{B K} \\ \mathbf{0} & \mathbf{A}-\mathbf{L} \mathbf{C}\end{array}\right]}_{\mathbf{A}_{e}} \underbrace{\left[\begin{array}{c}\mathbf{x} \\ \mathbf{e}\end{array}\right]}_{\mathbf{x}_{e}}+\underbrace{\left[\begin{array}{c}\mathbf{B S} \\ \mathbf{0}\end{array}\right]}_{\mathbf{B}_{e}} \mathbf{w} x ˙ e [ x ˙ e ˙ ] = A e [ A − BK 0 BK A − LC ] x e [ x e ] + B e [ BS 0 ] w
合成的系统可以描述为X ˙ e = A e X + B e u \dot X_e = A_e X + B_e u X ˙ e = A e X + B e u A e A_e A e det ( λ L − A e ) = det ( λ L − A + B K ) det ( λ L − A + L C ) \det(\lambda L - A_e) = \det(\lambda L - A+BK) \det(\lambda L-A+LC) det ( λ L − A e ) = det ( λ L − A + B K ) det ( λ L − A + L C )
Kalman滤波器原理以及在matalb中的实现
状态转移矩阵：
这里要改一下，改成估计量
x t − = F t x t − 1 + B t u t x_t^- = F_t x_{t-1} + B_t u_t x t − = F t x t − 1 + B t u t
状态转移矩阵:P t − = F P t − 1 F T + Q P_t^-=FP_{t-1}F^T+Q P t − = F P t − 1 F T + Q
协方差矩阵:
[ σ 11 σ 12 σ 12 σ 22 ] \begin{bmatrix}
\sigma_{11}&\sigma_{12}\\
\sigma_{12}&\sigma_{22}\\
\end{bmatrix} [ σ 11 σ 12 σ 12 σ 22 ]
卡尔曼方程≠状态观测器
x ^ \hat x x ^
概率函数相乘，多传感器信息融合
非线性控制理论
ARC
Barbalat’s 引理 lemma
V ≥ 0 V\geq0 V ≥ 0 V ˙ ≤ − g ( t ) \dot{V} \leq -g(t) V ˙ ≤ − g ( t ) g ( t ) ≥ 0 g(t)\geq 0 g ( t ) ≥ 0 g ˙ ( t ) ∈ L ∞ \dot{g}(t)\in L_{\infty} g ˙ ( t ) ∈ L ∞ g ˙ ( t ) \dot{g}(t) g ˙ ( t ) g ( t ) g(t) g ( t ) lim t − > ∞ g ( t ) = 0 \lim_{t->\infty} g(t)=0 lim t − > ∞ g ( t ) = 0 lim t − > ∞ e = 0 ( k ≠ 0 ) \lim_{t->\infty} e = 0 (k\neq0) lim t − > ∞ e = 0 ( k = 0 )
实部
判断方法：
以小车为例，讲卡尔曼滤波最优状态估计
在上图中，P是观测值的方差
R是观测器中，来自预估值的比例