现代控制理论 2-3 线性定常离散系统的分析(2)

β0
z n + an−1z n−1 + L + a1z + a0
= bn
+
N D
z z
⎧ β0 = b0 − a0bn
⎪ ⎪⎪ ⎨
β1 = b1 − a1bn M
⎪⎪βn−2 = bn−2 − an−2bn
⎪⎩ βn−1 = bn−1 − an−1bn
若 bn = 0
Y (z ) U (z )
=
N(z) D(z)
⎧x(k +1) = Gx(k )+ hu(k )
⎨ ⎩
y(k ) = cx(k )+ du(k )
⎪ ⎪
xn−1
(k
+ 1)
=
xn
(k
)
⎪⎩xn (k +1) = −a0 x1(k )− a1x2 (k )−L− an−1xn (k )+ β0u(k )
y(k ) = x1(k )
返回
⎡0
⎢ ⎢
0
G= 0.1998 0.2325
A = [0 1; -2 -3]; B = [0;1]; T = 1; [Phi, G] = c2d(A,B,T)
二二、、线线性性定定常常离离散散系系统统状状态态方方程程的的解解
迭代法(递推法) x(k +1) = Φ(T )x(k )+ G(T )u(k ) k = 0 : x(1) = Φ(T )x(0)+ G(T )u(0) k = 1: x(2) = Φ(T )x(1)+ G(T )u(1)
− +
0.8 z −2 0.1z −2
可控标准型
=
2
+
1
+
2.2z −1 0.7z −1
− +
z−2 0.1z
−2
=
2+
z2
2.2z −1 + 0.7z + 0.1
返回
⎡ x1(k
⎢ ⎣
x2
(k
+1)⎤ + 1)⎥⎦
=
⎡0 ⎢⎣− 0.1
−
1⎤ 0.7⎥⎦
⎡ ⎢ ⎣
x1(k x2 (k
)⎤ )⎥⎦
+
=
⎡2
⎢ ⎢ ⎢
s +1 −2
− +
s
1
+ 2
2
⎣s +1 s + 2
1 s +1 −1
− +
s
1
+ 2
2
⎤ ⎥ ⎥ ⎥
s +1 s + 2⎦
[ ] ( ) ( ) Φ t
= eAt = L-1
sΙ − A -1
=
⎡ 2e−t ⎢⎣− 2e−t
− +
e−2t 2e−2t
e−t − e−2t ⎤
−
e−t
βi = bi
U(z)
β z n−1 n−1
+
β zn−2 n−2
+L+
β1z
+
β0
z n + an−1z n−1 + L + a1z + a0
Y(z)
串联分解: 1
D(z)
N(z)
U(z)
1 zn +an−1zn−1 +L+a1z +a0
Q(z)
β zn−1 n−1
+βn−2zn−2
+L+β1z
+β0
友友矩矩阵阵 状态方程：
⎡0 1 0 L 0 ⎤
⎢ ⎢
0
0
1L
0
⎥ ⎥
⎡0⎤ ⎢⎢0⎥⎥
x1(k +1) = x2 (k ) x2 (k +1) = x3(k )
G = ⎢ M M M O M ⎥ h = ⎢M⎥
⎢ ⎢
0
0
0L
1
⎥ ⎥
⎢⎢0⎥⎥
⎢⎣− a0 − a1 − a2 L − an−1⎥⎦
⎢⎣1⎥⎦
G(z)
=
Y (z) U (z)
=
bn z n + bn−1z n−1 + L + b1z + b0 z n + an−1z n−1 + L + a1z + a0
( ) 综合 ( ) 除法:
=
bn
+
β z n−1 n−1 zn +
+
β zn−2 n−2
+L+
β1z
+
an−1z n−1 + L + a1z + a0
+
2e−2t
⎥ ⎦
( ) ( ) Φ T
= eA(T )
=Φ
t
t =T
=
⎡ 2e−T ⎢⎣− 2eT
− e−2T + 2e−2T
e−T − e−2T ⎤
− e−T
+
2e−2T
⎥ ⎦
⎡ ⎢ ⎣
x&1 x&2
⎤ ⎥ ⎦
=
⎡0 ⎢⎣− 2
1⎤ − 3⎥⎦
⎡ ⎢ ⎣
x1 x2
⎤ ⎥ ⎦
+
⎡0⎤ ⎢⎣1⎥⎦u
G=⎢ M
⎢ ⎢
0
⎢⎣− a0
1 0L 0 1L MM 0 0L − a1 − a2 L
系统矩阵
0⎤
0
⎥ ⎥
M ⎥,
1
⎥ ⎥
− an−1 ⎥⎦
h
=
⎡0 ⎤
⎢⎢0 ⎢M
⎥ ⎥⎥,
⎢⎣β
0
⎥ ⎦
c
= [1
0
L
0]
控制矩阵 输出矩阵
(2) 控制函数含u(k), u(k+1), … u(k+n)
y(k + n)+ an−1y(k + n −1)+L+ a1 y(k +1)+ a0 y(k ) = bnu(k + n)+ bn−1u(k + n −1)+L+ b1u(k +1)+ b0u(k )
( ) Φ t
=
⎡ 2e−t ⎢⎣− 2e−t
− e−2t + 2e−2t
e−t − e−2t ⎤
− e−t
+
2e−2t
⎥ ⎦
G
(T
)
=
T
∫0
Φ(t
)bdt
∫=
T ⎡ 2e−t − e−2t 0 ⎢⎣− 2e−t + 2e−2t
e−t − e−t
− e−2t ⎤⎡0⎤
+
2e−
2t
⎥ ⎦
⎢⎣1⎥⎦
dt
=
⎢⎡− ⎢⎣
e−T + e−T
1 e−2T 2 − e−2T
+
1⎤
2
⎥ ⎥⎦
∴
⎡ x1(k ⎢⎣x2 (k
+ 1)T + 1)T
⎤ ⎥⎦
=
⎡ 2e−T ⎢⎣− 2eT
− e−2T + 2e−2T
−
e−T e−T
− +
e −2T 2e−2T
⎤ ⎥ ⎦
⎡ x1(kT ⎢⎣x2 (kT
)⎤ )⎥⎦
+
⎢⎡− ⎢⎣
Y(z)
Q(z) U (z)
=
zn
+
an−1z n−1
1 +L+
a1z
+ a0
Y (z ) Q(z)
=
β
n−1
z
n
−1
+
L
+
β1z
+
β
0
⎧ ⎪
选取状态变量：⎪⎪⎨
x1(k ) = q(k ) x2(k ) = q(k +1) x3(k ) = q(k + 2)
⎪ ⎪ ⎪⎩
xn
(k
)
=
M
q(k
+
n
−
1)
第二章 线性系统的状态空间分析法 §1 线性系统的状态空间描述 §2 线性定常连续系统的分析 §3 线性定常离散系统的分析 §4 系统的传递函数矩阵
系统中各变量只在离散时刻取值，状态空间描述只 反映离散时刻的变量组之间的因果关系和转换关系。
一、线性定常离散系统的状态空间表达式
n×n
系统矩阵
n× p
控制矩阵
2.2]⎢⎡
⎣
x1 x2
(k (k
)⎤ )⎥⎦
+
2u(k
)
2.2
u(k )
1
x2 (k ) = x1(k +1) 1
x1(k )
−1
y(k )
z
z
− 0.7
− 0.1
2
3, 线性定常系统的离散化
x&(t) = Ax(t)+ Bu(t) y(t) = Cx(t)+ Du(t)
离散化:
x(k +1) = Φ(T )x(k )+ G(T )u(k ) y(k ) = Cx(k )+ Du(k )
⎧x(k +1) = Gx(k )+ Hu(k )
⎨ ⎩
y(k ) = Cx(k )+ Du(k )
单位延迟器
u(k )
H
D
x(k +1)
x(k )
z-1
C
G
y(k )
前页
1, 根据系统差分方程建立离散状态空间表达式
(1) 控制函数仅含 u(k)
y(k + n)+ an−1y(k + n −1)+L+ a1y(k +1)+ a0 y(k ) = β0u(k ) 选取 x1(k ) = y(k ), x2(k ) = y(k +1), L, xn (k ) = y(k + n −1)
前页
MMAATTLLAABB相相关关函函数数
Φ(T
)
=
⎡ 2e−T ⎢⎣− 2eT
− e−2T + 2e−2T
e−T − e−2T ⎤
− e−T
+ 2e−2T
⎥ ⎦
G(T ) =
⎢⎡− e−T + ⎢⎣ e−T
1 e−2T 2 − e−2T
+
1⎤
2
⎥ ⎥⎦
Phi = [ 2*exp(-T)-exp(-2*T), exp(-T)-exp(-2*T)] [ 2*exp(-2*T)-2*exp(-T), -exp(-T)+2*exp(-2*T)]
M
xn (k +1) = −a0x1(k )− a1x2 (k )−L− an−1xn (k )+ u(k )
输出方程： c = [β0 β1 L ] βn−1 可控标准型 y(k ) = β0x1(k )+ β1x2 (k )+L+ βn−1xn (k )
若 bn ≠ 0
G(z
)
=
Y U
(z) (z)
e−T + e−T
1
2 −
e−2T e−2T
+
1⎤
2
⎥u(kT
⎥⎦
)
MMAATTLLAABB相相关关函函数数
⎡ x&1 ⎢⎣ x&2
⎤ ⎥⎦
=
⎡0 ⎢⎣− 2
1⎤ − 3⎥⎦
⎡ ⎢⎣
x1 x2
⎤ ⎥⎦
+
⎡0⎤ ⎢⎣1⎥⎦u
求离散系统矩 阵和输入矩阵
表达式
A = [0 1; -2 -3]; B = [0;1]; syms T; [Phi, G] = c2d(A,B,T)
= Φ2(T )x(0)+ Φ(T )G(T )u(0) + G(T )u(1)
k −1
k = k −1: x(k ) = Φk (T )x(0)+ ∑Φk−1−i (T )G(T )u(i) i=0 k −1 y(k ) = CΦk (T )x(0)+ C∑ Φk−1−i (T )G(T )u(i)+ Du(k ) i=0
c = [1 0 L 0 0] d = h0
比较
2, 根据脉冲传递函数建立离散状态空间表达式
y(k + n)+ an−1 y(k + n −1)+L+ a1y(k +1)+ a0 y(k ) = bnu(k + n)+ ( bn−1u k + n −1)+L+ b1u(k +1)+ b0u(k )
零状态下 取z变换
β0
=
bn
+
N D
z z
零状态下 取z变换
G(s)
=
Y (z ) U (z)
=
bn z n + bn−1z n−1 + L + b1z + b0 z n + an−1z n−1 + L + a1z + a0
( ) 综合 ( ) 除法:
= bn
+
β z n−1 n−1
+
βn−2
z n−2
+L
+
β1z
+
=
bn
+
N (z ) D(z)
则 y = cx + bnu
返回
例：已知系统的差分方程,求状态空间表达式。
y(k )+ 0.7 y(k −1)+ 0.1y(k − 2) = 2u(k )+ 3.6u(k −1)− 0.8u(k − 2)
解:
Y (z) U (z)
=
2 + 3.6z−1 1+ 0.7z−1
⎪⎪hn = bn − anb0 − an−1h1
⎪⎩
−L − a1hn−1
1
⎡0 1 0 L 0 ⎤
⎢ ⎢
0
0
1L
0
⎥ ⎥
G=⎢ M M M
M ⎥,
⎢ ⎢
0
0
0L
1
⎥ ⎥
⎢⎣− a0 − a1 − a2 L − an−1⎥⎦
⎡ h1 ⎤
⎢ ⎢
h2
⎥ ⎥
h=⎢ M ⎥
⎢⎢hn−1
⎥ ⎥
⎢⎣ hn ⎥⎦
n×n
n ×1
⎧x(k +1) = Gx(k )+ Hu(k ) ⎧x(k +1) = Gx(k )+ hu(k )
⎨ ⎩
y(k ) = Cx(k )+ Du(k )
⎨ ⎩
y(k ) = cx(k)+ du(k )
q×n
输出矩阵
q× p
1× n
标量
前馈矩阵
k 表示 kT 时刻，T 为采样周期。
返回
向量动态方程的系统结构图
G= [ -exp(-T)+1/2*exp(-2*T)+1/2] [ exp(-T)-exp(-2*T)]
3
MMAATTLLAABB相相关关函函数数
求离散系统矩阵 和输入矩阵值
⎡ x&1
⎢ ⎣
x&2
⎤ ⎥ ⎦
=
⎡0 ⎢⎣− 2
1⎤ − 3⎥⎦
⎡ ⎢ ⎣
x1 x2
⎤ ⎥ ⎦
+
⎡0⎤ ⎢⎣1⎥⎦u
Phi = 0.6004 0.2325 -0.4651 -0.0972
⎧x1(k +1) = x2 (k ) ⎪⎪⎪⎨Mx2 (k +1) = x3(k )
⎪ ⎪
ห้องสมุดไป่ตู้
xn−1
(k
+ 1)
=
xn
(k
)
⎪⎩xn (k +1) = −a0 x1(k )− a1x2 (k )−L− an−1xn (k )+ β0u(k )
y(k ) = x1(k )
⎧x1(k +1) = x2 (k ) ⎪⎪⎪⎨Mx2 (k +1) = x3(k )
选取状态变量：
⎧x1(k ) = y(k )− h0u(k )
⎧h1 = b1 − a1b0
⎪ ⎪⎪ ⎨
x2 x3
(k (k
) )
= =
x1 x2
(k (k
+ 1) − + 1) −
h1u(k ) h2u(k )
⇒ ⎪⎪⎪⎨Mh2 = (b2 − a2b0 )− a1h1
⎪⎪L
⎪⎩xn (k ) = ( xn−1 k +1)− ( hn−1u k )
Φ(T ) = eAT = Φ(t )t=T
G
(T
)
=
∫T 0
Φ(τ
)Bdτ
比较
例：已知系统的状态方程,试将其离散化。
解:
⎡ ⎢⎣
x&1 x&2
⎤ ⎥⎦
=
⎡0 ⎢⎣− 2
1⎤ − 3⎥⎦
⎡ ⎢⎣
x1 x2
⎤ ⎥⎦
+
⎡0⎤ ⎢⎣1⎥⎦u
(sΙ
−
)A -1
=
(s
1
+1)(s
+
2)
⎡s ⎢ ⎣
+ 2
3
1⎤ s⎥⎦
⎡0⎤ ⎢⎣1⎥⎦u
(k
)
y(k ) = [−1
2.2]⎢⎡
⎣
x1 x2
(k (k
)⎤ )⎥⎦
+
2u(k
)
前页
2
⎡ ⎢ ⎣
x1(k x2 (k
+ 1)⎤ + 1)⎥⎦
=
⎡0 ⎢⎣− 0.1
−
1⎤ 0.7⎥⎦
⎡ ⎢ ⎣
x1 x2
(k (k
)⎤ )⎥⎦
+
⎡0⎤ ⎢⎣1⎥⎦u
(k
)
前页
标量系统结构图：
y(k ) = [−1