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第一章. 线性系统的状态空间描述. 目 录. 第一节 状态空间分析法 第二节 由系统框图求状态空间描述 第三节 由系统机理建立状态空间描述 第四节 由外部描述化为内部描述及其标准型式 第五节 由状态空间方程求传递函数 第六节 离散时间系统的状态空间描述 第七节 状态矢量的线性变换 第八节 MATLAB 应用. 系统描述中常用的基本概念 系统的外部描述 传递函数 (零初始条件) 系统的内部描述 状态空间表达式. 一、举例 例 1 求图示机械系统的状态空间表达式

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第一章

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MATLAB



1

-----


:


2 RLC

i(t)





  • 1.:

  • 2.

  • t0


3.

  • n

    n



1. MIMO rm


n

r

m


/


r=3m=2


2. SISO r=1m=1


3.

:

:


3


4


5


[1-2]

[1-3]

[1-4]


6


,



1)2)3)

7 RLC


1


2


8 Fm1m2y1y2




x3

x3

1.

2.

3.

abuiyxi


x3

x3

a0

b x1 = x2x1x3(x2x3)x2x3


SISO3-8

nSISO


bn

: mn

m<nd=0

m=nd=bn


Abcm<nbnu


I

II


1. I


  • y/b0y/b0.

  • A10


1

ABCD



2.

2

yy


n


IAbIIAc


u


  • z

1.

(1)


:


-

Abc

  • W(s):


3


.

2I



u0 u


-


3 ,


2.

(1)

Ac=AoTCc=boT


(2)

Ao=AcTCo=bcT

3. Jordan

Jordan


ci---

Jordan

(1)

D(s)


a. :

xiy


1


b. :


(2)

W(s)r1



:


Jordan



3



--

  • 1-1

  • 1-3

  • 1-4 (2)



1

T:


2.

SISO


Q(z)


-

(k+1)TkT kT kT

-


1-10

I


L

X(0)=X0=0

A

SI-A


[]

1dim(W(s))=mrdim()mr

2W(s)

ijj

3W(s)


1-4


1-5 W(s)

[]



(SI-A)-1

iR i (i=n-1,n-2,,0)


tr( ) :


1-6


T


T


[1-12]

1T

2T


AA

A

A

A

A

An

A .

A


I:

II:

1n A n

2An

3


4 A A P

5A


i

1 A

2


A ,

1-13A

[]

1)

A

2


.


TABC

1. A

1

A

T


1

A T

1

4)A

T


2

T-1


.

:

1):

2)T



3


(2)


T

m


T



()

T

m n-m

T


cTABC

aA

b T

.


:

1) .

2) T.


3


1-15


2. AI

(1) A

Vandemonde


.

1.


  • T


(2) A

1) m1P1

p2p3...pm


2) 51p1p2


(3) A

PR1PI1


MATLAB

1.

  • num=[0 bn-1 bn-2 b1 b0 ];

    • den=[1 an-1 an-2 a1 a0 ];

  • [ A,B,C,D ] = tf2ss ( num , den )


num=[0 0 10 10];

den=[1 6 5 10];

[A,B,C,D]=tf2ss(num,den)


[numden]=ss2tf(ABCDiu)

A=[0 1 0;0 0 1;-5.008 25.1026 5.03247];

B=[0;25.04;-121.005];

C=[1 0 0];

D=[0];

[num,den]=ss2tf(A,B,C,D)


[RPK] = residue (numden)

If there are no multiple roots,

num(s) R(1) R(2) R(n)

---------- = -------- + -------- + ... + -------- + K(s)

den(s) s - P(1) s - P(2) s - P(n)

If P(j) = ... = P(j+m-1) is a pole of multplicity m, then the expansion includes terms of the form

R(j) R(j+1) R(j+m-1)

-------- + ------------ + ... + ------------

s - P(j) (s - P(j))^2 (s - P(j))^m

[numden] = residue (RPK)


2.

A=[ 7 1 ; -4 3 ];

inva=inv(A)

syms s

A=[ 7 1 ; -4 3 ];

invas = inv( eye(2) * s-A );


D

3. [ V,D ] = eig(A) AV=VD

V=[p1p2.pn]

[ V,J ]=jordan(A) V \ A*V = J

VDJ

[Vr,Dr]=cdf2rdf(Vc,Dc)


4.

sys1 = ss(A1,B1,C1,D1);

sys2 = ss(A2,B2,C2,D2);

syss = series(sys1,sys2)

sysp = parallel(sys1,sys2)

sysf = feedback(sys1,sys2)


--

  • 1-5 (2)

  • 1-6 (4)

  • 1-7 (1) (3)


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