利用MATLAB设计状态观测器

利用MATLAB 设计状态观测器

本节将介绍用MATLAB 设计状态观测器的若干例子。我们将举例说明全维状态观测器和最小阶状态观测器设计的MATLAB 方法。

------------------------------------------------

[例1] 考虑一个调节器系统的设计。给定线性定常系统为

Cx

y Bu Ax x =+=& 式中

]01[,10,06.2010=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=C B A

且闭环极点为)2,1(==i s i μ，其中

4.28.1,4.28.121j j −−=+−=μμ

期望用观测-状态反馈控制，而不是用真实的状态反馈控制。观测器的期望特征值为

821−==μμ

试采用MATLAB 确定出相应的状态反馈增益矩阵K 和观测器增益矩阵e K 。

[解]对于题中给定的系统，可利用如下MATLAB Program 1来确定状态反馈增益矩阵K和观测器增益K。

矩阵

e

MATLAB Program 1

% Pole placement and design of observer ------

% ***** Design of a control system using pole-placement

% technique and state observer. Solve pole-placement

% problem *****

% ***** Enter matrices A,B,C,and D *****

A=[0 1;20.6 0];

B=[0;1]

C=[1 0];

D=[0];

% ***** Check the rank of the controllability matrix Q *****

Q=[B A*B];

Rank(Q)

ans=

2

% ***** Since the rank of the controllability matrix Q is 2, % arbitrary pole placement is possible *****

% ***** Enter the desired characteristic polynomial by

% defining the following matrix J and computingpoly(J) *****

J=[-1.8+2.4*i 0;0 -1.8-2.4*i];

Poly(J)

ans=

1.000 3.6000 9.0000

% ***** Enter characteristic polynomial Phi *****

Phi=polyvalm(poly(J),A);

% ***** State feedback gain matrix K can be given by ***** K=[0 1]*inv(Q)*Phi

K=

29.6000 3.6000

% ***** The following program determines the observer matrix Ke *****

% ***** Enter the observability matrix RT and check its rank *****

RT=[C’ A’*C’];

rank(RT)

ans=

2

% ***** Since the rank of the observability matrix is 2, design of

% the observer is possible *****

% **** Enter the desired characteristic polynomial by defining % the following matrix J0 and entering statement poly(JO) *****

JO=[-8 0;0 -8];

Poly(JO)

ans=

1 16 64

% ***** Enter characteristic polynomial Ph ***** Ph=polyvalm(ply(JO),A);

% ***** The observer gain matrix Ke is obtained from ***** Ke=Ph*(inv(RT’))*[0;1]

Ke=

16.0000

84.60000

求出的状态反馈增益矩阵K 为

[]6.36.29=K

观测器增益矩阵e K 为

⎥⎦

⎤⎢⎣⎡=6.8416e K 该观测-状态反馈控制系统是4阶的，其特征方程为

0=+−+−C K A sI BK

A sI e

通过将期望的闭环极点和期望的观测器极点代入上式，可得

576

4.3746.1306.19)8)(4.28.1)(4.28.1(2342++++=+++−+=

+−+−s s s s s j s j s C K A sI BK A sI e

这个结果很容易通过MATLAB得到，如MATLAB Program 2所示(MATLAB Program 2是

K已MATLAB Program1的继续。矩阵A、B、C、K和

e 在MATLAB Program 1中给定)。

MATLAB Program 2

% ------- Characteristic polynomial -------

% ***** The characteristic polynomial for the designed system

% is given by |sI-A+BK||sI-A+KeC| *****

% ***** This characteristic polynomial can be obtained by use of

% eigenvalues of A-BK and A-KeC as follows ***** X=[eig(A-B*K);eig(A-Ke*C)]

X=

-1.8000+2.4000i

-1.8000-2.4000i

-8.0000