modern control theory

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Experiment 4 Solving Similar Transformation and Canonical Form of State-Space Model and Its Realization Using Matlab

Objectives:

1. Grasping the expression of diagonal and Jordan canonical form and the methods of solutions to transfer matrices corresponding to the above matrices of LTI system by experiment.

2. Grasping the controllability, observability criterions and controllable, observable decompositions of the system by programming and debugging.

3. Reinforcing the understanding about the methods of transfer function to controllable and observable canonical form.

Requirements:

Answering the following question through an example:

The state-space model of same transfer function is unique or not.

Solution:

We choose the transfer function is:

1.diagonal canonical form:

execute the m-file:

num=[1]

den=[1 8.5 20 12.5]

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

G=ss(A,B,C,D);

[V,D]=eig(A)

G1=ss2ss(G,D)

Obtain:

D =

   -5.0000         0         0

         0   -2.5000         0

         0         0   -1.0000

a =

          x1     x2     x3

   x1   -8.5    -40  -62.5

   x2    0.5      0      0

   x3      0    0.4      0

b =

       u1

   x1  -5

   x2   0

   x3   0

c =

       x1  x2  x3

   y1  -0   0  -1

d =

       u1

   y1   0

2. Jordan canonical form

num=[1]

den=[1 8.5 20 12.5]

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

G=ss(A,B,C,D);

[V,J]=jordan(A)

G1=ss2ss(G,V)

Obtain:

V =

    2.5000    0.1667   -1.6667

   -0.5000   -0.1667    0.6667

    0.1000    0.1667   -0.2667

a =

           x1      x2      x3

   x1    -104  -648.8  -854.8

   x2      21   131.8   174.8

   x3    -4.2  -26.89  -36.29

b =

         u1

   x1   2.5

   x2  -0.5

   x3   0.1

c =

       x1  x2  x3

   y1   1   6   5

d =

       u1

   y1   0

Now, we should determine its controllability, observability and conduct the controllable, observable decomposition

3. Determine the controllability and observability

num=[1]

den=[1 8.5 20 12.5]

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

G=ss(A,B,C,D);

Tc=ctrb(A,B)

n=size(A);

if rank(Tc)==n(1)

    disp('The system is controlled')

else

    disp('The system is not controlled')

end

Obtain:

Tc =

    1.0000   -8.5000   52.2500

         0    1.0000   -8.5000

         0         0    1.0000

The system is controlled

num=[1]

den=[1 8.5 20 12.5]

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

G=ss(A,B,C,D)

To=obsv(A,C)

n=size(A);

if rank(To)==n(1)

    disp('The system is observability')

else

    disp('The system is not observability')

end

Obtain:

To =

     0     0     1

     0     1     0

     1     0     0

The system is observability

4. Controllable decomposition

num=[1];

den=[1 8.5 20 12.5];

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

G=ss(A,B,C,D);

[Ac,Bc,Cc,Tc,Kc]=ctrbf(A,B,C)

G1=ss2ss(G,Tc)

Obtain:

Tc =

     0     0    -1

     0    -1     0

     1     0     0

a =

         x1    x2    x3

   x1     0     1     0

   x2     0     0    -1

   x3  12.5    20  -8.5

b =

       u1

   x1   0

   x2   0

   x3   1

 c =

       x1  x2  x3

   y1  -1  -0   0

 d =

       u1

   y1   0

5. Observable decomposition

num=[1];

den=[1 8.5 20 12.5];

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

G=ss(A,B,C,D);

[Ao,Bo,Co,To] = obsvf(A,B,C)

G1=ss2ss(G,To)

Obtain:

To =

    -1     0     0

     0     1     0

     0     0    -1

a =

          x1     x2     x3

   x1   -8.5     20  -12.5

   x2     -1      0      0

   x3      0     -1      0

b =

       u1

   x1  -1

   x2   0

   x3   0

 c =

       x1  x2  x3

   y1  -0   0  -1

d =

       u1

   y1   0

Conclusion:

From the experiments, we can use MATLAB more and more. I hope all of us can obtain many things.

第二篇：现代控制理论英文词汇表

term

state

state variablesstate vectorstate space

state-space equationsstate of a system

state-space representationstate matrixinput matrixoutput matrix

direct transmission matrixtransition matrixstate variable feedbacktime domain

time-varying systemtime-invariant systemdimensions of the matrixeigenvalue of the matrixeigenvectors of the matrixdeterminant of the matrixdiagonal matrix

linear differential equationsnth-order systemsnth-order differential equationstate equationourput equationalgebraic equationtransfer functionnumerator

denominator

numerator polynomialdenominator polynomialnumerator polynomial in sdenominator polynomial in sinverted pendulumattitude controlspace boostercanonical form

diagonal canonical formJordan canonical formJordan block

controllable canonical formobservable canonical formmatrix exponential functionstate-transition matrixhomogeneous state equationsnonhomogeneous state equationscontrollability

completely state controllablecomplete state controllability

controllability matrixtranslation状态状态变量状态向量/状态矢量状态空间状态空间方程系统状态状态空间表示/状态空间表达式状态矩阵输入矩阵（控制矩阵）输出矩阵直接传递矩阵转移矩阵状态变量反馈时域时变系统时不变系统/非时变系统矩阵的维数矩阵的特征值矩阵的特征向量矩阵的行列式值对角线矩阵线性微分方程n阶系统n阶微分方程状态方程输出方程代数方程传递函数分子分母分子多项式分母多项式关于变量s的分子多项式关于变量s的分母多项式倒立摆姿态控制太空推进器标准型/规范型对角标准型/对角线标准型约当标准型约当块能控标准型能观标准型矩阵指数函数状态转移矩阵齐次状态方程非齐次状态方程能控性状态完全能控状态完全能控性能控性矩阵

controllable system

output controllability

stabilizability

observability

observable system

observability matrix

detectable

detectability

pole assignment

state variable feedbackobserver

full-state feedback control lawfull state observer

matrix

adjoint matrix

companion matrix

inverse matrix

conjugate matrix

single-input-single-output systemmulti-input-multi-output systemderivate

spring

dashpot

mass能控系统输出能控性可稳定性能观性能观系统能观性矩阵能检测能检测性极点配置状态变量反馈观测器全状态反馈控制律全维状态观测器矩阵伴随矩阵友矩阵逆矩阵共轭矩阵单输入单输出系统多输入多输出系统导数弹簧阻尼器质量块