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ZigangPan`LinearSystems`

dynamicextension

[system]

applies dynamic extension to the LTI system to determine whether it may admit vector relative degree after some steps of dynamic extension from the active inputs to the active outputs. It requires that system admit at least one active input and at least one active output. If the system does not admit vector relative degree and may be extended to admit vector relative degree, then the output is a 5-term list

{vectorrelativedegree,relativedegrees,highfrequencygain,systemextended,systemaddnew}

vectorrelativedegree

= True. The extended system is

systemextended

, the added dynamics in the extension process is

systemaddnew

, the individual relative degrees of each output is the list

relativedegrees

, and

highfrequencygain

is the high frequency gain matrix of the extended system. If the system does admit vector relative degree but the relative degrees are not uniform, then the output is again a 5-term list

{vectorrelativedegree,relativedegrees,highfrequencygain,systemextended,systemaddnew}

The function then applies dynamic extension to the system to yield uniform vector relative degree.

vectorrelativedegree

=True. The extended system is

systemextended

, the added dynamics in the extension process is

systemaddnew

, the individual relative degrees of each output of the extended system is the list

relativedegrees

, and

highfrequencygain

is the high frequency gain matrix of the extended system. If the system admits uniform vector relative degree to begin with, then the output is a 4-term list

{vectorrelativedegree,relativedegrees,highfrequencygain,systemextended}

It basically reports that the system admits vector relative degree by setting

vectorrelativedegree

= True, the list of individual relative degrees of each out is listed in

relativedegrees

, the high frequency gain matrix is

highfrequencygain

, the system itself is returned in

systemextended

. If the system can not be extended to admit vector relative degree, then the output is set to 'error'

Examples

(1)

Basic Examples

(1)

In[1]:=

system={{x1,x2,x3},{u1,u2,w1,w2},{y1,y2,z1,z2},{{-1,2,0,0,0,0,1},{-2,-1,1,1,1,1,0},{-1,-2,-3,-1,1,1,1},{1,0,0,0,0,1,0},{0,1,0,0,0,0,1},{1,0,0,0,0,0,0},{0,1,0,0,0,0,0}},{1,2},{1,2},{1,2},{3,4},{1,2},{3,4}};

In[2]:=

dynamicextension

[system]

System can not be extended to admit a vector relative degree!

Out[2]=

error

In[3]:=

system1={{x1,x2,x3},{u1,u2,w1,w2},{y1,y2,z1,z2},{{-1,2,0,0,0,0,1},{-2,-1,1,1,1,1,0},{-1,-2,-3,-1,1,1,1},{1,0,0,0,0,1,0},{0,0,1,0,0,0,1},{1,0,0,0,0,0,0},{0,0,1,0,0,0,0}},{1,2},{1,2},{1,2},{3,4},{1,2},{3,4}}

Out[3]=

{{x1,x2,x3},{u1,u2,w1,w2},{y1,y2,z1,z2},{{-1,2,0,0,0,0,1},{-2,-1,1,1,1,1,0},{-1,-2,-3,-1,1,1,1},{1,0,0,0,0,1,0},{0,0,1,0,0,0,1},{1,0,0,0,0,0,0},{0,0,1,0,0,0,0}},{1,2},{1,2},{1,2},{3,4},{1,2},{3,4}}

In[4]:=

dynamicextension

[system1]

Out[4]=

{True,{2,2},{{0,4},{-1,-4}},{{x4,x1,x2,x3},{u3,u4,u1,u2,w1,w2},{y3,y4,y1,y2,z1,z2},{{0,0,0,0,1,0,0,0,0,0},{0,-1,2,0,0,0,0,0,0,1},{1,-2,-1,1,0,2,1,1,1,0},{-1,-1,-2,-3,0,0,-1,1,1,1},{1,0,0,0,0,1,0,0,0,0},{0,0,0,0,0,1,0,0,0,0},{0,1,0,0,0,0,0,0,1,0},{0,0,0,1,0,0,0,0,0,1},{0,1,0,0,0,0,0,0,0,0},{0,0,0,1,0,0,0,0,0,0}},{1,2},{3,4},{1,2,3,4},{5,6},{1,2,3,4},{5,6}},{{x4},{u3,u4},{y3,y4},{{0,1,0},{1,0,1},{0,0,1}},{1,2},{1,2},{1,2},{},{1,2},{}}}

SeeAlso

calculaterelativedegree

RelatedGuides

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Guide for ZigangPan`LinearSystems`