ZigangPan/LinearSystems | Paclet Repository

ZigangPan`LinearSystems`

EZDCFAD

{r,highfrequencygain,systemaddI,systemaddO,transformation,zerodynamics}=EZDCFAD[system,θ]

calculates the extended zero dynamics canonical form of a LTI system with respect to active inputs and active outputs whenever it exists. It requires that system admits at least one active input and at least one active output, and the number of active inputs equals to the number of active outputs.If the system can not be dynamically extended to admit vector relative degree, or if it can only be dynamically extended to admit vector relative degree by a control law that depends on the parameter vector θ, then the function returns 'error'.If the system may be dynamically extended to admit vector relative degree by a control law that is independent of θ, then it is further extended as much as possible with known control laws to an LTI system with vector relative degree such that the relative degrees are close to each other. Furthermore, the outputs were integrated to lead to an LTI system with uniform vector relative degree. From that extended system, one further calculates the extended zero-dynamics canonical form.

= the uniform relative degree (scalar);

highfrequencygain

= high frequency gain matrix of the extended system;

systemaddI

= the LTI system that defines the dynamic feedback law for the active inputs, it is set to 'default' if it has transfer function IdentityMatrix[ma];

systemaddO

= the LTI system that defines the dynamic output law for the active outputs, it is set to 'default' if it has transfer function IdentityMatrix[ma];

transformation

= the state transformation xold = transformation . xnew that brings the extended system into extended zero-dynamics canonical form;

zerodynamics

= the extended zero-dynamics canonical form of the extended system.

Examples

(1)

Basic Examples

(1)

In[1]:=

system={{x1,x2,x3},{u1,u2,w1,w2},{y1,y2,z1,z2},{{-1,θ1,0,0,0,0,1},{-2,-1,1,1,1,1,0},{-1,-θ2,-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[2]:=

θ={θ1,θ2};

In[3]:=

EZDCFAD

[system,θ]

State transformation is xold = transformation.xnew

Out[3]=

{2,{{0,2θ1},{-1,-2θ2}},{{x4},{u3,u4},{y3,y4},{{0,1,0},{1,0,1},{0,0,1}},{1,2},{1,2},{1,2},{},{1,2},{}},default,{{-1,0,-1,-θ2},{1,0,0,0},{0,-1,0,1},{0,1,0,0}},{{x11,x21,x31,x41},{u3,u4,u1,u2,w1,w2},{y3,y4,y1,y2,z1,z2},{{-1,-θ1,0,θ1,0,0,0,0,0,1},{0,-3+θ2,1,0,0,0,-1,1,1,1},{1+3θ2,θ1+θ2-

θ2

,0,-θ1+θ2+

θ2

,-1,-2θ2,0,-2θ2,-2θ2,-1-θ2},{-3,-1+θ2,0,-1-θ2,0,2,0,2,2,1},{-1,0,-1,-θ2,0,1,0,0,0,0},{0,0,0,0,0,1,0,0,0,0},{1,0,0,0,0,0,0,0,1,0},{0,1,0,0,0,0,0,0,0,1},{1,0,0,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0,0,0}},{1,2},{3,4},{1,2,3,4},{5,6},{1,2,3,4},{5,6}}}

SeeAlso

ZDCF

▪

EZDCF

▪

rd0DMcompute

▪

uniformobservabilityindices

RelatedGuides

▪

Guide for ZigangPan`LinearSystems`