ZigangPan/MRRACLinearFullOrder

ZigangPan`MRRACLinearFullOrder`

costtocomefunctionLTIMIMO1

{estimatorsystem,estimatorIS,qbar,Π1,af1,qchk1}=costtocomefunctionLTIMIMO1[m,r,a1,achk,b,bchkb,dchk,d1,a2111,a2112,a2113,a212,e1,θ,pθ,γ,referencemodel,θchk0,xchk0,q0,Φ0,ρo,Δbar1i,βΔ,kccoeff]

: is output dimension.

: is the uniform vector relative degree of the design model.

: is the

nn

-dimensional system matrix.

achk

: is the

nmm

-dimensional matrix

: is the

nmm

-dimensional matrix.

bchkb

: is the

nm(p-m)

-dimensional matrix.

dchk

: is the

nmqchk

-dimensional matrix.

: is the

nqb

-dimensional matrix.

a2111

: is the

nmσm

-dimensional tensor.

a2112

: is the

nmσ(p-m)

-dimensional tensor.

a2113

: is the

nmσqchk

-dimensional tensor.

a212

: is the

nmσm

-dimensional tensor.

: is the

1qb

-dimensional matrix.

: is the vector of unknown parameters.

pθ

: is a pure function such that

Apply

[

pθ

,θ]≤1

for true values

: is the desired disturbance attenuation level.

referencemodel

: is a NLsystem model describing the reference model of the system.

θchk0

: is the initial estimate for

xchk0

: is the initial estimate for the system states (in the design model).

: is the positive definite weighting matrix for the initial parameter estimation error:

θ-θchk0

Φ0

: is the initial value for the matrix

ρo

: is a real value that is greater than 1.

Δbar1i

: is the initial value for

Δbar1

. (The program will yield all rational number coefficients in the controller, there is no approximation in the controller.)

βΔ

: is a nonnegative real value.

kccoeff

: is a real value that is greater than or equal to 1, and sets

Kc=kccoeff*γ^2*

[q0]

. When kccoeff=1, then Σ and sΣ are constant matrix and constant scalar, respectively. Returned variables:

estimatorsystem

: is a NLsystem model for the estimator dynamics.

estimatorIS

: is the initial state for the estimator dynamics.

qbar

: is the expression for the matrix

qbar

Π1

: is the numeric positive definite matrix

Π1

af1

: is the numeric Hurwitz matrix

af1

qchk1

: is the dimension of the measured disturbances with relative degree ≥

. If the inputs are of rational coefficients, then the returned variables are of rational coefficients.

Examples

(1)

Basic Examples

(1)

In[1]:=

Needs["ZigangPan`Examples`"]

assumePositive

::shdw

:Symbol assumePositive appears in multiple contexts {ZigangPan`Examples`,Global`}; definitions in context ZigangPan`Examples` may shadow or be shadowed by other definitions.

assumeReal

::shdw

:Symbol assumeReal appears in multiple contexts {ZigangPan`Examples`,Global`}; definitions in context ZigangPan`Examples` may shadow or be shadowed by other definitions.

In[2]:=

θbar={θ1};

In[3]:=

mimosystem={{x1,x2,x3},{u1,u2,w1,w2},{y1,y2,z1,z2},{{0,0,1,1,θ1,0,0},{0,0,0,-θ1,-θ1^2,0,0},{0,0,0,1,0,0,0},{1,0,0,0,0,1,0},{θ1,1,0,0,0,0,1},{1,0,0,0,0,0,0},{θ1,1,0,0,0,0,0}},{1,2},{1,2},{1,2},{3,4},{1,2},{3,4}};systemcheck[mimosystem]

Out[3]=

True

In[4]:=

$Assumptions=$Assumptions&&assumePositive[θbar]&&assumeReal[Join[mimosystem〚1〛,mimosystem〚2〛,mimosystem〚3〛]];

In[5]:=

{r,highfrequencygain,systemaddI,systemaddO,transformation,zerodynamics}=EZDCFAD[mimosystem,θbar]

System already has uniform vector relative degree!

State transformation is xold = transformation.xnew

Out[5]=

2,{{1,θ1},{θ1,0}},default,{{x4},{u3,u4},{y3,y4},{{0,1,0},{1,0,0},{0,0,1}},{1,2},{1,2},{1,2},{},{1,2},{}},0,1,

θ1

,0,{0,-θ1,0,0},0,0,0,

θ1

,{1,0,0,0},{x11,x21,x31,x41},{u1,u2,w1,w2},{y1,y2,z1,z2,y3,y4},0,1,

θ1

,0,0,0,1,0,{0,0,0,0,1,θ1,0,0},{0,0,0,1,0,0,0,0},{0,0,0,0,θ1,0,0,0},0,1,

θ1

,0,0,0,1,0,{0,0,1,0,0,0,0,1},0,1,

θ1

,0,0,0,0,0,{0,0,1,0,0,0,0,0},{1,0,0,0,0,0,0,0},{0,0,1,0,0,0,0,1},{1,2},{5,6},{1,2},{3,4},{1,2,5,6},{3,4}

In[6]:=

systemoperation[zerodynamics,{"View"}];

Active							a1	a2
	Type		s1	s2	s3	s4	c1	c2	d1	d2
		names	x11	x21	x31	x41	u1	u2	w1	w2
	s1	x11	0	1	1 θ1	0	0	0	1	0
	s2	x21	0	0	0	0	1	θ1	0	0
	s3	x31	0	0	0	1	0	0	0	0
	s4	x41	0	0	0	0	θ1	0	0	0
	m1	y1	0	1	1 θ1	0	0	0	1	0
	m2	y2	0	0	1	0	0	0	0	1
	c1	z1	0	1	1 θ1	0	0	0	0	0
	c2	z2	0	0	1	0	0	0	0	0
a1	m3	y3	1	0	0	0	0	0	0	0
a2	m4	y4	0	0	1	0	0	0	0	1

In[7]:=

{systemobservable,nO,observabilityindices,transformation1,systemextended}=uniformobservabilityindices[zerodynamics];

State transformation is xold = transformation.xnew

System already has uniform observability indices!

In[8]:=

systemextended=FullSimplify[systemextended];

In[9]:=

systemoperation[systemextended,{"View"}];

Active							a1	a2
	Type		s1	s2	s3	s4	c1	c2	d1	d2
		names	x1	x2	x3	x4	u1	u2	w1	w2
	s1	x1	0	1	1 θ1	0	0	0	1	0
	s2	x2	0	0	0	0	1	θ1	0	0
	s3	x3	0	0	0	1	0	0	0	0
	s4	x4	0	0	0	0	θ1	0	0	0
	m1	y1	0	1	1 θ1	0	0	0	1	0
	m2	y2	0	0	1	0	0	0	0	1
	c1	z1	0	1	1 θ1	0	0	0	0	0
	c2	z2	0	0	1	0	0	0	0	0
a1	m3	y3	1	0	0	0	0	0	0	0
a2	m4	y4	0	0	1	0	0	0	0	1

In[10]:=

designmodel=systemoperation[systemoperation[systemextended,{"Drop outputs",{y1,y2}}],{"View"}];

Active							a1	a2
	Type		s1	s2	s3	s4	c1	c2	d1	d2
		names	x1	x2	x3	x4	u1	u2	w1	w2
	s1	x1	0	1	1 θ1	0	0	0	1	0
	s2	x2	0	0	0	0	1	θ1	0	0
	s3	x3	0	0	0	1	0	0	0	0
	s4	x4	0	0	0	0	θ1	0	0	0
	c1	z1	0	1	1 θ1	0	0	0	0	0
	c2	z2	0	0	1	0	0	0	0	0
a1	m1	y3	1	0	0	0	0	0	0	0
a2	m2	y4	0	0	1	0	0	0	0	1

In[11]:=

designmodelr=systemoperation[designmodel,{"State transformation",Inverse[{{1,0,0,0},{0,0,1,0},{0,1,0,0},{0,0,0,1}}]}]

State transformation is xold = transformation.xnew

Out[11]=

{x11,x21,x31,x41},{u1,u2,w1,w2},{z1,z2,y3,y4},0,

θ1

,1,0,0,0,1,0,{0,0,0,1,0,0,0,0},{0,0,0,0,1,θ1,0,0},{0,0,0,0,θ1,0,0,0},0,

θ1

,1,0,0,0,0,0,{0,1,0,0,0,0,0,0},{1,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,1},{1,2},{3,4},{1,2},{3,4},{3,4},{1,2}

In[12]:=

systemoperation[designmodelr,{"View"}];

Active							a1	a2
	Type		s1	s2	s3	s4	c1	c2	d1	d2
		names	x11	x21	x31	x41	u1	u2	w1	w2
	s1	x11	0	1 θ1	1	0	0	0	1	0
	s2	x21	0	0	0	1	0	0	0	0
	s3	x31	0	0	0	0	1	θ1	0	0
	s4	x41	0	0	0	0	θ1	0	0	0
	c1	z1	0	1 θ1	1	0	0	0	0	0
	c2	z2	0	1	0	0	0	0	0	0
a1	m1	y3	1	0	0	0	0	0	0	0
a2	m2	y4	0	1	0	0	0	0	0	1

In[13]:=

θ=Table[ToExpression[StringJoin["θ",ToString[i]]],{i,1,2}];

In[14]:=

θrules={θ21/θ1}

Out[14]=

θ2

θ1



In[15]:=

θbarvalue={θ11}

Out[15]=

{θ11}