ZigangPan/MRRACLinearFullOrder

ZigangPan`MRRACLinearFullOrder`

robustMRACdesignMIMOFullOrder1

{controller,controllerIS}=robustMRACdesignMIMOFullOrder1[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,xd0,Yi,γs,βs]

: 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 Φ.

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

Σn

and

sΣ

are constant matrix and constant scalar, respectively.

xd0

: is the initial state for the reference model.

: is the initial value for

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

γs

: is a list of length

of positive definite matrices (formula that may depend on

) used in the backstepping design as the additive value function.

βs

: is a list of length

of positive definite matrices (formula that may depend on

) used in the backstepping design as the negative drift. Returned variables:

controller

: is the NLsystem model for the robust adaptive controller.

controllerIS

: is the initial condition for the robust adaptive controller. If the inputs are of rational coefficients, then the returned variables are of rational coefficients. Design is guaranteed to work for minimum phase MIMO systems with additional measured disturbance as additional measurement outputs and additional control inputs in addition to the ones with uniform vector relative degree

Examples

(1)

Basic Examples

(1)

In[1]:=

Needs["ZigangPan`Examples`"]

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}

In[16]:=

θvalues=θ/.θrules/.θbarvalue