ZigangPan/MRRACLinearFullOrder

ZigangPan`MRRACLinearFullOrder`

robustMRACdesignMIMOFullOrder0

{controller,controllerIS,af}=robustMRACdesignMIMOFullOrder0[m,r,a,b,dchk,d,a2111,a2113,a212,e,c,b0,c13,c12,q,θ,pθ,γ,referencemodel,θchk0,xchk0,q0,Φ0,ρo,Δ1i,βΔ,kccoeff,xd0]

m

: is output dimension.

r

: is the uniform vector relative degree of the design model, must equal to 0.

a

: is the

nn

-dimensional system matrix.

b

: is the

nm

-dimensional matrix.

dchk

: is the

nqchk

-dimensional matrix.

d

: is the

nqb

-dimensional matrix.

a2111

: is the

nσm

-dimensional tensor.

a2113

: is the

nσqchk

-dimensional tensor.

a212

: is the

nσm

-dimensional tensor.

e

: is the

mqb

-dimensional matrix.

c

: is the

mn

-dimensional matrix.

b0

: is the

mm

-dimensional matrix.

c13

: is the

mσqchk

-dimensional tensor.

c12

: is the

mσm

-dimensional tensor.

q

: is the positive definite weighting matrix for the tracking error.

θ

: is the

σ

-dimensional vector of unknown parameters.

pθ

: is a pure function such that

Apply

[

pθ

,θ]≤1

for true values

θ

.

γ

: is the desired disturbance attenuation level.

referencemodel

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

θchk0

: is the initial estimate for

θ

.

xchk0

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

q0

: 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.

Δ1i

: is the initial value for

Δ1

. (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*

Tr

[q0]

. When

kccoeff=1

, then Σ and sΣ are constant matrix and constant scalar, respectively.

xd0

: is the initial state for the reference model. Returned variables:

controller

: is the NLsystem model for the robust adaptive controller.

controllerIS

: is the initial condition for the robust adaptive controller.

af

: is the numeric Hurwitz matrix

af

. 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 additioanl measured disturbance as additional measurement outputs and with uniform vector relative degree zero and the system must be in strict observer canonical form.

Examples

(1)

Basic Examples

(1)

In[1]:=

system1={{x1,x2,x3},{u1,u2,w1,w2,wchk},{y1,y2,z1,z2,wchk},{{0,1,0,2θ1,2θ2,0,0,1},{0,0,1,0,0,0,0,0},{0,-θ3,0,0,0,0,0,0},{1,0,0,θ1,θ2,1,0,0},{0,0,0,-θ2,θ1,0,1,0},{1,0,0,θ1,θ2,0,0,0},{0,0,0,-θ2,θ1,0,0,0},{0,0,0,0,0,0,0,1}},{1,2},{1,2},{1,2},{3,4,5},{1,2,5},{3,4}};systemcheck[system1]

Out[1]=

True

In[2]:=

θbar={θ1,θ2,θ3};

In[3]:=

$Assumptions=$Assumptions&&assumeReal[Join[system1〚1〛,system1〚2〛,system1〚3〛]]&&assumePositive[θbar]

Out[3]=

x1∈&&x2∈&&x3∈&&u1∈&&u2∈&&w1∈&&w2∈&&wchk∈&&y1∈&&y2∈&&z1∈&&z2∈&&wchk∈&&θ1>0&&θ2>0&&θ3>0

In[4]:=

EZDCFAD[system1,θbar]

Out[4]=

0,{{θ1,θ2},{-θ2,θ1}},default,default,{{1,0,0},{0,1,0},{0,0,1}},{x1,x2,x3},{u1,u2,w1,w2,wchk,y1,y2},{y1,y2,z1,z2,wchk},-

2

θ1

2

θ1

+

2

θ2

-

2

θ2

2

θ1

+

2

θ2

,1,0,0,0,-

2

θ1

2

θ1

+

2

θ2

-

2

θ2

2

θ1

+

2

θ2

,0,1,

2

θ1

2

θ1

+

2

θ2

+

2

θ2

2

θ1

+

2

θ2

,0,{0,0,1,0,0,0,0,0,0,0},{0,-θ3,0,0,0,0,0,0,0,0},{1,0,0,θ1,θ2,1,0,0,0,0},{0,0,0,-θ2,θ1,0,1,0,0,0},{1,0,0,θ1,θ2,0,0,0,0,0},{0,0,0,-θ2,θ1,0,0,0,0,0},{0,0,0,0,0,0,0,1,0,0},{1,2,6,7},{1,2},{1,2},{3,4,5},{1,2,5},{3,4}

In[5]:=

strictOCFAD[system1]

State transformation is xold = transformation.xnew

Out[5]=

{True,True,3,{3,0},{{1,0,0},{0,1,0},{-θ3,0,1}},{{x11,x21,x31},{u1,u2,w1,w2,wchk},{y1,y2,z1,z2,wchk},{{0,1,0,2θ1,2θ2,0,0,1},{-θ3,0,1,0,0,0,0,0},{0,0,0,2θ1θ3,2θ2θ3,0,0,θ3},{1,0,0,θ1,θ2,1,0,0},{0,0,0,-θ2,θ1,0,1,0},{1,0,0,θ1,θ2,0,0,0},{0,0,0,-θ2,θ1,0,0,0},{0,0,0,0,0,0,0,1}},{1,2},{1,2},{1,2},{3,4,5},{1,2,5},{3,4}}}

In[6]:=

{strictform,systemobservable,nO,observabilityindices,transformation,systeminobservercanonicalform}=%

Out[6]=

{True,True,3,{3,0},{{1,0,0},{0,1,0},{-θ3,0,1}},{{x11,x21,x31},{u1,u2,w1,w2,wchk},{y1,y2,z1,z2,wchk},{{0,1,0,2θ1,2θ2,0,0,1},{-θ3,0,1,0,0,0,0,0},{0,0,0,2θ1θ3,2θ2θ3,0,0,θ3},{1,0,0,θ1,θ2,1,0,0},{0,0,0,-θ2,θ1,0,1,0},{1,0,0,θ1,θ2,0,0,0},{0,0,0,-θ2,θ1,0,0,0},{0,0,0,0,0,0,0,1}},{1,2},{1,2},{1,2},{3,4,5},{1,2,5},{3,4}}}

In[7]:=

designmodelr=rd0DMcompute[systeminobservercanonicalform,θbar]

State transformation is xold = transformation.xnew

Out[7]=

{{x11,x21,x31},{u1,u2,w1,w2,wchk,y1},{y1,y2,z1,z2,wchk},{{0,1,0,2θ1,2θ2,0,0,1,0},{0,0,1,θ1θ3,θ2θ3,θ3,0,0,-θ3},{0,0,0,2θ1θ3,2θ2θ3,0,0,θ3,0},{1,0,0,θ1,θ2,1,0,0,0},{0,0,0,-θ2,θ1,0,1,0,0},{1,0,0,θ1,θ2,0,0,0,0},{0,0,0,-θ2,θ1,0,0,0,0},{0,0,0,0,0,0,0,1,0}},{1,2},{1,2},{1,2},{3,4,5,6},{1,2,5},{3,4}}

In[8]:=

θ=Table[ToExpression[StringJoin["θ",ToString[i]]],{i,5}]

Out[8]=

{θ1,θ2,θ3,θ4,θ5}

In[9]:=

θrules={θ4θ1θ3,θ5θ2θ3}

Out[9]=

{θ4θ1θ3,θ5θ2θ3}

In[10]:=

designmodelr〚4〛〚All,4;;5〛={{2θ1,2θ2},{θ4,θ5},{2θ4,2θ5},{θ1,θ2},{-θ2,θ1},{θ1,θ2},{-θ2,θ1},{0,0}}

Out[10]=

{{2θ1,2θ2},{θ4,θ5},{2θ4,2θ5},{θ1,θ2},{-θ2,θ1},{θ1,θ2},{-θ2,θ1},{0,0}}

In[11]:=

systemoperation[designmodelr,{"View"}];

Active						a1	a2
	Type		s1	s2	s3	c1	c2	d1	d2	d3	d4
		names	x11	x21	x31	u1	u2	w1	w2	wchk	y1
	s1	x11	0	1	0	2θ1	2θ2	0	0	1	0
	s2	x21	0	0	1	θ4	θ5	θ3	0	0	-θ3
	s3	x31	0	0	0	2θ4	2θ5	0	0	θ3	0
a1	m1	y1	1	0	0	θ1	θ2	1	0	0	0
a2	m2	y2	0	0	0	-θ2	θ1	0	1	0	0
	c1	z1	1	0	0	θ1	θ2	0	0	0	0
	c2	z2	0	0	0	-θ2	θ1	0	0	0	0
	m3	wchk	0	0	0	0	0	0	0	1	0

In[12]:=

Mgrave={{1,0},{0,1},{θ3,0}};

In[13]:=

a={{0,1,0},{0,0,1},{0,0,0}};b={{0,0},{0,0},{0,0}};dchk={{1},{0},{0}};c={{1,0,0},{0,0,0}};b0={{0,0},{0,0}};

In[14]:=

d={{0,0,0},{0,0,1},{0,0,0}};

In[15]:=

mm={2θ1u1+2θ2u2,θ4u1+θ5u2-θ3y1,2θ4u1+2θ5u2+θ3*wchk}

Out[15]=

{2u1θ1+2u2θ2,-y1θ3+u1θ4+u2θ5,wchkθ3+2u1θ4+2u2θ5}

In[16]:=

a2111=D[D[mm,{θ}],{{y1,y2}}]

Out[16]=

{{{0,0},{0,0},{0,0},{0,0},{0,0}},{{0,0},{0,0},{-1,0},{0,0},{0,0}},{{0,0},{0,0},{0,0},{0,0},{0,0}}}

In[17]:=

a2113=D[D[mm,{θ}],{{wchk}}]

Out[17]=

{{{0},{0},{0},{0},{0}},{{0},{0},{0},{0},{0}},{{0},{0},{1},{0},{0}}}

In[18]:=

a212=D[D[mm,{θ}],{{u1,u2}}]

Out[18]=

{{{2,0},{0,2},{0,0},{0,0},{0,0}},{{0,0},{0,0},{0,0},{1,0},{0,1}},{{0,0},{0,0},{0,0},{2,0},{0,2}}}

In[19]:=

mc={θ1u1+θ2u2,-θ2u1+θ1u2}

Out[19]=

{u1θ1+u2θ2,u2θ1-u1θ2}

In[20]:=

c13=D[D[mc,{θ}],{{wchk}}]

Out[20]=

{{{0},{0},{0},{0},{0}},{{0},{0},{0},{0},{0}}}

In[21]:=

c12=D[D[mc,{θ}],{{u1,u2}}]

Out[21]=

{{{1,0},{0,1},{0,0},{0,0},{0,0}},{{0,1},{-1,0},{0,0},{0,0},{0,0}}}

In[22]:=

e={{1,0,0},{0,1,0}}

Out[22]=

{{1,0,0},{0,1,0}}

In[23]:=

q={{2,-1},{-1,1}};

In[24]:=

θbarrule={θ11,θ21,θ34};

In[25]:=

θtrue=θ/.θrules/.θbarrule

Out[25]=

{1,1,4,4,4}

In[26]:=

pθ=FormulaToFunction[θ,256/275*(4(θ1-1)^6+4(θ2-1)^6+(θ3/4-1)^4+(θ4/4-1)^4+(θ5/4-1)^4)]

Out[26]=

Function{θ1,θ2,θ3,θ4,θ5},

256

275

4

6

(-1+θ1)

+4

6

(-1+θ2)

+

4

-1+

θ3

4

+

4

-1+

θ4

4

+

4

-1+

θ5

4



In[27]:=

ρo=5;Δ1i=IdentityMatrix[3];βΔ=0;kccoeff=1;xd0={1,0};