ZigangPan/SystemofSystems | Paclet Repository

ZigangPan`SystemofSystems`

robustMRACdesignSystemofSystems

{picontroller,picontrollerIS}=robustMRACdesignSystemofSystems[{gy,gu},systemsdata,ub,γ,choice]

: is the (directed) graph representation of the composite system with the outputs y.

: is the (directed) graph representation of the composite system with the inputs ua.

and

must have the same vertex list.

systemdata

: is a list of length

= VertexCount[

], each containing the data for the subsystem

among the composite system.

: is the vector

that enters into every subsystem.

: is the desired disturbance attenuation level.

choice

: is a boolean variable, True for the optimal

ξc

design, and False for the suboptimal design. Returned variables:

picontroller

: is the NLsystem model for the robust adaptive controller.

picontrollerIS

: 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 the composite system if it is minimum phase and it satisfies the interconnection property. For subsystem

, the data required are as follows. If

=0, then the data is the list {

wchkj

bjl

's (l ∍. the edge



∈

, or l=j)},

bchkbj

achkjl

's (l ∍. the edge



∈

and l≠j)},

dchkj

a2111jl

's (l ∍. the edge



∈

or l=j)},

a2112j

a2113j

, {

a212jl

's (l ∍. the edge



∈

, or l=j)},

bj0

c12j

c13j

θj

pθj

qj0

Φj0

θchkj0

xchkj0

βΔj

Δinij1

kccoeffj

ρoj

Yinij

referencemodelj

xdj0

} where

: is output dimension.

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

wchkj

: is the

qchkj

-dimensional measured disturbance vector.

: is the

nOj×nOj

-dimensional system matrix.

bjl

: is the

nOj×ml

-dimensional matrix, l ∍. the edge



∈

, or l=j

bchkbj

: is the

nOj×pchk

-dimensional matrix.

achkjl

: is the

nOj×ml

-dimensional matrix, l ∍. the edge



∈

or l≠j

dchkj

: is the

nOj×qchkj

-dimensional matrix.

: is the

nOj×qbj

-dimensional matrix.

a2111jl

: is the

nOj×σj×ml

-dimensional tensor, l ∍. the edge



∈

or l=j

a2112j

: is the

nOj×σj×pchk

-dimensional tensor.

a2113j

: is the

nOj×σj×qchkj

-dimensional tensor.

a212jl

: is the

nOj×σj×ml

-dimensional tensor, l ∍. the edge



∈

, or l=j

: is the

mj×nOj

-dimensional matrix.

: is the

mj×qbj

-dimensional matrix.

bj0

: is the

mj×mj

-dimensional matrix.

c12j

: is the

mj×σj×mj

-dimensional tensor.

c13j

: is the

mj×σj×qchkj

-dimensional tensor.

θj

: is the

σj

-dimensional vector of unknown parameters for sj.

pθj

: is a pure function such that Apply[

pθj

θj

]≤1 for true values

θj

: is the

mj×mj

-dimensional positive definite weighting matrix for the tracking error.

qj0

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

θ-θchk0

Φj0

: is the initial value for the matrix

Φj

θchkj0

: is the initial estimate for

θj

xchkj0

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

βΔj

: is a nonnegative real value.

Δinij1

: is the initial value for

Δ1

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

kccoeffj

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

kccoeffj

^2*Tr[

qj0

]. When

kccoeffj=1

, then

Σj

and

sΣj

are constant matrix and constant scalar, respectively.

ρoj

: is a real value that is greater than 1.

Yinij

: is the initial value for

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

referencemodelj

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

xdj0

: is the initial state for the reference model. If

rj>0

, then the data is the list {

wchkj

aj1

bjl

's (l ∍. the edge slsj ∈ gu, or l=j)},

bchkbj

achkjl

's (l ∍. the edge slsj ∈

or l=j)},

dchkj

dj1

a2111jl

's (l ∍. the edge slsj ∈

or l=j)},

a2112j

a2113j

, {

a212jl

's (l ∍. the edge slsj ∈ gu, or l=j)},

ej1

θj

pθj

qj0

Φj0

θchkj0

xchkj0

βΔj

Δinij1

kccoeffj

ρoj

Yinij

referencemodelj

xdj0

γsj

βsj

} where

: is output dimension.

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

wchkj

: is the

qchkj

-dimensional measured disturbance vector.

aj1

: is the

nj×nj

-dimensional system matrix.

bjl

: is the

njmj×ml

-dimensional matrix, l ∍. the edge slsj ∈ gu, or l=j

bchkbj

: is the

njmj×pchk

-dimensional matrix.

achkjl

: is the

njmj×ml

-dimensional matrix, l ∍. the edge slsj ∈

or l=j

dchkj

: is the

njmj×qchkj

-dimensional matrix.

dj1

: is the

nj×qbj

-dimensional matrix.

a2111jl

: is the

njmj×σj×ml

-dimensional tensor, l ∍. the edge slsj ∈

or l=j

a2112j

: is the

njmj×σj×pchk

-dimensional tensor.

a2113j

: is the

njmj×σj×qchkj

-dimensional tensor.

a212jl

: is the

njmj×σj×ml

-dimensional tensor, l ∍. the edge slsj ∈ gu or l=j

ej1

: is the

1×qbj

-dimensional matrix.

θj

: is the

σj

-dimensional vector of unknown parameters for sj.

pθj

: is a pure function such that Apply[

pθj

θj

]≤1 for true values

θj

qj0

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

θj-θchkj0

Φj0

: is the initial value for the matrix

Φj

θchkj0

: is the initial estimate for

θj

xchkj0

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

βΔj

: is a nonnegative real value.

Δinij1

: is the initial value for

Δj1

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

kccoeffj

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

kccoeffj

^2*Tr[

qj0

]. When

kccoeff=1

, then

Σj

and

sΣj

are constant matrix and constant scalar, respectively.

ρoj

: is a real value that is greater than 1.

Yinij

: is the initial value for

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

referencemodelj

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

xdj0

: is the initial state for the reference model.

γsj

: is a list of length

of positive definite matrices (formula that may depend on

θchkj

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

βsj

: is a list of length

of positive definite matrices (formula that may depend on

θchkj

) used in the backstepping design as the negative drift.

Examples

(0)

SeeAlso

systemofsystems

▪

interconnectionpropertyQ

▪

EZDCFSS

RelatedGuides

▪

Guide to ZigangPan`SystemofSystems`

System 1.

In[3]:=

θbar1={θb11,θb12,θb13};

In[4]:=

system1={{x11,x12,x13,x14,x15},{u11,u12,u21,y31,y32,wg1,wg2,wg3,wg4,wg5,wg6},{y11,y12,z11,z12},{{0,1,0,θb11,0,2θb11,θb12,0,θb11,-θb12,0,0,0,0,0,θb11},{0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,-θb13,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{-1,0,0,-2,1,-θb11,-θb12,0,θb11,θb12,0,0,0,0,0,θb12},{0,0,0,-1/2,-1,0,0,1/6*θb12,θb11,-θb12,0,0,0,0,0,θb12},{1,0,0,0,0,θb11,θb12,0,0,0,1,0,0,0,0,0},{0,0,0,1,0,-θb12,θb11,0,0,0,0,1,0,0,0,0},{1,0,0,0,0,θb11,θb12,0,0,0,0,0,0,0,0,0},{0,0,0,1,0,-θb12,θb11,0,0,0,0,0,0,0,0,0}},{1,2},{1,2},{1,2},{3,4,5,6,7,8,9,10,11},{1,2},{3,4}}

Out[4]=

{x11,x12,x13,x14,x15},{u11,u12,u21,y31,y32,wg1,wg2,wg3,wg4,wg5,wg6},{y11,y12,z11,z12},{0,1,0,θb11,0,2θb11,θb12,0,θb11,-θb12,0,0,0,0,0,θb11},{0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,-θb13,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{-1,0,0,-2,1,-θb11,-θb12,0,θb11,θb12,0,0,0,0,0,θb12},0,0,0,-

,-1,0,0,

θb12

,θb11,-θb12,0,0,0,0,0,θb12,{1,0,0,0,0,θb11,θb12,0,0,0,1,0,0,0,0,0},{0,0,0,1,0,-θb12,θb11,0,0,0,0,1,0,0,0,0},{1,0,0,0,0,θb11,θb12,0,0,0,0,0,0,0,0,0},{0,0,0,1,0,-θb12,θb11,0,0,0,0,0,0,0,0,0},{1,2},{1,2},{1,2},{3,4,5,6,7,8,9,10,11},{1,2},{3,4}

In[5]:=

systemoperation[system1,{"View"}];

Active								a1	a2
	Type		s1	s2	s3	s4	s5	c1	c2	d1	d2	d3	d4	d5	d6	d7	d8	d9
		names	x11	x12	x13	x14	x15	u11	u12	u21	y31	y32	wg1	wg2	wg3	wg4	wg5	wg6
	s1	x11	0	1	0	θb11	0	2θb11	θb12	0	θb11	-θb12	0	0	0	0	0	θb11
	s2	x12	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
	s3	x13	0	-θb13	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	s4	x14	-1	0	0	-2	1	-θb11	-θb12	0	θb11	θb12	0	0	0	0	0	θb12
	s5	x15	0	0	0	- 1 2	-1	0	0	θb12 6	θb11	-θb12	0	0	0	0	0	θb12
a1	m1	y11	1	0	0	0	0	θb11	θb12	0	0	0	1	0	0	0	0	0
a2	m2	y12	0	0	0	1	0	-θb12	θb11	0	0	0	0	1	0	0	0	0
	c1	z11	1	0	0	0	0	θb11	θb12	0	0	0	0	0	0	0	0	0
	c2	z12	0	0	0	1	0	-θb12	θb11	0	0	0	0	0	0	0	0	0

In[6]:=

systemcheck[system1]

Out[6]=

True

In[7]:=

$Assumptions=True

Out[7]=

True

In[8]:=

$Assumptions=$Assumptions&&assumeReal[Join[θbar1,system1〚1〛,system1〚2〛,system1〚3〛]]

Out[8]=

θb11∈&&θb12∈&&θb13∈&&x11∈&&x12∈&&x13∈&&x14∈&&x15∈&&u11∈&&u12∈&&u21∈&&y31∈&&y32∈&&wg1∈&&wg2∈&&wg3∈&&wg4∈&&wg5∈&&wg6∈&&y11∈&&y12∈&&z11∈&&z12∈

In[9]:=

{r1,highfrequencygain1,systemaddI1,systemaddO1,transformation1,zerodynamics1}=EZDCFAD[system1,θbar1]

Wolfram Language Paclet Repository

System 1.

System 2.

System 3

Parallel interconnection.