Stability, Stabilizability and Control of certain classes ...

Positive Systems and Compartmental SystemsFeedback stabilization of Compartmental Systems

Stability, Stabilizability andControl of certain classes of

Positive Systems

Irene Zorzan

Ph.D. Advisor: Prof. Maria Elena Valcher

Ph.D. School in Information Engineering - University of Padova

February 22, 2018

Irene Zorzan Stability, stabilizability and control of Positive Systems 1/31


Outline of the talk

Positive Systems: definition and motivations

Control of Positive Multi-Agent SystemsOptimal control of Positive Bilinear SystemsStability and stabilizability of Compartmental Switched Systems

Feedback stabilization of Compartmental Systems



Positive Consensus ProblemOptimal Control of Positive Bilinear SystemsCompartmental Switched Systems

Positive Systems

Positive Systemsnonnegative state and output variablesfor every nonnegative initial condition

and every nonnegative input




Positive Systems



pressuresabsolute temperaturesconcentrations of substancespopulation levelsprobabilities




Positive Systems



pressuresabsolute temperaturesconcentrations of substancespopulation levelsprobabilities

economicssystem biologycongestion controlpharmacokineticspower networks




Positive Systems: state-space representation

Consider a continuous-time linear system described by

x(t) = Ax(t) +Bu(t) (1)






x(t) = Ax(t) +Bu(t) (1)

wherex is the n-dimensional state vector

u is the m-dimensional input vectorA, B are matrices of consistent dimensions






x(t) = Ax(t) +Bu(t) (1)

wherex is the n-dimensional state vectoru is the m-dimensional input vector

A, B are matrices of consistent dimensions






x(t) = Ax(t) +Bu(t) (1)

wherex is the n-dimensional state vectoru is the m-dimensional input vectorA, B are matrices of consistent dimensions






x(t) = Ax(t) +Bu(t) (1)


System (1) is a Positive System if and only if

A is a Metzler matrixA square matrix is Metzler if all its off-diagonal entries are nonnegative.

B is a positive matrixA matrix is positive if all its entries are nonnegative.






x(t) = Ax(t) +Bu(t) (1)


System (1) is a Positive System if and only ifA is a Metzler matrixA square matrix is Metzler if all its off-diagonal entries are nonnegative.

B is a positive matrixA matrix is positive if all its entries are nonnegative.




Positive consensus problem: set-up





1 2

i

...

... N

N identical Positive Single-Input Systems:

xi(t) = Axi(t) +Bui(t), i ∈ [1, N ]





1 2

i

...

... N


xi(t) = Axi(t) +Bui(t), i ∈ [1, N ]

Undirected, weighted and connected com-munication graph:

G = (V, E ,A)





1 2

i

...

... N


xi(t) = Axi(t) +Bui(t), i ∈ [1, N ]

Undirected, weighted and connected com-munication graph:

G = (V, E ,A)

Assume that A is non-Hurwitz and that (A,B) is stabilizable.




Positive consensus: problem formulation

1 2

i

...

... N

Each agent adopts the state-feedback con-trol law

ui(t) = K

N∑j=1

Aij [xj(t)− xi(t)]





1 2

i

...

... N


ui(t) = K

N∑j=1


where K ∈ R1×n is a matrix to be designed.





1 2

i

...

... N


ui(t) = K

N∑j=1



Positive consensus problem:determine a feedback matrix K ∈ R1×n such that:

i) positivity of the overall dynamics is preservedii) all the agents achieve consensus, namely for every i, j ∈ [1, N ]

it holds limt→+∞ xi(t)− xj(t) = 0





1 2

i

...

... N


ui(t) = K

N∑j=1




i) positivity of the overall dynamics is preserved

ii) all the agents achieve consensus, namely for every i, j ∈ [1, N ]it holds limt→+∞ xi(t)− xj(t) = 0





1 2

i

...

... N


ui(t) = K

N∑j=1




i) positivity of the overall dynamics is preservedii) all the agents achieve consensus, namely for every i, j ∈ [1, N ]

it holds limt→+∞ xi(t)− xj(t) = 0




Positive consensus problem: motivations

In several contexts where consensus problems arise, the positivityof the overall controlled system is an intrinsic feature that should betaken into consideration when formalizing the problem.





In several contexts where consensus problems arise, the positivityof the overall controlled system is an intrinsic feature that should betaken into consideration when formalizing the problem.This is the case, for instance, when dealing with

sensor networks for greenhouse monitoring

CO2 concentrationhumidity levellight intensity

fleet of cooperating hybrid vehicles

CO2 emission levelelectric/combustion based propulsion






sensor networks for greenhouse monitoringCO2 concentrationhumidity levellight intensity

fleet of cooperating hybrid vehicles

CO2 emission levelelectric/combustion based propulsion






sensor networks for greenhouse monitoringCO2 concentrationhumidity levellight intensity

fleet of cooperating hybrid vehiclesCO2 emission levelelectric/combustion based propulsion




Positive Bilinear Systems





Consider a Positive Bilinear System described by

x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vector

w is an arbitrary n-dimensional external disturbancey is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbance

y is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbancey is the p-dimensional output vector

ui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbancey is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputs

A is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbancey is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrix

Di, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbancey is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matrices

C is a p× n nonnegative matrix






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

wherex is the n-dimensional state vectorw is an arbitrary n-dimensional external disturbancey is the p-dimensional output vectorui, i ∈ [1,M ], are the scalar control inputsA is an n× n Metzler matrixDi, i ∈ [1,M ], are n× n diagonal matricesC is a p× n nonnegative matrix




Optimal control of Positive Bilinear Systems


x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)

Assume that the Metzler matrix A is non-Hurwitz.






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)


Optimal robust control problem:determine ui ∈ R+, i ∈ [1,M ], with the following characteristics:

i) satisfy the constraint∑M

i=1 ui < 1

ii) make the system asymptotically stableiii) maximize system robustness against the external disturbance






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)




i=1 ui < 1

ii) make the system asymptotically stable

iii) maximize system robustness against the external disturbance






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)




i=1 ui < 1

ii) make the system asymptotically stableiii) maximize system robustness against the external disturbance






x(t) =

(A+

M∑i=1

uiDi

)x(t) + w(t)

y(t) = Cx(t)




i=1 ui < 1

ii) make the system asymptotically stableiii) minimize either the L1-norm or the H∞-norm of the system




Positive Bilinear Systems: application areas

Positive Bilinear Systems have been fruitfully employed in a variety ofapplication areas:

system biology

a Positive Bilinear System can be adopted to describe thebehaviour of a group of self-replicating genotypes⇒ determing the optimal control means determing the best drugsconcentration profilenetwork theoryoptimal control issues concern the proper selection of leaders in adirected graph






system biologya Positive Bilinear System can be adopted to describe thebehaviour of a group of self-replicating genotypes⇒ determing the optimal control means determing the best drugsconcentration profile

network theoryoptimal control issues concern the proper selection of leaders in adirected graph






system biologya Positive Bilinear System can be adopted to describe thebehaviour of a group of self-replicating genotypes⇒ determing the optimal control means determing the best drugsconcentration profilenetwork theoryoptimal control issues concern the proper selection of leaders in adirected graph




Compartmental Systems: definition

Compartmental Systems represent physical systems whose describ-ing variables are intrinsically nonnegative and obey some conserva-tion law (e.g., mass, energy, fluid).




Compartmental Systems: definition

Compartmental Systems represent physical systems whose describ-ing variables are intrinsically nonnegative and obey some conserva-tion law (e.g., mass, energy, fluid).

A Positive Systemx(t) = Ax(t) +Bu(t)

is a Compartmental System if the Metzler matrix A is such that the en-tries of each of its columns sum up to a nonpositive number

1>nA ≤ 0,

i.e., A is a compartmental matrix.




Compartmental Systems: motivations

Compartmental Systems arise in the description of a good number ofdynamical processes:

liquids flowing in a network of interconnected tankstime evolution of temperatures in adjacent roomskinetics of substances in tracer experimentscell proliferation through mitosisphysiological processes, e.g., insulin secretion, glucoseabsorption






liquids flowing in a network of interconnected tanks

time evolution of temperatures in adjacent roomskinetics of substances in tracer experimentscell proliferation through mitosisphysiological processes, e.g., insulin secretion, glucoseabsorption






liquids flowing in a network of interconnected tankstime evolution of temperatures in adjacent rooms

kinetics of substances in tracer experimentscell proliferation through mitosisphysiological processes, e.g., insulin secretion, glucoseabsorption






liquids flowing in a network of interconnected tankstime evolution of temperatures in adjacent roomskinetics of substances in tracer experiments

cell proliferation through mitosisphysiological processes, e.g., insulin secretion, glucoseabsorption






liquids flowing in a network of interconnected tankstime evolution of temperatures in adjacent roomskinetics of substances in tracer experimentscell proliferation through mitosis

physiological processes, e.g., insulin secretion, glucoseabsorption






liquids flowing in a network of interconnected tankstime evolution of temperatures in adjacent roomskinetics of substances in tracer experimentscell proliferation through mitosisphysiological processes, e.g., insulin secretion, glucoseabsorption




Compartmental Switched Systems

Compartmental Switched Systems are adopted to describe the be-haviour, under different operating conditions, of systems modeling theexchange of material between different compartments.






A Compartmental Switched System is described by

x(t) = Aσ(t)x(t) +Bσ(t)u(t)






A Compartmental Switched System is described by

x(t) = Aσ(t)x(t) +Bσ(t)u(t)

and consists of:a family of compartmental models (the subsystems)(Ai, Bi), i ∈ [1,M ]

a switching function describing which of the subsystems is activeat every time instantσ : R+ → [1,M ]




Compartmental Switched Systems: motivations

Compartmental Switched Systems can be employed to describe:

a fluid network undergoing different open/closed configurations ofthe pipes connecting the tanksa thermal system whose heat transmission coefficients dependon the open/closed configurations of doors and windowsthe lung dynamics alternating between inspiration and expirationphases





Compartmental Switched Systems can be employed to describe:a fluid network undergoing different open/closed configurations ofthe pipes connecting the tanks

a thermal system whose heat transmission coefficients dependon the open/closed configurations of doors and windowsthe lung dynamics alternating between inspiration and expirationphases





Compartmental Switched Systems can be employed to describe:a fluid network undergoing different open/closed configurations ofthe pipes connecting the tanksa thermal system whose heat transmission coefficients dependon the open/closed configurations of doors and windows

the lung dynamics alternating between inspiration and expirationphases

Room 1 Room 2

Room 3

u2

u1α

β δ





Compartmental Switched Systems can be employed to describe:a fluid network undergoing different open/closed configurations ofthe pipes connecting the tanksa thermal system whose heat transmission coefficients dependon the open/closed configurations of doors and windowsthe lung dynamics alternating between inspiration and expirationphases

x1

x2 x3

x4




Compartmental Switched Systems: stabilizability

Compartmental Switched System with autonomous subsystems

x(t) = Aσ(t)x(t), σ : R+ → [1,M ]






x(t) = Aσ(t)x(t), σ : R+ → [1,M ]

Stabilizability:under what conditions on the matrices Ai, i ∈ [1,M ], for every pos-itive initial state x(0) > 0, there exists a switching function σ thatmakes x(t) asymptotically converge to zero?






x(t) = Aσ(t)x(t), σ : R+ → [1,M ]

Stabilizability:under what conditions on the matrices Ai, i ∈ [1,M ], for every pos-itive initial state x(0) > 0, there exists a switching function σ thatmakes x(t) asymptotically converge to zero?

IFF there exists a Hurwitz convex combination of Ai, i ∈ [1,M ]




Compartmental Switched Systems: stability


x(t) = Aσ(t)x(t), σ : R+ → [1,M ]






x(t) = Aσ(t)x(t), σ : R+ → [1,M ]

Asymptotic stability under arbitrary switching:under what conditions on the matrices Ai, i ∈ [1,M ], the state tra-jectory x(t) asymptotically converges to zero for every positive ini-tial state x(0) > 0 and for every switching function σ?






x(t) = Aσ(t)x(t), σ : R+ → [1,M ]

Asymptotic stability under arbitrary switching:under what conditions on the matrices Ai, i ∈ [1,M ], the state tra-jectory x(t) asymptotically converges to zero for every positive ini-tial state x(0) > 0 and for every switching function σ?

IFF all matrices Ai, i ∈ [1,M ], are Hurwitz




The non-autonomous case

Autonomous case:x(t) = Aσ(t)x(t)

Non-autonomous case:x(t) = Aσ(t)x(t) +Bσ(t)u(t)

Asympt. stable under arbitrary switchingIFF Ai is Hurwitz for every i ∈ [1,M ]








Feedback stabilization problem:under what conditions on the pairs (Ai, Bi), i ∈ [1,M ], thereexist feedback matrices Ki, i ∈ [1,M ], such that the control lawu(t) = Kix(t) makes the closed-loop system compartmental andasymptotically stable under arbitrary switching?








Feedback stabilization problem:under what conditions on the pairs (Ai, Bi), i ∈ [1,M ], thereexist feedback matrices Ki, i ∈ [1,M ], such that the control lawu(t) = Kix(t) makes the closed-loop system compartmental andasymptotically stable under arbitrary switching?

IFF ∃Ki, i ∈ [1,M ], s.t. Ai +BiKi is compartmental and Hurwitz



Mathematical preliminariesHurwitz stability of compartmental matricesThe case when A is irreducibleThe case when A is reducible

State-feedback stabilization problem





Consider a Compartmental System

x(t) = Ax(t) +Bu(t)






x(t) = Ax(t) +Bu(t)

and assume that the compartmental matrix A is non-Hurwitz.






x(t) = Ax(t) +Bu(t)

and assume that the compartmental matrix A is non-Hurwitz.

State-feedback stabilization problem:determine K ∈ Rm×n such that the state-feedback control lawu(t) = Kx(t) makes the closed-loop system

x(t) = (A+BK)x(t)compartmental and asymptotically stable.




Example: room temperature regulation

Room 1 Room 2

Room 3





Room 1 Room 2

Room 3

u2

u1





Room 1 Room 2

Room 3

u2

u1α

β γ





Room 1 Room 2

Room 3

u2

u1α

β γ

Assume that the system is thermally isolatedfrom the external environment.





Room 1 Room 2

Room 3

u2

u1α

β γ


xi = (ith room temp− desired temp) ≥ 0





Room 1 Room 2

Room 3

u2

u1α

β γ



Compartmental model describing tempera-tures evolution:x(t) =Ax(t) +Bu(t)

=

−(α+β) α βα −(α+γ) γβ γ −(γ+β)

x(t)+0 01 00 1

u(t)





Room 1 Room 2

Room 3

u2

u1α

β γ



Compartmental model describing tempera-tures evolution:x(t) =Ax(t) +Bu(t)

=

−(α+β) α βα −(α+γ) γβ γ −(γ+β)

x(t)+0 01 00 1

u(t)Room temperaturte regulation problem:determine a state-feedback control law that regulates all temperaturesby making use only of the temperature in Room 1.




Preliminary definitions

In the following we will denote by:

ei the ith canonical vector1n the n-dimensional vector with all entries equal to 1

Si ∈ R(n−1)×n the selection matrix obtained by removing the ithrow in the identity matrix In, i.e.,

Si =

[Ii−1 0 0

0 0 In−i

]

For any vector v ∈ Rn, Siv is the vector obtained from v byremoving the ith entry.For any matrix A ∈ Rn×m, SiA denotes the matrix obtained fromA by removing the ith row.





In the following we will denote by:ei the ith canonical vector

1n the n-dimensional vector with all entries equal to 1


Si =

[Ii−1 0 0

0 0 In−i

]






In the following we will denote by:ei the ith canonical vector1n the n-dimensional vector with all entries equal to 1


Si =

[Ii−1 0 0

0 0 In−i

]








Si =

[Ii−1 0 0

0 0 In−i

]For any vector v ∈ Rn, Siv is the vector obtained from v byremoving the ith entry.

For any matrix A ∈ Rn×m, SiA denotes the matrix obtained fromA by removing the ith row.







Si =

[Ii−1 0 0

0 0 In−i

]For any vector v ∈ Rn, Siv is the vector obtained from v byremoving the ith entry.For any matrix A ∈ Rn×m, SiA denotes the matrix obtained fromA by removing the ith row.




Preliminary definitions (cont’d)

A Metzler matrix A is reducible if there exists a permutation matrixΠ such that

Π>AΠ =

[A11 A12

0 A22

]where A11 and A22 are square nonvacuous matrices, otherwise itis irreducible.




Frobenius normal form

For every Metzler matrix A a permutation matrix Π can be found suchthat

Π>AΠ =

A11 A12 . . . A1s

0 A22 . . . A2s...

. . ....

0 . . . Ass

(2)

where each diagonal block Aii, of size ni × ni, is either scalar (ni = 1)or irreducible.




Frobenius normal form

For every Metzler matrix A a permutation matrix Π can be found suchthat

Π>AΠ =

A11 A12 . . . A1s

0 A22 . . . A2s...

. . ....

0 . . . Ass

(2)

where each diagonal block Aii, of size ni × ni, is either scalar (ni = 1)or irreducible. The block form (2) is usually known as Frobenius nor-mal form of A.




Communication classes

Compartmental matrix A ∈ Rn×n

A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann

Directed graph G(A) := V, E






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


set of vertices

1 2

...

j `

... n






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


set of edges

1 2

...

j `

... n






A =

a11 a12 . . . a1n

a21. . . aj`

......

. . ....

an1 . . . . . . ann


set of edges

1 2

...

j `

... n

aj` > 0






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


1 2

...

j `

... n






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


1 2

...

j `

... n

Vertex j and vertex ` can communicate






A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


1 2

...

j `

... n

Communication class:a set of vertices that communicate withevery other vertex in the class and withno other vertex.The corresponding subgraph is stronglyconnected.







1 2

...

j `

... n

Frobenius normal form of A

A =

A11 A12 . . . A1s

0 A22 . . . A2s...

. . ....

0 . . . . . . Ass







1 2

...

j `

... n

Frobenius normal form of A

A =

A11 A12 . . . A1s

0 A22 . . . A2s...

. . ....

0 . . . . . . Ass

C1

C2

Cs




Hurwitz stability


A =

a11 a12 . . . a1n

a21. . .

......

. . ....

an1 . . . . . . ann


1 2

...

j `

... n




Hurwitz stability


A =

a11 a12 a1j a1n

a21. . .

......

. . ....

an1 . . . anj ann


1 2

...

j `

... n

1>nAej < 0

Vertex j has outflow to the environment




Hurwitz stability


A =

a11 a12 a1j a1n

a21. . .

......

. . ....

an1 . . . anj ann


1 2

...

j `

... n

1>nAej < 0

Vertex j has outflow to the environment

Vertex ` is outflow connected




Hurwitz stability


A =

a11 a12 a1j a1n

a21. . .

......

. . ....

an1 . . . anj ann


1 2

...

j `

... n

1>nAej < 0

Lemma 1:A compartmental matrix A is Hurwitz ifand only if all compartments are outflowconnected.




The case when A is irreducible

Proposition 1:

Consider a Compartmental System and assume that the state-spacematrix A is irreducible and non-Hurwitz.





Proposition 1:

Consider a Compartmental System and assume that the state-spacematrix A is irreducible and non-Hurwitz.The state-feedback stabilization problem is solvable if and only if thereexist h ∈ [1, n] and v ∈ Rm such that

Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)





Proposition 1:


Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)

When so, any matrix K = εve>h with ε ∈ (0, 1) is a possible solution.





Proposition 1:


Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)


compartmental property





Proposition 1:


Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)



outflow to environment





Proposition 1:


Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)



graph (strong) connectedness






Proposition 1:


Sh (Aeh +Bv) ≥ 0 (3)

1>nBv < 0 (4)


Every compartment is outflow connected!




Remarks

When A is irreducible and the state-feedback stabilizationproblem is solvable, there always exists a solution K with aunique nonzero column.

However, not all columns (i.e., not allindices h ∈ [1, n]) play an equivalent role.

In the Single-Input case (m = 1), problem solvability does notdepend on the specific values of the entries of A and B but onlyon their nonzero pattern.

The nonzero pattern of a matrix A ∈ Rn×m is the setZP(A) := (i, j) ∈ [1, n]× [1,m] : [Aij ] 6= 0.




Remarks

When A is irreducible and the state-feedback stabilizationproblem is solvable, there always exists a solution K with aunique nonzero column. However, not all columns (i.e., not allindices h ∈ [1, n]) play an equivalent role.

In the Single-Input case (m = 1), problem solvability does notdepend on the specific values of the entries of A and B but onlyon their nonzero pattern.

The nonzero pattern of a matrix A ∈ Rn×m is the setZP(A) := (i, j) ∈ [1, n]× [1,m] : [Aij ] 6= 0.




Remarks

When A is irreducible and the state-feedback stabilizationproblem is solvable, there always exists a solution K with aunique nonzero column. However, not all columns (i.e., not allindices h ∈ [1, n]) play an equivalent role.

In the Single-Input case (m = 1), problem solvability does notdepend on the specific values of the entries of A and B but onlyon their nonzero pattern.The nonzero pattern of a matrix A ∈ Rn×m is the setZP(A) := (i, j) ∈ [1, n]× [1,m] : [Aij ] 6= 0.




The case when A is reducible

Proposition 2:Consider a Compartmental System and assume that the state-spacematrix A is reducible and non-Hurwitz.





Proposition 2:Consider a Compartmental System and assume that the state-spacematrix A is reducible and non-Hurwitz.Assume w.l.o.g. that A is in Frobenius normal form:

A =

A11 . . . 0 . . . A1s

0. . .

...... Arr Ars...

. . ....

0 . . . 0 Ass






A =

A11 . . . 0 . . . A1s

0. . .

...... Arr Ars...

. . ....

0 . . . 0 Ass

Aii, i ∈ [1, r], irreducible non-Hurwitz






A =

A11 . . . 0 . . . A1s

0. . .

...... Arr Ars...

. . ....

0 . . . 0 Ass


Aii, i ∈ [r + 1, s], irreducible Hurwitz






A =

A11 . . . 0 . . . A1s

0. . .

...... Arr Ars...

. . ....

0 . . . 0 Ass


Aii, i ∈ [r + 1, s], irreducible Hurwitz

Let Ωi, i ∈ [1, s], be the set of all column indices corresponding to Aii.




The case when A is reducible (cont’d)

If for every i ∈ [1, r] there exist ì ∈ Ωi and vi ∈ Rm such that

Sì (Aeì +Bvi) ≥ 0 (5)

1>nBvi < 0 (6)

then the state-feedback stabilization problem is solvable.






Sì (Aeì +Bvi) ≥ 0 (5)

1>nBvi < 0 (6)

then the state-feedback stabilization problem is solvable.Moreover, any matrix

K := ε∑r

i=1 vie>ì, ε ∈ (0, 1)

is a possible solution.






Sì (Aeì +Bvi) ≥ 0 (5)

1>nBvi < 0 (6)


K := ε∑r

i=1 vie>ì, ε ∈ (0, 1)








Sì (Aeì +Bvi) ≥ 0 (5)

1>nBvi < 0 (6)


K := ε∑r

i=1 vie>ì, ε ∈ (0, 1)



subgraph (strong) connectedness







Sì (Aeì +Bvi) ≥ 0 (5)

1>nBvi < 0 (6)


K := ε∑r

i=1 vie>ì, ε ∈ (0, 1)


Every compartment is outflow connected!




Remarks

Let r be the number of non-Hurwitz communication classes.

The previous proposition provides only a sufficient (but notnecessary) condition for the problem solvability.

If the feedbackstabilization problem is solvable, the minimum number of nonzerocolumns of any solution K is at least r but it might be greater.

Algorithm that allows to assess problem solvability andprovides a solution whenever it exists.

In the Single-Input case (m = 1) a necessary condition for theproblem solvability is that r = 1.




Remarks

Let r be the number of non-Hurwitz communication classes.

The previous proposition provides only a sufficient (but notnecessary) condition for the problem solvability. If the feedbackstabilization problem is solvable, the minimum number of nonzerocolumns of any solution K is at least r but it might be greater.

Algorithm that allows to assess problem solvability andprovides a solution whenever it exists.

In the Single-Input case (m = 1) a necessary condition for theproblem solvability is that r = 1.



Conclusions

Positive Systems provide an effective mathematical description ofa variety of real phenomena/processes.

Positivity constraint makes it possible to tackle many controlissues by resorting to ad hoc tools and techniques that in thegeneral case cannot be employed.

Think positive, be positive!



Thank you!

Prof. Maria Elena Valcher

Prof. Anders Rantzer


Stability, Stabilizability and Control of certain classes ...

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