A general necessary and sufficient condition for ...

A necessary condition for controllability of a networkKnown results for LTI networks with static linear couplings

A necessary and sufficient condition for LTI networksA little counting game

A general necessary and sufficient condition forcontrollability of networks of linear systems

Jochen Trumpf (joint work with Harry Trentelman)

in memory of Uwe Helmke

Sde Boker

March 2017

J. Trumpf, H.L. Trentelman Network controllability



Outline

1 A necessary condition for controllability of a network

2 Known results for LTI networks with static linear couplings

3 A necessary and sufficient condition for LTI networks

4 A little counting game




A network

Controllability: For all xs =

(x (1)

x (2)

)s

, xf =

(x (1)

x (2)

)f

and ts

there exist tf and u1|[ts ,tf ] such that x(tf ; xs , ts , u1) = xf .




A network and one of its internal signals


(x (1)

x (2)

)s

, xf =

(x (1)

x (2)

)f

and ts





A network and one of its node systems


(x (1)

x (2)

)s

, xf =

(x (1)

x (2)

)f

and ts





A first result

Theorem: If an i/o-coupled network of i/s/o systems iscontrollable then each of the node systems is controllable.

Remark: This does not quite follow from the fact thatcontrollability is hereditary under behavior projection.




LTI networks with static linear couplings

Node systems (i = 1, . . . ,N)

x (i) = α(i)x (i) + β(i)v (i), x (i) ∈ Rni , v (i) ∈ Rmi

w (i) = γ(i)x (i), w (i) ∈ Rpi

Static linear couplings with external input [and output]

v (i) =∑j

Aijw(j) + Biu, u ∈ Rm

y =∑i

Ciw(i), y ∈ Rp




LTI networks with static linear couplings

Node systems (i = 1, . . . ,N)

x (i) = α(i)x (i) + β(i)v (i), x (i) ∈ Rni , v (i) ∈ Rmi

w (i) = γ(i)x (i), w (i) ∈ Rpi

Static linear couplings with external input [and output]

v (i) =∑j

Aijw(j) + Biu, u ∈ Rm

Network system (full behavior B(x ,u,w ,v))

x = (α + βAγ) x + βBu,

w = γx

v = Aw + BuJ. Trumpf, H.L. Trentelman Network controllability



The homogeneous SISO case

Networkx = (α + βAγ) x + βBu

Homogeneous network

α(i) = α(0), β(i) = β(0), γ(i) = γ(0), i = 1, . . . ,N

α = I ⊗ α(0), β = I ⊗ β(0), γ = I ⊗ γ(0)

SISO network: single input single output agents[not necessarily single integrator]

Theorem: [Hara et al., 2007] Let rank(B) < N. Then thehomogeneous SISO network is controllable if and only if

1. The node dynamics is controllable and observable

2. The matrix pair (A,B) is controllable




The Fuhrmann-Helmke test

Networkx = (α + βAγ) x + βBu

Left coprime factorizationD(s)−1N(s) = γ (sI − α)−1 β

Theorem: [Fuhrmann/Helmke, 2014] Let the node systems becontrollable and observable. Then the network is controllable ifand only if (

D(s)−N(s)A −N(s)B)

is left prime.




A behavioral proof of the Fuhrmann-Helmke test



w = γx

v = Aw + Bu

Node systems (open network), left coprime factorizationw = D(s)−1N(s)v = γ (sI − α)−1 βv

Rearranging

D(s)w = N(s)(Aw +Bu) ⇔(D(s)−N(s)A −N(s)B

)(wu

)= 0

Fuhrmann-Helmke test ⇔ B(u,w) controllable




A behavioral proof of the Fuhrmann-Helmke test



w = γx

v = Aw + Bu

Node systems observable ⇒ x observable from (w , v) ⇒x observable from (u,w)

Observable i/s/o system:B(u,w) controllable ⇒ B(x ,u) controllable

[⇒B(x ,u,w ,v) controllable

]Conversely:B(x ,u,w ,v) controllable ⇒ B(u,w) controllable




A convenient network representation



w = γx

v = Aw + Bu

Node systems (open network)(α11 α12

0 α22

) (β1

β2

)(

0 γ2

)(γ2, α22) observable

l.c.f. D(s)−1N(s) = γ2 (sI − α22)−1

Playing with the system variables

γ =

(I 00 γ2

), A =

(0 A

), B = B




kernel representations are magical!




A kernel representation of the network i/o behavior


0 α22

) (β1

β2

)(

0 γ2

)γ =

(I 00 γ2

), w1 = x1

A =(0 A

), B = B

(γ2, α22) observable

l.c.f. D(s)−1N(s) = γ2 (sI − α22)−1

(X (s) Y (s)

)(sI − α22

γ2

)= I

Kernel representation of the (augmented) network i/o behavior

B(w1,w2,u) = Ker(

sI−α11 −α12(Y (s)+X (s)β2A)−β1A −α12X (s)β2B−β1B0 −D(s)+N(s)β2A N(s)β2B

)J. Trumpf, H.L. Trentelman Network controllability



The main result


0 α22

) (β1

β2

)(

0 γ2

)Network (closed loop)


w = γx

v = Aw + Bu

l.c.f. D(s)−1N(s) = γ2 (sI − α22)−1

(X (s) Y (s)

)(sI − α22

γ2

)= I

Theorem: [Generalized Fuhrmann-Helmke test] The network iscontrollable if and only if(sI − α11 −α12 (Y (s) + X (s)β2A)− β1A −α12X (s)β2B − β1B

0 −D(s) + N(s)β2A N(s)β2B

)is left prime.




Homogeneous networks

Node system (open network)

α11 = I ⊗ α(0)11

(sI − α11 −α12 (Y (s) + X (s)β2A)− β1A −α12X (s)β2B − β1B

0 −D(s) + N(s)β2A N(s)β2B

)

Counting rank

column rank ≤ N · rank(sI − α(0)11 ) +

∑i pi + rank(B)

number of rows = N · dim(x(0)1 ) +

∑i pi




Homogeneous networks

Let rank(B) < N and s ∈ σ(α(0)11 ).

Counting rank

column rank ≤ N · rank(sI − α(0)11 ) +

∑i pi + rank(B)

< N ·(dim(x

(0)1 )− 1

)+∑

i pi + N

number of rows = N · dim(x(0)1 ) +

∑i pi

Theorem: Let rank(B) < N. If the network is homogeneous andcontrollable then the node dynamics is observable.




Thank you.


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