[PPT] - ON NETWORKS OF INTERCONNECTED SYSTEMS Paul A. Fuhrmann Ben-Gurion PowerPoint Presentation

SLIDE 1

ON NETWORKS OF INTERCONNECTED SYSTEMS

Paul A. Fuhrmann Ben-Gurion University of the Negev (Joint work with U. Helmke) xi(t + 1) = αixi(t) + βivi(t) wi(t) = γixi(t), i = 1, . . . , N. vi(t) = N

j=1 Aijwj(t) + Biu(t)

y(t) = N

i=1 Ciwi(t) + Du(t)

Ben Gurion University, May 27, 2012 – p. 1/37

SLIDE 2

MOTIVATION

Study controllability and observability

properties of networks of, homogeneous or nonhomogeneous, node linear systems.

Allow flexibility in the nature (discrete time,

continuous time) and representation (first

rder, high order) of both the node systems as

well as the interconnections (constant, dynamic).

Checking controllability by use of the Kalman

controllability and observability matrices is

impractical. These we replace by compact

coprimeness conditions.

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SLIDE 3

EXAMPLES

Formation Control of Autonomous Distributed
Systems. (Common Dynamics), Flocks of

Birds.

Traffic Control; Truck Convoys.
(Interconnecting) Biological Systems; Cicadas.
Synchronization Phenomena; Ciliary Beating.
Control of Electrical Networks (Grids).

Ben Gurion University, May 27, 2012 – p. 3/37

SLIDE 4

DIFFICULTIES

Linearity is too restrictive.
(Ballistic) Controllability may not be the

correct concept.

Study self organizing systems (organizations).

A variational approach? What do we optimize? Efficiency? What is it?

Ben Gurion University, May 27, 2012 – p. 4/37

SLIDE 5

REFERENCES

P.A. Fuhrmann and U. Helmke, ”Strict equivalence, controllability and observability of networks of linear systems”, submitted to Mathematics of Control, Signals and Systems. P.A. Fuhrmann, “On strict system equivalence and similarity”, Int. J. Contr., 25, 5-10, (1977). H.H. Rosenbrock and A.C. Pugh, “Contributions to a hierarchical theory of systems”, Int. J. Contr., 19, 845-867, (1974). F.M. Callier, and C.D. Nahum, “Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime decomposition of a rational matrix”, IEEE Trans. Circuits and System,

Ben Gurion University, May 27, 2012 – p. 5/37

SLIDE 6

FROM POLYNOMIALS TO MODEL SPACES

Polynomial Arithmetic: D(z) ∈ F[z]m×m, det D(z) = 0 Geometry: Polynomial Model:      πD : F[z]m − → F[z]m πDf = Dπ−D−1f XD = Im πD ≃ F[z]m/D(z)F[z]m Rational Model:      D(σ) : z−1F[[z−1]]m − → z−1F[[z−1]]m D(σ)h = π−Dh, πDh = π−D−1π+Dh Ker D(σ) = Im πD = XD ≃ XD

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SLIDE 7

FACTORIZATIONS AND INVARIANT SUBSPACES

D(z) ∈ F[z]m×m nonsingular. V ⊂ XD&SDV ⊂ V ⇔ V = D1XD2 V ⊂ XD&SDV ⊂ V ⇔ V = XD2 D(z) = D1(z)D2(z), D1(z), D2(z) ∈ F[z]m×m CONNECTION BETWEEN ALGEBRA AND GEOMETRY DIRECT LINK TO GEOMETRIC CONTROL AND BEHAVIORS

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SLIDE 8

MODEL HOMOMORPHISMS

Let D1(z) ∈ F[z]m×m and D2(z) ∈ F[z]p×p be nonsingular. SD = σ|XD Z : XD1 − → XD2 is an F[z]-homomorphism, i.e. ZSD1 = SD2Z if and only if there exist N1(z), N2(z) ∈ F[z]p×m such that N2(z)D1(z) = D2(z)N1(z) Zh = π−N1h = N1(σ)h. INVERTIBILITY AND COPRIMENESS

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SLIDE 9

THE SHIFT REALIZATION

G(z) = V (z)T(z)−1U(z) + W(z) =   A B C D  

       A = ST Bξ = πTUξ, Cf = (V T −1f)−1 D = G(∞).

CAj−1Bξ = (V T −1πT zj−1πT Uξ)−1 = (V π−T −1zj−1Uξ)−1 = (zj−1(V T −1U + W)ξ)−1

Realization is reachable ⇔ T(z) and U(z) left coprime. Realization is observable ⇔ T(z) and V (z) right coprime.

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SLIDE 10

SYSTEM EQUIVALENCE

Rosenbrock, Fuhrmann G(z) = Vi(z)Ti(z)−1Ui(z) + Wi(z), i = 1, 2 (no coprimeness assumptions); Σi the associated shift realizations Pi = Ti(z) −Ui(z) Vi(z) Wi(z)

P1 ≃FSE P2 if Σ1 ≃ Σ2

P1 ≃FSE P2 ⇔ ∃M(z), X(z), N(z), Y (z), such that M(z) ∧L T2(z) = I, & N(z) ∧R T1(z) = I

  M(z) X(z) I     T1(z) −U1(z) V1(z) W1(z)   =   T2(z) −U2(z) V2(z) W2(z)     N(z) Y (z) I  

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SLIDE 11

NODE REPRESENTATIONS

Gi(z) = γi(zI − αi)−1βi = Qi(z)−1Pi(z) = P i(z)Qi(z)−1 = Vi(z)Ti(z)−1Ui(z) + Wi(z). Assume all representations are minimal, i.e., δ(Gi) = deg det Qi = deg det Qi = deg det Ti. This is equivalent to the controllability and

bservability of the shift realizations associated

with these representations, i.e., to the left coprimeness of Qi(z), Pi(z), the right coprimeness

f P i(z), Qi(z), the left coprimeness of Ti(z), Ui(z)

and the right coprimeness of Ti(z), Vi(z).

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SLIDE 12

INTERCONNECION MODEL

xi(t + 1) = αixi(t) + βivi(t) wi(t) = γixi(t), i = 1, . . . , N. vi(t) = N

j=1 Aijwj(t) + Biu(t)

y(t) = N

i=1 Ciwi(t) + Du(t)

x(t + 1) = αx(t) + βv(t) w(t) = γx(t) v(t) = Aw(t) + Bu(t) y(t) = Cw(t) + Du(t) x(t + 1) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) A := α + βAγ, B := βB, C := Cγ, D := D.

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SLIDE 13

Node transfer function G(z) := diag (G1(z), . . . , GN(z)) = γ(zI − α)−1β. Interconnection transfer function ϕ(z) = C(zI − A)−1B + D. Global network transfer function Φ(z) = C(zI − A)−1B + D; = Cγ(zI − α − βAγ)−1βB + D.

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SLIDE 14

MATRIX FRACTION DESCRIPTIONS (MFD)

Qi(σ)ξi = Pi(σ)vi wi = ξi vi(t) = N

j=1 Aijwj(t) + Biu(t)

y(t) = N

i=1 Ciwi(t) + Du(t)

Φ(z) = C(Q(z) − P(z)A)−1P(z)B + D. Here Q(z) = diag (Q1(z), . . . , QN(z)), etc.

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SLIDE 15

POLYNOMIAL MATRIX DESCRIPTION (PMD)

Ti(σ)ξi = Ui(σ)vi wi = Vi(σ)ξi + Wi(σ)vi vi(t) = N

j=1 Aijwj(t) + Biu(t)

y(t) = N

i=1 Ciwi(t) + Du(t)

For the case W(z) = 0: Φ(z) = CV (z)(T(z) − U(z)AV (z))−1U(z)B + D. Here T(z) = diag (T1(z), . . . , TN(z)), etc.

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SLIDE 16

EQUIVALENCE UNDER INTERCONNECTION

Given N FSE node systems in PMD form

T (ν)

i

(z) −U (ν)

i

(z) V (ν)

i

(z) W (ν)

i

(z)

, ν = 1, 2; i = 1, . . . , N.

Dynamic interconnection:

E(σ)v

= A(σ)w + B(σ)u y = C(σ)w + Du.        T (1) −U(1) V (1) W (1) −I E −A −B C D        ≃F SE        T (2) −U(2) V (2) W (2) −I E −A −B C D       

Ben Gurion University, May 27, 2012 – p. 16/37

SLIDE 17

STANDARD INTERCONNECTIONS

SERIES CONNECTION

A =

I
, B =
I
, C =
I
PARALLEL CONNECTION

A =

, B =
I

I

, C =
I

I

FEEDBACK CONNECTION

A =

I

I

, B =
I
, C =
I
Ben Gurion University, May 27, 2012 – p. 17/37

SLIDE 18

SERIES CONNECTION

Given systems Σi, i = 1, 2, with transfer functions Gi(z) = Ai Bi Ci Di

.

G1(z) ∈ F(z)k×m, G2(z) ∈ F(z)p×k. The series coupling Σ1 ∧ Σ2, is defined by:               

x(1)

t+1

= A1x(1)

t

+ B1ut y′

t

= C1x(1)

t

+ D1ut x(2)

t+1

= A2x(2)

t

+ B2u′

t

yt = C2x(2)

t

+ D2u′

t

u′

t

= y′

t.

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SLIDE 19

Schematically, we have the following diagram: G2 G1

✲ ✲ ✲

u y G2(z)G1(z) = A2 B2 C2 D2

×

A1 B1 C1 D1

=

  A1 B1 B2C1 A2 B2D1 D2C1 C1 D2D1   . Controllability and observability are difficult to deduce from this representation.

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SLIDE 20

SERIES CONNECTION AND COPRIMENESS

Let G1(z) ∈ F(z)k×m, G2(z) ∈ F(z)p×k have the

coprime matrix fraction representations G1(z) = DL1(z)−1NL1(z) = NR1(z)DR1(z)−1 G2(z) = DL2(z)−1NL2(z) = NR2(z)DR2(z)−1 Then δ(G2G1) ≤ δ(G1) + δ(G2).

The series coupling of the shift realizations

associated with NR2(z)DR2(z)−1 and NR1(z)DR1(z)−1 is controllable if and only if NR1(z) and DR2(z) are left coprime.

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SLIDE 21

SERIES COUPLING CONTINUED

The series coupling of the shift realizations

associated with DL2(z)−1NL2(z) and DL1(z)−1NL1(z) is observable if and only if NL2(z) and DL1(z) are right coprime.

We have the equality

δ(G2G1) = δ(G1) + δ(G2) if and only if NL2(z), DL1(z) are right coprime and NR1(z), DR2(z) are left coprime.

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SLIDE 22

SERIES COUPLING N SYSTEMS

Given the node systems Σi, i = 1, . . . , N. We define, inductively, N1,...,i(z) = Ni(z)N 1,...,i−1(z) D1,...,i(z) = D1,...,i−1(z)Di(z) Then a necessary and sufficient for Σ1 ∧ · · · ∧ ΣN to be controllable is that one of the following equivalent conditions is satisfied:

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SLIDE 23

N SYSTEMS CONTINUED

1. The polynomial matrix

    

−NR1(z) DR2(z) . . . . −NR(N−1)(z) DRN(z)

     is left prime.

2. For all i = 2, . . . , N we have the left

coprimeness of the polynomial matrices N1,...,i−1(z) and Di(z).

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SLIDE 24

PARALLEL CONNECTION

Given systems Σi, i = 1, 2, with transfer functions Gi(z) = Ai Bi Ci Di

∈ F[z]p×m, state spaces Xi,

i = 1, 2, . The parallel coupling, Σ1 ∨ Σ2, is defined by:                x(1)

t+1 = A1x(1) t

+ B1ut y(1)

t

= C1x(1)

t

+ D1ut x(2)

t+1 = A2x(2) t

+ B2ut y(2)

t

= C2x(2)

t

+ D2ut yt = y(1)

t

+ y(2)

t .

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SLIDE 25

Schematically, we have the following diagram G2 G1

✲ ✲ ✲ ✲ ✲ ✲

u y

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SLIDE 26

PARALLEL CONNECTION AND COPRIMENESS

G1(z) = NR1(z)DR1(z)−1 = DL1(z)−1NL1(z) G2(z) = NR2(z)DR2(z)−1 = DL2(z)−1NL2(z)

TFSAE:
1. The parallel coupling of the shift

realizations associated with NR2(z)DR2(z)−1 and NR1(z)DR1(z)−1 is controllable.

2. DR1(z), DR2(z) are left coprime.

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SLIDE 27

PARALLEL COUPLING CONTINUED

TFSAE:
1. The parallel coupling of the shift

realizations associated with DL2(z)−1NL2(z) and DL1(z)−1NL1(z) is

bservable.
2. DL1(z), DL2(z) are right coprime.
TFSAE:
1. The parallel coupling of the shift

realizations associated with the matrix fraction representations is minimal.

2. DL1(z), DL2(z) are right coprime and

DR1(z), DR2(z) are left coprime.

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SLIDE 28

PARALLEL COUPLING N SYSTEMS

Given nonsingular polynomial matrices Di(z) ∈ F[z]m×m are mutually left coprime if for each i, Di(z) is left coprime with Dµi = l.c.r.m.{Dj}j=i, the unique, up to a right unimodular factor, least common right multiple of all {Dj(z)}j=i. Given the node systems Σi, i = 1, . . . , N with Gi(z) = DLi(z)−1NLi(z) = NRi(z)DRi(z)−1. Σ1 ∨ · · · ∨ Σi+1 := (Σ1 ∨ · · · ∨ Σi) ∨ Σi+1. Then a necessary and sufficient for Σ1 ∨ · · · ∨ ΣN to be controllable is that the DRi(z) are mutually left coprime.

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SLIDE 29

MUTUAL RIGHT COPRIMENESS

Given nonsingular polynomial matrices Di(z) ∈ F[z]m×m, i = 1, . . . , s, then the Di(z) are mutually left coprime if and only if the following matrix     

−D1(z) D2(z) . . . . −Ds−1(z) Ds(z)

     . is left prime. A similar result holds for mutual right coprimeness.

Ben Gurion University, May 27, 2012 – p. 29/37

SLIDE 30

FEEDBACK CONNECTION

G(e1 + y2) = y1 K(e2 + y1) = y2. Schematically, we have the following diagram: G K

✲

y2 e2 y1 e1

✲ ✲ ✛ ✻ ❄ ✛ ✛ ✛ ✲

Ben Gurion University, May 27, 2012 – p. 30/37

SLIDE 31

FEEDBACK CONNECTION CONTINUED

G(z) ∈ F[z]p×m, K(z) ∈ F[z]m×p, proper and (I − G(z)K(z)) properly invertible and the following factorizations are coprime:

G(z) = Q(z)−1P(z) = P(z)Q(z)−1 K(z) = S(z)−1R(z) = R(z)S(z)−1. Φ =   I −G −K I  

−1 

 G K   =   G K     I −K −G I  

−

=   (I − GK)−1G G(I − KG)−1K (I − KG)−1KG (I − KG)−1K   .

Ben Gurion University, May 27, 2012 – p. 31/37

SLIDE 32

FEEDBACK

Φ(z) =

Q

−P −R S

−1

P R

=
P

R Q −R −P S

−1

Gf (z) =

I
Φ(z)

  I   = (I − G(z)K(z))−1G(z) = G(z)(I − K(z)G(z))−1 = S(z)(Q(z)S(z) − P(z)R(z))−1P(z) = P(z)(S(z)Q(z) − R(z)P(z))−1S(z) =

P


 Q −R −P S  

−1 

 I   =

I


 Q −P −R S  

−1 

 P   det   Q −P −R S   = det(QS − PR) = det(SQ − RP) = det   Q −R −P S  

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SLIDE 33

EQUIVALENCE

    Q −R −I −P S P     ≃FSE     Q −P −P −R S I     ≃FSE   QS − PR −P I  

I

I QS − PR −P I

=
Q

−P −P −R S I S R I

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SLIDE 34

REACHABILITY & OBSERVABILITY

TFSAE:

1. The shift realization associated with any of the

above representations is reachable.

2. P(z), S(z) are left coprime.
3. Q(z)S(z), P(z) are left coprime.
4. R(z)P(z), S(z) are left coprime.

TFSAE:

1. The shift realization associated with any of the

above representations is observable.

2. P(z), S(z) are right coprime.
3. S(z)Q(z), P(z) are right coprime.
4. P(z)R(z), S(z) are right coprime.

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SLIDE 35

FEEDBACK BY DYNAMIC COUPLING

Node: Qw = Pη + Pu Coupling: Sη = Rw y = w   I   y =   Q −P −P −R S I     w η u  

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SLIDE 36

SCALAR HOMOGENEOUS NETWORKS

ϕ(z) = C(zI − A)−1B + D G(z) = Q(z)−1P(z) Φ(z) = C(Q(z) − P(z)A)−1P(z)B + D G(z) = q(z)−1p(z)I h(z) = p(z)−1q(z) Φ(z) = C(Q(z) − P(z)A)−1P(z)B + D = C(q(z)I − p(z)A)−1p(z)B + D = C(p(z)−1q(z)I − A)−1B + D = (ϕ ◦ h)(z)

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SLIDE 37

SCALAR HOMOGENEOUS NETWORKS

THEOREM [Hara, Hayakawa and Sugata]: The shift realization (A, B, C) of Φ(z) is controllable (observable) if and only the realization (A, B, C) of the interconnection transfer function ϕ(z) is controllable (observable). In particular, controllability of (A, B, C) is independent of the choice of the node transfer function g(z), as long as g(z) is scalar rational and strictly proper.

q(z)I − p(z)A −p(z)B
left prime

⇔

zI − A −B
left prime

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