Decentralized and distributed control - Dynamic feedback ...sisdin.unipv.it/.../EECI_DEDICO/...controllers_Dynamic_Feedback.pdf · Decentralized and distributed control Dynamic feedback

Decentralized and distributed controlDynamic feedback control

M. Farina1 G. Ferrari Trecate2

1Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB)Politecnico di Milano, [email protected]

2Dipartimento di Ingegneria Industriale e dell’Informazione (DIII)Universita degli Studi di Pavia, Italy

[email protected]

EECI-HYCON2 Graduate School on Control 2015Supelec, France

Farina, Ferrari Trecate Decentralized and distributed control EECI-HYCON2 School 2015 1 / 46

[email protected]

[email protected]

Outline

1 Decentralized dynamic output feedbackPole placement through a specific channelA general method

2 Distributed dynamic output-feedback: a procedure based on DeDOf

3 Summary of methods for De/Di control not covered in this course


Outline





Outline





Decentralized dynamic output feedback



MIMO System

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N

N input/output channels

General MIMO system withB =

[B1 · · · BN

]

C = col(C1, . . . ,CN)

Decentralized dynamic output feedback (DeDOf)

𝒞i :{xri = Fixri + Giyi

ui = −Kxixri − Kyiyi

Controller 𝒞i connected tochannel i only

xri (t) ∈ Rnri state ofcontroller i

No setpoints: we will focus on stabilization of the origin of the closed-loopsystem



MIMO System

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N



[B1 · · · BN

]

C = col(C1, . . . ,CN)






No setpoints: we will focus on stabilization of the origin of the closed-loopsystem


Existence of DeDOf

Theorem (review)

Let Λdf be the set of DFMs of system Σ. Then

(i) There is a DeDOf that stabilizes Σ iff all 𝜆 ∈ Λdf havestrictly negative real part

(ii) There is a DeDOf such that the closed-loop system has aprescribed symmetric set of eigenvalues iff Λdf = ∅

Next: two methods for designing DeDOf

Pole placement through a specific channel (Corfmat & Morse, 1976)

A general procedure (Davison & Chang, 1990)


Existence of DeDOf

Theorem (review)

Let Λdf be the set of DFMs of system Σ. Then

(i) There is a DeDOf that stabilizes Σ iff all 𝜆 ∈ Λdf havestrictly negative real part

(ii) There is a DeDOf such that the closed-loop system has aprescribed symmetric set of eigenvalues iff Λdf = ∅

Next: two methods for designing DeDOf

Pole placement through a specific channel (Corfmat & Morse, 1976)

A general procedure (Davison & Chang, 1990)


DeDOf: pole placement through aspecific channel


DeDOf: pole placement through a specific channel

Goal

Assign prescribed eigenvalues using the controllers

ui = −Kyiyi , i = 1 : N, i = k (1)

𝒞k :

{xrk = Fkxrk + Gkyk

uk = −Kxkxrk − Kykyk(2)

Requirements

(a) N − 1 static controllers in (1) for making the closed-loopsystem

x =

⎛⎝A−

∑

i=1:N,i =k

BiKyiCi

⎞⎠ x + Bkuk

yk = Ckx

controllable and observable

(b) a dynamic controller on channel k that assigns all eigenvalues



Goal

Assign prescribed eigenvalues using the controllers

ui = −Kyiyi , i = 1 : N, i = k (1)

𝒞k :

{xrk = Fkxrk + Gkyk

uk = −Kxkxrk − Kykyk(2)

Requirements

(a) N − 1 static controllers in (1) for making the closed-loopsystem

x =

⎛⎝A−

∑

i=1:N,i =k

BiKyiCi

⎞⎠ x + Bkuk

yk = Ckx

controllable and observable

(b) a dynamic controller on channel k that assigns all eigenvaluesFarina, Ferrari Trecate Decentralized and distributed control EECI-HYCON2 School 2015 7 / 46


Remark

When requirement (a) is fulfilled one can use a standard CeDOf forfulfilling requirement (b)

Problem

When static controllers fulfilling requirement (a) exist ?



Remark

When requirement (a) is fulfilled one can use a standard CeDOf forfulfilling requirement (b)

Problem

When static controllers fulfilling requirement (a) exist ?



DFM revised

if 𝜆 ∈ 𝜎(A) is not a DFM, there is a partition 𝒟 and ℋ of {1, . . . ,N} suchthat 𝜆 can be made controllable and observable either from a singlechannel i ∈ ℋ, after closing suitable static feedbacks at channels i ∈ 𝒟, orfrom a single channel i ∈ 𝒟, after closing suitable static feedbacks atchannels i ∈ ℋ.

Issues

𝒟 and ℋ depend upon 𝜆

Even if the same partition 𝒟 and ℋ works for the non-DFM 𝜆1 and𝜆2, different feedbacks/channels might be needed.

Beside the absence of DFM, additional assumptions are needed foravoiding these issues



DFM revised

if 𝜆 ∈ 𝜎(A) is not a DFM, there is a partition 𝒟 and ℋ of {1, . . . ,N} suchthat 𝜆 can be made controllable and observable either from a singlechannel i ∈ ℋ, after closing suitable static feedbacks at channels i ∈ 𝒟, orfrom a single channel i ∈ 𝒟, after closing suitable static feedbacks atchannels i ∈ ℋ.

Issues

𝒟 and ℋ depend upon 𝜆

Even if the same partition 𝒟 and ℋ works for the non-DFM 𝜆1 and𝜆2, different feedbacks/channels might be needed.

Beside the absence of DFM, additional assumptions are needed foravoiding these issues



Theorem

Assume Σ is controllable and observable. For an arbitrary k ∈ 1 : N thereexist matrices Ki , i = k such that requirement (a) is fulfilled iff for allpartitions (𝒟,ℋ) of 1 : N a at least one of the following conditions isverified

(A,B𝒟) is controllable

(A,Cℋ) is observable

aHere sets 𝒟 and ℋ are not related to the controllability/observabilityproperties of channels.



Design algorithm

Given Σ verifying the assumptions of the above theorem

1 Choose k and static output feedbacks on channels i = k such thatthe system Σcl ,k obtained by closing these loops is observable andcontrollable from channel k

2 Design a CeDOf for channel k that stabilizes Σcl ,k .



Drawbacks

If the design of the dynamic controller is based on observers, theorder of the controller is equal to the order of the system (that can bevery high)

If some modes are weakly controllable/observable form channel k ,large controller gains are required

Disturbances need to propagate up to channel k before gettingattenuated by the dynamic controller



Drawbacks






Drawbacks






Assumptions of the theorem require more than the absence of DFM

Σ :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣1 01 00 1

⎤⎦[u1u2

]

[y1y2

]=

[1 2 00 0 1

]x

Σ has no DFM. However, for 𝒟 = {1} and ℋ = {2} the system(A,B1,C2), i.e.

Σ :

⎧⎪⎪⎨⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣110

⎤⎦ u1

y2 =[0 0 1

]x

is neither controllable nor observable. Then Σ cannot be madecontrollable and observable form a single channel.



Assumptions of the theorem require more than the absence of DFM

Σ :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣1 01 00 1

⎤⎦[u1u2

]

[y1y2

]=

[1 2 00 0 1

]x

Σ has no DFM. However, for 𝒟 = {1} and ℋ = {2} the system(A,B1,C2), i.e.

Σ :

⎧⎪⎪⎨⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣110

⎤⎦ u1

y2 =[0 0 1

]x

is neither controllable nor observable. Then Σ cannot be madecontrollable and observable form a single channel.



For instance, for u2 = −ky2y2, Σ becomes

Σ :

⎧⎪⎪⎨⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3− ky2

⎤⎦ x +

⎡⎣110

⎤⎦ u1

y1 =[1 2 0

]x

that is never controllable and observable. However, Σ can be stabilizedusing a suitable DeDOf (see later ...)


DeDOf: a general method



MIMO system

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N


general MIMO system withB =

[B1 · · · BN

]

C = col(C1, . . . ,CN)






Main idea

Sequential design of controllers 𝒞i . At each step one assigns as many aseigenvalues as possible



MIMO system

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N



[B1 · · · BN

]

C = col(C1, . . . ,CN)






Main idea




MIMO system

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N



[B1 · · · BN

]

C = col(C1, . . . ,CN)






Main idea




Let B i =[B1 . . . Bi

]and C i = col(C1, . . . ,Ci )

Theorem

If the DFM of Σ are all in the lhp, a DeDOf that stabilizes the system canbe computed according to the following algorithm

1 Design a CeDOf 𝒞1 for Σ1 = (A,B1,C 1) such that all eigenvalues ofthe closed-loop system are in the lhp, except for the CFMs of Σ1

2 for i = 2, . . . ,NI Let Σi be the system with input/output ui/yi obtained applying

regulators 𝒞1, . . . 𝒞i−1 to system Σi,ol = (A,B i ,C i ). Design a CeDOf 𝒞ifor Σi such that all eigenvalues of the closed-loop system are in thelhp, except for the CFMs of Σi,ol . This is possible for almost all Kxj ,Kyj , j = 1, . . . , i − 1.



Example: input decoupled system

System Σ

x1⌃1

⌃2

x2

x3⌃3

u1

u2

u3

y1

y2

y3 System Σ1 Design of 𝒞1




System Σ

x1⌃1

⌃2

x2

x3⌃3

u1

u2

u3

y1

y2

y3

System Σ1

x1⌃1

⌃2

x2

x3⌃3

u1 y1

Design of 𝒞1




System Σ

x1⌃1

⌃2

x2

x3⌃3

u1

u2

u3

y1

y2

y3

System Σ1

x1⌃1

⌃2

x2

x3⌃3

u1 y1

Design of 𝒞1

x1⌃1

⌃2

x2

x3⌃3

C1

u1 y1




System Σ2

x1⌃1

⌃2

x2

x3⌃3

C1

u2 y2

Design of 𝒞2 System Σ3




System Σ2

x1⌃1

⌃2

x2

x3⌃3

C1

u2 y2

Design of 𝒞2

x1⌃1

⌃2

x2

x3⌃3

C1

C2

u2 y2

System Σ3




System Σ2

x1⌃1

⌃2

x2

x3⌃3

C1

u2 y2

Design of 𝒞2

x1⌃1

⌃2

x2

x3⌃3

C1

C2

u2 y2

System Σ3

x1⌃1

⌃2

x2

x3⌃3

C1

C2

u3y3




Design of 𝒞3 and final system

x1⌃1

⌃2

x2

x3⌃3

C1

C2

C3

u3y3



Remarks

Pros:

At each step an observer-based CeDOf can be designed. Assigning alleigenvalues that are not CFM of Σi , at the end of the algorithmallows one to assign all eigenvalues that are not DFM of Σ

The theorem can be generalized to the case

yi = Cix +N∑

j=1

Dijuj



Remarks

Pros:

At each step an observer-based CeDOf can be designed. Assigning alleigenvalues that are not CFM of Σi , at the end of the algorithmallows one to assign all eigenvalues that are not DFM of Σ

The theorem can be generalized to the case

yi = Cix +N∑

j=1

Dijuj



Remarks

Cons:

The order of systems Σi is non-decreasing. It might happen that theorder of the final closed-loop system is very high

I However, there are theoretical results guaranteeing that stabilizationcan be achieved with controllers whose order sum up at most to n

The complexity of the controller on channel i depends on the channelordering



Remarks

Cons:

The order of systems Σi is non-decreasing. It might happen that theorder of the final closed-loop system is very high

I However, there are theoretical results guaranteeing that stabilizationcan be achieved with controllers whose order sum up at most to n

The complexity of the controller on channel i depends on the channelordering



Example

Σ :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣1 01 00 1

⎤⎦[u1u2

]

[y1y2

]=

[1 2 00 0 1

]x

We have seen before that this system cannot be stabilized through aspecific channel but it can be stabilized by suitable DeDOf that wecompute

using the general method

designing at each step an observer-based CeDOf (eigenvalues havebeen chosen randomly in the lhp)



Example

Σ :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x =

⎡⎣0 2 30 −2 40 0 3

⎤⎦ x +

⎡⎣1 01 00 1

⎤⎦[u1u2

]

[y1y2

]=

[1 2 00 0 1

]x

Order of the resulting closed-loop system

Step 1 : controller 𝒞1 on channel (u1, y1) will have order 3.Therefore Σ2 will have order 6.

Step 2 : controller 𝒞2 on channel (u2, y2) will have order 6.Therefore the final closed-loop system will have order3 + 3 + 6 = 12



Whole decentralized controller

𝒞 :

{xr = Fxr + Gy

u = −KxxrF = diag(F1,F2)

G = diag(G1,G2)

H = diag(H1,H2)




𝒞 :

{xr = Fxr + Gy


G = diag(G1,G2)

H = diag(H1,H2)

F =⎡⎢⎢⎢⎢⎣

-286.73 178.56 2163.25 0.00 0.00 0.00 0.00 0.00 0.00-287.61 172.80 2164.25 0.00 0.00 0.00 0.00 0.00 0.00-7.90 -15.81 3.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 2.00 -39.76 -282.33 185.36 2160.250.00 0.00 0.00 0.00 -2.00 -74.29 -282.33 185.36 2160.250.00 0.00 0.00 25655.08 -20551.94 -132.51 -25664.38 20558.28 -4458.570.00 0.00 0.00 4.40 8.79 -39.78 -286.73 178.56 2163.250.00 0.00 0.00 5.28 10.56 -74.21 -287.61 172.80 2164.250.00 0.00 0.00 7.90 15.81 0.15 -7.90 -15.81 3.00

⎤⎥⎥⎥⎥⎦




𝒞 :

{xr = Fxr + Gy


G = diag(G1,G2)

H = diag(H1,H2)

G =

⎡⎢⎢⎢⎢⎣

4.40 0.005.28 0.007.90 0.000.00 42.760.00 78.290.00 10.990.00 39.780.00 74.210.00 -0.15

⎤⎥⎥⎥⎥⎦




𝒞 :

{xr = Fxr + Gy


G = diag(G1,G2)

H = diag(H1,H2)

Kx =[-282.33 185.36 2160.25 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 25655.08 -20551.94 -124.53 -25664.38 20558.28 -4458.57

]

Closed-loop eigenvalues:-85.81 -77.04 -54.46 -1.38 -5.08 -37.08 -32.39 -13.16 -7.49 -19.82 -13.66 -7.99


Distributed dynamic output feedback


DiDOf based on DeDOf

MIMO system

Σ :

⎧⎪⎪⎨⎪⎪⎩

x = Ax +N∑

i=1

Biui

yi = Cix , i = 1 : N

N input/output channels.ui ∈ Rmi , yi ∈ Rpi


[B1 · · · BN

]

C = col(C1, . . . ,CN)

Main idea

Design method proposed in (Lavaei & Aghdam, 2008) for reducing thedesign of distributed controllers to the design of decentralized controllers



Distributed controller structure

Specify a block matrix 𝒦 with binary entries where block (i , j), i , j ∈ 1 : Nare mi × pi matrices with all entries equal to

1 if yj can be used for computing ui

0 otherwise

Example: system with 4 SISO channels

𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦

Special cases

𝒦 block-diagonal ⇒ decentralized control

𝒦 full ⇒ centralized control






0 otherwise


𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦

Special cases








0 otherwise


𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦

Special cases





Distributed controller

Associate to 𝒦 a matrix K (s) of dimension m × p, m =∑N

j=1mj ,

p =∑N

j=1 pj obtained by replacing nonzero blocks (i , j) of 𝒦 with anmi × pj matrix Kij(s) of transfer functions


𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦ K (s) =

⎡⎢⎢⎣

K11(s) 0 0 0K21(s) K22(s) 0 K24(s)K31(s) 0 K33(s) 0

0 K42(s) 0 K44(s)

⎤⎥⎥⎦

Problem

When there is a controller K (s) with the above structure that stabilizesthe closed-loop system ?

Next: constructive method structured in the sequence of 3 procedures



Distributed controller

Associate to 𝒦 a matrix K (s) of dimension m × p, m =∑N

j=1mj ,

p =∑N

j=1 pj obtained by replacing nonzero blocks (i , j) of 𝒦 with anmi × pj matrix Kij(s) of transfer functions


𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦ K (s) =

⎡⎢⎢⎣

K11(s) 0 0 0K21(s) K22(s) 0 K24(s)K31(s) 0 K33(s) 0

0 K42(s) 0 K44(s)

⎤⎥⎥⎦

Problem

When there is a controller K (s) with the above structure that stabilizesthe closed-loop system ?

Next: constructive method structured in the sequence of 3 procedures



Bipartite graph

An undirected graph 𝒢 with set of vertices V and set of edges E isbipartite if V can be partitioned in two sets U and L and each edgeconnects a node in U with a node in L.

Example

Not bipartite

1 2 3

4 5 6

Bipartite

1 2 3

4 5 6



Bipartite graph

An undirected graph 𝒢 with set of vertices V and set of edges E isbipartite if V can be partitioned in two sets U and L and each edgeconnects a node in U with a node in L.

Example

Not bipartite

1 2 3

4 5 6

Bipartite

1 2 3

4 5 6



Complete bipartite graph

An bipartite graph is complete if each vertex in U is connected to everyvertex in V .

Example

Not complete

1 2 3

4 5 6

Complete

1 2 3

4 5 6



Complete bipartite graph

An bipartite graph is complete if each vertex in U is connected to everyvertex in V .

Example

Not complete

1 2 3

4 5 6

Complete

1 2 3

4 5 6



Procedure 1

Build a bipartite graph 𝒢 from 𝒦 as follows

Define sets U and L with N elements and label the elements of bothsets with {1, . . . ,N}For any i , j ∈ 1 : N connect vertex i in U with vertex j in L with anedge if the (i , j) block entry of 𝒦 is not zero. Label the edge withKij(s)

Example

System with 4 SISO channels

𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦

Associated bipartite graph

3

if the (i, j) block entry of K is not a zero matrix, i.e.,if the output of the jth subsystem can contribute to theconstruction of the input of the ith subsystem. Denotethe gain of this edge with Kij(s).

As an example, consider a system consisting of four single-input single-output (SISO) subsystems with the followinginformation flow matrix:

K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)

The graph G corresponding to the matrix K given above isdepicted in Figure 1.

Fig. 1. The graph G corresponding to the matrix K given by (4).

The following procedure can be used to construct a bipartitegraph G with a decentralized structure from the graph G.Procedure 2: Partition the graph G into a set of complete

bipartite subgraphs such that each edge of the graph G appearsin only one of the subgraphs. It is to be noted that this partitionmay require some of the vertices of the graph G to appear inseveral subgraphs. Denote the resultant graph with G.As an example, consider again the graph G sketched in

Figure 1. The graph G for this graph can be considered asthe one depicted in Figure 2. It is obvious from Figure 2 that,in this particular example, vertices 2 and 3 of the first set ofvertices of G are repeated twice in G.

Fig. 2. A decentralized graph G obtained from the graph G in Figure 1.

It is to be noted that Procedure 2 will not necessarily resultin a unique decentralized graph G for a given graph G. Forinstance, a trivial partition for any given graph can be obtainedby considering each edge as a complete bipartite subgraph.The following procedure can be used to construct the matrix

function K(s) corresponding to the graph G.Procedure 3: Form a m £ r block diagonal matrix K(s),

where m and r are the number of vertices in sets 1 and 2 of G,respectively, and the number of the blocks on its main diagonalis equal to the number of partitioned subgraphs in G. Labelthe complete bipartite subgraphs of G as subgraphs 1 to ∫.Furthermore, label the vertices of subgraph l, l = 1, 2, ..., ∫,

as vertex 1, ..., ¥l in subset 1 (corresponding to set 1) andvertex 1, ..., ¥l in subset 2 (corresponding to set 2). The (l, l)block entry of K(s), l = 1, ..., ∫, is a matrix, whose (i, j)block entry is equal to the gain of the edge connecting vertexi of subset 1 to vertex j of subset 2 in subgraph l of G, for anyi 2 {1, ..., ¥l}, j 2 {1, ..., ¥l}. Denote the (l, l) block entry ofK(s) with Kl(s) 2 <ml£rl , for l = 1, 2, ..., ∫.Remark 1: It can be easily concluded from Procedures 1, 2

and 3, that there exists an onto mapping between the nonzeroblock entries of the matrix K(s) and those of the matrix K(s).For instance, using Procedure 3 and for a particular number-

ing of vertices in each subgraph of G in Figure 2, the followingblock diagonal matrix K(s) is obtained:

K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)

Theorem 1: Suppose that K(s) is derived from K(s) usingProcedures 1, 2 and 3. There exist constant matrices ©1 and©2 such that they satisfies the following equality:

K(s) = ©1K(s)©2 (6)Proof: It is straightforward to show (by using Procedures 1,

2 and 3) that the matrix K(s) can alternatively be constructedfrom K(s) through a sequence of L matrices, denoted byK1(s),K2(s), ...KL(s), where K(s) = K1(s) and K(s) =KL(s). In addition, Ki(s) is obtained from Ki°1(s), for anyi = 2, 3, ..., L, by one of the following two operations:1. Swapping either two columns or two rows of the matrix

Ki°1(s).2. Splitting one of the rows (or columns) of Ki°1(s)denoted by v, into two vectors v1 and v2, i.e. v =£

v1 v2

§(or v =

£v1; v2

§). Then, replacing that

row (or column) with£

v1 0§(or

£v1; 0

§), and in-

serting another row (or column) equal to v =£

0 v2

§

(or v =£

0; v2

§) into the matrix.

It is desired now to prove for any i = 1, ..., L ° 1, that thereexist matrices ©i1 and ©i2 such that Ki+1(s) = ©i1Ki(s)©i2 .In that case, the matrices ©1 and ©2 can be computed asfollows:

©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12

(7)As mentioned earlier, Ki+1(s) can be derived from Ki(s)using one of the two operations stated above. These operationslead to one of the following outcomes.1. Assume that Ki+1(s) is derived from Ki(s) by swap-ping its h’th and q’th columns. It is straightforward toshow that, in this case, the matrices ©i1 and ©i2 will beas follows:a) ©i1 is an identity matrix, whose dimension is thesame as the number of the rows of Ki(s).

b) ©i2 is derived from an identity matrix, whosedimension is equal to the number of the columns



Procedure 1

Build a bipartite graph 𝒢 from 𝒦 as follows

Define sets U and L with N elements and label the elements of bothsets with {1, . . . ,N}For any i , j ∈ 1 : N connect vertex i in U with vertex j in L with anedge if the (i , j) block entry of 𝒦 is not zero. Label the edge withKij(s)

Example

System with 4 SISO channels

𝒦 =

⎡⎢⎢⎣

1 0 0 01 1 0 11 0 1 00 1 0 1

⎤⎥⎥⎦

Associated bipartite graph

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12





Procedure 2: build a bipartite graph 𝒢 from 𝒢Split 𝒢 into a set of complete bipartite graph such that each edge of 𝒢appears only in one subgraph. If needed, some vertices of 𝒢 can beduplicated in several subgraphs.

Example (ctd.)

Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Remark

The result of procedure 2 is not unique (e.g. a graph 𝒢 where each edge isa subgraph is always a possibility)




Example (ctd.)

Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Remark





Example (ctd.)

Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Remark




Procedure 3: build a matrix K (s) from K (s)

Build an m × p block matrix K (s) from K (s) where m = |U|, p = |L| (Uand L are node sets of 𝒢) in the following way

Label the subgraphs of 𝒢 as 1 : N and the vertices of sets Ui , Li ofsubgraph i with {1, . . . , 𝜂ui} and {1, . . . , 𝜂li}, respectivelyK is block diagonal with N blocks and block Kt(s) ∈ R𝜂ut×𝜂lt has theelement (i , j) equal to the transfer matrix connecting vertex i of Ut tovertex j of Lt

Example (ctd.)

Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Matrix K (s)⎡⎢⎢⎢⎢⎢⎣K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

⎤⎥⎥⎥⎥⎥⎦



Procedure 3: build a matrix K (s) from K (s)

Build an m × p block matrix K (s) from K (s) where m = |U|, p = |L| (Uand L are node sets of 𝒢) in the following way

Label the subgraphs of 𝒢 as 1 : N and the vertices of sets Ui , Li ofsubgraph i with {1, . . . , 𝜂ui} and {1, . . . , 𝜂li}, respectivelyK is block diagonal with N blocks and block Kt(s) ∈ R𝜂ut×𝜂lt has theelement (i , j) equal to the transfer matrix connecting vertex i of Ut tovertex j of Lt

Example (ctd.)

Graph 𝒢

3



K =

2664

1 0 0 01 1 0 11 0 1 00 1 0 1

3775 (4)













K(s) =

0BBBBBB@

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

1CCCCCCA

(5)





v1 v2

§(or v =

£v1; v2



v1 0§(or

£v1; 0

§), and in-


0 v2

§

(or v =£

0; v2



©1 = ©11©21 · · · ©(L°1)1 , ©2 = ©(L°1)2©(L°2)2 · · · ©12



Matrix K (s)⎡⎢⎢⎢⎢⎢⎣K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

⎤⎥⎥⎥⎥⎥⎦Farina, Ferrari Trecate Decentralized and distributed control EECI-HYCON2 School 2015 34 / 46


Theorem

If K (s) is derived from K (s) using procedures 1 2 and 3, then there existconstant matrices Φ1 and Φ2 such that

K (s) = Φ1K (s)Φ2

Remarks

Matrices Φ1 and Φ2 are unique for a given K (s) and there is an efficientmethod for computing them



Example (ctd.)

Matrix K (s)⎡⎢⎢⎣K11(s) 0 0 0K21(s) K22(s) 0 K24(s)K31(s) 0 K33(s) 0

0 K42(s) 0 K44(s)

⎤⎥⎥⎦Matrix K (s)⎡⎢⎢⎢⎢⎢⎣

K11(s) 0 0 0K21(s) 0 0 0K31(s) 0 0 0

0 K22(s) K24(s) 00 K42(s) K44(s) 00 0 0 K33(s)

⎤⎥⎥⎥⎥⎥⎦

By direct calculation one has K (s) = Φ1K (s)Φ2 withMatrix ΦT

1⎡⎢⎢⎢⎢⎢⎣1 0 0 00 1 0 00 0 1 00 1 0 00 0 0 10 0 1 0

⎤⎥⎥⎥⎥⎥⎦

Matrix Φ2⎡⎢⎢⎣1 0 0 00 1 0 00 0 0 10 0 1 0

⎤⎥⎥⎦



Extended system

Consider the following system

Σ :

{˙x = Ax + Bu

y = C x

whereB = BΦ1, C = Φ2C

By construction, Σ has N input/output channels that are decoupled

Main theorem

The eigenvalues of the closed-loop system Σ under K (s) and of theclosed-loop system Σ under K (s) are identical.



Extended system

Consider the following system

Σ :

{˙x = Ax + Bu

y = C x

whereB = BΦ1, C = Φ2C

By construction, Σ has N input/output channels that are decoupled

Main theorem

The eigenvalues of the closed-loop system Σ under K (s) and of theclosed-loop system Σ under K (s) are identical.



Definition (generalization of DFM)

𝜆 ∈ 𝜎(A) is a Decentralized Overlapping Fixed Mode (DOFM) of Σ withrespect to the controller structure 𝒦 if it is a DFM of system Σ where

B = BΦ1, C = Φ2C

Theorem (eigenvalue assignment)

For a given symmetric set ℛ ⊆ C there is an LTI controller K (s) withstructure 𝒦 that places all eigenvalues of Σ in ℛ except for the DOFMsthat are not in ℛ.

Remarks

if ℛ is a symmetric collection of n complex numbers, we are solving aneigenvalue assignment problem





B = BΦ1, C = Φ2C



Remarks






B = BΦ1, C = Φ2C



Remarks




Corollary (stabilization)

The system Σ is stabilizable under an LTI controller K (s) with structure 𝒦iff all DOFM of Σ with respect to 𝒦 are in the lhp.

Remarks

From the previous results, for assigning the non-DOFM of Σ one can1 design the DeDOF controller K (s) using one of the methods seen

before so as to assign the non-DFM of Σ2 compute K (s) from K (s)

All results can be generalized to the case

yi = Cix +N∑

j=1

Dijuj





Remarks




yi = Cix +N∑

j=1

Dijuj





Remarks




yi = Cix +N∑

j=1

Dijuj



Example - system with 3 channels

Σ :

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

x =

⎡⎣1 0 10 1 21 2 3

⎤⎦ x +

⎡⎣0 1 00 0 11 0 1

⎤⎦⎡⎣u1u2u3

⎤⎦

⎡⎣y1y2y3

⎤⎦ =

⎡⎣1 0 00 0 11 2 2

⎤⎦ x

Consider the following controller with a BBD structure

𝒦 =

⎡⎣1 0 10 1 11 1 1

⎤⎦ K (s) =

⎡⎣K11(s) 0 K13(s)

0 K22(s) K23(s)K31(s) K32(s) K33(s)

⎤⎦

The goal is to design K (s) so as to place the dominant poles of theclosed-loop system at −1± i .



From procedures 1, 2 and 3 one can obtain

K (s) =

⎡⎢⎢⎢⎢⎣

K22(s) K23(s) 0 0K32(s) K33(s) 0 0

0 0 K11(s) 00 0 K31(s) 00 0 0 K13(s)

⎤⎥⎥⎥⎥⎦

and K (s) = Φ1K (s)Φ2 with

Matrix Φ1⎡⎣0 0 1 0 11 0 0 0 00 1 0 1 0

⎤⎦

Matrix Φ2⎡⎢⎢⎣

0 1 00 0 11 0 00 0 1

⎤⎥⎥⎦



Example

Using the general method presented before one obtains

K11(s) = K13(s) = K31(s) = 1

K22(s) = (−89900− 96100s − 34100s2 − 5480s3 − 409s4 − 11.5s5)/Den(s)

K23(s) = (−15700− 20500s − 8810s2 − 1730s3 − 160s4 − 5.69s5)/Den(s)

K32(s) = (−64500− 52500s − 16900s2 − 2740s3 − 220s4 − 7.05s5)/Den(s)

K33(s) = (−88000− 64500s − 19200s2 − 2880s3 − 219s4 − 6.7s5)/Den(s)

Den(s) = 53000 + 41488s + 13396s2 + 2269.3s3 + 210.44s4 + 9.95s5 + 0.18s6

Closed-loop eigenvalues are in −1± i , − 4, − 6, − 7 and −8.


A partial summary of methods for De/Dicontrol not covered in this course


Other De/Di control schemes

Decentralized synthesis

An ideal decentralized synthesis procedure designs regulators 𝒞i withoutusing knowledge of the whole system in any step.

For interconnected subsystems, 𝒞i should be based at most on theknowledge of Σi and its neighbors ⇒ some methods with this featurewill be discussed later on (“plug-and-play” control).

Only possible for a limited class of systems. In the literature there aremany ”practical” decentralized design procedures that involve somecentralized computation of limited complexity















Other design methods

Procedures for DeDOf developed in the 70’s and 80’s based ondynamic compensation (an alternative to observer-based design oflocal controllers)

Methods based on optimal control

De/Di DOf regulators for tracking

Synthesis of De/Di DOf based on LMI (Stankovic et al., 2007)I A complete characterization of LTI systems for which the computation

of stabilizing DeDOf is a convex problem has been given in (Rotkowitz& Lall, 2006)


























Bibliography

(Corfmat & Morse, 1976). J.P. Corfmat & A.S. Morse. Decentralizedcontrol of linear multivariable systems. Automatica, (12):5, pp. 479-495,1976

(Davison & Chang, 1990). E.J. Davison & T.N. Chang. Decentralizedstabilization and pole assignment for general proper systems. IEEETransactions on Automatic Control, (35):6, pp. 652-664, 1990

(Lavaei & Aghdam, 2008). J. Lavaei & A.G. Aghdam. Control ofcontinuous-time LTI systems by means of structurally constrainedcontrollers. Automatica, (44):1, pp. 141-148, 2008

(Stankovic et al., 2007). S.S. Stankovic, S.S., D.M. Stipanovic & D.D. andSiljak.Decentralized dynamic output feedback for robust stabilization of aclass of nonlinear interconnected systems. Automatica, (43):5, pp. 861-867,2007

(Rotkowitz & Lall, 2006). M. Rotkowitz & S. Lall. A characterization ofconvex problems in decentralized control IEEE Transactions on AutomaticControl, 51:(2), pp. 274286, 2006


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