About H2 Decentralized Control: Static and Dynamic Issues

Copyright © IFAC Large Scale Systems, Rio Patras. Greece. 1998

ABOUT H2 DECENTRALIZED CONTROL: STATIC AND DYNAMIC ISSUES

J. Bernussou' C. Courties' J .C. Geromel" • LAAS-CNRS, 7 av. du Colonel Roche, 31077 Toulouse Cedex 4, France. e-mail: [email protected], [email protected] .* LAC-DT-UNICAMP, CP 6101, 13081-970 Campinas, SP, Brazil

Abstract: The decentralized control problem with 11.2 norm minimization is addressed in the LMI formulation. The state control problem is given a necessary and sufficient condition for stability in terms of LMI so that the 11.2 problem can easily be solved. In the dynamic output control case the condition, still necessary and sufficient, is no longer a LMI but a BM!. A numerical procedure is given which enable, from an initial feasible solution, to solve a guaranteed cost problem. Copyright © 1998lFAC Keywords: 11.2 norm optimization, dynamic output control, decentralized control

nication supports the need for development of methodologies for large complex problems.

1. INTRODUCTION

In the seventies, a very popular approach was based on the use of the so called vector Lyapunov functions (Geromel et al., 1994) . When applied to interconnected systems, defined by a set of subsystems linked through an interconnection network, the main ideas beyond were to define scalar Lyapunov functions associated with the isolated subsystems and then use them to derive sufficient conditions for stability of the overall system. This last step was generally performed by constructing either scalar Lyapunov functions, linear combination of the subsystems functions or vector Lyapunov functions . The conditions were derived from M-matrices or comparison system results, and always appeared as weak interconnection norm conditions. In the same spirit, Lyapunov type designs have been provided where the local decentralized gains together with the local Lyapunov functions had to be found in order to end with stability (with performances) for the overall system. For instance, under the assumption of local controllability, LQ designs were developped at each subsystem level providing in

The decentralized control problem is a long lasting open problem which was addressed by a large community in the seventies. A confirmation of this assertion can be found in the books, survey paper and references therein which appeared at that time (Michel and Miller, 1977; Siljak, 1978; Sandell Jr et al., 1978). Since then, the interest apparently lowered although from time to time contributions are given to this problem which presents some interesting features as mentionned in (Sandell Jr et al., 1978) and summarized below. First, the decentralized control problem is basic in the domain of constrained information control problems, known to raise fundamental difficulties as compared with the standard, centralized, ones. From a pratical point of view, the interest of dealing with constrained information is manifold: control of geographically distributed pr
249

one step the local Riccati feedback as well as the Riccati matrix, used in a second stability test step as the local Lyapunov Matrix. Various examples of such designs can be found in (Siljak, 1978).

Ud will denote the set of decentralized controls for which the local input Ui is derived according to the local state (decentralized state control) or to the local measured output (dynamic decentralized output control) .

Another approach worth mentionning at that time, algebraic in nature, was based on the definition and study of the so called fixed modes. Necessary and sufficient conditions were derived for decentralized (or more generally structurally constrained) control, leading to non linear parametrical optimization problems to be solved to achieve the control. While the Lyapunov approach was developped through quite workable numerical algorithms, this last one suffered from the numerical complexity of the associated algorithms, especially for large scale problems (the natural domain of the decentralized control problem). (Davison, 1976; Davison and Chang, 1986) . Most of the results obtained in the aformentionned works (especially those using the Lyapunov approach) were state feedback results, some refinements concerning sub-optimal decentralized control are found in (Geromel and Bernussou, 1982; Makila and Toivonen, 1987), through the use of mathematical programming and parametrical optimization.

The problem to be addressed in the following is : (2 )

namely: find the control in the predefined class minimizing the 11.2 norm of the transfert between the controlled output z and the exogeneous signal w.

2. DECENTRALIZED STATE FEEDBACK In this section some results of (Geromel et al., 1994) are briefly summarized. With Ui = kiXi, or U = KdX , Kd = blocdiag{ki } , the problem (2) can be written as a parametrical optimization problem: min Tr [B~ .cOBI] Kd

.co > 0

In the last years, some contributions appeared using mainly algebraic and factorization approaches for the static output decentralized control problem (Duan, 1994; Leventides and Karkanias, 1995) and the dynamic one (Ravi et al., 1995; Sourlas and Manousiouthakis, 1995) . In the state space approach relatively few results appeared on the dynamic output decentralized control problem, the one when only the local measured output is used for synthesis of the local input (Date and Chow, 1989; Veillette et al., 1992) . Moreover, all the given results provide only sufficient condition for stability and performance.

where Lo is the observability grammian. To proceed further, let us perform an over bounding of the Lo grammian by a bloc diagonal Pd matrix such that: (4)

the structure of Pd (Q d) being that of blocdiag{ Ad. A guaranteed cost problem can then be defined: min Tr[WJ Qd , Rd

B~d)

W ( Bid Qd

The problem formulation can be stated as follows : let the linear time invariant interconnected system be described by Xi ::::: AiXi +

1

Yi ::::: C2iXi

t j~i

>0

AQd + Qd A ' + B 2dRc. + (

M

with M::::: (CIQd

R~B;d

M' )< -I

(5)

0

+ D I2 dRc.)

whose solution (W*, Qd' Rd) is such that with = KdX = RdQ~-lx one has IITzwll~ < Tr[W*]

AijXj + BliWi + B2i U i

+ D2li Wi

(3)

(A + B2dKd)' .co + .co (A + B 2d K d ) + (Cl + DI2dKd)' (Cl + D12dKd) ::::: 0

U

(1)

The fundamental and good feature in (5) is that it only involves linear matrix inequalities so that a now quite efficient numerical machinery can be processed even for large scale problem, which is, by nature, the case for decentralized control design for interconnected systems.

z::::: Clx+D12u

where (Xi, Ui, Yi) are the local state, control, measured output vectors associated with the ith subsystem, Wi are exogeneous noises, z is an overall controlled output.

A, Bid (likely but nor necessarily) = blocdiag{Bli }, B2d = blocdiag{B2i }, C2d = blocdiag{C2i } are the overall dynamic, input, output matrices, D 2i d = blocdiag{D2li} with the corresponding overall vectors Xl = [xL . .. , xN]' ul = [ui, . . . ,uN], y , w.

250

It is to be emphasized that the constraint (4) just follows the framework which has been widely used in the seventies where, as said in the introduction, the interconnected system analysis and control design were based on the existence of local Lyapunov functions and matrices, generally determined from the consideration of the isolated subsystems. In these works the analysis and synthesis

was achieved through only sufficient conditions , generally quite conservative due to the fact that the interconnection terms and the Aij matrices were treated as kind of perturbations. Indeed, the stability conditions were always given in terms of low interconnections ones.

and

The problem min IITzwll~ with respect to the unknowns A c, Bc , Cc can be written in terms of the parametric optimization problem:

In fact , the last linear matricial inequality in (5) is a necessary and sufficient condition for decentralized state control and existence of a composite Lyapunov function , sum of local ones:

(9)

or equivalently

N

v(x) = Xl Pdx =

L X~PdiXi

min Tr[WJ

~

i=l

The determination of the Lyapunov composite function results howewer from a centralized treatment, which may in some very large scale cases, restore some advantage for the former approach by decomposition over each isolated subsystem.

{

R

P= [ UI

Define T

{

z

= [~I ~],

Z(R , S,L , F)

= CXc>

T is regular. The first

~]

and

= A + [s

L']

[~ ~

r[;, ]

min Tr[W]

w

BI1 BIR+DI 1 21 FI) L

• S ••

The feedback connection of the system with the controller produces a linear system whose state space representation is written as:

z

[sVI v] S

=

The problem (10) is rewritten with respect to the optimization variables S, R, L , F, M as:

(7)

in order to minimize the 1-£2 norm of the Tzw transfert .

{

-1

and

Xc

+ Bw

P

= AS + SAl + B2L + LI B; G(R, F) = AIR+ RA+ FC2 +C~FI

= C1X + D12U

xc> =_Axc>

u] k

H(S,L)

(6)

The problem is to determine a strictly proper dynamic output compensator (of same order as (6)):

{

<0

Introducing the matrix functions:

= Ax + Bl W + B2U

= AXe + BeY U = CeXe

A

[~ J?,] respectively, the second one by [~ ~] TIO] and [ 0:1 .

Before dealing with the decentralized case some results are given on the non constrained one described by:

+ DZ1 W

pr

inequality is post (pre) multiplied by [ ~

3. DYNAMIC OUTPUT FEDBACK

Y = C2 X

A'

(10)

Let P and p- 1 be partitionned according to:

It is to be noticed that the problem (5) avoids the problem of initialization which was a key point of the algorithms dealing wich a parametric optimization (Geromel and Bernussou, 1982; Makila and Toivonen, 1987) where initial feasible (Le. stabilizing) feedback were to be known.

X

~~)

W BIP) PB P > 0

[

>0

R

(U)

H Z+M SC'1 +LIDI12 )

•

G

••

~

<0

-L

with the following change of variables (8)

L

= Cc V I,

F

= U Bc ,

M

= V Aeu

l

In the non constrained case the optimal solution of (ll)is obtained by fixing:

where

M

= -z - (SC~ + LDi2)Cl

(12)

which then inplies that the optimization problem is solved by using the LMI "machinery" . Note that

251

Let A, Bl = blocdiag(Bli be the overall dynamic and input matrices for the overall system with X h · an d W '_[ w Il , w 2I , . .. , W IN 1 as testate an d nOIse vectors.

the stabilizability condition by dynamic output feedback given by: H Z+M) G <0 ( Z' +M'

reduces (in the non constrained case) to the stabilizability and detect ability conditions H

< 0, G <

The problem defined by (2) can also be written in terms of the following parametric optimization problem:

°

by choosing (which is always possible) (15 )

Le. A~

M = -Z,

The "philosophy" behind this approach comes fundamentaly from (Gahinet and Apkarian, 1994; Scherer et al., 1997) see also (Geromel et al., 1997).

which can be written into an equivalent (in terms of the solution) problem:

In the constrained case and generally the ones when it is impossible to take:

(16 )

M=-Z+Mnew

in order to remove the non linear terms contained in Z to get a linear matrix inequality in terms of (R, S, L , F, M new ), a cross decomposition algorithm can be proposed as in (Geromel et al., 1997) .

As for the state feedback case, a bloc diagonal structure is taken for the P Lyapunov matrix, following the same idea of having a composite Lyapunov function which is the sum of the local ones. It is to be noticed that hence:

4. DECENTRALIZED DYNAMIC OUTPUT FEEDBACK

Considering the interconnected dynamical system described by (1) , the control is now generated though the output measurement in a decentralized manner such as: Xci {

=

AciXci

+ BciYi

R; Pi = [ U:

(13) Ui

Ui]

k;

':i]

[Si V-', Si

'

S.,>, R":-l

defining

= A;X; +

t

Aij Xj

Ti

= [ ~~ ~],

Td

= blocdiag(Ti )

Pre and post multiplication of the first inequality in (16) by respectively

+ ihiWi (14)

j#i

{ Z

1 , P ,.- =

matrices according

= CciXci

where dim Xci = dim Xi ( the local dynamic compensator is same dimenion as the corresponding subsystem) . The overall controlled system is given by: Xi

l

let us partition the Pi and Pito:

=C1X

[

where

~ ~]

and

[~~ ]

and

[~~ ]

and the second one by

x'

= [x~, x~, ... ,x~]

[~ ~]

and

one gets after straigtforward calculations the expression:

with the obvious notation Ccd

= blocdiag{ Cci}. 252

(17)

C'1

<0

-I

where

_ [Si A ij SiA ij R j ] [e]ij, i#j Aij Aij R j

Zi = Ai

+

, [Si li]

[

Ai B2i C2i 0

]' [ ] R.; F:

and

Fd

= blocdiag(Fi) , Ld = blocdiag(L;)

Assuming that a feasible solution in terms of the R d , Sd, L d , Fd , Mi matrices is known for the last inequality (17) then a stabilizing decentralized dynamic output controller is easily found:

Mni =Mi+Zi

• choose the Vi matrices from which Cci = L i (V;,)-I, i = 1,2, .. . , N • calculate the "corresponding" Ui matrices according to UiV;' = I-R;Si, i = 1,2, ... , N then Bci = Fi Ui- 1 • the matrices Aci are given by Aci = Ui- 1M:(V;')-1

Obviously the constrain in (18) can always be fulfilled choosing >. sufficiently large. The matricial inequality in (18) is still a BM!. We first pre and post multiply the matrix in -1

PI (Si

The problem for the determination of a feasible and then optimal solution for (17) is not easy because, unlike the state feedback case, the last matricial inequality in (17) is not an LMI, but a BMI (bilinear) with respect to the unknowns. As said in the preceding section a cross decomposition algorithm can be used to address the solution.

under

M~i

:

blocdiag{[e]ij}

[e]ij,}#i

<0

AijSj-1 AijRj ] A·.]· R J·

= [ A .]. ' S-1 j

and

With kci hold fixed Vi, (19) is a generalized eigenvalue problem which can be solved using the LMI machinery, with respect to the variables Sil, R;, Fi , Mi • If its optimal value is negative a feasible solution can be provided for the 11.2 optimization problem (17) . If not a dual P2 (Si , Ri 1 , L i , Mi ) problem is readily found by pre and post multiplication by

minA

"

, R;, F i , M i )

and get the problem

with

The first problem to be solved is to find a feasible solution for the matricial inequalities (17). Let us first consider the generalized eigenvalue problem:

[e] .. = [Hi - ASi

-

minA

5. A CROSS DECOMPOSITION ALGORITHM

under blOCdiag{[e]r'} < 0 .th [ ] _ SiAij SiAij Rj Wl e ij, j#i Aij Aij R;

[S~l ~]}

(18) by blOCdiag{

(18)

Mni Gi - AR.;

blOcdiag{ [

where

253

~ R~1 ] }

giving P2(Si, Ril, Li, Mi):

(19)

minA under with

blocdiag{[e)ij }

[e) i j.j# i -_

[ 8 iA ij 1 A ij

Ri

<0 iA j

i ] Ri8 1 Aij

(20 )

and

Here too with kf; hold fixed Vi , P2 (S;, Ri-l, L; , M; ) is a generalized eigenvalue problem, if its solution is negative an initial feasible point can be provided for the 11.2 optimization problem (17) . If not its solution can be used to give an initial point for PI and the PI , P2 problems can be solved iteratively. Running a numerical algorithm in such a cyclic way, provides a decreasing sequence of ). and a feasible point for (17) is achieved when ). becomes negative. It is not claimed that such a proccess will always converge to a feasible solution whenever the system is stabilizable by dynamic output decentralized control. Howewer, numerical experiments performed showed that such a type of algorithm presents a fairly good rate of success (De Oliveira et al. , n.d.). Problem (17), with an initial feasible solution, can be solved in much the same way as the generalized eigenvalue problem. The iterative algorithm, working on two "dual" LMI problems, provides a decreasing sequence in terms of Tr[WJ, the convergence being obtained at a local minimum in general.

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itive principal minors. Czech Math . Journal Vo!. 12, 382-400. Gahinet, P. and P . Apkarian (1994) . A linear inequality approach to Hoc control. International Journal of Robust and Nonlinear Control Vo!. 4, 421-448. Geromel, J .C. and J. Bernussou (1982). Optimal decentralized control of dynamic systems. A utomatica No. 18, 545- 557. Geromel, J.C. , J . Bernussou and P.L.D. Peres (1994) . Decentralized control through parameter space optimization. Automatica Vol. 30(10) , 1565-1578. Geromel, J .C., J .M. Bernussou and M.C . de Oliveira (1997) . H2 norm optimization with constrained dynamic output feedback controllers : decentralized and reliable control. In: Proc. 3ffh CDC. IEEE. San Diego. Leventides, T . and V. Karkanias (1995). Global asymptotic linearization of the pole placement map: a closed form solution for the constant output feedback problem. Automatica Vo!. 31(9) , 1303-1309. Makila, P .M. and H.T . Toivonen (1987) . Computational method for parametric Iq problems. IEEE Transactions on Automatic Control AC-32 , 658-671. De Oliveira, M.C. , J.C. Geromel and J. Bernussou (n.d.) . Design of dynamic output feedback decentralized controllers via a separation procedure. Accepted in IEEE Tran. Aut . Control. Sandell Jr, N.S., P. Varayia, M. Athans and M. Safonov (1978). Survey of decentralized control methods for large scale systems. IEEE Transactions on Automatic Control AC23, 108-128. Michel, A.M. and RK. Miller (1977) . Qualitative analysis of large scale dynamical systems. Vol. 134 of Mathematics in Science and Ingineering. Academic Press. New York. Ravi, M.S., J . Rosenthal and X.A. Wang (1995) . On decentralized dynamic pole placement and feedback stabilization. IEEE Transactions on Automatic Control AC-40, 16031614. Scherer, C., P. Gahinet and M. Chilali (1997). Multiobjective output-feedback control via lmi optimization. IEEE Transactions on Automatic Control AC-42(7) , 896-911. Sourlas, D.D. and V. Manousiouthalds (1995). Best achievable decentralized performance. IEEE Transactions on Automatic Control AC-40(1l), 1858-1871. Veillette, RJ., J.V. Medanic and W.R Perkins (1992) . Design of reliable control systems. IEEE Transactions on Automatic Control AC-37(3), 290-304. Siljak, D.D. (1978) . Large scale dynamic systems - Stability and structure. System Science and Ingineering. North-Holland. New York.