Partially Decentralized Control of Large Scale Systems with Prescribed Stability

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

PARTIALLY DECENTRALIZED CONTROL OF LARGE SCALE SYSTEMS WITH PRESCRIBED STABILITY D.P. Iracleous

A.T. Alexandridis

Department of Electrical and Computer Engineering University ofPatras Rion 26500, Patras, Greece

Abstract: This paper discusses the problem of assigning the eigenvalues of a mUlti-input linear large scale system (LSS) in such a way that a certain degree of decentralized state feedback controls is obtained. It is proven that for an n-order system with m independent inputs the problem is split into the following two sequential stages. Initially the n-m eigenvalues are assigned using an n-m order system. This assignment determines the non-square part of the state feedback gainmatrix. Next, an m-order system is used to assign the remaining m eigenvalues so that the square part of the state feedback gain-matrix takes on a diagonal form. Therefore, the state feedback gainmatrix leads to partially decentralized control schemes since each input is fed-back by a local state variable and n-m common state variables. For LSS, this means a drastic reduction of the measured states at each input, from n to n-m+ 1, without any risk on stability. Copyright © 1998 1FAC Keywords: Decentralized control, stability, large scale systems.

importance, however, fully decentralized feedback control schemes are difficult to be designed.

1. INTRODUCTION Demands on stability and robustness of multivariable linear systems often result in the design .of integrated state feedback controls. State feedback control requires that every controller has access to all the measurements taken from the system. For large scale systems (LSS), this requirement presents a need for long-distance communication that can be impractical and uneconomical. To reduce the measurements required for stable designs, decentralized state controllers [1-10] or output feedback controllers [11-12] are used. Decentralized control seems to be preferable for LSS such as electric power systems, transportation systems, large space structures and whole economic systems. Despite its

In the literature, necessary and sufficient conditions as well as many methods have been proposed for eigenvalue assignment by decentralized control [3-10] . In most cases the design is based on special structural characteristics of the LSS such as symmetrical interconnections [6,10] etc. In other cases additional constraints [7] or parametric approaches [8,9] are used to achieve the desired eigenvalue assignment. However, all these methods seems to be cumbersome to apply. In this paper an alternative procedure with main advantage the easy application is proposed. The

163

method achieves eigenvalue assignment and therefore, guarantees stability while provides a partially decentralized control design. It is proven that for an n-order system with m independent inputs (n>m), the n-order optimal eigenvalue assignment problem can be reduced to an m-input m-output eigenvalue assignment problem where the remaining n-m eigenvalues are assigned by any common technique. On the other hand, the reduced m-input system has a diagonal input matrix and therefore a diagonal state feedback gain matrix is adequate for the assignment of the m closed-loop eigenvalues. This significantly simplifies the complexity of the problem, since the n-m eigenvalues are assigned exactly at any desired stable positions to provide performance characteristics of the system while the m eigenvalues are assigned to achieve a partially decentralized scheme. To demonstrate the application of the proposed method a numerical example of a real LSS IS used from the power system control field .

Assume that (A, B) is a completely controllable pair. Applying on system (1) the state feedback control law U=

where u=[u,

(6)

U2 .. .

umf, and

(7) K=[K\ K 2 ] with K, the mxrn square part of the feedback gain-matrix, we obtain the following closedloop form

x = Acx ,

where

Ac

= A + BK .

(8)

Our purpose is to obtain a gain matrix K in such a way that K, be diagonal, i.e.,

_Ikl

K,-

o

k2

0

1

(9)

km

Simultaneously, both the diagonal K, and the K2 must be selected so that the closed-loop eigenvalues of the system (1) to be located at some desired positions. Therefore, every input u, is a function of a reduced order state vector, i.e. ui=k;xi+kjX" where i E {1 ,2, ... , m} and k, denotes the i row of K2 and XI=[Xm +', Xm_2, .. .,

2. PRELIMINARIES Let the time invariant mUlti-input linear system

xnf.

m

x=Ax+ " s .b .u ~IJI

Kx

X(O)

= Xo

(1)

This guarantees the desired stability and realizes a remarkable decentralization.

j=\

where x E Rn is the system state vector, Ui ER are the local inputs (m
Under these assumptions, we next propose a different approach for the eigenvalue assignment by state feedback which leads to a partially decentralized eigenvalue assignment procedure.

(2)

3. MAIN RESULTS

System (1) can be written in the form x=Ax+Bu

3.1 A two stage procedure for eigenvalue assignment To proceed with our approach we present the following theorem. Theorem 1: Tthere exists a state feedback gain-matrix K determined by the expression

(3)

where B is of the form (4)

whereBl= diag{ sI, s2, '" sm} and A is decomposed as follows (5)

which assigns the entire set of the n eigenvalues of the closed-loop system (8) exactly at the same positions where: i) the arbitrary mx(n-m) matrix L assigns the (n-m) eigenvalues of the matrix A22 + A21L and

where All is an mxrn submatrix., Al2 is an mx(nm) submatrix., A2l is an (n-m)xrn submatrix and A22 is an (n-m)x(n-m) submatrix.

164

ii) the arbitrary mxm matrix K, assigns the m eigenvalues of the matrix A , -+- B,K" where A , = All - LA'l' Then, K2 is a function of K, and L, i.e., K2 == K2(K" L) =

Lemma 1: The pair (An, A2J) is a completely controllable pair if and only if the pair (A , B) is a completely controllable pair. 2nd stage: Using the L determined from the 1st stage we construct the matrix A, = All - LA2 , • Then, the remaining m eigenvalues are assigned as the eigenvalues of the mxm matrixA J + B,K" by selecting an appropriate matrix K, . This is equivalent to the solution of an m reduced-order state feedback eigenvalue-assignrnent problem. However, since in this case, the input matrix is the m-order square matrix B, the assignment of the m eigenvalues is always possible since obviously holds that the pair (AJ, B,) is a completely controllable pair for any A ,. Consequently, the gain-matrix K is determined from (7) and (15).

B,-'(-A"L - B,K,L + LA 2 ,L - A\2 + LA 22 )

Proof Let T be the nxn matrix T=

[I

n-m

o

L]

-

(10)

Im

where In •m is the (n-m)-identity matrix and L is an arbitrary mx(n-m) constant matrix. Then, the inverse of T is

r' = [I n-m L] o

(11)

Im

Transforming the closed-loop matrix Ac = A + BK by using the similarity transformation TAcT" we arrive at TA cT" [

3.2 A simple decentralized eigenvalue assignment design procedure In this section a simple decentralized solution is proposed. Particularly, since the input matrix of A,+B,K, is the diagonal matrix B" the feedback gain-matrix KJ is an mxm square matrix. Therefore, if one selects K, to be diagonal, i.e.

=

A" - LA2l + BlK l

All L + BlK lL - LA2lL + A ll + B lK2 - LAn]

A 2l

An + A2lL

(12)

Defining (13)

(16)

where the scalar a is suitably selected to ensure the assignment of the eigenvalues in a desired region. The m eigenvalues of AJ+BJK, which are also the m closed-loop eigenvalues of Ac are, then, located at

the similarity transformation of Ac results in

0]

TAcT" = [A, + B,K, A 2,

A22

+A

(14)

2, L

where K2 is determined as follows K 2=B 1-' (-A"L - B,K,L + LA 2 ,L - A\2 + LA 22 ) (15)

(17)

Equation (14) shows that the feedback gainmatrix K given by (7), assigns all the closedloop eigenvalues of Ac (which is similar to TAcT") at the n-m eigenvalues of An+A2,L and 0 the m eigenvalues of AJ +BJKJ.

i.e. more to the left on the complex plane than the eigenvalues of AJ= All - LA2J which have been determined by L. In practical applications, however, the n-m eigenvalues which can be assigned exactly in any desired stable positions are the dominant eigenvalues of the closed-loop system. The m eigenvalues which are manipulated to provide decentralization and which are constrained to be located more to the left than the eigenvalues of AJ can be the non-dominant closed-loop eigenvalues.

Theorem 1 reveals that the assignment of the closed-loop eigenvalues of the system (3) can be achieved into two sequential stages: 1st stage: The (n-m) eigenvalues are assigned by selecting an appropriate matrix L. As indicated by the form of the matrix A22 + A 2,L , this selection of L is equivalent to the solution of an (n-m) reduced-order state feedback eigenvalue-assignment problem. Therefore, the assignment of the (n-m) eigenvalues is possible if the pair (An, A2J) is a completely controllable pair [13]. This is true for system (1), in accordance with the following lemma [13].

Moreover, if one can exploit the degrees of freedom of the gain-matrix L, which assigns the first n-m eigenvalues, in such a way that the m eigenvalues of A J= AJ+BJKJ be in locations distant a from the desired locations, i.e. l(AJ) = ~j,.JAc) + a (18)

165

then, exact optimal pole-placement can be achieved since equations (17) and (18) obviously lead to: Ades;re,AAc) = A(AJ+B}K}) .

Step 5: Terminate algorithm.

Remark 1. It is clear that in the general case KJ can be any stable diagonal matrix. However, the assignment of the m-eigenvalues in this case is not as obvious as in the proposed case (eq. 16).

4. EXAMPLE: A THREE MACHINE POWER SYSTEM CONTROL The linearized dynamical model of a three machine power system, taken from [14], is used to demonstrate the application of the proposed method. The system is a 9th order system with 5 is inputs. The state vector x=[~ ~2 Ecf3 Ed2 Cl.2 Wj ~3 "i2 bi3jT where W}, W2 and W3 are the angular velocities of the machines, E;2' E;3 are the q-axis components

3.3 The proposed design algorithm The procedure described above leads to the following algorithm: Initialize: (i) Enter system data A, B; (ii) calculate An, A}2, A2JA22 Derive L, (use a pole-placement technique to assign n-m eigenvalues of the subsystem (An

Step 1:

of voltage behind the transient reactance, E ~2 ' E~3

Av» . Step 2:

Calculate AJ = All - LAI2 and find its eigenvalues.

Step 3:

Determine the diagonal K J = -a B J-} (where a is selected so that the poles of AJ -aIm are shifted to desired locations).

Step 4:

are the d-axis components of voltage

behind the transient reactance and 612 , 613 are the angle differences of the machines 1-2 and 13, respectively. IS The input vector U=[Tml

EFV2 Trra

EFD3

Tm3

Y

where Tml , Tm2 , Tm3 are the mechanical torques applied to each machine and E FD2' E FD3 are the excitation voltages applied to machines 2 and 3 respectively.

Determine K2 from (15).

The system matrix is

A=

-0.5610

0.6793

0

0.4980

0

0.6099

0.5463

- 0.9520

- 0.7494

0

-13.7658

- 2.0723

3.6163

0

1.4409

1.1781

8.5472

- 3.3161

0

- 6.5352

0

-1.1714

2.2156

5.4592

5.6334

0

0.9552 -16.5675

0

0

0

0.4076

1.4111

- 4.2309

- 2.3385 10.117

0

2.9781

0

-10.6238

- 4.4063

3.9766

- 4.7247

- 5.2010

10.7116

0

- 15.5076

0

-12.6793

0

-150.1554

38.9205

42.4023

- 21.4333

0

- 3.8073

0

-13.1829

0

52.627

-156.9117

- 38.8349

68.5987

10000

0

- 10000

0

0

0

0

0

0

10000

0

0

0

- 10000

0

0

0

0

10-4

(l9a)

The system input matrix is

B=

0.5610

0

0

0

0

0

4.4210

0

0

0

0

0

2.0723

0

0

0

0

0

4.5035

0

0

0

0

0

4.4063

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.10-4

Let that we wish to assign the 4 (=n-m) closedloop eigenvalues of Ac at the values: {-0.0020 ± 0.0500 i, -0.0010 ± 0.0020i}_ Then, we determine the gain-matrix L for the subsystem (An. Av) as follows

(19b)

In this case we have an LSS with 9 states and 5 inputs (n=9, m=5). The open-loop eigenvalues are: {-0.0027 ±0.0346i, -0.0006 ± 0.0230i, -0.0002 ± O.OOOli, -0.0167, -0.0104, -0.0005}. 166

0.6528

- 0.2914

- 0.0106

0.0308

0.6528

- 0.2914

- 0.0106

0.0308

L= 0.8487

- 0.3788

- 0.0138

0.0400

1.3056

- 0.5827

- 0.0213

0.0615

0.3946

- 0.1134

- 0.0207

0.0075

(20)

Then, the submatrix AI AI=

- 0.0202

0.0010

- 0.0106

0.0005

- 0.0201

- 0.0005

- 0.0106

0.0008

0.0308

- 0 .0262

0.0005

- 0.0140

0 .0007

0.0400

- 0.0402

0.0024

- 0.0213

- 0.0008

0.0615

0 .0132

0.0009

- 0.0207

- 0.0007

0 .0070

0.0308

(21 ) [7]

has the following eigenvalues: {-0.0137± 0.0089 i, 0.0013, -0.0003, -0.0022}. In order to assign the m eigenvalues of Ac more to the left on the complex plane than the eigenvalues of AI we select a = -0.005. Finally, for this a the following state feedback gain matrix K=[KI K 2 ] is determined as follows

[8]

[9] r-89 1266

K

J 1

o -llJ097

-241278 -1110)0

11.6897

-4.4418

75.5201

-43.0045

2.4237

- !210l

7.4160

- 4.8233

7.8211

- lSlI]

233011

- 14.346\

7.4008

- lIDO]

19.2624

- 13.0910

- llJ474 3.!1582

- 01869

5.7280

- 5.1 574

(22) [10]

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167

[11]

[12]

[13]

[14]

controller synthesis for a class of large-scale systems with symmetrically interconnected subsystems, Automatica, Vo\. 27, 2, pp. 383-388, 1991. J.Lu, H. Chiang, J. Thorp, Eigenstructure assignment by decentralized feedback control, IEEE Trans. Autom. Control, Vo\. AC-38, No. 4, pp. 587-594, 1993. G.R. Duan, Eigenstructure assignment by decentralized output feedback. A complete parametric approach, IEEE Trans. Autom. Control, Vo\. AC-39, 5, pp. 1009-1014, 1994. M.S. Ravi, J. Rosenthal, X.A Wang, On decentralized dynamic pole placement and feedback stabilization, IEEE Trans Autom . Control, Vo\. AC-40, 9, pp. 1603-1614, 1995. G.H. Yang, S.Y. Zhang, Decentralized control of a class of large-scale systems with symmetrically interconnected subsystems, IEEE Trans Autom. Control, Vo\. AC-41, No. 5, pp. 710-713,1996. AT Alexandridis, P.N. Paraskevopoulos, A new approach to eigenstructure assignment by output feedback, IEEE Trans Autom. Control, Vol. AC-41, No. 7, pp. 1046-1050, 1996. D.P. Iracleous, AT. Alexandridis, Eigenstructure assignment by output feedback: A linear algebraic solution, Proc. IEEE 4th Mediterranean Symposium on Control and Automation, Chania, pp. 139143, June 1996. C.T. Chen, Linear System Theory and Design, Holt, Rinehart and Wiston Inc., 1984 P.M. Anderson, AA Fouad, Power System Control and Stability, The Iowa State University Press, Ames, Iowa, USA, 1977.