Technical Note: DLSDP: The decentralised linear systems design package

Technical Note: DLSDP:The Decentralised linear systems design package IVAN POPCHEV and SVETOSLAV SAVOV Institute o f Industrial Cybernetics and Robotics, Bulgarian Academy o f Science, Acad. G. Bonchev str., Bl.2, 1113 Sofia, Bulgaria

When control theory is a p p l i e d to solve p r o b l e m s o f large scale systems an important feature called decentralisation often arises. D u e to t h e nonclassical information pattern, it is in general impossible to exploit classical techniques for control

design. This paper describes several decentralised control design methods and a F O R T R A N package DLSDP based on t h e m . T h e o r g a n i s a t i o n o f the paper is as follows. Section 1 is an introduction to the decentralised control p r o b l e m . Classes o f systems under control considered in the p a p e r are p r e s e n t e d in section 2. Main p r o b l e m s closely related with decentralised control design are pointed out as well. Methods for design of stabilising control for systems with and without p u r e t i m e - d e l a y s , o p t i m a l control and robust u n d e r structural perturbations control are described and discussed in section 3. T h e F O R T R A N package features and software capabilities are discussed in section 4. T h e subroutines and t h e i r a l g o r i t h m s are described in detail. A numerical example, illustrating the solution of design problems b y D L S D P , is given in section 5. Key Words: Large scale systems, decentralised control, stability, optimal control, robust systems, time-delay, FORTRAN subroutines.

I. INTRODUCTION The problem of decentralised control (DC) design has been a subject of intensive research in the past 10-15 years. When control theory is applied to solve problems of large scale systems (LSS), e.g. power systems, socioeconomic systems, etc., an important feature called decentralisation often arises, since a distinctive peculiarity of many regulation problems is the presence of constraints on the structure of the control system. Decentralised systems (DS) have several control stations. At each station, the controller observes only local system outputs or states and controls only local inputs. All the controllers are involved, however, in controlling the same LSS. 1 It is important to know under what conditions there exists a set of local feedback control laws that will meet some requirements towards the overall system. Significant difficulties in the analysis and synthesis

of DS arise due to the non-classical information structure. When such constraints have to be faced, it is in general impossible to exploit standard techniques for control design, like LQG optimal control or pole assignment. This paper describes several DC design methods and a FORTRAN package DLSDP based on them. DLSDP contains a range of subroutines for DC design using the state space approach.

2. PRELIMINARY DEVELOPMENTS The aim of this section is to present classes of systems under control considered in the paper. Main problems closely related with DC design are pointed out as well. Consider the set of N linear dynamic interconnected systems: N

Yci(t) = Aiixi(t ) + ~ eli(t) Aiixj(t) + Biiui(t), j:l

j~i i = 1,2 . . . . . N

(1)

where Xi(t ) E R ni and Ui(t ) E R mi are the local subsystems state and control vectors. All matrixes Aij, i,j = 1 , 2 , . . . , N and Bii , i = 1, 2 . . . . , N are real constant and with compatible dimensions. Suppose that system (1) is of high order and is comprised of a large number of subsystems. Therefore, the design of a control strategy based on a centralised state feedback law is not justified (excessive computer memory required, much computer time needed, long and expensive interconnection lines between the central control station and the plant are necessary, etc.). In such cases the designer faces a real LSS problem. It is a matter of common knowledge that the nature of a LSS cannot be precisely known and since essential uncertainties reside in the interconnections, the terms eij(t) E [0, 1] (continuous functions of time, defining the connection of the ith subsystem with the jth one 2) are used to express the action of structural perturbations (SP) between the subsystems. Systems, retaining their properties, e.g. stability, for all values of eij(t) E [0, 1] are said to be robust under SP. Many technological processes and transport operations are characterised by the presence of pure time-delays in the local states and in the interconnections. Such time-delay systems (TDS) are described by: N

fCi(t) : Aiixi(t ) + Aiixi(t--7ii) + ~ Ai/x/(t) /-1

Accepted April 1985. Discussion closes March 1986.

52

Adv. Eng. Software, 1986, Vol. 8, No. 1

0141-1195/86/010052-07 $2.00 o 1986 Computational Mechanics Publications

depend on K, a natural approach a consists in computing the gradient matrix G of Xa with respect to K:

N

+ ~ Aqxj(t--rii)+Biiui(t) ]=1

i= 1,2,...,N

(2) where ri! > O, i,] = 1,2, ... , N are the pure time-delays. The basic aim of DC design is to find a set of local feedback control laws:

ui(t ) =Kixi(t),

i = 1, 2 . . . . . N

(3)

such that some overall system problems are solved. Let us discuss briefly what kinds of problems these could be. The most important property of every system under automatic control is its stability. But this is far from being the only one. It is not only necessary to produce, but what is more important is to do it effectively. To guarantee robustness under SP is a problem of vital importance especially for DS. The solution of DC problems for special types of LSS, e.g. TDS, has not only theoretical but first of all applied significance. So, the main DC design problems can be summarised as:

P1. P2. P3. P4.

Decentralised stabilisation Decentralised optimisation Design o f a robust DC Decentralised stabilisation o f TDS

Some other problems, associated with the plant specifically are possible as well. DLSDP solves problems P1-P4 at this stage of development.

O, if kq G = (gij), gij = Re(aXd/~kii) = {

0

vjwTbi/wTv, otherwise

(7) where b i is the ith column of B, v/is the jth component of the right eigenvector v and w T is the left eigenvector. Once G is calculated, a steepest descent search in direction --G can be performed to compute the new gain matrix K. The method possesses the following properties: PI.1 The iterative procedure is initialised with K E I~ P1.2 If, for some K, it turns out that Re(Xa) >~ 0 is a local minimum, K must be perturbed slightly to continue the iterations. P1.3 It is possible that the decrease in the real part of ~a is offset by an increase in the real part of another eigenvalue. Convergence of the method is proved. 4

1)2. Decentralised optimisation The gain matrix K for which (6) holds can be used as a first step control for those optimisation methods, requiring an initial stable closed loop system. Let ~ be the set of all stabilising K EI~. Problem P2 is formulated as follows: determine a gain matrix Kopt E Y. for which the linear quadratic performance index 0o

3. DC DESIGN METHODS Formulations and solutions of DC design problems, raised in section 2, will be presented. Some important features of the design methods will be discussed too.

P1. Decentralised stabilisation Let all subsystems (1) be ideally connected, i.e. eii= 1, i, ] = 2 . . . . . N, i :~]. Equation (1) can obviously be written as:

J(K) = f

( x r ( t ) [ a + K r R K ] x ( t ) } dr, Q = ccT>~ O,R > 0

¢J

0

(8) is minimised. Problem P2 can be solved by a method of standard mathematical programming, using a gradient technique and a feasible direction method. The gradient matrix of J(K) with respect to K is: 4 ~)J(K)/bK = 2(RK - - B r P ) L = D

Yc(t) = Ax(t) + Bu(t)

where x(t) E R n and u(t) E R m are the overall system state and control vectors: N

n = Z ni

N

and

i=1

m = ~ mi i=1

Let I~ be the set of all block diagonal gain matrixes in the state feedback"

= (K E R m xn/K = bl.diag.(Kb K~ . . . . . KN), g i @ R mixni, i = 1,2 . . . . . N )

(5)

Problem PI can be formulated as: find N local control laws (3), such that the condition for asymptotic stability:

o(A + B K ) C C - ,

KEK

(9)

(4)

(6)

holds, where o(.) and C- denote the spectrum of (.) and the open left half of the complex plane respectively. Provided that

(i) (Aii , Bii), i = I, 2 . . . . . N are controllable pairs; (ii) system (4) has no unstable fixed modes 1 problem P1 can be solved by an iterative procedure of minimising the real part of the dominant eigenvalue Xa of the closed loop system. Since all eigenvalues of A + BK

where P and L are the positively definite solutions of two Liapunov equations:

(A+BK)Tp+P(A+BK)+Q+KTRK=O

(10a)

L(A + BK)T + (A + BK) L + Vo = 0

(lOb)

where Vo = E(x(O)xT(O)~, x(O) is the initial state vector, and D = bl.diag. (D1, D2 . . . . . DN) is a matrix, which selects those entries of the gradient matrix corresponding to unconstrained entries of K and puts the others to zero. Provided that (i) (,4, C) is a completely observable pair and Vo > 0 convergence of the method is proved. 4 The following are important features: P2.1. The iterative procedure is initialised with K E ~. P2.2 The solution corresponds to a local minimum. P2.3 At every iteration stability of the system is guaranteed.

P3. Design o f a robust DC Consider the set of systems (1) subjected to SP. Problem P3 can be formulated like this: find N local control laws (3), such that the performance index

Adv. Eng. Software, 1986, Vol. 8, No. 1

53

N;

J z= ~

i=1

IxT(g) Q i x i ( t ) + llT([) Rib!i(t)l dt ,

o

Qi>~O,Ri>O,i

= 1.2 . . . . . N

(11)

is minimised and the overall system remains stable under SP. If: (i) ( A i i , Bii), i = 1, 2, . . . , N are stabilisable pairs (ii) (SQRT(Qi), Aii ), i = 1, 2 . . . . . N are detectable pairs (iii) the overall system has no unstable fixed modes (iv) neither of the local subsystems has fixed modes

problem P3 can be solved by ensuring an appropriate degree of stability ~i for every subsystem. The gain matrices K i are given by: K i = --R~'BTpi,

(12)

i= 1,2 ..... N

where Pi are the unique positively (semi)definite solutions o f N decoupled matrix algebraic Riccati equations: ( A i i + (3ill)Tel + P i ( A i i + (3ili) --PiBuRilBTpi

+ Qi = 0

(13)

The overall system is asymptotically stable, suboptimal and robust, iri s N

2 ~

N

(14)

~ Oiir?ils to the left of --[ISII. This method can be applied to solve problem P1, but now entirely in terms of local subsystems and pole assignment, thus giving to the designer one more degree of freedom. In this case all matrices corresponding to states with time-delays are zero matrices. Problem P4(P1) can be solved in this way, if:

(15)

]=1

(n i is the ith subsystem order), L2 = N N - L I , M= m (number of control inputs), M M = M x M, N M = N x M, NNN

M1 = ~

where yi(t)>>-Ixi(t)l, I'1 denotes norm of a vector and II "ll is the spectral norm of a matrix. According to the comparison principle z stability of (15) implies stability of the original system (2). The following result is important. The aggregated system (15) is stable, if 6 /a* > IlSl[

(16)

where/a* = m i n # i and S is a matrix, defined by i

[IA.iilI, for i = ] S = {Si]}, gO ~

54

[[Ai/ll + [IAi/II, otherwise

Adv. Eng. Software, 1986, Vol. 8, No. 1

(17)

2

mi

i-I

(m i is the number of control inputs of the ith subsystem) and NNN

M2 = ~

n i ×m i

i=1

ITMAX is an integer variable, specified by the user and is equal to the maximum number of iterations to be carried out, ITER is an output integer variable equal to the number of iterations carried out, IND is an output integer variable

equal to 1, if the respective problem is solved within the specified ITMAX and is equal to 0 otherwise, IER is an error indicator. Array input is columnwise.

method is used: at+X=nat, i f j t < j t - l a n d P t > O

n>l

a t +- ua t, otherwise 0 < u < 1 Subroutine D S T A B This subroutine solves problem P1. The algorithm of the iterative method is:

Step 1. Choose arbitrary initial K E I~ and a > 0. Set l = 1. Step 2. Compute h~ of A + B K I. If Re(h~} < 0 record K t as a decentralised stabilising gain and stop. Otherwise, compute G t and go to step 3. Step 3. Set l = 1 + 1. Update on K --K l = K z-1 -- at-iG t-1 Go to step 2. DSTAB calls five subroutines: MAX, GRAD, ORTH, ORTR, HQRV. Spectrum of A + B K t and corresponding left and right eigenvectors are computed by ORTH, ORTR and HQRV. MAX determines Re (ha} and the corresponding to it v and w T. GRAD computes the gradient matrix GJ(7). In order to guarantee a continuous decrease in the real part of ha, an adaptive method to compute the step size is used: a TM

= "/'fa t,

if Re {hi] - < Re (h/-1 ) 7r > 1

ceI "-- va t, otherwise 0 < u < 1

Input: A ( N N ) and B(NM) - arrays, containing overall system state and control matrices, G K ( N M ) - a n array, containing initial arbitrary block diagonal gain matrix, ALPHA - an arbitrary positive initial step size, PI and FNI - values of n and u, GAMA - a perturbation parameter, X - an indicator for local minimum, N, M, ITMAX. Output: G S ( N M ) - an array, containing the stabilising gain matrix, WR(N) and WI(N) -arrays, containing real and imaginary parts of the closed loop system eigenvalues, D E ( I T M A X ) - a n array, containing all intermediate dominant eigenvalues (real parts), ITER, IND, IER. Special care is taken in DSTAB to detect and overcome local minima. Obviously, the procedure reaches a local minimum, if IRe (h~t) - Re (h~-'}[ < X , for Re (ha} ~> 0 In such cases, the elements of the gain matrix K t are slightly perturbed to continue the iterations k~! +- k~, + GAMA

Both X and GAMA need to be sufficiently small. IER = s, if more than 30 iterations are necessary to compute the sth eigenvalue and IER = 0, if there is no error. Subroutine DOPT

This subroutine solves problem P2. The algorithm is: Step 1. Choose an arbitrary initial stabilising gain K E and a > 0. Select a value for the tolerance error e > 0 . S e t / = 1. Step 2. Compute D t. If IIDtI[ < e, record K t as an optimal solution and stop. Otherwise, go to step 3. Step 3. Set l = 1 + 1. Update on K - - K l = K t-1 -- a t - l D t - I Go to step 2. DOPT calls three subroutines-LYES, GRCRIT, MINV. The first one solves Liapunov equations (10). GRCRIT computes the gradient matrix D t (9) and uses solution pt of (10.1) to obtain the performance index value j l = trace (ptVo). In order to guarantee a continuous decrease of J and stability at each iteration, like in DSTAB an adaptive

Stability condition pt > 0 is checked at each iteration by computing (subroutine MINV) and evaluating the determinants (should be positive for stability) of all principal minors o f P l. Input: A(NN) and B(NM) - arrays, containing the overall system state and control miatrices, GS(NM) - an array, containing initial stabilising gain matrix, Q(NN) and R(MM) - arrays, containing appropriate weighting matrices, V 0 ( N N ) - an array, containing the initial state second order moment matrix, A L P H A - an arbitrary positive initial step size, PI, FNI and EPS - values n, u and e, ITMAX, N, M. Output: GOPT(NM) - an array, containing the optimal gain matrix, FGS and FGF - initial and optimal gain matrices norms, PIVS and P I V F - initial and final performance index values, D N O R M - norm of the gradient matrix D , D N O R M I ( I T M A X ) - an array, containing all intermediate values of DNORM, ITER, IND, IER. Variable EPS must be selected sufficiently small, to avoid premature termination of the procedure. IER = 1, if matrix A + BK t has opposite eigenvalues (solutions p t and L t are not unique then) and IER = 0, if there is no error. Subroutine DROB

This subroutine solves P3 by means of the following iterative algorithm: Step 1. Choose N N N arbitrary initial values for f3i ~> 0. Choose N N N arbitrary positive values for Afti. Set l = 1. Step 2. Solve equations (13) and check (14). If satisfied, record K~, i = 1, 2 . . . . . N N N as local robust feedback gains and stop. Otherwise, go to step 3. Step 3. Set l = l + l and / 3 ~ = ~ - x + A f l i , i = 1, 2 . . . . . NNN. Go to step 2. DROB calls two subroutines - RICV and MINV. The first one solves equations (13). MINV inverts local weighting matrices Ri, needed to compute K~ (12). DROB copies local and interconnection submatrices, needed to solve (13) and check (14). This is convenient for the user, since only overall system state, control and weighting matrices are supplied. Input: ALl(L1) and B L ( M 2 ) - a r r a y s , containing local state and control matrices, H(L2) - an array, containing interconnection matrix, Q(L 1) and R(M1) - arrays, containing local weighting matrices, NSIZE(NNN) and M S I Z E ( N N N ) - integer arrays, containing local subsystems state and control dimensions, B E T A S ( N N N ) an array, containing initial values of local degrees of stability (on return contains degrees of stability, for which the overall system is robust), D B E T A ( N N N ) an array, containing increments of degrees of stability, ITMAX, N N N . Output: P(NN) and G R ( N M ) - a r r a y s , containing the solution of the decomposed Riccati equation and robust gain matrix, ITER, IND, IER. IER = 1, if s o m e R i is a singular matrix, IER = 2, if some (Aii, Bii ) is not a stabilisable pair and/or (SQRT(Qi) , Aii ) is not a detectable pair and/or Qi is not a positively (semi)definite matrix and/or R i is not a positively definite

Adv. Eng. Software, 1986, Vol. 8, No. 1

55

matrix. IER = O, if there is no error. Arrays AL and BL are input subsystem by subsystem (subsystem 1 state (control) matrix, subsystem 2 state (control) matrix, etc.). The same refers to Q and R. Elements of NSlZE and MSIZE should correspond to this sequence (subsystem 1 state (control) dimension, subsystem 2 state (control) dimension, etc.). Array H is input like this: interconnection matrices A12, A 13 . . . . . A21, A23 . . . . , A N N N 1, A N N N 2 , etc.

Consider a linear interconnected system, comprised of three second order subsystems. The overall system state and control matrices are:

Input: AL(L1) and A L D ( L 1 ) - a r r a y s , containing local state matrices, corresponding to states without and with time-delays respectively, H(L2) and H D ( L 2 ) - arrays containing interconnection matrices, corresponding to states without and with time-delays respectively, N S l Z E ( N N N ) - an integer array, containing subsystems state dimensions, N N N . Output: S(NN2) - an array, containing matrix S, SNORM spectral norm of S, IER. IER = s, if more than 30 iterations are necessary to compute the sth eigenvalue and IER = 0, if there is no error. Subroutine DSLSA copies local state and control matrices and calls SPCF to perform the pole assignment. Input: AL(L1) and B L ( M 2 ) - arrays, containing local subsubsystems states (without time-delays) and control matrices, N S I Z E ( N N N ) and M S l Z E ( N N N ) - i n t e g e r arrays, containing local state and control dimensions, S P E C T R ( L 1 ) - a n array, containing local subsystems eigenvalues to be assigned. Output: G(NM) - an array, containing the stabilising gain matrix, IER = 1, if s o m e (Aii , Bii ) is not a completely controllable pair and IER = 0, if there is no error. Arrays AL(ALD), BL, NSIZE(MSIZE) are input in the same way as in DROB. Each ith block of array SPECTR of length n 2, i = 1,2 . . . . . N N N , is of the following form" , ~kisi, Oil, --(-Oil, G,)il, Oil . . . .

,

Oili, --COili, COili, Oil i]

where ki~, r = 1, 2 , . . . , s i , are s i eigenvalues with zero imaginary parts and the rest n i - - s i eigenvalues correspond to complex conjugate pairs. As it was pointed out in section 3, NORM and DSLSA can be used to solve problem P1 too. In this case ALD and HD are input as zero arrays. The subroutines presented here have been designed so that the user can easily integrate them into a package of his own. In cases when IND = 0 (problem not solved), the user has to increase the value of ITMAX, since all algorithms are convergent under some special conditions, already discussed.

0.2

0.13

1.3

-- 1.6

0.4

0.3

0.11

0.02

0.5

0.4

1.5

-1.6

0.3

0.2

0.1

0.1

1.3

1.4

0.31

0.4

0.2

0.01

0.12

1.2

0

0.9

0.11

1.17 --2.1 _

0.5

0

0

0

0.4

0

0

0

0

0.1

0.2

0

0

0.1

0.3

0

0

0

0

0.8

0

0

0.9

0.1 _ 0.13

B=

_0

i [

Adv. Eng. Software, 1986, Vol. 8, No. 1

--l.1

For this particular case N N N = 3, N = 6 , M = 4 , L I = 12, L2 = 24,M1 = 6 , M 2 = 8, NSIZE = [2, 2,2] and MSIZE = [1, 2, 1]. The eigenvalues of A are 3.71, 2.31, 1.37, 0.389, 0.061 and --2.25 (real parts). Therefore, the system is unstable. Taking the array, containing the initial arbitrary block diagonal gain matrix as

13I(=

i

o o 0 01

0

-3

0 0

-4

0

0

0

3

4

0

0

0

--2

-1

and A L P H A = 1.5, P I = 5 , FNI=0.5, GAMA=0.1, X = 0.001 after eight iterations we obtain by DSTAB the stabilising gain matrix 15.07 - 5 . 5 5 GS = I -

0

°

0

0

--4.17 --8.24

0

0

0

4.02 - 9 . 7 5

_0

0

0

0

0

-22.9 -1.3

with Re{X,~)=--0.178. Array DE [2.001, 1.95, 1.37. 1.05, 0.64, 0.22, 0.05, --0.178] illustrates the evolution of Re {Xd}. Using GS as an initial stabilising gain matrix we next optimise it by DOPT. In this case we assume Q = I, R = 1 , V0 = 1, ALPHA = 0.5, FNI = 0.5, PI = 1.05, EPS = 0.01. The convergence condition DNORM.LT.EPS is satisfied for 63 iterations and DNORM = 0.0087 is obtained. Intermediate values of DNORM are stored in array DNORMI and some of these are: DNORMI(1) = 90.43, DNORMI(15) = 18.43, DNORMl(45) = 2.08, DNORMI(63) = 0.0087. The performance index value has changed from PIVS = 609.3 to PIVF = 206.85. The optimal gain matrix is

EXAMPLE

All subroutines included in DLSDP have been thoroughly tested at the Institute of Industrial Cybernetics and Robotics for rather high dimension multi-input/nmltioutput systems, e.g. N = 20, M = 15. For a lack of space we give a simple numerical example just to illustrate the solutions of problems P1-P4 by DLSDP.

56

0.5

i

These subroutines solve problem P4. NORM calls three s u b r o u t i n e s - ORTR, ORTH and HQRV used to compute the spectral norms of matrices ~tii , i = 1, 2 , . . . , N N N Aij and fiki] , i, j = 1, 2 . . . . . N N N , i ~ j, copies local and interconnection submatrices, forms matrix S and computes its norm.

5. N U M E R I C A L

0.6

A=

Subroutines N O R M and D S L S A

[ ~ i l , ~ki2 . . . .

1.6

1

GOPT =

i-,4.93-7. 20 0

0

0

0

0

0

--1.98

--3.83

[ 5.75 --22.7 0

0

0

l

0 --6.88 --3,8

Values of initial (GS) and optimal (GOPT) gain matrices norms are FGS = 31.3 and FGF = 30.26 respectively. It should be underlined that in reality DC is always sub-

optimal, since the only optimal control is the centralised one. Therefore, it is important to determine how much the system performance is affected, when DC is applied. Experimental results for different pairs of weighting matrices Q and R are given in Table 1. It is evident that performance degradation in case of DC optimisation is insignificant, while DS poses many advantages with comparison to the centralised ones. For the solution of problem P3 by DROB we input the following arrays (besides NSIZE and MSIZE): A L = [1, 1.3, 1.6, --1.6, 1.5, 1.3,--1.6, 1.4, 1.2, 1.17,--1.1,--2.1], B L = [0.5, 0.4, 0.13, 0.t, 0.2, 0.3, 0.8, 0.9], H = [0.6, 0.4, 0.5, 0.3, 0.2, 0.11, 0.13, 0.02, 0.5, 0.1,0.4, 0.1,0.3, 0.31, 0.2, 0.4, 0.1,0.13, 0.2,0, 0.01, 0.9, 0.12,0.11], BETAS = [0, 0,0], DBETA = [0.1, 0.1, 0.1]. Q and R are assumed to be identity matrices and the respective arrays, containing them are Q = [ 1 , 0 , O, 1, 1 , 0 , 0 , 1 , 1 , O, 0 , 1 ] a n d R = [ 1 , 1 , 0 , 0 , 1 , 1 ] . Condition for robustness (14) was satisfied for three iterations and thus local degrees of stability, stored in array BETAS = [0.2, 0.2, 0.2] are obtained. The robust gain matrix is

V; -430

°"l; o°

0

0

--3.08 --53.9

25

--200.1

0

0

0

0

-]

0

0

0

o

J

--2.59 0.89

Robustness of the system was checked for different spectral norms of the interconnection matrix H which obviously corresponds to SP between the subsystems. Experimental results are given in Table 2, which illustrates the robust property of the overall system, retaining its stability under different SP. Let ,~ be a matrix, corresponding to time-delays in local states and interconnections:

--70.49

0

80.3

19.13 --44.2

0

0

0

0

-21.44

11.3

Next we input ALD and HD as zero arrays to illustrate the solution of problem PI by these subroutines. By NORM we obtain SNORM = 1.341. Array S P E C T R - [ - - 1 . 4 , 0, O, --1.43, --1.47, 0, 0, --1.5, --1.51,0, 0,--1.53] is input and by DSLSA we obtain the stabilising gain matrix: -3.8 --0.82

G=

L 0

0

0

0

0

0

0

0

0

0

0

oo

°o

--4.75

1.8

4J

with Re {;~a} = - 2 . 2 0 1 for the closed loop system A + B G . This gain matrix can be optimised by DOPT. It is evident that different solutions will be obtained for a different choice of ALPHA, PI, FNI, GAMA, X, EPS, GK, BETAS, DBETA, Q and R. This is illustrated s by the solutions of the same numerical example for different values of the above mentioned variables and arrays. Thus the designer can perform a set of experiments and choose the 'best' solution in accordance with his specific needs.

0.41

0.13

0.12

0.53

0.73--

0.53

0.53

0.32

0.41

0.62

0.11

0.87

0.13

0.32

0.43

0.02

O. 1

0.99

0.01

--0.11

0.75

0.03

0

6. CONCLUSIONS

0.2

0

0.01

0

0.31

1.2

0.01

0.2

0.1

0.99

1.5 _

Several DC design methods are presented. A FORTRAN package solving different control problems in a fully decentralised way is described. The methods have rather simple software realisation and all subroutines included are based on numerically stable algorithms. Only one-dimensional arrays are used and calculations are done with double precision. A numerical example, illustrating the package itself, is solved too. Experiments performed with DLSDP show rapid convergence of the iterative algorithms for high dimension systems.

Comparison between centralised and DC optimization

Performance index J Weighting matrices Q

R

Centralised optimisation

DC optimization

101 251 1001 I

I 1 ! 51

230.14 315.83 411.57 523.62

232.43 318.17 416.28 528.09

Table 2.

I3° i94° °li 0

G=

-0.87

_0.1

Table 1.

Array ALD contains the diagonal blocks and is input like AL. Array HD contains the off-diagonal blocks and is input like H. By subroutine NORM we obtain the norm of matrix S - SNORM = 3.781. Therefore, all eigenvalues of the local state matrices Aii, i = l, 2 , . . . , N N N must be assigned to the left of --3.781. Local spectra assignment is done by subroutine DSLSA. We input array SPECTR = [ - 4 , 0, 0, --3.89, - 3 . 8 3 , 0, 0, --3.85, - 3 . 8 1 , 0 , 0, -3.99]. The block diagonal gain matrix, assigning the local subsystems eigenvalues at the desired positions and stabilising the overall TDS is:

Robustness o f the overall system under SP

Interconnection matrix H norm

Real part of hd

0.637 0.542 0.417 0.383 0.109

-0.813 -2.13 -0.65 -0.73 --0.14

REFERENCES

1 Wang, S. H. and Davison, E. J. On the stabilisation of decentralised control systems, 1EEE Trans. Automat. Contr. 1973, 5,473 2 Siljak,D. D. Large Scale Systems: Stability and Structure, NorthHolland, New York, 1978 3 Armentano, V. A., Singh, M. G. A new approach to the decentralised controller initialisation problem, Proc. 8th Trien. Worm Congr., IFAC, Kyoto, Japan, 1981 4 Geromel, J. C., Bernussou, J. An algorithm for optimal decentralised regulation of linear quadratic interconnected systems, Automatics 1979, 15,489 5 Singh, M. G. Decentralized Control, North-Holland, Amsterdam, 1981, pp. 13-35 6 Mori, T., Fukuma, N. and Kuwahara, M. Simple stability criteria for single and composite linear systems with time-delays, Int. J.

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(bntr. 1981,6, 1175 Petkov, P. H., Hristov, N. D. and Konstantinov, M. M. t,'ORTRAN Programs ./or Analysis arid Synthesis of Linear MMtil,ariable Systems (in Bulgarian), Technika, Sofia, 1983

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Adv. Eng. Software, 1986, Vol. 8, No. 1

I'opchev, I. P., Savoy, S. G. l)ecentralised stabilis~tion, opt imisation and robust control design for large scale systems, l#'~c, flrd InterHational 6bn/~'rc~lce o~ S.v,~'/cmsk'tLgiHeeciH*~,,I)ay t~n, 1!SA, 1984