Structural modelling and optimisation of large scale systems P.P. Groumpos Indexing term.?: Structural opiimisation and confro/, Decentralised optimisation, Large scale systems Abstract: As engineering systems become more complex, the need for ‘optimal’ designs becomes apparent and very important. A centralised approach can be economically unfeasible, owing to computer limitations, or because it is too complex and cumbersome to be implemented. In this paper, an approach using a suboptimal con- troller is proposed leading to a structural optim- isation problem of large scale systems (LSS). The mathematical formulation of this problem is pro- vided. A new structural controller is developed, and a step-by-step algorithm to design it is pro- vided. An overall cost is defined and can be com- puted from the proposed algorithm. Two simple examples illustrating the applicability and useful- ness of the new structural optimisation approach are included. I 1 Introduction In recent years, an increasing amount of research has been devoted to the study of large scale and complex systems 11-15]. A complex system is often made up of a few and/or many interrelated subsystems. When a complex system is analysed, there is a need to investigate the impact of any of the subsystems on the overall system performance. In addition, each subsystem should have some information (total or partial) regarding the behav- iour and objectives of the overall and/or other subsystem if it is to meet its own objective. A very interesting and challenging problem of these systems is their structural analysis and optimisation. Although studies in large scale systems (LSS) [I 1, [3- 61, multilevel hierarchical systems [2], and multilevel decision support problems (DSP) [SI, have been per- formed during the last two decades, it is interesting to make some brief historical remarks about the importance of the interaction of subsystems as part of the LSS inves- tigated in societal and physical systems. It is reported that, as early as 1830, Gauss, and later in 1843, his student, Gerling, worked successfully in solving simplified systems of equations with a predominant prin- cipal diagonal [l]. The classical competitive equilibrium problem in economics has been studied extensively since the 18th century. In 1874, Leon Walras [4], the founder of the Lausanne School, proposed a vigorous formula- tion of a competitive economy and brought about full 0 IEE, 1994 Paper 9441D (C8, CY), received 12th May 1992 The author is with the Laboratory of Automation and Robotics, Department of Electrical Engineering, University of Patras, Rion 26001, Greece IEE Proc.-Control Theory Appl., Vol. 141, No. I, January 1994 recognition of general equilibrium theory. His method was iterative which might never converge to any equi- librium set price, and Walras was aware of this [4]. He argued then that supply and demand for any given com- modity are influenced by changes in its own price rather than by changes in all other prices. In this way, when the change of prices is determined, the dominance of the price in its own market overrides the cross-coupling effects from other markets; after each step, the prices get closer to their equilibrium values. All of this was done in an ad hoc way, and it was not until the work of Paul Samuelson in 1942, that the competitive market was explicitly treated as a dynamic system. Here the con- vergence problem was recognised as a stability problem, and a whole host of results (such as D stability) from the well known theories of differential equations were made available to economists to resolve the problem of the competitive equilibrium. It is obvious then that a form of structural optimisation, diagonal dominance, was used as early as the 18th century. In the engineering field, Norm Wiener with his work in the early 1940s on cybernetics 1161, laid the ground- work for serious research in the area of large and complex systems. However, it was late in the 1960s that the first series of studies [2] were conducted at the Systems Research Center of Case Western Reserve Uni- versity by Mesavovic and his coworkers. Since then, numerous studies and contributions have been made in the field of LSS and multilevel hierarchical systems. However, the importance of structural analysis and optim- isation of LSS has not yet fully been understood or inves- tigated. 2 The challenge of structural optimisation of large scale systems The control and subsequent optimisation of any system is a problem that demands careful consideration. This is even more the case when a control law is to be designed which would also satisfy some performance of an LSS. Kuppuraju et al. [9] points out some of the challenging questions to this problem, while Groumpos [lS] men- tions seven reasons which make the problem of control and optimisation of LSS a very difficult one. Let us briefly discuss some of them. (i) Large complex systems are made up of interrelated subsystems. Knowledge of the overall system is usually inexact, and the models representing it are not very precise ones. (ii) There is a structure of semiautonomous controllers functioning to meet ‘local’ objectives as well as contribu- tions to ‘overall’ objectives. Some decisions can be made concurrently while some can be made sequentially. In addition, for these decisions to be made, the constraints 1
Transcript
Structural modelling and optimisation of large scale systems
P.P. Groumpos
Indexing term.?: Structural opiimisation and confro/, Decentralised optimisation, Large scale systems
Abstract: As engineering systems become more complex, the need for ‘optimal’ designs becomes apparent and very important. A centralised approach can be economically unfeasible, owing to computer limitations, or because it is too complex and cumbersome to be implemented. In this paper, an approach using a suboptimal con- troller is proposed leading to a structural optim- isation problem of large scale systems (LSS). The mathematical formulation of this problem is pro- vided. A new structural controller is developed, and a step-by-step algorithm to design it is pro- vided. An overall cost is defined and can be com- puted from the proposed algorithm. Two simple examples illustrating the applicability and useful- ness of the new structural optimisation approach are included.
I
1 Introduction
In recent years, an increasing amount of research has been devoted to the study of large scale and complex systems 11-15]. A complex system is often made up of a few and/or many interrelated subsystems. When a complex system is analysed, there is a need to investigate the impact of any of the subsystems on the overall system performance. In addition, each subsystem should have some information (total or partial) regarding the behav- iour and objectives of the overall and/or other subsystem if it is to meet its own objective. A very interesting and challenging problem of these systems is their structural analysis and optimisation.
Although studies in large scale systems (LSS) [I 1, [3- 61, multilevel hierarchical systems [2], and multilevel decision support problems (DSP) [SI, have been per- formed during the last two decades, it is interesting to make some brief historical remarks about the importance of the interaction of subsystems as part of the LSS inves- tigated in societal and physical systems.
It is reported that, as early as 1830, Gauss, and later in 1843, his student, Gerling, worked successfully in solving simplified systems of equations with a predominant prin- cipal diagonal [l]. The classical competitive equilibrium problem in economics has been studied extensively since the 18th century. In 1874, Leon Walras [4], the founder of the Lausanne School, proposed a vigorous formula- tion of a competitive economy and brought about full
0 IEE, 1994 Paper 9441D (C8, CY), received 12th May 1992 The author is with the Laboratory of Automation and Robotics, Department of Electrical Engineering, University of Patras, Rion 26001, Greece
IEE Proc.-Control Theory Appl., Vol. 141, No. I , January 1994
recognition of general equilibrium theory. His method was iterative which might never converge to any equi- librium set price, and Walras was aware of this [4]. He argued then that supply and demand for any given com- modity are influenced by changes in its own price rather than by changes in all other prices. In this way, when the change of prices is determined, the dominance of the price in its own market overrides the cross-coupling effects from other markets; after each step, the prices get closer to their equilibrium values. All of this was done in an ad hoc way, and it was not until the work of Paul Samuelson in 1942, that the competitive market was explicitly treated as a dynamic system. Here the con- vergence problem was recognised as a stability problem, and a whole host of results (such as D stability) from the well known theories of differential equations were made available to economists to resolve the problem of the competitive equilibrium. It is obvious then that a form of structural optimisation, diagonal dominance, was used as early as the 18th century.
In the engineering field, Norm Wiener with his work in the early 1940s on cybernetics 1161, laid the ground- work for serious research in the area of large and complex systems. However, it was late in the 1960s that the first series of studies [2] were conducted at the Systems Research Center of Case Western Reserve Uni- versity by Mesavovic and his coworkers. Since then, numerous studies and contributions have been made in the field of LSS and multilevel hierarchical systems. However, the importance of structural analysis and optim- isation of LSS has not yet fully been understood or inves- tigated.
2 The challenge of structural optimisation of large scale systems
The control and subsequent optimisation of any system is a problem that demands careful consideration. This is even more the case when a control law is to be designed which would also satisfy some performance of an LSS. Kuppuraju et al. [9] points out some of the challenging questions to this problem, while Groumpos [lS] men- tions seven reasons which make the problem of control and optimisation of LSS a very difficult one. Let us briefly discuss some of them.
(i) Large complex systems are made up of interrelated subsystems. Knowledge of the overall system is usually inexact, and the models representing it are not very precise ones.
(ii) There is a structure of semiautonomous controllers functioning to meet ‘local’ objectives as well as contribu- tions to ‘overall’ objectives. Some decisions can be made concurrently while some can be made sequentially. In addition, for these decisions to be made, the constraints
1
associated with each subsystem and the overall system are different, and often not known to all subsystems.
(iii) The problems associated with each subsystem and the overall system are numerous and multidisciplinary. Some are: stability, controllability, speed of response, choice of identification of critical parameters to one sub- system but not to the overall system. For example, a control law designed only on local subsystem informa- tion, which might meet local objective functions, could lead to an overall unstable system.
(iv) All of the information required for making a deci- sion may not be available [SI. This is true in LSS because the transference of the whole information vector throughout the LSS is often impractical, uneconomical and sometimes impossible. In some other cases, there might be more information than needed about a local subsystem, not related to the local decision process. The problem here is how does a decision maker (controller) decide which information is relevant and which is not.
(v) In Reference 9 we see that there are multiple meas- ures of merit, and some of them may not be equally important to the final overall objective. There exists a qualitative goal of efficient, secure and reliable operations expressed in terms of a number of more specific, import- ant quantative performance measures often described as 'local' versus 'global', which are possibly incommensurate and conflicting. Moreover, today we do not have well formulated and accurate objective functions for address- ing this particular problem.
The above observations and those made [9] and [l5] point out that the problem of structural optimisation of LSS is in its infancy. It is appropriate to mention a few of today's approaches that address this problem. Late in the 1960s, and early in the 1970s, the theory of hierarchical multilevel systems proposed a number of techniques [2]. The LSS was classified as a stratified structure, a multi- layer structure or a multiechelon structure, with the two- level structure being the most often considered. The main objective here has been to coordinate harmoniously the overall LSS. To accomplish this, three coordination prin- ciples have been proposed : (i) the interaction prediction, (ii) the interaction decoupling, and (iii) the interaction estimation. Since then, there have been a number of similar approaches such as those cited [6-91.
The decentralised techniques discussed [4, 5, 111 address the question about the conditions under which there exist a set of appropriate local control laws, whether it will meet local performance criteria and control the overall system. Although this general decentralised syn- thesis technique is constructive, when the actual imple- mentation of these controllers is considered further, constraints and limitations emerge. In practical terms, it is generally impossible to design and connect all decen- tralised controllers of the LSS system simultaneously [l5]. Furthermore, there is no way of accessing the overall performance of the LSS.
Another approach is the pertubational approach [lo] in which the subsystems are first decoupled, and then optimal control laws are designed optimising only local objectives. This approach assumes that goal harmony among the subsystems is the overall objective, and that the subsystems will cooperate in the minimisation of an overall objective, which is the summation of all the local objectives. However, this is not very realistic or practical. With the pertubational approach, a global corrective control law must be designed to take into consideration the neglected interconnections when the decoupled local
2
optimal control laws were designed. The combined local plus global corrective control law is a suboptimal one, and a suboptimality index can be determined. However, there appears to be no overall objective function that can be optimised.
Other approaches are the dynamical hierarchical control [3], the multilevel hierarchical one [4, 141, and the multitime scale approach, [12] in which singular per- tubation theory is used.
From all these approaches, we can conclude that the optimisation problem of LSS is still an open one. In the research reported here, a new approach is used.
3 Mathematical modelling
The BAS-decentralised LSS model consists of N sub- systems, s,, s,, . . . , s, uncoupled from each other and interconnected with a central common subsystem, So. The mathematical model in state space description of such a system in a decomposition form is as follows:
s o : i o ( t ) = Ao(t)xo(t) + c A0j ' )X j t ) + Bo(t)uo(t); j = I
xo(t ) = xoo (14 where xi(t), xo(t), ui(t) and uo(t) are the n,-dimensional, no-dimensional state vectors, r,-dimensional, ro- dimensional control vectors for subsystems Si and So, respectively. Then the matrices A i l ) , A,(t)BAt) and Bo(?) describe the dynamics and control distribution for Si and S o , respectively. The matrices Aio(t) and A,,@) represent the interconnections (or information transfer) from Si to So and S j to So, respectively.
The BAS decentralised LSS described in eqn. 1 can be written in the standard overall description as follows: I:"]
= 1 i N ( t )
X O ( t )
Since the plant dynamics A(t) depicts an arrow structure, the system of eqn. 2 is referred to the block-arrow struc- ture (BAS) LSS model. The control problem associated with the BAS-LSS model in eqn. 2 in an oversimplified manner is as follows. Given eqn. 2 and some performance measure, determine partially decentralised controllers u,(t) and uo(t), utilising the same information exchange as the BAS model such that the state trajectories xi(t) for i = 1, 2, ..., N and xo(t) behave in an acceptable way.
I E E Proc.-Control Theory Appl., Vol. 141, No. I , Jonuary 1994
For the BAS-LSS model which is linear time varying, we assume that the initial time t o , the initial state x(to) and the final time t, E (to, 00) are all given. Our goal is to determine a BAS controller.
u(t) = [uT(t), uT(t), . . . , u%), 4 t ) l ' which approximately minimises the following objective function :
4 x O 3 t o , t f , u,)
= xT(t , )Hx(tf)
+ l > x T ( t ) Q W ( r ) + uT(t)R(r)u(tl1 dt (3)
where t, is fixed, x(t,) is free, and u(t) is free and u(t) is unbounded. The matnces H , Q(t) and R(t) are symmetric, n x n and r x r dimensional, positive semidefinite, semi- definite and definite, respectively. It should be pointed out here that the matrices H and Q(t) since associated with the state, assume a BAS form, while R(t) without loss of generality is assumed to be blocked diagonal [17, 181. In other words, the objective here is to find a BAS controller of the following form:
which approximately minimises eqn. 3. In addition, the BAS controller of eqn. 4 is to have the following four highly desired technical features:
(i) utilisation of the interconnections of the BAS-LSS model
(ii) parallel implementation for online computations (iii) ability to provide means of determining the associ-
ated overall cost and (iv) structure flexibility in the sense that the addition
and/or deletion of a subsystem, S i , with the BAS model specifications and constraints does not require the solu- tion of the whole problem from the beginning.
The linear control problem, as posed above, is a sub- optimal control problem because of the restrictions imposed on the controller u(t), i.e. to assume a BAS struc- ture, eqn. 4. The optimal centralised solution would produce a full feedback matrix G(t).
The following BAS controller was developed [17, 181.
ui(t) = Gi(t)xi(t) + Gio(t)xo(t) i = 1, 2, . . . , N ( 5 4 N
uo(t) = Goi(t)xi(t) + Go(t)xo(t) (56)
where the feedback gain matrices G,(t), Gio(t), G o i t ) and Go(t) are ri x n i , ri x uo , ro x ni and ro x no dimensional, respectively. This decomposition of A t ) and uo(t) admits the corresponding utilisation of the interconnections between the subsystems in a BAS form. Since any con- troller is an input to the system and not a property of the system, without loss of generality the control uo(t) is a separate controller [17, 181:
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Now eqns. 5a, 5b and 5c can be integrated to a single control vector u(t).
= G(t)x(t) (6) To accomplish the above goal we first form the following partitioned partially decentralised controller structures:
where each can be associated with the autonomous sub- system pair ( S i , So), i.e.
AiO(t)
It is easy to see that the integration of eqn. 7 into a single control vector u(t) gives the desired BAS controller of eqn. 6. It should be noted here that the partitioned par- tially decentralised controllers of eqn. 7 possess the desired technical features (i), (ii) and (iv).
Having established the system modelling and control decomposition, a similar decomposition is required for the BAS quadratic cost functional of eqn. 3. This is easily accomplished by assuming that the appropriate block weighting elements of H and Q are also separable. In other words, we have
N N
H , = E H i and Qo(t) = ,I Q&(t) (9) j = 1 I = I
such that
are positive semidefinite and positive definite respectively, and constitute the corresponding weighting matrices for the partitioned autonomous subsystem pairs (Si, So), i = 1, 2, ..., N,eqn. 8.
Since the control problem of time invariant LSS is most often considered, and to provide some illustrative examples and compare them with other approaches of today, we will consider the solution to a linear time invariant BAS-LSS model, but before we do so, let us provide some justification for the proposed structural BAS modelling and control of LSS.
4 Why the structural-BAS approach?
As we said in Sections 1 and 2, there is a need to consider structural LSS models and to address the control and optimisation problem associated with them. A very satis- factory structural LSS model is the BAS-LSS model [l5, 17, 181.
3
the dynamically interconnected system has not been referred to as a BAS model.
In Reference 12 a model for composite LSS that has a timescale property which can be arranged in a two level
,= 1 j # t
+ Ao(t)xo(t) + B,(t)uo(t) ( I l b ) In a matrix overall form, the above expanded system is
"\
rather than the uncoupled system from which other sub- systems (SI, S , , . , . , s,) occupy the lower hierarchical level. The So subsystems are considered slower than the N subsystems. However, the system dynamics matrix A assumes a BAS-LSS model. In the same work [12], an example is given for power systems (load frequency control), which demonstrates the applicability of the BAS-LSS model to real life complex systems. However, the controller developed is determined using the singular pertubation theory, and is applicable only to time invari- ant systems. The proposed structural BAS modelling and control approach is more general and applicable both to time-varying and time-invariant systems.
In Reference 19, p. 108 and p. 137, a similar model is considered, and is referred to as having an arrow shape or a block angular structure. It is also stated that the structural BAS model can be obtained through a simple permutation matrix operation. In Reference 20, the con- trollability, observability and stabilisation with a decen- tralised feedback controller for a BAS-LSS model have been investigated.
Another good reason for considering the new approach is that the proposed structural BAS approach seems to address the structural optimisation problem of LSS while meeting the four desired technical features in a unique way.
The approach of considering the partitioned partially decentralised controller structures of eqn. 7 with the
hierarchy has been considered and analysed. The model assumed there is not as general as the BAS model pro- posed here, in that it assumes the So subsystem, having its own controller uJtl. is in the second level of hierarchv
and eqn. 12 can be represented in the following compact form:
(13) i ( t ) = A ( t ) i ( t ) + B(t)ii(t)
where Z(c) and &t) are the expanded versions of A(t) and B(t), from eqn. 12. The expanded system 1 can be obtained from the original system in such a way as to - include all the information concerning 1. The expansion 1 is accomplished using a linear operator T(t ) , defined as follows:
i ( t ) = T(t ) x ( t ) (14) where T is an ( n + N x no) x ( n + no) matrix with full column rank. When this is accomplished, we say that includes the system 2. It can be shown that the control as well as the stabilitygalysis of 1 can be carried out on the expanded system 2 and the solutions mapped onto the original space of E.
Let us define that what we mean by a sy2tem 1 includes the system e. Let us denote J and J as th_e quadratic performance indices associated with systems, respectively.
De$nition I : We say that the pair a, 1) includes the pair (1, J ) if there exists a matrix T(t ) and its generalised inverse T'(t) such that, for any initial state x(to) of 1, the
IEE Pruc.-Control Theory Appl., Vol. 141, Nu. I , January 1994
and
Let us refer to model of eqn. 2 as mode] of eqn, 12 ,,E, Let us define
and the expanded
4
choice
?(to) = T(t )x( to) (1 5) of the initial state i ( t o ) o f x implies
x [ t ; x( to)] = T'(t)i(t, X,,) (16) For the linear time invariant case, the transformation T and its generalised inverse T' are constant matrices and then eqns. 15 and 16 can be simplified. For this case, the following theorem holds:
Theorem I ; For the linear time invariant case of eqns. 2 and 13, the expanded system (E, J ) includes the original system (c, J ) if there exists a constant matrix T and its generalised inverse T' such that M T = 0 for a constant complementary matrix M , such that
A = T A T ' + M and B = T B
Proof: The state trajectory of 1 is
x( t ) = Q(t - to)x(to) + r Q ( t - T)Bu(T) d? (17) 10
where Q(t - to) = is the state transition matrix of
The state trajectory of 1 is c.
i ( t ) = @(t - to)
= i ( t 0 ) + @)(t - T ) B U ( ? ) dz (18)
- where @t - to) = ez(t-ro) is the state transition matrix of
Using the hypothesis of the theorem, and that of eqn. 15 which also holds for T being constant; eqn. 18
1: c. becomes
i ( t ) = [exp [TAT' + M ] ( t - t o )Tx( to )
+ {exp [TAT' + M ] ( t - T)}TBu(T) d z 1: Let us expand exp [TAT' + M ] in series
exp (TAT' + M )
= I + (TAT' + M )
+ I /2[ (TAT' + MXTAT' + M ) ] + . ' . Now multiply eqn. 20 by T and we have
[exp (TAT' + M ) ] T
= T + (TAT' + M ) T
+ 1/2[(TAT' + M)(TAT' + M ) ] T + ... = T + ( T A + M T )
+ 1 / 2 [ ( T A T ' + M ) ( T A + M T ) ] + . . .
Since M T = 0, eqn. 21 reduces to
[exp (TAT' + M)]' = T[I + A + 1/2A2 + . . . ]
= T exp ( A )
Now eqn. 19 becomes
{exp I 4 t - to)l}x(to)
+ I+P CA(t - 411Bu(4 dT
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Therefore i ( t ) = Tx( t ) for all time, and this completes the proof.
The proposed structural BAS approach is also consti- tuted with the general philosophy of the control of LSS through a hierarchically ordered subsystem approach [14 ] . Given a LSS which is sparse, it is transformed to a state space model in which the A and B matrices appear as lower block triangular form. This transformation effect- ively corresponds to transforming the LSS to a structural hierarchically ordered system. The overall optimisation is then determined as a sequence of optimisation problems starting with the first subsystem on the top of the hier- archy and proceeding to the bottom of the hierarchy to produce an overall controller for the LSS. In building the overall computer, which is the union of the subsystem controllers, the resulted gain matrix K of the LSS takes also the same lower block triangularform of the A and B matrices. Both offline and online computational require- ments are reduced by the proposed 'piece-by-piece' hier- archical ordered approach when compared to the centralised optimal linear quadratic Gaussian (LQG) design. These savings are the result of the fact that the overall gain matrix is of lower block triangular form. The overall controller is suboptimal but stable. The structural BAS approach also exploits the structure of the state model, and proposes an overall controller which takes the same form of the A matrix, This is addressed in the following Section.
5 The structural BAS-controller
In Section 3, a time-varying case was presented. However, the linear time invariant control problem will be discussed. The approach followed to develop an approximate controller of the form of eqn. 6 is that of the direct arguments approach [21 ] . This approach has been chosen in References 17 and 18 because it advant- ageously utilises the linearity and the structure of the system as well as some properties of quadratic forms. Furthermore, it automatically provides conditions that this structural BAS controller must satisfy (given an expression for the associated cost of implementing the new control algorithm), and it clearly identifies the effect of the interconnections in the solution.
The linear time invariant BASLSS control problem is as follows :
i ( t ) = Ax(t) + Bu(r) x(t,) = xo (24)
where the overall state matrix A assumes a BAS form, and B is a block diagonal matrix, the associated quadra- tic cost function
J [ x o , t o , 00, u(t)] = [x'(t)Qx(t) + u'(t)Ru(t)] d t (25)
is to be approximated by a time invariant BAS controller of the form
i:
Since Q assumes a BAS form, eqn. 25 is referred to as a BAS quadratic cost functional.
Now we seek conditions under which the BAS quadra- tic cost functional of eqn. 25 is finite, and hence the infin- ite interval problem is well defined. This cost functional is finite if both lim Ilx(t)ll and is lirn ilu(t)II as t + a, are
5
zero. This limiting property of the state x(t) and control ~ ( t ) as t -+ to is assured by the following proposition.
Proposition 1 : The set of time invariant partially decen- tralised controllers uAt), u,(t) and their associated trajec- tories xi(t), x,(t) for which the BAS quadratic cost functional eqn. 25 is finite and nonempty if (i) each of the N autonomous ( S i , So ) pairs given by the time invariant version of eqn. 8 are completely controllable and (ii) the overall time invariant system of eqn. 2 is BAS stabilisable (property of the system which admits the possibility of selecting a BAS matrix P such that the BAS matrix A + BP is a stability matrix). One solution now to our control problem is given by the following theorem [18] :
Theorem 2 : Given the time invariant version BAS decen- tralised LSS dynamic system of eqn. 2 with the condi- tions in Proposition 1 holding and the infinite interval BAS quadratic cost functional of eqn. 25, a suboptimal time invariant feedback BAS decentralised large scale controller
~ ( t ) = - R - ' B T K x ( t )
- _ -
X
K 1 . 0
0 . KN
1'
where the feedback gain matrix K is real, positive defin- ite, and obtained by integrating into a single matrix the partitions
which is the solution to the following algebraic Riccati equation:
K . Ki, Ai Ai, Q Qio
+ [ K f b Kbl[A,i A , l + [ Q f b Q.,l
with Di = Bi R;'B' and Do = Bo RO'B;. The trajectory of the closed-loop time invariant BAS decentralised large scale system satisfies
X( t) = Ax( t )
A101
LA,,. . . . . . A" O N
where 2, = [ A , - DiKi]; Ai, = [A,, - DiKio]; AOi = [Aoi - Do K,,]; and
The cost which the suboptimal controller of eqn. 26 achieves is given by
J ( X , , to, a) = xT(to)CK + Llx(t , ) 00, a) (30)
where L satisfies the Lyapunov equation
A T L + L A = - E (31)
with
Eio i , j = 1,2, . . . , N (denoting index) E = ( E ; E , ]
where
Eii = 0
E i j = K i , A",j + A;j KT, r 7
E, , = K i ] j = 1 j + i
and
proof for the above can be found in Reference 18. The results developed in the above theorem [lg],
have been contingent upon the assumption that lim,-m ilx(t)ll = 0, that is, u p o ~ the assumption that the resulting closed loop matrix A of eqn. 29 is a stability matrix. Therefore to accept the above BAS control scheme of eqn. 27, the stability of the closed loop system of eqn. 29 must be addressed.
Stability is an extremely important qualitative pro- perty of dynamical systems, and it can be considered as a necessary condition of acceptability of any control design, which satisfies some objective functional.
Here, sufficient conditions are given for the stability of the closed loop linear time invariant BAS decentralised large scale system of eqn. 29. These conditions are derived from the Lyapunov stability theory and from analysing the strength of the interconnections of the closed loop system of eqn. 29.
For a time invariant linear dynamical system
i ( t ) = A x ( T )
the matrix A is a stability matrix, if and only if, for any real positive definite symmetric matrix N , the solution for the real symmetric matrix P of the Lyapunov equation.
A T P + P A = - N (32) is positive definite.
Focussing now on the problem at hand, eqn. 32 above could very well be applied to determine the stability of the linear time invariant system of eqn. 29. In this case though, its usefulness as a practicaL tool is limited due to the fact that the system of eqn. 29 is of large dimension- ality. However, by changing its hypothesis slightly, and by using the partially decentralised control scheme devel- oped above, the following condition which is simpler but sufficient for the stability of the system of eqn. 29 can easily be established.
Theorem 3 : Let 0 = (Q + K B R - ' B T K - E ) be a com- posite real symmetric matrix where the matrices Q, R , B, K , and E are as defined previously. Then the linear time
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invariant closed loop BAS decentralised large scale system of eqn. 29 is asymptotically stable if the composite matrix 0 which satisfies the Lyapunov equation
ATK + R A = -0 is positive definite or if
L,JQ + K B R - I B ' K ) > AmaX(E) (33) where A m i n ( . ) and A m a x ( . ) indicate the minimum and maximum eigenvalues, respectively, of the arguments.
The proof for this can be found in Reference 18. Having established the above theorems, the following
structural BAS optimisation approach is proposed. Step I: Given a LSS, obtain an equivalent BAS model,
through simple similarity transformations. S t e p 2: Solve N independent algebraic Riccati equa-
tions, eqn. 28, for
Then integrate them into a single BAS feedback gain matrix
. . . . . . N
Kg0 C K ' , j = I
Step 3: Compute the closed loop matrix 2 = [ A - B R - ~ B ~ K ~ .
Step 4 : Using the results from Step 2 and Step 3,
Step 5: Compute the composite symmetric matrix = [ Q + K B R - ' B T K - E ] . Step 6 : If 0 is positive definite or if A,,(Q
+ K B R - ' B T K ) > Amax(€) continue, or else select differ- ent positive definite weighting matrices Q and/or R and go to Step 2.
Step 7: Determine the partially decentralised control law given by eqn. 27.
Step 8 : Solve the Lyapunov eqn. 31 for the unique matrix L.
Step 9 : Compute the associated overall cost J ( x o , t o , CO) = x:(K + L)xo .
Let us discuss the significance of this structural BAS optimisation approach. First of all, we can see that the four desired technical features are met, and in a dynamic optimisation way. The parallel implementation feature is a very important one, since most of today's techniques, if used on the BAS-LSS model, cannot use such a feature. The ability to determine the overall cost J , eqn. 30, is also unique to this approach.
The feedback gain matrix K is independent of the state x(t) and the initial time t o and hence, the suboptimal BAS control law, eqn. 27 can be determined offline. Then each subsystem SI, S , , ..., S , can set up its own feedback controller if it has access to its own local states and the states of subsystem So. Direct state information from other subsystems is not required. The only subsystem that would require access to the states of the N sub- systems is the subsystem So, but this is consistent with the BAS approach.
For the subsystem of eqn. 24, determination of the optimal cost would require the computation of the optimal feedback gain matrix K , which would be a full matrix. This would require the solution of an algebraic
compute the matrix E.
I E E Proc-Control Theory Appl., Vol. 141, No. I , January 1994
Riccati equation with n(n + 1)/2 unknown, where n = If=, n, + no. On the other hand, the proposed BAS approach would require the computation of only N separate and independent algebraic Riccati equations, each with (ni + no)(n, + ni + 1)/2 unknowns for i = 1, 2, . . . , N . Moreover, since the algebraic Riccati equations are separate and independent from each other, parallel processing can be used to obtain their solutions.
The proposed BAS control scheme takes into account the effect of the interconnections between subsystems So and the other N subsystems S , , S, , . . , , SN in a very neat and unique way (with the two-at-a-time solution). Fur- thermore, if we are to compare it with other approaches, such as the multilevel [4] or the pertubational [lo], the BAS controller, eqn. 26 can be expressed as
u(t) = uyt) + U"t)
where U'([) and ug(t) can be identified as local and global controllers as has been used in the other approaches [2, 4, 5, 10, 12, 191. The global controller ug(t) modifies the effect of the interconnections, and in all of today's approaches is computed after the solution to complete autonomous decentralised system assumption is com- pleted. This is the main reason why the other methods cannot implement the overall control design in a parallel way as with the proposed scheme here. In our approach, the 'local' and 'global' controllers can be computed simultaneously, utilising the information between So and Si. with the autonomous subsystem pair formulation, eqns. 7 and 8.
6 Illustrative examples
Let us demonstrate the applicability and unique features of the proposed structural BAS approach by considering simple numerical examples.
Example 1 : Consider the third-order system
The problem here is to find a BAS control u(t) such that J [ x o , 0, CO, u(t)] is approximately minimised, and compare this solution to the optimal centralised and the total decentralised approaches.
7
Solution: The step-by-step solution to obtain the BAS controller is provided in Reference 18. The BAS controller is found to be
-6.60343 0 uBAs(t) = -KBAsx( t ) = -
0 I 'I 2o 0
and the associated BAS cost (Step 9 of the algorithm) is
where xo indicates an initial condition of the overall state vector, and not just the state of the So subsystem. The optimal feedback control gain matrix K* and the optimal control law u*(t) is found to be
6.61087 0.49825 2.84452 x,(t)
2.84452 0.340 7.75793 xo(t)
0.49825 2.52182 0.340 ][ x,(t)]
and the associated optimal cost is
J*(xo, 0, m) = x,'K*xo
Now when the complete decentralised approach is used, the decentralised control law is
u,(t) = - K , x ( t ) = -
and again the associated complete decentralised cost is
J&o, 0, a) = -6 K D xo
For an initial condition for the state vector of xo = [ l 1 119 the associated cost and the performance degradation AJ, compared to the optimal solution are
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x(t)
9
7 Discussion of results
There are several remarks which can be made, based on the results from the above two examples, and from results of a number of other simulation studies. The BAS and decentralised approaches result in suboptimal control laws. The BAS and decentralised controllers assume a specific structure for the feedback gain K , which is not a full matrix, as in the case of the optimal centralised approach. The suboptimal controllers could lead to insta- bility problems as is the case with the complete decentral- ised case, in Example 2. The problem of how to compare different control approaches is a difficult one. The per- formance degradation is one possible solution, but perhaps not the best one, since we usually use the steady solutions. By comparing the closed loop eigenvalues of the above two examples, it is clear that we should also try to use them to judge a recommended control law. Based on the results, we see that the BAS approach pro- vides a control law with better performance degradation than the complete decentralised approach. In addition, in the case where the decentralised approach leads to an unstable closed loop system (Example 2), the BAS approach provides a stable overall system. In Example 2, the eigenvalues when using the BAS approach are much closer to the eigenvalues of the optimal centralised ones than the complete decentralised approach. However, this is not the case in Example 1, where the decentralised approach provides eigenvalues closer to the optimal cen- tralised approach than the BAS approach.
Computationally the decentralised approach would require the solution of the least number of simultaneous nonlinear algebraic equations, while the optimal central- ised approach would require the most, and the BAS approach would require an intermediate number. Specifically for Example 2, we have the following compu- tational requirements. Required solution of simultaneous nonlinear algebraic equations: Complete decentralised:
In the cases of complete decentralised and BAS approaches, parallel processing could be used in solving separate and independent algebraic Riccati equations.
From a number of our simulation studies, the off diagonal block submatrices of the optimal feedback gain, K * strongly depend on the strength of the intercon- nections Ai , and A o i . In an example where the plant matrix was a 10 x 10 matrix with three subsystems, each of third order and the So subsystem of first order and the interconnection being less than unity (in other words, Ai, < 1 and Aoi < I), the off diagonal block submatrices of K* were relatively small. The optimal centralised cost was 0.321, while the BAS-K feedback gain would produce a cost of 0.318, with a performance degradation of I AJI = 0.9% which is very small. The closed loop eigenvalues of the BAS approach were very close to the optimal centralised approach and stable. The complete decentralised approach lead to an unstable closed loop system.
There is an obvious question of why use the central- ised and not the BAS approach when parallel processing can be utilised and there are computational advantages. Specifically, for the centralised approach of this example to determine the optimal K * , the solution of lO(10 + 1)/ 2 = 55 simultaneous nonlinear algebraic Riccati equa-
10
{2(2 + 2)/2 + 3(3 + 1)/2 + 2(2 + 1)/2} = 12
tions is required. On the other hand, the BAS approach would require four parallel processors each to solve 4(4 + 1)/2 = 10 simultaneous nonlinear algebraic Riccati equations. This parallel implementation, when closed loop stability is attained, and a level of performance degradation is accepted, is the biggest advantage of the proposed BAS control scheme. One last remark wrt the structural flexibility of the BAS approach: compared to the centralised approach, in which the addition or dele- tion of a subsystem requires the solution of the control from the beginning, the BAS approach needs the addition or deletion of a solution to a much smaller problem.
8 Summary and closing remarks
In this paper, the challenge of structural optimisation of LSS has been addressed. A structural two level LSS has been considered and mathematically modelled from the state-space point of view. This system has been referred to as a BAS decentralised LSS. An objective function which explores the structure of the model is developed and a structural suboptimal BAS controller has been developed. This controller is to have four highly desired technical features which have been defined here.
The overall cost of the BAS approach also follows a structural pattern. It consists of two parts: the first is due to the autonomous partially decentralised controllers, when the pairs ( S i , S o ) are considered, and the second is due to both the system and the BAS quadratic cost func- tional decompositions, taking into consideration the interconnections between So and the N subsystems. This clear identification of the overall cost has not been seen in any other decentralised approach, and is the result of using a direct approach in developing the proposed BAS controller.
The proposed BAS approach explores the structural, state information pattern of the plant model in a unique way and produces a BAS controller in which the ‘control information’ actions follow the same structure as the plant model. This is accomplished by utilising the ‘state control information’ in a unique way, i.e. minimising the problem by appropriately weighting the information transfer. The proposed BAS controller provides a way of weighting its global part into the optimisation problem, a highly desired feature but not present in other LSS control approaches.
In conclusion, there is a need for more structural analysis and design of LSS where the structure of the system is better utilised.
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