Control of Large Scale Systems

1
Control of Large Scale Systems

• Jari Hätönen,
• April 2, 2003
• Department of Automatic Control and Systems
  Engineering,
• University of Sheffield, UK
• Systems Engineering Laboratory
• University of Oulu,
• Finland

2
Introduction

• The design of large (complex) systems commonly
  requires the division of the large system into
  smaller subsystems
• For each subsystem it is necessary to define the
  subsystem model, the objective of the subsystem,
  and the constraints present in the subsystem

3
Introduction

• The overall structure resulting from the
  interconnections of the subsystems can be very
  complex in this talk only Two-level
  hierarchical systems are considered
• In hierarchical systems each subsystem has its
  own decision unit and control unit
• The decision unit and the control unit are
  responsible for making the subsystem to achieve
  its objectives

4
A Two-Level Hierarchical System
Coordinator
Upper level
Lower-level decision making
1
2
N

Lower level
Process level
1
2
N

5
Introduction

• The hierarchical system theory has a strong
  connection with organisational theory!!!
• Also connections with economical models can be
  found, i.e. a market driven economy can be
  considered as a two-level hierarchical systems
  where the prices of the products are the
  coordination variables determined by the
  government of pure competition.

6
Introduction

• The degree of interconnectedness (the more
  interconnected, the more difficult it is to
  obtain overall balance) is highly dependent on
  system design.
• For example in chemical unit processes buffer
  tanks can be used to cut the physical
  interconnection between two units (resulting in
  higher cost)

7
Why hierarchical systems are so important and
common?

• The system design is easier to control (module
  thinking)
• Subsystem allow specialisation, i.e. each
  subsystem is only responsible for its own task
  and does not require information how the overall
  system works.
• Maybe evolution also encourages hierarchical
  systems (i.e. the brain, pre-historic tribes etc).

8
Why hierarchical systems are so important and
common?

• They allow a certain degree of fault tolerance,
  i.e. if a sub-system breaks down, it can be
  easily replaced.
• However, the coordinator is the weak point, i.e.
  if it stops working, the system stops
  functioning.
• Interesting implications to warfare (i.e. Hussein
  and his closest allies were the first ones to be
  attacked).

9
Hierarchical systems and dynamics at different
time-scales

• Large plants have typically subsystems that have
  dynamics at different scales (i.e. in a paper
  machine the paper quality is kept fixed for a
  week, but the paper machine dynamics excited by
  disturbances have dynamics of few seconds).
• Consequently it is natural to take the slow
  dynamics as the upper-level and the fast dynamics
  as the lower-level.

10
Hierarchical systems and dynamics at different
time-scales

• Coordination variables can selected to be for
  example the constant set-points for the
  lower-level decision units, that classically are
  PID-controllers.
• The coordination variables (constant set-points)
  can be selected to be a solution of suitable
  (static) optimisation problem.

11
PROCESS MODEL
Coordinator
Upper level
Lower-level decision making
1
2
N

Lower level
Process level
1
2
N

12
Mathematical preliminaries

• For each subsystem there exists a mapping

where the triplet (Ii,Oi,fi) defines the
input-output model for subsystem i

• The input and output domains are further divided
  into

13
Mathematical preliminaries

• The set Mi are the free inputs of the systems and
  Xi are the interconnected input, i.e the set Xi
  is determined by the behaviour of other
  subsystems
• The Zi is set of interconnected outputs, i.e.
  they are used as inputs in other subsystems. The
  set Yi is the set of free outputs.

14
Free and interconnected variables
fi
fi
15
An example
m3
y3
y2
m1
m2
y1
z21
x31
f1
f2
f3
z11
x21
z31
x11
x32
z12
z22
x12
16
A general two-level hierarchical system
y
m
f
z
x
C
17
A general two-level hierarchical system

• Furthermore, it can be shown that

where each Cij is a matrix where each element is
either zero or one (a connection matrix)
18
A general two-level hierarchical system
F
y
y
m
f
z
x
z
x
C
For mathematical tractability it has to assumed
that there exists
19
Comments on the overall model F

• The whole point is that in practise it can be
  impossible (or impractical) to form explicitly F
  because it is implicitly defined by the
  constraint zC(x) .
• This is especially true if the number of
  subsystems is large or the subsystem models are
  complex.

20
DECISION UNITS
Coordinator
Upper level
Lower-level decision making
1
2
N

Lower level
Process level
1
2
N

21
Decision units

• For each subsystem i there is a decision unit,
  whose objective is to control the subsystem
  according to its own objectives by manipulating
  the input variables mi.
• In this talk it is assumed that the objective of
  the subsystem is to minimise a real-valued cost
  function.
• The upper level decision unit tries to affect the
  lower level decision units so that the overall
  cost function would be minimised, which in this
  talk is the sum of individual cost functions.

22
The cost functions for lower-level units

• More precisely, each subunit attempt to minimise
  the cost function

or equivalently by using the subsystem model
where
23
COORDNINATION
Coordinator
Lower-level decision making
1
2
N

Lower level
Process level
1
2
N

24
The cost function for the upper-level

• In a similar fashion there exists an overall cost
  function

• Using the overall process model this can be
  equivalently written as

and in this talk it is assumed that (is this
always the best choice?)
25
The upper-level decision problem

• The objective of the coordinator is to affect the
  lower-level decision making so that the overall
  cost function G is minimised.
• This optimisation problem can be equivalently
  written as a constrained optimisation problem

26
The upper-level decision problem

• The construction of the overall optimisation
  problem G requires the overall system model F,
  but F is not explicitly available.
• Consequently the coordinator cannot check using G
  if the system has reached its objectives.
• Idea modify the overall optimisation problem so
  that it can be divided into independent
  sub-problems, and the coordinator can manipulate
  the lower-level decision making so that the
  overall optimality would be achieved.

27
Modification

• Lets define new modified system descriptions

where
is an external coordination variable

• In a similar fashion let

28
The modified sub-system decision process

• The decision unit i has to control the sub-system
  i so that

is being minimised
with a fixed ?

• If the solution exists it is called the ?-optimal
  solution (m(?),x(?))
• The objective of the coordinator is to find a ?
  so that the overall cost function is minimised

29
How the modification should be done?

• Whether or not the overall objective is achieved
  depends on how the modification is done. One
  straightforward possibility is to select

In other words the modified cost function is
equal to the original cost function if the
interconnection equation xK(m) is satisfied.
30
Coordinability

• Using the modification in the previous slide
  coordinability can be defined as
• The overall optimisation problem has a solution
• For each ? the the sub-system optimisation
  problem has a solution, i.e.
• There exists (at least one) so that

31
Coordination

• In practise it is impossible to know immediately
  the correct coordination parameter
• An iterative process is needed where the
  coordination parameter is updated so that
  improvement in the overall objective is achieved.
• In this case the coordinator needs a coordination
  strategy which tells how to update

32
Coordination algorithm

• An initial guess is made for
• Sub-system decision units solve their
  optimisation problems, resulting in (m(?),x(?))
• If ? gives the optimal solution, stop. Otherwise
  update ? the following way

and go to Step 2
33
Coordination algorithm
No
Is optimality achieved
Select ?
Coordinator
Yes
Sub-system decision unit
Solve m(?),x(?)
m(?)
m(?)
Process level
34
Initial thoughts on decomposition

• As was defined earlier, the overall optimisation
  problem can be written as

• This cost function is separable in the sense that
  each term Gi(mi,xi) contains only variables from
  the sub-system i
• However, the constraint equation xK(m) makes the
  variables dependent, and the problem is not
  decomposable.

35
The balancing principle

• In the balance principle the interactions are
  removed in order to get a truly decomposable
  system
• The sub-system optimisation is done as a function
  of mi and xi
• As a result the optimal control policy (m,x) does
  not satisfy the constrain xK(m) and balancing
  is needed.

36
The balancing principle

• The sub-system variables are modified in the
  following way

where
if and only if (the balance
condition)
37
The balancing principle

• In the balance principle only the cost function
  is modified and the sub-system model fi remains
  the same.
• The cost function modification is called the
  zero-sum modification because if the system is in
  balance, the effect of the modification
  disappears and the overall performance is just

38
Sub-system decision process with balancing

• For each subsystem i and given ? find optimal
  pair (m(?),x(?)) so that

• Define now

39
Coordination in balancing

• The modified overall cost function can be written
  as

40
Coordination in balancing

• Suppose that the overall optimisation problem has
  a solution and there exits a so that
  the solution is in balance,
  i.e.

then (the proof is trivial due the
zero-sum modification)
41
Coordination in balancing

• On the other hand for it is true that

42
Coordination in balancing

• Consequently for all

and the optimisation problem for the
coordinator becomes

• In practise it often impossible to solve the
  maximisation problem explicitly the best one
  can do is to resort to gradient search

43
Coordination in balancing

• In summary
• Set k0
• Make an initial guess
• Solve the sub-system optimisation problems with
• If the system is balance, stop. If not calculate
  the gradient of and
  calculate

?0?k
?k
and set kk1 and go to 3.
44
Some remarks on balancing and prediction

• During the iterative process the algorithm gives
  values for ? that do not result in balance if
  the balance equations describe for example flows
  in a chemical process, the inputs the algorithm
  gives during iterations cannot be used because
  they do not fulfil physical constraints not
  suitable for on-line applications!!!
• An alternative method called the prediction
  method will always give a ? that satisfies the
  constraints. However, it has its own weaknesses
  and is rarely used in real life.
• Hybrid methods exists that mix ingredients from
  the balance method and predictive method.

45
Further remarks on balancing

• Already in 1970s it was suggested that the
  balancing principle could be solved by using a
  bargaining process.
• Preliminary convergence analysis was done by
  resorting to game theory.
• This can be seen as the first attempt to define
  agents

46
Balancing wiyth Langrange techniques

• Consider now the more general optimisation problem

where
and V,W are
are real Banach spaces

• The original optimisation problem is recovered if

47
The Langrange function

• Define the Langrange function

where w is the Langrange multiplier and belongs
to the dual space W of W.

• It is easy to show that if there exists

so that
and
then
is the minimising solution
48
The saddle point theorem

• Suppose the Langrange function has a saddle point
  so that

then
and
Proof. Omitted
49
The dual function

• Consider now the dual function

• Properties of the dual function
• is concave and bounded (requires some
  additional assumptions)
• It can be shown that

50
The maximisation theorem

• If is a saddle point of the
  Langrange function L, then

Proof. Omitted.
51
Cost function modification with
Balancing/Langrange

• The overall cost function is modified to be

• Changing the summation order it can be shown that

52
Cost function modification with
Balancing/Langrange

• In other words each term Li depends only on the
  variables related to the sub-system i plus the
  Langrange multiplier.
• This is modification is a zero-sum modification
  because when the constraint

are met, the effect of the Langrange modifier on
the overall cost function disappears.
53
A toy example

• Consider the system

with a coupling

• The cost function is

54
A toy example

• It takes two minutes to show that the optimal
  input is

• The Langrange modified cost function becomes

55
A toy example

• This results in the decomposed cost functions

• The optimal control actions become

56
A toy example

• The gradient of is just

and the update law becomes (coordination
algorithm)
where k0.1 (a sophisticated guess)
57
A toy example

• It can be shown that the optimal

• In the following material the results from
  simulation of the balance method with Langrange
  techniques are given.
• The initial guess in the simulations is

58
A toy example
59
A toy example
60
A toy example
61
Conclusions

• A two-level hierarchical systems theory offers a
  very general method to optimise the running of
  hierarchical systems.
• A can be applied on a wide range of applications,
  examples being economics, organisation theory and
  large-scale processing plants.

62
Conclusions

• The theory can be seen as a starting point for
  decentralised control.
• The main idea is to divide the system into
  specialised subsystems and optimise
  independently the running of these sub-systems.
• In order to have harmony a coordinator is needed
  that affects the decision making of lower-level
  decision systems so that an overall balance
  (harmony, satisfaction level etc.) is achieved.

63
Conclusions

• In this talk the balancing method was analysed in
  detail (not suitable for on-line applications)
• The balancing was implemented using the
  well-known Langrange principle.
• Unfortunately, the theory is quite mathematical,
  and utilises the theory of constrained
  optimisation in general Banach spaces.
• Hard for engineers to digest low success in the
  industry

64
Conclusions

• Also other numerical approaches exists for
  large-scale systems a typical example is a
  transportation problem where the special
  structure of the problem is utilised so that
  efficient numerical methods can be found to solve
  the optimisation problem.