1 Control of Large Scale Systems Jari Hätönen *,** April 2, 2003 * Department of Automatic Control and Systems Engineering, University of Sheffield, UK.

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

1 2 N

Upp

er le

vel

Low

er le

vel

Lower-leveldecision making

Processlevel1 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

1 2 N

Upp

er le

vel

Low

er le

vel

Lower-leveldecision making

Processlevel1 2 N…

…

12

Mathematical preliminaries

For each subsystem there exists a mapping

iii OIf :

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

The input and output domains are further divided into

iii

iii

ZYO

OMI

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

fiii Ii ii Oo

fi

ii Mm ii Yy

ii Xx ii Zz

15

An example

f1 f2 f3

m1 m2m3 y3

z11 x21 z31

y2y1

z22

z21 x31

x11

x12

z12

x32

16

A general two-level hierarchical system

fm y

C

zx

)(

),(),(

zCx

xmfzy

17

A general two-level hierarchical system

Furthermore, it can be shown that

N

ijiji zCx

1

where each Cij is a matrix where each elementis either zero or one (a connection matrix)

18

A general two-level hierarchical system

f

Czx

y y

z

x

m F

For mathematical tractability it has to assumed that there exists

))(),(),(()(

),,()(:

mSmKmPmF

zxymFmF

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 z=C(x) .

This is especially true if the number of subsystems is large or the subsystem models are complex.

20

DECISION UNITS

Coordinator

1 2 N

Upp

er le

vel

Low

er le

vel

Lower-leveldecision making

Processlevel1 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

iiiii ZYXMg :

or equivalently by using the subsystem model

iii XMG :

where)),(),,(,,(),( 22

iiiiiiiiiiii xmfxmfxmgxmG

23

COORDNINATION

Coordinator

1 2 N

Low

er le

vel

Lower-leveldecision making

Processlevel1 2 N…

…

24

The cost function for the upper-level

In a similar fashion there exists an overall cost function

ZYXMg :

Using the overall process model this can be equivalently written as

))(),(),(,()( mSmPmKmgmG

and in this talk it is assumed that (is this alwaysthe best choice?)

N

iiii mKmGmG

1

))(,()(

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

N

iiii

mKxFDm

xmG1)(

)(),(min

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

Let’s define new modified system descriptions

iiiii

iiiii

ZYXMg

ZYXMf

:~:

~

where is an external coordination variable

In a similar fashion let

),,(~

),,,(~

,,(~),,(~

,21 iiiiiiiiiiii xmfxmfxmgxmG

28

The modified sub-system decision process

The decision unit i has to control the sub-system i so that ),,(

~ iii xmG is being minimisedwith a fixed γ

If the solution exists it is called the γ-optimal solution (m(γ),x(γ))

The objective of the coordinator is to find a γ so that the overall cost function is minimised

))(()(min)(

mGmGFDm

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

))(,()),(,(~

))(,()),(,(~

mKmGmKmG

mKmfmKmf

iiiiii

iiiiii

In other words the modified cost function is equal to theoriginal cost function if the interconnection equation x=K(m)is satisfied.

30

Coordinability

Using the modification in the previous slide coordinability can be defined as:

1. The overall optimisation problem has a solution

2. For each γ the the sub-system optimisation problem has a solution, i.e.

3. There exists (at least one) so that

)()(min)(

mGmGFDm

)),(),((~

),,(~

min)(),(

iiiiifDxm

xmGxmGiii

))(()( mGmG

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

1. An initial guess is made for

2. Sub-system decision units solve their optimisation problems, resulting in (m(γ),x(γ))

3. If γ gives the optimal solution, stop. Otherwise update γ the following way:

))(),(,( xmand go to Step 2

33

Coordination algorithm

Select γ

Pro

cess

leve

lC

oord

inat

orS

ub-s

yste

mde

cisi

on u

nit

Solve m(γ),x(γ)

m(γ)

m(γ)

))(),(,( xmNo

Yes

Is optimality achieved

34

Initial thoughts on decomposition

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

N

iiii

mKxFDm

xmG1)(

)(),(min

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 x=K(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 x=K(m) and “balancing” is needed.

36

The balancing principle

The sub-system “variables” are modified in the following way:

)),,(),,(,,(~:),,(~

),,(),,,(:),,,,(~),(:),,(

~

21

iiiiiiiiiiii

iiiiiiiiiiiii

iiiiii

xmfxmfxmgxmG

zxzyxmgzyxmg

xmfxmf

where

N

iiii zxzx

1

0),,(:),,(

if and only if (the balance condition)

N

ijiji zCx

1

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

N

iiii xmGmG

1

),(~

)(

38

Sub-system decision process with balancing

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

)),(),((),,(~

min)(),(

iiiiiifDxm

xmGxmGiii

Define now

),,(~

min)()(),(

iiifDxm

i xmGiii

39

Coordination in balancing

The modified overall cost function can be written as

N

iiiiii

N

iiiiiiiiii

N

iiii

xmfx

xmfxmfxmg

xmG

1

2

1

21

1

)),,(,(

)),(),,(,,(

),,(~

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.

m ))(),(( xm

))(()( mKx

then (the proof is trivial due the zero-summodification)

N

ii

mG

1

))(()(

41

Coordination in balancing

On the other hand for it is true that

))((

)(),(min

),,(~

min

),,(~

min)(

1)()(),(

1)()(),(

1)(),(

mG

mGxmG

xmG

xmG

N

iiii

mKxfDxm

N

iiii

mKxfDxm

ii

N

ii

fDxm

42

Coordination in balancing

Consequently for all

)()( and the optimisation problem for the coordinatorbecomes

)(max

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 summary1. Set k=0

2. Make an initial guess

3. Solve the sub-system optimisation problems with

4. If the system is balance, stop. If not calculate the gradient of and calculate

γ0=γk

)( k)(11 kkkk

and set k=k+1 and go to 3.

γk

)( k

44

Some remarks on balancing and prediction

During the iterative process the algorithm gives values for γ 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 γ 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 )(min

0)(

vQvR

VSv

The original optimisation problem is recovered if )()(and,

1

mKxvRGQXMVN

ii

where WVRVQ :,: and V,W areare real Banach spaces

47

The Langrange function

Define the Langrange function

)(,)(),( ** vRwvQwvL

where w* is the Langrange multiplier and belongsto the dual space W* of W.

It is easy to show that if there exists **ˆ,ˆ WwVv so that

)ˆ,ˆ()ˆ,(min ** wvLwvLSv

and 0)ˆ( vR then v̂ is the minimising solution

48

The saddle point theorem

Suppose the Langrange function has a saddle point so that**)ˆ,ˆ( WSwv **),( WSwv

)ˆ,()ˆ,ˆ(),ˆ( *** wvLwvLwvL

then 0)(ˆ 1 vRVvSv and

)ˆ()(min1

vQvQSSv

Proof. Omitted

49

The dual function

Consider now the dual function

),(min)( ** wvLwSv

Properties of the dual function

1. is concave and bounded (requires some additional assumptions)

2. It can be shown that

)()(

)0)((wand*

1**

1

vQw

vRVvSWSSv

50

The maximisation theorem

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

)ˆ()ˆ()(max **

**vQww

Ww

**)ˆ,ˆ( WSwv

Proof. Omitted.

51

Cost function modification with Balancing/Langrange

The overall cost function is modified to be

N

i

N

jjjjijii

N

iiii

xmfCx

xmGxmL

1 1

2

1

),(,

),(),,(

Changing the summation order it can be shown that

N

jiiiijj

iiiiiiii

N

iiii

xmfC

xxmGxmL

xmLxmL

1

2

1

),(,

,),(:),,(

),(),,(

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

N

jjjjiji xmfCx

1

2 ),(

are met, the effect of the Langrange modifier on the overall cost function disappears.

53

A toy example

Consider the system

),(2

),(2

222222

111111

umPumy

umPumy

with a coupling 1221 , mumu The cost function is

22

21

22

2 )2()1(),(1

yymmymG

54

A toy example

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

10

17,

5

1(ˆ m

The Langrange modified cost function becomes

)()(

)2()1(),(

122211

22

21

22

2

1

mumu

yymmymG

55

A toy example

This results in the decomposed cost functions

21222

222

12112

12

)2(),(

)1(),(

2

11

muymymG

muymymG

The optimal control actions become

2

1

2

2

1

2

1

1

4

8

24

410,

2

4

24

410

u

m

u

m

56

A toy example

The gradient of is just )(

12

21)(mu

mu

and the update law becomes (coordination algorithm)

12

2111

)(

)(

)1(

)1(

mu

muk

k

k

k

k

where k=0.1 (a sophisticated guess)

57

A toy example

It can be shown that the optimal

)5

4,

5

1(ˆ

In the following material the results from simulation of the balance method with Langrange techniques are given.

The initial guess in the simulations is

)10,10()0(ˆ

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.