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 y2 m3 y3 m1 y1 m2 z21 x31 f1 z11 x21 f2 f3 z31 x11 x32 z12 x12 z22

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 x z 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 z=C(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 • Let’s 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 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

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 x=K(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 γ 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(γ),x(γ)) • If γ gives the optimal solution, stop. Otherwise update γ the following way: and go to Step 2

33. Coordination algorithm Is optimality achieved No Select γ Coordinator Yes Solve m(γ),x(γ) Sub-system decision unit m(γ) m(γ) 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 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: 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 γ find optimal pair (m(γ),x(γ)) 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 k=0 • 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 γ0=γk γk and set k=k+1 and go to 3.

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 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.