Large-Scale Systems

1. Large-Scale Systems Mojtaba Hajihasani Mentor: Dr. Twohidkhah

2. Contents • Introduction • Large-Scale Systems Modeling • Aggregation Methods • Perturbation Methods • Structural Properties of Large Scale Systems • Hierarchical Control of Large-Scale Systems • Coordination of Hierarchical Structures • Hierarchical Control of Linear Systems • Decentralized Control of Large-Scale Systems • Distributed Control of Large-Scale System • MPC of Large-Scale System

3. Introduction

4. Introduction • Many technology and societal and environmental processes which are highly complex, "large" in dimension, and uncertain by nature. • How large is large? • if it can be decoupled or partitioned into a number of interconnected subsystems or "small-scale“ systems for either computational or practical reasons • when its dimensions are so large that conventional techniques of modeling, analysis, control, design, and computation fail to give reasonable solutions with reasonable computational efforts.

5. Introduction • Since the early 1950s, when classical control theory was being established, • These procedures can be summarized as follows: • Modeling procedures • Behavioral procedures of systems • Control procedures • The underlying assumption: "centrality“ • A notable characteristic of most large-scale systems is that centrality fails to hold due to either the lack of centralized computing capability or centralized information, e.g. society, business, management, the economy, the environment, energy, data networks, aeronautical systems, power networks, space structures, transportation, aerospace, water resources, ecology, robotic systems, flexible manufacturing systems, and etc.

6. Large-Scale Systems Modeling Aggregation Methods Perturbation Methods

7. Introduction • In any modeling task, two often conflicting factors prevail: • "simplicity“ • "accuracy" • The key to a valid modeling philosophy is: • The purpose of the model • The system's boundary • A structural relationship • A set of system variables • Elemental equations • Physical compatibility • Elemental, continuity, and compatibility equations should be manipulated • The last step to a successful modeling

8. Introduction • The common practice has been to work with simple and less accurate models. There are two different motivations for this practice: • (i) the reduction of computational burden for system simulation, analysis, and design; • (ii) the simplification of control structures resulting from a simplified model. • Until recently there have been only two schemes for modeling large-scale systems • Aggregate method: economy • Perturbation Method: Mathematics

9. Aggregation Method • A "coarser" set of state variables. • For example, behind an aggregated variable, say, the consumer price index, numerous economic variables and parameters may be involved. • The underlying reason: retain the key qualitative properties of the system, such as stability. • In other words, the stability of a system described by several state variables is entirely represented by a single variable-the Lyapunov function.

10. General Aggregation • where C is an l x n (l < n) constant aggregation matrix and l x 1 vector z is called the aggregation of x • aggregated system • where the pair (F,G) satisfy the following, so-called dynamic exactness (perfect aggregation) conditions: L l

11. General Aggregation • Error vector is defined as e(t) = z(t)-C.x(t), • dynamic behavior is given by e(t) = F.e(t)+(FC-CA)x(t)+(G - CB)u(t), • reduces to e(t) = F.e(t) if previous conditions hold. • Example:Consider a third-order unaggregated system described by It is desired to find a second-order aggregated model for this system.

12. General Aggregation • SOLUTION: λ(A} = {-0.70862, -6.6482, -4.1604}, the first mode is the slowest of all three. • Aggregation matrix C can be • The aggregated model becomes • The resulting error vector e(t) satisfies • An alternative choice of C • This scheme leads to dynamically inexact aggregation also. • Modal Aggregation • Balanced Aggregation

13. Perturbation Methods • The basic concept behind perturbation methods is the approximation of a system's structure through neglecting certain interactions within the model which leads to lower order. • There are two basic classes: • weakly coupled models • strongly coupled models • Example of weakly coupled: chemical process control and space guidance: different subsystems are designed for flow, pressure, and temperature control

14. weakly coupled models • Consider the following large-scale system split into k linear subsystems • where ε is a small positive coupling parameter, xi and ui are ith subsystem state and control vectors. • when k = 2, has been called the ε-coupled system. It is clear that when ε = 0 the ε-coupled system decouples into two subsystems, • which correspond to two approximate aggregated models one for each subsystem.

15. Perturbation Method & Decentralized Control • In view of the decentralized structure of large-scale systems, these two subsystems can be designed separately in a decentralized fashion shown in Figure. • There has been no hard evidence that two reducing model method are the most desirable for large-scale systems.

16. StructuralProperties ofLarge-Scale Systems

17. Structural Properties of Large-Scale Systems • Stability • Controllability • Observability • When the stability of large-scale system is of concern, one basic approach, consisting of three steps, has prevailed "composite system method“: • decompose a given large-scale system into a number of small-scale subsystems • Analyze each subsystem using the classical stability theories and methods • combine the results leading to certain restrictive conditions with the interconnections and reduce them to the stability of the whole • One of the earliest efforts regarding the stability of composite systems: using the theory of the vector Lyapunov function • The bulk of research in the controllability and observabilityof largescale systems falls into four main problems: • controllability and observability of composite systems, • controllability (and observability) of decentralized systems, • structural controllability, • controllability of singularly perturbed systems.

18. HierarchicalControl ofLarge-Scale Systems Coordination of Hierarchical Structures Hierarchical Control of Linear Systems

19. Hierarchical Structures • The idea of "decomposition" was first treated theoretically in mathematical programming by Dantzig and Wolfe. • The coefficient matrices of such large linear programs often have sparse matrices. • The "decoupled" approach divides the original system into a number of subsystems involving certain values of parameters. Each subsystem is solved independently for a fixed value of the so-called "decoupling" parameter, whose value is subsequently adjusted by a coordinator in an appropriate fashion so that the subsystems resolve their problems and the solution to the original system is obtained. • This approach, sometimes termed as the "multilevel" or "hierarchical” approach.

20. Hierarchical Structures • There is no uniquely or universally accepted set of properties associated with the hierarchical systems. However, the following are some of the key properties: • decision-making components • The system has an overall goal • exchange information (usually vertically) • As the level of hierarchy goes up, the time horizon increases

21. Coordination of Hierarchical Structures • Most of hierarchically controlled are essentially a combination of two distinct approaches: • the model-coordination method (or "feasible" method) • The goal-coordination method (or "dualfeasible” method) • These methods are described for a two-subsystem static optimization (nonlinear programming) problem.

22. Model Coordination Method • static optimization problem • where x is a vector of system (state) variables, u is a vector of manipulated (control) variables, and y is a vector of interaction variables between subsystems. • objective function be decomposed into two subsystems: • by fixing the interaction variables • Under this situation the problem may be divided into the following two sequential problems: • First-Level Problem-Subsystem • Second-Level Problem

23. Model Coordination Method • The minimizations are to be done, respectively, over the following feasible sets: • A system can operate with these intermediate values with a near-optimal performance.

24. Goal Coordination Method • In the goal coordination method the interactions are literally removed by cutting all the links among the subsystems. • Let yi be the outgoing variable from the ith subsystem, while its incoming variable is denoted by zi. Due to the removal of all links between subsystems, it is clear that yi ≠zi. • In order to make sure the individual sub problems yield a solution to the original problem, it is necessary that the interaction-balance principle be satisfied, i.e., the independently selected yi and zi actually become equal. • By introducing the z variables, the system's equations are given by

25. Goal Coordination Method • The set of allowable system variables is defined by • objective function • Expanding the penalty term: • First-level problem • Second-level problem

26. Goal Coordination Method • It will be seen later that the coordinating variable a can be interpreted as a vector of Lagrange multipliers and the second-level problem can be solved through well-known iterative search methods, such as the gradient, Newton's, or conjugate gradient methods.

27. Hierarchical Control of Linear Systems • The goal coordination formulation of multilevel systems is applied to large-scale linear continuous-time systems. • A large-scale dynamic interconnected system • It is assumed that the system can be decomposed into N interconnected subsystems Si

28. Hierarchical Control of Linear Systems • The objective, in an optimal control sense • Through the assumed decomposition of system into N interconnected subsystems • The above problem, known as a hierarchical (multilevel) control

29. Linear System Two-Level Coordination • Consider a large-scale linear time-invariant system: • decompose into • interaction vector • The original system's optimal control problem

30. Linear System Two-Level Coordination • The "goal coordination" or "interaction balance" approach as applied to the "linear-quadratic” problem is now presented. • The global problem SG is replaced by a family of N subproblems coupled together through a parameter vector α= (α1, α2, ... , αN) and denoted by Si (α). • The coordinator, in turn, evaluates the next updated value of α

31. Linear System Two-Level Coordination • where εl is the lth iteration step size, and the update term dl, as will be seen shortly, is commonly taken as a function of "interaction error":

32. Reference • M. Jamshidi, “Large-Scale Systems: Modeling, Control and Fuzzy Logic”, Prentice Hall PTR, New Jersey, 1997.

33. Thanks for your attention!