Mojtaba HajihasaniMentor: Dr. Twohidkhah

ContentsIntroductionLarge-Scale Systems Modeling

Aggregation MethodsPerturbation Methods

Structural Properties of Large Scale SystemsHierarchical Control of Large-Scale Systems

Coordination of Hierarchical StructuresHierarchical Control of Linear Systems

Decentralized Control of Large-Scale SystemsDistributed Control of Large-Scale SystemMPC of Large-Scale System

IntroductionMany 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.

IntroductionSince the early 1950s, when classical control

theory was being established,These procedures can be summarized as follows:

Modeling proceduresBehavioral procedures of systemsControl 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.

Aggregation Methods

Perturbation Methods

IntroductionIn any modeling task, two often conflicting factors

prevail:"simplicity“"accuracy"

The key to a valid modeling philosophy is:The purpose of the modelThe system's boundaryA structural relationshipA set of system variablesElemental equationsPhysical compatibilityElemental, continuity, and compatibility equations

should be manipulatedThe last step to a successful modeling

IntroductionThe 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 systemsAggregate method: economyPerturbation Method: Mathematics

Aggregation MethodA "coarser" set of state variables.For example, behind an

aggregated variable, say, theconsumer 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.

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:

Ll

General AggregationError 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.

General AggregationSOLUTION: 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 AggregationBalanced Aggregation

Perturbation MethodsThe 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 forflow, pressure, and temperaturecontrol

weakly coupled modelsConsider 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 approximateaggregated models one for each subsystem.

Perturbation Method & Decentralized ControlIn 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.

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 observability of

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.

Coordination of Hierarchical Structures

Hierarchical Control of Linear Systems

Hierarchical StructuresThe 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.

Hierarchical StructuresThere 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 componentsThe system has an overall goalexchange information

(usually vertically)As the level of hierarchy

goes up, the time horizon increases

Coordination of Hierarchical StructuresMost 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.

Model Coordination Methodstatic 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 variablesUnder this situation the problem may be divided

into the following two sequential problems:First-Level Problem-SubsystemSecond-Level Problem

Model Coordination MethodThe minimizations are to be done,

respectively, over the following feasible sets:

A system can operatewith these intermediatevalues with a near-optimalperformance.

Goal Coordination MethodIn 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

Goal Coordination MethodThe set of allowable system variables is

defined by

objective function

Expanding the penalty term:

First-level problemSecond-level problem

Goal Coordination MethodIt 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 conjugategradient methods.

Hierarchical Control of Linear SystemsThe 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

Hierarchical Control of Linear SystemsThe 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

Linear System Two-Level CoordinationConsider a large-scale linear time-invariant

system:

decompose into

interaction vector

The original system's optimal control problem

Linear System Two-Level CoordinationThe "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

Linear System Two-Level Coordinationwhere 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":

ReferenceM. Jamshidi, “Large-Scale Systems:

Modeling, Control and Fuzzy Logic”, Prentice Hall PTR, New Jersey, 1997.

Thanks for your attention!