LSS-MPC
Mojtaba Hajihasani
Mentor: Dr. Twohidkhah
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
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.
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.
Aggregation Methods
Perturbation Methods
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
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
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.
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 l
L
where the pair (F,G) satisfy the following, so-called dynamic
exactness (perfect aggregation) conditions:
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.
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
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
weakly coupled models
Consider the following large-scale system split into k linear subsystems
where ε is a small positive coupling parameter, x i
subsystem state and control vectors.
and u i are i th 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.
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.
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 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.
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
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.
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
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.
Goal Coordination Method
In the goal coordination method the interactions are
literally removed by cutting all the links among the subsystems.
Let y i be the outgoing variable from the ith subsystem, while its incoming variable is denoted by z i . Due to the removal of all links between subsystems, it is clear that y i ≠z i .
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 y i and z i actually become equal.
By introducing the z variables, the system's equations are given by
Goal Coordination Method
The set of allowable system variables is defined by
objective function
Expanding the penalty term:
First-level problem
Second-level problem
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.
Hierarchical Control of Linear Systems
The goal coordination formulation of multilevel systems is applied to large-scale linear continuoustime systems.
A large-scale dynamic interconnected system
It is assumed that the system can be decomposed into
N interconnected subsystems S i
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
Linear System Two-Level
Coordination
Consider a large-scale linear time-invariant system:
decompose into
interaction vector
The original system's optimal control problem
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 S
G
is replaced by a family of N subproblems coupled together through a parameter vector α= (α
1
, α
2
, ... , α
N
) and denoted by S i
(α).
The coordinator, in turn, evaluates the next updated value of α
Linear System Two-Level Coordination
where ε l is the lth iteration step size, and the update term d l , as will be seen shortly, is commonly taken as a function of "interaction error":
Reference
M. Jamshidi, “Large-Scale Systems: Modeling,
Control and Fuzzy Logic”, Prentice Hall PTR, New
Jersey, 1997.
Thanks for your attention!
