Transcript 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
where the pair (F,G) satisfy the following, so-called dynamic
exactness (perfect aggregation) conditions:
L
l
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, 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.
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 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 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 Si
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 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 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":
Reference
M. Jamshidi, “Large-Scale Systems: Modeling,
Control and Fuzzy Logic”, Prentice Hall PTR, New
Jersey, 1997.
Thanks for your
attention!