control of technical systems

7/30/2019 Control of Technical Systems

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Mostaghim,

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Organic Computing

Chapter 5: Control of technical Systems


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Overview

This chapter focuses on the basics of control of technical systems:

Basics of control engineering

Adaptive controllers

Uncertainties in design

Robustness

Reliability


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Basics

Process

Input Output

If we have an exact knowledge of the process (controlled variable behavior),

we can use a Feedforward Control.

OutputProcess

Input Control

System

Feedforward is being used to maintain some desired state of the system.

Everything is predefined

The control system responds to a known disturbance.

The control system does not observe the output of the process it is controlling.

A Process:

The main goal of control systems is to improve the behavior of the system

Disturbance


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Feedforward

Example:

The task is to control the system

such that the shafts rotate with equal speeds in spite of different possible

friction levels (disturbances).


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Feedforward

A desired speed should be achieved.

The torque to accelerate the mass to a desired speed can be easily computed if

we assume a frictionless system.

T

SpeedProcess

Speed

referenceControl

System

Computes the correct Torque value

Disturbance


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Speedreference

Feedforward vs. Feedback

Unfortunately the real world is full of non-linearities that

limit the effectiveness of such feed forward control schemes:

Different friction levels come from different sources and vary over time,

therefore some feedback is introduced:

SpeedProcess

Control

System

Disturbance

Computes the correct Torque value

to control the speed.


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Feedback

Feedback Control:

In every feedback loop,

information about the result of a

transformation or an action is sent

back to the input of the system inthe form of input data.

ProcessOutput

Disturbance

Control

System

Input

Feedback

OpinionFeedback systems also appear

in non-technical systems.


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Positive vs. Negative Feedback

Positive Feedback: If the feedback to the system influences the output of theprocess in the same direction as the preceding observed changes, it is

positive feedback - its effects are cumulative: there is exponential growth or

decline.

Time

Ou

tput Explosion

Blocking

Start

Positive Feedback: exponential growth or decline

(diverging behavior)


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Positive vs. Negative Feedback

Negative Feedback: If the feedback to the system influences the output of theprocess in the opposite direction, it is negative feedback - its effects stabilize

the system: there is maintenance of the equilibrium.

Time

Output

Start

Goal

Start

Negative Feedback: Maintenance of equilibrium and convergence


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Adaptive systems and control systems

An adaptive system is a system that is able to adapt its behavior according tochanges in its environment or in parts of the system itself.

Control systems utilize feedback loops in order to sense conditions in their

environment and adapt accordingly.

The aim of control engineering, beside the others, is to determine:

the model of the controller

the parameters of the controller

It is desirable that the controllers adapt their parameters to the changing

environment parameters

Adaptive Controllers


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Adaptive Control

Adaptive Controllers:

In Adaptive Control, the controller parameters are variable and there is a

mechanism to adjust them online based on the signals in the system.

Example: A robot carrying an unknown load (a load of uncertain mass properties).

There are two ways to construct the adaptive controllers:

1- Model-reference adaptive control

2- Self-tuning method


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Adaptive Control

Model-reference adaptive control:

requires a reference model for the controller in order to compute

the error of the systems output and the adaptation law sets the

parameters of the controller so that the error is minimized.

Process

Disturbance

Control

System

Ref.

Adaptation

law

Reference

model

error

Estimates the parameters


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Adaptive Control

Self-tuning adaptive control:

Based on the input and output of the process, the estimator

estimates parameters of the controller.

Process

Disturbance

ControlSystemRef.

Estimator

The selected Parameters must be robust with respect to the changes

caused by the disturbance.

This is an on-line estimator


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Robustness

Searching for robust solutions:

In many real world applications the adaptation is not possible:

1- the environment changes too quickly

2- the environment cannot be monitored closely enough

3- the changes happen after the commitment to a particular solution has been

made.

In such cases, one must search for solutions that perform well in all possible

future scenarios.

The property of being insensitive to the slight changes of the

environment or noise in the decision variables is called

Robustness.


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x

f(x)

Robustness

Example:

A change affects the

quality of the solution onthe thin peak much more

that the solution on the

plateau.

We consider the disturbance to appear indesign variables.


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Uncertainties in the design

Different kinds of uncertainties:

1. Aleatory Uncertainty

describes the inherent stochastic variation of the physical system.

such variation is usually caused by the random nature of the input data.

Variations can occur in the form of manufacturing tolerances or

uncontrollable variations in the external environment.

They are usually modeled as random phenomena characterized by probability

distributions. The probability distributions are constructed using the relativefrequency of the occurrence of events, which requires large amounts of

information. Most often such information does not exist and designers usually

make assumptions on the characteristics (mean, variances, correlation

coefficients) of the random phenomena causing the variation.


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2. Epistemic Uncertainty

Epistemic uncertainty also known as subjective uncertainty arises due to

ignorance lack of knowledge

incomplete information

decision making (due to trade-offs)

In engineering systems, the epistemic uncertainty can be

Parametric: uncertain parameters for which the available information is

inadequate

Model-based: improper model of the systems usually due to the lack of

knowledge about the physics of the system


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3. Numerical Uncertainty (Error)

commonly associated with the numerical models used for modeling and

simulation.

typical examples:

- round-off errors

- truncation errors- error associated with the solution of ODEs and PDEs which typically uses

discretization schemes.


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Definitions of Robustness

Possible definitions for a Robust Solution assuming a certain range of

uncertainties:

1- Maximize the minimum possible outcome:

This is appropriate for problems like when:

- an investment strategy is sought that in no possible way leads to bankruptcy.

- flight control strategies must not crash the airplane.

2- Trade-off between quality and variance:If the focus is on small variance, one might explicitly look at the trade-off

between quality and variance.


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x

f(x)

Robustness

3- Maximize the expected performance:

The expected performance (effective quality) can be calculated as follows:

+

+=+= d)f(x).()E(f(x(x)feff

p

Probability density function of disturbance

Because fis not known in a closed form, feffcannot be easily computed.

But it can be estimated by methods like Monte Carlo integration.

Monte Carlo Integration:Sampling over a number of realization of.Each sample corresponds to one fitness evaluation.


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Sampling methods

=

+=n

1i

0n

10eff )f(x)(xf

Integral Approximation:For a given point x0 the integral can be estimated by evaluating a set ofn

samples xi = x0+ in the neighborhood ofx0:

Sampling methods:

1- Random Method

2- Antithetic3- Stratified

4- Latin Hypercube


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Sampling Methods

1- Random Method

2- Antithetic

Random Method:

Samples points at random

Antithetic:

produces pairs of disturbances which lead to

negatively correlated estimation.

For uniformly distributed disturbances, the

first vector is selected at random (), thesecond is then chosen as -.


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Sampling Methods

3- Stratified

4- Latin Hypercube

Stratified:

Divides the space of possible disturbances

into possible equal properties.

Draws one disturbance from every region.

Latin Hypercube:

In order to draw k samples, every dimension is

divided into k parts.

k samples are chosen such that each quantile in

each dimension is covered by exactly one

sample.


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x

f(x)

Robustness vs. Reliability

Robust designs are designs at which

the variation in the performance

functions is minimal.

Reliable designs are designs at which

the chance of failure of the system is

low.

x

f(x)

constraints

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Reliability

x1

x2

Deterministic Optimum

Feasible region

Reliable solution A

Infeasible regionInfeasible region

Example :

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How to compute the reliability of a solution?

A desired reliability value R can be measured for solution A:

The probability of having an feasible solution created through uncertainties

from solution A is:

( ) JjRdxgP jj ,,2,10, K=

Where gj denotes thejth

constraint (e.g. the distance from some infeasible region).x and dare the vectors of uncertain and deterministic design parameters.

The quantity Rj is the required reliability (within [0, 1] )

for satisfying the jth constraint.

A computational method is required to estimate the probability.

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How to compute the reliability?

Simulation method

A set of Ndifferent parameter

settings is created by following theknown distribution of variation ofx.

For each sample each constraint gjisevaluated and checked for

possible constraint violation.

Ifrcases (ofN) satisfy all gjconstraints, we can replace

with RNr

From [H. Agrawal 2004]

( ) JjRdxgP jj ,,2,10,K

=

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How to compute the reliability?

This is a simple method and works well when the desired reliabilityR is not

very close to one.

A major drawback is that the sample sizeNneeded for finding rmust belarge enough, such that at least one infeasible case is present in the sample

Computationally expensive

Biased Monte-Carlo simulation can be used to solve this.

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Output from

observer

Controller in the generic architecture

The task of controller is to :

Select an observation model

Control the SuOC

Input

Control

System

Goals. SuOC

Process

Observer

Output

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Model Predictive Control (MPC)

Predictive control is a form of control which incorporates the prediction of a

system behavior into its formulation.

The prediction serves to estimate the future values of system variablesbased on the available information on the current status of the system.

The prediction is used to optimize necessary control actions: meaning to drive

or maintain the system in a state which satisfies the objectives.

MPC is used when the future behavior of the system might be quite different

from that currently perceived. In particular, the current model for prediction

might be just a currently feasible simplified abstraction of a much more

complex system, i.e. the model has to be updated whenever the realbehavior of the system deviates too much from the predictions.

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MPC

Prediction and control horizon in MPC:

control horizon

time

FuturePast

Input

time

now

desired

output

measured

output

predicted

outputReference

trajectory

prediction horizon

output

Process

Input output

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MPC

Procedure in predictive controller:

1. Sample the output of the process.

2. Check for necessary updates of the model due to deviations between

observed and previously predicted behavior.

3. Use the model of the process to predict its future behavior over a prediction

horizon, when the control action is applied for a control horizon.4. Calculate the optimal control sequence (Input to the process) that minimizes

the error between Input, output and reference.

5. Apply the input to the process and repeat the procedure for the next sampling

time.

Process

Disturbance

Predictive

controller

desired output

(reference)

model

outputInput

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MPC

MPC is a particularly attractive option, if the system behavior over time is not

adequately modeled by one linear model, but may be approximated with a

certain accuracy by a sequence of linear models M1, M2, , Mi,

Without using prediction, it would not be feasible to detect online whether it is

necessary to transform model Mi

into model Mi

+1.

Process

Disturbance

Predictive

controller

desired output

(reference)

model

outputInput

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Distributed control

A distributed control system (DCS) refers to a control system, in which the

controller elements are not centralized but are distributed throughout the

system.

In each subsystem,

local control inputs are computed using the measurements and reduced-order

models.

the sub-system is controlled by one or more controllers. The entire system

may be networked for communication and monitoring.

Process

Dist.

Control

Process

Dist.

Control

Process

Dist.

ControlProcess

Dist.

Control

Central Control Distributed Control

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Distributed control

In large-scale applications, it is useful (or sometimes necessary) to have

distributed or decentralized control schemes, as the global measurement of

the system parameters is not possible.

The main challenge in distributed control is to achieve some degree ofcoordination among the controllers.

In distributed control, we have the same problems of emergent phenomena as

outlined for self-organizing systems.

Distributed control systems (DCSs) are used in many industrial applications to

monitor and control distributed systems, like:

Electrical power grids

Traffic signals

Sensor networks Economic systems

Large scale distributed telescopes (in astronomy)

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Organic Computing and Controllers

In OC, we have different kinds of controllers:

Central, distributed, and Multi-level.

Model-predictive controllers might help to efficiently control the behavior ofhighly complex systems.

observer controller

SuOC

observer controller

SuOC

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O CSuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

SuOC

O C

central distributed Multi-level

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Summary

In this chapter we had a very brief view on different aspects of controlengineering:

Basics in control

In dynamic environments, adaptive controllers can adjust the controllerparameters to achieve desirable controller model and parameters. An essential task of control engineering is to guarantee robust and reliable

system behavior. Model-predictive controllers incorporate a prediction of the future system

behavior and support the stepwise linear (or simplified) control of highlycomplex systems.

Distributed control is an essential aspect dealing with large scaleapplications, but has to cope with the problems of emergence.

All these issues can be used effectively in OC systems. So far, control engineering does not provide adequate answers to the key

problems addressed in OC systems related to controlled self-organization.

Next:

As learning plays an important role in organic computing, the next chaptersurveys different machine learning methods.

control of technical systems

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