controller design for multivariable nonlinear control systems based on multi objective evolutionary...

Controller Design forController Design for Multivariable Nonlinear Control Systems Multivariable Nonlinear Control Systems Based on Multi Objective Evolutionary Based on Multi Objective Evolutionary

Techniques.Techniques.

Presented by: Presented by: Mahdi EftekhariSupervisor: Supervisor: Prof. S. D. Katebi

Dept. of Computer Science and Engineering Dept. of Computer Science and Engineering

Shiraz UniversityShiraz University

In the name of God

ContentsContents

Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works

Nonlinear ControlNonlinear Control

Most practical dynamic systems exhibit Most practical dynamic systems exhibit nonlinear behavior.nonlinear behavior.

The theory of nonlinear systems is not as well The theory of nonlinear systems is not as well advanced as the linear systems theory.advanced as the linear systems theory.

A general and coherent theory dose not exist A general and coherent theory dose not exist for nonlinear design and analysis. for nonlinear design and analysis.

Nonlinear systems are dealt with on the case Nonlinear systems are dealt with on the case by case bases.by case bases.

Nonlinear DesignNonlinear Design

Most Nonlinear Design techniques are Most Nonlinear Design techniques are based on:based on: Linearization of some formLinearization of some form

Quasi–Linearization : Quasi–Linearization : Linearization around Linearization around the operating conditionsthe operating conditions

Extension of linear techniquesExtension of linear techniques

Rosenbrock:Rosenbrock: extended Nyquist techniques to extended Nyquist techniques to MIMO Systems in the form of Inverse Nyquist MIMO Systems in the form of Inverse Nyquist ArrayArray

MacFarlane:MacFarlane: extended Bode to MIMO in the extended Bode to MIMO in the form of characteristic lociform of characteristic loci

Soltine:Soltine: extends feedback linearization extends feedback linearization Astrom:Astrom: extends Adaptive Control extends Adaptive Control Katebi:Katebi: extends SIDF to Inverse Nyquist Array extends SIDF to Inverse Nyquist Array Others…..Others…..

ContentsContents


Multi-Objective Multi-Objective OptimizationOptimization

MOOMOO Optimization deals with the problem of Optimization deals with the problem of

searching feasible solutions over a set of searching feasible solutions over a set of possible choices to optimize certain criteriapossible choices to optimize certain criteria.

MOO implies that there are more than one MOO implies that there are more than one criterion and they must be treated criterion and they must be treated simultaneouslysimultaneously

Formulation of MOOFormulation of MOO Single objectiveSingle objective

Straight forward extension to MOOStraight forward extension to MOO

i

n

maximize ( )Subject to; g ( ) 0, i=1,2,...m

x R , f ( ) Objective, ( ) Inequality Constraints

| ( ) 0, 1,2,..., , 0

S= feasible area in decision space

i

Z f xx

x g xnS x R g x i m xi

1 1 2 2

i

1 1 2 2

( ),.... ( )}maximize { ( ),

Subject to; g ( ) 0 i=1,2,...m

| ( ), ( ),..., ( ),

| ( ) 0, 1,2,..., , 0

Z= feasible region in the criterion space

q q

qq q

f x z f xZ z f x z

x

Z z R z f x z f x z f x x S

nS x R g x i m xi

Solution Of MOOSolution Of MOO

Several numerical techniquesSeveral numerical techniquesGradient basedGradient based

Steepest decentSteepest decentNon-gradient basedNon-gradient based

Hill-climbingHill-climbingnonlinear programmingnonlinear programmingnumerical search (Tabu, random,..)numerical search (Tabu, random,..)We focus on Evolutionary techniquesWe focus on Evolutionary techniquesGA,GP, EP, ESGA,GP, EP, ES

GA at a glanceGA at a glance

Wide rang Applications of MOOWide rang Applications of MOO

Design, modeling and planning Design, modeling and planning Urban transportation. Urban transportation. Capital budgeting Capital budgeting Forest managementForest management Reservoir management Reservoir management Layout and landscaping of new cities Layout and landscaping of new cities Energy distribution Energy distribution Etc…Etc…

MOO and Control DesignMOO and Control Design

Any Control systems design can be formulated as Any Control systems design can be formulated as MOOMOO

Ogata, 1950s; optimization of ISE, ISTE (analyticOgata, 1950s; optimization of ISE, ISTE (analytic)) Zakian, 1960s;optimazation of time response parameters Zakian, 1960s;optimazation of time response parameters

(numeric);(numeric); Clark, 1970s, LQR, LQG (analytic)Clark, 1970s, LQR, LQG (analytic) Doyle and Grimble, 1980s, (analytic)Doyle and Grimble, 1980s, (analytic) MacFarlane, 1990s, loop shaping (grapho-analytic)MacFarlane, 1990s, loop shaping (grapho-analytic)

Whidborn,2000s, suggest GA for solution of all the Whidborn,2000s, suggest GA for solution of all the aboveabove

H

ContentsContents


Types of NonlinearitiesTypes of Nonlinearities Implicit:Implicit: friction changes with speed in a nonlinear friction changes with speed in a nonlinear

manner manner Explicit Explicit

Single-valued : Single-valued : eg. dead-zone, hard limit, saturation in op eg. dead-zone, hard limit, saturation in op Amp.Amp.

Multi-valued Multi-valued eg. Hysteresis in mechanical systems eg. Hysteresis in mechanical systems

22 5V x xx

Methods For Nonlinear Systems DesignMethods For Nonlinear Systems Design

Build Prototype and test Build Prototype and test (expensive)(expensive) Computer simulation Computer simulation (trial and error)(trial and error) Closed form Solutions Closed form Solutions (only for rare cases)(only for rare cases) Lyapunov’s Direct Method Lyapunov’s Direct Method (only Stability)(only Stability) Series–Expansion solution Series–Expansion solution (only implicit)(only implicit) Linearization around the operating conditions Linearization around the operating conditions

(only small changes)(only small changes) Quasi–Linearization: Quasi–Linearization: (Describing Function)(Describing Function)

Exponential Input Describing Function Exponential Input Describing Function (EIDF)(EIDF)

One particular form of Describing function is EIDFOne particular form of Describing function is EIDF

Assuming an exponential waveform at the input of a Assuming an exponential waveform at the input of a single value nonlinear element and minimizing the single value nonlinear element and minimizing the integral-squared errorintegral-squared error

Then Then

Where applicable, EIDF facilitate the study of the Where applicable, EIDF facilitate the study of the transient response in nonlinear systems transient response in nonlinear systems

Output Amp.

Input Amp.EIDF

EIDF DerivationEIDF Derivation

Single value nonlinear Single value nonlinear elementelement

ErrorError ISEISE

( ) . ( ) [ ( )]e t NE x t y x t

2 2 2 2

0 0 0 0

( ) ( ) 2. ( ). [ ( )] [ ( )]e t dt NE x t dt NE x t y x t dt y x t dt

Example of EIDFExample of EIDF

2( ) (1 )2

DEIDF EE E

ContentsContents


A general MIMO nonlinear SystemA general MIMO nonlinear System

Close loop Transfer functionClose loop Transfer function

( )GNCY RI GCN

1 1

mod m n

Y output vectore mR input vectore nG linear model n mN nonlinear elC controller matrix n n

Nonlinear Multivariable systemsNonlinear Multivariable systems Block diagram of 2-input 2-output feedback system. Belongs Block diagram of 2-input 2-output feedback system. Belongs

to a special configuration with a class of separable, single to a special configuration with a class of separable, single value Nonlinear systemvalue Nonlinear system

C11 G11

C22 G22

C12 G12

C21 G21

N11

N22

N12

N21

ProblemsProblems

The behavior of multi-loop nonlinear systems The behavior of multi-loop nonlinear systems is not as well understood as the single-loop is not as well understood as the single-loop systems systems

Generally, extensions of single-loop Generally, extensions of single-loop techniques can result in methods that are valid techniques can result in methods that are valid for multi-loop systems for multi-loop systems

Cross coupling and Loop interaction pose Cross coupling and Loop interaction pose major difficulties in MIMOmajor difficulties in MIMO

ContentsContents


Design procedureDesign procedure

Replace: Nonlinear elements EIDFS

The structure of controller is chosen

Time domain objectives are formulated

MOGA is applied to solve MOO

End

Start

Rise time, settling time,…

Time Domain objectivesTime Domain objectives

Find a set of M admissible points Find a set of M admissible points Such that;Such that;

is real number, p is a real vector is real number, p is a real vector and is real function of P and is real function of P (controller parameter) and time (controller parameter) and time

Any value of p which satisfies the above Any value of p which satisfies the above inequalities characterizes an acceptable inequalities characterizes an acceptable designdesign

, 1, 2,...j MPj

( , ) , ( 1,.... , 1,.... )ji ip t j M i n

i 1 2( , ,..., )np p p

i

Time domain Time domain specificationsspecifications

In a control systems represents functionals In a control systems represents functionals Such as:Such as:

Rise time, settling time, overshoot, steady state Rise time, settling time, overshoot, steady state error, loops interaction (For multivariable error, loops interaction (For multivariable systems), ISE, ITSE.systems), ISE, ITSE.

For a given time response which is provided by For a given time response which is provided by the SIMULINK, these are calculated numerically the SIMULINK, these are calculated numerically based on usual formulabased on usual formula

i

ContentsContents


ExampleExample A 2 by 2 Uncompensated A 2 by 2 Uncompensated SystemSystem

Nonlinear elements are replaced byNonlinear elements are replaced bythe EIDF gain and the place of the the EIDF gain and the place of the

compensator is decidedcompensator is decided

Design in time domainDesign in time domain

Structure of the compensator is now decideStructure of the compensator is now decide We started with simplest diagonal and constant We started with simplest diagonal and constant

controllerscontrollers The desired time domain specifications are now given The desired time domain specifications are now given

to the MOGA programto the MOGA program MOGA is initialized randomly and the parameter MOGA is initialized randomly and the parameter

limits are setlimits are set MOGA searches the space of the controller MOGA searches the space of the controller

parameters to find at least one set that satisfy all the parameters to find at least one set that satisfy all the specified objectivesspecified objectives

The evolved controller and its The evolved controller and its performanceperformance

Design criterion in time domain are metDesign criterion in time domain are metName of objectivesName of objectives Desired Desired

objectivesobjectivesResulted Resulted objectivesobjectives

Rise time1Rise time1 1515 11.031811.0318

Rise time 2Rise time 2 22 1.16911.1691

Over shoot1Over shoot1 0.50.5 0.03970.0397


settling1settling1 1515 12.189512.1895


Steady state1Steady state1 0.10.1 00


Interaction 1Interaction 122 5%5% 0.89 %0.89 %

Interaction 2Interaction 211 5%5% 0.03 %0.03 %

Time responsesTime responses

Conflicting objectivesConflicting objectives

It is observed after 50 generation of MOGA It is observed after 50 generation of MOGA with a population size of 50with a population size of 50

That although trade-off have been made That although trade-off have been made between the objectivesbetween the objectives

But due to conflict, all the required design But due to conflict, all the required design criterion are not metcriterion are not met

Alternative: we decided to use a more Alternative: we decided to use a more sophisticated controllersophisticated controller

Diagonal dynamic compensatorDiagonal dynamic compensator

DSC

BSA

Design criterion in time domain are metDesign criterion in time domain are metName of objectivesName of objectives Desired Desired

objectivesobjectivesResulted Resulted objectivesobjectives


Rise time 2Rise time 2 22 0.74280.7428







Interaction 1Interaction 122 5%5% 0.23%0.23%

Interaction 2Interaction 211 5%5% 0.09%0.09%

ResponsesResponses

More sophisticated controllerMore sophisticated controller

Responses from time domain and conflicting Responses from time domain and conflicting objectivesobjectives

Characteristics of Characteristics of responsesresponses

Name of Name of objectivesobjectives

Desired Desired objectivesobjectives

Resulted Resulted objectivesobjectives


Rise time 2 2 0.8598

Over shoot1Over shoot1 0.20.2 00

Over shoot2 0.2 0.0004


settling2 3 1.8500



Interaction Interaction 1122

1%1% 0%0%


1%1% 0%0%

Name of Name of objectivesobjectives

Desired Desired objectivesobjectives

Resulted Resulted objectivesobjectives


Rise time 2 2 0.8895

Over shoot1Over shoot1 0.20.2 00

Over shoot2 0.1 0.0001


settling2 3 2.2815




1%1% 0.1%0.1%


1%1% 0%0%

Analysis and SynthesisAnalysis and Synthesis

EIDF accuracy is investigatedEIDF accuracy is investigated

Convergence of MOGA and aspects Convergence of MOGA and aspects of local minima is also look into.of local minima is also look into.

EIDF AccuracyEIDF Accuracy The response of compensated system withThe response of compensated system with EIDF in place and the actual nonlinearities are EIDF in place and the actual nonlinearities are

comparedcompared When the basic assumption of exponential input is satisfied When the basic assumption of exponential input is satisfied

EIDF is very accurateEIDF is very accurate

MOGAMOGA ObservationsObservations

1.1. The range of controller parameters The range of controller parameters should be chosen carefully (domain should be chosen carefully (domain knowledge is useful)knowledge is useful)

2.2. The Parameters of MOGA such as X-over and The Parameters of MOGA such as X-over and mutation rates should be initially of nominal mutation rates should be initially of nominal vale (Pc=0.7, Pm=0.01)vale (Pc=0.7, Pm=0.01)

3.3. If a premature convergence occurs then these If a premature convergence occurs then these values have to be investigatedvalues have to be investigated

ContentsContents


ConclusionsConclusions A new technique based on MOGA for design of A new technique based on MOGA for design of

controller for MIMO nonlinear systems were controller for MIMO nonlinear systems were describeddescribed

The EIDF linearization facilitate the time response The EIDF linearization facilitate the time response synthesissynthesis

Based on the domain knowledge the designer is able Based on the domain knowledge the designer is able to effect trade off between the conflicting objectives to effect trade off between the conflicting objectives and also modifies the structure of the controller, if and also modifies the structure of the controller, if and when necessary.and when necessary.

Time domain approach is more explicit with regards Time domain approach is more explicit with regards to the system time performanceto the system time performance

ConclusionConclusion The approach was shown to be effective and has The approach was shown to be effective and has

several advantages over other techniquesseveral advantages over other techniques1.1. The easy formulation of MOGAThe easy formulation of MOGA

2.2. Provides degree of freedom for the designerProvides degree of freedom for the designer

3.3. Acceptable computational demandAcceptable computational demand

4.4. Accurate and multiple solutionsAccurate and multiple solutions

5.5. Very suitable for the powerful MATLAB environment Very suitable for the powerful MATLAB environment Several other examples with different linear and Several other examples with different linear and

nonlinear model have been solved and will be nonlinear model have been solved and will be included in the thesisincluded in the thesis

ContentsContents


Future ResearchFuture Research Different MIMO nonlinear configuration exist, further Different MIMO nonlinear configuration exist, further

works may be undertaken for other configurationworks may be undertaken for other configuration The class of nonlinearity considered here only The class of nonlinearity considered here only

encompass the memory less (single value) elements.encompass the memory less (single value) elements. As the EIDF is not applicable to the multi-valued As the EIDF is not applicable to the multi-valued

nonlinearities, theoretical works are required to nonlinearities, theoretical works are required to extend the design to those class on nonlinearities.extend the design to those class on nonlinearities.

Several explicit parallel version of MOGA exist,Several explicit parallel version of MOGA exist, For higher dimensional systems parallel algorithms For higher dimensional systems parallel algorithms

may become necessary.may become necessary. Application of other evolutionary algorithms such as Application of other evolutionary algorithms such as

EP, ES, GP and swarm optimization is another line of EP, ES, GP and swarm optimization is another line of further researchfurther research

Question TimeQuestion Time

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