I n the name of God Evolutionary Based Nonlinear Multivariable Control System Design Presented by: M. Eftekhari Supervisor : DR. S. D. Katebi Dept. of

IIn the name of Godn the name of God

Evolutionary Based Nonlinear Multivariable Control System Design

Presented by: M. EftekhariSupervisor : DR. S. D. Katebi

Dept. of Computer Science and Engineering Shiraz University

ContentsContents

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

Nonlinear ControlNonlinear Control

Most practical dynamic systems exhibit nonlinear Most practical dynamic systems exhibit nonlinear behavior.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 for A general and coherent theory dose not exist for nonlinear design and analysis. nonlinear design and analysis.

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

Nonlinear DesignNonlinear Design

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

Quasi–Linearization Quasi–Linearization Linearization around the operating Linearization around the operating

conditionsconditions

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 form extended Bode to MIMO in the form of characteristic lociof 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


Types of NonlinearitiesTypes of Nonlinearities

Implicit:Implicit: friction changes with speed in a nonlinear friction changes with speed in a nonlinear mannermanner

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 systemseg. 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 DerivationSingle value nonlinear elementSingle value nonlinear element

ErrorError

ISEISE

)]([)(.)( txytxNEte

0 0 0

22

0

22 )]([)]([).(.2)()( dttxydttxytxNEdttxNEdtte

Example of EIDFExample of EIDF

2( ) (1 )

2

DEIDF E

E E

ContentsContents


Multi-Objective OptimizationMulti-Objective OptimizationMOOMOO

Optimization deals with the problem of searching Optimization deals with the problem of searching feasible solutions over a set of possible choices to feasible solutions over a set of possible choices to optimize certain criteriaoptimize certain criteria.

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

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 DesignAny Control systems design can be formulated Any Control systems design can be formulated

as MOOas MOO

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

parameters (numeric);parameters (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 Whidborn,2000s, suggest GA for solution of all the aboveall the above

H

ContentsContents


UncertaintyUncertainty

Structured (parametric): is caused by Structured (parametric): is caused by undesired parameter changes. undesired parameter changes.

Unstructured: is due to un-modeled Unstructured: is due to un-modeled dynamicsdynamics

Uncertainty can influence and degrade Uncertainty can influence and degrade the plant operation in 2 waysthe plant operation in 2 ways

InfluenceInfluence of Uncertainty of Uncertainty

MultiplicativeMultiplicative: G(s)=G: G(s)=G00(s)(1+w(s)(1+wmm(s).(s).ΔΔ(s))(s))

G(s): Perturbed Transfer FunctionG(s): Perturbed Transfer Function

GG00(s): Nominal Transfer Function(s): Nominal Transfer Function

ΔΔ(s): disturbance (perturbation)(s): disturbance (perturbation)

WWmm(s): weighting, upper bound(s): weighting, upper bound AdditiveAdditive: G(s)=G: G(s)=G00(s)+w(s)+waa(s).(s).ΔΔ(s)(s) A good design must be robust in the presence of A good design must be robust in the presence of

uncertainty and undesired perturbation uncertainty and undesired perturbation of the plant parametersof the plant parameters

MeasureMeasureof performance and robustnessof performance and robustness

A SISO block diagramA SISO block diagram

Sensitivity is:Sensitivity is:

Complementary Sensitivity is:Complementary Sensitivity is: T+S=IT+S=I

1

1TfCS

G C

C GR(t)

y(t)

CG

CGT

1

n

d

Robust Performance & robust Robust Performance & robust stabilitystability

Robust PerformanceRobust Performance

Robust stabilityRobust stability

Objective:Objective:

Find WFind W11 and W2 such that objective is satisfied under constraints and W2 such that objective is satisfied under constraints . .

S+T=IS+T=I W1 and w2 effect a trade-off between S and TW1 and w2 effect a trade-off between S and T

1 . 1w S for RP

1 2min W S W T

2 w 1 T for RS

=H=H∞ ∞ NormNorm

Defined as;Defined as; HH∞∞ Norm: Norm:

where where

( ) max ( )F s F j

pp

pjFjF

)(lim)(max

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

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

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

Little is known about Stability and Robustness Little is known about Stability and Robustness for the case of MIMO nonlinear systems for the case of MIMO nonlinear systems

Absolute StabilityAbsolute Stability

Rosenbrock 1965, purposed Multivariable Circle Rosenbrock 1965, purposed Multivariable Circle Criterion as a test for Bounded Input Bounded OutputCriterion as a test for Bounded Input Bounded Output

stability.stability. If applied for controller design, it will result in a very If applied for controller design, it will result in a very

conservative controllerconservative controller Several similar methods for the absolute stability test Several similar methods for the absolute stability test

exist, but non suitable for design.exist, but non suitable for design.

ContentsContents


Design procedureDesign procedure

Replace: Nonlinear elements EIDFS

The structure of controller is chosen

Time domain objectives are formulated

Another objective to ensure RP and RS

MOGA is applied to solve MOO

End

Start

1 2w S w T

Rise time, settling time,…

Uncertainty exist due to EIDF linearizationUncertainty exist due to EIDF linearization

Assuming linear model is accurateAssuming linear model is accurate Uncertainty exist due to linearizing the nonlinear Uncertainty exist due to linearizing the nonlinear

elementelement Modeling the Modeling the AdditiveAdditive UncertaintyUncertainty

Where K is a Where K is a vectorvector sampled from EIDF gains. n is an sampled from EIDF gains. n is an integer.integer.

)(__

EIDFijij Kmeann

)( EIDFij

nij KSD

npp GG

__

nij

nij

nij

nij

n

UsingUsing HH∞∞ Weighted Sensitivity Weighted Sensitivity

= uncertain plant = Nominal plant= uncertain plant = Nominal plant

1 pGIS

ijijijp CnGjiG ..),(____

pG pG

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 andis real number, p is a real vector and

is real function of P (controller parameter) and is real function of P (controller parameter) and t (time)t (time)

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

, 1, 2,...jP j M

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

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

i

Time domain specificationsTime domain specifications

In a control systems represents functionalsIn a control systems represents functionalsSuch 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 For a given time response which is provided by the SIMULINK, these are calculated by the SIMULINK, these are calculated numerically based on usual formulanumerically based on usual formula

i

Frequency Domain ObjectivesFrequency Domain Objectives

may represent any specification in the frequency domain may represent any specification in the frequency domain such as bandwidth, GM, PM etc.such as bandwidth, GM, PM etc.

In order to make the design more robust, we used the followingIn order to make the design more robust, we used the following

measure of stability (noise rejection)measure of stability (noise rejection) performance index (disturbance rejection)performance index (disturbance rejection)

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

i

11

12

( ) norm of the sensetivity function

( ) norm of complemetary sent.

I GNC H

GNC I GNC H

21

Mixed Time and Frequency Mixed Time and Frequency Domain optimizationDomain optimization

Using ITSE or ISE as performance Indices, With RP and Using ITSE or ISE as performance Indices, With RP and RS Constraints. RS Constraints.

Other objectives have also been testdOther objectives have also been testd

Eg. Weighted sumEg. Weighted sum

1 2

1 2

1

2

min( ), st.

min( ),st.

ISE

ISTE

1 2 1 2 st., T + S = 1 and 1obj S T

ContentsContents


Example 1Example 1

A 2 by 2 Uncompensated SystemA 2 by 2 Uncompensated System

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

Name of Name of objectivesobjectives

Desired Desired objectivesobjectives

Resulted 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 %

Design criterion in time domain are metDesign criterion in time domain are met

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 Name of objectivesobjectives











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

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

ResponsesResponses

Making the design robustMaking the design robust

In addition to the time domain specificationsIn addition to the time domain specificationsThe criterion for robust stability and robust The criterion for robust stability and robust

performance is now added to the objectivesperformance is now added to the objectivesThe controllers are modified by MOGA to The controllers are modified by MOGA to

take care of these additional objectivestake care of these additional objectivesIn addition to robust stability to the design, In addition to robust stability to the design,

the overall performance has also improved.the overall performance has also improved.

ResponsesResponsesName of Name of objectivesobjectives


Rise time1Rise time1 1.28551.2855

Rise time 2Rise time 2 0.97700.9770

Over shoot1Over shoot1 00

Over shoot2Over shoot2 0.00080.0008

settling1settling1 1.41521.4152

settling2settling2 2.09832.0983

Steady state1Steady state1 00

Steady state2Steady state2 00

Interaction Interaction 1122

0.23%0.23%


0.09%0.09%

Plant uncertainty taken into accountPlant uncertainty taken into account

Still a more comprehensive controller is requiredStill a more comprehensive controller is required

Responses from time and frequency domain Responses from time and frequency domain objectives for uncertain plantobjectives for uncertain plant

Time Domain objectives Mixed Time and Frequency domain objectives

Characteristics of responsesCharacteristics of responsesName of Name of objectivesobjectives





Over shoot1Over shoot1 0.20.2 00







1%1% 0%0%


1%1% 0%0%

Name of Name of objectivesobjectives





Over shoot1Over shoot1 0.20.2 00







1%1% 0.1%0.1%


1%1% 0%0%

Analysis and SynthesisAnalysis and Synthesis

EIDF accuracy is investigatedEIDF accuracy is investigated

Robust stability and robust Robust stability and robust performancesperformances

are examinedare examined

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 comparedEIDF in place and the actual nonlinearities are compared 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

Test for Robust stabilityTest for Robust stability

The compensated system is subjected to a disturbanceThe compensated system is subjected to a disturbance It is seen that instability do not occur and system recovers to its It is seen that instability do not occur and system recovers to its

optimal operating conditionsoptimal operating conditions

Test for Robust performanceTest for Robust performance

The system is subjected to step input of The system is subjected to step input of differing magnitude, the time domain differing magnitude, the time domain specifications do not changespecifications do not change

MOGAMOGA

ObservationsObservations1.1. 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 The Parameters of MOGA such as X-over and mutation rates should be initially of and mutation rates should be initially of nominal vale (Pc=0.7, Pm=0.01)nominal 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 synthesis and the extension of robustness in the synthesis and the extension of robustness in the frequency domain to the MIMO nonlinear systemsfrequency domain to the MIMO nonlinear systems

As the MOGA design progresses the designer obtain As the MOGA design progresses the designer obtain more knowledge about the systemmore knowledge about the system

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.

ConclusionsConclusionsTime domain approach is more explicit with Time domain approach is more explicit with

regards to the system time performanceregards to the system time performanceBy combining the time and the frequency By combining the time and the frequency

domain objectives design robustness is domain objectives design robustness is guaranteedguaranteed

Taking ISTE or ISE as objectives, Taking ISTE or ISE as objectives,

subject to S<Msubject to S<M1 1 and T<Mand T<M22 can also guarantee can also guarantee

the robustness.the robustness.

ConclusionConclusion

The design technique described is compared [Katebi] and The design technique described is compared [Katebi] and contrasted with other methods for MIMO nonlinear systemscontrasted with other methods for MIMO nonlinear systems

The approach was shown to be effective and has several The approach was shown to be effective and has several advantages over other techniquesadvantages over other techniques

1.1. The easy formulation of MOGAThe easy formulation of MOGA2.2. Provides degree of freedom for the designerProvides degree of freedom for the designer3.3. Acceptable computational demandAcceptable computational demand4.4. Accurate and multiple solutionsAccurate and multiple solutions5.5. Very suitable for the powerful MATLAB environment Very suitable for the powerful MATLAB environment

Several other examples with different linear and nonlinear Several other examples with different linear and nonlinear model have been solved and will be included in the thesismodel have been solved and will be included in the thesis

ContentsContents


Future ResearchFuture Research

Different MIMO nonlinear configuration exist, further works Different MIMO nonlinear configuration exist, further works may be undertaken for other configurationmay be undertaken for other configuration

The class of nonlinearity considered here only encompass the The class of nonlinearity considered here only encompass the memoryless (single value) elements.memoryless (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 extend the nonlinearities, theoretical works are required to extend the design to those class on nonlinearities.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 may For higher dimensional systems parallel algorithms may

become necessary.become necessary. Application of other evolutionary algorithms such as EP, ES, Application of other evolutionary algorithms such as EP, ES,

GP and swarm optimization is another line of further researchGP and swarm optimization is another line of further research

Question TimeQuestion Time

Documents

I n the name of God Evolutionary Based Nonlinear Multivariable Control System Design Presented by: M. Eftekhari Supervisor : DR. S. D. Katebi Dept. of