Controller Design for Controller Design for Multivariable Nonlinear Control Multivariable Nonlinear Control Systems Systems Based on Multi Objective Based on Multi Objective Evolutionary Techniques. Evolutionary Techniques. Presented by: Presented by: Mahdi Eftekhari Supervisor: Supervisor: Prof. S. D. Katebi Dept. of Computer Science and Dept. of Computer Science and Engineering Engineering Shiraz University Shiraz University In the name of God
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Controller Design for Multivariable Nonlinear Control Systems Based on Multi Objective Evolutionary Techniques. Presented by: Presented by: Mahdi Eftekhari.
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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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
Whidborn,2000s, suggest GA for solution of all the Whidborn,2000s, suggest GA for solution of all the aboveabove
H
ContentsContents
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
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 time1Rise time1 22 1.47511.4751
Rise time 2 2 0.8598
Over shoot1Over shoot1 0.20.2 00
Over shoot2 0.2 0.0004
settling1settling1 33 1.65221.6522
settling2 3 1.8500
Steady state1Steady state1 0.010.01 00
Steady state2Steady state2 0.010.01 00
Interaction Interaction 1122
1%1% 0%0%
Interaction Interaction 2211
1%1% 0%0%
Name of Name of objectivesobjectives
Desired Desired objectivesobjectives
Resulted Resulted objectivesobjectives
Rise time1Rise time1 22 1.38971.3897
Rise time 2 2 0.8895
Over shoot1Over shoot1 0.20.2 00
Over shoot2 0.1 0.0001
settling1settling1 33 1.64741.6474
settling2 3 2.2815
Steady state1Steady state1 0.010.01 00
Steady state2Steady state2 0.010.01 00
Interaction Interaction 1122
1%1% 0.1%0.1%
Interaction Interaction 2211
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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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
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
Introduction Multi-objective optimization Nonlinear systems Nonlinear Multivariable systems Implementation Results Conclusions Future works
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