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

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

Nonlinear Control Most practical dynamic systems exhibit nonlinear behavior. Most practical dynamic systems exhibit nonlinear behavior. The theory of nonlinear systems is not as well advanced as the linear systems theory. The theory of nonlinear systems is not as well advanced as the linear systems theory. A general and coherent theory dose not exist for nonlinear design and analysis. A general and coherent theory dose not exist for nonlinear design and analysis. Nonlinear systems are dealt with on the case by case bases. Nonlinear systems are dealt with on the case by case bases.

Nonlinear Design Most Nonlinear Design techniques are based on: Most Nonlinear Design techniques are based on: Linearization of some form Linearization of some form Quasi–Linearization Quasi–Linearization Linearization around the operating conditions Linearization around the operating conditions

Extension of linear techniques Rosenbrock: extended Nyquist techniques to MIMO Systems in the form of Inverse Nyquist Array Rosenbrock: extended Nyquist techniques to MIMO Systems in the form of Inverse Nyquist Array MacFarlane: extended Bode to MIMO in the form of characteristic loci MacFarlane: extended Bode to MIMO in the form of characteristic loci Soltine: extends feedback linearization Soltine: extends feedback linearization Astrom: extends Adaptive Control Astrom: extends Adaptive Control Katebi: extends SIDF to Inverse Nyquist Array Katebi: extends SIDF to Inverse Nyquist Array Others….. Others…..

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

Types of Nonlinearities Implicit: friction changes with speed in a nonlinear manner Implicit: friction changes with speed in a nonlinear manner Explicit Explicit Single-valued : eg. dead-zone, hard limit, saturation in op Amp. Single-valued : eg. dead-zone, hard limit, saturation in op Amp. Multi-valued Multi-valued eg. Hysteresis in mechanical systems eg. Hysteresis in mechanical systems

Methods for nonlinear systems Design Build Prototype and test (expensive) Build Prototype and test (expensive) Computer simulation (trial and error) Computer simulation (trial and error) Closed form Solutions (only for rare cases) Closed form Solutions (only for rare cases) Lyapunov’s Direct Method (only Stability) Lyapunov’s Direct Method (only Stability) Series–Expansion solution (only implicit) Series–Expansion solution (only implicit) Linearization around the operating conditions (only small changes) Linearization around the operating conditions (only small changes) Quasi–Linearization: (Describing Function) Quasi–Linearization: (Describing Function)

Exponential Input Describing Function (EIDF) One particular form of Describing function is EIDF One particular form of Describing function is EIDF Assuming an exponential waveform at the input of a single value nonlinear element and minimizing the integral-squared error Assuming an exponential waveform at the input of a single value nonlinear element and minimizing the integral-squared error Then Then Where applicable, EIDF facilitate the study of the transient response in nonlinear systems

EIDF Derivation Single value nonlinear element Error Error ISE ISE

Example of EIDF

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

Multi-Objective Optimization MOO Optimization deals with the problem of searching feasible solutions over a set of possible choices to optimize certain criteria Optimization deals with the problem of searching feasible solutions over a set of possible choices to optimize certain criteria. MOO implies that there are more than one criterion and they must be treated simultaneously MOO implies that there are more than one criterion and they must be treated simultaneously

Formulation of MOO Single objective Single objective Straight forward extension to MOO Straight forward extension to MOO

Solution Of MOO Several numerical techniques Several numerical techniques Gradient based Steepest decent Non-gradient based Hill-climbing nonlinear programming numerical search (Tabu, random,..) We focus on Evolutionary techniques GA,GP, EP, ES

GA at a glance

Wide rang Applications of MOO Design, modeling and planning Design, modeling and planning Urban transportation. Urban transportation. Capital budgeting Capital budgeting Forest management Forest 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 Design Any Control systems design can be formulated as MOO Any Control systems design can be formulated as MOO Ogata, 1950s; optimization of ISE, ISTE (analytic ) Ogata, 1950s; optimization of ISE, ISTE (analytic ) Zakian, 1960s;optimazation of time response parameters (numeric); Zakian, 1960s;optimazation of time response 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 all the above Whidborn,2000s, suggest GA for solution of all the above

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

Uncertainty Structured (parametric): is caused by undesired parameter changes. Structured (parametric): is caused by undesired parameter changes. Unstructured: is due to un-modeled dynamics Unstructured: is due to un-modeled dynamics Uncertainty can influence and degrade the plant operation in 2 ways Uncertainty can influence and degrade the plant operation in 2 ways

Influence of Uncertainty Influence of Uncertainty Multiplicative: G(s)=G 0 (s)(1+w m (s).Δ(s)) Multiplicative: G(s)=G 0 (s)(1+w m (s).Δ(s)) G(s): Perturbed Transfer Function G 0 (s): Nominal Transfer Function Δ(s): disturbance (perturbation) W m (s): weighting, upper bound Additive: G(s)=G 0 (s)+w a (s).Δ(s) Additive: G(s)=G 0 (s)+w a (s).Δ(s) A good design must be robust in the presence of uncertainty and undesired perturbation A good design must be robust in the presence of uncertainty and undesired perturbation of the plant parameters

Measure of performance and robustness A SISO block diagram A SISO block diagram Sensitivity is: Sensitivity is: Complementary Sensitivity is: Complementary Sensitivity is: T+S=I T+S=I CG R(t) y(t) n d

Robust Performance & robust stability Robust Performance Robust Performance Robust stability Robust stability Objective: Objective: Find W 1 and W2 such that objective is satisfied under constraints. Find W 1 and W2 such that objective is satisfied under constraints.S+T=I W1 and w2 effect a trade-off between S and T W1 and w2 effect a trade-off between S and T

=H ∞ Norm Defined as; Defined as; H ∞ Norm: H ∞ Norm:where

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

A general MIMO nonlinear System Close loop Transfer function Close loop Transfer function

Nonlinear Multivariable systems Block diagram of 2-input 2-output feedback system. Belongs to a special configuration with a class of separable, single value Nonlinear system Block diagram of 2-input 2-output feedback system. Belongs to a special configuration with a class of separable, single value Nonlinear system C11G11 C22G22 C12 G12 C21G21 N11 N22 N12 N21

Problems The behavior of multi-loop nonlinear systems is not as well understood as the single-loop systems The behavior of multi-loop nonlinear systems is not as well understood as the single-loop systems Generally, extensions of single-loop techniques can result in methods that are valid for multi-loop systems Generally, extensions of single-loop techniques can result in methods that are valid for multi-loop systems Cross coupling and Loop interaction pose major difficulties in MIMO Cross coupling and Loop interaction pose major difficulties in MIMO Little is known about Stability and Robustness Little is known about Stability and Robustness for the case of MIMO nonlinear systems

Absolute Stability Rosenbrock 1965, purposed Multivariable Circle Criterion as a test for Bounded Input Bounded Output Rosenbrock 1965, purposed Multivariable Circle Criterion as a test for Bounded Input Bounded Outputstability. If applied for controller design, it will result in a very conservative controller If applied for controller design, it will result in a very conservative controller Several similar methods for the absolute stability test exist, but non suitable for design. Several similar methods for the absolute stability test exist, but non suitable for design.

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

Design 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 Rise time, settling time,…

Uncertainty exist due to EIDF linearization Assuming linear model is accurate Assuming linear model is accurate Uncertainty exist due to linearizing the nonlinear element Uncertainty exist due to linearizing the nonlinear element Modeling the Additive Uncertainty Modeling the Additive Uncertainty Where K is a vector sampled from EIDF gains. n is an integer. Where K is a vector sampled from EIDF gains. n is an integer.

Using H ∞ Weighted Sensitivity Using H ∞ Weighted Sensitivity = uncertain plant = Nominal plant = uncertain plant = Nominal plant

Time 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 and is 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 characterizes an acceptable design

Time domain specifications In a control systems represents functionals In a control systems represents functionals Such as: Such as: Rise time, settling time, overshoot, steady state error, loops interaction (For multivariable systems), ISE, ITSE. Rise time, settling time, overshoot, steady state error, loops interaction (For multivariable systems), ISE, ITSE. For a given time response which is provided by the SIMULINK, these are calculated numerically based on usual formula For a given time response which is provided by the SIMULINK, these are calculated numerically based on usual formula

Frequency Domain Objectives may represent any specification in the frequency domain such as bandwidth, GM, PM etc. may represent any specification in the frequency domain such as bandwidth, GM, PM etc. In order to make the design more robust, we used the following In 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)

Mixed Time and Frequency Domain optimization Using ITSE or ISE as performance Indices, With RP and RS Constraints. Using ITSE or ISE as performance Indices, With RP and RS Constraints. Other objectives have also been testd Other objectives have also been testd Eg. Weighted sum

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

Example 1 A 2 by 2 Uncompensated System A 2 by 2 Uncompensated System

Nonlinear elements are replaced by the EIDF gain and the place of the compensator is decided

Design in time domain Structure of the compensator is now decide Structure of the compensator is now decide We started with simplest diagonal and constant controllers We started with simplest diagonal and constant controllers The desired time domain specifications are now given to the MOGA program The desired time domain specifications are now given to the MOGA program MOGA is initialized randomly and the parameter limits are set MOGA is initialized randomly and the parameter limits are set MOGA searches the space of the controller parameters to find at least one set that satisfy all the specified objectives MOGA searches the space of the controller parameters to find at least one set that satisfy all the specified objectives

The evolved controller and its performance

Name of objectives Desired objectives Resulted objectives Rise time Rise time Over shoot Over shoot settling settling Steady state Steady state Interaction 1  2 5% 0.89 % Interaction 2  1 5% 0.03 % Design criterion in time domain are met

Time responses

Conflicting objectives It is observed after 50 generation of MOGA with a population size of 50 It is observed after 50 generation of MOGA with a population size of 50 That although trade-off have been made between the objectives That although trade-off have been made between the objectives But due to conflict, all the required design criterion are not met But due to conflict, all the required design criterion are not met Alternative: we decided to use a more sophisticated controller Alternative: we decided to use a more sophisticated controller

Diagonal dynamic compensator

Design criterion in time domain are met Name of objectives Desired objectives Resulted objectives Rise time Rise time Over shoot Over shoot settling settling Steady state Steady state Interaction 1  2 5%0.23% Interaction 2  1 5%0.09%

Responses

Making the design robust In addition to the time domain specifications In addition to the time domain specifications The criterion for robust stability and robust performance is now added to the objectives The criterion for robust stability and robust performance is now added to the objectives The controllers are modified by MOGA to take care of these additional objectives The controllers are modified by MOGA to take care of these additional objectives In addition to robust stability to the design, the overall performance has also improved. In addition to robust stability to the design, the overall performance has also improved.

Responses Name of objectives Resulted objectives Rise time Rise time Over shoot1 0 Over shoot settling settling Steady state1 0 Steady state2 0 Interaction 1  % Interaction 2  %

Plant uncertainty taken into account Still a more comprehensive controller is required Still a more comprehensive controller is required

Responses from time and frequency domain objectives for uncertain plant Time Domain objectivesMixed Time and Frequency domain objectives

Characteristics of responses Name of objectives Desired objectives Resulted objectives Rise time Rise time Over shoot Over shoot settling settling Steady state Steady state Interaction 1  2 1%0% Interaction 2  1 1%0% Name of objectives Desired objectives Resulted objectives Rise time Rise time Over shoot Over shoot settling settling Steady state Steady state Interaction 1  2 1%0.1% Interaction 2  1 1%0%

Analysis and Synthesis EIDF accuracy is investigated EIDF accuracy is investigated Robust stability and robust performances Robust stability and robust performances are examined are examined Convergence of MOGA and aspects of local minima is also look into. Convergence of MOGA and aspects of local minima is also look into.

EIDF Accuracy The response of compensated system with The response of compensated system with EIDF in place and the actual nonlinearities are compared When the basic assumption of exponential input is satisfied EIDF is very accurate When the basic assumption of exponential input is satisfied EIDF is very accurate

Test for Robust stability The compensated system is subjected to a disturbance The compensated system is subjected to a disturbance It is seen that instability do not occur and system recovers to its optimal operating conditions It is seen that instability do not occur and system recovers to its optimal operating conditions

Test for Robust performance The system is subjected to step input of differing magnitude, the time domain specifications do not change The system is subjected to step input of differing magnitude, the time domain specifications do not change

MOGA Observations Observations 1. The range of controller parameters should be chosen carefully (domain knowledge is useful) 2. The Parameters of MOGA such as X-over and mutation rates should be initially of nominal vale (Pc=0.7, Pm=0.01) 3. If a premature convergence occurs then these values have to be investigated

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

Conclusions A new technique based on MOGA for design of controller for MIMO nonlinear systems were described A new technique based on MOGA for design of controller for MIMO nonlinear systems were described The EIDF linearization facilitate the time response synthesis and the extension of robustness in the frequency domain to the MIMO nonlinear systems The EIDF linearization facilitate the time response synthesis and the extension of robustness in the frequency domain to the MIMO nonlinear systems As the MOGA design progresses the designer obtain more knowledge about the system As the MOGA design progresses the designer obtain more knowledge about the system Based on the domain knowledge the designer is able to effect trade off between the conflicting objectives and also modifies the structure of the controller, if and when necessary. Based on the domain knowledge the designer is able to effect trade off between the conflicting objectives and also modifies the structure of the controller, if and when necessary.

Conclusions Time domain approach is more explicit with regards to the system time performance Time domain approach is more explicit with regards to the system time performance By combining the time and the frequency domain objectives design robustness is guaranteed By combining the time and the frequency domain objectives design robustness is guaranteed Taking ISTE or ISE as objectives, Taking ISTE or ISE as objectives, subject to S<M 1 and T<M 2 can also guarantee the robustness.

Conclusion The design technique described is compared [Katebi] and contrasted with other methods for MIMO nonlinear systems The design technique described is compared [Katebi] and contrasted with other methods for MIMO nonlinear systems The approach was shown to be effective and has several advantages over other techniques The approach was shown to be effective and has several advantages over other techniques 1. The easy formulation of MOGA 2. Provides degree of freedom for the designer 3. Acceptable computational demand 4. Accurate and multiple solutions 5. Very suitable for the powerful MATLAB environment Several other examples with different linear and nonlinear model have been solved and will be included in the thesis Several other examples with different linear and nonlinear model have been solved and will be included in the thesis

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

Future Research Different MIMO nonlinear configuration exist, further works may be undertaken for other configuration Different MIMO nonlinear configuration exist, further works may be undertaken for other configuration The class of nonlinearity considered here only encompass the memoryless (single value) elements. The class of nonlinearity considered here only encompass the memoryless (single value) elements. As the EIDF is not applicable to the multi-valued nonlinearities, theoretical works are required to extend the design to those class on nonlinearities. As the EIDF is not applicable to the multi-valued nonlinearities, theoretical works are required to 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 may become necessary. For higher dimensional systems parallel algorithms may become necessary. Application of other evolutionary algorithms such as EP, ES, GP and swarm optimization is another line of further research Application of other evolutionary algorithms such as EP, ES, GP and swarm optimization is another line of further research

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