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

1. In the name of God Controller Design for Multivariable Nonlinear Control Systems Based on Multi Objective Evolutionary Techniques. Presented by: Mahdi Eftekhari Supervisor: Prof. S. D. Katebi Dept. of Computer Science and Engineering Shiraz University

2. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

3. Nonlinear Control • Most practical dynamic systems exhibit nonlinear behavior. • 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. • Nonlinear systems are dealt with on the case by case bases.

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

5. Extension of linear techniques • 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 • Soltine: extends feedback linearization • Astrom: extends Adaptive Control • Katebi: extends SIDF to Inverse Nyquist Array • Others…..

6. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

7. Multi-Objective OptimizationMOO • 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

8. Formulation of MOO • Single objective • Straight forward extension to MOO

9. Solution Of MOO • 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

10. GA at a glance

11. Wide rang Applications of MOO • Design, modeling and planning • Urban transportation. • Capital budgeting • Forest management • Reservoir management • Layout and landscaping of new cities • Energy distribution • Etc…

12. MOO and Control Design • Any Control systems design can be formulated as MOO • Ogata, 1950s; optimization of ISE, ISTE (analytic) • Zakian, 1960s;optimazation of time response parameters (numeric); • Clark, 1970s, LQR, LQG (analytic) • Doyle and Grimble, 1980s, (analytic) • MacFarlane, 1990s, loop shaping (grapho-analytic) • Whidborn,2000s, suggest GA for solution of all the above

13. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

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

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

16. Exponential Input Describing Function (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 • Then Where applicable, EIDF facilitate the study of the transient response in nonlinear systems

17. EIDF Derivation Single value nonlinear element • Error • ISE

18. Example of EIDF

19. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

20. A general MIMO nonlinear System • Close loop Transfer function

21. 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 C11 N11 G11 C21 N21 G21 C12 N12 G12 C22 N22 G22

22. Problems • 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 • Cross coupling and Loop interaction pose major difficulties in MIMO

23. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

24. Design procedure Start Replace: Nonlinear elements EIDFS The structure of controller is chosen Rise time, settling time,… Time domain objectives are formulated MOGA is applied to solve MOO End

25. Time Domain objectives • Find a set of M admissible points • Such that; • is real number, p is a real vector and is real function of P (controller parameter) and time • Any value of p which satisfies the above inequalities characterizes an acceptable design

26. Time domain specifications • In a control systems represents functionals Such as: • 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

27. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

28. Example A 2 by 2 Uncompensated System

29. Nonlinear elements are replaced bythe EIDF gain and the place of the compensator is decided

30. Design in time domain • Structure of the compensator is now decide • We started with simplest diagonal and constant controllers • The desired time domain specifications are now given to the MOGA program • 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

31. The evolved controller and its performance

32. Design criterion in time domain are met

33. Time responses

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

35. Diagonal dynamic compensator

36. Design criterion in time domain are met

37. Responses

38. More sophisticated controller

39. Responses from time domain and conflicting objectives

40. Characteristics of responses

41. Analysis and Synthesis • EIDF accuracy is investigated • Convergence of MOGA and aspects of local minima is also look into.

42. EIDF Accuracy • 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

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

44. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

45. Conclusions • A new technique based on MOGA for design of controller for MIMO nonlinear systems were described • The EIDF linearization facilitate the time response synthesis • 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. • Time domain approach is more explicit with regards to the system time performance

46. Conclusion • The approach was shown to be effective and has several advantages over other techniques • The easy formulation of MOGA • Provides degree of freedom for the designer • Acceptable computational demand • Accurate and multiple solutions • 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

47. Contents • Introduction • Multi-objective optimization • Nonlinear systems • Nonlinear Multivariable systems • Implementation • Results • Conclusions • Future works

48. Future Research • Different MIMO nonlinear configuration exist, further works may be undertaken for other configuration • The class of nonlinearity considered here only encompass the memory less (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. • Several explicit parallel version of MOGA exist, • 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

49. Question Time Thank you for attention