Computation of Limit Cycles for

1. Computation of Limit Cycles for Uncertain Nonlinear Fractional-order Systems using Interval Constraint Propagation P. S. V. Nataraj & Rambabu Kalla Systems & Control Engineering IIT Bombay (India)

2. Outline • Problem Formulation • Interval Arithmetic • Constraint Satisfaction Problems • Interval Constraint Satisfaction Problems • Interval Constraint Propagation • HC4 filter (with an example)‏ • Outline of the Algorithm (Flow Chart) • Theoretical Properties • Application- Gas Turbine Plant • Computation of Limit Cycles • Conclusions

3. Problem formulation for Limit Cycle Computation Nonlinear Feedback System • Linear Fractional Order plant p(s,qg ) • Nonlinear element with describing function N(a,ω,qn ) • Example nonlinearity as relay with hysteresis type (Nonlinearity with memory)

4. Problem formulation for Limit Cycle Computation contd. • Variables • Specifications • Constraint • Specifications • Initial search domain • Specifications

5. Tools for problem solving We use the following tools to solve the problem of computing the Limit Cycles for Uncertain Nonlinear fractional-order system Tool of Interval arithmetic Tool of Interval Constraint Propagation Brief review of above mentioned tools follows.

6. Another Arithmetic ParadigmInterval Arithmetic Well known computing paradigms are Fixed & Floating point computer arithmetic paradigms. Today, researchers are seriously considering another computer arithmetic paradigm—namely, interval arithmetic. Interval arithmetic started with the work of Moore in 1966. Interval arithmetic is a natural tool to deal with problems involving uncertainty such as Robust systems analysis & Control.

7. Interval Arithmetic In interval arithmetic, if we add two intervals, we get, [a,b] + [c,d] = [a+c, b+d]. Then the lower bound of the result is rounded down to (a+c)- and the upper bound rounded up to (b+d)+. In this way, the computed result C = [(a+c)-,(b+d)+] is an interval that is known to contain the correct result.

8. Virtues of Interval Arithmetic Methods based on interval mathematics, in particular interval-Newton methods, can enclose any and all solutions to a nonlinear equation system. Also can find the global optimum of a nonlinear function. These methods provide a mathematical and also computational guarantee of reliability.

9. Constraint Satisfaction Problems A Constraint Satisfaction Problem is characterized by : • A set of variables , • For each variable with domain with possible values for that variable, and • A set of constraints, i.e., relations, that are assumed to hold between the values of the variables. The constraint satisfaction problem is to find, for each from to a value in for so that all constraints are satisfied. Applications : AI & Operations Research

10. Interval Constraint Satisfaction Problem • Set of variables • Set of Constraints • Initial search domain or box Example :

11. Solving an ICSP • Constraint Propagation (Pruning) Propagating through the constraint tree to reduce the search domain by throwing out the infeasible region • Constraint branching (Splitting)‏ Creating sub problems from the main problem and solving those sub problems.

12. HC4 Filter (For Pruning) • Forward Evaluation • Backward Propagation Example :

13. +10 y +1 -1 +1 +10 x -10 -1 -10

14. HC4 Filter (Tree Construction)‏ = + 1 ^ ^ X Y [-10,10] 2 [-10,10] 2

15. HC4 Filter (Forward Evaluation)‏ = [0,200] + 1 [0,100] [0,100] ^ ^ X Y [-10,10] 2 [-10,10] 2

16. HC4 Filter (Backward Propagation)‏ = [0,200] [1,1] + [1,1] 1 X1 + [0,100] = [1,1] X1 = [-99,1] X1 = [-99,1] ∩ [0,100] [0,100] [0,100] Y1 = [0,1] ^ X1 = [0,1] ^ X1 = [0,1] X Y [-10,10] 2 2 [-1,+1] [-10,10] [-1,+1] Y1 + [0,1] = [1,1] Y1 = [0,1] Y1 = [0,1] ∩ [0,100] Y1 = [0,1]

17. +10 y +1 -1 +1 +10 x -10 -1 -10

18. Algorithm

19. No need for approximation of FO system. • Reliable in the face of computational errors. We do not miss out any limit cycle points in the given search domain. • Accuracy guaranteed within user’s specification. The maximum error in the computed LC points cannot be more than the accuracy tolerance specified. • Computationally efficient (HC4 filter)‏ • Errors resulting in the limit cycle locus due to approximate nature of describing function method cannot be avoided. Theoretical Properties

20. Gas Turbine Engine • A critical system in aircraft is the gas turbine engine. • A gas turbine engine provides thrust under all • conditions enveloping flight spectrum of altitude • And speed. • It is a is a complex machine consisting of a number of rotating and stationary components having aerodynamic and thermodynamic properties.

21. Schematic of Gas Turbine Compressor Variable Geometry ( VG) Nozzle actuators Main Burners Afterburner flow Distributor Reheat Nozzle Hydromec-hanical Systems MainFuel VG Manual Fuel Control Linkage Fuel in PLA Gearbox Digital Electronic Control Unit

22. Various Gas Turbine Configurations • A Single Spool Turbojet Power Plant consists of an intake, a compressor, a combustion chamber, a turbine and a propelling nozzle. • A Twin Spool turbofan power plant consists of an intake, a low pressure compressor, high pressure compressor, a combustion chamber, a high pressure turbine, a low pressure turbine, a mixer, and a propelling nozzle.

23. HPC FLOW IN N O Z Z L E HPT FLOW OUT Combustor LPT Variable Geometry Fuel Flow LPC: Low Pressure Compressor HPC: High Pressure Compressor LPT: Low Pressure Turbine HPT: High Pressure Turbine Schematic of a Twin Spool Gas Turbine Engine LPC 23

24. Application to a Twin Spool Gas Turbine Plant • Operating Regime : 90% to 93% high pressure spool speed demand. • Input : Fuel rate to the gas turbine • Output : High pressure spool speed • Steady state values for i/p & o/p are 0.2442 Kg/sec & 20,620 rpm for 90% HP spool speed.

25. Identification at 90% HP spool speed • Input : PRBS signal • Sampling Time : 0.01 sec • Method : Output-Error Identification • Model Orders : OE221 to OE999 • Identification- fractional and integer order models are obtained using output-error identification technique.

26. FO Model Identification • FO model Structure used • OE identification technique with GL approximation and N = 25.

27. Gas Turbine Plant Cont. (Input Perturbation, 90% HP spool speed )

28. Gas Turbine Plant Cont. (Output, 90% HP spool speed)

29. FO Model Identified (90% HP spool speed) • FO Model Identified is

30. Model Validation (Input, 90% HP spool speed)

31. Model Validation (90% HP spool speed) Respective MSEs for FO & IO are 2.34e-04 & 2.57e-03

32. FO Model Identified (93% HP spool speed) • Similarly at another operating regime of 93% HP spool speed, the FO Model identified is

33. Combined FO Model for 90-93% HP spool speed • Combined model for 90-93% HP spool speed obtained by identification is

34. Computation of Limit Cycles for combined FO Model for GT • Gas turbine model • Nonlinear Element (Relay D with hysteresis H) • Specifications

35. Search domain and accuracy • Initial search domain • Limit Cycles are computed to an accuracy of

36. Limit Cycle Locus with variation in b1

37. Limit Cycle Locus with variation in a1

38. Limit Cycle Locus with variation in a2

39. Limit Cycle Locus with variation in a3

40. Limit Cycle Locus with variation in α1

41. Limit Cycle Locus with variation in α2

42. Limit Cycle Locus with variation in D

43. Limit Cycle Locus with variation in H

44. Analysis of Limit Cycles for the combined FO model of the Gas Turbine • The limit cycle frequency decreases with a1, α1, D but increases with b1, a2, α2 and H. • The limit cycle amplitude decreases with a2, α2 but increases with b1, a1, α1, H, D. • Both the limit cycle frequency and amplitude increase with b1, H. • Note that since a3 parameter has been fixed at unity and so does not vary, there is no variation in the limit cycle frequency and amplitude for this parameter.

45. Analysis of Limit Cycles for the combined FO model of the Gas Turbine

46. Example-2 • Model • Specifications

47. Limit Cycle Locus with variation in q1

48. Limit Cycle Locus with variation in q2

49. Limit Cycle Locus with variation in q3

50. Limit Cycle Locus with variation in q4