Nonlinear stability and control study of highly maneuverable high performance aircraft

APPENDIX A MATLAB and C Program Comparisons NONLINEAR STABILITY AND CONTROL STUDY OF HIGHLY MANEUVERABLE HIGH-PERFORMANCE AIRCRAFT.

NASA Project Progress Report.

by ZION HALEVY ** E.C.E, Oregon State University, Corvallis Or.

JULY 16, 1991 1. SUMMARY : The object of this report is to summarize part of the programming work done on nonlinear adaptive control model of aircraft operating in highly nonlinear regimes with large values of angle of attack.

Two models (Cho+and Stalford*models) were simulated on PC-MATLAB and converted to "C". Comparison of the simulation time indicated that performance using the "C" algorithm was increased by two orders of magnitude. Thus enabling detailed parametric analysis in short time span. The fundamental structure of C programs can be also utilized extended to future simulator models.

* * On sabbatical leave from ADA, Israel.

+ Refers to the full nonlinear longitudinal airframe model (Section 2.1 of Annual Report) * Refers to approximate nonl1_ear model of Reference 3.

-1- 2. CONTENTS.

page 3. MATLAB simulations ..............................

3.1 Cho model .....................................

3.2 Stalford model ..............................

4. "C" programs ....................................

4.1 Cho model .....................................

4.2 Stalford model ..............................

6. Conclusions and recommendations for future work... 8 9. Appendices ...................................... I0 -2- 3. MATLAB SIMULATIONS.

3.1 Cho model [2] An original MATLAB program using an Euler integration scheme (BLINI.M ,COF.M -written by Cho 1991 based on Cao et al 1990) was analyzed and restructured in order to enable general purpose use and faster execution time.

The changes performed on the original programs are as follows: a) Removal of the close-loop BARMA controller (Cho 1990) from the main program model BLINI.M and writing of a general purpose structure controller sub program.

b) Modification and improvement of the structure of the program, in particular modification of the sub program COF.M (calculate seven aerodynamic coefficients) which was called at each iteration.

The final programs named ZBLINI.M and ZCOF2.M are in the attached disk (see Appendix 3).

3.2 Stalford model. [3] A model taken from Stalford 1989 was written in PC-MATLAB with the same general purpose structure noted above . This work was done jointly with J. Dory.

The final programs named ZRUN5.M ,ZIN5.M and ZMAIN5.M using ODE45.M (Runge-Kutta 4th and 5th order integration function for numerical solution of ordinary differential equations) are on the attached disk.

The results from the MATLAB simulation ( Appendix 2) indicate that the numerical model is identical to that presented by Stalford 1989 -3- 4. "C" PROGRAMS.

4.1 Cho model The final MATLAB programs (ZBLINI.M ,ZCOF2.M) was converted to the "C" language (ZBLI3.C ,ZCOF3.C and ZBLI3.EXE -executable program).

The final programs are in the attached disk (see Appendix 3).

4.2 Stalford model.

The final MATLAB programs (ZRUN5.M ,ZIN5.M,ZMAIN5.M, ODE45.M ) was converted to "C" language (ZRUN5.C, ZIN5.C, ZMAIN5.C, ODE45.C and ZRUN5.EXE -executable program) .

* xxx.M / xxx. C extensions denote MATLAB and "C" programs files respectively.

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5. Comparison between MATLAB and "C" proqrams

Matlab is an interactive program that aid with fast performence scientific and engineering numerical calculation.

Matlab allows you to solve numerical and simulation problems in a fraction of the time it would take to write in other languages like "C" or Fortran.

Furthermore, problem solutions are expressed in almost exactly the same way as they are written mathematically.

It is a complete integrated system including graphics, programmable macros,an interpreter and analytical commands.

"C" and Turbo C++ has fast and efficient compiler that enables transfer application programs to other systems.

The MATLAB is an interpreter and hasn't a compiler like "C",so the "C" language is more efficient .

The MATLAB simulations included in this report were obtained using PC-MATLAB version 3.2 .

5.1 Cho model.

Comparison of performance are summarized in Table i.

and Appendix 1 consists of examples plots from both MATLAB "C" in the same conditions as follow: Euler integration step= 0.05 [sec] (the desired accuracy).

tfinal= I0 [sec] (final value of t).

tO= 0 [sec] (initial value of t).

-5- TABLE i: Cho model.

CONTROLLER FLIGHT TIME SIMULATION TIME PROGRAM INPUT dh [SEC] [SEC] i0.0 1068 VARIABLE I. BLINI.M ,COF.M (Cho) (original prog.)

i0.0 580 VARIABLE 2. BLINI.M, ZCOF2.M (Cho) (Improve COF.M) CONSTANT i0.0 3.ZBLINI.M,COF.M -I.0 (Remove controller from main prog, [deg] original COF.M) i0.0 364 4.ZBLINI.M, ZCOF2.M CONSTANT -i.0 (Remove controller from main prog, [deg] improve COF.M) CONSTANT 10.0 6 5. C PROGRAMS ZBLI3.C ,ZCOF3.C -i.0 (Remove controller from main prog. [deg] improve COF.M) -6- 5.2 Stalford model.

Comparison of performance are summarized in Table 2.

Appendics 2 consists of examples plots from both MATLAB and "C" in the same conditions as follow: ODE45 step integration= 0.i [sec].

ODE45 tolerance= l.e-4 [sec] (the desired accuracy).

tfinal=T FINAL=I3.6565 [sec] (final value of t).

tO= 0.45,T FINAL [sec] (initial value of t).

TABLE 2 :Stalford model.

CONTROLLER FLIGHT TIME SIMULATION TIME PROGRAM INPUT dh [SEC] [SEC] VARIABLE 1.MATLAB programs: 7.5 1866 STEP ZRUN5.M ,ZMAIN5.M ZIN5.M .

(ZIN5.M) 2. C PROGRAMS: VARIABLE 7.5 6 STEP ZRUN5.C ,ZMAIN5.C (ZIN5.M) ZIN5.C j (and ZRUN5.EXE) -7- 6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK.

6.1 Conclusions Camprison of MATLAB and "C" simulation reveal that performance was increased by two orders of magnitude by the "C" simulation .

MATLAB is a "package" for development stage but for detailed parametric analysis the "C" performance better.

6.2 recommendations.

a) Using of PC-MATLAB version 3.5 (for 386 computer) for first step developing of the simulation.

We used in PC-MATLAB version 3.2 for our simulation but the new version 3.5 (for 386 computer) is faster at least four times (it hasn't limits imposed by 16 bit nature of 80286/8086 ) We used also PC-386/20 Epson computer with 80287 numeric coprocessor chip.

b) Using of MATLAB MEX-files enables to combine "C" and MATLAB programs MEX-file produce from compiled "C" linked into .EXE files and renames to .MEX extension.

So it is possible to call your own "C" programs from MATLAB as if they were built-in MATLAB function.

Speed improvement of up to a factor of 25 are possible in this way c) Using of "C" (turbo C++ Ver 1.0 ) simulation for parametric analysis.

d) Using at Cho model in ODE45.M Runge-Kutta integration function instead of Euler integration scheme to get better accuracy of results.

-8- 7. ACKNOWLEDGEMENT.

I would like to thank Prof. R.R. Mohler,Principal Investigator and R.R Zakrzewski for the assistance during this work.

8. REFERENCES.

I. J. Cao H. stalford "analytical aerodynamic model of a high alpha research vehicle wind-tunnel model" NASA NAGI-959 1990 2. Cho model- MATLAB programs: BLINI.M ,COF.M (written by Cho based on [I]) .

3. H. Stalford E. Hoffman "Maximum principle solutions for time-optimal half-loop maneuvers of high alpha fighter aircraft" ACC ,1989 (p. 2453-2458).

4. C. Moler "PC-MATLAB for MS-DOS personal computers" User's Guide ver. 3.2 1987 5. B.W Kernighan D.M Ritchie "The C programming language" 6. Turbo C++ manual User's Guide of Borland -9-

9. APPENDICES

i. Cho model : examples plot ( Fig 1 - Fig 6 ).

2. Stalford model : examples plot ( Fig 7 - Fig i0 ).

3. Programs description of the attached disk.

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APPENDIX 3 The programs description in the attached disk : i. Cho model [21.

a) MATLAB programs.

BLINI.M- Original main program model (including the Cho close loop controller) [2].

COF.M - Original function called by main program (BLINI.M) to calculate seven aerodynamic coefficients [2] ZBLINI.M-The final main program after the changes performed on BLINI.M (see 3.1).

ZCOF2.M- The final function after the changes performed on COF.M (see 3.1).

b) "C" programs ZBLI3.C- The final MATLAB program ZBLINI.M converted to "C" ZCOFS.C- The final MATLAB program ZCOF2.M converted to "C".

ZBLI3.PRJ-"C" project (link) file for ZBLI3.C .

ZCOF3.H- "C" header file for ZCOF3.C (prototype).

ZBLI3.EXE-The final executable program for Cho model.

RES.RES- The output results file (alpa,V,q,teta,dh,t).

ZLCI.M -The program used for graphic program (at MATLAB).

to load RES.RES.

ZPCC5.M -The graphic program called by ZLCI.M.

ZPC51.M -The graphic program called by ZLCI.M.

2. Stalford model [3].

a) MATLAB programs.

ZRUN5.M- The macro running program that give initial values to the model function (ZMAIN5.M) and also graphic program (zpS.m).

-21-

ZMAIN5.M-The function describes nonlinear aircraft

equations in the form to be used by ODE45.M

(integrating routine) called by ZRUN5.M.

ZIN5.M -The function describes input controller called by

ZRUN5.M , the output of the function is the value

of the control (the elevetor angle).

ODE45.M -Integration of a system of ordinary differential

equations called by ZRUN5.M .The function is using

4th and 5th order- Runge-Kutta formulae .

ZP5.M -The graphic program called by ZRUN5.M.

b) "C" proqrams .

ZRUN5.C- The final MATLAB program ZRUN5.M converted to "C" ZMAIN5.C-The final MATLAB program ZMAIN5.M converted to llCl! , ZIN5.C- The final MATLAB program ZIN5.M converted to "C" ODE45.C- The final MATLAB program ODE45.M converted to "C" ZRUN5.PRJ-"C" project (link) file for ZRUN5.C ZMAIN5.H- "C" header file for ZMAIN5.C (prototype).

ZMAIN51.H- "C" header file for ZMAIN5.C (decleration).

ZIN5.H- "C" header file for ZIN5.C (prototype).

ODE45.H- "C" header file for ODE45.C (prototype).

UTIL.C-Utility program for ODE45.C .

UTIL.H- "C" header file for UTIL.C (prototype).

ZRUN5.EXE-The final executable program for Cho model.

RES.RES- The output results file( alfa, V, q, teta, delta_h, tn).

ZLC.M -The program used for graphic program (at MATLAB) to load RES.RES.

ZP5.M -The graphic program called by ZLC.M.

ZP51.M -The graphic program called by ZLC.M.

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APPENDIX B Project Publications PROJECT PUBLICATIONS (Supported Wholly or in Part by NASA Grant) ° R.R. Zakrzewski, RoR. Mohler, "On Nonlinear Model Algorithm Controller Design," Proceedings, IFIP Conf. Sys. Modeling and Optimiz., Zurich, 1991 (to appear).

. R.R. Mohler, V. Rajkumar, R.R. Zakrzewski, "Nonlinear Time-Series Based Adaptive Control Applications," Proceedings, IEEE Conf. Decision & Control, Brighton, 1991 (to appear).

, R.R. Mohler, Nonlinear Systems: VoL 2 Applications to Bilinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.

4.

R.R. Mohler, V. Rajkumar, R.R. Zakrzewski, "On Discrete Nonlinear Self-Tuning Control," Proceedings, Korean Control Conf., Seoul, 1991 (to appear).