Fine Tuning with Sigmoid Functions in RobustFixed Point Transformation

Krisztian Kosi(Phd student)

Doctoral School of Applied InformaticsObuda University

Budapest, HungaryEmail: kosi.krisztian@phd.uni-obuda.hu

Janos F. Bito, Jozsef K. TarInstitute of Applied Mathematics

Obuda UniversityBudapest, Hungary

Email:bito@uni-obuda.hu, tar.jozsef@nik.uni-obuda.hu

Abstract—In this paper a novel implementation of the adaptivecontrollers designed by the use of Robust Fixed Point Trans-formation is studied. Instead guaranteeing global stability theso designed controllers work smoothly in a bounded regionof operation. Both the limits of this region as well as theperformance of the controller depends on the basic componentof the RFPT-based design, i.e. on the properties of a sigmoidfunction that can be defined in various manners. In this casea special, easily parameterizable sigmoid was chosen that iswidely used in the daily engineering practice, a truncated linearfunction. This function has a single parameter, its slope, thatin the same time determines the width of the window withinwhich the adaptive nature of the controller is guaranteed. It wasfound that by varying this parameter either the precision of thecontroller or the frequency of the necessary adaptive tuning canbe improved. This statement is substantiated by simulations.

I. INTRODUCTION

In the field of nonlinear control the most popular and mostwidely used controller designing methods even in our days arebased on Lyapunov’s 2nd or “direct” method he invented in1892 and published in his PhD Thesis in connection with thesatbility of motion of dynamical systems [1], [2]. For instancein the design of nonlinear observers in fuzzy model referencecontrollers [3], in guaranteeing and analyzing the stability offuzzy controllers designed for Multiple Input- Multiple Outpu(MIMO) systems [4], though in well defined special subjectareas as model-based control of electric motors its use can beabandonaed ([5]).

The idea of the fixed point transformations based adaptivecontrollers was outlined in 2007 [6]. An especially advanta-geous variant of such transformations (called “Robust FixedPoint Transformations (RFPT)”) was introduced in 2008 [7].The motivation in this new approach was the intent of theevasion of the mathematical complexity of the design methodsusing Lyapunov’s “direct” or 2𝑛𝑑 method [1] that became themost popular design tool from the sixties of the past centurysince Lyapunov’s results became available in English (e.g. [2]).In the design of classical adaptive controllers (e.g. [8],[9]), thedesign for controlling robotic manipulation (e.g. [10], [11],[12], [13]), teleoperation applications [14], control of servomanipulators [15] Lyapunov’s 2𝑛𝑑 method traditionally playsa significant role.

Lyapunov’s great merit is providing us with a mathematicaltool by the use of which the global stability of a controllercan be guaranteed without knowing the details of the solutionof the equations of motion of the controlled system for whichgenerally no closed form analytical solution exists. Though itis easy to understand the essence of Lyapunov’s 2𝑛𝑑 method,its application is rather an art requiring exceptionally goodskills on behalf of the designer. Another significant motivatingelement was the fact that though global stability is a significantachievement, the designer normally has to guarantee finedetails of the controlled motion that are not evidently revealedor addressed by Lyapunov’s approach. On the contrary, thenew design ab ovo concentrates on the realization of someprescribed tracking error relaxation at the cost of giving up theneed of global stability. In the same time it worths noting thatglobal stability guarantees much more than the real practicalneeds. Each “Modern Robust Controller” necessarily has someworking boundary within which it can reliably work, andnormally this range is defined by cuts [16].

The RFPT-based adaptive controllers’ applicability was ex-tensively investigated via simulations for control of robots[17], platoons [18], chaos synchronization [19], underactuatedClassical Mechanical systems [20], strongly nonlinear chem-ical processes (e.g. [21], [22]). In the practice, for properlydealing with uncertainties of not statistical nature the theoryof fuzzy sets can be successfully used in various control tasks(e.g. [23], [24]), the applicability of RFPT-based adaptationin the correction of the output of a fuzzy controller was alsosuccessfully investigated in [25].

The RFPT-based controller has the advantage that theycontain the program excerpts of the kinematically prescribedtrajectory tracking strategy, the approximate model of thesystem under control, as well as the blocks of various model-independent observers in different blocks that can be con-nected by various information channels that make this con-struction compatible with the modern design approach basedon cognitive infocommunication (e.g. [26]).

The adaptive controllers designed by the use of RFPT showcertain robustness with respect to the uncertainties and errorsof the available system model in the phase of the design.Robustness is achieved by the inclusion of a sigmoid function

– 411 –

8th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 23–25, 2013 • Timisoara, Romania

978-1-4673-6400-3/13/$31.00 ©2013 IEEE

in the controller’s structure that determines the details of theiterative learning of the adaptation process. The range ofsaturation of this function also defines a bounded region ofconvergence for the iteration. Fine tuning of the parametersof these sigmoid functions allows the designer to optimize thecontroller for different purposes.

In order to keep the system in the range of stable operationcomplementary tuning methods for certain parameters of thiscontroller were introduced in [27] and [28]. Furthermore, thebehavior of the controller outside the region of convergencewas investigated in [29] and [30]. It was found that undercertain conditions it behaves like a sliding mode controllerwith large chattering that was reduced and later was made tocease. In the previous applications in the place of the sigmoidfunction the 𝑡𝑎𝑛ℎ(𝑥) and 𝜎(𝑥) := 𝑥

1+∣𝑥∣ functions were used.Presently the application of the truncated linear function isinvestigated since this function can easily be realized in thepractice. The controlled system (i.e. our paradigm) used forthe simulations was a 2 DoF system, two mass points coupledby nonlinear springs. In the sequel at first the basics of theRFPT-based design is briefly summarized then the paradigmused for the simulations is outlined. The simulation resultswill be compared with previous results.

II. THE SYSTEM MODEL USED IN THE SIMULATIONS

The paradigm used for the simulations was a 2 DoF system:two mass-points coupled by nonlinear damped springs invertical direction. In the equations of motion of the controlledsystem (1) the parameters were 𝑚1 = 20 𝑘𝑔, 𝑚2 = 30 𝑘𝑔point-like masses attached to the springs, the gravitationalacceleration was 𝑔 = 9.81𝑚/𝑠2, the lengths of the springs atzero force were 𝐿1 = 0.4𝑚, 𝐿2 = 0.8𝑚 with stiffness values𝑘1 = 120𝑁/𝑚, 𝑘2 = 200𝑁/𝑚 and damping coefficients𝑏1 = 0.6𝑁𝑠/𝑚, and 𝑏2 = 0.4𝑁𝑠/𝑚, respectively. Its roughmodel in (2) had the parameters as follows: ��1 = 40 𝑘𝑔,��2 = 40 𝑘𝑔, 𝑔 = 11𝑚/𝑠2, ��1 = 0.3𝑚, ��2 = 0.3𝑚, 𝑘1 =260𝑁/𝑚, 𝑘2 = 260𝑁/𝑚, ��1 = 1𝑁𝑠/𝑚, and ��2 = 1𝑁𝑠/𝑚.The nonlinearities of the springs in the controlled system andthe approximate model were evidently different. The controlsignals were the 𝑄1 and 𝑄2 forces that were calculated for themasses 𝑚1 and 𝑚2 by the use of the rough model. Trackingof the nominal trajectories (3rd order spline functions) wasdefined by (3) with a time-constant of desired error relaxationΛ = 20/𝑠. The numerical simulations were made by Eulerintegration of 1𝑚𝑠 time-resolution using the free softwareSCILAB. The adaptive parameter settings were 𝐵 = −1,𝐾 = 106, and the values 𝐴𝑖 ∈ {10−7.5, 10−6.5, 10−5.5} wereapplied with the tuning solution published at [28]: insteadusing a single 𝐴𝑐 parameter the 𝑟(𝑖)𝑛+1 responses were averagedwith some 𝑤𝑖 ≥ 0 “voting weights” with the constraint∑

𝑖 𝑤𝑖 = 1. The tuning used in [28] systematically movedthe maximum weight to the best choice.

𝑚1(𝑞1 − 𝑔) + 𝑘1 ⋅ (𝑞1 − 𝐿1)3 −

𝑘2 ⋅ (𝑞2 − 𝑞1 − 𝐿2)3+ 𝑏1𝑞1 = 𝑄1

𝑚2(𝑞2 − 𝑔) + 𝑘2 ⋅ (𝑞2 − 𝑞1 − 𝐿2)3+ 𝑏2𝑞2 = 𝑄2

(1)

��1(𝑞1 − 𝑔) + 𝑘1 ⋅(𝑞1 − ��1

)5

−𝑘2 ⋅

(𝑞2 − 𝑞1 − ��2

)5

+ ��1𝑞1 = 𝑄1

��2(𝑞2 − 𝑔) + 𝑘2 ⋅(𝑞2 − 𝑞1 − ��2

)5

+ ��2𝑞2 = 𝑄2

(2)

𝑞𝑑𝑖 (𝑡) := 𝑞𝑁𝑖 (𝑡) + 3Λ2

(𝑞𝑁𝑖 (𝑡)− 𝑞𝑖(𝑡)

)+

+3Λ(𝑞𝑁𝑖 (𝑡)− 𝑞𝑖(𝑡)

)+

+Λ3∫ 𝑡

0

(𝑞𝑁𝑖 (𝜏)− 𝑞𝑖(𝜏)

)𝑑𝜏

(3)

III. THE BASICS OF THE RFPT-BASED DESIGN

The basic equations of RFPT for a multiple input - multipleoutput system are given in (4, 5). In the case our paradigm, the“system response in control cycle 𝑛+ 1” i.e. 𝑟𝑛+1 physicallymeans the array [𝑞1, 𝑞2], ℎ𝑛+1 := 𝑞𝑑𝑛+1 − 𝑞𝑛 denotes the“response” error, 𝑒𝑛+1 := ℎ𝑛+1

∣ℎ𝑛+1∣ , the three control parametersare𝐾𝑐, 𝐴𝑐 and 𝐵𝑐, and 𝜎() corresponds to a sigmoid function.Normally the 𝐵 = ±1 possibilities are viable depending onthe design of the controller. To each control cycle an iterationbelongs.

𝑟𝑛+1 = (1 +𝐵𝑛+1)𝑟𝑛 +𝐾𝑐𝑒𝑛+1 (4)

𝐵𝑛+1 = 𝐵𝑐𝜎(𝐴𝑐∥ℎ𝑛+1∥) (5)

The RFPT-based control is evidently an iterative “learning”method. It compares the realized output with the desiredone, and computes the transformation of the input. If thecontroller works well then this iterative task generates aCauchy sequence. Its appropriate limit point corresponds tothe solution of the control task.

IV. THE SIGMOID FUNCTION

The sigmoid function must produce output between ±1 withthe restrictions that 𝜎(0) = 0, 𝑑𝜎

𝑑𝑥 ∣𝑥=0 = 1 [7]. For sigmoidfunction now the following construction was used (6):

𝜎 :=

⎧⎨⎩

−1 𝑖𝑓 𝑥 < −𝜀𝑥𝜀 ; 𝑖𝑓 − 𝜀 ≤ 𝑥 ≤ 𝜀1 𝑖𝑓 𝑥 > 𝜀

(6)

This choice differs from the originally used sigmoids that therestriction 𝑑𝜎

𝑑𝑥 ∣𝑥=0 = 1 not necessarily is met. In this mannerthe width of the unsaturated region is not uniquely determinedby parameter 𝐴𝑐 so the introduction of this new parameter 𝜀introduces new possibilities in the design.

V. FINE TUNING RESULTS

The results are compared with the previous ones taken from[30] where 𝜎(𝑥) := 𝑥

1+∣𝑥∣ was in use. If 𝜀 = 1 the function in(6) produces results that are very similar to that obtained by theoriginal sigmoid function. In the here presented simulationsthe option 𝜀 = 2.4 was chosen. Finding a balance betweensmoothness and precision, parameter 𝜀 = 1.6 was chosen.The trajectories were approximately the same, just smalldifferences are in the that look like that of the original casethe original case was a bit more precise Fig. 1,2.

K. Kósi et al. • Fine Tuning with Sigmoid Functions in Robust Fixed Point Transformation

– 412 –

Fig. 1. Trajectories for the truncated linear system (𝜀 = 2.4)

Fig. 2. Trajectories for the original case, taken from [30]

It can be noted that the phase trajectories are a bit betterin the original case.

Fig. 3. Phase trajectories for the truncated linear system (𝜀 = 2.4)

Fig. 4. Phase trajectories for the original case, taken from [30]

The exerted force was approximately the same for both the𝜀 = 2.4 and the original case, because the same masses weremoved along almost the same paths.

Fig. 5. Exerted force (Q) for the truncated linear system (𝜀 = 2.4)

Fig. 6. Exerted force (Q) for the original case, taken from [30]

The accelerations of the original system were a bit moreprecise then that belonging to the truncated linear functionwith 𝜀 = 2.4.

Fig. 7. Accelerations for the truncated linear system (𝜀 = 2.4)

– 413 –

8th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 23–25, 2013 • Timisoara, Romania

Fig. 8. Accelerations for the original case, taken from [30]

Fig. 9. Accelerations for the truncated linear system (zoomed) (𝜀 = 2.4)

Fig. 10. Accelerations for the original case (zoomed), taken from [30]

The tracking error was smaller in the original system thanthat of the truncated linear system, but it was acceptabledifference for that case.

Fig. 11. Tracking error for the truncated linear system (𝜀 = 2.4)

Fig. 12. Tracking error for the original case, taken from [30]

The variation of the voting weights were much moresmoother in the case of the truncated linear system than inthe original case. It means that the controller works muchsmoother.

Fig. 13. Voting weights for the truncated linear system (𝜀 = 2.4)

Fig. 14. Voting weights for the original case, taken from [30]

The response error is better in the original system, but thedifference is in the acceptable range.

K. Kósi et al. • Fine Tuning with Sigmoid Functions in Robust Fixed Point Transformation

– 414 –

Fig. 15. Response error for the truncated linear system (𝜀 = 2.4)

Fig. 16. Response error for the original case, taken from [30]

The “Required Accelerations” (i.e. the adaptively distortedones) were approximately same for both cases.

Fig. 17. The “Required Acceleration” for the truncated linear system (𝜀 =2.4)

Fig. 18. The “Required Acceleration” for the original case, taken from [30]

The response error was higher than in the original case,but it is less when truncated linear sigmoid function withparameter 𝜀 = 2.4 applied.

Fig. 19. Response error for the truncated linear system (𝜀 = 1.6)

The tracking error was higher than in the original case,but it was less when truncated linear sigmoid function withparameter 𝜀 = 2.4 was applied.

Fig. 20. Tracking error for the truncated linear system (𝜀 = 1.6)

The variation of the voting weights was much smootherthan in the original case, but the truncated linear function withparameter 𝜀 = 2.4 yielded smoother result.

Fig. 21. Voting weights for the truncated linear system (𝜀 = 1.6)

The results show that by sacrificing some precision a con-troller can be obtained that works smoother than the originalone.

– 415 –

8th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 23–25, 2013 • Timisoara, Romania

VI. CONCLUSION

On the basis of the compared results it can stated thatthe introduction of a new parameter into a new (truncatedlinear) sigmoid function allowed the extension of the range ofadaptivity in certain cases, therefore it provides the designerwith richer possibilities as

∙ Increasing the precision of the controller, or∙ Smoothing the operation of the, or∙ Finding a balance between smoothness and precision.

The one of the important fact is that this solution is easy toimplement in hardware.

ACKNOWLEDGMENT

The authors thankfully acknowledge the grant provided bythe National Development Agency in the frame of the projectsTAMOP-4.2.2/B-10/1-2010-0020 (Support of the scientifictraining, workshops, and establish talent management systemat the Obuda University) and TAMOP-4.2.2.A-11/1/KONV-2012-0012: Basic research for the development of hybrid andelectric vehicles - The Project is supported by the HungarianGovernment and co-financed by the European Social Fund.

REFERENCES

[1] A.M. Lyapunov, A general task about the stability of motion. (in Russian),PhD Thesis, 1892.

[2] A.M. Lyapunov, Stability of motion. Academic Press, New-York andLondon, 1966.

[3] M. A. Khanesar, M. Teshnehlab, O. Kaynak, Model Reference FuzzyControl of Nonlinear Dynamical Systems Using an Optimal Observer,Acta Polytechnica Hungarica, Vol. 8, No.4, pp 35-54, 2011.

[4] R-E. Precup1, M-L. Tomescu, E. M. Petriu, S. Preitl, C-A. Dragos, StableDesign of a Class of Nonlinear Discrete-Time MIMO Fuzzy ControlSystems, Acta Polytechnica Hungarica, Vol. 9, No.2, pp 57-76, 2012.

[5] R. Sadouni, A. Meroufel, Indirect Rotor Field-oriented Control (IRFOC)of a Dual Star Induction Machine (DSIM) Using a Fuzzy Controller, ActaPolytechnica Hungarica, Vol. 9, No.4, pp 177-192, 2012.

[6] J.K. Tar, I.J. Rudas and K.R. Kozłowski, Fixed Point Transformations-Based Approach in Adaptive Control of Smooth Systems. Lecture Notesin Control and Information Sciences 360 (Eds.: M. Thoma and M.Morari), Robot Motion and Control 2007 (Ed.: Krzysztof R. Kozłowski),Springer Verlag London Ltd., pp. 157–166, 2007.

[7] J.K Tar, J.F. Bito, I.J Rudas, K.R. Kozlowski J.A. Tenreiro Machado,Possible adaptive control by tangent hyperbolic fixed point transforma-tions used for controlling the Φ6-type Van der Pol oscillator, Proc. of the6th IEEE International Conference on Computational Cybernetics (ICCC2008), November 27-29, 2008, Stara Lesna, Slovakia, 15-20.

[8] Jean-Jacques E. Slotine, W. Li, Applied Nonlinear Control, Prentice HallInternational, Inc., Englewood Cliffs, New Jersey, 1991.

[9] R. Isermann, K.H. Lachmann, and D. Matko, Adaptive Control Systems,New York DC, Prentice-Hall, USA, 1992.

[10] R.M. Murray, Z. Li, S.S. Sastry, A mathematical introduction to roboticmanipulation, CRC Press, New York, 1994.

[11] Charles C. Nguyen, Sami S. Antrazi, Zhen-Lei Zhou, Charles E.Campbell Jr., Adaptive control of a stewart platform-based manipulator,Journal of Robotic Systems, Volume 10, Number 5, 1993, pp. 657–687,1993.

[12] R. Kamnik, D. Matko and T. Bajd, Application of Model ReferenceAdaptive Control to Industrial Robot Impedance Control, Journal ofIntelligent and Robotic Systems, vol. 22, pp. 153–163, 1998.

[13] J. Somlo, B. Lantos, P.T. Cat, Advanced robot control, Akademiai Kiado,Budapest, 2002.

[14] K. Hosseini–Suny, H. Momeni, and F. Janabi-Sharifi, Model ReferenceAdaptive Control Design for a Teleoperation System with Output Pre-diction, J Intell Robot Syst, DOI 10.1007/s10846-010-9400-4, pp. 1–21,2010.

[15] C.J. Khoh and K.K. Tan, Adaptive robust control for servo manipulators,Neural Comput & Applic, vol. 12, pp. 178–184, 2005.

[16] L. Kovacs, Modern Robust Control in Patophysiology From Theory toApplication, IEEE 11th International Symposium on Applied MachineIntelligence and Informatics (SAMI 2013), pp. 13. 2013.

[17] ) J.K. Tar, J.F. Bito, Krzysztof R. Kozłowski, J.A. Tenreiro Machado,Application of Robust Fixed Point Transformations for TechnologicalOperation of Robots, Lecture at the Seventh International Workshop onRobot Motion and Control, June 1-3, 2009, Czerniejewo, Poland; issued inLecture Notes in Control and Information Sciences 396 (Eds: M. Thoma,F. Allgver, M. Morari) - Robot Motion and Control 2009 (Ed. KrzysztofR. Kozowski), Springer-Verlag Berlin Heidelberg, Chapter 9, pp. 93–101.

[18] J.K. Tar, Decentralized Control of Platoons Based on a Novel AdaptiveControl of Lucid Geometric Interpretation, in the Proc. of the 9th WSEASIntl. Conf. on Mathematical and Computational Methods in Science andEngineering” (MACMESE’07), Trinidad and Tobago Islands, November5-7 2007, WSEAS Press, pp. 185–190.

[19] T.A. Varkonyi, J.K. Tar, I.J. Rudas, Robust Fixed Point Transformationsin Chaos Synchroniztion, in Proc. of the 11th International Symposiumof Hungarian Researchers on Computational Intelligence and Informatics,Budapest, November 18-20, 2010, Obuda University, Hungary, pp. 219–224

[20] T.A. Varkonyi, J.K. Tar, I.J. Rudas, Fuzzy Parameter Tuning in theStabilization of an RFPT-based Adaptive Control for an UnderactuatedSystem, Proc. of the 2011 IEEE 12th International Symposium onComputational Intelligence and Informatics (CINTI), 21-22 Nov. 2011,Budapest, pp. 591–596, 2011.

[21] K. Kosi, J.F. Bito, J.K. Tar, On the Effects of Strong Asymmetries onthe Adaptive Controllers Based on Robust Fixed Point Transformations,IEEE 10th Jubilee International Symposium on Intelligent Systems andInformatics (SISY 2012), September 20-22 2012, Subotica, Serbia, pp.259–264.

[22] J. K.Tar, I. J. Rudas, L. Ndai, K. Kosi, Adaptive Controllability of theBrusselator Model with Input Coupling, 4th IEEE International Sympo-sium on Logistics and Industrial Informatics (LINDI 2012), September5-7 2012, Smolenice, Slovakia, pp. 157–162.

[23] J. Tick and J. Fodor, Fuzzy Implications and Inference Processes,COMPUTING AND INFORMATICS, 24:(6) pp. 591–602, 2005.

[24] J. Tick, Fuzzy extension to P-graph based workflow models, IEEE 7thInternational Conference on Computational Cybernetics (ICCC 2009),Palma de Mallorca, Spain, 26-29 November, 2009. pp. 109–112, 2009.

[25] T.A. Varkonyi, Fuzzyfied Robust Fixed Point Transformations. In: Pro-ceedings of the 16th International Conference on Intelligent EngineeringSystems, INES 2012, Lisbon, Portugal, 1315 June 2012, pp. 457-462,2012.

[26] A. Csapo, P. Baranyi, A Unified Terminology for the Structure andSemantics of CogInfoCom Channels, Acta Polytechnica Hungarica, Vol.9, No.1, pp 85-105, 2012.

[27] J.K. Tar, Towards Replacing Lyapunov’s “Direct” Method in AdaptiveControl of Nonlinear Systems, (invited plenary lecture), Prorc. of the3rd Conference in Mathematical Methods in Engineering, 21–24 October2010, Coimbra, Portugal, Paper 11 (CD issue) 2010.

[28] J.K. Tar, L. Nadai, I.J. Rudas, T.A. Varkonyi, RFPT-based AdaptiveControl Stabilized by Fuzzy Parameter Tuning, Proc. of the 9th EuropeanWorkshop on Advanced Control and Diagnosis, ACD 2011. Budapest,Hungary, pp. 1–8. Paper 6, 2011.

[29] T.A. Varkonyi, J.K. Tar, I.J. Rudas, I. Kromer, VS-type Stabilizationof MRAC Controllers Using Robust Fixed Point Transformations, Proc.of the 7th IEEE International Symposium on Applied ComputationalIntelligence and Informatics, May 24-26, 2012, Timisoara, Romania, pp.389–394, 2012.

[30] K. Kosi, Sz. Hajdu, J.F. Bito, J.K. Tar, Chaos Formation and Reductionin Robust Fixed Point Transformations Based Adaptive Control, 4th IEEEInternational Conference on Nonlinear Science and Complexity (NSC2012), August 6-11 2012, Budapest, Hungary, pp. 211-216.

K. Kósi et al. • Fine Tuning with Sigmoid Functions in Robust Fixed Point Transformation

– 416 –