Fine Tuning with Sigmoid Functions in Robust Fixed Point Transformation
Kriszti´an K´osi (Phd student)
Doctoral School of Applied Informatics Obuda University´
Budapest, Hungary
Email: kosi.krisztian@phd.uni-obuda.hu
J´anos F. Bit´o, J´ozsef K. Tar Institute of Applied Mathematics
Obuda University´ Budapest, Hungary
Email:
bito@uni-obuda.hu, tar.jozsef@nik.uni-obuda.hu
Abstract—In this paper a novel implementation of the adaptive controllers designed by the use of Robust Fixed Point Trans- formation is studied. Instead guaranteeing global stability the so designed controllers work smoothly in a bounded region of operation. Both the limits of this region as well as the performance of the controller depends on the basic component of the RFPT-based design, i.e. on the properties of a sigmoid function that can be defined in various manners. In this case a special, easily parameterizable sigmoid was chosen that is widely used in the daily engineering practice, a truncated linear function. This function has a single parameter, its slope, that in the same time determines the width of the window within which the adaptive nature of the controller is guaranteed. It was found that by varying this parameter either the precision of the controller or the frequency of the necessary adaptive tuning can be improved. This statement is substantiated by simulations.
I. INTRODUCTION
In the field ofnonlinear controlthe most popular and most widely used controller designing methods even in our days are based on Lyapunov’s 2nd or “direct” method he invented in 1892 and published in his PhD Thesis in connection with the satbility of motion of dynamical systems [1], [2]. For instance in the design of nonlinear observers in fuzzy model reference controllers [3], in guaranteeing and analyzing the stability of fuzzy controllers designed for Multiple Input- Multiple Outpu (MIMO) systems [4], though in well defined special subject areas as model-based control of electric motors its use can be abandonaed ([5]).
The idea of the fixed point transformations based adaptive controllers was outlined in 2007 [6]. An especially advanta- geous variant of such transformations (called “Robust Fixed Point Transformations (RFPT)”) was introduced in 2008 [7].
The motivation in this new approach was the intent of the evasion of the mathematical complexity of the design methods using Lyapunov’s “direct” or 2𝑛𝑑 method [1] that became the most popular design tool from the sixties of the past century since Lyapunov’s results became available in English (e.g. [2]).
In the design of classical adaptive controllers (e.g. [8],[9]), the design for controlling robotic manipulation (e.g. [10], [11], [12], [13]), teleoperation applications [14], control of servo manipulators [15] Lyapunov’s 2𝑛𝑑 method traditionally plays a significant role.
Lyapunov’s great merit is providing us with a mathematical tool by the use of which the global stability of a controller can be guaranteed without knowing the details of the solution of the equations of motion of the controlled system for which generally no closed form analytical solution exists. Though it is easy to understand the essence of Lyapunov’s2𝑛𝑑 method, its application is rather an art requiring exceptionally good skills on behalf of the designer. Another significant motivating element was the fact that though global stability is a significant achievement, the designer normally has to guarantee fine details of the controlled motion that are not evidently revealed or addressed by Lyapunov’s approach. On the contrary, the new design ab ovo concentrates on the realization of some prescribed tracking error relaxation at the cost of giving up the need of global stability. In the same time it worths noting that global stability guarantees much more than the real practical needs. Each “Modern Robust Controller” necessarily has some working boundary within which it can reliably work, and normally 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], underactuated Classical Mechanical systems [20], strongly nonlinear chem- ical processes (e.g. [21], [22]). In the practice, for properly dealing with uncertainties of not statistical nature the theory of fuzzy sets can be successfully used in various control tasks (e.g. [23], [24]), the applicability of RFPT-based adaptation in the correction of the output of a fuzzy controller was also successfully investigated in [25].
The RFPT-based controller has the advantage that they contain the program excerpts of the kinematically prescribed trajectory tracking strategy, the approximate model of the system 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 based on cognitive infocommunication (e.g. [26]).
The adaptive controllers designed by the use of RFPT show certain robustness with respect to the uncertainties and errors of the available system model in the phase of the design.
Robustness is achieved by the inclusion of a sigmoid function
in the controller’s structure that determines the details of the iterative learning of the adaptation process. The range of saturation of this function also defines a bounded region of convergence for the iteration. Fine tuning of the parameters of these sigmoid functions allows the designer to optimize the controller for different purposes.
In order to keep the system in the range of stable operation complementary tuning methods for certain parameters of this controller were introduced in [27] and [28]. Furthermore, the behavior of the controller outside the region of convergence was investigated in [29] and [30]. It was found that under certain conditions it behaves like a sliding mode controller with large chattering that was reduced and later was made to cease. In the previous applications in the place of the sigmoid function the𝑡𝑎𝑛ℎ(𝑥)and𝜎(𝑥) := 1+∣𝑥∣𝑥 functions were used.
Presently the application of the truncated linear function is investigated since this function can easily be realized in the practice. The controlled system (i.e. our paradigm) used for the simulations was a 2 DoF system, two mass points coupled by nonlinear springs. In the sequel at first the basics of the RFPT-based design is briefly summarized then the paradigm used for the simulations is outlined. The simulation results will be compared with previous results.
II. THESYSTEMMODELUSED IN THESIMULATIONS
The paradigm used for the simulations was a 2 DoF system:
two mass-points coupled by nonlinear damped springs in vertical direction. In the equations of motion of thecontrolled system (1) the parameters were 𝑚1 = 20𝑘𝑔, 𝑚2 = 30𝑘𝑔 point-like masses attached to the springs, the gravitational acceleration was 𝑔= 9.81𝑚/𝑠2, the lengths of the springs at zero force were𝐿1= 0.4𝑚,𝐿2= 0.8𝑚with stiffness values 𝑘1 = 120𝑁/𝑚, 𝑘2 = 200𝑁/𝑚 and damping coefficients 𝑏1 = 0.6𝑁𝑠/𝑚, and 𝑏2 = 0.4𝑁𝑠/𝑚, respectively. Its rough model in (2) had the parameters as follows: 𝑚˜1 = 40𝑘𝑔,
˜
𝑚2 = 40𝑘𝑔, ˜𝑔 = 11𝑚/𝑠2, 𝐿˜1 = 0.3𝑚, 𝐿˜2 = 0.3𝑚,𝑘˜1 = 260𝑁/𝑚,𝑘˜2= 260𝑁/𝑚,˜𝑏1= 1𝑁𝑠/𝑚, and˜𝑏2= 1𝑁𝑠/𝑚. The nonlinearities of the springs in the controlled system and the approximate model were evidently different. The control signals were the𝑄1and𝑄2forces that were calculated for the masses 𝑚1 and𝑚2 by the use of the rough model. Tracking of the nominal trajectories (3rd order spline functions) was defined by (3) with a time-constant of desired error relaxation Λ = 20/𝑠. The numerical simulations were made by Euler integration of 1𝑚𝑠 time-resolution using the free software SCILAB. The adaptive parameter settings were 𝐵 = −1, 𝐾= 106, and the values𝐴𝑖∈ {10−7.5,10−6.5,10−5.5} were applied with the tuning solution published at [28]: instead using a single𝐴𝑐parameter the𝑟𝑛+1(𝑖) responses were averaged with some 𝑤𝑖 ≥ 0 “voting weights” with the constraint
∑
𝑖𝑤𝑖 = 1. The tuning used in [28] systematically moved the maximum weight to the best choice.
𝑚1(¨𝑞1−𝑔) +𝑘1⋅(𝑞1−𝐿1)3− 𝑘2⋅(𝑞2−𝑞1−𝐿2)3+𝑏1𝑞˙1=𝑄1
𝑚2(¨𝑞2−𝑔) +𝑘2⋅(𝑞2−𝑞1−𝐿2)3+𝑏2𝑞˙2=𝑄2 (1)
˜
𝑚1(¨𝑞1−𝑔) + ˜˜ 𝑘1⋅( 𝑞1−𝐿˜1
)5
−
˜𝑘2⋅(
𝑞2−𝑞1−𝐿˜2
)5
+ ˜𝑏1𝑞˙1=𝑄1
˜
𝑚2(¨𝑞2−˜𝑔) + ˜𝑘2⋅(
𝑞2−𝑞1−𝐿˜2
)5
+ ˜𝑏2𝑞˙2=𝑄2
(2)
¨
𝑞𝑑𝑖(𝑡) := ¨𝑞𝑖𝑁(𝑡) + 3Λ2(
𝑞𝑁𝑖 (𝑡)−𝑞𝑖(𝑡)) +3Λ( +
˙
𝑞𝑁𝑖 (𝑡)−𝑞˙𝑖(𝑡)) + +Λ3∫𝑡
0
(𝑞𝑖𝑁(𝜏)−𝑞𝑖(𝜏)) 𝑑𝜏
(3)
III. THE BASICS OF THERFPT-BASEDDESIGN
The basic equations of RFPT for a multiple input - multiple output system are given in (4, 5). In the case our paradigm, the
“system response in control cycle𝑛+ 1” i.e.𝑟𝑛+1 physically means the array [¨𝑞1,𝑞¨2], ℎ𝑛+1 := ¨𝑞𝑑𝑛+1 −𝑞¨𝑛 denotes the
“response” error,𝑒𝑛+1:= ∣ℎℎ𝑛+1𝑛+1∣, the three control parameters are𝐾𝑐,𝐴𝑐and𝐵𝑐, and𝜎()corresponds to a sigmoid function.
Normally the 𝐵 =±1 possibilities are viable depending on the design of the controller. To each control cycle an iteration belongs.
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 desired one, and computes the transformation of the input. If the controller works well then this iterative task generates a Cauchy sequence. Its appropriate limit point corresponds to the solution of the control task.
IV. THESIGMOIDFUNCTION
The sigmoid function must produce output between±1with the restrictions that 𝜎(0) = 0, 𝑑𝜎𝑑𝑥∣𝑥=0 = 1 [7]. For sigmoid function now the following construction was used (6):
𝜎:=
⎧⎨
⎩
−1𝑖𝑓 𝑥 <−𝜀
𝑥𝜀;𝑖𝑓 −𝜀≤𝑥≤𝜀
1𝑖𝑓 𝑥 >𝜀 (6)
This choice differs from the originally used sigmoids that the restriction 𝑑𝜎𝑑𝑥∣𝑥=0= 1not necessarily is met. In this manner the width of the unsaturated region is not uniquely determined by parameter 𝐴𝑐 so the introduction of this new parameter 𝜀 introduces new possibilities in the design.
V. FINETUNINGRESULTS
The results are compared with the previous ones taken from [30] where𝜎(𝑥) := 1+∣𝑥∣𝑥 was in use. If𝜀= 1the function in (6) produces results that are very similar to that obtained by the original sigmoid function. In the here presented simulations the option 𝜀 = 2.4 was chosen. Finding a balance between smoothness and precision, parameter 𝜀 = 1.6 was chosen.
The trajectories were approximately the same, just small differences are in the that look like that of the original case the original case was a bit more precise Fig. 1,2.
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 better in 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 the 𝜀= 2.4 and the original case, because the same masses were moved 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 more precise then that belonging to the truncated linear function with𝜀= 2.4.
Fig. 7. Accelerations for the truncated linear system (𝜀= 2.4)
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 than that of the truncated linear system, but it was acceptable difference 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 more smoother in the case of the truncated linear system than in the original case. It means that the controller works much smoother.
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 the difference is in the acceptable range.
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 distorted ones) 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 with parameter𝜀= 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 with parameter𝜀= 2.4 was applied.
Fig. 20. Tracking error for the truncated linear system (𝜀= 1.6)
The variation of the voting weights was much smoother than in the original case, but the truncated linear function with parameter𝜀= 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 original one.
VI. CONCLUSION
On the basis of the compared results it can stated that the introduction of a new parameter into a new (truncated linear) sigmoid function allowed the extension of the range of adaptivity in certain cases, therefore it provides the designer with 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 to implement in hardware.
ACKNOWLEDGMENT
The authors thankfully acknowledge the grant provided by the National Development Agency in the frame of the projects T ´AMOP-4.2.2/B-10/1-2010-0020 (Support of the scientific training, workshops, and establish talent management system at the ´Obuda University) and T ´AMOP-4.2.2.A-11/1/KONV- 2012-0012: Basic research for the development of hybrid and electric vehicles - The Project is supported by the Hungarian Government and co-financed by the European Social Fund.
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