[ieee 2012 ieee control and system graduate research colloquium (icsgrc) - shah alam, selangor,...

Analyzing Large Dynamic Set-point Change Tracking of MRAC by Exploiting Fuzzy Logic based

Automatic Gain Tuning

R. Karthikeyan, Rahul kumar yadav, Shikha Tripathi, Hemanth kumar.G Electronics and Communication Engineering Department,

Amrita Vishwa Vidyapeetham, Amrita School of Engineering, Bangalore, Karnataka, India - 560035

[email protected], [email protected], {t_shikha, g_hemanth}@blr.amrita.edu

Abstract—A Model reference adaptive control (MRAC) belongs to the class of adaptive servo system in which the desired performance is expressed with the help of a reference model. MRAC aims to create a closed loop controller with parameters that can change the response of the system to mimic a desired response. However, analysis unravels that there is a tolerance band for the set point change which defines the effectiveness of a particular adaptive gain (γ). Any changes in the set point which is beyond this band calls for a γ-readjustment. We propose a method which aims to overcome this pitfall in conventional MRAC by fusing fuzzy logic to dynamically vary γ. In essence, a fixed γ which fails to stabilize the system response in the advent of a large change in the average value of the set point shall be empowered with fuzzy logic to do the needful. Also, this concept never requires human interference for gain adjustment. A second order Linear Time Invariant system has been considered for all illustration. The results show considerable improvement in performance over the existing conventional MRAC system.

Keywords-Model Reference Adaptive Control (MRAC); Fuzzy logic; Adaptive control; set point tracking.

I. INTRODUCTION Non linear control technology aims to implement high performance control systems when plant dynamics are poorly known or not known at all [1]. It spans to include four classes of controllers namely, Robust controllers, Adaptive controllers, Fuzzy logic controllers and Neural controllers. MRAC belongs to the adaptive control genre, which consists of adjustable parameters and a mechanism to adjust such parameters to maintain consistent system performance in presence of uncertainty or unknown variation in plant parameters. More elaborately, for a first order system given by (1), .uBxAx += (1) where x stands for the measured output and u is the control variable, an MRAC with a finely tuned gain γ, is capable of driving the closed loop system described by a model given by (2), where xm stands for model's response and r, the reference signal.

.rBxAx mmmm += (2) In this paper, we use fuzzy logic as an inference engine and a carefully designed rule-base, which are typically IF-THEN rules [2]. In our study, we found that a particular gain value γ can yield effective set point tracking only if the average change in the set point or reference signal is within a band that a given γ can handle. Violation of the above value requires another γ to allow the needed tracking. This task of redesign can get sufficiently tedious and time consuming. To overcome this problem, exploitation of fuzzy inference can provide potential solution. To the best of our knowledge, no work has reported such a technique. Besides, trying to improve an MRAC performance in its native state has been relatively ignored in the decade gone by, which saw more of hybridizing intelligent control techniques with MRAC. Moreover, an explicit study of fuzzy logic handling any input size has not been elaborated so far. It shall be worthwhile to note that, while MRAC drives the plant's response in a manner that makes it mimic the output of the model, the incorporated fuzzy logic control claims the responsibility of providing the needed gain, allowing the plant to follow the specified model. Also, no necessity of tuning the MRAC in the advent of large step input is the merit of this fuzzy-MRAC hybrid genre. Standard metrics of performance evaluation have been provided for proving the efficacy of our study and solution. The organization of the paper is as follows. Section 2 presents a description in brief about some of the related work proposed thus far. Section 3 brings out a short discussion on constructing MRAC layout for a second order system, using MIT rule. Section 4 introduces the details of the study of the proposed problem and the solution. The simulation result of our proposed technique is presented in Section 5. We conclude the paper by posting our conclusion and few remarks along with the directions for future work in Section 6.

2012 IEEE Control and System Graduate Research Colloquium (ICSGRC 2012)

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II. BRIEF REVIEW OF RELATED WORKS

One of the earliest related works by Cheung [3] addresses the issue of tuning the adaptation gains of MRAC systems using fuzzy logic technique. The technique was implemented and applied to control a vehicle suspension system under severe operating conditions and the results exhibited both good ride and handling performance as required in vehicles. But this work did not show any simulation results on system’s response for dynamically varying large changes in the set point of the system, under the MIT-based MRAC control. It shows load changes separately which does not dynamically vary during a single course of the system. It focuses on presenting a methodology of tuning the adaptation gains by fuzzy logic technique. Another work proposed by Yin & Lee [4], aims to incorporate fuzziness in MRAC, not for gain scheduling, but to represent the unknown parameters using the fuzzy bases function expansion (FBFE) and then identify the coefficients of the FBFE. With its aid, the unknown plant parameters are estimated with good precision. The fuzzy-MRAC created, in essence, has been shown to perform better than conventional MRAC. However, their investigation does not explicitly expose or promote any idea on exploiting such a novel method for ensuring good response irrespective of the size of the input. One of the recent work presented by Shao et al. [5], proposes a Neurofuzzy control based Model reference adaptive inverse control for induction machines, which are highly nonlinear and has time-varying parameters. It aims to resolve the shortage of MRAC by using fuzzy logic and neural network, based on rotor field orientation motion model of the induction motor. However, this exceptional proposition lacks the demonstration of a detailed analysis of the drawback that we shall discuss in the succeeding sections. The work by Liu et al. [6] is a significant proposition in that it aims to fuse Neurofuzzy networks with MRAC to implement a successful nonlinear MRAC. To validate the effectiveness, varying set point changes has been considered, but the case of system response in the advent of large change in average value of the step size has not been demonstrated. Evidently, the investigation that prevailed in the last decade focused on hybridizing intelligent control systems with conventional MRAC and exploiting it on SISO or MIMO systems, thereby showing the effectiveness of the created novel idea, only for a specific application. A void created by the absence of a generic explicit study of automatic gain scheduling to handle any input size has been relatively ignored. Our proposition aims to fill this gap between current trends in MRAC investigation and former findings pertaining to MIT-rule based MRAC. The succeeding sections justify the claim.

III. MRAC USING MIT-RULE FOR A SECOND ORDER SYSTEM

The classical approach to MRAC design of a second-order linear time-invariant system is presented below. The architecture is obtained based on the MIT rule. MRAC for first-order LTI system such as (1) can be found in most of the textbooks on adaptive control (e.g. [1]). MRAC begins by defining the tracking error e, which is the difference between the plant output and the reference model output as shown in (3).

.modelplant yye −= (3)

The error enables one to define a cost function, usually denoted by J, which depends on parameter θ adapted within the controller. The cost function dictates how the parameters are to be updated. Equation (4) shows a typical cost function.

).(21)( 2 θθ eJ = (4)

Equation (5) gives the update law.

.δθδγ

δθδγθ eeJ

dtd −=−= (5)

The above relation of dθ/dt and J(θ) is known as the MIT rule and this rule provides the adaptive nature to the controller. The structure of MRAC as a function of θ1 and θ2 is displayed in Fig.1. The control law is defined in (6).

.21 plantyru θθ −= (6)

We now use (3) to derive the useful expression of the given error in (7).

.modmod rGuGyye elplantelplant −=−= (7)

Gplant and Gmodel represent the plant and the model transfer functions, respectively. The change of error with respect to the control parameters, known as sensitivity derivative, is obtained in (8) and (9), by using the model parameters, instead of that of the plant.

,01

201

1r

asasasae

mm

mm

+++

=∂∂θ

(8)

.01

201

2plant

mm

mm yasas

asae++

+=

∂∂θ

(9)

Now the MIT rule can be applied to obtain the update rule for each θ. This is illustrated in (10) and (11).


77

,01

201

1

1 erasas

asaee

dtd

mm

mm⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

−=∂∂−= γθ

γθ (10)

.01

201

2

2 eyasas

asaee

dtd

plantmm

mm⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

−=∂∂−= γθ

γθ (11)

The expressions in (10) and (11) directly translate to the MRAC architecture shown in Fig.2. The reference model should be designed next. For the purpose of illustration, we chose the model to have a Ts = 3 seconds and a ζ = 0.707, which is an industry accepted standard. The acceptable system response for gains is presented in Fig.3. It is worthwhile to note that increasing γ makes the system respond much faster, but the system threatens to become unstable. While smaller γ leads to longer adaptation time. The gain has to be manually tuned, based on the application.

Fig. 1 Generalized Structure of MRAC.

Fig. 2 MRAC layout for second order system.

Fig. 3 System response for various γ

IV. PROPOSED TECHNIQUE AND ANANLYSIS

A detailed analysis of the control strategy exploiting MRAC reveals that MIT rule by itself does not guarantee convergence or stability. An MRAC designed using MIT rule is very sensitive to the amplitudes of the signals. And thus, as a general rule the value of γ is kept small. Tuning of the same is crucial to the adaptation rate and stability of the controller. However, as the signals in a system increase, the likelihood of the system entering nonlinear regions of operation increases. For very large reference signals, this nonlinear operation will occur for almost every physical system. This is where the drawback in the conventional MIT-based MRAC will show up, which we aim to eliminate by including fuzzy logic control. Since conventional MRAC completely fails to handle a plant of nonlinear behavior, we consider exploiting fuzzy logic to vary the gain so as to help prevent instability in the control strategy of MRAC. To validate its efficacy we study the system response when subjected to large variation in the set-point within the ambit of a single course of the system's study. The succeeding subsection exhibits the fore said.

A. Potential problem in MIT based MRAC Consider the response of the second order system, presented in Fig.2, when subjected to multiple fluctuations in the set-point with γ = 0.01. This is shown in Fig.4. Evidently, large oscillations seen with γ being the same at t = 400 units, while reasonably acceptable response for t ≈ 100 units unfolds the fact that, around the set-point about which the γ was designed, there exist a tolerance band which defines the effectiveness of the chosen γ. The near marginal stability is demonstrated in Fig.5 for the same γ.

Fig. 4 Response of the second order system for

γ = 0.01.


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Fig. 5 Near marginal stability during the course of set-point

tracking for slightly larger step, with γ = 0.01. Clearly, a marginal increase in the set-point fluctuation induces large oscillations in the transient period of MRAC control (around t = 300 units). The instability aspect is exhibited in Fig.6, showing that further increase in the input size drives the system to fluctuate about the declared reference point. One can now see that on adjusting γ to 0.005, the tolerance band seems to have increased, as the revised γ is worthy of handling an even greater size of reference amplitude. Fig.7 illustrates this fact. This behavior of MIT-MRAC with respect to the gain is a matter of concern. It is known that a lower γ leads to longer adaptation time which in turn leads to extended settling time of the system's response.

Fig. 6 Failure of γ = 0.01 to handle the change in the set-point

beyond its tolerance band.

Fig. 7 Response with revised γ = 0.005.

Evidently, one cannot simultaneously handle the large variation in set-point value and its high frequency of occurrence, with a conventional MRAC. Our proposed technique addresses this drawback of the conventional MRAC & provides a feasible solution. This is presented in the succeeding subsection.

B. Proposed solution We aim to resolve the above mentioned problem by incorporating fuzziness in MRAC. The layout of this fuzzy-MRAC is realized in next section. We shall now turn our focus to the exploited fuzzy control for the purpose of automatic gain scheduling. The error in response between the model and the plant, given in (1), and the time rate of change of this error ė, are the parameters fed to the fuzzy logic block for the purpose of decision making on gain selection. Since these parameters are continuously varying, the γ value also varies throughout the course of control, till it arrives at a gain that suppresses the error to zero. Being able to make γ as γ(e,t) is the crux of the concept of handling large input size. The γ-updation derives more from art than from scientific basis, and is mathematically stated by (12).

γγγ Δ±= previousrevised (12)

The factor γΔ± , which we shall call as the correction factor for γ-revision, takes care of updating old γ to obtain a new gain as demanded by the control. The fuzzy control has been realized using the following structure of rules. IF (e is x) and (ė is k) THEN γrevised is m, where x, k, and m represents suitable linguistic variables. The designed rules are intuitive and are based on the following idea: [IF e is present and ė is lowered, THEN decrease γ] [IF e is present and ė is higher, THEN increase γ] [IF e is zero, THEN no modification required in γ] With a particular rule, Defuzzification provides the necessary numerical value to γrevised . “Centriod” is the method chosen for defuzzification in our simulation. The implementation of the proposed solution is presented in the next section along with the analysis of the associated statistics.

V. EXPERIMENTATION AND STATISTICS The experimentation of the proposed solution yielded promising results. Fig.8 shows the block diagram of the implementation. The MRAC block has been enclosed within the subsystem “MRAC”, due to space constraint. The fuzzy controller takes the error and the change in error, to yield a suitable gain in the specified range.


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Fig. 8 Fuzzy-MRAC structure.

This helps the MRAC provide the necessary control for large reference changes, that a mono-gain MRAC can't handle. The architecture of the membership function of gain is presented in Fig.9. The linguistic rule base designed for the fuzzy inference engine is presented in Table I. The acronyms used are VL-Very Low, L-Low, ME-MEdium, MO-MOre, H-High, and VH-Very High.

TABLE I. LINGUISTIC RULE BASE OF THE INFERENCE ENGINE

ė e VL L ME H VH VL VL VL ME ME H L L L ME H H ME ME ME MO H H H H H H VH VH VH M H VH VH VH

The design of the membership function chosen for change in error and the gain are presented Fig.10 and Fig.11, respectively. Rigorous efforts toward the choice of the range of action of the fore mentioned parameters can help suppress occasional oscillations & should certainly yield more appealing results, one of which is shown in Fig.13.

Fig. 9 Membership function of gain used for the control action

Fig. 10 Membership function of error used for the control

action

Fig. 11 Membership function of change in error, used for the

control action. Noteworthy remarks highlighting the areas that our proposition improves is included in Fig.13. The response of MIT-MRAC for γ=0.01 is shown in Fig.12. Table II presents the performance-indices as the standard metrics to validate the efficacy of our experimentation.

Fig. 12 System response for MIT-MRAC with γ = 0.01.

It can be inferred from this table that though a marginal & insignificant degradation in ITAE has suppressed ISE by a significant order. Clearly, apart from the adaptive nature that MRAC provides, the fuzzy logic makes the entire set-up double adaptive in that the MRAC can now handle a signal of any amplitude. The constructed severely varying conditions show the efficacy of the concept.

Fig. 13 System response for the considered design of the

membership functions and the rule-base.


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TABLE II. VALIDATION USING PERFORMANCE INDICES

Performance- Index MIT-MRAC (γ = 0.01)

Fuzzy-MRAC (automatic- tuning)

ISE 6762 4984 ITAE 2.76E+005 2.77E+005

The displayed results justify that fuzzy logic empowers MIT-MRAC.

VI. CONCLUSION AND FUTURE WORK The system proposed in this paper aims to fuse fuzzy logic with conventional MIT-rule based MRAC to handle large dynamic set-point changes. As far as the author's knowledge goes, this has not been explored so far. The result obtained by using this concept has demonstrated significant potential in handling great fluctuations in the input size. The concept also relieves the operator from manually tuning the gain, either during the course of control action, or in the advent of a large step size. Elimination of gain selection goes a long way in saving operator's time and operation cost. However, only a deliberately designed rule-base of the exploited inference engine can obtain the desired response. It becomes explicit from the obtained results that frequency of the reference fluctuation becomes a bottleneck for this genre of hybrid. The problem lies in the fact that, during the course of control with larger gain, the overshoots though of lower magnitude become apparent. Due to the nature of inferred control, damping occurs at a slower but acceptable rate. This shows up as slightly long settling time. It shall be interesting to determine

if the current advances in the area of MRAC should intersect with the presented fuzzy-MRAC to suppresses the adaptation time and also improve the overall response of conventional MRAC. Our future work shall explore the fore mentioned scalability, along with the investigation of the avenues that should further improve the parameters that govern the accuracy of the MRAC’s response.

REFERENCES [1] J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood cliffs,

NJ:Prentice-Hall, 1991. [2] T. J. Ross, Fuzzy Logic with Engineering Applications, Wiley and Sons,

2008. [3] J. Y. M Cheung, “A fuzzy logic model reference adaptive

controller,” ,” IEE colloquim on Adaptive controllers in Practice, pp. 1-6, 1996.

[4] T. K. Yin and C. G. S. Lee, “Fuzzy model-reference adaptive control,” IEEE Trans. Syst., Man, Cybern., Vol. 25, pp. 1606-1615, Dec. 1995.

[5] Z. Shao, Y. Zhan and Y. Guo, “Fuzzy neural network-based model-reference adaptive inverse control for induction machines,” IEEE Int.confrence on Applied superconductivity and electromagnetic devices, Chengudu, China, Sept.25-27, 2009.

[6] X. J. Liu, F. L. Rosano, and C. W. Chan, “Model-reference adaptive control based on Neurofuzzy networks,” IEEE Trans. Syst., Man, Cybern., Vol. 34, pp. 1094-6977, June, 2004.

[7] S. Sastry and M. Bodson, Adaptive Control: stability, Convergence and Robustness. Upper Saddle River, NJ: Prentice-Hall, 1989.

[8] Y, D. Landau, Adaptive Control: The Model Reference Approach. New York: Marcel Dekker, 1979.

[9] S. Tong, J. Tang, and T. Wang, “Fuzzy adaptive control of multivariable nonlinear systems,” IEEE Trans. Fuzzy Sets, Syst., Vol. 111, pp. 153-167, 2000.

[10] X.-J. Liu and X. Zhou, “Structure analysis of fuzzy controller with Gaussian membership function,” Proc. Of 14th IFAC World Congr., Vol. K, pp. 201-206, Beijing, China, July 5-9, 1999.


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