INTERACTIVE SOFTWARE FOR MODELING AND CONTROL OF NONLINEARSYSTEMS: A T-S FUZZY APPROACH

Michael Klug∗ Eugenio B. Castelan† Daniel Coutinho†

∗Instituto Federal de Santa Catarina – IFSC,Rua Pavao, 1337, Costa e Silva, 89220− 200, Joinville, SC, Brasil.

† Departamento de Automacao e Sistemas – DASUniversidade Federal de Santa Catarina – UFSC

88040-900, Florianopolis, SC, Brasil.

Email: michael.klug@ifsc.edu.br,eugenio.castelan@ufsc.br,daniel.coutinho@ufsc.br

Abstract— This paper describes a Matlab-based interactive tool focused on the modeling and control ofnonlinear systems using a Takagi-Sugeno (T-S) fuzzy approach. Noticeably the modeling of dynamical nonlinearsystems plays a fundamental role in the field of automatic control, since it allows to apply well-established analysisand control synthesis tools based on the Lyapunov theory. The interactive tool presented in this work is dedicatedto aid control practitioners to easily derive a fuzzy state-space representation which is locally equivalent to theoriginal nonlinear system and also to obtain a controller meeting desired closed-loop requirements.

Keywords— Nonlinear systems, T-S Fuzzy Systems, Modeling, Control Synthesis.

1 Introduction

The mathematical modeling of dynamical sys-tems is an important research field in automaticcontrol (Guzman et al., 2008). An analyticalmodel is essential for running simulations and de-signing model-based control systems despite inmany cases not representing every aspect of re-ality. Nevertheless, obtaining at least an approxi-mate representation is a good step for deriving acontroller performing in general better than thoseobtained with non-formal methods purely basedon empirical knowledge of the process to be con-trolled. In addition, if the plant model accuratelydescribes the system dynamics (e.g., through anexact modeling approach), then the closed-loopstability of the original system can be guaran-teed. Furthermore, the dynamics of practical con-trol systems are inherently nonlinear and thus theuse of certain approximate strategies not consi-dering some important aspects of the nonlinearbehavior (for instance, based on local linear ap-proximations) might result in undesirable effectssuch as poor performance or even closed-loop ins-tability (Khalil, 2003).

In the last decade, Takagi-Sugeno (T-S) fuzzymodels have been extensively investigated, be-cause they are able to exactly describe nonli-near dynamics yet keeping model simplicity (Feng,2010; Tanaka and Wang, 2001). Basically, T-Sfuzzy models describe nonlinear systems in termsof local linear models that are smoothly connectedby means of nonlinear fuzzy membership func-tions (MFs) allowing the development of stabi-lity analysis and control design approaches thatcan be systematically and numerically solved. Asa result, there exist numerous successful applica-tions of T-S model-based control techniques whichhave been recently reported in specialized litera-

ture (Chang and Yang, 2014; Figueredo et al.,2014; Qiu et al., 2013; Guerra et al., 2012; Tognettiet al., 2013; Klug et al., 2013; Klug et al., 2015).Many of them are based on well established Lya-punov and LMI-based tools for parameter varyingcontrol systems (Mozelli and Palhares, 2011).

Two important issues must be consideredwhen dealing with T-S fuzzy control techniqueswith respect to (w.r.t.) the original nonlinearsystem: (i) the use of convex methods to dealwith fuzzy based stability conditions can only gua-rantee the local stability of the original nonli-near closed-loop system, since it is required thatthe premise variables are bounded in some cho-sen compact set (for a more detailed explanation,see Klug et al. (2015)); and (ii) the trade-off be-tween model accuracy and the number of fuzzyrules may lead to computationally untractablecontrol design conditions which can be mitigatedby means of either approximate models as des-cribed in (Teixeira and Zak, 1999) or nonlinearT-S fuzzy models as proposed in the references(Dong et al., 2010; Klug et al., 2013; Figueredoet al., 2013) with the advantage of not compro-mising the model exactness.

In this paper, it is introduced a newly de-veloped Matlab-based interactive software for themodeling and control of nonlinear systems usingT-S fuzzy techniques taking the aforementionedissues into account. The proposed softwareis implemented using a graphical user interface(GUI) in which changes on the model and con-trol design parameters are immediately graphi-cally shown to the user, which is a powerfulframework for learning and design. Particularly,we focus on the two main Matlab-based applica-tions dedicated to modeling and control design.The reader is cordially invited to visit the website (at www.mathworks.com/discovery/matlab-

apps.html) for downloading, installing and run-ning the proposed Matlab-App in order to evalu-ate the interactive features of the Fuzzy T-S Mo-deling tool.

2 T-S Fuzzy Modeling

It is well known that nonlinear control de-sign for complex systems is a hard task. Manyof the usual techniques such as backstepping, sli-ding mode and feedback linearization, require thatthe system equations are represented in a particu-lar form or even satisfy a certain set of propertieslimiting their application in practical situations.Moreover, the control design procedure is of dif-ficult generalization and systematization. In thisscenario, an interactive Matlab-based App is con-sidered to represent a general class of nonlinearsystems by means of T-S fuzzy models.

As previously discussed, T-S fuzzy modelingis basically an interpolation of local linear mo-dels by means of (nonlinear) membership func-tions (MFs) allowing to use several results fromthe robust control theory of Linear ParameterVarying (LPV) systems. Then, the Linear Ma-trix Inequality (LMI) framework can be easily ap-plied to derive a reliable numerical solution for thecontrol design problem. However, obtaining T-Sfuzzy models through analytical equations and de-signing a controller which meets a certain closed-loop performance may not be so clear to studentsand control engineers which are not familiarizedwith this setup.

In light of the above, the main objective ofthe proposed interactive software is to automatethe process of obtaining an equivalent model ofthe original nonlinear system and then designinga controller. To this end, consider the followingclass of nonlinear systems:

xk+1 = f(xk) +H(xk)uk (1)

where xk ∈ <n and uk ∈ <m are the state and thecontrol input, respectively. The functions f(·) :<n → <n, with f(0) = 0, and H(·) : <n → <n×mare continuous and bounded for all xk. Now, webriefly present the two methodologies (approxi-mate and exact models) that were used and itsrelations with the exactness and number of rulesof the representation.

2.1 Approximate Modeling

In this T-S fuzzy modeling technique, the con-trol designer must determine r operating points(e.g., based on physical knowledge of the systemdynamics) which will be associated with local li-near models representing approximately the beha-vior of the nonlinear plant in the neighborhood ofthese points. Hence, the local models are inter-

connected by a certain membership function (res-pecting a convex sum property).

For obtaining the local models, notice that ifthe operating point of the system is also an equi-librium point, we can simply apply the linear ap-proximation by Taylor series expansion. Other-wise, considering x0 as a selected operating point,we have the following problem:f(xk) +H(xk)uk≈ Axk +Buk ∀ uk , xk ≈ x0

f(x0) +H(x0)u = Ax0 +Buk ∀ uk , xk = x0

where the optimal solution is described as(Teixeira and Zak, 1999):B = H(x0)

a(i) = ∇f(i)(x0) +f(i)(x0)− xT0∇f(i)(x0)

‖x0‖22x0

(2)for x0 6= 0, with:

∇f(i)(x) =[∂f(i)(x)/∂x(1) · · · ∂f(i)(x)/∂x(n)

]′,

A =[a(1) · · · a(n)

]′and ‖x0‖2 = x

′

0x0.

Thus, the following T-S fuzzy model can beobtained

xk+1 =

r∑i=1

αk(i) Aixk +Biuk (3)

with αk(1), . . . , αk(r) being membership functionschosen by the control designer generally with oneof the following shapes: triangular, trapezoidal,gaussian or bell.

2.2 Exact Modeling

The exact T-S Fuzzy modeling requires somealgebraic manipulations and the reader may fo-llow the step-by-step procedure as described inthe work of Klug and Castelan (2011). In simpleterms, suppose that the nonlinear function f(xk)in (1) can be rewritten as follows:

f(xk) = fa(xk) +G(xk)ϕ(xk) (4)

where ϕ(xk) ∈ <s is a sector bounded nonlinearvector function, i.e., ϕ(.) ∈ [Ω2 Ω1] (Khalil, 2003).The functions fa(·) : <n → <n, with fa(0) =0, and G(·) : <n → <n×s are continuous andbounded for all xk.

From (4), the ith element of fa(xk) is com-puted as

fa(xk)(i) =

n∑j=1

fij(xk)xk(j).

Applying a similar procedure to H(xk)uk andG(xk)ϕk and replacing it in (1), leads to the fol-lowing equivalent ith system dynamics:

xk+1(i) =

n∑j=1

fij(xk)xk(j) +

m∑κ=1

hiκ(xk)uk(κ)

+

s∑l=1

gis(xk)ϕk(s), i = 1, ..., n (5)

Now, every nonlinearity fij(xk), hiκ(xk) andgis(xk) can be represented in a compact set ofstate variables X ⊂ <n by (also known as Sec-tor Nonlinearity Approach - SNA, or max/min)

π(xk) =

2∑`=1

ω`(xk)υ` (6)

with

υ1 = maxxk∈X

π(xk) , υ2 = minxk∈X

π(xk),

ω1(xk) =π(xk)− υ2

υ1 − υ2, ω2(xk) =

υ1 − π(xk)

υ1 − υ2,

where the tuple (π, ω, υ) represents either each ele-ment of (f , δ, a), (h, β, b) and (g, γ, c). In the se-quel, replacing (6) in (5), and conveniently chan-ging the index of summations, its possible to ob-tain

xk+1 =

2∑p11=1

...

2∑pnn=1

2∑q11=1

...

2∑qnm=1

2∑r11=1

...

2∑rns=1

αp,q,r(xk)(Apxk + Bquk + Grϕk) (7)

with

αp,q,r = δ11p11 ...δnnpnnβ11q11 ...βnmqnmγ11r11 ...γnsqns

Ap =

a11p11 · · · a1np1n...

. . ....

an1pn1 · · · annpnn

Bq =

b11q11 · · · b1mq1m...

. . ....

bn1qn1 · · · bnmqnm

Gr =

c11r11 · · · c1sq1s...

. . ....

cn1qn1 · · · cnsqns

Then, aggregating the summations and per-forming a mesh transformation (modifying ϕ to acone sector nonlinearity ϕ(.) ∈ [0 Ω]), we have thefollowing nonlinear T-S fuzzy model (referred asN-Fuzzy)

xk+1 =2w∑i=1

αk(i) Aixk +Biuk +Giϕk (8)

with αi = αp,q,r and w = nn+ nm+ ns.It is important to highlight that does not nece-

ssarily exist a nonlinearity for each position of thestate and control input matrices, as represented in(7). In this case, linear or null terms are not con-sidered in the summations reducing the numberof rules in a ratio of 2nl, with nl representing thenumber of nonlinearities. Furthermore, the vectorϕk eliminates functions that should also be repre-sented by summations, which also contributes toa more compact representation. In addition, no-tice that the classical T-S fuzzy modeling is as aparticular case of the N-Fuzzy model with ϕk = 0.

3 Interactive Software

In this section we present the interactive soft-ware that automates the modeling process andcontrol design of nonlinear systems by using T-S fuzzy models. Currently, there are two sepa-rate applications: the first one is dedicated toobtaining the T-S fuzzy model, while the secondone is related to the control design. In a futurerelease the applications will be assembled into aunique one allowing users to quickly obtain con-troller gains from the simple insertion of the non-linear system and a few steps to follow. For ad-vanced user this enables the rapid implementationin real systems, significantly reducing the designphase.

Notice that modeling and control of nonlinearsystems are key elements in automatic control,and are fundamental parts in the curricula of con-trol engineering courses.

Three different languages are available forusers (English, French and Portuguese), as wellas other features, e.g. predefined examples,open/save systems, export to latex. For now, themodeling application was published as an Matlabapp1, available from version R2012b or newer. Itcan also be run from an executable file, providedthe user has installed the Matlab Compiler Run-time (MCR). In a future release the authors intendto launch it as a standalone application.

3.1 Modeling Application

This application uses the theoretical basispresented in the previous section. Basically,the programming code uses symbolic math andstring handling, among other packages/toolboxesof Matlab environment. An initial window allowsthe user to select the type of modeling (which sub-divides the app into two distinct modules: appro-ximate or exact) and the desired language. Fora more detailed explanation of how the softwareworks the reader can consult the user guide thatis available with the application.

Approximate Modeling Module:

In this part, the application allows to handlewith the model shown in (3), where the local ma-trices are computed by (2). The main window isobserved in Figure 1 with the following subdivi-sions (a Menu and five Panels):

1. Menu: the user can access pre-defined exam-ples, save and open a nonlinear system, aswell as change the language of the program.

2. Nonlinear System Panel: the user enters theanalytic equations of the nonlinear system,

1Because of its powerful library and coverage in the sci-entific community

Figure 1: Approximate T-S Modeling Program Window

using by default pre-declared states x1 andx2.

3. Region of Operation Panel: the user can con-figure the region of interest (via slide bars)and the desired number of operation points.

4. Fuzzy Rules Panel: the user can visualize thefuzzy rules.

5. Membership Functions Panel: it allows theuser to change and visualize the shape of themembership functions (not all types are al-ready programmed).

6. Open Loop Simulation: it allows the user toexecute a comparative simulation with theoriginal nonlinear plant and the respective T-S fuzzy model.

The utilization is intuitive. Once the non-linear system is inserted and the operation re-gion configured, the approximate fuzzy modelis obtained by pressing the button “Acquire TSModel”. Then, the user can view the fuzzy rulesand export them in various formats (including la-tex).

Intending to put in evidence how each subsys-tem contributes for the global fuzzy T-S model,each membership function is represented with thesame color as the associated operation point.

At last, once the nonlinear system is modeled,the user can run an open loop simulation settingup the desired initial condition and the numberof samples. In a square is represented the ini-tial condition, and the trajectory for the original

nonlinear system and for the fuzzy model are re-presented by “+” and “”, respectivelly.

Exact Modeling Module:

Let us firstly notice that this module requiresmore user familiarity with the considered exactfuzzy modeling technique and more step configu-rations than in the previous module.

The main window is observed in Figure 2. TheMenu and four Panels are identical to the onesdescribed previously: Panels 2, 5, 6 and 7 in ac-cordance with the previous Panels 2, 4, 5 and 6.The following two Panels have specific functions:

3. Validity Domain and Nonlinear TreatmentPanel: the user sets via slide bars the desireddomain of validity, where the model convexityis guaranteed. Then, the fuzzification is madeseparately for each nonlinearity, by choosingthe desired approach (max/min, sector or n-fuzzy) each time.

4. Modeling and Verification Panel: once thesystem is inserted and configured, the usershould check if the rewritten equation of thesystem is correct. Thus, the user should pressthe button ”Obtain Model”, enabling to ex-port the resulting T-S fuzzy model in variousformats (including latex).

The procedure for obtaining the exact fuzzyT-S model is based on the individual treatment ofnonlinearities of the system. After the nonlinearsystem is inserted the user must press the button“Acquire Terms”. In this way the program will

Figure 2: Exact T-S Modeling Program Window

make a separation between the linear terms (whichdo not require treatment) and nonlinear terms (re-quiring treatment). For each nonlinearity, the usermust choose the desired method (max/min, sectoror n-fuzzy) and define the premise variable, whichwill be pre-filled. Remember that for the max/minmethod a state variable must be isolated (e.g. ifthe nonlinearity is x3

1, the premise variable will bedefined as x2

1; if the nonlinearity is x1x2, the pre-misse variable can be defined as x1 or x2). Oncethe premise variable is defined, the user shouldpress the button “Set PV / Plot”, verify the graphand press the button “Next”. Repeat the proce-dure for the other nonlinearities. Finally, press“Finish”.

Similarly to the approximating modelingapplication the user can visualize the membershipfunctions and run an open loop simulation.

3.2 Control Application

To use this program it is necessary to haveobtained the T-S fuzzy model with the modelingsoftware. The program is able to load .mat files(a Matlab format), and then, compute the con-troller gains based on a selected control law andin a few optimization settings. The user must haveinstalled the Yalmip Interface, and solvers Sedumiand SPDT3.

Until now the following controllers are availa-ble: state feedback, state and sector nonlinearityfeedback, and dynamic output feedback, depend-ing on whether the system is represented (classi-cal or N-fuzzy) and based in the works Klug andCastelan (2011), Klug and Castelan (2012) andKlug et al. (2014). In future releases the authors

intend to allow that users can enter their own con-trol laws through inserting their LMIs, then in-creasing the controllers library.

The Figure 3 shows the program window. Theuser can select a objective for the optimization:just feasibility or enlargement of stability region(using a unit ball or shape approach). Additiona-lly, an inclusion condition can be insert to ensurethat closed-loop trajectory do not evolve outsidethe local domain.

4 Conclusions

A new interactive software for modeling andcontrol of nonlinear systems using a T-S fuzzy ap-proach has been described. The purpose is to as-sist researchers and engineers to design, in a fewsteps, a reasonable controller for a known nonli-near system. It also allows users to quickly obtainresults for comparison with their own techniques.From another point of view, the software interac-tivity enhance the learning process by exploitingthe advantages of immediately seeing the effects ofchanges that can never be shown in static pictures.Three languages are available as well as other in-teresting features, such as: approximate and ex-act modeling, predefined examples, open/save sys-tems, export to latex, simulation. Currently, inthe part related to control design, the users canchoose between state feedback, state and sectornonlinearity feedback, and dynamic output feed-back. Some new tools and bug fixes are underdevelopment and will be presented in future re-leases.

Figure 3: Control Design Program Window

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