Fuzzy-chaos hybrid controller for controlling of ... - Semantic Scholar

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Fuzzy-Chaos Hybrid Controller for Controlling of Nonlinear Systems Lanka Udawatta, Keigo Watanabe, Kazuo Kiguchi, and Kiyotaka Izumi

Abstract—A novel concept for controlling of nonlinear systems using chaos and fuzzy model-based regulators is presented. In the control of such systems, we employ two phases, the first of which uses open-loop control forming a chaotic attractor or using chaotic inherent features in a system itself. Once the system states reach a predefined convex domain, open-loop control is cut off and a fuzzy model-based controller is employed under state feedback control in the second phase. The relaxed stability conditions and linear matrix inequalities (LMIs)-based design for a fuzzy regulator is introduced to construct a fuzzy attractive domain, in which a global solution is obtained so as to achieve the desired stability condition of the closed-loop system. The proposed controller architecture has been tested using three nonlinear systems: the Henon map, the Lorenz attractor, and a two-link manipulator. The simulation results show the effectiveness of the proposed controller. Index Terms—Chaotic systems, convex domains, fuzzy modelbased control, linear matrix inequalities (LMIs), Lyapunov stability, nonlinear dynamics.

I. INTRODUCTION

I

N THE field of controlling nonlinear dynamical systems that show chaotic features, the approaches adopted so far can be categorized into two groups in large. One approach is to curb the chaotic dynamics of a system in order to reduce the system to a manageable level, so that conventional nonlinear control methodologies can be employed. The other aggressive approach is to intentionally activate chaotic dynamics in a normal nonlinear dynamic system to exploit some of the latent characteristics of chaotic attractors. The latter approach is illustrated in [1]–[6] where it has been stated that in a nonlinear dynamical system, a chaotic attractor can be formed by an appropriate usage of open-loop control. When it comes to employ deterministic chaos in developing control algorithms, there are tremendous advantages such as low energy consumption, robustness of the controller performance, information security and simplicity of employing chaos whenever it has chaotic attractive features in the original system itself [1], [3], [7]. This chaotic dynamics can be manipulated by the open-loop control signal so that the resultant chaotic attractor is formed to drive the system states toward a desired predefined region in the state space in the first phase of control. Once the states are attracted to the desired attractive region by the chaotic attractor, the open-loop excitation is cutoff and the second phase of control with state feedback control can be brought into effect. This will help a smooth transition of states to any given point in the state space. This basic principle has been made as the base in this approach, giving rise to a versatile controller design. Manuscript received August 3, 2000; revised January 18, 2001 and November 13, 2001. The authors are with the Graduate School of Science and Engineering, Saga University, 840-8502 Saga, Japan (e-mail: [email protected]). Publisher Item Identifier S 1063-6706(02)04825-7.

While most of the researches done so far for controlling and analyzing chaos have used conventional control techniques [8]–[10], there has been recent interest in controlling chaos by various artificial intelligence approaches [11]–[13] and some applications via chaos are available, based on practical aspects [5], [14], [15]. Joo et al. [11] explained and showed that hybrid state-space fuzzy model-based controller with dual-rate sampling can be employed for digital control of chaotic systems. Here, to obtain continuous-time optimal state-feedback gains, the constructed discrete-time fuzzy system is then converted into a continuous-time system. Controlling of chaos by a genetic algorithm (GA)-based reinforcement learning neural networks was recently developed by Lin and Jou [7]. It proposes that a temporal difference (TD)- and GA-based reinforcement neural learning scheme can be used for controlling chaotic dynamics based on small perturbations. In particular, controlling of robot manipulators with the aid of chaos phenomena has been presented in [5], [14], and [15]. The concept explained in [1]–[3] was successfully applied to control underactuated manipulators via chaos [5]. In [15], nonlinear behavior of manipulators with free-joints and control techniques via averaging method were discussed. Recently, a unified approach to controlling chaos via a linear matrix inequality (LMI)-based fuzzy control system design is introduced in [16] to realize stabilization, synchronization, and chaotic model following control. Apart from that, research over the past two decades shows a rapid development of fuzzy model-based control (FMC) theory which brings up scientist into a new era in controlling nonlinear systems [17]–[21]. In fact, the FMC can be applied very well to nonlinear dynamical systems [20]–[22]; this attempt also implies the ability of controlling chaotic systems via fuzzy regulators. Recent advances in LMI theory allowed handling nonlinear control system problems via semidefinite programming [13], [20], [23], and it is categorized as one of the promising techniques for analyzing stability criteria with rapidly growing enhanced computing facilities. Even though it is an interesting research topic to employ inherent chaotic dynamics to develop control algorithms, the optimization algorithm proposed by Vincent in [1] and [2] for finding the largest level curve in the closed-loop system will generally have multiple local minima. Therefore, to find a global minimum, almost all such local minima have to be identified with trial and error. That implies the complexity and inefficiency of the approach in [1]. In this paper, a novel algorithm is presented in order to eliminate the drawbacks of [1] by introducing a fuzzy model-based controller by employing powerful LMIs available to date for stability analysis. Here, the second phase of control is carried out by constructing a convex fuzzy attractive domain while eliminating the local minima problem with the help of evolutionary computation [3], [4], [24] or recently developed semidefinite programming algorithms.

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Fig. 2. Fig. 1.

Convex fuzzy attractive domain.

Fuzzy-chaos hybrid control scheme.

The purpose of this paper is twofold: utilization of inherent chaotic attractive features or employment of intended deterministic chaos and eliminate the shortcomings of the approach [1] are focused. Here, we introduce two phases of control. First phase uses an open-loop controller as studied in [1] and [3]. Once the system has entered a specified area, open-loop control is cut off and the proposed second phase of control scheme is adopted. In the proposed second phase, an FMC is employed under state feedback. For this purpose, the state feedback gain scheduling of the control system in the second phase is achieved by solving a set of LMIs via an optimization technique based on evolutionary computation. In addition, the proposed method has the advantage of solving LMIs either using evolutionary computation as proposed in [4] or recently developed powerful semidefinite programming tools based on convex optimization algorithms available to date in mathematical literature. Therefore, the designer has the flexibility of selecting one of the above methods that lead to obtain a global solution in the optimization process when it comes to stability analysis. The rest of the paper is organized as follows: In Section II, concept of the proposed control algorithm is discussed in details. Stability and gain scheduling of the closed-loop system are presented in Section III. A systematic approach for constructing the fuzzy attractive domain is described in Section IV, especially when it comes to multidimensional cases. In Section V, three design examples: the Henon map, the Lorenz attractor, and a two-link manipulator with simulation results are presented. Finally, concluding remarks are given in Section VI. II. CONCEPT OF THE CONTROLLER Inherent chaotic characteristics can be useful in moving a system to various points in state space confining to specific do. If the original system does not have chaotic atmain tractive features, a suitable open-loop control input will be introduced to create chaos in order to fulfill the desired control objective. In this concept, this feature is promoted to drive the with system states to a predefined convex domain aid of an appropriate open-loop control excitation on the nonlinear system. The domain must be a convex region which consists of a well-defined fuzzy rule-base and the construction details are given in the next section. Once it reaches to the predefined fuzzy attractive domain, open-loop input is cut off and a fuzzy model-based controller is employed under state feedback control to achieve desired target. This design concept is schematically given in Fig. 1 for a two-dimensional (2-D) case. to . Here, it is intended to drive the system states from

In the first phase of control, chaotic attractive features drive the to . Then, from to is systematically system from achieved via a fuzzy model-based regulator. The feedback controller design is based on multiple linearizations around a single equilibrium point, i.e., so-called off-equilibrium linearizations. It is known that the method will significantly improve the transient dynamics of the control system for a general control problem [17]. Rather, it is interesting to note that such a technique is useful for constructing a globally stable fuzzy attractive domain without trial and error. A. Off-Equilibrium Linearization Nonlinear dynamic continuous-time systems (CSs) can be described by nonlinear differential equations [or difference equations for discrete-time systems (DSs)] as for CS for DS

(1)

is the state vector and gives the control where (or input vector of the systems. The equilibrium points fixed points) of the dynamic system satisfy for CS for DS

(2)

Here, it is proposed to select a suitable set of off-equilibrium points such that all the subsystems compose a convex region which keeps the equilibrium point approximately on the center of gravity of the convex region as shown in Fig. 2. Neglecting higher order terms, we obtain a linas folearized model around any arbitrary point lows:

(3) where

B. Fuzzy Models and Regulators The dynamics of the nonlinear system are approximated near . Then, (3) can be rewritten in an arbitrary point the form

(4)

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where . need not be an equiNote here that an arbitrary point . librium point Fuzzy models (due to Takagi–Sugeno) consist of a set of IF–THEN rules for the above approximated systems. The th plant rule of each subsystems for both continuous-time and discrete-time fuzzy systems is given by

1) Stabilization of a Prespecified Equilibrium Point: In this case, the fuzzy-chaos hybrid control can be implemented by IF

THEN

ELSE (11)

IF

is

and

and

is

THEN

(5)

and where is the number of fuzzy rules and are the fuzzy sets. The state vector is , and the output vector is given by input vector is . and are the system parameter matrices is the offset term of the th fuzzy model. For a given and are the premise variables (or antecedent state, inputs). Subjecting to the parallel distributed compensation, we can design the following fuzzy regulators: Regulator Rule IF is

is an open-loop input to make the original nonwhere is the prespecified equilibrium linear system chaotic and point which would be stabilized. Moreover, if the condition of is satisfied, then fuzzy model-based regand this is the first ulator will be the controller in time that satisfies the condition of applying the second phase of control. 2) Stabilization of Any Equilibrium Point: If the stabilized equilibrium point is arbitrary among all the equilibrium points, the above fuzzy-chaos hybrid control can be modified as follows: IF

THEN

ELSE and

and

is (12)

THEN (6) is a state reference trajecfor the fuzzy models (5), where is the local tory, is the corresponding input trajectory, and feedback gain matrix. Thus, the fuzzy regulator rules have linear state-feedback laws in the consequent parts and the overall fuzzy regulator can be reduced to

denotes the rule number that has largest rule conis the corresponding feedback gain matrix, and is an equilibrium point existing in the fuzzy atth rule. tractive domain constructed by using the

where fidence,

III. CONSTRUCTION OF THE FUZZY ATTRACTIVE DOMAIN A. Requirement

(7) where (8) (9) (10)

for all , in which of membership of

in

denotes the confidence (or grade) .

C. Fuzzy-Chaos Hybrid Controller In order to control the original nonlinear system with a chaotic input in the open-loop system and a fuzzy controller in the closed-loop system, a fuzzy-chaos hybrid control scheme is proposed here. Such a control scheme can be considered in two cases, depending on the choice of equilibrium points as the reference.

Before solving LMIs for gains, the fuzzy attractive domain should be constructed using a set of well-defined off-equilibrium points as shown in Fig. 2. It is necessary to introduce a systematic design procedure to construct a given fuzzy attractive domain to cope up the second phase of control in the closed-loop mode. This section is focused to construct a fuzzy attractive domain for given system, especially when it comes to multidimensional cases. As far as higher dimensional cases are concerned, the construction of the convex region of the fuzzy attractive domain around a fixed-point becomes more complex. In the previous section, it is proposed to select a suitable set of off-equilibrium points such that all the selected points compose a convex rewhich keeps the equilibrium point . gion More generally, -dimensional state-space system needs at least off-equilibrium points which represent the convex region to ensure the stability of a particular subsystem. For examples, 2-D state-space model needs three off-equilibrium points such that the equilibrium point lies on the center of mass of an equilateral triangle keeping its corners on the three off-equilibrium points while three-dimensional (3-D) state-space model leads to a tetrahedron keeping the four off-equilibrium points of its corners [25], [26].

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Note: According to the size of the fuzzy attractive domain, one can define the maximum and minimum values for in searching the common and matrices, which guarantee the Lyapunov stability criteria of the closed-loop system as discussed in Section IV, such that

for and . The above objective function (13) can be maximized by one of the available optimization techniques such as gradient descent method, evolutionary computational algorithms, etc. Fig. 3.

Schematic diagram of the higher dimensional fuzzy attractive domain.

IV. STABILITY ANALYSIS AND GAIN SCHEDULING B. Implementation

A. Stability of the Open-Loop Controller

represents the As explained in Section II, the number number of state variables of the actual dynamical system. of Assume that the number of off-equilibrium points a multidimensional system with -number of state variables ) are defined in a convex region ( performing a fuzzy attractive domain as shown in Fig. 2. Here, represent the off-equilibrium points of the convex domain. A suitable region associated with a set of off-equilibrium points such that all the points compose a convex region , which keeps the equilibrium point nearly in the center of gravity of the convex region, is achieved by the following method. Taking the off-equilibrium points as s, as shown in Fig. 3 and distances from the equilibrium point as , respecto the points and tively. The Euclidean distance between the points is denoted by . To keep the equilibrium point nearly in the center of gravity of the convex region, the following objective function will be maximized:

Let the trajectory and initial condition the linearized equation (2). Define

(13) where

Here, represents the th Cartesian variable of the corresponding point . Furthermore

where, , i.e.,

is the th Cartesian variable of the equilibrium point , and

be a solution to (1) with be the eigenvalues of (14)

The dynamical system trajectory tive if the Lyapunov exponents earized model are such that

will be chaotic and attracof the above lin-

one of the Lyapunov exponents (15) where (16) Since the control technique of the first phase is realized through , robustness of the system is guarthe chaotic attractor anteed even for large disturbances. Chaotic attractors have particular domains such that the system states always drive toward the safe attractive region. Therefore, this advantage can not be gained by classical control schemes like PI, PID, and others. B. Stability of the Closed-Loop Controller The stability of a nonlinear control system is systematically checked by the well-known Lyapunov stability theorems. Here, the stability of the closed-loop system of (5) is explained by ensuring the stability. Off-equilibrium models , , associated with the convex domain are schematically given in Fig. 2. Tanaka et al. [20] have postulated that the equilibrium of the fuzzy system described by the Takagi–Sugeno fuzzy model (5) is asymptotically stable in the large if there exists a common positive–definite (PD) matrix and semidefinite matrix such that for a decay rate problem in CS (17)

(18) where,

and

denote the mean values of and , respectively.

such where feedback gains are denoted by , if and . Note that that as far as the speed response is concerned, it is important to

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set the largest Lyapunov exponent (decay rate) and the for all trajectories is equivcondition alent to (17) and (18). For a discrete-fuzzy control system, the system will asympand a totically stable if there exist a common PD matrix common positive–semidefinite (PSD) matrix such that (19)

(20) . where Notice that if is large, it might be difficult to find the common PD matrix and a common PSD matrix . In such improves situations, this relaxed stability condition with performance when it determines the common and in order to find the gains [20]. Note also that if the initial values of the states are selected randomly such that they are in the chaotic , the proposed method guarantee the stability attractor of the open-loop and it leads to the stability of the total system. C. Systematic Design Procedure via LMIs

subjected to

(21)

(22) (23) and . where is given by For a discrete system, the decay rate . The following LMIs are equivalent to (19) and (20): subjected to

(24) (25)

(26) and .

V. DESIGN EXAMPLES A. Henon Map 1) System Modeling: In this example, the Henon map is presented so as to illustrate the proposed design procedure. The nonlinear dynamic equations of the Henon map are given by

(27)

Engineering design often involves several objectives. Therefore, determination of feedback gains of the closed-loop system can be carried out by either evolutionary computational based algorithm as proposed in [4] or recently developed interior-point methods using LMI tool up to date. Moreover, the following LMIs are presented to fulfill a systematic design and the system designer has the flexibility of including and optimizing other constraints when it comes to evolutionary computation. For a continuous dynamical system, (17) and (18) can be rearranged to formulate the following optimization problem:

where lected as

0

Fig. 4. The uncontrolled Henon map starting from ( 0:3; 0:0).

and

is se-

For simplicity, the uncontrolled Henon map, starting from is given in Fig. 4. Here, the points 1, 2, 3, 4, and 5 denote the first consequent five points of the chaotic attractor, respectively. Control input to the system (27), was selected as in (28) in implementing the desired control algorithm:

(28) Fixed points of the system of difference equations (27) are satisfied as the equation (2), resulting two fixed-points and . Therefore, we can design two convex regions that correspond to two fixed-points. 2) Construction of Two-Fuzzy Attractive Domains: Here, we selected a triple point subregion for such that it surrounds the fixed-point as shown in Fig. 5. Assume lies on the center of mass that the equilibrium point , of an equilateral triangle having the coordinates and in order to determine the common and . In order to construct the fuzzy attractive domain that corresponds to a particular fixed point, it is necessary to determine the maximum size of the convex region such that there exit common PD matrix and PSD matrix . By varying the length of the triangle as shown in Fig. 5, it is possible to change the size of the fuzzy attractive domain with respect to a particular fixed point. Note here that, from the previous as, , and sumptions, , if and are given. By , it is possible to increase the size increasing the width of the convex domain of the first fixed-point and stability details are given in Table I.

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0

Fig. 6. Trajectory of the controlled Henon map starting from 1 ( 0:3; 0).

Fig. 5. Triple-point subsystem (i

=1

;

2; 3) and its membership functions.

TABLE I DESIGN PARAMETERS FOR VARIOUS x

0

with the LMIs of (24)–(26) and it is solved by using an optimization technique based on evolutionary computation [4]. Here we for the first fixed-point guaranobtain the common and and as follows at teeing the stability. We obtained the :

x

Similarly, and matrices associated with the second fixed: point could be obtained as follows at

The gains , and were obtained as follows, providing the global stability of the fuzzy control system for the first fuzzy attractive domain: According to the results given in Table I, the value was selected to illustrate the design procedure and three linearized models corresponded to it are given as follows: Similarly, the gains , and were obtained for the second fuzzy attractive domain as follows:

The same procedure was repeated to select the next subdomain for around the second fixedand three linearized models for the point same size of the triangular fuzzy domain as above are given as follows:

4) Simulation and Results: Based on the methodology, as proposed in Section II, the Henon map is controlled by a fuzzychaos hybrid controller. Since the system has been already a chaotic, it can be used for the first phase of control with no . Once the system states reach to one of the input above two subdomains, fuzzy controller will drive the system toward the fixed point. For example, by using the above gains , the fuzzy controller was constructed from the following IF–THEN rule base: IF IF IF

is is is

AND AND AND

is is is

THEN THEN THEN (29)

The points , , on the Henon map are shown in Fig. 6. 3) Calculation of Feedback Gains: Gain scheduling of the above problem can be formulated as an optimization problem

and represent the words positive small, poswhere itive and positive big, respectively. In order to verify the above design procedure in Sections II and III, the proposed fuzzychaos hybrid controller was applied to the chaotic system (28). Here, we allow the system to drive its states toward one of the

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Fig. 7.

Time response of the variables.

Fig. 8.

Time response: robustness over large disturbance.

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above two fixed points chaotically. The rule base (12) was employed here to construct the controller as follows:

IF

B. Lorenz Attractor 1) Dynamical System: The system of equations of Lorenz attractor is given by the following differential equations:

THEN

ELSE (30) (32) is the gain which corresponds to . Fig. 6 shows the resulting trajectory of the chaotic system . controlled by the proposed controller starting from Fig. 7 shows the resulting time response for each variable of the controlled Henon map. 5) Robustness of the Controller: One of the objectives of a control system is to provide disturbance rejection in order to maintain good performance even in the presence of external disturbances. Introducing two disturbances and to the system (28), it can be rearranged as follows: where

, and have the values , , and Here, the terms , respectively. Phase trajectory of the uncontrolled Lorenz attractor starting ) is shown in Fig. 9. System of differential from ( equations in (32) has three fixed-points numerically (0, 0, 0), (8.4853, 8.4853, 27) and ( 8.4853, 8.4853, 27). The control input was selected to control the system shown in (32) and it is given as in (33) in implementing the desired control system

(33) (31) Fig. 8 shows the response for a large disturbance at and were set to 1.63 and Here, the values of respectively.

s. 0.59,

, and represent the states of the variables , where and respectively, i.e., the system state vector is defined by . 2) Construction of the Fuzzy Attractive Domain: The fixed-point (8.4853, 8.4853, 27) was selected as the reference point in order to illustrate the design procedure. Therefore,

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Fig. 10.

3-D fuzzy attractive domain and membership for variable x.

Fig. 9. 3-D phase trajectory of the uncontrolled Lorenz attractor starting from (1,1,1).

Four linearized models corresponded to the selected fixed-point are given as follows:

TABLE II COORDINATES OF THE 3-D FUZZY ATTRACTIVE DOMAIN

the convex region which corresponds to this fixed-point was constructed using four linearized models for such that it surrounds the fixed-point (8.4853, 8.4853, 27) as explained in Section IV with aid of the optimization function (13). The equilibrium point is kept the center of mass of a tetrahedron having the coordinates of and as shown in Table II in order to corners and . Based on the methodology determine the common as proposed in Section II, the Lorenz attractor is controlled by the fuzzy-chaos hybrid controller. Since the system has been already a chaotic, it can be used for the first phase of control and a simplified version of the control with no input algorithm is presented in the next subsection. Once the system states reach to the fuzzy attractive domain, fuzzy controller will drive the system toward the fixed point. Constructed fuzzy attractive domain using the proposed concept explained in Section II and it is shown in Fig. 10. As an example, triangular (i.e., ) is also given fuzzy memberships for the variable and represent the words in Fig. 10. Here, the terms positive big, positive, and negative, respectively. The rule base can be simply implemented as follows: for variable IF

is THEN IF is THEN IF is THEN IF is THEN

AND

is

AND

AND is

is

AND

, is

,

AND

is

AND

is

,

AND

is

AND

is

, (34)

These four linearized models were used to determine the gains of the fuzzy attractive domain in the second phase of control as explained in Section II. 3) Stabilization of a Prespecified Equilibrium Point: Simplified version of the total controller is presented as follows: IF

THEN

ELSE (35) The optimization algorithm presented in (21)–(23) was apfor in plied to the system order to determine the desired gains of the feedback controller. The LMIs were solved by using an optimization technique based on evolutionary computation as explained in [4]. The following common and were obtained:

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in which denotes the mass of th link and is the th link and represent the Coriolis, centrifugal, vislength. cous friction and gravity terms, respectively

Fig. 11.

Here, the terms and are given by , and , respectively. 2) Design of Open-Loop Controller: Here, we designed the open-loop controller that gives satisfactory performance for constructing a chaotic attractor in the first phase. It can be ) constructed by applying a periodic torque ( and the position of the attracto the joint-1 keeping tive domain will be changed according to the dc component ( ), amplitude ( ) and the frequency ( ). The manipulator parameters are given follows:

3-D view of the controlled Lorenz attractor.

The following gains were obtained at

:

Note that the triangular fuzzy membership functions were selected similar to the previous case in keeping the desired width of each triangle. The controlled 3-D view of the Lorenz attractor starting from (1, 1, 1) is presented in Fig. 11 under the proposed control scheme. Fig. 12 shows the resulting time response for each variable of the controlled Lorenz system. C. Two-Link Robot Arm 1) Manipulator Modeling: In this example, two-link manipulator in Fig. 13 was taken into consideration in order to employ of the manipudeterministic chaos. Joint torque vector lator is given in the dynamic equation (36) as follows:

As an example, Table III gives different values for and for constructing corresponding attractors in Fig. 14. In this deand sign, we set the joint-1 torque with (i.e., ) in order to drive the system state vector toward the equilibrium point ( ) ). forming a chaotic attractor around ( 3) Stabilization of a Prespecified Equilibrium Point: In this case, convex domain around the equilibrium point was constructed as proposed in Section III using the objective function in (13). Gain scheduling of the closed-loop controller could be carried out using the LMIs [20]. Fuzzy attractive domain was constructed with five off-equi) and the points are given librium points , ( in Table IV. Rest of the design procedure of the closed-loop controller is similar to the first two examples given above. After the optimization process based on evolutionary computation and , the following gains for determining the common were obtained:

(36) where denotes the joint angle of th link and matrix given by

is the inertia

Fig. 15 shows the resulting phase trajectories of the controlled . system starting from

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Fig. 12.

Time response of the variables Lorenz attractor.

Fig. 13.

Two-link manipulator. TABLE III

DIFFERENT

A; B , AND ! VALUES FOR CONSTRUCTING DIFFERENT ATTRACTORS

Fig. 15. Controlled phase trajectories of stabilization of a prespecified equilibrium point.

4) Stabilization of an Arbitrary Point: It was supposed to control the tip position in the Cartesian from (0.3, 1.8) to (0.2, 0.5). As proposed in Section II, the open-loop controller can be constructed by applying a periodic torque ) to the joint-1 keeping the second joint ( ) in order to construct the as underactuated condition ( open-loop controller. Same procedure was repeated to construct the closed-loop controller as discussed in the previous section. Feedback gains of the fuzzy controller are given as follows:

Fig. 14.

Construction of different attractors.

TABLE IV COORDINATES OF THE FOUR-DIMENSIONAL (4-D) FUZZY ATTRACTIVE DOMAIN

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REFERENCES

Fig. 16.

Controlled phase trajectories for the stabilization of the selected point.

Controlled phase trajectories of the two-link manipulator for the stabilization of the selected point are given in Fig. 16. VI. CONCLUDING REMARKS A new concept for controlling chaotic systems using fuzzy model-based regulators has been presented to employ and utilize inherent chaotic features of nonlinear dynamical systems. This approach is different from the others in regarding well-defined stability criteria for the closed-loop stability with aid of a fuzzy model-based regulator. Without deforming chaotic features in a nonlinear system, a systematic design procedure has been presented. Designer has the choice of selecting one of the optimization algorithms mentioned in the paper when it comes to construct the fuzzy attractive domain, in which a global solution is obtained so as to achieve the desired stability condition of the closed-loop system. In addition to that, system designer has the flexibility of including and optimizing other constraints if it uses evolutionary computation in solving LMIs. Simulation results given by three different examples showed the effectiveness of the proposed controller and the ability of employing and utilizing deterministic chaos. Furthermore, this method has the advantages such as low energy consumption in the first phase of control, robust performance and simplicity of employing chaos. According to the results, the performance of the proposed fuzzy-chaos hybrid controller confirms the applicability of the proposed algorithm. The major drawback of this method is that it cannot be applied to all the nonlinear systems available unless it shows chaotic attractive features or possibility of employing deterministic chaos. There is a possibility that a long waiting period will exist until the system states reach to the convex domain. However, this can be minimized by enlarging the convex fuzzy attractive domain, provided that it satisfies the stability requirement. These two disadvantages cannot be eliminated whenever it employs chaos in this manner, not only in this algorithm, but also in the past work done up until now.

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