Robust Switched Control Design for Nonlinear Systems Using Fuzzy Models

1 Department of Academic Areas of Jataı́, Federal Institute of Education, Science and Technology of Goiás (IFG), Campus Jataı́, 75804-020 Jataı́, GO, Brazil 2 Department of Electrical Engineering, Univ Estadual Paulista, Campus of Ilha Solteira (UNESP), 15385-000 Ilha Solteira, SP, Brazil 3 Department of Academic Areas of Januária, Federal Institute of Education, Science and Technology of Norte of Minas Gerais (IFNMG), Campus Januária, 39480-000 Januária, MG, Brazil

Results on switching laws based on the premise variable can be seen in [9,10,17].In [9,10], a switched fuzzy system was used to represent the nonlinear dynamical model of a hovercraft vehicle and to design a switching fuzzy controller.Then, in [10] smoothness conditions were established, which avoid the phenomenon of discontinuity in the control signal.The problem of dynamic output feedback H ∞ control was addressed in [17].Switching laws based on the values of the membership functions are considered in [11,12,[14][15][16], where the switched control scheme presented in [14] is an extension of the parallel distributed compensation (PDC).A dynamic output feedback controller, which is based on switched dynamic parallel distributed compensation, was proposed in [15].
Switching laws based on the plant state vector were proposed, for instance, in [13,18].The control design presented in [13] uses local state feedback gains obtained from the solution of an optimization problem that assures a guaranteed cost performance.LMIs conditions for robust switched fuzzy parallel distributed compensation controller design and a H ∞ criterion were obtained in [18].The procedure to design switching controllers described in [18] was based on the switched quadratic Lyapunov function proposed in [19].
This paper proposes a new method of switched control for some classes of uncertain nonlinear systems described by Takagi-Sugeno fuzzy models.This new control law, which also depends on the state variables, generalizes the results given in [8], which considered only linear plants.The proposed controller chooses a gain from a set of gains by means 2 Mathematical Problems in Engineering of a suitable switching law that returns the smallest value of the Lyapunov function time derivative.The proposed methodology enables us to design the set of gains based on LMIs and on the parallel distributed compensation, as proposed, for instance, in [20][21][22][23][24][25][26].
The main advantage of this new procedure is its practical application because it eliminates the need to find the explicit expressions of the membership functions, which can often have long and/or complex expressions or may not be known due to the uncertainties.Furthermore, for certain classes of nonlinear systems, the switched controller can operate even with an uncertain reference control signal.Additionally, with the proposed methodology the closed-loop systems usually present a settling time that is smaller than those obtained with fuzzy controllers, without using switching, that are widely studied in the literature.Moreover, performance indices such as decay rate and constraints on the plant's input and output can be added in the control design procedure.
Simulation results of the control of a ball-and-beam system and of a magnetic levitator are presented to compare the performance of the proposed control law with the traditional PDC fuzzy control law [20,22].The computational implementations were carried out using the modeling language YALMIP [27] with the solver LMILab [28].
The paper is organized as follows.Section 2 presents the preliminary results on Takagi-Sugeno fuzzy model, fuzzy regulator design, and stability of the Takagi-Sugeno fuzzy systems via LMIs.Section 3 offers a new switching control method for some classes of nonlinear systems described by Takagi-Sugeno fuzzy models.Some examples to illustrate the performance of the new proposed method are given in Section 4. Finally, Section 5 draws the conclusions.
For convenience, in some places, the following notation is used:

Takagi-Sugeno Fuzzy Systems and Fuzzy Regulator
Consider the Takagi-Sugeno fuzzy model as described in [29][30][31]: where    is the fuzzy set  of the rule ,  ∈   and  ∈   , () ∈ R  is the state vector, () ∈ R  is the input vector, () ∈ R  is the output vector,   ∈ R × ,   ∈ R × ,   ∈ R × , and  1 (), . . .,   () are premise variables that in this paper are the state variables.
From [20], ẋ () given in (2) can be written as follows: where   (()) is the normalized weight of each local model system   () +   () that satisfies (1).Assuming that the state vector () is available, from the Takagi-Sugeno fuzzy model ( 2), the control input of fuzzy regulators via parallel distributed compensation has the following structure [20]: Similar to (3), one can consider the control law [20] From ( 5), (3), and (1), one obtains 2.1.Stability of Takagi-Sugeno Fuzzy Systems via LMIs.The following theorem, whose proof can be seen in [20], guarantees the asymptotic stability of the origin of the system (6).
Theorem 1.The equilibrium point  = 0 of the continuoustime fuzzy control system given in (6) is asymptotically stable in the large if there exist a common symmetric positive definite matrix  ∈ R × and   ∈ R × such that, for all ,  ∈ K  , the following LMIs hold: excepting the pairs (i,j) such that     = 0, for all .If (7) are feasible, the controller gains are given by   =    −1 ,  ∈ K  .
Remark 2. In this paper, for simplicity, the new design method of the controller gains was based on Theorem 1.However, the proposed methodology does not exclude the use of other relaxed control design methods also based on LMIs, for plants described by Takagi-Sugeno fuzzy models, as those presented in [20,22,23,[32][33][34][35].
Suppose that (7) are feasible and let   =    −1 ,  ∈ K  , be the gains of the controller given in (5), and  =  −1 is obtained from the conditions of Theorem 1.Then, define the switched controller by Therefore, from (1), the controlled system ( 8) and ( 9) can be written as follows: Theorem 3. Assume that the conditions of Theorem 1, related to the system (8) with the control law (5), hold and obtain   =    −1 ,  ∈ K  and  =  −1 .Then, the switched control law (9) makes the equilibrium point  = 0, of the system (8), asymptotically stable in the large.
Proof.Consider a quadratic Lyapunov candidate function  =   .Define V   and V   as the time derivatives of  for the system (8), with the control laws (5) and (9), respectively.Then, from (10), Thus, note that, from (1) and ( 9), min Therefore, from (11) and the laws given in ( 9) and ( 5) observe that Then, V   ≤ V   .Furthermore, from Theorem 1 V   < 0 for  ̸ = 0. Thus, the proof is concluded.
Remark 4. Theorem 3 shows that if the conditions of Theorem 1 are satisfied, then V   (()) < 0 for all () ̸ = 0 and thus V   (()) < 0 for () ̸ = 0, ensuring that the equilibrium point  = 0 of the controlled system ( 8) and ( 9) is asymptotically stable in the large.Thus, Theorem 1 can be used to project the gains  1 ,  2 , . . .,   and the matrix  =  −1 of the switched control law (9).Additionally, note that the switched control law (9) does not use the membership functions   ,  ∈ K  , which would be necessary to implement the control law (5) and may thus offer a relatively simple alternative for implementing the controller.

Case 2: Fuzzy System with Nonlinearity in the Matrix 𝐵(𝛼).
In this case a fuzzy system similar to (3) will be considered, with   ,  ∈ K  , defined in (1); namely, Let V ∈ R  be the time derivative of the control input vector  ∈ R  .Define  + and V  , such that ẋ + () = u  () = V  (),  ∈ K  .Thus one obtains the following system: or equivalently [36] ẋ () =  ()  () + V () , where After the aforementioned considerations, note that the system ( 16) is similar to the system (8) and therefore the control problem falls into Case 1.Thus, one can adopt the procedure stated in Case 1 for designing a switched control law V() = −  (),   ∈ R ×+ .

Case 3: Fuzzy System with Uncertainty in the Control
Signal.In this case, it is assumed that the plant given by ẋ = (, ) has an equilibrium point  =  0 and the respective control input is  =  0 , such that ( 0 ,  0 ) = 0. Suppose that  0 is known,  0 is uncertain, but 0 <  0 ∈ [ 0 min ,  0 max ], where  0 min and  0 max are known, and the plant can be described by the Takagi-Sugeno fuzzy system (1)-(3), where () = ()− 0 , () is the state vector of the plant and () = () −  0 ,  is the control input of the plant.

Mathematical Problems in Engineering
Now consider that () can be written as follows: where  is a known constant matrix and (()) > 0, for all , is an uncertain nonlinear function.Thus, the system (18) can be written as follows: Assume that the gains   =    −1 ,  ∈ K  , and the matrix  =  −1 have been obtained using the vertices of the polytope of the system (18) in the LMIs (7) from Theorem 1, as proposed in [20].Now, given a constant  > 0, define the control law as where Within this context the following theorem is proposed.

Example of Case 1.
To illustrate this case, presented is the control design of a ball-and-beam system, in Figure 1, whose mathematical model [38, page 26] is given by the following equations: where  is the position of the ball;  is the angle of the beam relative to the ground;  is the torque applied to the beam and the control input;  = 9.81 m/s 2 is the acceleration of the gravity; and  =  2 /(  + 2 ) is an uncertain parameter of the system which depends on the mass , the radius , and the moment of inertia   of the ball. , the system (26) can be written as follows: or equivalently where Note that, for implementing the switched controller (9), the controller gains will be designed using the generalized form proposed in [22], and therefore the following domain will be considered for the system (28) and ( 29): After the calculations the following maximum and minimum values of the functions  23 and  24 were obtained: as the membership functions of the system ( 28) and ( 29), and their local models: ] . (41) The goal of the simulation is to keep the ball at the origin (, ) = (0, 0).Considering,  = 0.7, the initial condition , and the equilibrium point   = [0 0 0 0]  , the simulation of the controlled systems ( 28), ( 29), ( 9), ( 41) and ( 28), ( 29), ( 5), ( 31)-( 41) presented the responses shown in Figures 2 and 3. Note that the controller gains have been found using the generalized form proposed in [22].However, the switched controller   given in (9) does not use the membership functions and therefore it is not necessary to find and implement such functions.Thus, an advantage of this new methodology is that one can eliminate all the steps of the project given in ( 32)-( 39) that are needed to find the membership functions, which can sometimes have long and/or complex expressions or may not be known due to the uncertainties and so their practical implementations are not possible, as is the case of this example.

Example of Case 2.
To illustrate this case, consider the control system design of a magnetic levitator presented in Figure 4, whose mathematical model [38, page 24] is given by where  = 0.05 Kg is the mass of the ball;  = 9.8 m/s 2 is the gravity acceleration;  = 0.460 H,  = 2 m −1 , and  = 0.001 Ns/m are positive constants;  is the electric current; and  is the position of the ball.Define the state variables  1 =  and  2 = ẏ .Then, (42) can be written as follows [39]: Consider that during the required operation, where The objective of the paper is to design a controller that keeps the ball in a desired position  =  1 =  0 , after a transient response.Thus, the equilibrium point of the system (43) is From the second equation in (43), observe that, in the equilibrium point, ẋ 2 = 0 and  =  0 , where Note that the equilibrium point is not in the origin = [0 0]  .Thus, for the stability analysis the following change of coordinates is necessary: that is, Therefore, ẋ 1 = ẋ 1 and ẋ 2 = ẋ 2 and from (45),  2 =  + 2(1 +  0 ) 2 /.Hence, the system (43) can be written as Finally, from (48) it follows that where (50) Now, define  3 and V such that ẋ 3 = u = V; that is,  3 = .Thus, considering (50), the system (49) can be given by After this adjustment it is seen that the problem falls into Case 1.Thus, the procedure stated in Case 1 can be used for designing a switched control law V() = −  (),   ∈ R 3 .
Note that in this case it is not possible to obtain the membership functions, since the mass is uncertain, but the proposed method overcomes this problem, because it does not depend on such functions.Observe also that even with uncertainty in the reference control signal (because  =  2 −  2 0 and  2 0 given in (45) is uncertain considering that  is uncertain), the proposed methodology was efficient and provided an appropriate transient response, as shown in Figure 7.
Remark 7. In a control design it is important to assure stability and usually other indices of performance for the controlled system, such as the settling time (related to the decay rate), constraints on input control and output signals.The proposed methodology allows specifying these performance indices, without changing the LMIs given in [40] or their relaxations as presented, for instance, in [20,23], by adding a new set of LMIs.

Conclusions
This paper proposed a new switched control design method for some classes of uncertain nonlinear plants described by Takagi-Sugeno fuzzy models.The proposed controller is based on LMIs and the gain is chosen by a switching law that returns the smallest time derivative value of the Lyapunov function.An advantage of the proposed methodology is that it does not change the LMIs given in the control design methods commonly used for plants described by Takagi-Sugeno fuzzy models as proposed, for instance, in [20,22,23,34].Furthermore, it eliminates the need to obtain the explicit expressions of the membership functions, to implement the control law.This fact is relevant in cases where the membership functions depend on uncertain parameters or are difficult to implement.Simulating the implementation of this new procedure in the control design of a ball-andbeam system and of a magnetic levitator, the controlled system presented an appropriate transient response, as seen in Figures 2, 3, 5, 6, and 7. Thus, the authors consider that the proposed method can be useful in practical applications for the control design of uncertain nonlinear systems.