About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 731298, 17 pages
http://dx.doi.org/10.1155/2012/731298
Research Article

Design of a Takagi-Sugeno Fuzzy Regulator for a Set of Operation Points

1Department of Electrical Engineering, Universidade Estadual Paulista (UNESP), Campus de Ilha Solteira, 15385-000 Ilha Solteira, SP, Brazil
2Department of Academic Areas of Jataí, Federal Institute of Education (IFG), Science and Technology of Goiás, Câmpus Jataí, 75804-020 Jataí, GO, Brazil

Received 2 December 2011; Revised 19 March 2012; Accepted 5 April 2012

Academic Editor: Pedro Ribeiro

Copyright © 2012 Máira P. A. Santim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper proposes a new design method based on linear matrix inequalities (LMIs) for tracking constant signals (regulation) considering nonlinear plants described by the Takagi-Sugeno fuzzy models. The procedure consists in designing a single controller that stabilizes the system at operation points belonging to a certain range or region, without the need of remaking the design of the controller gains at each new chosen equilibrium point. The control system design of a magnetic levitator illustrates the proposed methodology.

1. Introduction

In recent years the design of tracking control systems for nonlinear plants described by Takagi-Sugeno fuzzy models [1] has been the subject of several studies [210]. For the tracking problem the goal is to make the tracking error (difference between the output and desired output) equal to zero, ensuring the asymptotic stability of the equilibrium point. The linear matrix inequality (LMI) formulation [11] has emerged recently as a useful tool for solving a great number of practical control problems [9, 1217]. The advantage is that LMIs, when feasible, can be easily solved using available software [18, 19]. Furthermore, the procedure based on LMIs can also consider other design specifications regarding plant uncertainties, such as decay rate (related to the setting time) and output and input constraints [20].

An interesting method for the design of tracking control systems using LMIs was studied in [2]. The tracking process uses the concept of virtual desired variable, and the design is divided into two steps: first determine the virtual desired variables of the system; then determine the control gains based on LMIs, for the stabilization of the system. In [4] is proposed a design method for tracking system with disturbance rejection applied to a class of nonlinear systems using fuzzy control. The method is based on the minimization of the norm between the reference signal and the tracking error signal, where the tracking error signal is the difference between the reference input signal and the output signal. In [6] is addressed the speed tracking control problem of permanent magnet synchronous motors with parameter uncertainties and load torque disturbance. Fuzzy logic systems are used to approximate the nonlinearities, and an adaptive backstepping technique is employed to construct the controllers. The proposed controller guarantees the convergence of the tracking error to a small neighborhood of the origin and achieves good tracking performance. A similar study is presented in [7], where a robust reference-tracking control problem for nonlinear distributed parameter systems with time delays, external disturbances, and measurement noises is studied; the nonlinear distributed parameter systems are measured at several sensor locations for output-feedback tracking control. A fuzzy-spatial state-space model derived via finite-difference approach was introduced to represent the nonlinear distributed parameter time-delayed system.

In this context there exist many other researches. In [21] a neural network-based approach was developed which combines control performance with the Takagi-Sugeno fuzzy control for the purpose of stabilization and stability analysis of nonlinear systems. In [22] an analytical solution was derived to describe the wave-induced flow field and surge motion of a deformable platform structure controlled with fuzzy controllers in an oceanic environment. In the controller design procedure, a parallel distributed compensation scheme was utilized to construct a global fuzzy logic controller by blending all local state feedback controllers, and the Lyapunov method was used to carry out stability analysis of a real system structure.

This paper proposes a new control methodology for tracking constant signals for a class of nonlinear plants. This method is based on LMIs and uses the Takagi-Sugeno fuzzy models to accurately describe the nonlinear model of the plant. The main idea of the method was to add in the domain of the nonlinear functions of the plant the coordinate of the equilibrium point that we desire to track. An application of the methodology in the control of a magnetic levitator, given in [23], is presented.

The main advantage of this new procedure is its practical application because the designer chooses the desired region of the equilibrium points and designs a single set of gains of the regulator that guarantees asymptotic stability of the system at any equilibrium point previously chosen in the region. This region is flexible and can be specified by the designer. The project considers that the change from an operating point to another occurs after large time intervals, such that in the instants of the changes the system is practically in steady-state. In addition, this new methodology allows the use of well-known LMIs-based design methods, for the design of fuzzy regulators for plants described by the Takagi-Sugeno fuzzy models, for instance presented in [11, 14, 15, 2428], which allows the inclusion of the specification of performance indices such as decay rate and constraints on the plant input and output.

2. Preliminary Results

2.1. The Takagi-Sugeno Fuzzy Regulator

As described in [1], the Takagi-Sugeno fuzzy model is as follows.Rule𝑖:If𝑧1(𝑡)is𝑀𝑖1,,𝑧𝑝(𝑡)is𝑀𝑖𝑝,theṅ𝑥(𝑡)=𝐴𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡),𝑦(𝑡)=𝐶𝑖𝑥(𝑡),(2.1) where 𝑖=1,2,,𝑟, 𝑀𝑖𝑗, 𝑗=1,2,,𝑝 is the fuzzy set 𝑗 of Rule 𝑖, 𝑥(𝑡)𝑛 is the state vector, 𝑢(𝑡)𝑚 is the input vector, 𝑦(𝑡)𝑞 is the output vector, 𝐴𝑖𝑛×𝑛, 𝐵𝑖𝑛×𝑚, 𝐶𝑖𝑞×𝑛, and 𝑧1(𝑡),,𝑧𝑝(𝑡) are premise variables, which in this paper are the state variables.

As in [24], ̇𝑥(𝑡) given in (2.1) can be written as follows:̇𝑥(𝑡)=𝑟𝑖=1𝛼𝑖𝐴(𝑥(𝑡))𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡),(2.2) where 𝛼𝑖(𝑥(𝑡)) is the normalized weight of each local model system 𝐴𝑖𝑥(𝑡)+𝐵𝑖𝑢(𝑡) that satisfies the following properties:𝛼𝑖(𝑥(𝑡))0,for𝑖=1,2,,𝑟,𝑟𝑖=1𝛼𝑖(𝑥(𝑡))=1.(2.3)

Considering the Takagi-Sugeno fuzzy model (2.1), the control input of fuzzy regulators via parallel distributed compensation (PDC) has the following structure [24]:Rule𝑗:If𝑧1(𝑡)is𝑀𝑗1,,𝑧𝑝(𝑡)is𝑀𝑗𝑝,then𝑢(𝑡)=𝐹𝑗𝑥(𝑡).(2.4)

Similar to (2.2), it can be concluded that𝑢(𝑡)=𝑟𝑗=1𝛼𝑗(𝑥(𝑡))𝐹𝑗𝑥(𝑡).(2.5)

From (2.5), (2.2) and observing that 𝑟𝑖=1𝛼𝑖(𝑥(𝑡))=1, we obtain thaṫ𝑥(𝑡)=𝑟𝑟𝑖=1𝑗=1𝛼𝑖(𝑥(𝑡))𝛼𝑗𝐴(𝑥(𝑡))𝑖𝐵𝑖𝐹𝑗𝑥(𝑡).(2.6) Defining𝐺𝑖𝑗=𝐴𝑖𝐵𝑖𝐹𝑗,(2.7) then (2.6) can be written as follows:̇𝑥(𝑡)=𝑟𝑟𝑖=1𝑗=1𝛼𝑖(𝑥(𝑡))𝛼𝑗(𝑥(𝑡))𝐺𝑖𝑗𝑥(𝑡).(2.8)

2.2. Stability of the Takagi-Sugeno Fuzzy Systems via LMIs

The following theorem, whose proof can be seen in [24], guarantees the asymptotic stability of the origin of the system (2.8).

Theorem 2.1. The equilibrium point of the continuous time fuzzy control system given in (2.6) is asymptotically stable in the large if a common symmetric positive definite matrix 𝑋𝑛×𝑛(𝑋0) and𝑀𝑖𝑛×𝑚,𝑖=1,2,,𝑟 exists such that the following LMIs are satisfied: 𝑋𝐴𝑇𝑖+𝐴𝑖𝑋𝐵𝑖𝑀𝑖𝑀𝑇𝑖𝐵𝑇𝑖𝐴0,𝑖+𝐴𝑗𝐴𝑋+𝑋𝑖+𝐴𝑗𝑇𝐵𝑖𝑀𝑗𝐵𝑗𝑀𝑖𝑀𝑇𝑖𝐵𝑇𝑗𝑀𝑇𝑗𝐵𝑇𝑖0,𝑖<𝑗,(2.9) for all 𝑖,𝑗=1,2,,𝑟, excepting the pairs (𝑖,𝑗) such that 𝛼𝑖(𝑥(𝑡))𝛼𝑗(𝑥(𝑡))=0, for all 𝑥(𝑡). If there exists such a solution, the controller gains are given by 𝐹𝑖=𝑀𝑖𝑋1,𝑖=1,2,,𝑟.

In a control design it is important to assure stability and usually other indices of performance for the controlled system, such as the response speed, restrictions on input control, and output signals. The speed of the response is related to the decay rate of the system (2.6) or largest Lyapunov exponent, which is defined as the largest 𝛽>0 such thatlim𝑡𝑒𝛽𝑡𝑥(𝑡)=0(2.10) holds for all trajectories 𝑥(𝑡).

As in [11, page 66], one can use a quadratic Lyapunov function 𝑉(𝑥(𝑡))=𝑥(𝑡)𝑇𝑃𝑥(𝑡) to establish a lower bound for the decay rate of system (2.6). The condition ̇𝑉(𝑥(𝑡))2𝛽𝑉(𝑥(𝑡)) for all trajectories 𝑥(𝑡) assures that the system has a decay rate greater or equal to 𝛽. This condition is considered in Theorem 2.2, whose proof can be found, for instance, in [24].

Theorem 2.2. The equilibrium point of the continuous time fuzzy control system given in (2.8) is globally asymptotically stable, with decay rate greater or equal to 𝛽, if there exists a positive definite symmetric matrix 𝑋𝑛×𝑛(𝑋0) and matrices 𝑀𝑖𝑛×𝑚,𝑖=1,2,,𝑟, such that the following LMIs are satisfied: 𝑋𝐴𝑇𝑖+𝐴𝑖𝑋𝐵𝑖𝑀𝑖𝑀𝑇𝑖𝐵𝑇𝑖𝐴+2𝛽𝑋0,𝑖+𝐴𝑗𝐴𝑋+𝑋𝑖+𝐴𝑗𝑇𝐵𝑖𝑀𝑗𝐵𝑗𝑀𝑖𝑀𝑇𝑖𝐵𝑇𝑗𝑀𝑇𝑗𝐵𝑇𝑖+4𝛽𝑋0,𝑖<𝑗,(2.11) for all 𝑖,𝑗=1,2,,𝑟, excepting the pairs (𝑖,𝑗) such that 𝛼𝑖(𝑥(𝑡))𝛼𝑗(𝑥(𝑡))=0, for all 𝑥(𝑡). If there exists this solution, the controller gains are given by 𝐹𝑖=𝑀𝑖𝑋1,𝑖=1,2,,𝑟.

3. Magnetic Levitator

Currently, magnetic suspension systems are mainly used in applications where the reduction of friction force due to mechanical contact is essential. They are usually found in high-speed trains, gyroscopes, and accelerometers [23, page 23].

This paper considers the mathematical model of a magnetic levitator to illustrate the proposed control design method. Figure 1 shows the basic configuration of a magnetic levitator whose mathematical model [23, page 24] is given by𝑚̈𝑦=𝑘̇𝑦+𝑚𝑔𝜆𝜇𝑖22(1+𝜇𝑦)2,(3.1) where 𝑚 is the mass of the ball; 𝑔 is the gravity acceleration; 𝜇 and 𝑘 are positive constants; 𝑖 is the electric current; and𝑦 is the position of the ball.

731298.fig.001
Figure 1: Magnetic levitator.

Define the state variable 𝑥1=𝑦 and 𝑥2=̇𝑦. Then, (3.1) can be written as follows [29]: ̇𝑥1=𝑥2,̇𝑥2𝑘=𝑔𝑚𝑥2𝜆𝜇𝑖22𝑚1+𝜇𝑥12.(3.2)

Consider that, during the required operation, [𝑥1𝑥2]𝑇𝐷, where𝐷=𝑥1𝑥2𝑇20𝑥10.15.(3.3) The paper aims to design a controller that keeps the ball in a desired position 𝑦=𝑥1=𝑦0, after a transient response. Thus, the equilibrium point of the system (3.2) is 𝑥𝑒=[𝑥1𝑒𝑥2𝑒]𝑇=[𝑦00]𝑇.

From the second equation ̇𝑥2 in (3.2), observe that, in the equilibrium point, ̇𝑥2=0 and 𝑖=𝑖0, where𝑖20=2𝑚𝑔𝜆𝜇1+𝜇𝑦02.(3.4)

Note that the equilibrium point is not in the origin [𝑥1𝑥2]𝑇=[00]𝑇. Thus, the following change of coordinates is necessary for the stability analysis:𝑥1=𝑥1𝑦0,𝑥2=𝑥2,𝑢=𝑖2𝑖20,𝑥1=𝑥1+𝑦0,𝑥2=𝑥2,𝑖2=𝑢+𝑖20.(3.5) Therefore, ̇𝑥1=̇𝑥1, ̇𝑥2=̇𝑥2, and from (3.4), 𝑖2=𝑢+(2𝑚𝑔/𝜆𝜇)(1+𝜇𝑦0)2.

Hence, the system (3.2) can be written as follows:̇𝑥1=𝑥2,̇𝑥2𝑘=𝑔𝑚𝑥2𝜆𝜇𝑢+(2𝑚𝑔/𝜆𝜇)1+𝜇𝑦02𝑥2𝑚1+𝜇1+𝑦02,(3.6) and also, after some simple calculations, bẏ𝑥1=𝑥2,̇𝑥2=𝑔𝜇𝜇𝑥1+2𝜇𝑦0+2𝑥1+𝜇1+𝑦02𝑥1𝑘𝑚𝑥2𝜆𝜇𝑥2𝑚1+𝜇1+𝑦02𝑢.(3.7) Finally, from (3.7) it follows thaṫ𝑥1̇𝑥2=𝑓0121𝑥1,𝑦0𝑘𝑚𝑥1𝑥2+0𝑔21𝑥1,𝑦0𝑢,(3.8) where𝑓21𝑥1,𝑦0=𝑔𝜇𝜇𝑥1+2𝜇𝑦0+2𝑥1+𝜇1+𝑦02,𝑔(3.9)21𝑥1,𝑦0=𝜆𝜇𝑥2𝑚1+𝜇1+𝑦02.(3.10)

4. Regulator Design for an Operating Point

The goal of the design in this subsection is to keep the ball in a given position 𝑥1=𝑦0. In the first design, 𝑦0=0.1 m, and in the second, 𝑦0=0.05 m. Table 1 presents the parameters of the plant (3.8)–(3.10), for the controller design.

tab1
Table 1: Parameters of the plant (3.8)–(3.10) [29].

First, assume that 𝑦0=0.1 m and consider the following domain during the operation𝐷1=𝑥1𝑥2𝑇20.1𝑥10.05,𝑦0=0.1.(4.1)

Now, from the generalized form proposed in [26], it is necessary to obtain the maximum and minimum values of the functions 𝑓21 and 𝑔21 in the domain 𝐷1. After the calculations, it follows that𝑎211=max𝑥1𝐷1𝑓21𝑥1𝑎=43.1200,212=min𝑥1𝐷1𝑓21𝑥1𝑏=28.9941,211=max𝑥1𝐷1𝑔21𝑥1𝑏=5.4438,212=min𝑥1𝐷1𝑔21𝑥1=9.2000.(4.2)

Thus, the nonlinear the function 𝑓21 can be represented by a Takagi-Sugeno fuzzy model, considering that there exists a convex combination with membership functions 𝜎211(𝑥1) and 𝜎212(𝑥1) and constant values 𝑎211 and 𝑎212 given in (4.2) such that [26]𝑓21𝑥1=𝜎211𝑥1𝑎211+𝜎212𝑥1𝑎212,(4.3) with0𝜎211𝑥1,𝜎212𝑥11,𝜎211𝑥1+𝜎212𝑥1=1.(4.4) Therefore, from (4.3) and (4.4) note that𝜎211𝑥1=𝑓21𝑥1𝑎212𝑎211𝑎212,𝜎212𝑥1=1𝜎211𝑥1.(4.5)

Similarly, from (4.2) there exist 𝜉211(𝑥1) and 𝜉212(𝑥1) such that𝑔21𝑥1=𝜉211𝑥1𝑏211+𝜉212𝑥1𝑏212,(4.6) with0𝜉211𝑥1,𝜉212𝑥11,𝜉211𝑥1+𝜉212𝑥1=1.(4.7) Hence, from (4.6) and (4.7) observe that𝜉211𝑥1=𝑔21𝑥1𝑏212𝑏211𝑏212,𝜉212𝑥1=1𝜉211𝑥1.(4.8)

Recall that 𝜉211(𝑥1)+𝜉212(𝑥1)=1. Therefore, from (4.3) it follows that𝑓21𝑥1=𝜉211𝑥1+𝜉212𝑥1𝜎211𝑥1𝑎211+𝜎212𝑥1𝑎212=𝜎211𝑥1𝜉211𝑥1𝑎211+𝜎211𝑥1𝜉212𝑥1𝑎211+𝜎212𝑥1𝜉211𝑥1𝑎212+𝜎212𝑥1𝜉212𝑥1𝑎212.(4.9)

Similarly, from (4.6) and 𝜎211(𝑥1)+𝜎212(𝑥1)=1, we obtain𝑔21𝑥1=𝜎211𝑥1+𝜎212𝑥1𝜉211𝑥1𝑏211+𝜉212𝑥1𝑏212=𝜎211𝑥1𝜉211𝑥1𝑏211+𝜎211𝑥1𝜉212𝑥1𝑏212+𝜎212𝑥1𝜉211𝑥1𝑏211+𝜎212𝑥1𝜉212𝑥1𝑏212.(4.10)

Now, define𝛼1𝑥1=𝜎211𝑥1𝜉211𝑥1,𝛼2𝑥1=𝜎211𝑥1𝜉212𝑥1,𝛼3𝑥1=𝜎212𝑥1𝜉211𝑥1,𝛼4𝑥1=𝜎212𝑥1𝜉212𝑥1,(4.11) as the membership functions of the system (3.8)–(3.10), and their local models𝐴1=𝐴2=𝑎012110.02,𝐴3=𝐴4=𝑎01212,𝐵0.021=𝐵3=0𝑏211𝑇,𝐵2=𝐵4=0𝑏212𝑇,(4.12) where 𝑎211 and 𝑎212, 𝑏211 and 𝑏212 are the maximum and minimum values of the functions 𝑓21(𝑥1) and 𝑔21(𝑥1), respectively, as described in (4.2).

Therefore, the system (3.8)–(3.10), with the control law (2.5), can be represented as a Takagi-Sugeno fuzzy model, given in (2.6) and (2.8), with 𝑟=4:̇𝑥(𝑡)=44𝑖=1𝑗=1𝛼𝑖(𝑥(𝑡))𝛼𝑗(𝑥(𝑡))𝐺𝑖𝑗𝑥(𝑡),where𝐺𝑖𝑗=𝐴𝑖𝐵𝑖𝐹𝑗.(4.13)

Thus, using the LMIs (2.9) from Theorem 2.1, we obtain the following controller gains:𝐹1=,𝐹38.30984.76102=,𝐹26.01503.19373=,𝐹37.27964.84234=.23.32733.0979(4.14)

Considering the initial condition 𝑥0=[0.041]𝑇 and 𝑦0=0.1 m for the system (3.2) (for the system (3.8)–(3.10), the initial condition is 𝑥0=𝑥0[𝑦00]𝑇=[0.061]𝑇), the simulation of the controlled system (3.8)–(3.10), (2.5), and (4.14) presented the responses shown in Figures 2 and 3. Note that 𝑦()=𝑦0, as desired.

fig2
Figure 2: Position 𝑦(𝑡)=𝑥1(𝑡) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0=0.1 m.
fig3
Figure 3: Control signal (𝑢(𝑡)=𝑖(𝑡)2𝑖20) and electrical current (𝑖(𝑡)) of the controlled system for 𝑦0=0.1 m.

Now, suppose that the wanted position of the magnetic levitator is 𝑦0=0.05 m. Thus, for the design of the control law, consider that in the required operation the domain is𝐷2=𝑥1𝑥2𝑇20.05𝑥10.1,𝑦0=0.05.(4.15)

For the generalized form, as proposed in [26], it is necessary to find the maximum and minimum values of the functions 𝑓21 and 𝑔21 in the domain 𝐷2. The obtained values were the following:𝑎211=max𝑥1𝐷2𝑓21𝑥1𝑎=41.1600,212=min𝑥1𝐷2𝑓21𝑥1𝑏=27.8343,211=max𝑥1𝐷2𝑔21𝑥1𝑏=5.4438,212=min𝑥1𝐷2𝑔21𝑥1=9.2000.(4.16)

Considering the same procedure adopted in (4.3)–(4.12) and from the condition given in Theorem 2.1, we obtain the following controller gains:𝐹1=,𝐹17.55141.77472=,𝐹12.08561.19223=,𝐹16.29111.80984=.10.02551.1717(4.17) For the initial condition 𝑥0=[0.11] and 𝑦0=0.05 m (for the system (3.8)–(3.10), the initial condition is 𝑥0=[0.051]𝑇), the simulation of the controlled system (3.8)–(3.10), (2.5), and (4.17) presented the responses given in Figures 4 and 5.

fig4
Figure 4: Position 𝑦(𝑡)=𝑥1(𝑡) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0=0.05 m.
fig5
Figure 5: Control signal (𝑢(𝑡)=𝑖(𝑡)2𝑖20) and electrical current (𝑖(𝑡)) of the controlled system for 𝑦0=0.05 m.

Note that the gains of the controllers (4.14) and (4.17) change according to the change of 𝑦0. It happens because each time we change the value of 𝑦0, the operation point changes. Thus, the local models and membership functions also change. Therefore, following the presented control design method, it is necessary to design a new regulator when the value of 𝑦0 changes, which makes difficult the practical implementation in cases where the system can work in different operating points. To solve this problem, in the next section we present a method for designing a single Takagi-Sugeno fuzzy controller via LMIs, for all of the range of known values of 𝑦0, related to the operation of the system.

5. Regulator Design for a Set of Operation Points

Before presenting the method, it is necessary to understand the following property.

Property 1. Let 𝐼1𝑛1 and 𝐼0𝑛0 be compact subsets such that 𝐼=𝐼1×𝐼0, 𝑓𝐼𝑛𝑡 a continuous function and 𝑛𝑡=𝑛1+𝑛0. If for some given 𝑦0𝐼0, 𝑀=max𝑦𝐼1{𝑓(𝑦,𝑦0)} and 𝑚=min𝑦𝐼1{𝑓(𝑦,𝑦0)}, then 𝑀max(𝑦,𝑦0)𝐼{𝑓(𝑦,𝑦0)} and 𝑚min(𝑦,𝑦0)𝐼{𝑓(𝑦,𝑦0)}.

Proof. Suppose, by contradiction, that 𝑀>max(𝑦,𝑦0)𝐼{𝑓(𝑦,𝑦0)}. Then, this implies that 𝑓(𝑦,𝑦0)<𝑀 for all (𝑦,𝑦0)𝐼 which is an absurd because 𝐼 is compact. Thus, there exists (𝑦,𝑦0)𝐼 such that 𝑀𝑓(𝑦,𝑦0).
Similarly it is shown that 𝑚min(𝑦,𝑦0)𝐼{𝑓(𝑦,𝑦0)}.

Property 1 is important to justify the proposed methodology. For instance, suppose that the plant can work in the region 𝑥1[0,0.15] and that we want the asymptotic stability of operating points [𝑥1𝑥2]𝑇=[𝑦00]𝑇, where 𝑦0 is a known constant and 𝑦0𝐼0=[0.04,0.11]. Thus the range of 𝑥1=𝑥1𝑦0 for all 𝑦0𝐼0 is 𝐼1=[0.11,0.11]. So we could get the gains of regulator for all 𝑦0𝐼0, where 𝑦0 will be considered as a new variable for the specification of the domain 𝐷3 of the nonlinear functions 𝑓21 and 𝑔21:𝐷3=𝑥1𝑥2𝑦0𝑇30.11𝑥10.11,0.04𝑦00.11.(5.1)

From Property 1, (4.1), (4.15), and (5.1) note thatmax(𝑥1,𝑦0)𝐷1,𝐷2𝑓21𝑥1,𝑦0max(𝑥1,𝑦0)𝐷3𝑓21𝑥1,𝑦0,min(𝑥1,𝑦0)𝐷1,𝐷2𝑓21𝑥1,𝑦0min(𝑥1,𝑦0)𝐷3𝑓21𝑥1,𝑦0,max(𝑥1,𝑦0)𝐷1,𝐷2𝑔21𝑥1,𝑦0max(𝑥1,𝑦0)𝐷3𝑔21𝑥1,𝑦0,min(𝑥1,𝑦0)𝐷1,𝐷2𝑔21𝑥1,𝑦0min(𝑥1,𝑦0)𝐷3𝑔21𝑥1,𝑦0.(5.2)

Indeed, after the calculations, considering (3.9), (3.10), Table 1, and (5.1), we obtain𝑎211=max(𝑥1,𝑦0)𝐷3𝑓21𝑥1,𝑦0𝑎=51.4116,212=min(𝑥1,𝑦0)𝐷3𝑓21𝑥1,𝑦0𝑏=25.1427,211=max(𝑥1,𝑦0)𝐷3𝑔21𝑥1,𝑦0𝑏=4.4367,212=min(𝑥1,𝑦0)𝐷3𝑔21𝑥1,𝑦0=12.4392.(5.3)

Based on the same procedure adopted in (4.3)–(4.12) (now for the domain 𝐷3) and from the LMIs of Theorem 2.1, the controller gains are the following:𝐹1=,𝐹23.54252.44472=,𝐹14.13281.39353=,𝐹22.23832.56294=.10.80401.2964(5.4)

For numerical simulation, at 𝑡=0 s, the initial conditions are 𝑥0=[0.041]𝑇 and 𝑦0=0.1 m. Thus, 𝑥(0)=𝑥(0)[𝑦00]𝑇=[0.061]𝑇. In 𝑡=1 s, from Figure 6, the system is practically at the point 𝑥(1)=[𝑥1(1)𝑥2(1)]𝑇=[0.10]𝑇. After changing 𝑦0 from 0.1 m to 0.05 m at 𝑡=1 s, we can see that the system is practically at the point 𝑥(2)=[0.050]𝑇 at 𝑡=2 s from Figure 6. After changing again 𝑦0 from 0.05 m to 0.08 m at 𝑡=2 s, we can see from Figure 6 that 𝑥()=[0.080]𝑇. Figures 6 and 7 illustrate the system response.

fig6
Figure 6: Position (𝑦(𝑡)=𝑥1(𝑡)) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0[0.04,0.11], considering 𝑦0=0.1 m, 0.05 m, and 0.08 m for 𝑡[0,1),𝑡[1,2), and 𝑡2 s, respectively.
fig7
Figure 7: Control signal (𝑢(𝑡)=𝑖(𝑡)2𝑖20) and electrical current (𝑖(𝑡)) for 𝑦0[0.04,0.11], considering 𝑦0=0.1 m, 0.05 m, and 0.08 m for 𝑡[0,1),𝑡[1,2), and 𝑡2 s, respectively.

Note that the control law, given in (2.5) with 𝑟=4, uses a single set of gains presented in (5.4). However, the membership functions 𝛼𝑖(𝑥(𝑡)),𝑖=1,2,3,4, specified in (3.9), (3.10), (4.3)–(4.11), and (5.3), are functions of 𝑦0, and so they must be updated each time that there is a change in the value of 𝑦0. Finally, observe that 𝑥=[𝑥1𝑥2]𝑇[𝑦00]𝑇 also must be changed in the control law (2.5), when 𝑦0 is modified.

Remark 5.1. In the example of the levitator, the range considered for the desired position point 𝑦0 was [0.04,0.11] and the domain of 𝑥1 was [0,0.15]. Thus, in general, the restriction for the proposed method is only that the region containing the desired equilibrium points must be contained in the domain of the state variables of the system. This region is flexible and can be chosen by the designer.

5.1. Regulator Design for a Set of Points of Operation with Rate of Decay

Usually, in control system designs, it is important to consider the stability and other performance indices for the controlled system, such as response speed, input constraint, and output constraint. The proposed methodology allows the specification of these performance indices, without changing the LMIs given in [11] (the same presented in Theorems 2.1 and 2.2), or their relaxations presented, for instance, in [14, 24], by adding a new set of LMIs.

Now, a decay rate will be specified for designing the new control gains for the magnetic levitator. Thus, it was considered (4.3)–(4.12), (5.3), and LMIs (2.11) from Theorem 2.2, with decay rate 𝛽=0.8, and the obtained controller gains are the following:𝐹1=,𝐹36.28293.95092=,𝐹21.84872.31213=,𝐹36.62584.26624=.17.99362.1555(5.5)

The numerical simulation supposes the same condition of the last simulation; that is, initially it was considered the initial condition 𝑥0=[0.041]𝑇 and 𝑦0=0.1 m. In 𝑡=1 s the system is practically at the point 𝑥0=[0.10]𝑇, and then, 𝑦0 is changed from 0.1 m to 0.05 m. At 𝑡=2 s the system is almost at the point 𝑥0=[0.050]𝑇, and now 𝑦0 is changed from 0.05 m to 0.08 m. Figures 8 and 9 illustrate the response of the system.

fig8
Figure 8: Position (𝑦(𝑡)=𝑥1(𝑡)) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0[0.04,0.11], considering 𝑦0=0.1 m, 0.05 m, and 0.08 m for 𝑡[0,1),𝑡[1,2), and 𝑡2 s, respectively, and decay rate 𝛽=0.8.
fig9
Figure 9: Control signal (𝑢(𝑡)=𝑖(𝑡)2𝑖20) and electrical current (𝑖(𝑡)) for 𝑦0[0.04,0.11], considering 𝑦0=0.1 m, 0.05 m, and 0.08 m for 𝑡[0,1),𝑡[1,2), and 𝑡2 s, respectively, and decay rate 𝛽=0.8.

As can be seen in Figure 8, using a decay rate greater than or equal to 𝛽=0.8, the response of the system was adequate and faster, when compared with Figure 6. However, from (5.4) and (5.5), note that the controller gains are now greater and consequently the control signal and the electric current are also greater, as can be seen by comparing Figures 7 and 9.

Remark 5.2. The proposed methodology can also be applied when the plant has known parameters belonging to a given region. In this case, one must consider these parameters as new variables in the domain of the nonlinearities and obtain the maximum and minimum values of the nonlinearities, in the region of operation. In the example of the levitator, we can consider, for instance, that the mass m is a known constant parameter, belonging to the range 𝑚[𝑚min,𝑚max] with 𝑚min and 𝑚max known constants. Thus, the nonlinearities (3.9) and (3.10) are now given by 𝑓21(𝑥1,𝑦0,𝑚) and 𝑔21(𝑥1,𝑦0,𝑚), respectively. Note that, from (3.4), in this case 𝑖0 also depends on the mass 𝑚 and therefore must be updated when the mass changes. The design considers that the change of the mass occurs after large time intervals, such that in the instants of the changes the system is practically in steady state.

6. Conclusions

In this paper we proposed a new design method of regulators with operating points belonging to a given region, which allows the tracking of constant signals for nonlinear plants described by the Takagi-Sugeno fuzzy models. The design is based on LMIs, and an application in the control design of a magnetic levitator illustrated the proposed procedure.

An advantage of the proposed methodology is that it does not change the LMIs given in the control design methods usually adopted for plants described by the Takagi-Sugeno fuzzy models, for instance, as proposed in [4, 11, 14, 15, 2428, 3032]. Furthermore, it allows to choose an equilibrium point of the system in a region of values previously established without needing of remaking the design of the controller gains for each new chosen equilibrium point. Moreover, the simulation of the application of this new control design method in a magnetic levitator presented an appropriate transient response, as can be seen in Figures 8 and 9. Thus, the authors think that the proposed method can be useful in practical applications of nonlinear control systems.

Acknowledgment

The authors gratefully acknowledge the financial support by CAPES, FAPESP, and CNPq from Brazil.

References

  1. T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. View at Scopus
  2. K. Y. Lian and J. J. Liou, “Output tracking control for fuzzy systems via output feedback design,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 628–639, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676–697, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Q. Andrea, J. O. P. Pinto, E. Assunção, M. C. M. Teixeira, and L. J. Galotto, “Controle ótimo H de sitemas não lineares com modelos fuzzy Takagi-Sugeno,” Sba: Controle & Automação, vol. 19, pp. 256–269, 2008 (Portuguese).
  5. P. C. Chen, C. W. Chen, and W. L. Chiang, “GA-based fuzzy sliding mode controller for nonlinear systems,” Mathematical Problems in Engineering, vol. 2008, Article ID 325859, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Yu, J. Gao, Y. Ma, and H. Yu, “Adaptive fuzzy tracking control for a permanent magnet synchronous motor via backstepping approach,” Mathematical Problems in Engineering, vol. 2010, Article ID 391846, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. T. Chang and B. S. Chen, “A fuzzy approach for robust reference-tracking-control design of nonlinear distributed parameter time-delayed systems and its application,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 6, Article ID 5510143, pp. 1041–1057, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. K.-S. Shih, H. S. Li, and S.-H. Tsai, “New nonlinear controller for a class of chaotic systems based on adaptive backstepping fuzzy-immune control,” Mathematical Problems in Engineering, vol. 2011, Article ID 439324, 20 pages, 2011. View at Publisher · View at Google Scholar
  9. C.-W. Chen, C.-W. Shen, C.-Y. Chen, and M.-J. Cheng, “Stability analysis of an oceanic structure using the Lyapunov method,” Engineering Computations, vol. 27, no. 2, pp. 186–204, 2010. View at Publisher · View at Google Scholar
  10. C.-Y. Chen, J.-W. Lin, W.-I. Lee, and C.-W. Chen, “Fuzzy control for an oceanic structure: a case study in time-delay TLP system,” Journal of Vibration and Control, vol. 16, no. 1, pp. 147–160, 2010. View at Publisher · View at Google Scholar
  11. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, Society for Industrial and AppliedMathematics, Philadelphia, Pa, USA, 1994.
  12. D. D. Šiljak and D. M. Stipanović, “Robust stabilization of nonlinear systems: the LMI approach,” Mathematical Problems in Engineering. Theory, Methods and Applications, vol. 6, no. 5, pp. 461–493, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. R. Moreira, E. I. Mainardi Jr, T. T. Esteves et al., “Stabilizability and disturbance rejection with state-derivative feedback,” Mathematical Problems in Engineering, vol. 2010, Article ID 123751, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. M. C. M. Teixeira, E. Assunção, and R. G. Avellar, “On relaxed LMI-based designs for fuzzy regulators and fuzzy observers,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 5, pp. 613–623, 2003. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Sala and C. Ariño, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya's theorem,” Fuzzy Sets and Systems, vol. 158, no. 24, pp. 2671–2686, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C.-W. Chen, “Application of fuzzy-model-based control to nonlinear structural systems with time delay: an LMI method,” Journal of Vibration and Control, vol. 16, no. 11, pp. 1651–1672, 2010. View at Publisher · View at Google Scholar
  17. K. Yeh, C.-Y. Chen, and C.-W. Chen, “Robustness design of time-delay fuzzy systems using fuzzy Lyapunov method,” Applied Mathematics and Computation, vol. 205, no. 2, pp. 568–577, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox-For Use with Matlab, The Math Works, City, State, USA, 1995.
  19. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11/12, no. 1–4, pp. 625–653, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. C. M. Teixeira, A. D. S. Lordelo, and E. Assunção, On LMI Based Design of SPR Systems and Output Variable Structure Controllers, World Scientific Publishing Company, Singapore, Singapore, 2000.
  21. C.-W. Chen, “Stability analysis and robustness design of nonlinear systems: an NN-based approach,” Applied Soft Computing Journal, vol. 11, no. 2, pp. 2735–2742, 2011. View at Publisher · View at Google Scholar · View at Scopus
  22. C. W. Chen, “Modeling and fuzzy PDC control and its application to an oscillatory TLP structure,” Mathematical Problems in Engineering, vol. 2010, Article ID 120403, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. H. J. Marquez, Nonlinear Control Systems—Analysis and Design, Wiley, New Jersey, NJ, USA, 2003.
  24. K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Transactions on Fuzzy Systems, vol. 6, no. 2, pp. 250–265, 1998. View at Scopus
  25. M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 133–142, 1999. View at Publisher · View at Google Scholar · View at Scopus
  26. T. Taniguchi, K. Tanaka, H. Ohtake, and H. O. Wang, “Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 4, pp. 525–538, 2001. View at Publisher · View at Google Scholar · View at Scopus
  27. L. A. Mozelli, R. M. Palhares, F. O. Souza, and E. M. A. M. Mendes, “Reducing conservativeness in recent stability conditions of TS fuzzy systems,” Automatica, vol. 45, no. 6, pp. 1580–1583, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. V. F. Montagner, R. C. L. F. Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for quadratic stabilizability and H∞ control of Takagi-Sugeno fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 863–873, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. R. Cardim, Projeto de Controladores Baseados em LMIs: realimentação derivativa e sistemas chaveados utilizando estrutura variável, Ph.D. thesis, Faculdade de Engenharia, Unversidade Estadual Paulista, Ilha Solteira, Brazil, 2009.
  30. T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form,” Automatica, vol. 40, no. 5, pp. 823–829, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. M. Klug, E. B. Castelan, and V. J. S. Leite, “A dynamic compensator for parameter-varying systems subject to actuator limitations applied to a T-S fuzzy system,” in Proceedings of the 18th IFAC World Congress, vol. 18, pp. 14495–14500, Milan, Italy, 2011.
  32. V. F. Montagner, R. C. L. F. Oliveira, V. J. S. Leite, and P. L. D. Peres, “LMI approach for H linear parameter-varying state feedback control,” Control Theory and Applications, IEE Proceedings, vol. 152, no. 2, pp. 195–201, 2005.