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

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 [2–10]. 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, 12–17]. 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 ℋ_∞ 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 ℋ_∞ 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, 24–28], 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)is𝑀^𝑖1,…,𝑧_𝑝(𝑡)is𝑀^𝑖𝑝,then𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡),(2.1) where 𝑖=1,2,…,𝑟, 𝑀^𝑖𝑗, 𝑗=1,2,…,𝑝 is the fuzzy set 𝑗 of Rule 𝑖, 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the input vector, 𝑦(𝑡)∈ℝ^𝑞 is the output vector, 𝐴_𝑖∈ℝ^𝑛×𝑛, 𝐵_𝑖∈ℝ^𝑛×𝑚, 𝐶_𝑖∈ℝ^𝑞×𝑛, and 𝑧₁(𝑡),…,𝑧_𝑝(𝑡) are premise variables, which in this paper are the state variables.

As in [24], 𝑥̇(𝑡) given in (2.1) can be written as follows:𝑥̇(𝑡)=^𝑟_𝑖=1𝛼_𝑖(𝑥(𝑡))𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),(2.2) where 𝛼_𝑖(𝑥(𝑡)) is the normalized weight of each local model system 𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡) that satisfies the following properties:𝛼_𝑖(𝑥(𝑡))≥0,for𝑖=1,2,…,𝑟,^𝑟_𝑖=1𝛼_𝑖(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)is𝑀^𝑗1,…,𝑧_𝑝(𝑡)is𝑀^𝑗𝑝,then𝑢(𝑡)=−𝐹_𝑗𝑥(𝑡).(2.4)

Similar to (2.2), it can be concluded that𝑢(𝑡)=−^𝑟_𝑗=1𝛼_𝑗(𝑥(𝑡))𝐹_𝑗𝑥(𝑡).(2.5)

From (2.5), (2.2) and observing that ∑^𝑟𝑖=1𝛼_𝑖(𝑥(𝑡))=1, we obtain that𝑥̇(𝑡)=^𝑟_𝑟𝑖=1_𝑗=1𝛼_𝑖(𝑥(𝑡))𝛼_𝑗(𝑥(𝑡))𝐴_𝑖−𝐵_𝑖𝐹_𝑗𝑥(𝑡).(2.6) Defining𝐺_𝑖𝑗=𝐴_𝑖−𝐵_𝑖𝐹_𝑗,(2.7) then (2.6) can be written as follows:𝑥̇(𝑡)=^𝑟_𝑟𝑖=1_𝑗=1𝛼_𝑖(𝑥(𝑡))𝛼_𝑗(𝑥(𝑡))𝐺_𝑖𝑗𝑥(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 𝑋∈ℝ^𝑛×𝑛(𝑋≻0) and𝑀_𝑖∈ℝ^𝑛×𝑚,𝑖=1,2,…,𝑟 exists such that the following LMIs are satisfied: 𝐴𝑋𝐴^𝑇𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀_𝑖−𝑀^𝑇𝑖𝐵^𝑇𝑖≺0,_𝑖+𝐴_𝑗𝑋+𝑋𝐴_𝑖+𝐴_𝑗^𝑇−𝐵_𝑖𝑀_𝑗−𝐵_𝑗𝑀_𝑖−𝑀^𝑇𝑖𝐵^𝑇𝑗−𝑀^𝑇𝑗𝐵^𝑇𝑖≼0,𝑖<𝑗,(2.9) for all 𝑖,𝑗=1,2,…,𝑟, excepting the pairs (𝑖,𝑗) such that 𝛼_𝑖(𝑥(𝑡))𝛼_𝑗(𝑥(𝑡))=0, for all 𝑥(𝑡). If there exists such a solution, the controller gains are given by 𝐹_𝑖=𝑀_𝑖𝑋⁻¹,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 thatlim_𝑡→∞𝑒^𝛽𝑡‖𝑥(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 𝑉̇(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 𝑋∈ℝ^𝑛×𝑛(𝑋≻0) and matrices 𝑀_𝑖∈ℝ^𝑛×𝑚,𝑖=1,2,…,𝑟, such that the following LMIs are satisfied: 𝐴𝑋𝐴^𝑇𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀_𝑖−𝑀^𝑇𝑖𝐵^𝑇𝑖+2𝛽𝑋≺0,_𝑖+𝐴_𝑗𝑋+𝑋𝐴_𝑖+𝐴_𝑗^𝑇−𝐵_𝑖𝑀_𝑗−𝐵_𝑗𝑀_𝑖−𝑀^𝑇𝑖𝐵^𝑇𝑗−𝑀^𝑇𝑗𝐵^𝑇𝑖+4𝛽𝑋≼0,𝑖<𝑗,(2.11) for all 𝑖,𝑗=1,2,…,𝑟, excepting the pairs (𝑖,𝑗) such that 𝛼_𝑖(𝑥(𝑡))𝛼_𝑗(𝑥(𝑡))=0, for all 𝑥(𝑡). If there exists this solution, the controller gains are given by 𝐹_𝑖=𝑀_𝑖𝑋⁻¹,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 by𝑚𝑦̈=−𝑘𝑦̇+𝑚𝑔−𝜆𝜇𝑖²2(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.

Figure 1 

Magnetic levitator.

Define the state variable 𝑥₁=𝑦 and 𝑥₂=𝑦̇. Then, (3.1) can be written as follows [29]: 𝑥̇‌₁=𝑥₂,𝑥̇₂=𝑔−𝑘𝑚𝑥₂−𝜆𝜇𝑖²2𝑚1+𝜇𝑥₁².(3.2)

Consider that, during the required operation, [𝑥₁𝑥₂]^𝑇∈𝐷, where𝐷=𝑥₁𝑥₂^𝑇∈ℝ²∶0≤𝑥₁≤0.15.(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 𝑥_𝑒=[𝑥_1𝑒𝑥_2𝑒]^𝑇=[𝑦₀0]^𝑇.

From the second equation 𝑥̇‌₂ in (3.2), observe that, in the equilibrium point, 𝑥̇‌₂=0 and 𝑖=𝑖₀, where𝑖²⁰=2𝑚𝑔𝜆𝜇1+𝜇𝑦⁰².(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:⎧⎪⎨⎪⎩𝑥𝑥₁₂=𝑥₁−𝑦₀,𝑢==𝑖₂𝑥₂,−𝑖²⁰,⟹⎧⎪⎨⎪⎩𝑥₁=𝑥₁+𝑦₀,𝑥𝑖₂₂=𝑥₂,=𝑢+𝑖²⁰.(3.5) Therefore, 𝑥̇₁=𝑥̇₁, 𝑥̇₂=𝑥̇₂, and from (3.4), 𝑖²=𝑢+(2𝑚𝑔/𝜆𝜇)(1+𝜇𝑦₀)².

Hence, the system (3.2) can be written as follows:𝑥̇₁=𝑥₂,𝑥̇₂=𝑔−𝑘𝑚𝑥²−𝜆𝜇𝑢+(2𝑚𝑔/𝜆𝜇)1+𝜇𝑦⁰²2𝑚1+𝜇𝑥₁+𝑦₀²,(3.6) and also, after some simple calculations, by𝑥̇₁=𝑥₂,𝑥̇₂=𝑔𝜇𝜇𝑥¹+2𝜇𝑦₀+21+𝜇𝑥₁+𝑦₀²𝑥₁−𝑘𝑚𝑥²−𝜆𝜇2𝑚1+𝜇𝑥₁+𝑦₀²𝑢.(3.7) Finally, from (3.7) it follows that𝑥𝑥̇̇¹₂=𝑓𝑥0₁,𝑦₀−1₂₁𝑘𝑚𝑥¹𝑥₂+𝑔₂₁𝑥0₁,𝑦₀𝑢,(3.8) where𝑓₂₁𝑥₁,𝑦₀=𝑔𝜇𝜇𝑥¹+2𝜇𝑦₀+21+𝜇𝑥₁+𝑦₀²𝑔,(3.9)₂₁𝑥₁,𝑦₀=−𝜆𝜇2𝑚1+𝜇𝑥₁+𝑦₀².(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.

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=𝑥₁𝑥₂^𝑇∈ℝ²∶−0.1≤𝑥₁≤0.05,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 that𝑎𝑎₂₁₁=max_{𝑥1∈𝐷1}𝑓₂₁𝑥₁=43.1200,₂₁₂𝑏=min_{𝑥1∈𝐷1}𝑓₂₁𝑥₁=28.9941,₂₁₁𝑏=max_{𝑥1∈𝐷1}𝑔₂₁𝑥₁=−5.4438,₂₁₂=min_{𝑥1∈𝐷1}𝑔₂₁𝑥₁=−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]𝑓₂₁𝑥₁=𝜎₂₁₁𝑥₁𝑎₂₁₁+𝜎₂₁₂𝑥₁𝑎₂₁₂,(4.3) with0≤𝜎₂₁₁𝑥₁,𝜎₂₁₂𝑥₁≤1,𝜎₂₁₁𝑥₁+𝜎₂₁₂𝑥₁=1.(4.4) Therefore, from (4.3) and (4.4) note that𝜎₂₁₁𝑥₁=𝑓²¹𝑥₁−𝑎₂₁₂𝑎₂₁₁−𝑎₂₁₂,𝜎₂₁₂𝑥₁=1−𝜎₂₁₁𝑥₁.(4.5)

Similarly, from (4.2) there exist 𝜉₂₁₁(𝑥₁) and 𝜉₂₁₂(𝑥₁) such that𝑔₂₁𝑥₁=𝜉₂₁₁𝑥₁𝑏₂₁₁+𝜉₂₁₂𝑥₁𝑏₂₁₂,(4.6) with0≤𝜉₂₁₁𝑥₁,𝜉₂₁₂𝑥₁≤1,𝜉₂₁₁𝑥₁+𝜉₂₁₂𝑥₁=1.(4.7) Hence, from (4.6) and (4.7) observe that𝜉₂₁₁𝑥₁=𝑔²¹𝑥₁−𝑏₂₁₂𝑏₂₁₁−𝑏₂₁₂,𝜉₂₁₂𝑥₁=1−𝜉₂₁₁𝑥₁.(4.8)

Recall that 𝜉₂₁₁(𝑥₁)+𝜉₂₁₂(𝑥₁)=1. Therefore, from (4.3) it follows that𝑓₂₁𝑥₁==𝜎𝜉₂₁₁₂₁₁𝑥₁+𝜉₂₁₂𝑥₁𝜎₂₁₁𝑥₁𝑎₂₁₁+𝜎₂₁₂𝑥₁𝑎₂₁₂𝑥₁𝜉₂₁₁𝑥₁𝑎₂₁₁+𝜎₂₁₁𝑥₁𝜉₂₁₂𝑥₁𝑎₂₁₁+𝜎₂₁₂𝑥₁𝜉₂₁₁𝑥₁𝑎₂₁₂+𝜎₂₁₂𝑥₁𝜉₂₁₂𝑥₁𝑎(4.₂₁₂9.)

Similarly, from (4.6) and 𝜎₂₁₁(𝑥₁)+𝜎₂₁₂(𝑥₁)=1, we obtain𝑔₂₁𝑥₁==𝜎𝜎₂₁₁₂₁₁𝑥₁+𝜎₂₁₂𝑥₁𝜉₂₁₁𝑥₁𝑏₂₁₁+𝜉₂₁₂𝑥₁𝑏₂₁₂𝑥₁𝜉₂₁₁𝑥₁𝑏₂₁₁+𝜎₂₁₁𝑥₁𝜉₂₁₂𝑥₁𝑏₂₁₂+𝜎₂₁₂𝑥₁𝜉₂₁₁𝑥₁𝑏₂₁₁+𝜎₂₁₂𝑥₁𝜉₂₁₂𝑥₁(𝑏4.1₂₁₂0.)

Now, define𝛼𝛼₁₂𝑥₁=𝜎₂₁₁𝑥₁𝜉₂₁₁𝑥₁,𝛼₃𝑥₁=𝜎₂₁₁𝑥₁𝜉₂₁₂𝑥₁,𝛼₄𝑥₁=𝜎₂₁₂𝑥₁𝜉₂₁₁𝑥₁,𝑥₁=𝜎₂₁₂𝑥₁𝜉₂₁₂𝑥₁,(4.11) as the membership functions of the system (3.8)–(3.10), and their local models𝐴₁=𝐵𝐴₂=𝑎0−01₂₁₁.02,𝐴³=𝐴₄=𝑎0−01₂₁₂.02₁,=𝐵₃=0𝑏₂₁₁^𝑇,𝐵₂=𝐵₄=0𝑏₂₁₂^𝑇,(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:𝑥̇(𝑡)=⁴_4𝑖=1_𝑗=1𝛼_𝑖(𝑥(𝑡))𝛼_𝑗(𝑥(𝑡))𝐺_𝑖𝑗𝑥(𝑡),where𝐺_𝑖𝑗=𝐴_𝑖−𝐵_𝑖𝐹_𝑗.(4.13)

Thus, using the LMIs (2.9) from Theorem 2.1, we obtain the following controller gains:𝐹𝐹₁=−38.3098−4.7610₂,𝐹=−26.0150−3.1937₃,𝐹=−37.2796−4.8423₄,=−23.3273−3.0979.(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.

(a)

(b)

Figure 2 

Position 𝑦(𝑡)=𝑥1(𝑡) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0=0.1 m.

(a)

(b)

Figure 3 

Control signal (𝑢(𝑡)=𝑖(𝑡)2−𝑖20) 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=𝑥₁𝑥₂^𝑇∈ℝ²∶−0.05≤𝑥₁≤0.1,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:𝑎𝑎₂₁₁=max_{𝑥1∈𝐷2}𝑓₂₁𝑥₁=41.1600,₂₁₂𝑏=min_{𝑥1∈𝐷2}𝑓₂₁𝑥₁=27.8343,₂₁₁𝑏=max_{𝑥1∈𝐷2}𝑔₂₁𝑥₁=−5.4438,₂₁₂=min_{𝑥1∈𝐷2}𝑔₂₁𝑥₁=−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:𝐹𝐹₁=−17.5514−1.7747₂,𝐹=−12.0856−1.1922₃,𝐹=−16.2911−1.8098₄,=−10.0255−1.1717.(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.

(a)

(b)

Figure 4 

Position 𝑦(𝑡)=𝑥1(𝑡) and velocity (𝑥2(𝑡)) of the controlled system for 𝑦0=0.05 m.

(a)

(b)

Figure 5 

Control signal (𝑢(𝑡)=𝑖(𝑡)2−𝑖20) 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 𝐼₁⊂ℝ^𝑛1 and 𝐼₀⊂ℝ^𝑛0 be compact subsets such that 𝐼=𝐼₁×𝐼₀, 𝑓∶𝐼⊂ℝ^𝑛𝑡→ℝ a continuous function and 𝑛_𝑡=𝑛₁+𝑛₀. If for some given 𝑦₀∈𝐼₀, 𝑀=max_𝑦∈𝐼1{𝑓(𝑦,𝑦₀)} and 𝑚=min_𝑦∈𝐼1{𝑓(𝑦,𝑦₀)}, then 𝑀≤max_{(𝑦,𝑦0)∈𝐼}{𝑓(𝑦,𝑦₀)} and 𝑚≥min_{(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^∗,𝑦^∗0)∈𝐼 such that 𝑀≤𝑓(𝑦^∗,𝑦^∗0).
Similarly it is shown that 𝑚≥min_{(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 𝑦₀∈𝐼₀=[0.04,0.11]. Thus the range of 𝑥₁=𝑥₁−𝑦₀ for all 𝑦₀∈𝐼₀ is 𝐼₁=[−0.11,0.11]. So we could get the gains of regulator for all 𝑦₀∈𝐼₀, where 𝑦₀ will be considered as a new variable for the specification of the domain 𝐷₃ of the nonlinear functions 𝑓₂₁ and 𝑔₂₁:𝐷₃=𝑥₁𝑥₂𝑦₀^𝑇∈ℝ³∶−0.11≤𝑥₁≤0.11,0.04≤𝑦₀≤0.11.(5.1)

From Property 1, (4.1), (4.15), and (5.1) note thatmmainx_{(𝑥1,𝑦0)∈𝐷1,𝐷2(𝑥1,𝑦0)∈𝐷1,𝐷2}𝑓₂₁𝑥₁,𝑦₀≤max_{(𝑥1,𝑦0)∈𝐷3}𝑓₂₁𝑥₁,𝑦₀,max_{(𝑥1,𝑦0)∈𝐷1,𝐷2}𝑓₂₁𝑥₁,𝑦₀≥min_{(𝑥1,𝑦0)∈𝐷3}𝑓₂₁𝑥₁,𝑦₀,min_{(𝑥1,𝑦0)∈𝐷1,𝐷2}𝑔₂₁𝑥₁,𝑦₀≤max_{(𝑥1,𝑦0)∈𝐷3}𝑔₂₁𝑥₁,𝑦₀,𝑔₂₁𝑥₁,𝑦₀≥min_{(𝑥1,𝑦0)∈𝐷3}𝑔₂₁𝑥₁,𝑦₀.(5.2)

Indeed, after the calculations, considering (3.9), (3.10), Table 1, and (5.1), we obtain𝑎𝑎₂₁₁=max_{(𝑥1,𝑦0)∈𝐷3}𝑓₂₁𝑥₁,𝑦₀=51.4116,₂₁₂𝑏=min_{(𝑥1,𝑦0)∈𝐷3}𝑓₂₁𝑥₁,𝑦₀=25.1427,₂₁₁𝑏=max_{(𝑥1,𝑦0)∈𝐷3}𝑔₂₁𝑥₁,𝑦₀=−4.4367,₂₁₂=min_{(𝑥1,𝑦0)∈𝐷3}𝑔₂₁𝑥₁,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:𝐹𝐹₁=−23.5425−2.4447₂,𝐹=−14.1328−1.3935₃,𝐹=−22.2383−2.5629₄,=−10.8040−1.2964.(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.

(a)

(b)

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 𝑡≥2 s, respectively.

(a)

(b)

Figure 7 

Control signal (𝑢(𝑡)=𝑖(𝑡)2−𝑖20) 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 𝑡≥2 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 𝛼_𝑖(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:𝐹𝐹₁=−36.2829−3.9509₂,𝐹=−21.8487−2.3121₃,𝐹=−36.6258−4.2662₄,=−17.9936−2.1555.(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.

(a)

(b)

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 𝑡≥2 s, respectively, and decay rate 𝛽=0.8.

(a)

(b)

Figure 9 

Control signal (𝑢(𝑡)=𝑖(𝑡)2−𝑖20) 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 𝑡≥2 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∈[𝑚_min,𝑚_max] with 𝑚_min and 𝑚_max 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, 24–28, 30–32]. 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.