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

Authors: Máira P. A. Santim; Marcelo C. M. Teixeira; Wallysonn A. de Souza; Rodrigo Cardim; Edvaldo Assunção
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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.(2.1)Rulei:Ifz1(t)isM1i,…,zp(t)isMpi,thenẋ(t)=Aix(t)+Biu(t),y(t)=Cix(t),
where i=1,2,…,r, Mji, j=1,2,…,p is the fuzzy set j of Rule i,  x(t)∈ℝn is the state vector, u(t)∈ℝm is the input vector, y(t)∈ℝq is the output vector, Ai∈ℝn×n, Bi∈ℝn×m, Ci∈ℝq×n,  and z1(t),…,zp(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:(2.2)ẋ(t)=∑i=1rαi(x(t))(Aix(t)+Biu(t)),
where αi(x(t)) is the normalized weight of each local model system Aix(t)+Biu(t) that satisfies the following properties:(2.3)αi(x(t))≥0,fori=1,2,…,r,∑i=1rαi(x(t))=1.Considering the Takagi-Sugeno fuzzy model (2.1), the control input of fuzzy regulators via parallel distributed compensation (PDC) has the following structure [24]:(2.4)Rulej:Ifz1(t)isM1j,…,zp(t)isMpj,thenu(t)=-Fjx(t).Similar to (2.2), it can be concluded that(2.5)u(t)=-∑j=1rαj(x(t))Fjx(t).From (2.5), (2.2) and observing that ∑i=1rαi(x(t))=1, we obtain that(2.6)ẋ(t)=∑i=1r∑j=1rαi(x(t))αj(x(t))[Ai-BiFj]x(t).
Defining(2.7)Gij=Ai-BiFj,
then (2.6) can be written as follows:(2.8)ẋ(t)=∑i=1r∑j=1rαi(x(t))αj(x(t))Gijx(t).


### 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∈ℝn×n(X≻0) andMi∈ℝn×m,i=1,2,…,r exists such that the following LMIs are satisfied:
(2.9)XAiT+AiX-BiMi-MiTBiT≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT≼0,i<j,
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=MiX-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 that(2.10)lim⁡t→∞eβt‖x(t)‖=0
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∈ℝn×n(X≻0) and matrices Mi∈ℝn×m,i=1,2,…,r,  such that the following LMIs are satisfied:
(2.11)XAiT+AiX-BiMi-MiTBiT+2βX≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT+4βX≼0,i<j,
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=MiX-1,i=1,2,…,r.


## 2.1.  The Takagi-Sugeno Fuzzy Regulator

As described in [1], the Takagi-Sugeno fuzzy model is as follows.(2.1)Rulei:Ifz1(t)isM1i,…,zp(t)isMpi,thenẋ(t)=Aix(t)+Biu(t),y(t)=Cix(t),
where i=1,2,…,r, Mji, j=1,2,…,p is the fuzzy set j of Rule i,  x(t)∈ℝn is the state vector, u(t)∈ℝm is the input vector, y(t)∈ℝq is the output vector, Ai∈ℝn×n, Bi∈ℝn×m, Ci∈ℝq×n,  and z1(t),…,zp(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:(2.2)ẋ(t)=∑i=1rαi(x(t))(Aix(t)+Biu(t)),
where αi(x(t)) is the normalized weight of each local model system Aix(t)+Biu(t) that satisfies the following properties:(2.3)αi(x(t))≥0,fori=1,2,…,r,∑i=1rαi(x(t))=1.Considering the Takagi-Sugeno fuzzy model (2.1), the control input of fuzzy regulators via parallel distributed compensation (PDC) has the following structure [24]:(2.4)Rulej:Ifz1(t)isM1j,…,zp(t)isMpj,thenu(t)=-Fjx(t).Similar to (2.2), it can be concluded that(2.5)u(t)=-∑j=1rαj(x(t))Fjx(t).From (2.5), (2.2) and observing that ∑i=1rαi(x(t))=1, we obtain that(2.6)ẋ(t)=∑i=1r∑j=1rαi(x(t))αj(x(t))[Ai-BiFj]x(t).
Defining(2.7)Gij=Ai-BiFj,
then (2.6) can be written as follows:(2.8)ẋ(t)=∑i=1r∑j=1rαi(x(t))αj(x(t))Gijx(t).


## 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∈ℝn×n(X≻0) andMi∈ℝn×m,i=1,2,…,r exists such that the following LMIs are satisfied:
(2.9)XAiT+AiX-BiMi-MiTBiT≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT≼0,i<j,
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=MiX-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 that(2.10)lim⁡t→∞eβt‖x(t)‖=0
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∈ℝn×n(X≻0) and matrices Mi∈ℝn×m,i=1,2,…,r,  such that the following LMIs are satisfied:
(2.11)XAiT+AiX-BiMi-MiTBiT+2βX≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT+4βX≼0,i<j,
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=MiX-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. Figure1 shows the basic configuration of a magnetic levitator whose mathematical model [23, page 24] is given by(3.1)mÿ=-kẏ+mg-λμi22(1+μy)2,
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 variablex̅1=y and x̅2=ẏ. Then, (3.1)  can be written as follows [29]:
(3.2)x̅̇1=x̅2,x̅̇2=g-kmx̅2-λμi22m(1+μx̅1)2.Consider that, during the required operation,[x̅1x̅2]T∈D, where(3.3)D={[x̅1x̅2]T∈R2:0≤x̅1≤0.15}.
The paper aims to design a controller that keeps the ball in a desired position y=x̅1=y0, after a transient response. Thus, the equilibrium point of the system (3.2) is x̅e=[x̅1ex̅2e]T=[y00]T.From the second equationx̅̇2 in (3.2), observe that, in the equilibrium point, x̅̇2=0 and i=i0, where(3.4)i02=2mgλμ(1+μy0)2.Note that the equilibrium point is not in the origin[x̅1x̅2]T=[00]T. Thus, the following change of coordinates is necessary for the stability analysis:(3.5){x1=x̅1-y0,x2=x̅2,u=i2-i02,⟹{x̅1=x1+y0,x̅2=x2,i2=u+i02.
Therefore, ẋ1=x̅̇1, ẋ2=x̅̇2, and from (3.4), i2=u+(2mg/λμ)(1+μy0)2.Hence, the system (3.2) can be written as follows:(3.6)ẋ1=x2,ẋ2=g-kmx2-λμ(u+(2mg/λμ)(1+μy0)2)2m(1+μ(x1+y0))2,
and also, after some simple calculations, by(3.7)ẋ1=x2,ẋ2=gμ(μx1+2μy0+2)(1+μ(x1+y0))2x1-kmx2-λμ2m(1+μ(x1+y0))2u.
Finally, from (3.7) it follows that(3.8)[ẋ1ẋ2]=[01f21(x1,y0)-km][x1x2]+[0g21(x1,y0)]u,
where(3.9)f21(x1,y0)=gμ(μx1+2μy0+2)(1+μ(x1+y0))2,(3.10)g21(x1,y0)=-λμ2m(1+μ(x1+y0))2.


## 4.  Regulator Design for an Operating Point

The goal of the design in this subsection is to keep the ball in a given positionx̅1=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].
m0.05Kgg9.8m/s2k0.001Ns/mλ0.460Hμ2m−1First, assume thaty0=0.1 m and consider the following domain during the operation(4.1)D1={[x1x2]T∈R2:-0.1≤x1≤0.05,y0=0.1}.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 that(4.2)a211=max⁡x1∈D1{f21(x1)}=43.1200,a212=min⁡x1∈D1{f21(x1)}=28.9941,b211=max⁡x1∈D1{g21(x1)}=-5.4438,b212=min⁡x1∈D1{g21(x1)}=-9.2000.Thus, the nonlinear the functionf21 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](4.3)f21(x1)=σ211(x1)a211+σ212(x1)a212,
with(4.4)0≤σ211(x1),σ212(x1)≤1,σ211(x1)+σ212(x1)=1.
Therefore, from (4.3) and (4.4) note that(4.5)σ211(x1)=f21(x1)-a212a211-a212,σ212(x1)=1-σ211(x1).Similarly, from (4.2) there exist ξ211(x1) and ξ212(x1) such that(4.6)g21(x1)=ξ211(x1)b211+ξ212(x1)b212,
with(4.7)0≤ξ211(x1),ξ212(x1)≤1,ξ211(x1)+ξ212(x1)=1.
Hence, from (4.6) and (4.7) observe that(4.8)ξ211(x1)=g21(x1)-b212b211-b212,ξ212(x1)=1-ξ211(x1).Recall thatξ211(x1)+ξ212(x1)=1. Therefore, from (4.3) it follows that(4.9)f21(x1)=(ξ211(x1)+ξ212(x1))(σ211(x1)a211+σ212(x1)a212)=σ211(x1)ξ211(x1)a211+σ211(x1)ξ212(x1)a211+σ212(x1)ξ211(x1)a212+σ212(x1)ξ212(x1)a212.Similarly, from (4.6) and σ211(x1)+σ212(x1)=1, we obtain(4.10)g21(x1)=(σ211(x1)+σ212(x1))(ξ211(x1)b211+ξ212(x1)b212)=σ211(x1)ξ211(x1)b211+σ211(x1)ξ212(x1)b212+σ212(x1)ξ211(x1)b211+σ212(x1)ξ212(x1)b212.Now, define(4.11)α1(x1)=σ211(x1)ξ211(x1),α2(x1)=σ211(x1)ξ212(x1),α3(x1)=σ212(x1)ξ211(x1),α4(x1)=σ212(x1)ξ212(x1),
as the membership functions of the system (3.8)–(3.10), and their local models(4.12)A1=A2=[01a211-0.02],A3=A4=[01a212-0.02],B1=B3=[0b211]T,B2=B4=[0b212]T,
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:(4.13)ẋ(t)=∑i=14∑j=14αi(x(t))αj(x(t))Gijx(t),whereGij=Ai-BiFj.Thus, using the LMIs (2.9) from Theorem 2.1, we obtain the following controller gains:(4.14)F1=[-38.3098-4.7610],F2=[-26.0150-3.1937],F3=[-37.2796-4.8423],F4=[-23.3273-3.0979].Considering the initial conditionx̅0=[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=x̅0-[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.Positiony(t)=x̅1(t) and velocity (x̅2(t)) of the controlled system for y0=0.1 m.
(a)(b)Control signal (u(t)=i(t)2-i02) and electrical current (i(t)) of the controlled system for y0=0.1 m.
(a)(b)Now, suppose that the wanted position of the magnetic levitator isy0=0.05 m. Thus, for the design of the control law, consider that in the required operation the domain is(4.15)D2={[x1x2]T∈R2:-0.05≤x1≤0.1,y0=0.05}.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:(4.16)a211=max⁡x1∈D2{f21(x1)}=41.1600,a212=min⁡x1∈D2{f21(x1)}=27.8343,b211=max⁡x1∈D2{g21(x1)}=-5.4438,b212=min⁡x1∈D2{g21(x1)}=-9.2000.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:(4.17)F1=[-17.5514-1.7747],F2=[-12.0856-1.1922],F3=[-16.2911-1.8098],F4=[-10.0255-1.1717].
For the initial condition x̅0=[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.Positiony(t)=x̅1(t) and velocity (x̅2(t)) of the controlled system for y0=0.05 m.
(a)(b)Control signal (u(t)=i(t)2-i02) and electrical current (i(t)) of the controlled system for y0=0.05 m.
(a)(b)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.
LetI1⊂ℝn1 and I0⊂ℝn0 be compact subsets such that I=I1×I0,  f:I⊂ℝnt→ℝ a continuous function and nt=n1+n0. If for some given y0∈I0,  M=maxy∈I1{f(y,y0)} and m=miny∈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, thatM>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 M≤f(y*,y0*).
Similarly it is shown thatm≥min(y,y0)∈I{f(y,y0)}.Property1 is important to justify the proposed methodology. For instance, suppose that the plant can work in the region x̅1∈[0,0.15] and that we want the asymptotic stability of operating points [x̅1x̅2]T=[y00]T, where y0 is a known constant and y0∈I0=[0.04,0.11]. Thus the range of x1=x̅1-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:(5.1)D3={[x1x2y0]T∈R3:-0.11≤x1≤0.11,0.04≤y0≤0.11}.From Property1, (4.1), (4.15), and (5.1) note that(5.2)max⁡(x1,y0)∈D1,D2{f21(x1,y0)}≤max⁡(x1,y0)∈D3{f21(x1,y0)},min⁡(x1,y0)∈D1,D2{f21(x1,y0)}≥min⁡(x1,y0)∈D3{f21(x1,y0)},max⁡(x1,y0)∈D1,D2{g21(x1,y0)}≤max⁡(x1,y0)∈D3{g21(x1,y0)},min⁡(x1,y0)∈D1,D2{g21(x1,y0)}≥min⁡(x1,y0)∈D3{g21(x1,y0)}.Indeed, after the calculations, considering (3.9), (3.10), Table 1, and (5.1), we obtain(5.3)a211=max⁡(x1,y0)∈D3{f21(x1,y0)}=51.4116,a212=min⁡(x1,y0)∈D3{f21(x1,y0)}=25.1427,b211=max⁡(x1,y0)∈D3{g21(x1,y0)}=-4.4367,b212=min⁡(x1,y0)∈D3{g21(x1,y0)}=-12.4392.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:(5.4)F1=[-23.5425-2.4447],F2=[-14.1328-1.3935],F3=[-22.2383-2.5629],F4=[-10.8040-1.2964].For numerical simulation, att=0 s, the initial conditions are x̅0=[0.041]T and y0=0.1 m. Thus, x(0)=x̅(0)-[y00]T=[-0.061]T. In t=1 s, from Figure 6, the system is practically at the point x̅(1)=[x̅1(1)x̅2(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 x̅(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 x̅(∞)=[0.080]T. Figures 6 and 7 illustrate the system response.Position(y(t)=x̅1(t)) and velocity (x̅2(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)Control signal(u(t)=i(t)2-i02) 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.
(a)(b)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=[x̅1x̅2]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 pointy0 was [0.04,0.11] and the domain of x̅1 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:(5.5)F1=[-36.2829-3.9509],F2=[-21.8487-2.3121],F3=[-36.6258-4.2662],F4=[-17.9936-2.1555].The numerical simulation supposes the same condition of the last simulation; that is, initially it was considered the initial conditionx̅0=[0.041]T and y0=0.1 m. In t=1 s the system is practically at the point x̅0=[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 x̅0=[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.Position(y(t)=x̅1(t)) and velocity (x̅2(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)Control signal(u(t)=i(t)2-i02) 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.
(a)(b)As can be seen in Figure8, 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 massm 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.


## 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:(5.5)F1=[-36.2829-3.9509],F2=[-21.8487-2.3121],F3=[-36.6258-4.2662],F4=[-17.9936-2.1555].The numerical simulation supposes the same condition of the last simulation; that is, initially it was considered the initial conditionx̅0=[0.041]T and y0=0.1 m. In t=1 s the system is practically at the point x̅0=[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 x̅0=[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.Position(y(t)=x̅1(t)) and velocity (x̅2(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)Control signal(u(t)=i(t)2-i02) 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.
(a)(b)As can be seen in Figure8, 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 massm 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, 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.



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*Source: 731298-2012-05-30.xml*