Robust PID Controller

8/20/2019 Robust PID Controller

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Robust PID Controller Designfor Nonlinear Systems

Part II

Amin Salar

8700884

Final Project

Nonlinear Control Course – Dr H.D. Taghirad


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About the Project In part one we discussed about auto‐tuning techniques for PID controllers, and a number of

frequency response methods have been introduced, such as:

• Prototype Frequency Response Approach (Nonparametric)

• Parametric Approach

•

Using Finite Number of Frequency Response Data In the first two topics which have been mentioned above we explained two ways for finding PID

parameters to control nonlinear systems, after searching through the internet we found that like

authors in [1] said, most of the tuning methods mentioned in papers and articles are only suitable

to linear systems. They introduced a fuzzy modeling approach for nonlinear ones.

So in part 2 of the project we will explain this technique that is one of the few main ways for

designing PID controllers for nonlinear systems, and we will show the results of our simulations

with simulink software.

Abstract The design problem of proportional and proportional‐plus‐integral (PI) controllers design for

nonlinear systems via fuzzy modeling approach is studied in this chapter. First, the Takagi–Sugeno (T–S) fuzzy model with parameter uncertainties is used to approximate the nonlinear systems. Then

a numerically tractable algorithm based on the technique of iterative linear matrix inequalities is

developed to design a proportional (static output feedback) controller for the robust stabilization

of the system in T–S fuzzy model. Third, we transform the problem of PI controller design to that of

proportional controller design for an augmented system and thus bring the solution of the former

problem into the configuration of the developed algorithm. Finally, the proposed method is applied

to the design of robust stabilizing controllers for the excitation control of power systems.

Simulation results show that the transient stability can be improved by using a fuzzy PI controller,

compared to the conventional PI controller designed by using linearization method around the

steady state.

I. INTRODUCTION

Despite the developments of various kinds of modern or postmodern control theories, such as LQG

or LQR optimal control, control, and analysis and synthesis, classical proportional‐plus‐integral (PI)

or proportional‐plus‐integral‐ derivative (PID) controllers are widely used in industry due to their

relatively simple structure, ready‐in‐hand implementation, and perhaps, being easily understood.

Therefore, it is often the case that in practical applications one first considers PID controllers unless

evidence shows that they are insufficient to meet specifications. Because of the popularity of PID

controllers in the real world, many approaches have been developed to determine the parameters

of PID controllers. The first systematic tuning method for PID parameters was proposed by Ziegler

and Nichols in the early 1940s. Then came the well‐known formulae such as the Cohen–Coon

method, integral absolute error (IAE) optimum method, integral time‐weighted absolute error (ITAE) optimum method, internal model control (IMC) method, and relay auto‐tuning method.

Recently, many modified tuning methods have also been proposed associated with different

performance specifications or different methods used accordingly.

Feng Zheng, Qing‐Guo Wang, Tong Heng Lee, and Xiaogang Huang in [1], declared that all of the

aforementioned tuning methods are only suitable to linear systems and introduced a fuzzy

modeling approach for nonlinear ones (which we are going to explain) but in some articles


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including [2] authors introduced a way for “Robust self ‐tuning PID controller for nonlinear systems”

which I have explained completely in the first part of my project.

But it is true that compared with the voluminous references on tuning PID controllers for linear

systems, few results have been reported in the literature on PID controller tuning for nonlinear

systems. Indeed, many industrial processes can be approximated sufficiently well in concerned

operating region of state space by linear systems. However, many other plants exist whose

dynamics must be described by nonlinear systems. Therefore, it is highly desirable to develop

effective methods to determine the parameters of PID controllers for nonlinear systems. As I said

so apart from the above works, the only reference discussing this problem is the monograph by

Freeman and Kokotovic [3], where backstepping and control Lyapunov function methods are used

to design PI control, but it is supposed that all state variables are available. In this chapter as told

before, we will try to take a different approach to design PI controllers for nonlinear systems. As

shown later, our controllers will use output variables only or partial state variables.

As in the design methodologies of many other types of controllers, a crucial step for the design of

PID controllers is the modeling of the nonlinear plants. Since the 1980s, fuzzy technique has been

widely adopted to model complex nonlinear plants. The so‐called T–S or T–S–K fuzzy model, first

proposed by Takagi and Sugeno [4] and further developed by Sugeno and Kang [5], is one of most

successful models in this direction. The basic idea in this approach is first to decompose the model

of a nonlinear system or other kinds of complex systems into linear systems in accordance with the

cases for which linear models are suitable to describe and then to aggregate (fuzzy blend) each

individual model (linear model) into a single nonlinear model in terms of their membership

function. Thus, the relatively complex consequence part allows the number of fuzzy rules (local

models) to be quite small in many applications. Consequently, the T–S fuzzy model is less prone to

the curse of dimensionality than other fuzzy models. As is well known, the key problem in this

approach is to what degree the nonlinear system can be approximated by a convex (fuzzy) blending

of several linear systems. Theoretical justification of T–S fuzzy model as a universal approximator

has been given by Wang and Mendel [6]. Now T–S model has found wide applications in the control

of complex systems, e.g., in the control of robot manipulators and time delay systems. In this chapter, we will apply T–S fuzzy modeling approach to the design of robust PI controllers for

nonlinear systems. This chapter is organized as follows. The problem formulation is introduced in

Section II. In Section III, proportional (static output feedback) controller design for nonlinear

systems using T–S fuzzy models is presented. PI controller design for nonlinear systems is given in

Section IV. A numerical example is provided in Section V to show the design procedure and the

effectiveness of the proposed method. Finally, concluding remarks are drawn in Section VI.

II. PROBLEM FORMULATION Consider the following uncertain nonlinear system:

, Δ, , , (1) Where vector of state variables vector of control inputs vector of outputs vector of uncertain parameters which is restricted to a prescribed bounding set nonlinear functions of (x,t), representing the nominal model of the system under consideration, are continuously differentiable with respect to (x,t)


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Δ represents the model uncertainties constant matrix Ignoring the uncertainty term, we obtain the nominal model of the system (1) as follows: , ,

(2),(3)

To deal with the problem for the system (2) to run at different operating points or in different

conditions, Takagi and Sugeno [4] proposed an effective method to represent the system (2). The

main feature of a T–S fuzzy model is to express the join dynamics of each fuzzy implication (rule) by

a linear system. The i th rule of the T–S fuzzy model is of the following form.

Plant

Rule

i

(i=1,…,p): IF θ1 is μi1 and … and θp is μip THEN Where θ j(x) (j=1,…,p) are the premise variables, which are functions of state variables x; μij (i=1,…r

,j=1,…,p ) are fuzzy sets; r is the number of the IF–THEN rules; p is the number of the premise

variables; and Ai and Bi are constant matrices of compatible dimensions. It is assumed that the

premise variables are independent of the input variables u(t). The overall fuzzy model is achieved

by fuzzy blending (aggregation) of each individual rule (model) as follows:

∑ ∑ (4) Where θ = [θ1 ,…, θp], and ωi: Rp ‐‐> [0,1], i=1,…,r, are the membership function of the system belonging to plant rule i.

Define ∑ The system (4) admits the following form:

∑

(5)

with the constraints

0, ∑ 1 (6) Stripping off the fuzzy cover of system (5), one can see that the model (5) is nothing but the convex combination of several linear systems. It has been shown [6], [7] that a nonlinear system can be

approximated by means of the above fuzzy basis functions to desired accuracy inside an arbitrarily

large compact subspace of Rn.

Correspondingly, we can model uncertain nonlinear system (1) as the following uncertain fuzzy

system:

Plant

Rule

i

(i=1,…,r): IF θ1 is μi1 and … and θp is μip THEN Δ,, (7) where ΔAi represents the uncertainties in system matrix. We assume that ΔAi admits the following

form: Δ, , Τ, , 1 , 2 , … , (8) where Di and Ei are known real constant matrices of dimensions nxnDi and nEixn, respectively, and

the uncertainty Ti(x,t,ξ), an unknown matrix‐valued function of (x,t,ξ), belongs to the following

bounded set: Γ , , : |, ,| 1 1 , 2 , … , (9) Fuzzy blending of each individual model yields the overall fuzzy model as follows:


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∑ Δ, , (10) Here, the output equation is also added into the overall model for the convenience of later citation.

We assume that all the triples (Ai, Bi, C), ‐=1,2,…,r , are controllable and observable.

It should be noted that the parameter uncertainty structure in (8) and (9) describes how the

uncertainties enter the system model and has been widely used in the study of the problem of

robust stability and stabilization of uncertain linear systems and it can represent parameter

uncertainty in many physical cases. Actually, any norm‐bounded parameter uncertainty can be

expressed in the form of (8) and (9). Our objective in this chapter is to design a PI controller of the

following form: (11) such that the closed loop system (10) and (11) is asymptotically stable, where FP and FI are constant

matrices.

Controller (11) is an ideal PI controller. Readers are referred to [8] for how to change the ideal PI

controllers into practical ones and the relationships between the parameters of the two kinds of

controllers.

III.

PROPORTIONAL

CONTROL

In this section, we first study the stabilization of system (10) by a proportional controller, i.e., by

the following controller: (12) where F is a constant matrix. To do this, we need the following proposition.

Proposition

1: System (10) is asymptotically stabilized via controller (12) if there exists a matrix such

that the following matrix inequalities hold: Δ Δ 0 , , Γ , 1,2, … , (13) Proof: First notice that system (10) is asymptotically stabilized via controller (12) if there exists a

matrix P>0 such that Δ Δ 0 , , Γ , 1,2, … , (14) To show this claim, we choose V(x):=x

TP‐1x(t) as a Lyapunov function candidate for system (10).

Since Гi is a closed set and r is finite, there exists a positive definite matrix Q 0 such that Δ Δ , , Γ , 1,2, … , if matrix inequality (14) holds. Then, noticing the fact that αi(x)≥0 , i=1,2,..,r , and ∑ 1 , we have

Δ Δ Thus follows the claim.

Now, following the same procedure as the proof of [9, Theorem 1], we can prove that matrix

inequality (14) holds if and only if matrix inequality (13) holds. This completes the proof.


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Theorem

1: Consider the uncertain nonlinear system (10). Suppose there exist positive definite

matrices P, a matrix F, and positive numbers εi , i=1,2,…,r , such that the following matrix

inequalities hold: 0 1 , 2 , … ,

(15)

Then, system (10) can be robustly stabilized by controller u(t)=Fy(t).

To prove this theorem, we need the following lemma.

Lemma

1:

Let D, E, and T be real matrices of appropriate dimensions with T satisfying ||T||≤1.

Then for any real number ε>0, we have (16) Proof: The assumption ||T||≤1 leads to I-TT

T≥0, which again yields the following inequality: 0

by applying Schur complements [10]. Then, for any two real numbers ε1 and ε2 we will have 0 (17) If ε1ε2=1, the inequality (17) will be of the following form:

0 which is equivalent to the inequality (16).

From (8) and (9) and applying Lemma 1, we have Δ Δ , , , , where εi, i=1,2,…,r , are real positive numbers. Thus, it follows that matrix inequalities (13) hold if

so do matrix inequalities (15). Applying Proposition 1, we arrive at this theorem.

One cannot directly use matrix inequalities (15) to calculate the required feedback matrix due to

the existence of the term ‐PBBTP in (15), which makes the solution of (15) very complicated.

Neither is it convex, nor can it be transformed into a convex problem in the space of unknown

parameters (P,F, εi, i=1,2,…,r). In [9], an iterative linear matrix inequality (ILMI) algorithm was developed to solve the similar problem. Following the same idea as in [9], we can also develop an

ILMI algorithm to solve matrix inequalities (15), which is summarized in the sequel.

Algorithm 1 (ILMI Algorithm for Proportional Nonlinear Controllers):

• Initial data: systems state space parameters (Ai, Bi, Ci, Di, Ei, i=1,2,…,r)

• Step 1. Choose P0>0. Set j=1 and X1=P0.

• Step 2. Solve the following optimization problem for P j, F, and α j .

OP1: Minimize subject to the following LMI constraints

Σ 0 0

0 (18) 0 1 , 2 , … , (19) Where Σ

Denote by α* j the minimized value of α j.

• Step 3.If α* j≤0, the obtained matrices (Pj, F, εi, i=1,2,…,r) solve the problem. Stop. Otherwise go to

Step 4.

• Step 4.Solve the following optimization problem for Pj, F and εi, i=1,2,…,r .


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OP2: Minimize tr(P j) subject to LMI constraints (18) and (19) with α j =α

* j, where tr stands for the

trace of a square matrix. Denote by P* j the optimal P j.

•

Step

5. If ||Xj‐ P* j ||


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Δ, , i.e.

Δ , , Where 0 0, 0 Δ , , , ,

0 0 Define

0

0 1 And denote : 0, : 0 , : , : Then the problem of PI controller design for system (10) is reduced to that of proportional

controller design for the following system:

Δ , ,

Thus, the feedback matrices FP and FI can be calculated by applying Algorithm 1 to system (22).

In the case where all Bi , i=1,2,…,r , are equal, the following kind of controller can be also designed

by applying Algorithm 1 and Theorem 2:

V. NUMERICAL EXAMPLE: EXCITATION CONTROL OF POWER SYSTEMS Most existing excitation controllers of power systems are designed by application of linear control

theory to the approximate linearized models of power systems. Thus, they work well only when the

disturbance caused by faults in the systems is relatively small. Under large disturbance, the systems

might be out of synchronization. In this section, we will apply the results obtained in the previous

sections to the design of excitation controllers for a thermal turbine synchronous generator. The

simplified dynamical model of the single‐machine infinite‐bus power system with a silicon‐

controlled rectifier direct excitor is as follows: 2


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1 Where

c os . sin where δ is the angular position of the rotor of generator (G) with respect to a synchronously

rotating reference, which is selected here to be the infinite bus; ω is the angular velocity of the

rotor; Pe and Pm are the active power and mechanical power of G, respectively, Eq is the

electromotive force (EMF) in the q‐axis of G; E’q is transient EMF in the q‐axis of G; Ef is the

equivalent EMF in the excitation winding of G; xd is the d‐axis reactance of G; x’d is the d‐axis

transient reactance of G; xT and xL are the reactances of the transformer and the transmission line,

respectively xd∑ = xd + xT + xL, x’d∑ = x’d + xT + xL , xad is the mutual reactance between the excitation

winding and the stator winding; Rf is the resistance of the excitation winding; kA is the gain of the

amplifier, Vs is the bus voltage; u is the control input; and h(t) denotes the external disturbance.

Suppose δ0 is the value of δ under steady operating condition. For this δ0 , let sin 1 cos

Define new state variables and control input as Δ The dynamical model of the power system (23) can be rewritten as 2 : sin sin cos :

cos

Δ : (24) The parameters of the system are as follows: f 0 = 50 Hz ω0 = 1 p.u. δ0 = 60

o H = 8 s D = 0.8 Pm = 0.79 p.u.

Vs = 1 p.u. xd =1.5 p.u. x’d = 0.3 p.u. xad = 1.3 p.u. xL = (0.8+ΔxL) p.u. ΔxL = 0.0008 p.u.

xT = 0.01 p.u. kA = 10 Td0 = 3 s Rf = 0.0045 p.u.

These parameters yield

E’q0 = 1.2723 p.u. u0 = 7.2942*10^‐4 p.u.


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Notice that all parameters except xL are supposed to be known here. The parameter xL is supposed

to have some perturbation. The reason why we choose the perturbation in xL to illustrate the

robustness of our controller is that xL is one of the system parameters which are most uncertain

and difficult to measure.

Generally, the state variable δ is difficult to measure. Hence, the system output equation is as

follows:

0 1 00 0 1 The physical limit of the excitation voltage leads to the constraint on the control input Δu as follows: | Δ| 3.0 . . Or equivalently 0.0018 . . Δ 0.00031 . . The local fuzzy models of the system are obtained through the linearization of the system (24)

around the points x1 = [‐30, 0, 0]T , x2 = [0, 0, 0]

T and x3 = [+30, 0, 0]

T, respectively. In terms of IF–

THEN rules, the fuzzy models admit the following form:

Rule 1. IF x1 (or δ ) is small (i.e., x1 is about ‐30o , or equivalently, δ is about 30

o ) THEN Δ Δ

Rule 2. IF x1 (or δ ) is middle (i.e., x1 is about 0o , or equivalently, δ is about 60

o ) THEN Δ Δ

Rule 3. IF x1 (or δ ) is large (i.e., x1 is about +30o , or equivalently, δ is about 90

o ) THEN Δ Δ

where ΔAi (i=1,2,3 ) account for the parameter perturbation in xL . By using Teixeira–Zak’s formula

[12]

|||| 0 0

1,2,3, 1,2,3 matrices Ai and Bi are obtained as follows (noticing that we set ΔxL=0 while solving Ai and Bi ): 0 314.16 00.1002 0.1 0.05630.2519 0 0.6937

0 314.16 00.1009 0.1 0.09750.3121 0 0.6937 0 314.16 00.085 0.1 0.11260.3441 0 0.6937


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00962.963 There are many options to assign membership functions. For the sake of convenience in

computation, we select triangular function as our membership functions, which are

1 63 3 6 6 3 0 3

0 3 3

2

3

3 2

1 2

1

Figure

1

Membership

functions

of

the

fuzzy

models

for

power

system

(24).

Since the right hand side of the first equation of the power system (24) includes no uncertainties

and it is also linear function of the state variables, we let the first row of ΔAi be zero. To account for

the parameter perturbation and approximation error caused by linearization, we let the second

and third rows be 0.1% times the corresponding elements of ΔAi i.e. Δ 0 0 00.1002 0.1 0.05630.2519 0 0.6937 1 0

Δ 0 0 00.1009 0.1 0.09750.3121 0 0.6937 1 0

Δ 0 0 00.085 0.1 0.11260.3441 0 0.6937 1 0 Since B1=B2=B3, we can use Theorem 2 combined with Algorithm 1 to calculate the feedback

matrices of proportional and PI controllers, which are as follows. For proportional controller 0.0398 0.0071


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0.0353 0.0065 0.0212 0.0027

For PI controller 0.0163 0.0021 0.0037 0.0125 0.0171 0.0028 0.0028 0.0125 0.0241 0.0033 0.005 0.0125


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Simulations

with

Simulink

1) With Proportional control and switching A1, A2 and A3, according to x1 we have:

x2 & x3 scope

x scope

u scope

u

MATLAB

Function

Switch3

Switch 2

Switch 1

A3x3

x' = Ax+Bu

y = Cx+Du

A3x2

x' = Ax+Bu

y = Cx+Du

A3x1

x' = Ax+Bu

y = Cx+Du

A2x3

x' = Ax+Bu

y = Cx+Du

A2x2

x' = Ax+Bu

y = Cx+Du

A2x1 scope

A2x1

x' = Ax+Bu

y = Cx+Du

A1x3

x' = Ax+Bu

y = Cx+Du

A1x2

x' = Ax+Bu

y = Cx+Du

A1x1

x' = Ax+Bu

y = Cx+Du


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X1: Yellow X2: Pink X3: Blue

Figure 2) X1: Yellow X2: Pink X3: Blue for P controller with switching


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The input u:

Figure

4)

input

u

for

P

controller

with

switching


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2) With Proportional plus Integral control and switching A1, A2 and A3, according to x1 we have:

y

MATLAB

Function

x2 & x3 scope

x scope

u scope

alfaiFij *(y/s)Matrix

Multiply

alfa 3Fi3

alfa 2Fi2

alfa 1Fi1

alfa

MATLAB

Function

Switch3

Switch 2

Switch1

Pu

MATLAB

Function

Integrator 2

1s

Integrator 1

1

s

Fi3

[.005 -.0125 ]

Fi2

[-.0028 -.0125 ]

Fi1

[-.0037 -.0125 ]

A3 x3

x' = Ax+Bu

y = Cx+Du

A3 x2

x' = Ax+Bu

y = Cx+Du

A3 x1

x' = Ax+Bu

y = Cx+Du

A2 x3

x' = Ax+Bu

y = Cx+Du

A2 x2

x' = Ax+Bu

y = Cx+Du

A2 x1

x' = Ax+Bu

y = Cx+Du

A1x3

x' = Ax+Bu

y = Cx+Du

A1x2

x' = Ax+Bu

y = Cx+Du

A1x1

x' = Ax+Bu

y = Cx+Du


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The input u:

Figure 7) input u for P controller with switching


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3) With Proportional control and without switching A1, A2 and A3 we have:

x

u

MATLAB

Function

Pu

x' = Ax+Bu

y = Cx+Du

A1x3

x' = Ax+Bu

y = Cx+Du

A1x2

x' = Ax+Bu

y = Cx+Du

A1x1


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Figure

8)

X1:

Yellow

X2:

Pink

X3:

Blue

for

P

controller

without

switching


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The input u:

Figure 9) input u for P controller without switching


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4) With Proportional plus Integral control and without switching A1, A2 and A3 we have:

MATLAB

Function

y

x scopeu scope

Matrix

MultiplyalfaiFij*(y/s)

alfa3Fi3

alfa2Fi2

alfa1Fi1

MATLAB

Function

alfa

MATLABFunction

Pu

1

s

Integrator2

1

s

Integrator1

[.005 -.0125]

Fi3

[-.0028 -.0125]

Fi2

[-.0037 -.0125]

Fi1

x' = Ax+Bu

y = Cx+Du

A1x3

x' = Ax+Bu

y = Cx+Du

A1x2

x' = Ax+Bu

y = Cx+Du

A1x1


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Figure 10) X1: Yellow X2: Pink X3: Blue for PI controller without switching


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The input u:

Figure 11) input u for PI controller without switching


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VI. CONCLUDING REMARKS As you saw the responses in switching mode have some jumps and nonsmoothnesses because of

switching phenomenon but they finally became stable, the results without switching obviously

shows more smoothness in responses. The design problem of proportional and PI controllers for

nonlinear systems has been studied in this chapter. First, Takagi–Sugeno (T–S) fuzzy model with

parameter uncertainties is used to model the nonlinear systems. The introduction of parameter

uncertainties in the T–S fuzzy model can account for both approximation error of fuzzy model and

the actual parameter uncertainties in the practical process. Second, an algorithm based on ILMI is

developed to design a proportional controller to robustly stabilize the system which can be

expressed by a T–S fuzzy model. Third, we transform the problem of PI controller design to that of

proportional controller design for an augmented system and thus bring the solution of the former

problem into the configuration of the developed algorithm. The characteristics of the developed

method is that it is numerically tractable and can be applied to general multi‐variable nonlinear

systems. Finally, the proposed method is applied to the robust stabilizing controller design for the

excitation control of power systems. Simulation results show that the transient stability can be

improved by using a fuzzy PI controller when large faults appear in the system, compared to the

conventional PI controller designed by using linearization method around the steady state. Notice

that the controllers proposed in this paper use only output variables (in the case of common

feedback matrix corresponding to all the fuzzy rules) or output variables plus some other state

variables needed in the fuzzy premise (in the case of different feedback matrix corresponding to

each fuzzy rule accordingly). In the latter case, the state variables needed in the fuzzy premise

should be available normally.

VII. References [1] Feng Zheng, Qing-Guo Wang, Tong Heng Lee and Xiaogang Huang “Robust PI Controller Design for Nonlinear

Systems via Fuzzy Modeling Approach” IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—

PART A: SYSTEMS AND HUMANS, VOL. 31, NO. 6, NOVEMBER 2001

[2] K.K. Tan*, S. Huang, R. Ferdous “Robust self-tuning PID controller for nonlinear systems” Journal of Process

Control 12 (2002) 753–761

[3] R. A. Freeman and P. K. Kokotovic, “Robust Nonlinear Control Design”. Cambridge, MA: Birkhaüser, 1996

[4] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE

Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, 1985.

[5] M. Sugeno and G. T. Kang, “Structure identification of fuzzy model,” Fuzzy Sets Syst., vol. 28, pp. 15–33, 1988.

[6] L. X.Wang and J. M. Mendel, “Fuzzy basis functions, universal approximatiors and orthogonal least-squarelearning,” IEEE Trans. Neural Networks, vol. 3, pp. 807–814, Sept. 1992.

[7] T. A. Johansen and B. A. Foss, “Constructing NARMAX models using ARMAX models,” Int. J. Control, vol. 58, pp. 1125–1153, 1993.

[8] “PID Controllers: Theory, Design, and Tuning”. Research Triangle Park, NC: Instrument Society of America, 1995.

[9] Y.-Y. Cao, J. Lam, and Y.-X. Sun, “Static output feedback stabilization: An ILMI approach,” Automatica, vol. 34,

pp. 1641–1645, 1998.

[10] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory.

Philadelphia, PA: SIAM, 1994.

[11] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, “LMI Control Toolbox: The Math Works Inc.”, 1995.


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[12] M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzymodels,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 133–142, Apr. 1999.

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