Robust adaptive fuzzy control for permanent ... - Wiley Online Library
Robust Adaptive Fuzzy Control for Permanent Magnet Synchronous Servomotor Drives Yansheng Yang, 1,† Changjiu Zhou 2, * 1 Navigation College, Dalian Maritime University (DMU), Dalian, P.R. China 2 School of Electrical and Electronic Engineering, Singapore Polytechnic, 500 Dover Road, Singapore 139651, Republic of Singapore
A novel robust adaptive fuzzy control (RAFC) algorithm for the permanent magnet (PM) synchronous servomotor drives with uncertain nonlinearities and time-varying uncertainties is presented in this article. Takagi–Sugeno-type fuzzy logic systems are used to approximate uncertain functions. The RAFC algorithm is designed by use of the input-to-state stability (ISS) approach and small gain theorem. The closed-loop system is proven to be semiglobally uniformly ultimately bounded. In addition, the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be removed and the adaptive mechanism with only one learning parameterization can be achieved. The proposed methodology is applied to design the position control of the PM synchronous servomotor drives. Simulation results show the effectiveness of the proposed control scheme. © 2005 Wiley Periodicals, Inc.
1.
INTRODUCTION
Recently, controlling permanent magnet (PM) synchronous servomotor drives, which consist of magnetic materials, semiconductor power devices, and so on, has been an interesting research line in motion-control application in the low to medium power range. The desirable features of the PM synchronous servomotor are its compact structure, high air-gap flux density, high power density, high torque-toinertia ratio, and high torque capability. Moreover, compared with an induction servomotor, the PM synchronous servomotor has such advantages as higher efficiency, owing to the absence of rotor losses and lower no-load current below the rated speed, and its decoupling control performance is much less sensitive to the parameter variations of the motor.1 To achieve fast four-quadrant operation and *Author to whom all correspondence should be addressed: e-mail: [email protected]. † e-mail: [email protected]. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 20, 153–171 (2005) © 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). • DOI 10.1002/int.20060
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smooth starting and acceleration, the field-oriented control, or vector control, is used in the design of the PM synchronous servomotor drives. However, the control performance of PM synchronous servomotor drives is still influenced by the uncertainties of the plant, which usually features unpredictable plant parameter variations, external load disturbance, and unmodeled and nonlinear dynamics. There has been tremendous interest in designing a controller for systems having uncertain nonlinear systems, and much significant research attention has been attracted in the past decade. Most results addressing this problem are available in the control literature, for example, Ref. 2 and references therein. Applications of these approaches are quite limited because they rely on the exact knowledge of the plant nonlinearities. To relax some of the exact model-matching restrictions, several adaptive schemes have recently been introduced to solve the problem of linearly parameterized uncertainties, which are referred to as structured uncertainties. Unfortunately, in the industrial control environment, there are some controlled systems that are characterized not only by the unstructured uncertainties, but are also represented by all the terms that cannot be modeled or are not repeatable. A solution to those problems was first presented by Wang 3 ; the fuzzy systems were proposed to uniformly approximate the uncertain nonlinear functions in the designed system by use of the universal approximation properties of certain classes of fuzzy systems presented in Refs. 4–7, and a Lyapunov based learning law was used. Then several stable adaptive fuzzy controllers that ensure the stability of the overall system were developed by Wang,8 Su and Stepanenko,9 Spooner and Passino,10 and Fischle and Schroder.11 The more application-motivated problem of adaptive fuzzy control for the uncertain nonlinear systems has gradually gained much attention (see, e.g., Refs. 12 and 13). However, there is a substantial restriction in the aforementioned works: Many parameters need to be tuned in the learning laws when there are many state variables in the designed system and many rule bases have to be used in the fuzzy system for approximating the nonlinear uncertain functions, so that the learning times tend to become unacceptably long for the systems of higher order, and a time-consuming process is unavoidable when the fuzzy logic controllers are implemented. This problem has been pointed out by Fischle and Schroder 11 and first researched by Yang and Ren.14 In this article, we present a novel robust adaptive fuzzy controller for a class of continuous uncertain systems with uncertain nonlinearities and timevarying uncertainties, and Takagi–Sugeno-type fuzzy logic systems 15 are used to approximate the unknown unstructured uncertain nonlinear functions in the system, and the adaptive mechanism with minimal learning parameterizations is proposed by use of input-to-state stability (ISS) theory first proposed by Sontag 16 and the nonlinear small gain approach given by Jiang et al.17 The main features of the algorithm proposed in the article are: (i) the closed-loop system is proven to be semiglobally uniformly ultimately bounded; (ii) only one function needs to be approximated by the T-S fuzzy systems, and no matter how many states in the designed system are investigated and how many rules in the fuzzy system are used, only one parameter needs to be adapted on-line, such that the burdensome computation of the algorithm can be lightened increasingly and it is convenient
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to implement this algorithm in real-time application; (iii) the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be avoided. The rest of this article is organized as follows. In Section 2, we give some needed definitions of input-state stability, small gain theorem, and preliminary results. Section 3 contains the description of position control of PM synchronous servomotor drives. In Section 4, a systematic procedure for the synthesis of robust adaptive fuzzy control (RAFC) is developed. In Section 5, the application examples for a PM synchronous servomotor drive system by use of the RAFC are shown and numerical simulation results are presented. Concluding remarks are given in Section 6. 2.
MATHEMATICAL PRELIMINARIES
Notation: Throughout this article, let 7 •7 be any suitable norm. The vector norm of x 僆 R n is Euclidean, that is, 7 x7 2 ⫽ ~ x T x! and the matrix norm of A 僆 R n⫻m is defined by 7A7 2 ⫽ l max ~AT A!, where l max(min) (•) denotes the operation of taking the maximum (minimum) eigenvalue. The vector norm over the space defined by stacking the matrix columns into a vector, so that it is compatible with the vector norm, that is, 7Ax7 ⱕ 7A7•7 x7. For any piecewise continuous function u : R ⫹ r R m, 7u 7` denotes sup $6u~t !6, t ⱖ 0%, which stands for L ` supremum norm, and for any pair of times 0 ⱕ t1 ⱕ t2 , the truncation u @t1 , t2 # is a function defined on R ⫹ that is equal to u~t ! on @t1 , t2 # and is zero outside the interval. In particular, u @0, t# is the usual truncated function u t . 2.1.
ISS and Small Gain Theorem
The concepts of ISS and ISS–Lyapunov function proposed in Refs. 16, 18, and 19 have recently been used in various control problems. To ease the discussion of the design of the RAFC scheme, the variants of those notions are reviewed in the following. First, we begin with the definitions of class K, K` , and KL functions that are standard in the stability literature (see Ref. 20). A class K-function g is a continuous, strictly increasing function from R ⫹ into R ⫹ and g~0! ⫽ 0. It is of class K` if additionally g~s! r ` as s r `. A function b : R ⫹ ⫻ R ⫹ r R ⫹ is of class KL if b~•, t ! is of class K for every t ⱖ 0 and b~s, t ! r 0 as t r `. Definition 1. For the system x_ ⫽ f ~ x, u!, it is said to be input-to-state practically stable (ISpS) if there exists a function g of class K, called the nonlinear L ` gain, and a function b of class KL such that, for any initial condition x~0!, each measurable essentially bounded control u~t ! defined for all t ⱖ 0 and a nonnegative constant d, the associated solutions x~t ! are defined on @0,`! and satisfy 7 x~t !7 ⱕ b~7 x~0!7, t ! ⫹ g~7u t 7` ! ⫹ d
(1)
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When d ⫽ 0 in Equation 1, the ISpS property collapses to the input-to-state stability (ISS) property introduced in Ref. 18. Definition 2. A C 1 function V is said to be an ISpS–Lyapunov function for the system x_ ⫽ f ~ x, u! if
•
there exist functions a1 , a2 of class K` such that a1 ~7 x7! ⱕ V~ x! ⱕ a2 ~7 x7!,
•
∀x 僆 R n
(2)
there exist functions a3 , a4 of class K and a constant d ⬎ 0 such that ]V~ x! ]x
f ~ x, u! ⱕ ⫺a3 ~7 x7! ⫹ a4 ~7 u7! ⫹ d
(3)
When Equation 3 holds with d ⫽ 0, V is referred to as an ISS–Lyapunov function. Then it holds that one may pick a nonlinear L ` gain g in Equation 1 of the form, which is given in Ref. 21: g~s! ⫽ a1⫺1 ° a2 ° a3⫺1 ° a4 ,
∀s ⬎ 0
(4)
The following proposition establishes equivalence between ISpS and the existence of ISpS–Lyapunov function by Ref. 19. Proposition 1. The system x_ ⫽ f ~ x, u! is ISpS if and only if there exists an ISpS–Lyapunov function. Consider the stability of the closed-loop interconnection of two systems as shown in Figure 1. A trivial refinement of the proof of the generalized small gain theorem given in Ref. 17 yields the following variant, which is suited for our applications presented in this article.
Figure 1.
Feedback connection of interconnection systems.
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Theorem 1. Consider a system in composite feedback form of two ISpS systems (cf. Figure 1): S zv I : S vzI :
再 再
x_ ⫽ f ~ x, v!
(5)
zI ⫽ H~ x!
y_ ⫽ g~ y, z! I
(6)
v ⫽ K~ y, z! I
In particular, there exist two constants d1 ⬎ 0, d2 ⬎ 0, and let bv and bj be of class KL, and gz and gv of class K, such that, for each v in the L ` supremum norm, each zI in the L ` supremum norm, each x 僆 R n and each y 僆 R m , all the solutions X~ x; v, t ! and Y~ y; z,I t ! are defined on @0,`! and satisfy, for almost all t ⱖ 0 7H~X~ x; v, t !!7 ⱕ bv ~7 x7, t ! ⫹ gz ~7vt 7` ! ⫹ d1
(7)
7K~Y~ y; z,I t !!7 ⱕ bj ~7y7, t ! ⫹ gv ~6 zI t 7` ! ⫹ d2
(8)
Under these conditions, if gz ~gv ~s!! ⬍ s ~resp. gv ~gz ~s!! ⬍ s!
∀s ⬎ 0
(9)
the solution of the composite systems 5 and 6 is ISpS. 2.2.
Takagi–Sugeno ~T-S! Fuzzy Systems
In this subsection, we briefly describe the structure of fuzzy systems to be used in this article. Let R denote the real numbers, R n the real n-vectors, and R n⫻m the real n ⫻ m matrices. Let S be a compact simply connected set in R n . With map f : S r R m , define C m ~S! to be the function space such that f is continuous. A fuzzy system can be employed to approximate the function f ~ x! in order to design the adaptive fuzzy robust control law. The configuration of Takagi–Sugeno-type fuzzy logic system (T-S fuzzy system)15 and its approximation theorem are discussed in the following. Consider a T-S fuzzy system to uniformly approximate a continuous multidimensional function y ⫽ f ~ x! that has a complicated formulation, where x is an input vector with n independent x ⫽ ~ x 1 , x 2 , . . . , x n ! T . The domain of x i is ui ⫽ @a i , bi # . It follows that the domain of x is Q ⫽ u1 ⫻ u2 ⫻ {{{ ⫻ un ⫽ @a 1 , b1 # ⫻ @a 2 , b2 # ⫻ {{{ ⫻ @a n , bn # To construct a fuzzy system, the interval @a i , bi # is divided into Ni subintervals: a i ⫽ C0i ⬍ C1i ⬍ . . . CNi n⫺1 ⬍ CNi n ⫽ bi ,
1ⱕiⱕn
On each interval ui ~1 ⱕ i ⱕ n!, Ni ⫹ 1~Ni ⬎ 0! continuous input fuzzy sets, denoted by Aij ~0 ⱕ j ⱕ Ni !, are defined to fuzzify x i . The membership function of Aij is denoted by m ij ~ xi !, which can be represented by triangular, trapezoid, generalized bell, or Gaussian type and so on.
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Generally, the T-S fuzzy system can be constructed by the following K~K ⬎ 1! fuzzy rules: R i : If x 1 is Aih1 AND x 2 is Aih2 AND . . . AND x n is Aihn THEN yi is a 1i x 1 ⫹ {{{ ⫹ a ni x n ,
i ⫽ 1,2, . . . , K
where a ji , j ⫽ 1,2, . . . , n, i ⫽ 1,2, . . . , K are the unknown constants. The product fuzzy inference is employed to evaluate the ANDs in the fuzzy rules. After being defuzzified by a typical center average defuzzifier, the output of the fuzzy system is K
F~ x! ⫽
( yi ji ~ x! ⫽ j~ x!A x x
(10)
i⫽1
where yi ⫽ a 1i x 1 ⫹ {{{ ⫹ a ni x n and ji ~ x! ⫽ ) nj⫽1 m ihj ~ x j !/ ( Ki⫽1 @ ) nj⫽1 m ihj ~ x j !# , which is called a fuzzy base function. And j~ x! ⫽ @j1 ~ x!, j2 ~ x!, . . . , jn ~ x!# , x ⫽ @x 1 , x 2 , . . . , x n # T,
Ax ⫽
冤
a 11
a 21
J
a n1
a 12
a 22
J
a n2
I
I
I
I
a 1K
a 2K
J
a nK
冥
The following lemma 3 gives that the above fuzzy logic system has been shown to be capable of uniformly approximating any well-defined nonlinear function over a compact set to any degree of accuracy. Lemma 1. Suppose that the input universal of discourse U is a compact set in R r . Then, for any given real continuous function f ~ x! on U and ∀« ⬎ 0, there exists a fuzzy system F~ x! in the form of Equation 10 such that sup 7 f ~ x! ⫺ F~ x!7 ⫽ sup 7 f ~ x! ⫺ j~ x!A Z Z7 ⱕ « x僆U
3.
(11)
x僆U
POSITION CONTROL OF PM SYNCHRONOUS SERVOMOTOR DRIVES
In this article, we consider the position control design for PM synchronous servomotor drives. Before designing the controllers, it is of interest to describe the dynamics of PM synchronous servomotor drives. The mathematical model 11 of PM synchronous servomotor drives can be described as follows:
冦
x_ 1 ⫽ x 2 x_ 2 ⫽ ⫺ y ⫽ x1
1 1 cT f ~ x2 ! ⫹ u ⫹ TL J J J
(12)
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where x 1 is the angular position (in rad), x 2 is the speed (in ras/s), u is the current reference (in A), TL is the load torque, J is the moment of inertia, and cT is the control gain coefficient. We assume that the dynamics of the current control loop can be neglected. f ~ x 2 ! is the nonlinear friction function, which is modeled as f ~ x 2 ! ⫽ a{arctan~ b{x 2 !
(13)
where a and b are the parameters. Next, we will discuss the more general problem. Choose the system’s model as n-order differential equations, that is,
冦
x_ i ⫽ x i⫹1 ,
1 ⱕ i ⱕ n ⫺1
x_ n ⫽ f ~ x! ⫹ g~ x!u ⫹ D~ x, t !
(14)
y ⫽ x1
where u 僆 R and y 僆 R represent the control input and the output of the system, respectively. x ⫽ ~ x 1 , x 2 , . . . , x n ! T 僆 R n is comprised of the states that are assumed to be available, the integer n denotes the dimension of the system. f ~ x! is an unknown smooth uncertain system function with f ~0! ⫽ 0, and g~ x! is an unknown smooth uncertain input gain function. f ~ x! and g~ x! may be nonrepeatable nonlinear functions. D~ x, t ! is a time-varying uncertainty coming from modeling errors and external disturbances of the system. Throughout the article, the following assumptions are introduced on System 14. Assumption 1. There exists an unknown positive constant p * such that ∀~t, x! 僆 R ⫹ ⫻ R n 6D~ x, t !6 ⱕ p * f~ x! where f~{! is a known nonnegative smooth function. Assumption 2. The sign of g~ x! is known, and there exists a constant bmin ⬎ 0 such that 6g~ x!6 ⱖ bmin , ∀x 僆 R n . The above assumption implies that smooth function g~ x! is strictly either positive or negative. From now onward, without loss of generality, we shall assume g~ x! ⱖ bmin ⬎ 0, ∀x 僆 R n . Assumption 1 is reasonable because g~ x! being away from zero is the controllable conditions of System 14. It should be emphasized that the lower bound bmin is only required for analytical purposes; its true value is not necessarily known. The primary goal of this article is to track a given reference signal yd ~t ! while keeping the states and control bounded. That is, the output tracking error e1 ⫽ y~t ! ⫺ yd ~t ! should be as small as possible. The given bounded reference signal yd ~t ! is generated from the following smooth model:
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x_ di ⫽ fdi ~ x d !,
1ⱕiⱕm
yd ⫽ xdi ,
nⱖm
(15)
where x d ⫽ @x d1 , x d2 , . . . , x dm # T 僆 R m are the states, yd 僆 R is the system output, and fdi ~•!s are known nonlinear functions. Assume that the states of the reference model remain bounded, that is, x d 僆 Vd , ∀t ⱖ 0. We can define the tracking error vector e~t ! ⫽ x~t ! ⫺ x d ~t !. Equation 14 can be transformed into
再
e_ i ⫽ ei⫹1 ,
1 ⱕ i ⱕ n ⫺1
(16)
e_ n ⫽ f ~ x! ⫹ g~ x!u ⫺ x dn ~t ! ⫹ D~ x, t !
In this article, we present a method for the adaptive robust control design for System 16 in the presence of unstructured uncertainties. Our design objective is to find an RAFC u~t ! with the following form: x_ ⫽ Ã~ x, j~ x!, e!,
x僆R
(17)
u~t ! ⫽ u~ x, j~ x!, e!
(18)
where j~ x! is the known fuzzy base function, which is designed in such a way that all the solutions of the closed-loop system 16, 17, and 18 are uniformly ultimately bounded. Furthermore, the output tracking error e1 ~t ! can be steered to a small neighborhood of origin. 4.
DESIGN OF THE ROBUST ADAPTIVE FUZZY CONTROLLER 4.1.
Controller Design Procedure
Using the pole-placement approach, we consider a term k T x where k T ⫽ @k 1 , k 2 , . . . , k n # , the k i s are chosen such that all roots of polynomial s n ⫹ k n s n⫺1 ⫹ {{{ ⫹ k 1 lie in the left-half complex plane, leading to the exponentially stable dynamics. Then Equation 16 can be transformed into e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ f ~ x! ⫺ x dn ~t ! ⫹ k T e ⫹ D~ x, t !#
(19)
where
A⫽
冤
0
1
J
0
0
0
J
0
I
I
I
I
0
0
J
1
⫺k 1
⫺k 2
J
⫺k n
冥 冤冥 0 0
,
B⫽ I 0 1
Because A is stable, a positive definite solution P ⫽ P T of the Lyapunov equation A T P ⫹ PA ⫹ Q ⫽ 0 always exists and Q ⬎ 0 is specified by the designer.
(20)
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For this control problem, if both functions f ~ x! and g~ x! in Equation 19 are available for feedback, the technique of the feedback linearization can be used to design a well-defined controller, which is usually given in the form of u ⫽ g⫺1 ~ x!~⫺k T e ⫺ f ~ x! ⫹ x dn ~t ! ⫹ u s ! for some auxiliary control input u s with g~ x! being nonzero for all time, such that the resulting closed-loop system can be shown to achieve a satisfactory tracking performance. However, in many practical control systems, plant uncertainties that contain structured (or parametric) uncertainties and unstructured uncertainties (or nonrepeatable uncertainties) are inevitable. Hence both f ~ x! and g~ x! may not be available directly in the robust control design. Obtaining a simple control algorithm as above is impossible. Moreover, if any adaptation scheme is implemented to estimate f ~ x! and g~ x! as f Z ~ x! and g~ [ x!, respectively, the simple control algorithm aforementioned can be also used for substituting f Z ~ x! and g~ [ x! for f ~ x! and g~ x!, so the extra precaution is required to guarantee that g~ [ x! ⫽ 0 for all time. At the present stage, no effective method is available in the literature. In this article, we develop a semiglobally stable adaptive fuzzy robust controller that does not require estimating the unknown function g~ x!, and therefore avoids the possible controller singularity problem. In this article, the effects due to plant uncertainties and external disturbances will be considered simultaneously. In the philosophy of our tracking controller design it is expected that the T-S fuzzy approximators equipped with adaptive algorithms are introduced first to learn the behaviors of uncertain dynamics. Here, only uncertain function f ~ x! needs to be considered. For f ~ x! is an unknown continuous function; by Lemma 1, the T-S fuzzy system f Z ~ x, A x ! with input vector x 僆 Ux for some compact set Ux 債 R n is proposed here to approximate the uncertain term f ~ x! where A x is a matrix containing the approximating parameters. Then f ~ x! can be expressed as f ~ x! ⫽ j~ x!A x x ⫹ « ⫽ j~ x!A x e ⫹ j~ x!A x x d ~t ! ⫹ «
(21)
where x ⫽ e ⫹ x d ~t ! and « is a parameter with respect to approximating accuracy. Substituting Equation 21 into 19, we get e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ Bj~ x!A x e
(22)
where d ⫽ ⫺x dn ~t ! ⫹ k T e ⫹ j~ x!A x x d ~t ! ⫹ « ⫹ D~ x, t !. T m ⫺1 m Let cu ⫽ 7A x 7 ⫽ l1/2 max ~A x A x !, such that A x ⫽ cu A x and 7A x 7 ⱕ 1. It follows that Equation 22 reduces to e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ cu Bj~ x!Amx e
(23)
To design the robust adaptive fuzzy controller easily by use of the small gain theorem, the following output equation can be obtained by comparing Equation 23 with Equation 5: zI ⫽ H~e! ⫽e
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Then the feedback equation is given as follows: S vzI : v ⫽ K~ z! I ⫽ Amz zI ⫽ Amx e So Equation 23 can be rewritten as Equations 5 and 6: S zv I :
再
e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ Bcu j~ x!v zI ⫽ H~e! ⫽ e
(24)
I ⫽ Amx zI ⫽ Amx e S vzI : v ⫽ K~ z!
(25)
Then the feedback connection using Equations 24 and 25 can be implemented using the block diagram shown in Figure 2. From Figure 2, we observe that the system S Izv should be made to satisfy the ISpS condition of the system through designing the controller u ⫽ U~e!. In Equation 24, cu is an unknown, and there exist some parameters with boundedness. According to this property, an adaptive fuzzy robust tracking control algorithm will be proposed that not only gives the controller u ⫽ U~e! to make the system S Izv meet the ISpS condition but also the on-line adaptive law for cu and the other parameters in Equation 24. Based on the aforementioned condition, we can get 7d7 ⱕ d1 ⫹ ~7k 7 ⫹ 7A x 7 7j~ x!7!7xd ~t !7 ⫹ 7k 7 7 x7 ⫹ p * f~ x! ⫹ « ⱕ uc~ x! (26) T where c~ x! ⫽ 1 ⫹ 7 x7 ⫹ 7j~ x!7 ⫹ f~ x!, 7A x 7 ⫽ l1/2 max ~A x A x !, 7 x dn ~t !7 ⱕ * d1 , and u ⫽ max~d1 ,7k 7 7xd ~t !7,7A x 7 7xd ~t !7, p , «!. u denotes the largest term with unknown constant in all boundedness.
Figure 2.
Feedback connection of the fuzzy system.
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4.2.
Stability Analysis
In this subsection, we will prove the main result proposed in this article. Theorem 2. Consider System 19; suppose that Assumptions 1 and 2 are satisfied and the f ~ x! can be approximated by the T-S fuzzy system. If we pick g ⬍ 1, which is the gain of S Izv , there exists a positive constant r and l min ~Q! ⬎ 2 in Equation 20, and then a tracking-based robust adaptive fuzzy control scheme is proposed as follows: u ⫽ ⫺lq~ Z x!B T Pe where q~ x! ⫽
冉
1 1 j~ x!j T ~ x! ⫹ c 2 ~ x! 4g 2 4r 2
(27)
冊
and the adaptive law for lZ is now chosen as l^ Z ⫽ G@q~ x!e T PBB T Pe ⫺ s~ lZ ⫺ l 0 !#
(28)
which can make all the solutions ~e~t !, l! Z of the derived closed loop system uniformly ultimately bounded. Furthermore, given any m ⬎ 0, we can tune our controller parameters such that the output error e1 ⫽ y~t ! ⫺ yd ~t ! satisfies lim tr` 6e1 ~t !6 ⱕ m. In Equation 28, G ⬎ 0 is an updating rate, s and l 0 are design constants, which are chosen by the designer, respectively. Proof. The proof of Theorem 2 can be divided into two. First, let the constant g ⬎ 0 and set v ⫽ Amx e be the input of the system S Izv , to prove the satisfaction of ISpS for the system S Izv by use of the adaptive fuzzy robust tracking controller, and then to prove uniform ultimate boundedness of the composite of two systems with the feedback system v ⫽ Amx e by use of small gain theorem. Choose the Lyapunov function as V⫽
1 1 T e Pe ⫹ bmin G ⫺1 lD 2 2 2
(29)
where lD ⫽ ~l ⫺ l!. Z The time derivative of V along the error trajectory 24 is V^ ⫽
1 T T e ~A P ⫹ PA!e ⫺ bmin G ⫺1 lD lZ^ ⫹ e T PB@ g~ x!u ⫹ d # ⫹ e T PBcu j~ x!v 2 (30)
We deal with relative items in Equation 30, substitute Equation 27 into the relative items above, and obtain Z x!q~ x!e T PBB T Pe e T PBg~ x!u ⫽ ⫺lg~ ⱕ ⫺bmin lq~ Z x!e T PBB T Pe
(31)
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and e T PBcu j~ x!v ⫺ g 2 7v7 2 ⫹ g 2 7v7 2 ⫽ ⫺g 2 7v ⫺ ⫹ ⱕ
cu T j ~ x!B T Pe7 2 2g 2
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2 (32)
and substituting Equation 26 into the relative items of Equation 30, by means of the Schwarz inequality, there exists a nonnegative constant r; it yields e T PBd ⱕ 7e T PB7 7d7 ⱕ e T PB7uc~ x! ⱕ
u2 2 c ~ x!e T PBB T Pe ⫹ r 2 4r 2
(33)
We can get e T PBcu j~ x!v ⫹ e T PBd ⱕ
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2 ⫹
u2 2 c ~ x!e T PBB T Pe ⫹ r 2 4r 2
ⱕ bmin lq~ x!e T PBB T Pe ⫹ g 2 7v7 2 ⫹ r 2 ⱕ bmin lq~ Z x!e T PBB T Pe ⫹ bmin lq~ D x!e T PBB T Pe ⫹ g 2 7v7 2 ⫹ r 2
(34)
⫺1 2 ⫺1 2 cu , bmin u !. max~bmin
where l ⫽ Substitute Equation 34 into Equation 30 such that 1 D x!e T PBB T Pe ⫺ l! Z^ ⫹ g 2 7v7 2 ⫹ r 2 V^ ⱕ ⫺ e T Qe ⫹ bmin G ⫺1 l~Gq~ 2
(35)
Using Equation 28, we get 1 D lZ ⫺ l 0 ! ⫹ g 2 7v7 2 ⫹ r 2 V^ ⱕ ⫺ e T Qe ⫹ sbmin l~ 2 1 1 ⱕ ⫺ e T Qe ⫺ sbmin lD 2 ⫹ g 2 7v7 2 ⫹ d1 2 2 where d1 ⫽ _12 bmin s~l ⫺ l 0 ! 2 ⫹ r 2 . If picking l min ~Q! ⬎ 2, we get V^ ⱕ ⫺7e7 2 ⫹ 2g 2 7v7 2 ⫹ d1
(36)
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By Definition 2, we propose the robust adaptive fuzzy tracking controller such that the requirement of ISpS for system S Izv can be satisfied with the functions a3 ~s! ⫽ s 2 and a4 ~s! ⫽ 2g 2 s 2 of class K` . By Definition 1 and Equation 3, we can get a gain function gz ~s! of system S Izv as follows: gz ~s! ⫽ a1⫺1 ° a2 ° a3⫺1 ° a4 ,
∀s ⬎ 0
where a1 ~z! ⱕ V~z! ⱕ a2 ~z!. For the subsystem S v Iz , it is a static system such that we have 7v7 ⱕ 7Amx 7 7 z7 I ⫽ g1 7 z7 I ⫽ g1 7e7
(37)
Then the gain function gv for the system S v Iz is gv ~s! ⫽ g1 s. According to the requirement gz ~gv ~s!! ⬍ s of small gain Theorem 1, we can get gg1 ⬍ 1. Because g1 ⫽ 7Amx 7 ⱕ 1, when choosing g ⬍ 1, the condition of the small gain Theorem 1 can be satisfied, so that it can be proven that the composite closed-loop system is ISpS. Therefore, a direct application of Proposition 1 yields that the composite closed-loop system has bounded solutions over [0,`). More precisely, there exist a class KL-function b and a positive constant d such that 7e~t !, l~t Z !7 ⱕ b~7e~0!, l~0!7, Z t! ⫹ d This, in turn, implies that the tracking error e~t ! is bounded over [0,`). By Proposition 1, there exists an ISpS–Lyapunov function for the composite closed-loop system. By substituting Equation 36 into Equation 35, the ISpS–Lyapunov function is satisfied as follows: 1 1 V^ ⱕ ⫺ e T Qe ⫺ sbmin lD 2 ⫹ g12 g 2 7e7 2 2 2 1 1 ⱕ ⫺ e T ~Q ⫺ 2In⫻n !e ⫺ sbmin lD 2 ⫹ d1 2 2 ⱕ ⫺c1 V ⫹ d1
(38)
It shows that the solutions of the composite closed-loop system are uniformly ultimately bounded, and implies that, for any m 1 ⬍ ~d1 /c1 !1/2 , there exists a constant T ⬎ 0 such that 7e1 ~t !7 ⱕ m 1 for all t ⱖ t0 ⫹ T. Note that ~d1 /c1 !1/2 can be made arbitrarily small if the design parameters l 0 , s, and r are chosen 䡲 appropriately. Remark 1. Because the function approximation property of fuzzy systems is only guaranteed within a compact set, the stability result proposed in this article is semiglobal in the sense that, for any compact set, there exists a controller with a sufficiently large number of fuzzy rules such that all the closed-loop signals are bounded when the initial states are within this compact set. In practical applications, the number of fuzzy rules usually cannot be chosen too large due to the possible computation problem. This implies that the fuzzy system approximation capability is limited; that is, the approximating accuracy « in Equation 21 for the estimated function f ~ x! will be greater when a small number of fuzzy rules is chosen. As an
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alternative solution, we could choose appropriately the design parameters l 0 , s, r to improve both stability and performance of the closed-loop systems.
5.
APPLICATION OF ROBUST ADAPTIVE FUZZY CONTROL FOR PM SYNCHRONOUS SERVOMOTOR DRIVES
The control objective in this article is to make the position of the PM synchronous servomotor track a perfect reference signal. The reference model is chosen so as to represent somewhat realistic performance requirements as y] d ⫽ 400w ⫺ 400yd ⫺ 40 y_ d
(39)
where w is the controller input. Without loss of generality, we assume that the function f ~ x 2 ! in Equation 12 can be defined in the function f ~ x 1 , x 2 !, which is unknown with a continuous complicated formulation system function; the T-S fuzzy system can be constructed to approximate the function f ~ x 1 , x 2 ! by the nine fuzzy IF-THEN rules in Table I. In Table I, we select yi ⫽ a 1i x 1 ⫹ a 2i x 2 , i ⫽ 1,2, . . . ,9. P denotes the fuzzy set “ Positive,” Z for “Zero,” and N for “Negative”. They can be characterized by the membership functions as follows: m positive ~ x! ⫽
1 1 ⫹ exp~⫺4~ x ⫺ p/2!!
m zero ~ x! ⫽ exp~⫺x 2 ! m negative ~ x! ⫽
1 1 ⫹ exp~⫺4~ x ⫹ p/2!!
Using the center average defuzzifier and the product inference engine, the fuzzy system is obtained as follows: f ~ x! ⫽ j~ x!A x x ⫹ «
(40)
Table I. Fuzzy IF-THEN rules. x2 x1
N
Z
P
N Z P
y1 y4 y7
y2 y5 y8
y3 y6 y9
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where
Ax ⫽
冤 冥 a 11
a 21
I
I
a 19
a 29
j~ x! can be defined as Equation 12. Let k 1 ⫽ 1, k 2 ⫽ 25, and Q ⫽ diag(2,2); then the solution of Lyapunov expression 20 is obtained by P⫽
冋
25.08
1.00
1.00
0.09
册
If the gain is g ⫽ 0.5 and design constant is r ⫽ 0.5, the robust adaptive fuzzy control scheme can be obtained for PM synchronous servomotor drives as follows:
冉( 9
u ⫽ ⫺l~1.00e Z 1 ⫹ 0.08e2 !
ji2 ~ x! ⫹ c 2 ~ x!
i⫽1
冊
(41)
where e1 ⫽ x 1 ⫺ yd , e2 ⫽ x 2 ⫺ y_ d , c~ x! ⫽ ~1 ⫹ ~ x 12 ⫹ x 22 ! 1/2 ⫹ ~ ( 9i⫽1 ji2 ~ x!! 1/2 !,
冋
lZ^ ⫽ 0.01 ~1.00e1 ⫹ 0.08e2 ! 2
冉( 9
冊
册
ji2 ~ x! ⫹ c 2 ~ x! ⫺ 0.01~ lZ ⫺ 0.1!
i⫽1
The simulation results are shown in Figures 3– 6 using the Matlab toolbox. The plant parameters are chosen as cT ⫽ 10 Nm/A, J ⫽ 0.1 kgm 2, a ⫽ 5 Nm, and b ⫽ 5 s/rad. To reveal the control performance of the proposed robust adaptive fuzzy control, there are two kinds of reference signal: w~t ! used to train the controller u~t !. The kind of reference signal w~t ! and corresponding simulation results are shown in Table II. In Table II, the sinusoidal signal is to change its value sinusoidally in the interval ~⫺p, p! with a frequency of 0.5 Hz. The band-limited white noise signal has the parameter with noise power 1 and sample time 1. We give the initial error p between position and reference signal in the simulation. From Figures 3 and 6, we can see that fairly good tracking performance is obtained. The boundedness of control signal u and adaptive parameter lZ are shown in Figures 4 and 6, respectively.
Table II. Corresponding simulation results.
Simulation 1 Simulation 2
Reference signal
Figures for simulation results
Sinusoidal signal Band-limited white noise signal
Figures 3, 4 Figures 5, 6
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Figure 3. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Servomotor drive position y (solid line) and reference signal yd (dashed line); (b) Tracking error between real position and reference signal.
Figure 4. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Control u; (b) Adaptive parameter l. Z
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Figure 5. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Servomotor drive position y (solid line) and reference signal yd (dashed line); (b) Tracking error between real position and reference signal.
Figure 6. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Control u; (b) Adaptive parameter l. Z
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6.
CONCLUDING REMARKS
The design of robust adaptive fuzzy tracking control for a class of uncertain nonlinear systems has been presented and then applied to PM synchronous servomotor drives. In accordance with the unstructured state-dependent (or nonrepeatable) unknown system and gain nonlinear functions, and Takagi–Sugeno-type fuzzy logic systems have been used to approximate unknown system function and a robust adaptive fuzzy tracking control (RAFC) algorithm, that can guarantee the closedloop in the presence of nonrepeatable uncertainties is semiglobally uniformly ultimately bounded, and the tracking error of the system can be steered to a small neighborhood around e~t ! ⫽ 0, has been achieved by use of the input-to-state stability and general small gain approach. The outstanding features of the algorithm proposed in this article are that (1) it can avoid the possible controller singularity problem in some of existing adaptive control schemes with feedback linearization techniques; (2) the adaptive mechanism can be obtained, which has only one learning parameterization to be adapted on-line, no matter how many states in the system are investigated and how many rules in the fuzzy system are used. Thus, the computation load of the algorithm can be reduced and it is easier to implement this algorithm in real-time applications. To verify the effectiveness of the proposed algorithm, it has been applied to PM synchronous servomotor drive system. In accordance with unknown parameters of PM synchronous servomotor drive model and unknown structure of uncertain function of the system, the T-S fuzzy system is used to approximate the uncertain function; then a RAFC scheme for the position control of PM synchronous servomotor is designed. Simulation results have shown the effectiveness of the proposed control scheme. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant No. 60474014, the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20020151005, and the Science Foundation of the National Ministry of Communications of China under Grant No. 95-06-02-22.
References 1. 2. 3. 4. 5. 6. 7.
Leonnard W. Microcomputer control of high dynamic performance AC drives, A survey. Automatica 1986;22:1–19. Kokotovic P, Arcak M. Constructive nonlinear control: A historical perspective. Automatica 2001;37:637– 662. Wang LX. A course in fuzzy systems and control. New York: Prentice-Hall Inc.; 1997. Wang LX. Fuzzy systems are universal approximators. In: Proc IEEE Int Conf on Fuzzy Systems, San Diego, CA, 1992. pp 1163–1170. Lee CC. Fuzzy logic in control systems: Fuzzy logic controller—Parts I & II. IEEE Trans Syst Man Cybern 1990;20:404– 435. Ying H. Sufficient conditions on general fuzzy systems as function approximators. Automatica 1994;30:521–525. Wang LX, Mendel JM. Fuzzy basis functions universal approximation, and orthogonal least-squares learning. IEEE Trans Neural Netw 1992;3:807–814.
ROBUST ADAPTIVE FUZZY CONTROL FOR PM SERVOMOTOR 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
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Wang LX. Stable adaptive fuzzy control of nonlinear systems. IEEE Trans Fuzzy Syst 1993;1:146–155. Su CY, Stepanenko Y. Adaptive control of a class of nonlinear systems with fuzzy logic. IEEE Trans Fuzzy Syst 1994;2:285–294. Spooner JT, Passino KM. Stable adaptive control using fuzzy systems and neural networks. IEEE Trans Fuzzy Syst 1996;4:339–359. Fischle K, Schroder D. An improved stable adaptive fuzzy control method. IEEE Trans Fuzzy Syst 1999;7:27– 40. Yang YS, Zhou C, Jia XL. Robust adaptive fuzzy control and its application to ship roll stabilization. Inform Sci 2002;142:177–194. Yang YS, Zhou C, Ren JS. Model reference adaptive robust fuzzy control for ship steering autopilot with uncertain nonlinear systems. Appl Soft Comput J 2003;3:305–316. Yang YS, Ren JS. Adaptive fuzzy robust tracking controller design via small gain approach and its application. IEEE Trans Fuzzy Syst 2003;11:783–795. Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 1985;15:116–132. Sontag ED. Smooth stabilization implies coprime factorization. IEEE Trans Autom Control 1989;34:1411–1428. Jiang ZP, Teel A, Praly L. Small-gain theorem for ISS systems and applications. Math Control Signals Syst 1994;7:95–120. Sontag ED. Further results about input-state stabilization. IEEE Trans Autom Control 1990;35:473– 476. Sontag ED, Wang Y. On characterizations of input-to-state stability property with respect to compact sets. In: Prep IFAC NOLCOS’95, Tahoe City, 1995. pp 226–231. Khalil HK. Nonlinear systems. New York: Macmillan; 1996. Sontag ED. On the input-to-state stability property. Eur J Control 1995;1:24–36.
A novel robust adaptive fuzzy control (RAFC) algorithm for the permanent magnet (PM) synchronous servomotor drives with uncertain nonlinearities and time-varying uncertainties is presented in this article. Takagi–Sugeno-type fuzzy logic systems are used to approximate uncertain functions. The RAFC algorithm is designed by use of the input-to-state stability (ISS) approach and small gain theorem. The closed-loop system is proven to be semiglobally uniformly ultimately bounded. In addition, the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be removed and the adaptive mechanism with only one learning parameterization can be achieved. The proposed methodology is applied to design the position control of the PM synchronous servomotor drives. Simulation results show the effectiveness of the proposed control scheme. © 2005 Wiley Periodicals, Inc.
1.
INTRODUCTION
Recently, controlling permanent magnet (PM) synchronous servomotor drives, which consist of magnetic materials, semiconductor power devices, and so on, has been an interesting research line in motion-control application in the low to medium power range. The desirable features of the PM synchronous servomotor are its compact structure, high air-gap flux density, high power density, high torque-toinertia ratio, and high torque capability. Moreover, compared with an induction servomotor, the PM synchronous servomotor has such advantages as higher efficiency, owing to the absence of rotor losses and lower no-load current below the rated speed, and its decoupling control performance is much less sensitive to the parameter variations of the motor.1 To achieve fast four-quadrant operation and *Author to whom all correspondence should be addressed: e-mail: [email protected]. † e-mail: [email protected]. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 20, 153–171 (2005) © 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). • DOI 10.1002/int.20060
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smooth starting and acceleration, the field-oriented control, or vector control, is used in the design of the PM synchronous servomotor drives. However, the control performance of PM synchronous servomotor drives is still influenced by the uncertainties of the plant, which usually features unpredictable plant parameter variations, external load disturbance, and unmodeled and nonlinear dynamics. There has been tremendous interest in designing a controller for systems having uncertain nonlinear systems, and much significant research attention has been attracted in the past decade. Most results addressing this problem are available in the control literature, for example, Ref. 2 and references therein. Applications of these approaches are quite limited because they rely on the exact knowledge of the plant nonlinearities. To relax some of the exact model-matching restrictions, several adaptive schemes have recently been introduced to solve the problem of linearly parameterized uncertainties, which are referred to as structured uncertainties. Unfortunately, in the industrial control environment, there are some controlled systems that are characterized not only by the unstructured uncertainties, but are also represented by all the terms that cannot be modeled or are not repeatable. A solution to those problems was first presented by Wang 3 ; the fuzzy systems were proposed to uniformly approximate the uncertain nonlinear functions in the designed system by use of the universal approximation properties of certain classes of fuzzy systems presented in Refs. 4–7, and a Lyapunov based learning law was used. Then several stable adaptive fuzzy controllers that ensure the stability of the overall system were developed by Wang,8 Su and Stepanenko,9 Spooner and Passino,10 and Fischle and Schroder.11 The more application-motivated problem of adaptive fuzzy control for the uncertain nonlinear systems has gradually gained much attention (see, e.g., Refs. 12 and 13). However, there is a substantial restriction in the aforementioned works: Many parameters need to be tuned in the learning laws when there are many state variables in the designed system and many rule bases have to be used in the fuzzy system for approximating the nonlinear uncertain functions, so that the learning times tend to become unacceptably long for the systems of higher order, and a time-consuming process is unavoidable when the fuzzy logic controllers are implemented. This problem has been pointed out by Fischle and Schroder 11 and first researched by Yang and Ren.14 In this article, we present a novel robust adaptive fuzzy controller for a class of continuous uncertain systems with uncertain nonlinearities and timevarying uncertainties, and Takagi–Sugeno-type fuzzy logic systems 15 are used to approximate the unknown unstructured uncertain nonlinear functions in the system, and the adaptive mechanism with minimal learning parameterizations is proposed by use of input-to-state stability (ISS) theory first proposed by Sontag 16 and the nonlinear small gain approach given by Jiang et al.17 The main features of the algorithm proposed in the article are: (i) the closed-loop system is proven to be semiglobally uniformly ultimately bounded; (ii) only one function needs to be approximated by the T-S fuzzy systems, and no matter how many states in the designed system are investigated and how many rules in the fuzzy system are used, only one parameter needs to be adapted on-line, such that the burdensome computation of the algorithm can be lightened increasingly and it is convenient
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to implement this algorithm in real-time application; (iii) the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be avoided. The rest of this article is organized as follows. In Section 2, we give some needed definitions of input-state stability, small gain theorem, and preliminary results. Section 3 contains the description of position control of PM synchronous servomotor drives. In Section 4, a systematic procedure for the synthesis of robust adaptive fuzzy control (RAFC) is developed. In Section 5, the application examples for a PM synchronous servomotor drive system by use of the RAFC are shown and numerical simulation results are presented. Concluding remarks are given in Section 6. 2.
MATHEMATICAL PRELIMINARIES
Notation: Throughout this article, let 7 •7 be any suitable norm. The vector norm of x 僆 R n is Euclidean, that is, 7 x7 2 ⫽ ~ x T x! and the matrix norm of A 僆 R n⫻m is defined by 7A7 2 ⫽ l max ~AT A!, where l max(min) (•) denotes the operation of taking the maximum (minimum) eigenvalue. The vector norm over the space defined by stacking the matrix columns into a vector, so that it is compatible with the vector norm, that is, 7Ax7 ⱕ 7A7•7 x7. For any piecewise continuous function u : R ⫹ r R m, 7u 7` denotes sup $6u~t !6, t ⱖ 0%, which stands for L ` supremum norm, and for any pair of times 0 ⱕ t1 ⱕ t2 , the truncation u @t1 , t2 # is a function defined on R ⫹ that is equal to u~t ! on @t1 , t2 # and is zero outside the interval. In particular, u @0, t# is the usual truncated function u t . 2.1.
ISS and Small Gain Theorem
The concepts of ISS and ISS–Lyapunov function proposed in Refs. 16, 18, and 19 have recently been used in various control problems. To ease the discussion of the design of the RAFC scheme, the variants of those notions are reviewed in the following. First, we begin with the definitions of class K, K` , and KL functions that are standard in the stability literature (see Ref. 20). A class K-function g is a continuous, strictly increasing function from R ⫹ into R ⫹ and g~0! ⫽ 0. It is of class K` if additionally g~s! r ` as s r `. A function b : R ⫹ ⫻ R ⫹ r R ⫹ is of class KL if b~•, t ! is of class K for every t ⱖ 0 and b~s, t ! r 0 as t r `. Definition 1. For the system x_ ⫽ f ~ x, u!, it is said to be input-to-state practically stable (ISpS) if there exists a function g of class K, called the nonlinear L ` gain, and a function b of class KL such that, for any initial condition x~0!, each measurable essentially bounded control u~t ! defined for all t ⱖ 0 and a nonnegative constant d, the associated solutions x~t ! are defined on @0,`! and satisfy 7 x~t !7 ⱕ b~7 x~0!7, t ! ⫹ g~7u t 7` ! ⫹ d
(1)
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When d ⫽ 0 in Equation 1, the ISpS property collapses to the input-to-state stability (ISS) property introduced in Ref. 18. Definition 2. A C 1 function V is said to be an ISpS–Lyapunov function for the system x_ ⫽ f ~ x, u! if
•
there exist functions a1 , a2 of class K` such that a1 ~7 x7! ⱕ V~ x! ⱕ a2 ~7 x7!,
•
∀x 僆 R n
(2)
there exist functions a3 , a4 of class K and a constant d ⬎ 0 such that ]V~ x! ]x
f ~ x, u! ⱕ ⫺a3 ~7 x7! ⫹ a4 ~7 u7! ⫹ d
(3)
When Equation 3 holds with d ⫽ 0, V is referred to as an ISS–Lyapunov function. Then it holds that one may pick a nonlinear L ` gain g in Equation 1 of the form, which is given in Ref. 21: g~s! ⫽ a1⫺1 ° a2 ° a3⫺1 ° a4 ,
∀s ⬎ 0
(4)
The following proposition establishes equivalence between ISpS and the existence of ISpS–Lyapunov function by Ref. 19. Proposition 1. The system x_ ⫽ f ~ x, u! is ISpS if and only if there exists an ISpS–Lyapunov function. Consider the stability of the closed-loop interconnection of two systems as shown in Figure 1. A trivial refinement of the proof of the generalized small gain theorem given in Ref. 17 yields the following variant, which is suited for our applications presented in this article.
Figure 1.
Feedback connection of interconnection systems.
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Theorem 1. Consider a system in composite feedback form of two ISpS systems (cf. Figure 1): S zv I : S vzI :
再 再
x_ ⫽ f ~ x, v!
(5)
zI ⫽ H~ x!
y_ ⫽ g~ y, z! I
(6)
v ⫽ K~ y, z! I
In particular, there exist two constants d1 ⬎ 0, d2 ⬎ 0, and let bv and bj be of class KL, and gz and gv of class K, such that, for each v in the L ` supremum norm, each zI in the L ` supremum norm, each x 僆 R n and each y 僆 R m , all the solutions X~ x; v, t ! and Y~ y; z,I t ! are defined on @0,`! and satisfy, for almost all t ⱖ 0 7H~X~ x; v, t !!7 ⱕ bv ~7 x7, t ! ⫹ gz ~7vt 7` ! ⫹ d1
(7)
7K~Y~ y; z,I t !!7 ⱕ bj ~7y7, t ! ⫹ gv ~6 zI t 7` ! ⫹ d2
(8)
Under these conditions, if gz ~gv ~s!! ⬍ s ~resp. gv ~gz ~s!! ⬍ s!
∀s ⬎ 0
(9)
the solution of the composite systems 5 and 6 is ISpS. 2.2.
Takagi–Sugeno ~T-S! Fuzzy Systems
In this subsection, we briefly describe the structure of fuzzy systems to be used in this article. Let R denote the real numbers, R n the real n-vectors, and R n⫻m the real n ⫻ m matrices. Let S be a compact simply connected set in R n . With map f : S r R m , define C m ~S! to be the function space such that f is continuous. A fuzzy system can be employed to approximate the function f ~ x! in order to design the adaptive fuzzy robust control law. The configuration of Takagi–Sugeno-type fuzzy logic system (T-S fuzzy system)15 and its approximation theorem are discussed in the following. Consider a T-S fuzzy system to uniformly approximate a continuous multidimensional function y ⫽ f ~ x! that has a complicated formulation, where x is an input vector with n independent x ⫽ ~ x 1 , x 2 , . . . , x n ! T . The domain of x i is ui ⫽ @a i , bi # . It follows that the domain of x is Q ⫽ u1 ⫻ u2 ⫻ {{{ ⫻ un ⫽ @a 1 , b1 # ⫻ @a 2 , b2 # ⫻ {{{ ⫻ @a n , bn # To construct a fuzzy system, the interval @a i , bi # is divided into Ni subintervals: a i ⫽ C0i ⬍ C1i ⬍ . . . CNi n⫺1 ⬍ CNi n ⫽ bi ,
1ⱕiⱕn
On each interval ui ~1 ⱕ i ⱕ n!, Ni ⫹ 1~Ni ⬎ 0! continuous input fuzzy sets, denoted by Aij ~0 ⱕ j ⱕ Ni !, are defined to fuzzify x i . The membership function of Aij is denoted by m ij ~ xi !, which can be represented by triangular, trapezoid, generalized bell, or Gaussian type and so on.
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Generally, the T-S fuzzy system can be constructed by the following K~K ⬎ 1! fuzzy rules: R i : If x 1 is Aih1 AND x 2 is Aih2 AND . . . AND x n is Aihn THEN yi is a 1i x 1 ⫹ {{{ ⫹ a ni x n ,
i ⫽ 1,2, . . . , K
where a ji , j ⫽ 1,2, . . . , n, i ⫽ 1,2, . . . , K are the unknown constants. The product fuzzy inference is employed to evaluate the ANDs in the fuzzy rules. After being defuzzified by a typical center average defuzzifier, the output of the fuzzy system is K
F~ x! ⫽
( yi ji ~ x! ⫽ j~ x!A x x
(10)
i⫽1
where yi ⫽ a 1i x 1 ⫹ {{{ ⫹ a ni x n and ji ~ x! ⫽ ) nj⫽1 m ihj ~ x j !/ ( Ki⫽1 @ ) nj⫽1 m ihj ~ x j !# , which is called a fuzzy base function. And j~ x! ⫽ @j1 ~ x!, j2 ~ x!, . . . , jn ~ x!# , x ⫽ @x 1 , x 2 , . . . , x n # T,
Ax ⫽
冤
a 11
a 21
J
a n1
a 12
a 22
J
a n2
I
I
I
I
a 1K
a 2K
J
a nK
冥
The following lemma 3 gives that the above fuzzy logic system has been shown to be capable of uniformly approximating any well-defined nonlinear function over a compact set to any degree of accuracy. Lemma 1. Suppose that the input universal of discourse U is a compact set in R r . Then, for any given real continuous function f ~ x! on U and ∀« ⬎ 0, there exists a fuzzy system F~ x! in the form of Equation 10 such that sup 7 f ~ x! ⫺ F~ x!7 ⫽ sup 7 f ~ x! ⫺ j~ x!A Z Z7 ⱕ « x僆U
3.
(11)
x僆U
POSITION CONTROL OF PM SYNCHRONOUS SERVOMOTOR DRIVES
In this article, we consider the position control design for PM synchronous servomotor drives. Before designing the controllers, it is of interest to describe the dynamics of PM synchronous servomotor drives. The mathematical model 11 of PM synchronous servomotor drives can be described as follows:
冦
x_ 1 ⫽ x 2 x_ 2 ⫽ ⫺ y ⫽ x1
1 1 cT f ~ x2 ! ⫹ u ⫹ TL J J J
(12)
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where x 1 is the angular position (in rad), x 2 is the speed (in ras/s), u is the current reference (in A), TL is the load torque, J is the moment of inertia, and cT is the control gain coefficient. We assume that the dynamics of the current control loop can be neglected. f ~ x 2 ! is the nonlinear friction function, which is modeled as f ~ x 2 ! ⫽ a{arctan~ b{x 2 !
(13)
where a and b are the parameters. Next, we will discuss the more general problem. Choose the system’s model as n-order differential equations, that is,
冦
x_ i ⫽ x i⫹1 ,
1 ⱕ i ⱕ n ⫺1
x_ n ⫽ f ~ x! ⫹ g~ x!u ⫹ D~ x, t !
(14)
y ⫽ x1
where u 僆 R and y 僆 R represent the control input and the output of the system, respectively. x ⫽ ~ x 1 , x 2 , . . . , x n ! T 僆 R n is comprised of the states that are assumed to be available, the integer n denotes the dimension of the system. f ~ x! is an unknown smooth uncertain system function with f ~0! ⫽ 0, and g~ x! is an unknown smooth uncertain input gain function. f ~ x! and g~ x! may be nonrepeatable nonlinear functions. D~ x, t ! is a time-varying uncertainty coming from modeling errors and external disturbances of the system. Throughout the article, the following assumptions are introduced on System 14. Assumption 1. There exists an unknown positive constant p * such that ∀~t, x! 僆 R ⫹ ⫻ R n 6D~ x, t !6 ⱕ p * f~ x! where f~{! is a known nonnegative smooth function. Assumption 2. The sign of g~ x! is known, and there exists a constant bmin ⬎ 0 such that 6g~ x!6 ⱖ bmin , ∀x 僆 R n . The above assumption implies that smooth function g~ x! is strictly either positive or negative. From now onward, without loss of generality, we shall assume g~ x! ⱖ bmin ⬎ 0, ∀x 僆 R n . Assumption 1 is reasonable because g~ x! being away from zero is the controllable conditions of System 14. It should be emphasized that the lower bound bmin is only required for analytical purposes; its true value is not necessarily known. The primary goal of this article is to track a given reference signal yd ~t ! while keeping the states and control bounded. That is, the output tracking error e1 ⫽ y~t ! ⫺ yd ~t ! should be as small as possible. The given bounded reference signal yd ~t ! is generated from the following smooth model:
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x_ di ⫽ fdi ~ x d !,
1ⱕiⱕm
yd ⫽ xdi ,
nⱖm
(15)
where x d ⫽ @x d1 , x d2 , . . . , x dm # T 僆 R m are the states, yd 僆 R is the system output, and fdi ~•!s are known nonlinear functions. Assume that the states of the reference model remain bounded, that is, x d 僆 Vd , ∀t ⱖ 0. We can define the tracking error vector e~t ! ⫽ x~t ! ⫺ x d ~t !. Equation 14 can be transformed into
再
e_ i ⫽ ei⫹1 ,
1 ⱕ i ⱕ n ⫺1
(16)
e_ n ⫽ f ~ x! ⫹ g~ x!u ⫺ x dn ~t ! ⫹ D~ x, t !
In this article, we present a method for the adaptive robust control design for System 16 in the presence of unstructured uncertainties. Our design objective is to find an RAFC u~t ! with the following form: x_ ⫽ Ã~ x, j~ x!, e!,
x僆R
(17)
u~t ! ⫽ u~ x, j~ x!, e!
(18)
where j~ x! is the known fuzzy base function, which is designed in such a way that all the solutions of the closed-loop system 16, 17, and 18 are uniformly ultimately bounded. Furthermore, the output tracking error e1 ~t ! can be steered to a small neighborhood of origin. 4.
DESIGN OF THE ROBUST ADAPTIVE FUZZY CONTROLLER 4.1.
Controller Design Procedure
Using the pole-placement approach, we consider a term k T x where k T ⫽ @k 1 , k 2 , . . . , k n # , the k i s are chosen such that all roots of polynomial s n ⫹ k n s n⫺1 ⫹ {{{ ⫹ k 1 lie in the left-half complex plane, leading to the exponentially stable dynamics. Then Equation 16 can be transformed into e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ f ~ x! ⫺ x dn ~t ! ⫹ k T e ⫹ D~ x, t !#
(19)
where
A⫽
冤
0
1
J
0
0
0
J
0
I
I
I
I
0
0
J
1
⫺k 1
⫺k 2
J
⫺k n
冥 冤冥 0 0
,
B⫽ I 0 1
Because A is stable, a positive definite solution P ⫽ P T of the Lyapunov equation A T P ⫹ PA ⫹ Q ⫽ 0 always exists and Q ⬎ 0 is specified by the designer.
(20)
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For this control problem, if both functions f ~ x! and g~ x! in Equation 19 are available for feedback, the technique of the feedback linearization can be used to design a well-defined controller, which is usually given in the form of u ⫽ g⫺1 ~ x!~⫺k T e ⫺ f ~ x! ⫹ x dn ~t ! ⫹ u s ! for some auxiliary control input u s with g~ x! being nonzero for all time, such that the resulting closed-loop system can be shown to achieve a satisfactory tracking performance. However, in many practical control systems, plant uncertainties that contain structured (or parametric) uncertainties and unstructured uncertainties (or nonrepeatable uncertainties) are inevitable. Hence both f ~ x! and g~ x! may not be available directly in the robust control design. Obtaining a simple control algorithm as above is impossible. Moreover, if any adaptation scheme is implemented to estimate f ~ x! and g~ x! as f Z ~ x! and g~ [ x!, respectively, the simple control algorithm aforementioned can be also used for substituting f Z ~ x! and g~ [ x! for f ~ x! and g~ x!, so the extra precaution is required to guarantee that g~ [ x! ⫽ 0 for all time. At the present stage, no effective method is available in the literature. In this article, we develop a semiglobally stable adaptive fuzzy robust controller that does not require estimating the unknown function g~ x!, and therefore avoids the possible controller singularity problem. In this article, the effects due to plant uncertainties and external disturbances will be considered simultaneously. In the philosophy of our tracking controller design it is expected that the T-S fuzzy approximators equipped with adaptive algorithms are introduced first to learn the behaviors of uncertain dynamics. Here, only uncertain function f ~ x! needs to be considered. For f ~ x! is an unknown continuous function; by Lemma 1, the T-S fuzzy system f Z ~ x, A x ! with input vector x 僆 Ux for some compact set Ux 債 R n is proposed here to approximate the uncertain term f ~ x! where A x is a matrix containing the approximating parameters. Then f ~ x! can be expressed as f ~ x! ⫽ j~ x!A x x ⫹ « ⫽ j~ x!A x e ⫹ j~ x!A x x d ~t ! ⫹ «
(21)
where x ⫽ e ⫹ x d ~t ! and « is a parameter with respect to approximating accuracy. Substituting Equation 21 into 19, we get e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ Bj~ x!A x e
(22)
where d ⫽ ⫺x dn ~t ! ⫹ k T e ⫹ j~ x!A x x d ~t ! ⫹ « ⫹ D~ x, t !. T m ⫺1 m Let cu ⫽ 7A x 7 ⫽ l1/2 max ~A x A x !, such that A x ⫽ cu A x and 7A x 7 ⱕ 1. It follows that Equation 22 reduces to e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ cu Bj~ x!Amx e
(23)
To design the robust adaptive fuzzy controller easily by use of the small gain theorem, the following output equation can be obtained by comparing Equation 23 with Equation 5: zI ⫽ H~e! ⫽e
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Then the feedback equation is given as follows: S vzI : v ⫽ K~ z! I ⫽ Amz zI ⫽ Amx e So Equation 23 can be rewritten as Equations 5 and 6: S zv I :
再
e_ ⫽ Ae ⫹ B@ g~ x!u ⫹ d # ⫹ Bcu j~ x!v zI ⫽ H~e! ⫽ e
(24)
I ⫽ Amx zI ⫽ Amx e S vzI : v ⫽ K~ z!
(25)
Then the feedback connection using Equations 24 and 25 can be implemented using the block diagram shown in Figure 2. From Figure 2, we observe that the system S Izv should be made to satisfy the ISpS condition of the system through designing the controller u ⫽ U~e!. In Equation 24, cu is an unknown, and there exist some parameters with boundedness. According to this property, an adaptive fuzzy robust tracking control algorithm will be proposed that not only gives the controller u ⫽ U~e! to make the system S Izv meet the ISpS condition but also the on-line adaptive law for cu and the other parameters in Equation 24. Based on the aforementioned condition, we can get 7d7 ⱕ d1 ⫹ ~7k 7 ⫹ 7A x 7 7j~ x!7!7xd ~t !7 ⫹ 7k 7 7 x7 ⫹ p * f~ x! ⫹ « ⱕ uc~ x! (26) T where c~ x! ⫽ 1 ⫹ 7 x7 ⫹ 7j~ x!7 ⫹ f~ x!, 7A x 7 ⫽ l1/2 max ~A x A x !, 7 x dn ~t !7 ⱕ * d1 , and u ⫽ max~d1 ,7k 7 7xd ~t !7,7A x 7 7xd ~t !7, p , «!. u denotes the largest term with unknown constant in all boundedness.
Figure 2.
Feedback connection of the fuzzy system.
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ROBUST ADAPTIVE FUZZY CONTROL FOR PM SERVOMOTOR
4.2.
Stability Analysis
In this subsection, we will prove the main result proposed in this article. Theorem 2. Consider System 19; suppose that Assumptions 1 and 2 are satisfied and the f ~ x! can be approximated by the T-S fuzzy system. If we pick g ⬍ 1, which is the gain of S Izv , there exists a positive constant r and l min ~Q! ⬎ 2 in Equation 20, and then a tracking-based robust adaptive fuzzy control scheme is proposed as follows: u ⫽ ⫺lq~ Z x!B T Pe where q~ x! ⫽
冉
1 1 j~ x!j T ~ x! ⫹ c 2 ~ x! 4g 2 4r 2
(27)
冊
and the adaptive law for lZ is now chosen as l^ Z ⫽ G@q~ x!e T PBB T Pe ⫺ s~ lZ ⫺ l 0 !#
(28)
which can make all the solutions ~e~t !, l! Z of the derived closed loop system uniformly ultimately bounded. Furthermore, given any m ⬎ 0, we can tune our controller parameters such that the output error e1 ⫽ y~t ! ⫺ yd ~t ! satisfies lim tr` 6e1 ~t !6 ⱕ m. In Equation 28, G ⬎ 0 is an updating rate, s and l 0 are design constants, which are chosen by the designer, respectively. Proof. The proof of Theorem 2 can be divided into two. First, let the constant g ⬎ 0 and set v ⫽ Amx e be the input of the system S Izv , to prove the satisfaction of ISpS for the system S Izv by use of the adaptive fuzzy robust tracking controller, and then to prove uniform ultimate boundedness of the composite of two systems with the feedback system v ⫽ Amx e by use of small gain theorem. Choose the Lyapunov function as V⫽
1 1 T e Pe ⫹ bmin G ⫺1 lD 2 2 2
(29)
where lD ⫽ ~l ⫺ l!. Z The time derivative of V along the error trajectory 24 is V^ ⫽
1 T T e ~A P ⫹ PA!e ⫺ bmin G ⫺1 lD lZ^ ⫹ e T PB@ g~ x!u ⫹ d # ⫹ e T PBcu j~ x!v 2 (30)
We deal with relative items in Equation 30, substitute Equation 27 into the relative items above, and obtain Z x!q~ x!e T PBB T Pe e T PBg~ x!u ⫽ ⫺lg~ ⱕ ⫺bmin lq~ Z x!e T PBB T Pe
(31)
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and e T PBcu j~ x!v ⫺ g 2 7v7 2 ⫹ g 2 7v7 2 ⫽ ⫺g 2 7v ⫺ ⫹ ⱕ
cu T j ~ x!B T Pe7 2 2g 2
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2 (32)
and substituting Equation 26 into the relative items of Equation 30, by means of the Schwarz inequality, there exists a nonnegative constant r; it yields e T PBd ⱕ 7e T PB7 7d7 ⱕ e T PB7uc~ x! ⱕ
u2 2 c ~ x!e T PBB T Pe ⫹ r 2 4r 2
(33)
We can get e T PBcu j~ x!v ⫹ e T PBd ⱕ
cu2 T e PBj~ x!j T ~ x!B T Pe ⫹ g 2 7v7 2 4g 2 ⫹
u2 2 c ~ x!e T PBB T Pe ⫹ r 2 4r 2
ⱕ bmin lq~ x!e T PBB T Pe ⫹ g 2 7v7 2 ⫹ r 2 ⱕ bmin lq~ Z x!e T PBB T Pe ⫹ bmin lq~ D x!e T PBB T Pe ⫹ g 2 7v7 2 ⫹ r 2
(34)
⫺1 2 ⫺1 2 cu , bmin u !. max~bmin
where l ⫽ Substitute Equation 34 into Equation 30 such that 1 D x!e T PBB T Pe ⫺ l! Z^ ⫹ g 2 7v7 2 ⫹ r 2 V^ ⱕ ⫺ e T Qe ⫹ bmin G ⫺1 l~Gq~ 2
(35)
Using Equation 28, we get 1 D lZ ⫺ l 0 ! ⫹ g 2 7v7 2 ⫹ r 2 V^ ⱕ ⫺ e T Qe ⫹ sbmin l~ 2 1 1 ⱕ ⫺ e T Qe ⫺ sbmin lD 2 ⫹ g 2 7v7 2 ⫹ d1 2 2 where d1 ⫽ _12 bmin s~l ⫺ l 0 ! 2 ⫹ r 2 . If picking l min ~Q! ⬎ 2, we get V^ ⱕ ⫺7e7 2 ⫹ 2g 2 7v7 2 ⫹ d1
(36)
ROBUST ADAPTIVE FUZZY CONTROL FOR PM SERVOMOTOR
165
By Definition 2, we propose the robust adaptive fuzzy tracking controller such that the requirement of ISpS for system S Izv can be satisfied with the functions a3 ~s! ⫽ s 2 and a4 ~s! ⫽ 2g 2 s 2 of class K` . By Definition 1 and Equation 3, we can get a gain function gz ~s! of system S Izv as follows: gz ~s! ⫽ a1⫺1 ° a2 ° a3⫺1 ° a4 ,
∀s ⬎ 0
where a1 ~z! ⱕ V~z! ⱕ a2 ~z!. For the subsystem S v Iz , it is a static system such that we have 7v7 ⱕ 7Amx 7 7 z7 I ⫽ g1 7 z7 I ⫽ g1 7e7
(37)
Then the gain function gv for the system S v Iz is gv ~s! ⫽ g1 s. According to the requirement gz ~gv ~s!! ⬍ s of small gain Theorem 1, we can get gg1 ⬍ 1. Because g1 ⫽ 7Amx 7 ⱕ 1, when choosing g ⬍ 1, the condition of the small gain Theorem 1 can be satisfied, so that it can be proven that the composite closed-loop system is ISpS. Therefore, a direct application of Proposition 1 yields that the composite closed-loop system has bounded solutions over [0,`). More precisely, there exist a class KL-function b and a positive constant d such that 7e~t !, l~t Z !7 ⱕ b~7e~0!, l~0!7, Z t! ⫹ d This, in turn, implies that the tracking error e~t ! is bounded over [0,`). By Proposition 1, there exists an ISpS–Lyapunov function for the composite closed-loop system. By substituting Equation 36 into Equation 35, the ISpS–Lyapunov function is satisfied as follows: 1 1 V^ ⱕ ⫺ e T Qe ⫺ sbmin lD 2 ⫹ g12 g 2 7e7 2 2 2 1 1 ⱕ ⫺ e T ~Q ⫺ 2In⫻n !e ⫺ sbmin lD 2 ⫹ d1 2 2 ⱕ ⫺c1 V ⫹ d1
(38)
It shows that the solutions of the composite closed-loop system are uniformly ultimately bounded, and implies that, for any m 1 ⬍ ~d1 /c1 !1/2 , there exists a constant T ⬎ 0 such that 7e1 ~t !7 ⱕ m 1 for all t ⱖ t0 ⫹ T. Note that ~d1 /c1 !1/2 can be made arbitrarily small if the design parameters l 0 , s, and r are chosen 䡲 appropriately. Remark 1. Because the function approximation property of fuzzy systems is only guaranteed within a compact set, the stability result proposed in this article is semiglobal in the sense that, for any compact set, there exists a controller with a sufficiently large number of fuzzy rules such that all the closed-loop signals are bounded when the initial states are within this compact set. In practical applications, the number of fuzzy rules usually cannot be chosen too large due to the possible computation problem. This implies that the fuzzy system approximation capability is limited; that is, the approximating accuracy « in Equation 21 for the estimated function f ~ x! will be greater when a small number of fuzzy rules is chosen. As an
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alternative solution, we could choose appropriately the design parameters l 0 , s, r to improve both stability and performance of the closed-loop systems.
5.
APPLICATION OF ROBUST ADAPTIVE FUZZY CONTROL FOR PM SYNCHRONOUS SERVOMOTOR DRIVES
The control objective in this article is to make the position of the PM synchronous servomotor track a perfect reference signal. The reference model is chosen so as to represent somewhat realistic performance requirements as y] d ⫽ 400w ⫺ 400yd ⫺ 40 y_ d
(39)
where w is the controller input. Without loss of generality, we assume that the function f ~ x 2 ! in Equation 12 can be defined in the function f ~ x 1 , x 2 !, which is unknown with a continuous complicated formulation system function; the T-S fuzzy system can be constructed to approximate the function f ~ x 1 , x 2 ! by the nine fuzzy IF-THEN rules in Table I. In Table I, we select yi ⫽ a 1i x 1 ⫹ a 2i x 2 , i ⫽ 1,2, . . . ,9. P denotes the fuzzy set “ Positive,” Z for “Zero,” and N for “Negative”. They can be characterized by the membership functions as follows: m positive ~ x! ⫽
1 1 ⫹ exp~⫺4~ x ⫺ p/2!!
m zero ~ x! ⫽ exp~⫺x 2 ! m negative ~ x! ⫽
1 1 ⫹ exp~⫺4~ x ⫹ p/2!!
Using the center average defuzzifier and the product inference engine, the fuzzy system is obtained as follows: f ~ x! ⫽ j~ x!A x x ⫹ «
(40)
Table I. Fuzzy IF-THEN rules. x2 x1
N
Z
P
N Z P
y1 y4 y7
y2 y5 y8
y3 y6 y9
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ROBUST ADAPTIVE FUZZY CONTROL FOR PM SERVOMOTOR
where
Ax ⫽
冤 冥 a 11
a 21
I
I
a 19
a 29
j~ x! can be defined as Equation 12. Let k 1 ⫽ 1, k 2 ⫽ 25, and Q ⫽ diag(2,2); then the solution of Lyapunov expression 20 is obtained by P⫽
冋
25.08
1.00
1.00
0.09
册
If the gain is g ⫽ 0.5 and design constant is r ⫽ 0.5, the robust adaptive fuzzy control scheme can be obtained for PM synchronous servomotor drives as follows:
冉( 9
u ⫽ ⫺l~1.00e Z 1 ⫹ 0.08e2 !
ji2 ~ x! ⫹ c 2 ~ x!
i⫽1
冊
(41)
where e1 ⫽ x 1 ⫺ yd , e2 ⫽ x 2 ⫺ y_ d , c~ x! ⫽ ~1 ⫹ ~ x 12 ⫹ x 22 ! 1/2 ⫹ ~ ( 9i⫽1 ji2 ~ x!! 1/2 !,
冋
lZ^ ⫽ 0.01 ~1.00e1 ⫹ 0.08e2 ! 2
冉( 9
冊
册
ji2 ~ x! ⫹ c 2 ~ x! ⫺ 0.01~ lZ ⫺ 0.1!
i⫽1
The simulation results are shown in Figures 3– 6 using the Matlab toolbox. The plant parameters are chosen as cT ⫽ 10 Nm/A, J ⫽ 0.1 kgm 2, a ⫽ 5 Nm, and b ⫽ 5 s/rad. To reveal the control performance of the proposed robust adaptive fuzzy control, there are two kinds of reference signal: w~t ! used to train the controller u~t !. The kind of reference signal w~t ! and corresponding simulation results are shown in Table II. In Table II, the sinusoidal signal is to change its value sinusoidally in the interval ~⫺p, p! with a frequency of 0.5 Hz. The band-limited white noise signal has the parameter with noise power 1 and sample time 1. We give the initial error p between position and reference signal in the simulation. From Figures 3 and 6, we can see that fairly good tracking performance is obtained. The boundedness of control signal u and adaptive parameter lZ are shown in Figures 4 and 6, respectively.
Table II. Corresponding simulation results.
Simulation 1 Simulation 2
Reference signal
Figures for simulation results
Sinusoidal signal Band-limited white noise signal
Figures 3, 4 Figures 5, 6
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Figure 3. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Servomotor drive position y (solid line) and reference signal yd (dashed line); (b) Tracking error between real position and reference signal.
Figure 4. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Control u; (b) Adaptive parameter l. Z
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Figure 5. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Servomotor drive position y (solid line) and reference signal yd (dashed line); (b) Tracking error between real position and reference signal.
Figure 6. Simulation results for the RAFC algorithm when employed for a PM synchronous servomotor drive: (a) Control u; (b) Adaptive parameter l. Z
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6.
CONCLUDING REMARKS
The design of robust adaptive fuzzy tracking control for a class of uncertain nonlinear systems has been presented and then applied to PM synchronous servomotor drives. In accordance with the unstructured state-dependent (or nonrepeatable) unknown system and gain nonlinear functions, and Takagi–Sugeno-type fuzzy logic systems have been used to approximate unknown system function and a robust adaptive fuzzy tracking control (RAFC) algorithm, that can guarantee the closedloop in the presence of nonrepeatable uncertainties is semiglobally uniformly ultimately bounded, and the tracking error of the system can be steered to a small neighborhood around e~t ! ⫽ 0, has been achieved by use of the input-to-state stability and general small gain approach. The outstanding features of the algorithm proposed in this article are that (1) it can avoid the possible controller singularity problem in some of existing adaptive control schemes with feedback linearization techniques; (2) the adaptive mechanism can be obtained, which has only one learning parameterization to be adapted on-line, no matter how many states in the system are investigated and how many rules in the fuzzy system are used. Thus, the computation load of the algorithm can be reduced and it is easier to implement this algorithm in real-time applications. To verify the effectiveness of the proposed algorithm, it has been applied to PM synchronous servomotor drive system. In accordance with unknown parameters of PM synchronous servomotor drive model and unknown structure of uncertain function of the system, the T-S fuzzy system is used to approximate the uncertain function; then a RAFC scheme for the position control of PM synchronous servomotor is designed. Simulation results have shown the effectiveness of the proposed control scheme. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant No. 60474014, the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20020151005, and the Science Foundation of the National Ministry of Communications of China under Grant No. 95-06-02-22.
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