Robust stability analysis and fuzzy-scheduling control for nonlinear ...
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Robust Stability Analysis and Fuzzy-Scheduling Control for Nonlinear Systems Subject to Actuator Saturation Yong-Yan Cao, Senior Member, IEEE, and Zongli Lin, Senior Member, IEEE
Abstract—Takagi–Sugeno (TS) fuzzy models can provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear inputoutput submodels. In this paper, the TS fuzzy modeling approach is utilized to carry out the stability analysis and control design for nonlinear systems with actuator saturation. The TS fuzzy representation of a nonlinear system subject to actuator saturation is presented. In our TS fuzzy representation, the modeling error is also captured by norm-bounded uncertainties. A set invariance condition for the system in the TS fuzzy representation is first established. Based on this set invariance condition, the problem of estimating the domain of attraction of a TS fuzzy system under a constant state feedback law is formulated and solved as a linear matrix inequality (LMI) optimization problem. By viewing the state feedback gain as an extra free parameter in the LMI optimization problem, we arrive at a method for designing state feedback gain that maximizes the domain of attraction. A fuzzy scheduling control design method is also introduced to further enlarge the domain of attraction. An inverted pendulum is used to show the effectiveness of the proposed fuzzy controller. Index Terms—Actuator saturation, fuzzy control, linear matrix inequality (LMI), nonlinear systems, robust control, uncertainty.
I. INTRODUCTION
F
UZZY logic control [27] is an effective approach to designing nonlinear control systems, especially in the absence of complete knowledge of the plant. It has found successful applications not only in consumer products but also in industrial processes (see, e.g., [8], [15], [18], [10], and the references therein). Recently, a conceptually simple nonlocal approach to fuzzy control design was proposed for nonlinear systems [20], [21], [25], [24], [7]. The procedure is as follows. First, the nonlinear plant is represented by a so-called Takagi–Sugeno (TS) type fuzzy model. In this type of fuzzy model, local dynamics in different state-space regions are represented by linear models. The overall model of the system is obtained by fuzzy “blending” of these local models. The control design is carried out based on the fuzzy model by the so-called parallel distributed compensation (PDC) scheme [21], [25]. For each local linear model, a linear feedback control is
Manuscript received March 13, 2002; revised June 7, 2002 and July 8, 2002. This work was supported in part by the United States office of Naval Research Young Investigator Program under Grant N00014-99-1-0670. The authors are with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2002.806317
designed. The resulting overall controller, which is nonlinear in general, is again a fuzzy blending of the individual linear controllers. Actuator saturation can severely degrade the closed-loop system performance and sometimes even make the otherwise stable closed-loop system unstable. The analysis and synthesis of control systems with actuator saturation nonlinearities have been receiving increasing attention recently (see, e.g., [2], [11], [17], and the references therein). Very often, actuator saturation is dealt with by either designing low gain control laws that, for a given bound on the initial conditions, avoid the saturation limits, or estimating the domain of attraction in the presence of actuator saturation. In this paper, we will utilize the TS fuzzy modeling approach to analyze the domain of attraction of nonlinear systems with actuator saturation. In our analysis procedure, a given nonlinear system with actuator saturation is first represented by a set of TS models with actuator saturation. The system dynamics is captured by a set of fuzzy implications which characterize local relations in the state space. The main feature of the TS fuzzy model is to express the dynamics corresponding to each fuzzy rule by a linear state space system model with input saturation. The overall fuzzy model of the system is obtained by fuzzy “blending” of those individual models with actuator saturation. In [23], actuator saturation constraint was dealt with by designing low gain control laws that, for a given bound on the initial conditions, avoid the saturation limits. It is known that low-gain controllers that avoid saturation will often result in low levels of performance. This paper takes the fuzzy control approach to dealing with stability analysis and control design of nonlinear systems with actuator saturation. A TS fuzzy model with actuator saturation and norm-bounded uncertainties is proposed to represent the original nonlinear systems with actuator saturation. A set invariance condition for the system in the TS fuzzy representation is first established. Based on this set invariance condition, the problem of estimating the domain of attraction of a TS fuzzy system under a constant state feedback law is formulated and solved as a linear matrix inequality (LMI) optimization problem. By viewing the state feedback gain as an extra free parameter in the LMI optimization problem, we arrive at a method for designing state feedback gain that maximizes the domain of attraction. The paper is organized as follows. In Section II, the TS fuzzy model with actuator saturation is first introduced and a set invariance condition is derived using Lyapunov function based approach. In Section III, a robust stability condition is given for the
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TS fuzzy model with actuator saturation and modeling uncertainties. In Section IV, a robust state feedback fuzzy scheduling control law for the uncertain fuzzy systems with actuator saturation is proposed based on the parallel distributed compensation. In Section V, an inverted pendulum subject to actuator saturation is used to demonstrate the effectiveness of our analysis and design method. The paper is concluded in Section VI. Notation: The following notation will be used throughout the set of nonthe paper. denotes the set of real numbers, the dimensional Euclidean space, negative real numbers, the set of all real matrices. In the sequel, and if not explicitly stated, matrices are assumed to have compatis used to deible dimensions. The notation note a symmetric positive–definite (positive–semidefinite, negative–definite, negative–semidefinite, respectively) matrix.
where is the fuzzy set and is the number of IF–THEN rules are the premise variable. It is and assumed in this paper that the premise variables do not explicitly ), depend on the input variables . Then, given a pair ( the resulting fuzzy system model is inferred as the weighted average of the local models and has the form (4) where
II. PROBLEM STATEMENT AND PRELIMINARIES A. Problem Statement Consider a nonlinear system described by (1) and is sufficiently smooth in and where affine in . The control input is subject to actuator saturation. Our goal is to design a state feedback controller
and where is the grade of membership of in . In this paper, we assume that all membership functions are continuous and piecewise continuously differentiable. We belongs to also note that the time-varying parameter vector a convex polytope , where (5)
(2) such that the origin of the closed-loop system is asymptotically stable with a domain of attraction as large as possible, where the is the standard saturation function of function appropriate dimensions defined as follows:
with . Here, we have slightly abused the notation by using to denote both the scalar valued and the vector valued saturation functions. Also, note that it is without loss of generality to assume unity saturation level. The nonunity saturation level can be absorbed into the input by applying the following substitution:
and for Therefore, when , the fuzzy model (4) reduces to its th linear time-invariant . It is clear that as “local” model, i.e., varies inside the polytope , the system matrices of (4) vary inside a corresponding polytope whose vertices consist of local system matrices (6) denotes the convex hull. where Based on the PDC [21], [25], we consider the following fuzzy control law for the fuzzy model (4): IF THEN
is
and
and
is (7)
The overall state feedback fuzzy control law is represented by where , and is the saturation amplitude of the th input. We will develop a simplified model for which control design is easier. Such a simplified model is labeled as the design model. In many situations, there may be human experts who can provide a linguistic description of the system in terms of IF–THEN rules. For example, Takagi and Sugeno [20] proposed an approach to modeling the nonlinear process. This method is further developed by Sugeno and Kang [19]. This type of models are referred as TS or Takagi–Sugeno–Kang (TSK) fuzzy models. The fuzzy model is described by fuzzy IF–THEN rules, which represent local linear input-output relations of a nonlinear system [21], [22], [3], [24]. The th rules of the fuzzy models are of the following form IF THEN
is
and
and
is (3)
(8) Because the fuzzy model (4) is subject to input saturation, in general, global stabilizing controllers do not exist. The aim of this paper is to design local linear state feedback law (7) or a time-varying parameter-dependent linear state feedback law (8) such that the origin of the closed-loop system with actuator saturation is asymptotically stable in a region as large as possible. For simplicity, we will first consider the following linear constant feedback control law: (9) in This control law can also be obtained by setting (7). Control law (9) is a constant feedback law, while (8) is a time-varying feedback law. Control law (8) is the so-called fuzzy scheduling controller.
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Proof: Choose a Lyapunov function
B. Set Invariance Analysis for Fuzzy Systems Let be the th row of the matrix . We define the symmetric polyhedron
Then
If the control does not saturate for all , that , then (4) under (9) admits the following linear is representation:
By Lemma 1, we have
(10) Let
be a positive-definite matrix and define . For a positive number , denote the ellipsoid
An ellipsoid is said to be contractively invariant if for all Thus, if an ellipsoid is contractively invariant, it is inside the is inside if and domain of attraction. An ellipsoid only if
Let be the set of diagonal matrices whose diagonal , then elements are either 1 or 0. For example, if
There are 2 elements in . Suppose that each element of is labeled as 2 , and denote . is also an element of if . Clearly, be given. For an , Lemma 1 [11]: Let , then if Consequently,
can be rewritten as
where , . Lemma 2 [5]: Suppose that matrices , and a positive–semidefinite matrix and , then given. If
. By (12), we have
Thus, if
, then for , i.e., is a contractively invariant set. This also implies that the closed-loop system (4) under state feedback (9) is asymptotcontained in the domain ically stable at the origin with of attraction. Remark 1: Theorem 3 gives a condition for the reto be inside the domain of attraction for the gion closed-loop system (4) under a linear constant feedback control , Theorem 3 recovers law (9). For the special case of the set invariance condition for linear time-invariant systems subject to actuator saturation [11]. With all the ellipsoids satisfying the set invariance condition of Theorem 3, we may choose the “largest” one to obtain the least conservative estimate of the domain of attraction. As in [11], we will measure the largeness of the ellipsoids with respect be a prescribed bounded to a shape reference set. Let which contains convex set containing origin. For a set origin, define Obviously, if are the ellipsoid
, then
are
. Two typical types of (13)
and the polyhedron (14)
(11) For the fuzzy system subject to actuator saturation (4) and a given linear control law (9), we have the following set invariance condition. Theorem 3: For a given fuzzy system (4) and a given state is a contracfeedback control matrix , the ellipsoid tively invariant set of the closed-loop system under linear state feedback control law (9) if there exists a matrix such that the following matrix inequalities hold:
are a priori given points in . where With the aforementioned reference sets, we can choose an from all that satisfy the condition such that the quanis maximized. This problem can be formulated tity into the following optimization problem:
s.t. inequalities
(12)
(15)
. Consequently, the closed-loop system is and contained in the asymptotically stable at the origin with domain of attraction.
where denotes the th row of . In what follows, we will show that the optimization problem (15) can be solved as an LMI optimization problem.
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In the case where the shape reference set Constraint a) is equivalent to
In the case where to
is given by (14),
(16) is given by (13), Constraint a) is equivalent
If we require , then we can recover the design algorithm which constrains the control law to be unsaturated [1], [23]. The unsaturated control algorithm can be described as
s.t.
or
(17)
(22)
(18)
Note that the constraints in (22) imply that and hence the control will never reach saturation limits. This will lead to a very conservative control law. In (21), we permit the control to be saturated and, thus, resulting in a larger domain of attraction. It is also known that low-gain controllers that avoid saturation will often result in low levels of performance [17].
Let
Then, (16) and (17) can be written as the following LMIs:
III. ROBUST STABILITY OF FUZZY SYSTEMS WITH UNCERTAINTY
and (19) respectively. Also, Constraint b) is equivalent to (20) Constraint c) is equivalent to
In Section II, we consider the stability of the fuzzy system (4) rather than the original nonlinear system (1). It is an obvious fact that the closed-loop stability of fuzzy system (4) cannot guarantee that of nonlinear system (1). As discussed in [4], we can present a fuzzy model with norm-bounded uncertainty to analyze the stability of the original nonlinear system (1). A TS fuzzy model with uncertainty is composed of plant rules that can be represented as [28], [14], [16] IF
Also, let the th row of be , i.e., . The optimization problem (15) can then be reduced to the following one with LMI constraints:
s.t.
or
THEN
is
and
and
is (23)
are real-valued time-varying matrices of apwhere , and propriate dimensions. Fuzzy model (23) is an extension of local fuzzy model (3). We assume that the time-varying uncertainties enter the system matrices in the following manner: (24)
(21) such that If we would like to design a control law the domain of attraction of the closed-loop system is as large as in (20). With the sopossible, we only need to replace ), the state feedback control matrix such that lution ( the origin of the system (4) is stabilized with a domain of attraction as large as possible with respect to a given shape reference can then be obtained as
In optimization problem (21), the amplitude of control law (9) is not constrained, i.e., there is no control amplitude constraint on the control law. On the other hand, to avoid the controller gain , being too large, we may constrain it to be bounded by , which is equivalent to the following LMIs: i.e.,
where
denotes th row of
.
and are some constant matrices of compatible diwhere and are real-valued matrix funcmensions and tions of compatible dimensions representing time-varying parameter uncertainties. Such uncertainties arise in the fuzzy representation (3) of the original nonlinear system (1). The uncertainties are assumed to be norm-bounded and be given by (25) , and are known constant matrices with comwhere are unknown nonpatible dimensions and linear time-varying matrix functions satisfying (26) are Lebesgue meaIt is assumed that the elements of surable. This type of uncertainty is an effective representation of some nonlinear uncertainties (see [6], [13], [26], and the references therein). By including these uncertainties, the fuzzy model (23) is expected to better represent (1).
CAO AND LIN: ROBUST STABILITY ANALYSIS AND FUZZY-SCHEDULING CONTROL
Now, given a pair follows:
, the final fuzzy system is inferred as
(27) (28) where
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By (29), we have
Thus, is a contractively invariant set. This also implies that the closed-loop system (27) under state feedback (9) is contained in the asymptotically stable at the origin with domain of attraction. It is easy to see that matrix inequality (29) can be transformed to the following LMIs:
(30) and represents the difference between (4) and , we can make the and (1). By suitably selecting original nonlinear system (1) completely included in the differential inclusions system (27) [1], [14], [4]. Theorem 4: For a given uncertain fuzzy system (27) and a is a given state feedback control matrix , the ellipsoid contractively invariant set of the closed-loop system under linear state feedback control law (9) if there exist matrices and such that (29), as shown at the bottom of the page, , where ’s represent blocks that are holds, and readily inferred by symmetry. Proof: Choose a Lyapunov function
Then
By Lemma 1, we have for all
,
As in Section II-B, we can formulate the following LMI optimization problem to maximize the domain of attraction of the closed-loop system:
s.t.
or
(31)
IV. ROBUST FUZZY SCHEDULING CONTROL LAW DESIGN As shown in Section II, the approach to fuzzy scheduling involves the design of several LTI controllers for a parameterized family of LTI fuzzy models. The resulting controller (8) is the interpolation of these gains. Note that in (9) is a constant matrix, while the control gain in (8) is a matrix function of time-varying . It is reasonable to expect that this membership function kind of control laws can result in a larger domain of attraction and better performance. With control law (8), the closed-loop system can be rewritten as
where
By Lemma 1, we have that for any matrix of the same di, and the equation at the mensions of such that , bottom of the next page holds true, where , for all 2 . If we let
(29)
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then
Then, by Lemma 1 (32)
where
Remark 2: It is easy to verify that the closed-loop system described by (32) can be further simplified if (23) possesses a for all . In this case, common input matrix , namely the closed-loop system can be simplified as
(34) for all
for all
Theorem 5: For a given uncertain fuzzy system (27), suppose that the local state feedback control matrices , are given. The ellipsoid is a contractively invariant set of the closed-loop system under the fuzzy sched, uling state feedback law (8) if there exist matrices , such that (33), as shown at the and . bottom of the page, holds, and . We Proof: Choose a Lyapunov function note that
implies
since
and
. Let
. It is easy to see that
and
2
if
. Note that
and
This implies that if the matrix inequalities (33) hold, then we have
and, hence, for all . That is, is contractively invariant. Corollary 6: For the special case of for all , the elis a contractively invariant set of the closed-loop lipsoid system under the fuzzy scheduling state feedback control law and such (8), if there exist matrices that (35), as shown at the bottom of the next page, holds, and . In what follows, we will present a less conservative set invariance condition.
(33)
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Proof: Choose the Lyapunov function as
Note that (32) can be rewritten as
Then, the equation at the bottom of the page holds true. It is easy if to see that
(38)
Let and
for
(39) for We then have
where Similar to the proof of Theorem 5, we can find that matrix inequality (38) holds if (36) holds. In what follows, we will prove that (39) holds if (37) holds. Note that
and
Hence
Theorem 7: For a given uncertain fuzzy system (27), suppose that the local state feedback control matrices , are known. The ellipsoid is a contractively invariant set of the closed-loop system under the fuzzy schedand uling control law (8), if there exist matrices , such that (36) and (37), shown at . the bottom of the page, hold, and
and
Then, if (37) holds. Remark 3: In comparison with Theorem 5, the number of 2 . matrix inequalities in Theorem 7 is reduced by
(35)
(36)
(37)
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In comparison with Corollary 6, another 2 inequalities can be removed for the special case of Let
matrix .
The control objective is to balance the inverted pendulum for the approximate range . The actuator is sub. That is, ject to saturation and the saturation level is we would like to design a state feedback control law (45)
Denote the th row of the matrix as are equivalent to the following LMIs:
. Then, (36) and (37)
such that the inverted pendulum can be balanced in the approx]. imate range [ We linearize the plant (43)–(44) at the origin and design a based on the linearizing constant linear control law model. It is easy to get the linearizing model
(40) and
(46) By placing the closed-loop eigenvalues at { 2 [25]
2}, we have (47)
(41) respectively. Then, we have the following theorem. Theorem 8: For a given uncertain fuzzy system (27), the fuzzy scheduling state feedback control law (8) such that the closed-loop system is robustly stable at the origin with a domain of attraction as large as possible can be solved by
where ( tion problem:
s.t.
) is a solution to the following LMI optimiza-
or and
We use Theorem 3 and optimization problem (21) to estimate through estimating the the permissible balancing range of domain of attraction of the closed-loop system. To apply the optimization method introduced in Section II-B, we set . Solving (21), we obtain
which corresponds to an angle much larger than . By applying the previous controller to the true plant (43)–(44), how, the constant conever, we find that when troller (47) fails to balance the pendulum. This is because of the very large modeling error between linear system (46) and original nonlinear system (43)–(44). Now, we take the fuzzy-scheduling control design approach proposed in this paper. We follow [25] approximate the system by the following two-rule fuzzy model:
(42)
IF
is about
IF
is about
THEN THEN
(48)
where V. AN EXAMPLE Consider the problem of balancing and swing-up of an inverted pendulum on a cart. The equations of motion for the pendulum are [25] (43)
and . Membership functions for Rule 1 and Rule 2 are set as follows:
(44) denotes the angle (in radians) of the pendulum from where is the angular velocity, m s is the the vertical, is the mass gravity constant, is the mass of the pendulum, of the cart, 2 is the length of the pendulum, is the force applied and . to the cart (in Kilo-Newtons), kg, kg, m in the We choose simulations.
which are shown in Fig. 1. The following fuzzy-scheduling control law is used to balance the plant: IF
is about
IF
is about
THEN THEN
(49)
CAO AND LIN: ROBUST STABILITY ANALYSIS AND FUZZY-SCHEDULING CONTROL
Fig. 1.
Membership functions of the two-rule model.
By placing the eigenvalues of both at { 2 2}, we have
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Fig. 2. Angle response using linear and two-rule fuzzy control: dashed curves-linear constant feedback control; solid curves-fuzzy control.
and
(50) In the absence of saturation, this control law can balance the as pendulum for initial conditions the control law proposed in [25]. In the presence of saturation, if applying our optimization method by solving (42) to estimate the balancing, we obtain
which still corresponds to an angle larger than . This implies that control law (50) should be able to balance the pendulum in ] even in the presence of satuthe required range [ ration. However, simulation indicates that the fuzzy-scheduling control law (8) can only balance the pendulum for initial condiin the presence of saturation. tions Fig. 2 shows the response of the pendulum system using linear , and fuzzy scheduling controls for initial conditions . The solid curves indicate responses 40 , 45 , 84 , and with the fuzzy scheduling controller. The dashed curves show those with the linear constant controller. From the curves, we can also find that the fuzzy scheduling controller can lead to fast responses. We next use Theorem 7 and (42) to design a controller such that the balancing range is as large as possible. Solving (42), we obtain the following result:
Fig. 3. State and input with different initial conditions: solid curves-84 ; dotted–dashed curves-45 ; dashed curves-20 .
Because the modeling error is not considered, the stability region of the fuzzy system may not be inside the stability region of the original nonlinear system. In what follows, we will analyze the modeling error between above fuzzy model and the actual pendulum model (43)–(44). With the fuzzy rules shown in (48), the resulting fuzzy system is given by
(51) corresponds to an angle of 128 . By Note that applying the above controller to the true plant (43)–(44) with , 45 and 20 ( ), the system initial conditions responses are shown in Fig. 3.
(52) The difference between (44) and (52) is
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where
Now, we use the following uncertain fuzzy model to analyze the stability of the original system IF THEN
is about
IF
is about
Fig. 4. State and input responses with different initial conditions: solid curves-84 ; dotted–dashed curves-69 ; dashed curves-45 .
THEN
(53)
In the aforementioned uncertain fuzzy models, we assume that the uncertainties that describe the modeling errors are in the form of
where
with
, then we gain
With the fixed controller (47), the optimization problem (31) has no solution. When the controller is not fixed, we obtain the following result:
, and
The parameters , , , , , and are to be determined. Hence, the uncertain nonlinear fuzzy system is
(54) i.e.,
(55) , If we set then they can be chosen as
If we constrain
and
,
Simulation results with this controller are shown in Fig. 4. This simulation shows that, by optimizing the feedback gain as in Section IV, even a constant controller is able to balance the pendulum in range of [ 84 , 84 ] and, more importantly, the estimated range of [ 69 , 69 ] is indeed inside the actual range of [ 84 , 84 ]. VI. CONCLUSION This paper has presented a stability analysis and design method for a class of nonlinear systems with actuator saturation. TS fuzzy models with actuator saturation and norm-bounded uncertainty are first extended to describe the nonlinear systems subject to actuator saturation. A set invariance condition for the TS model is then established. Based on this set invariance condition, an LMI-based optimization approach is proposed to estimate the domain of attraction of the fuzzy system under given feedback laws. A fuzzy scheduling controller design method is then developed to enlarge the domain of attraction of the closed-loop system. Finally, the design methodology is illustrated by its application to the stabilization of an inverted pendulum on a cart. REFERENCES [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [2] D. S. Bernstein and A. N. Michel, “A chronological bibliography on saturating actuators,” Int. J. Robust Nonlinear Control, vol. 5, pp. 375–380, 1995.
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[27] L. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. 3, pp. 28–44, Feb. 1973. [28] J. Zhao, V. Wertz, and R. Gorez, “Linear TS models based robust stabilizing controller design,” Proc. 34th IEEE Conf. Decision Control, pp. 255–260, 1995.
Yong-Yan Cao (M’97–SM’00) was born in Hunan, China, in 1968. He received the B.E. and M.Sc. degrees in electrical engineering from Wuhan University of Science and Technlogy, China, and the Ph.D. degree in industrial automation from Zhejiang University, China, in 1990, 1993, and 1996, respectively. From 1996 to 1997, he was with the Institute of Industrial Process Control, Zhejiang University, China, as a Postdoctoral Researcher. From 1997 to 1998, he was a Visiting Research Associate in the Department of Mechanical Engineering, The University of Hong Kong. From 1998 to 1999, he held an Associate Professor position at Zhejiang University. He then spent a year-and-a-half with the Department of Measurement and Control, Duisburg University, Germany, as an Alexander von Humboldt Research Fellow. In May 2000, he joined the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, as a Research Scientist. He has published more than 50 papers in some reputed journals and conferences such as the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Automatica, and Systems and Control Letters. He also has been active both in his own research and in his service as a Reviewer for many journals and conferences. His current research interests include robust and nonlinear control, constrained control, time-delay systems, sampled-data systems, and fuzzy control. Dr. Cao currently serves as an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society.
Zongli Lin (S’89–M’90–SM’93) was born in Fuqing, Fujian, China, in 1964. He received the B.S. degree in mathematics and computer science from Amoy University, Xiamen, China, the M.E. degree in automatic control from Chinese Academy of Space Technology, Beijing, China, and the Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, in 1983, 1989, and 1994, respectively. From July 1983 to July 1986, he worked as a Control Engineer at the Chinese Academy of Space Technology, China. In January 1994, he joined the Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, as a Visiting Assistant Professor, where he became an Assistant Professor in September 1994. Since July 1997, he has been with the Department of Electrical and Computer Engineering at University of Virginia, Charlottesville, where his is currently an Associate Professor. His current research interests include nonlinear control, robust control, control of systems with saturating actuators and modeling, and control of magnetic bearing systems. In these areas, he has published several papers. He is also the author of the book Low Gain Feedback (London, U.K.: Springer-Verlag, 1998) and a coauthor of the recent book Control Systems with Actuator Saturation: Analysis and Design (Boston, MA: Birkhäuser, 2001). Dr. Lin currently serves as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. Previously, he was an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society. He is also a member of the IEEE Control Systems Society’s Technical Committee on Nonlinear Systems and Control, and heads its Working Group on Control with Constraints. He is the recipient of a U.S. Office of Naval Research Young Investigator Award.
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Robust Stability Analysis and Fuzzy-Scheduling Control for Nonlinear Systems Subject to Actuator Saturation Yong-Yan Cao, Senior Member, IEEE, and Zongli Lin, Senior Member, IEEE
Abstract—Takagi–Sugeno (TS) fuzzy models can provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear inputoutput submodels. In this paper, the TS fuzzy modeling approach is utilized to carry out the stability analysis and control design for nonlinear systems with actuator saturation. The TS fuzzy representation of a nonlinear system subject to actuator saturation is presented. In our TS fuzzy representation, the modeling error is also captured by norm-bounded uncertainties. A set invariance condition for the system in the TS fuzzy representation is first established. Based on this set invariance condition, the problem of estimating the domain of attraction of a TS fuzzy system under a constant state feedback law is formulated and solved as a linear matrix inequality (LMI) optimization problem. By viewing the state feedback gain as an extra free parameter in the LMI optimization problem, we arrive at a method for designing state feedback gain that maximizes the domain of attraction. A fuzzy scheduling control design method is also introduced to further enlarge the domain of attraction. An inverted pendulum is used to show the effectiveness of the proposed fuzzy controller. Index Terms—Actuator saturation, fuzzy control, linear matrix inequality (LMI), nonlinear systems, robust control, uncertainty.
I. INTRODUCTION
F
UZZY logic control [27] is an effective approach to designing nonlinear control systems, especially in the absence of complete knowledge of the plant. It has found successful applications not only in consumer products but also in industrial processes (see, e.g., [8], [15], [18], [10], and the references therein). Recently, a conceptually simple nonlocal approach to fuzzy control design was proposed for nonlinear systems [20], [21], [25], [24], [7]. The procedure is as follows. First, the nonlinear plant is represented by a so-called Takagi–Sugeno (TS) type fuzzy model. In this type of fuzzy model, local dynamics in different state-space regions are represented by linear models. The overall model of the system is obtained by fuzzy “blending” of these local models. The control design is carried out based on the fuzzy model by the so-called parallel distributed compensation (PDC) scheme [21], [25]. For each local linear model, a linear feedback control is
Manuscript received March 13, 2002; revised June 7, 2002 and July 8, 2002. This work was supported in part by the United States office of Naval Research Young Investigator Program under Grant N00014-99-1-0670. The authors are with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2002.806317
designed. The resulting overall controller, which is nonlinear in general, is again a fuzzy blending of the individual linear controllers. Actuator saturation can severely degrade the closed-loop system performance and sometimes even make the otherwise stable closed-loop system unstable. The analysis and synthesis of control systems with actuator saturation nonlinearities have been receiving increasing attention recently (see, e.g., [2], [11], [17], and the references therein). Very often, actuator saturation is dealt with by either designing low gain control laws that, for a given bound on the initial conditions, avoid the saturation limits, or estimating the domain of attraction in the presence of actuator saturation. In this paper, we will utilize the TS fuzzy modeling approach to analyze the domain of attraction of nonlinear systems with actuator saturation. In our analysis procedure, a given nonlinear system with actuator saturation is first represented by a set of TS models with actuator saturation. The system dynamics is captured by a set of fuzzy implications which characterize local relations in the state space. The main feature of the TS fuzzy model is to express the dynamics corresponding to each fuzzy rule by a linear state space system model with input saturation. The overall fuzzy model of the system is obtained by fuzzy “blending” of those individual models with actuator saturation. In [23], actuator saturation constraint was dealt with by designing low gain control laws that, for a given bound on the initial conditions, avoid the saturation limits. It is known that low-gain controllers that avoid saturation will often result in low levels of performance. This paper takes the fuzzy control approach to dealing with stability analysis and control design of nonlinear systems with actuator saturation. A TS fuzzy model with actuator saturation and norm-bounded uncertainties is proposed to represent the original nonlinear systems with actuator saturation. A set invariance condition for the system in the TS fuzzy representation is first established. Based on this set invariance condition, the problem of estimating the domain of attraction of a TS fuzzy system under a constant state feedback law is formulated and solved as a linear matrix inequality (LMI) optimization problem. By viewing the state feedback gain as an extra free parameter in the LMI optimization problem, we arrive at a method for designing state feedback gain that maximizes the domain of attraction. The paper is organized as follows. In Section II, the TS fuzzy model with actuator saturation is first introduced and a set invariance condition is derived using Lyapunov function based approach. In Section III, a robust stability condition is given for the
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TS fuzzy model with actuator saturation and modeling uncertainties. In Section IV, a robust state feedback fuzzy scheduling control law for the uncertain fuzzy systems with actuator saturation is proposed based on the parallel distributed compensation. In Section V, an inverted pendulum subject to actuator saturation is used to demonstrate the effectiveness of our analysis and design method. The paper is concluded in Section VI. Notation: The following notation will be used throughout the set of nonthe paper. denotes the set of real numbers, the dimensional Euclidean space, negative real numbers, the set of all real matrices. In the sequel, and if not explicitly stated, matrices are assumed to have compatis used to deible dimensions. The notation note a symmetric positive–definite (positive–semidefinite, negative–definite, negative–semidefinite, respectively) matrix.
where is the fuzzy set and is the number of IF–THEN rules are the premise variable. It is and assumed in this paper that the premise variables do not explicitly ), depend on the input variables . Then, given a pair ( the resulting fuzzy system model is inferred as the weighted average of the local models and has the form (4) where
II. PROBLEM STATEMENT AND PRELIMINARIES A. Problem Statement Consider a nonlinear system described by (1) and is sufficiently smooth in and where affine in . The control input is subject to actuator saturation. Our goal is to design a state feedback controller
and where is the grade of membership of in . In this paper, we assume that all membership functions are continuous and piecewise continuously differentiable. We belongs to also note that the time-varying parameter vector a convex polytope , where (5)
(2) such that the origin of the closed-loop system is asymptotically stable with a domain of attraction as large as possible, where the is the standard saturation function of function appropriate dimensions defined as follows:
with . Here, we have slightly abused the notation by using to denote both the scalar valued and the vector valued saturation functions. Also, note that it is without loss of generality to assume unity saturation level. The nonunity saturation level can be absorbed into the input by applying the following substitution:
and for Therefore, when , the fuzzy model (4) reduces to its th linear time-invariant . It is clear that as “local” model, i.e., varies inside the polytope , the system matrices of (4) vary inside a corresponding polytope whose vertices consist of local system matrices (6) denotes the convex hull. where Based on the PDC [21], [25], we consider the following fuzzy control law for the fuzzy model (4): IF THEN
is
and
and
is (7)
The overall state feedback fuzzy control law is represented by where , and is the saturation amplitude of the th input. We will develop a simplified model for which control design is easier. Such a simplified model is labeled as the design model. In many situations, there may be human experts who can provide a linguistic description of the system in terms of IF–THEN rules. For example, Takagi and Sugeno [20] proposed an approach to modeling the nonlinear process. This method is further developed by Sugeno and Kang [19]. This type of models are referred as TS or Takagi–Sugeno–Kang (TSK) fuzzy models. The fuzzy model is described by fuzzy IF–THEN rules, which represent local linear input-output relations of a nonlinear system [21], [22], [3], [24]. The th rules of the fuzzy models are of the following form IF THEN
is
and
and
is (3)
(8) Because the fuzzy model (4) is subject to input saturation, in general, global stabilizing controllers do not exist. The aim of this paper is to design local linear state feedback law (7) or a time-varying parameter-dependent linear state feedback law (8) such that the origin of the closed-loop system with actuator saturation is asymptotically stable in a region as large as possible. For simplicity, we will first consider the following linear constant feedback control law: (9) in This control law can also be obtained by setting (7). Control law (9) is a constant feedback law, while (8) is a time-varying feedback law. Control law (8) is the so-called fuzzy scheduling controller.
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Proof: Choose a Lyapunov function
B. Set Invariance Analysis for Fuzzy Systems Let be the th row of the matrix . We define the symmetric polyhedron
Then
If the control does not saturate for all , that , then (4) under (9) admits the following linear is representation:
By Lemma 1, we have
(10) Let
be a positive-definite matrix and define . For a positive number , denote the ellipsoid
An ellipsoid is said to be contractively invariant if for all Thus, if an ellipsoid is contractively invariant, it is inside the is inside if and domain of attraction. An ellipsoid only if
Let be the set of diagonal matrices whose diagonal , then elements are either 1 or 0. For example, if
There are 2 elements in . Suppose that each element of is labeled as 2 , and denote . is also an element of if . Clearly, be given. For an , Lemma 1 [11]: Let , then if Consequently,
can be rewritten as
where , . Lemma 2 [5]: Suppose that matrices , and a positive–semidefinite matrix and , then given. If
. By (12), we have
Thus, if
, then for , i.e., is a contractively invariant set. This also implies that the closed-loop system (4) under state feedback (9) is asymptotcontained in the domain ically stable at the origin with of attraction. Remark 1: Theorem 3 gives a condition for the reto be inside the domain of attraction for the gion closed-loop system (4) under a linear constant feedback control , Theorem 3 recovers law (9). For the special case of the set invariance condition for linear time-invariant systems subject to actuator saturation [11]. With all the ellipsoids satisfying the set invariance condition of Theorem 3, we may choose the “largest” one to obtain the least conservative estimate of the domain of attraction. As in [11], we will measure the largeness of the ellipsoids with respect be a prescribed bounded to a shape reference set. Let which contains convex set containing origin. For a set origin, define Obviously, if are the ellipsoid
, then
are
. Two typical types of (13)
and the polyhedron (14)
(11) For the fuzzy system subject to actuator saturation (4) and a given linear control law (9), we have the following set invariance condition. Theorem 3: For a given fuzzy system (4) and a given state is a contracfeedback control matrix , the ellipsoid tively invariant set of the closed-loop system under linear state feedback control law (9) if there exists a matrix such that the following matrix inequalities hold:
are a priori given points in . where With the aforementioned reference sets, we can choose an from all that satisfy the condition such that the quanis maximized. This problem can be formulated tity into the following optimization problem:
s.t. inequalities
(12)
(15)
. Consequently, the closed-loop system is and contained in the asymptotically stable at the origin with domain of attraction.
where denotes the th row of . In what follows, we will show that the optimization problem (15) can be solved as an LMI optimization problem.
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In the case where the shape reference set Constraint a) is equivalent to
In the case where to
is given by (14),
(16) is given by (13), Constraint a) is equivalent
If we require , then we can recover the design algorithm which constrains the control law to be unsaturated [1], [23]. The unsaturated control algorithm can be described as
s.t.
or
(17)
(22)
(18)
Note that the constraints in (22) imply that and hence the control will never reach saturation limits. This will lead to a very conservative control law. In (21), we permit the control to be saturated and, thus, resulting in a larger domain of attraction. It is also known that low-gain controllers that avoid saturation will often result in low levels of performance [17].
Let
Then, (16) and (17) can be written as the following LMIs:
III. ROBUST STABILITY OF FUZZY SYSTEMS WITH UNCERTAINTY
and (19) respectively. Also, Constraint b) is equivalent to (20) Constraint c) is equivalent to
In Section II, we consider the stability of the fuzzy system (4) rather than the original nonlinear system (1). It is an obvious fact that the closed-loop stability of fuzzy system (4) cannot guarantee that of nonlinear system (1). As discussed in [4], we can present a fuzzy model with norm-bounded uncertainty to analyze the stability of the original nonlinear system (1). A TS fuzzy model with uncertainty is composed of plant rules that can be represented as [28], [14], [16] IF
Also, let the th row of be , i.e., . The optimization problem (15) can then be reduced to the following one with LMI constraints:
s.t.
or
THEN
is
and
and
is (23)
are real-valued time-varying matrices of apwhere , and propriate dimensions. Fuzzy model (23) is an extension of local fuzzy model (3). We assume that the time-varying uncertainties enter the system matrices in the following manner: (24)
(21) such that If we would like to design a control law the domain of attraction of the closed-loop system is as large as in (20). With the sopossible, we only need to replace ), the state feedback control matrix such that lution ( the origin of the system (4) is stabilized with a domain of attraction as large as possible with respect to a given shape reference can then be obtained as
In optimization problem (21), the amplitude of control law (9) is not constrained, i.e., there is no control amplitude constraint on the control law. On the other hand, to avoid the controller gain , being too large, we may constrain it to be bounded by , which is equivalent to the following LMIs: i.e.,
where
denotes th row of
.
and are some constant matrices of compatible diwhere and are real-valued matrix funcmensions and tions of compatible dimensions representing time-varying parameter uncertainties. Such uncertainties arise in the fuzzy representation (3) of the original nonlinear system (1). The uncertainties are assumed to be norm-bounded and be given by (25) , and are known constant matrices with comwhere are unknown nonpatible dimensions and linear time-varying matrix functions satisfying (26) are Lebesgue meaIt is assumed that the elements of surable. This type of uncertainty is an effective representation of some nonlinear uncertainties (see [6], [13], [26], and the references therein). By including these uncertainties, the fuzzy model (23) is expected to better represent (1).
CAO AND LIN: ROBUST STABILITY ANALYSIS AND FUZZY-SCHEDULING CONTROL
Now, given a pair follows:
, the final fuzzy system is inferred as
(27) (28) where
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By (29), we have
Thus, is a contractively invariant set. This also implies that the closed-loop system (27) under state feedback (9) is contained in the asymptotically stable at the origin with domain of attraction. It is easy to see that matrix inequality (29) can be transformed to the following LMIs:
(30) and represents the difference between (4) and , we can make the and (1). By suitably selecting original nonlinear system (1) completely included in the differential inclusions system (27) [1], [14], [4]. Theorem 4: For a given uncertain fuzzy system (27) and a is a given state feedback control matrix , the ellipsoid contractively invariant set of the closed-loop system under linear state feedback control law (9) if there exist matrices and such that (29), as shown at the bottom of the page, , where ’s represent blocks that are holds, and readily inferred by symmetry. Proof: Choose a Lyapunov function
Then
By Lemma 1, we have for all
,
As in Section II-B, we can formulate the following LMI optimization problem to maximize the domain of attraction of the closed-loop system:
s.t.
or
(31)
IV. ROBUST FUZZY SCHEDULING CONTROL LAW DESIGN As shown in Section II, the approach to fuzzy scheduling involves the design of several LTI controllers for a parameterized family of LTI fuzzy models. The resulting controller (8) is the interpolation of these gains. Note that in (9) is a constant matrix, while the control gain in (8) is a matrix function of time-varying . It is reasonable to expect that this membership function kind of control laws can result in a larger domain of attraction and better performance. With control law (8), the closed-loop system can be rewritten as
where
By Lemma 1, we have that for any matrix of the same di, and the equation at the mensions of such that , bottom of the next page holds true, where , for all 2 . If we let
(29)
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then
Then, by Lemma 1 (32)
where
Remark 2: It is easy to verify that the closed-loop system described by (32) can be further simplified if (23) possesses a for all . In this case, common input matrix , namely the closed-loop system can be simplified as
(34) for all
for all
Theorem 5: For a given uncertain fuzzy system (27), suppose that the local state feedback control matrices , are given. The ellipsoid is a contractively invariant set of the closed-loop system under the fuzzy sched, uling state feedback law (8) if there exist matrices , such that (33), as shown at the and . bottom of the page, holds, and . We Proof: Choose a Lyapunov function note that
implies
since
and
. Let
. It is easy to see that
and
2
if
. Note that
and
This implies that if the matrix inequalities (33) hold, then we have
and, hence, for all . That is, is contractively invariant. Corollary 6: For the special case of for all , the elis a contractively invariant set of the closed-loop lipsoid system under the fuzzy scheduling state feedback control law and such (8), if there exist matrices that (35), as shown at the bottom of the next page, holds, and . In what follows, we will present a less conservative set invariance condition.
(33)
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63
Proof: Choose the Lyapunov function as
Note that (32) can be rewritten as
Then, the equation at the bottom of the page holds true. It is easy if to see that
(38)
Let and
for
(39) for We then have
where Similar to the proof of Theorem 5, we can find that matrix inequality (38) holds if (36) holds. In what follows, we will prove that (39) holds if (37) holds. Note that
and
Hence
Theorem 7: For a given uncertain fuzzy system (27), suppose that the local state feedback control matrices , are known. The ellipsoid is a contractively invariant set of the closed-loop system under the fuzzy schedand uling control law (8), if there exist matrices , such that (36) and (37), shown at . the bottom of the page, hold, and
and
Then, if (37) holds. Remark 3: In comparison with Theorem 5, the number of 2 . matrix inequalities in Theorem 7 is reduced by
(35)
(36)
(37)
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In comparison with Corollary 6, another 2 inequalities can be removed for the special case of Let
matrix .
The control objective is to balance the inverted pendulum for the approximate range . The actuator is sub. That is, ject to saturation and the saturation level is we would like to design a state feedback control law (45)
Denote the th row of the matrix as are equivalent to the following LMIs:
. Then, (36) and (37)
such that the inverted pendulum can be balanced in the approx]. imate range [ We linearize the plant (43)–(44) at the origin and design a based on the linearizing constant linear control law model. It is easy to get the linearizing model
(40) and
(46) By placing the closed-loop eigenvalues at { 2 [25]
2}, we have (47)
(41) respectively. Then, we have the following theorem. Theorem 8: For a given uncertain fuzzy system (27), the fuzzy scheduling state feedback control law (8) such that the closed-loop system is robustly stable at the origin with a domain of attraction as large as possible can be solved by
where ( tion problem:
s.t.
) is a solution to the following LMI optimiza-
or and
We use Theorem 3 and optimization problem (21) to estimate through estimating the the permissible balancing range of domain of attraction of the closed-loop system. To apply the optimization method introduced in Section II-B, we set . Solving (21), we obtain
which corresponds to an angle much larger than . By applying the previous controller to the true plant (43)–(44), how, the constant conever, we find that when troller (47) fails to balance the pendulum. This is because of the very large modeling error between linear system (46) and original nonlinear system (43)–(44). Now, we take the fuzzy-scheduling control design approach proposed in this paper. We follow [25] approximate the system by the following two-rule fuzzy model:
(42)
IF
is about
IF
is about
THEN THEN
(48)
where V. AN EXAMPLE Consider the problem of balancing and swing-up of an inverted pendulum on a cart. The equations of motion for the pendulum are [25] (43)
and . Membership functions for Rule 1 and Rule 2 are set as follows:
(44) denotes the angle (in radians) of the pendulum from where is the angular velocity, m s is the the vertical, is the mass gravity constant, is the mass of the pendulum, of the cart, 2 is the length of the pendulum, is the force applied and . to the cart (in Kilo-Newtons), kg, kg, m in the We choose simulations.
which are shown in Fig. 1. The following fuzzy-scheduling control law is used to balance the plant: IF
is about
IF
is about
THEN THEN
(49)
CAO AND LIN: ROBUST STABILITY ANALYSIS AND FUZZY-SCHEDULING CONTROL
Fig. 1.
Membership functions of the two-rule model.
By placing the eigenvalues of both at { 2 2}, we have
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Fig. 2. Angle response using linear and two-rule fuzzy control: dashed curves-linear constant feedback control; solid curves-fuzzy control.
and
(50) In the absence of saturation, this control law can balance the as pendulum for initial conditions the control law proposed in [25]. In the presence of saturation, if applying our optimization method by solving (42) to estimate the balancing, we obtain
which still corresponds to an angle larger than . This implies that control law (50) should be able to balance the pendulum in ] even in the presence of satuthe required range [ ration. However, simulation indicates that the fuzzy-scheduling control law (8) can only balance the pendulum for initial condiin the presence of saturation. tions Fig. 2 shows the response of the pendulum system using linear , and fuzzy scheduling controls for initial conditions . The solid curves indicate responses 40 , 45 , 84 , and with the fuzzy scheduling controller. The dashed curves show those with the linear constant controller. From the curves, we can also find that the fuzzy scheduling controller can lead to fast responses. We next use Theorem 7 and (42) to design a controller such that the balancing range is as large as possible. Solving (42), we obtain the following result:
Fig. 3. State and input with different initial conditions: solid curves-84 ; dotted–dashed curves-45 ; dashed curves-20 .
Because the modeling error is not considered, the stability region of the fuzzy system may not be inside the stability region of the original nonlinear system. In what follows, we will analyze the modeling error between above fuzzy model and the actual pendulum model (43)–(44). With the fuzzy rules shown in (48), the resulting fuzzy system is given by
(51) corresponds to an angle of 128 . By Note that applying the above controller to the true plant (43)–(44) with , 45 and 20 ( ), the system initial conditions responses are shown in Fig. 3.
(52) The difference between (44) and (52) is
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where
Now, we use the following uncertain fuzzy model to analyze the stability of the original system IF THEN
is about
IF
is about
Fig. 4. State and input responses with different initial conditions: solid curves-84 ; dotted–dashed curves-69 ; dashed curves-45 .
THEN
(53)
In the aforementioned uncertain fuzzy models, we assume that the uncertainties that describe the modeling errors are in the form of
where
with
, then we gain
With the fixed controller (47), the optimization problem (31) has no solution. When the controller is not fixed, we obtain the following result:
, and
The parameters , , , , , and are to be determined. Hence, the uncertain nonlinear fuzzy system is
(54) i.e.,
(55) , If we set then they can be chosen as
If we constrain
and
,
Simulation results with this controller are shown in Fig. 4. This simulation shows that, by optimizing the feedback gain as in Section IV, even a constant controller is able to balance the pendulum in range of [ 84 , 84 ] and, more importantly, the estimated range of [ 69 , 69 ] is indeed inside the actual range of [ 84 , 84 ]. VI. CONCLUSION This paper has presented a stability analysis and design method for a class of nonlinear systems with actuator saturation. TS fuzzy models with actuator saturation and norm-bounded uncertainty are first extended to describe the nonlinear systems subject to actuator saturation. A set invariance condition for the TS model is then established. Based on this set invariance condition, an LMI-based optimization approach is proposed to estimate the domain of attraction of the fuzzy system under given feedback laws. A fuzzy scheduling controller design method is then developed to enlarge the domain of attraction of the closed-loop system. Finally, the design methodology is illustrated by its application to the stabilization of an inverted pendulum on a cart. REFERENCES [1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [2] D. S. Bernstein and A. N. Michel, “A chronological bibliography on saturating actuators,” Int. J. Robust Nonlinear Control, vol. 5, pp. 375–380, 1995.
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Yong-Yan Cao (M’97–SM’00) was born in Hunan, China, in 1968. He received the B.E. and M.Sc. degrees in electrical engineering from Wuhan University of Science and Technlogy, China, and the Ph.D. degree in industrial automation from Zhejiang University, China, in 1990, 1993, and 1996, respectively. From 1996 to 1997, he was with the Institute of Industrial Process Control, Zhejiang University, China, as a Postdoctoral Researcher. From 1997 to 1998, he was a Visiting Research Associate in the Department of Mechanical Engineering, The University of Hong Kong. From 1998 to 1999, he held an Associate Professor position at Zhejiang University. He then spent a year-and-a-half with the Department of Measurement and Control, Duisburg University, Germany, as an Alexander von Humboldt Research Fellow. In May 2000, he joined the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, as a Research Scientist. He has published more than 50 papers in some reputed journals and conferences such as the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Automatica, and Systems and Control Letters. He also has been active both in his own research and in his service as a Reviewer for many journals and conferences. His current research interests include robust and nonlinear control, constrained control, time-delay systems, sampled-data systems, and fuzzy control. Dr. Cao currently serves as an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society.
Zongli Lin (S’89–M’90–SM’93) was born in Fuqing, Fujian, China, in 1964. He received the B.S. degree in mathematics and computer science from Amoy University, Xiamen, China, the M.E. degree in automatic control from Chinese Academy of Space Technology, Beijing, China, and the Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, in 1983, 1989, and 1994, respectively. From July 1983 to July 1986, he worked as a Control Engineer at the Chinese Academy of Space Technology, China. In January 1994, he joined the Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, as a Visiting Assistant Professor, where he became an Assistant Professor in September 1994. Since July 1997, he has been with the Department of Electrical and Computer Engineering at University of Virginia, Charlottesville, where his is currently an Associate Professor. His current research interests include nonlinear control, robust control, control of systems with saturating actuators and modeling, and control of magnetic bearing systems. In these areas, he has published several papers. He is also the author of the book Low Gain Feedback (London, U.K.: Springer-Verlag, 1998) and a coauthor of the recent book Control Systems with Actuator Saturation: Analysis and Design (Boston, MA: Birkhäuser, 2001). Dr. Lin currently serves as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. Previously, he was an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society. He is also a member of the IEEE Control Systems Society’s Technical Committee on Nonlinear Systems and Control, and heads its Working Group on Control with Constraints. He is the recipient of a U.S. Office of Naval Research Young Investigator Award.
