Stable fuzzy control system design with pole ... - Semantic Scholar
Computers in Industry 51 (2003) 1–11
Stable fuzzy control system design with pole-placement constraint: an LMI approach Sung Kyung Honga,*, Yoonsu Namb a
School of Mechanical & Aerospace Engineering, Sejong University, 98 Kunja-Dong, Kwangjin-Ku, Seoul 143-747, South Korea b Department of Mechanical & Mechatronics Engineering, Kangwon National University, 192-1 Hoya-Dong, Kwangjin-Ku, Chunchon, Kangwon-Do 200-701, South Korea Received 15 November 2001; accepted 24 March 2003
Abstract In this paper, the synthesis of an Linear Matrix Inequality (LMI)-based stable fuzzy control system with pole-placement constraint is presented. The requirements of stability and pole-placement region are formulated based on the Lyapunov direct method. By recasting these constraints into LMIs, we formulate an LMI feasibility problem for the design of the fuzzy state feedback control system that guarantees stability and satisfies desired transient responses. This theoretical approach is applied to a nonlinear magnetic bearing system concerning the issue of rotor position control. Simulation results show that the proposed LMI-based design methodology yields better performance than those of a linear local controller or single objective controller. In addition, it is observed that the proposed fuzzy state feedback controller provides superior stability robustness against parameter variations. # 2003 Elsevier Science B.V. All rights reserved. Keywords: LMI; Fuzzy control system
1. Introduction In recent years, there have been a number of studies of the Takagi–Sugeno (TS) fuzzy model [1], which provides an effective representation of nonlinear systems with the aid of fuzzy sets, fuzzy rules, and a set of local linear models. Once the fuzzy model is obtained, control design is carried out via the so called Parallel Distributed Compensator (PDC) approach [1–5], which employs multiple linear controllers that correspond * Corresponding author. Tel.: þ82-2-3408-3772; fax: þ82-2-3408-3333. E-mail addresses: [email protected] (S.K. Hong), [email protected] (Y. Nam).
to the locally linear plant models with automatic scheduling performed by fuzzy rules. For this TS model based fuzzy control system (shortly FCS), Tanaka and Sugeno [6] and Wang et al. [7] proved the stability by finding a common symmetric positive definite matrix P for the r subsystems and suggested the idea of using Linear Matrix Inequality (LMI) for finding the common P matrix. By introducing the stability issue in FCS, their works have been considered very important results and some refining efforts have been pursued thereafter. However, the design process presented in [6,7] involves an iterative process. That is, for each rule a controller is designed based on consideration of local performance only, then LMI-based stability analysis is carried out to check the
0166-3615/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0166-3615(03)00057-5
2
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
global stability condition. In the case that the stability conditions are not satisfied, the controller for each rule should be redesigned. To overcome such a defect, Zhao et al. [8] pointed out that it is more desirable to directly design a FLC (instead of iterative process in [6,7]) which guarantees global stability by recasting to LMI problems. On the other hand, Hong and Langari [9] dealt with the method of LMI based direct FLC design with imposing H1 objective. They, however, focused on single objective such as stability and robustness, respectively, and did not consider imposing other important requirements such as transient performance issues at the same time. Generally, such single objective designs focused on either stability issue or robustness issue does not directly deal with the desired dynamic characteristic of the closedloop system, which is commonly expressed in terms of transient responses. In contrast, satisfactory time response and closed-loop damping can be enforced by constraining the closed-loop poles to lie in a suitable sub-region of the left-half plane [10]. Motivated by the LMI formulation of pole placement constraint of the conventional state feedback case in [10], there has been needs in the TS model based FCS case for the way of simultaneously guaranteeing global stability and adequate transient behaviors. In this paper, our main focus is on (1) the extension of the previous LMI-based direct design methodology for the stable FCS by imposing the additional requirement of the closed-loop pole location, and (2) the demonstration of the usefulness of the proposed design methodology via applying it to the regulation problem of the industrial nonlinear magnetic bearing system. The same model that has been studied previously in [5,9] is used. Simulation results show that the proposed LMI-based design methodology yields better performance than those of a linear local controller or single objective controller. In addition, it is observed that the proposed fuzzy state feedback controller provides superior stability robustness against parameter variations. This paper is organized into five sections. The next section introduces the background materials concerning TS fuzzy model and model-based fuzzy controller. Section 3 describes the formulations of the LMI-based fuzzy state feedback controller for the stability and the closed-loop pole location requirements. In Section 4, simulation results are presented by the application of
the proposed methodology to the nonlinear magnetic bearing system. Concluding remarks are given in Section 5.
2. TS fuzzy model and control 2.1. Affine TS fuzzy model It has been shown that fuzzy systems are universal approximator, i.e. they are capable of approximating any real continuous function on a compact set with arbitrary accuracy. To this end, the so called TS fuzzy model proposed by Takagi and Sugeno [1] offers an effective way to represent nonlinear dynamic systems in terms of a set of fuzzy If-Then rules whose consequent parts represent locally linear input-output relations that characterize a general nonlinear system. In general, an nth order SISO nonlinear system can be represented by the following linearized state space form with the bias term d induced from the model linearization [3]: x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ d
(1)
When d ¼ 0, this model is called a linear model. Otherwise, we call it an affine model. The continuous fuzzy dynamic model is described by fuzzy If-Then rules to express local linear input–output relations of nonlinear systems around each operating point by above linear local model. The ith rule of this fuzzy model is of the following form: If x1 ðtÞ is Li1 and xn ðtÞ is Ln1 and uðtÞ is Mi ; Then x_ ðtÞ ¼ Ai xðtÞ þ Bi uðtÞ þ di
(2)
where i ¼ 1; 2; . . . ; r and r is the number of rules and Lij and Mi are fuzzy sets centered at the ith operating point. The rules in effect suggest what form the plant model takes (in terms of Ai, Bi, and di), depending on the region of operation and the value of inputs. The global model is composed as a concatenation of the local models, and can be seen as a smoothed piecewise approximation of a nonlinear function. The truth value of the ith rule in the set li is usually obtained as the product of the membership grades in the premise part: li ðx; uÞ ¼ mLii1 ðx1 Þ mLi2 ðx2 Þ mLin ðx2 Þ mMi ðuÞ
(3)
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
Takagi and Sugeno [1] define the inference in the rule base as the weighted average of each rule’s consequents: Pr x_ ¼
x þ Bi u i¼1 li ðA Pi r i¼1 li
þ di Þ
(4)
It follows that for a given x and u, the global model is a convex linear combination of the local models. 2.2. TS model-based fuzzy control The concept of PDC, following the terminology [1,2], is utilized to design fuzzy state-feedback controllers on the basis of the TS fuzzy models (4). Linear control theory can be used to design the consequent parts of the fuzzy control rules, because the consequent parts of TS fuzzy models are described by linear state equations. If we compute the control input u to be ui ðtÞ ¼ ~ ui ðtÞ k0i
(5)
3
local sub-model of the system to be controlled. The resulting total control action is Pr li ðKi x k0i Þ (9) u ¼ i¼1 Pr i¼1 li Note that the resulting fuzzy controller (9) is nonlinear in general since the coefficient of the controller depends nonlinearly on the system input and output via the fuzzy weights. Substituting (9) into (4), the closed-loop FCS can be represented by Pr Pr li lj ðAi þ Bi Kj Þx i¼1 Pj¼1 x_ ¼ (10) r Pr i¼1 j¼1 li lj Also, the system (10) can be rewritten as " # r X 1 X li li ðAi þ Bi Ki Þ þ 2 li lj Gij x x_ ¼ W i¼1 i<j where Gij ¼ 12 ðAi þ Bi Kj Þ þ ðAj þ Bj Ki Þ for
i < j;
and r X r X li lj
in order to cancel the bias term di, then the equation of the consequent part of TS fuzzy model (2) is described by
W¼
x_ ðtÞ ¼ Ai xðtÞ þ Bi ~ ui ðtÞ;
3. Design requirements and LMI formulation
i ¼ 1; 2; . . . ; r
(6)
Based on the revised piecewise linear model (6), we determine state feedback controller described by ~ uðtÞ ¼ Ki xðtÞ
(7)
where Ki is n-vector of feedback gains to be chosen for ith operating point via proper design methodologies. It should be noted that the value of the control input actually used in the fuzzy rules would be derived from Eq. (5). Hence, a set of r control rules takes the following form: If x1 ðtÞ is Li1 and xn ðtÞ is Ln1 and uðtÞ is Mi ; Then ui ðt þ 1Þ ¼ Ki xðtÞ k0i
(8)
where the index t þ 1 in the consequent part is introduced to distinguish the previous control action in the antecedent part in order to avoid algebraic loops. Each of the rules can be viewed as describing a local statefeedback controller associated with the corresponding
(11)
i¼1 j¼1
To determine fuzzy state feedback controller described by uðtÞ ¼ Kfuzzy ðx; uÞ x
(12)
the following design requirements are considered in this study. Stabilization: Design a controller such that the closed-loop FCS is asymptotically stable, i.e. lim xðtÞ ¼ 0
t!1
(13)
for all initial condition x(0). Pole placement: Design a controller such that the closed-loop eigenvalues of FCS are located in a prescribed sub-region (D) in the left half plane to prevent too fast controller dynamics and achieve desired transient behavior, i.e. sðAi þ Bi Kj Þ D for all initial condition x(0).
(14)
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S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
3.1. Stabilization A sufficient quadratic stability condition derived by Tanaka and Sugeno [6] for ensuring stability of (10) is given as follows: Theorem 1. The FCS (10) is quadratically stable for some stable feedback Ki (via PDC scheme) if
Multiplying the inequality on the left and right by Qs (such a congruence preserves the inequality), we get an LMI in Yi and Qs. The following theorem provides a FCS with guaranteed stability. Theorem 3. The FCS (11) is stabilizable in PDC scheme if there exists a common Qs < 0 and Yi such that the following LMI condition hold:
Ai Qs þ Qs ATi þ Bi Yi þ YiT BTi < 0; 1 2 ½Ai Qs
i ¼ 1; 2; . . . ; r
þ Qs ATi þ Bi Yj þ YjT BTi þ 12 ½Aj Qs þ Qs ATj þ Bj Yi þ YiT BTj < 0; i < j r
(18)
there exists a common positive definite matrix P such that
Given a solution ðQs ; Yi Þ, the fuzzy state feedback gain is obtained:
ðAi þ Bi Kj ÞT P þ PðAi þ Bi Kj Þ < 0; i; j ¼ 1; 2; . . . ; r
Ki ¼ Yi Q1 s (15)
which is an LMI in P when Ki’s are predetermined. This theorem reduces to the Lyapunov stability theorem for linear system when r ¼ 1. Applying Theorem 1 to the revised FCS (11), we have the following revised sufficient condition for the fuzzy control system. Theorem 2. The FCS (11) is quadratically stable for some state feedback Ki (via PDC scheme) if there exists a common positive definite matrix P such that ðAi þ Bi Ki ÞT P þ PðAi þ Bi Ki Þ < 0; GTij P þ PGij < 0;
i ¼ 1; 2; . . . ; r i<j r (16)
i.e. a common P has to exist for both ðAi þ Bi Kj Þ and Gij. However, our objective is to design the gain matrix Ki such that conditions (16) are satisfied. That is, Ki’s are not pre-determined matrices any longer, but matrix variables. This is the quadratic stabilizability problem and can be recast as an LMI feasibility problem. With linear fractional transformation Qs P1 and Yi Ki Qs , we may rewrite (16):
(19)
Consequently, we can synthesize directly a fuzzy state feedback (via PDC scheme) which guarantees the stability for the nonlinear system by solving a set of simultaneous Lyapunov inequalities. 3.2. Pole placement In the synthesis of control system, meeting some desired performances should be considered in addition to stability. Generally, stability condition (Theorems 1 and 2) does not directly deal with the transient responses of the closed-loop system. In contrast, a satisfactory transient response of a system can be guaranteed by confining its poles in a prescribed region. For many practical problems, exact pole assignment may not be necessary: it suffices to locate the closed-loop poles in a prescribed sub-region in the complex left half plane. This section discusses a Lyapunov characterization of pole clustering regions in terms of LMls. For this purpose, we introduce the following LMI-based representation of stability regions. Motivated by Chilali and Gahinet [10] and Gutman’s theorem for LMI region [11], we consider circle LMI region D Dq;r ¼ fx þ jy 2 C : ðx þ qÞ2 þ y2 < r 2 g
1 1 1 ðAi þ Bi Yi1 Qs ÞT Q1 s þ Qs ðAi þ Bi Yi Qs Þ < 0; 1 1 1 1 1 T 1 1 1 1 1 2 ½ðAi þ Bi Yj Qs Þ þ ðAj þ Bj Yi Qs Þ Qs þ 2 Qs ½ðAi þ Bi Yj Qs Þ 1 1 þðAj þ Bj Yi Qs Þ < 0;
(20)
i ¼ 1; 2; . . . ; r (17) i<j r
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
5
Theorem 5. The FCS (11) is D-stablizable in PDC scheme if and only if there exists a positive symmetric matrix Qp and Yi such that the following LMI condition holds rQp
qQp þ Qp ATi þ YiT BTi
qQp þ Ai Qp þ Bi Yi
rQp
!
i¼j Fig. 1. Circular region (D) for pole location.
centered at (q, 0) and with radius r > 0, where the characteristic function is given by r z þ q fD ðzÞ ¼ (21) z þ q r As shown in Fig. 1, if l ¼ Bon þ jod is a complex pole lying in Dqr with damping ratio z, undamped natural frequency on, damped natural frequency od, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi then B > 1 ðr 2 =q2 Þ, on < q þ r, and od < r. Therefore, this circle region puts a lower bound on both exponential decay rate and the damping ratio of the closed-loop response, and thus is very common in practical control design. An extended Lyapunov Theorem for the FCS (11) is developed with above definition of an LMI-based circular pole region as below [10]. Theorem 4. The FCS (11) is D-stable (all the complex poles lying in LMI region D) for some state feedback Ki if and only if there exists a positive symmetric matrix Qp such that qQp þ Qp ðAi þ Bi Kj ÞT rQp qQp þ ðAi þ Bi Kj ÞQp rQp
!
Stable fuzzy control system design with pole-placement constraint: an LMI approach Sung Kyung Honga,*, Yoonsu Namb a
School of Mechanical & Aerospace Engineering, Sejong University, 98 Kunja-Dong, Kwangjin-Ku, Seoul 143-747, South Korea b Department of Mechanical & Mechatronics Engineering, Kangwon National University, 192-1 Hoya-Dong, Kwangjin-Ku, Chunchon, Kangwon-Do 200-701, South Korea Received 15 November 2001; accepted 24 March 2003
Abstract In this paper, the synthesis of an Linear Matrix Inequality (LMI)-based stable fuzzy control system with pole-placement constraint is presented. The requirements of stability and pole-placement region are formulated based on the Lyapunov direct method. By recasting these constraints into LMIs, we formulate an LMI feasibility problem for the design of the fuzzy state feedback control system that guarantees stability and satisfies desired transient responses. This theoretical approach is applied to a nonlinear magnetic bearing system concerning the issue of rotor position control. Simulation results show that the proposed LMI-based design methodology yields better performance than those of a linear local controller or single objective controller. In addition, it is observed that the proposed fuzzy state feedback controller provides superior stability robustness against parameter variations. # 2003 Elsevier Science B.V. All rights reserved. Keywords: LMI; Fuzzy control system
1. Introduction In recent years, there have been a number of studies of the Takagi–Sugeno (TS) fuzzy model [1], which provides an effective representation of nonlinear systems with the aid of fuzzy sets, fuzzy rules, and a set of local linear models. Once the fuzzy model is obtained, control design is carried out via the so called Parallel Distributed Compensator (PDC) approach [1–5], which employs multiple linear controllers that correspond * Corresponding author. Tel.: þ82-2-3408-3772; fax: þ82-2-3408-3333. E-mail addresses: [email protected] (S.K. Hong), [email protected] (Y. Nam).
to the locally linear plant models with automatic scheduling performed by fuzzy rules. For this TS model based fuzzy control system (shortly FCS), Tanaka and Sugeno [6] and Wang et al. [7] proved the stability by finding a common symmetric positive definite matrix P for the r subsystems and suggested the idea of using Linear Matrix Inequality (LMI) for finding the common P matrix. By introducing the stability issue in FCS, their works have been considered very important results and some refining efforts have been pursued thereafter. However, the design process presented in [6,7] involves an iterative process. That is, for each rule a controller is designed based on consideration of local performance only, then LMI-based stability analysis is carried out to check the
0166-3615/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0166-3615(03)00057-5
2
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
global stability condition. In the case that the stability conditions are not satisfied, the controller for each rule should be redesigned. To overcome such a defect, Zhao et al. [8] pointed out that it is more desirable to directly design a FLC (instead of iterative process in [6,7]) which guarantees global stability by recasting to LMI problems. On the other hand, Hong and Langari [9] dealt with the method of LMI based direct FLC design with imposing H1 objective. They, however, focused on single objective such as stability and robustness, respectively, and did not consider imposing other important requirements such as transient performance issues at the same time. Generally, such single objective designs focused on either stability issue or robustness issue does not directly deal with the desired dynamic characteristic of the closedloop system, which is commonly expressed in terms of transient responses. In contrast, satisfactory time response and closed-loop damping can be enforced by constraining the closed-loop poles to lie in a suitable sub-region of the left-half plane [10]. Motivated by the LMI formulation of pole placement constraint of the conventional state feedback case in [10], there has been needs in the TS model based FCS case for the way of simultaneously guaranteeing global stability and adequate transient behaviors. In this paper, our main focus is on (1) the extension of the previous LMI-based direct design methodology for the stable FCS by imposing the additional requirement of the closed-loop pole location, and (2) the demonstration of the usefulness of the proposed design methodology via applying it to the regulation problem of the industrial nonlinear magnetic bearing system. The same model that has been studied previously in [5,9] is used. Simulation results show that the proposed LMI-based design methodology yields better performance than those of a linear local controller or single objective controller. In addition, it is observed that the proposed fuzzy state feedback controller provides superior stability robustness against parameter variations. This paper is organized into five sections. The next section introduces the background materials concerning TS fuzzy model and model-based fuzzy controller. Section 3 describes the formulations of the LMI-based fuzzy state feedback controller for the stability and the closed-loop pole location requirements. In Section 4, simulation results are presented by the application of
the proposed methodology to the nonlinear magnetic bearing system. Concluding remarks are given in Section 5.
2. TS fuzzy model and control 2.1. Affine TS fuzzy model It has been shown that fuzzy systems are universal approximator, i.e. they are capable of approximating any real continuous function on a compact set with arbitrary accuracy. To this end, the so called TS fuzzy model proposed by Takagi and Sugeno [1] offers an effective way to represent nonlinear dynamic systems in terms of a set of fuzzy If-Then rules whose consequent parts represent locally linear input-output relations that characterize a general nonlinear system. In general, an nth order SISO nonlinear system can be represented by the following linearized state space form with the bias term d induced from the model linearization [3]: x_ ðtÞ ¼ AxðtÞ þ BuðtÞ þ d
(1)
When d ¼ 0, this model is called a linear model. Otherwise, we call it an affine model. The continuous fuzzy dynamic model is described by fuzzy If-Then rules to express local linear input–output relations of nonlinear systems around each operating point by above linear local model. The ith rule of this fuzzy model is of the following form: If x1 ðtÞ is Li1 and xn ðtÞ is Ln1 and uðtÞ is Mi ; Then x_ ðtÞ ¼ Ai xðtÞ þ Bi uðtÞ þ di
(2)
where i ¼ 1; 2; . . . ; r and r is the number of rules and Lij and Mi are fuzzy sets centered at the ith operating point. The rules in effect suggest what form the plant model takes (in terms of Ai, Bi, and di), depending on the region of operation and the value of inputs. The global model is composed as a concatenation of the local models, and can be seen as a smoothed piecewise approximation of a nonlinear function. The truth value of the ith rule in the set li is usually obtained as the product of the membership grades in the premise part: li ðx; uÞ ¼ mLii1 ðx1 Þ mLi2 ðx2 Þ mLin ðx2 Þ mMi ðuÞ
(3)
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
Takagi and Sugeno [1] define the inference in the rule base as the weighted average of each rule’s consequents: Pr x_ ¼
x þ Bi u i¼1 li ðA Pi r i¼1 li
þ di Þ
(4)
It follows that for a given x and u, the global model is a convex linear combination of the local models. 2.2. TS model-based fuzzy control The concept of PDC, following the terminology [1,2], is utilized to design fuzzy state-feedback controllers on the basis of the TS fuzzy models (4). Linear control theory can be used to design the consequent parts of the fuzzy control rules, because the consequent parts of TS fuzzy models are described by linear state equations. If we compute the control input u to be ui ðtÞ ¼ ~ ui ðtÞ k0i
(5)
3
local sub-model of the system to be controlled. The resulting total control action is Pr li ðKi x k0i Þ (9) u ¼ i¼1 Pr i¼1 li Note that the resulting fuzzy controller (9) is nonlinear in general since the coefficient of the controller depends nonlinearly on the system input and output via the fuzzy weights. Substituting (9) into (4), the closed-loop FCS can be represented by Pr Pr li lj ðAi þ Bi Kj Þx i¼1 Pj¼1 x_ ¼ (10) r Pr i¼1 j¼1 li lj Also, the system (10) can be rewritten as " # r X 1 X li li ðAi þ Bi Ki Þ þ 2 li lj Gij x x_ ¼ W i¼1 i<j where Gij ¼ 12 ðAi þ Bi Kj Þ þ ðAj þ Bj Ki Þ for
i < j;
and r X r X li lj
in order to cancel the bias term di, then the equation of the consequent part of TS fuzzy model (2) is described by
W¼
x_ ðtÞ ¼ Ai xðtÞ þ Bi ~ ui ðtÞ;
3. Design requirements and LMI formulation
i ¼ 1; 2; . . . ; r
(6)
Based on the revised piecewise linear model (6), we determine state feedback controller described by ~ uðtÞ ¼ Ki xðtÞ
(7)
where Ki is n-vector of feedback gains to be chosen for ith operating point via proper design methodologies. It should be noted that the value of the control input actually used in the fuzzy rules would be derived from Eq. (5). Hence, a set of r control rules takes the following form: If x1 ðtÞ is Li1 and xn ðtÞ is Ln1 and uðtÞ is Mi ; Then ui ðt þ 1Þ ¼ Ki xðtÞ k0i
(8)
where the index t þ 1 in the consequent part is introduced to distinguish the previous control action in the antecedent part in order to avoid algebraic loops. Each of the rules can be viewed as describing a local statefeedback controller associated with the corresponding
(11)
i¼1 j¼1
To determine fuzzy state feedback controller described by uðtÞ ¼ Kfuzzy ðx; uÞ x
(12)
the following design requirements are considered in this study. Stabilization: Design a controller such that the closed-loop FCS is asymptotically stable, i.e. lim xðtÞ ¼ 0
t!1
(13)
for all initial condition x(0). Pole placement: Design a controller such that the closed-loop eigenvalues of FCS are located in a prescribed sub-region (D) in the left half plane to prevent too fast controller dynamics and achieve desired transient behavior, i.e. sðAi þ Bi Kj Þ D for all initial condition x(0).
(14)
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S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
3.1. Stabilization A sufficient quadratic stability condition derived by Tanaka and Sugeno [6] for ensuring stability of (10) is given as follows: Theorem 1. The FCS (10) is quadratically stable for some stable feedback Ki (via PDC scheme) if
Multiplying the inequality on the left and right by Qs (such a congruence preserves the inequality), we get an LMI in Yi and Qs. The following theorem provides a FCS with guaranteed stability. Theorem 3. The FCS (11) is stabilizable in PDC scheme if there exists a common Qs < 0 and Yi such that the following LMI condition hold:
Ai Qs þ Qs ATi þ Bi Yi þ YiT BTi < 0; 1 2 ½Ai Qs
i ¼ 1; 2; . . . ; r
þ Qs ATi þ Bi Yj þ YjT BTi þ 12 ½Aj Qs þ Qs ATj þ Bj Yi þ YiT BTj < 0; i < j r
(18)
there exists a common positive definite matrix P such that
Given a solution ðQs ; Yi Þ, the fuzzy state feedback gain is obtained:
ðAi þ Bi Kj ÞT P þ PðAi þ Bi Kj Þ < 0; i; j ¼ 1; 2; . . . ; r
Ki ¼ Yi Q1 s (15)
which is an LMI in P when Ki’s are predetermined. This theorem reduces to the Lyapunov stability theorem for linear system when r ¼ 1. Applying Theorem 1 to the revised FCS (11), we have the following revised sufficient condition for the fuzzy control system. Theorem 2. The FCS (11) is quadratically stable for some state feedback Ki (via PDC scheme) if there exists a common positive definite matrix P such that ðAi þ Bi Ki ÞT P þ PðAi þ Bi Ki Þ < 0; GTij P þ PGij < 0;
i ¼ 1; 2; . . . ; r i<j r (16)
i.e. a common P has to exist for both ðAi þ Bi Kj Þ and Gij. However, our objective is to design the gain matrix Ki such that conditions (16) are satisfied. That is, Ki’s are not pre-determined matrices any longer, but matrix variables. This is the quadratic stabilizability problem and can be recast as an LMI feasibility problem. With linear fractional transformation Qs P1 and Yi Ki Qs , we may rewrite (16):
(19)
Consequently, we can synthesize directly a fuzzy state feedback (via PDC scheme) which guarantees the stability for the nonlinear system by solving a set of simultaneous Lyapunov inequalities. 3.2. Pole placement In the synthesis of control system, meeting some desired performances should be considered in addition to stability. Generally, stability condition (Theorems 1 and 2) does not directly deal with the transient responses of the closed-loop system. In contrast, a satisfactory transient response of a system can be guaranteed by confining its poles in a prescribed region. For many practical problems, exact pole assignment may not be necessary: it suffices to locate the closed-loop poles in a prescribed sub-region in the complex left half plane. This section discusses a Lyapunov characterization of pole clustering regions in terms of LMls. For this purpose, we introduce the following LMI-based representation of stability regions. Motivated by Chilali and Gahinet [10] and Gutman’s theorem for LMI region [11], we consider circle LMI region D Dq;r ¼ fx þ jy 2 C : ðx þ qÞ2 þ y2 < r 2 g
1 1 1 ðAi þ Bi Yi1 Qs ÞT Q1 s þ Qs ðAi þ Bi Yi Qs Þ < 0; 1 1 1 1 1 T 1 1 1 1 1 2 ½ðAi þ Bi Yj Qs Þ þ ðAj þ Bj Yi Qs Þ Qs þ 2 Qs ½ðAi þ Bi Yj Qs Þ 1 1 þðAj þ Bj Yi Qs Þ < 0;
(20)
i ¼ 1; 2; . . . ; r (17) i<j r
S.K. Hong, Y. Nam / Computers in Industry 51 (2003) 1–11
5
Theorem 5. The FCS (11) is D-stablizable in PDC scheme if and only if there exists a positive symmetric matrix Qp and Yi such that the following LMI condition holds rQp
qQp þ Qp ATi þ YiT BTi
qQp þ Ai Qp þ Bi Yi
rQp
!
i¼j Fig. 1. Circular region (D) for pole location.
centered at (q, 0) and with radius r > 0, where the characteristic function is given by r z þ q fD ðzÞ ¼ (21) z þ q r As shown in Fig. 1, if l ¼ Bon þ jod is a complex pole lying in Dqr with damping ratio z, undamped natural frequency on, damped natural frequency od, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi then B > 1 ðr 2 =q2 Þ, on < q þ r, and od < r. Therefore, this circle region puts a lower bound on both exponential decay rate and the damping ratio of the closed-loop response, and thus is very common in practical control design. An extended Lyapunov Theorem for the FCS (11) is developed with above definition of an LMI-based circular pole region as below [10]. Theorem 4. The FCS (11) is D-stable (all the complex poles lying in LMI region D) for some state feedback Ki if and only if there exists a positive symmetric matrix Qp such that qQp þ Qp ðAi þ Bi Kj ÞT rQp qQp þ ðAi þ Bi Kj ÞQp rQp
!
