- Email: [email protected]
servo control design
Fuzzy Sets and Systems 158 (2007) 2288 – 2305 www.elsevier.com/locate/fss
Affine TS-model-based fuzzy regulating/servo control design Shinq-Jen Wu∗ Department of Electrical Engineering, Da-Yeh University, Chang-Hwa,Taiwan, ROC Received 2 August 2006; received in revised form 14 May 2007; accepted 16 May 2007 Available online 24 May 2007
Abstract An affine T-S fuzzy system is derived naturally from linearizing a nonlinear system or from some data-driven identification techniques, for example, cluster-based algorithms or soft-computation learning structures. Little research is proposed for the intrinsic analysis of an affine-type fuzzy system. Further, the controllers to regulate or to achieve servo control of affine TS-based nonlinear systems are also few. Both affine-type fuzzy regulation and servo control design scheme are theoretically derived. A simple Lyapunovbased stability criterion and some extra conditions are proposed to guarantee the global stability of the generated closed-loop fuzzy systems. The exponential stability of the feedback fuzzy system is ensured under some conditions. The performance of the proposed fuzzy controllers and fuzzy servo controllers is examined by three case studies. Simulation results show that the proposed controllers can stabilize these affine fuzzy systems and the proposed servomechanism can adapt itself to various incoming signals in very short time spans. © 2007 Elsevier B.V. All rights reserved. Keywords: Linear T-S system; Affine T-S system; Completely controllable; Completely observable; Optimal control
1. Introduction The research in fuzzy modelling and fuzzy control has come of age [2,24,17,29]. There are two model-based approaches to theoretically construct a T-S fuzzy system of a nonlinear system. One is from local linear approximation, which generates linear consequence with a constant term included in each rule, called an affine T-S fuzzy system. Mollov and coauthors propose multi-step linearization along the predicted input–output trajectory to decrease the linearization error for long-range fuzzy model predictive control [20,19]. The other is via a sector nonlinearity concept [27,23,26] which, in general, results in constant-free linear consequence for each rule, called a linear T-S fuzzy system [33]. Both fuzzy systems are demonstrated to be universal approximations to any smooth nonlinear system [33,28,41]. However, it is impractical to theoretically convert a mathematical model into a T-S fuzzy model if the nonlinear system is too complex to describe. More and more researchers attempt to learn fuzzy modelling from input–output data [40,7]. Grisales and coauthors propose two cluster-based algorithms to construct the fuzzy subsystem for each output: Gustafson–Kessel and robust parallel-competitive-agglomerative algorithms
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E-mail addresses: [email protected], [email protected] (S. Wu). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.05.012
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[9,10]. We previously proposed two self-organization neural-fuzzy networks to learn affine and linear T-S fuzzy systems, respectively [34,38]. An affine T-S fuzzy model is much preferred over a linear type in providing one more adjustable parameter for soft-computation (hybrid neural fuzzy and evolution) learning [34,38]. The linear T-S fuzzy system is the most popular fuzzy model due to its further intrinsic analysis: the linear matrix inequality (LMI)-based fuzzy controller is to minimize the upper bound of the performance index [28]; structureoriented and switching fuzzy controllers are developed for more complicated systems [26,25,22]; the optimal fuzzy control technique is used to minimize the performance index from local-concept or global-concept approaches [35–37]; Yang and coauthors use an input-free T-S fuzzy system to approximate an uncertain nonlinear state function and then adopt hybrid sliding-mode, adaptive and backstepping control techniques to control a strick-feedback uncertainty nonlinear system [39]. Via fuzzy static output feedback, Lo and Lin reform a robust H∞ quadratic tracking regulating issue into bilinear matrix inequalities [18]. As for the tracking control problem, it is a common issue in real world but it is very difficult to solve. Little research is devoted to linear T-S fuzzy tracking control problems. Cuevas and Toledo deal with a chaotic synchronization problem to obtain two Lorenz’ attractors [8]. Uang and Hung address a model-following control issue [32]. Chen and coauthors reform the H∞ tracking problem into a LMI problem [31] and adopt this technique to derive the reference-tracking control design of interconnected systems [30]. They further use a T-S fuzzy model to describe a fuzzy stochastic moving-average model with control input, which is really a local linear system approximating a nonlinear stochastic system. Then they derive the minimum-variance and model-reference tracking controllers [6] where the fuzzy decentralized concept (one-to-one rule correspondence between fuzzy system and fuzzy controller) is introduced directly. However, no theoretical demonstration exists for such correspondence. For affine T-S fuzzy systems, few regulating controllers with no corresponding servo controller are proposed. Hsiao and coauthors propose a hybrid compensation controller [13]. Kim and coauthors synthesize the affine-type fuzzy controller via the convex optimization technique and recast it into an LMI problem [14,15]. They further specialize in an affine T-S fuzzy system with a constant input matrix and transform the regulating problem into bilinear matrix inequalities [16]. Bergsten and coauthors try to derive an affine-type observer; but the constant-term consequence in an affine T-S fuzzy system is just a trivial term for observer derivation and the simulation is in fact a typical lineartype formulation [1]. Based on a clustering technique, Grisales and coauthors identify the affine T-S fuzzy rule for each output, and then propose discrete static-gain (for zero reference) and integral-gain (for constant reference) biascompensation-based fuzzy controllers to control a FAMIMO biological benchmark, which is a continuous-flow aerated bio-reactor for waste-water treatment [10]. Nounou and the coauthor propose three fuzzy model predictive control algorithms [21]. Chang and coauthors propose LMI-based fuzzy controllers for continuous and discrete affine T-S fuzzy systems with time-delay or constraint [3–5]. Though both consequent parts (linear and affine type) are represented by linear state equations, there exists a constant singleton in the fuzzy rule consequence for an affine T-S fuzzy model. This constant term could be a bias for a biotechnological process [10], an offset in model prediction control of a chemical process [20,19,21], a system’s parameter (characteristics) [38], or an external disturbance. Even if the fuzzy subsystem is linear the entire affine fuzzy system is highly nonlinear, and we cannot transform it to a linear type by shift-operation directly. Chang and coauthors propose a PDE-based LMI approach by lumping both input- and constant-terms together. In our previous papers [35], we proposed a simple local-based control scheme for linear-type fuzzy system. However, the stability criterion is not serious enough; the case for matrix-multiplication uncertainty is missed (Ai Aj − Aj Ai is neither positive semi-definite nor negative semi-definite.). Here I propose four different criteria to guarantee the stability. Besides, readers always confused our local-concept approach with the PDC technique: in our approach, we denote that under energy consideration there exists ith-to-ith correspondence between system rules and control rules . However, the PDC technique only denotes that the controller shares the same fuzzy sets with the fuzzy system in the premise parts; the ith system rule can be correspondent to the jth control rule [28]. To clarify this point, we here provide a more completely theoretical derivation to reinforce our technique. (Global effect can be achieved by local-concept approach.) We shall develop the relative controlling and servo controlling techniques for affine TS-model-based nonlinear systems. The technical contributions of this paper can be described as follows. The regulating control and servo control for affine TS-based nonlinear systems is theoretically derived. Lyapunov-based criterion and some extra conditions to ensure the stability of generated closed-loop fuzzy regulating/servo systems are proposed. Several nonlinear systems are concerned to examine the performance of those regulating and servo controllers.
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2. Affine-type fuzzy controller design The following affine T-S fuzzy system is considered to describe a nonlinear physical system: ˙ R i : If x1 is T1i , . . . , xn is Tni , then X(t) = Ai X(t) + Bi u(t) + Di , i = 1, . . . , r, Y (t) = CX(t),
(1)
where R i denotes the ith rule of the fuzzy model; x1 , . . . , xn are system states; T1i , . . . , Tni are the input fuzzy terms in the ith rule; X(t) = [x1 , . . . , xn ]T ∈ n is the state vector, Y (t) = [y1 , . . . , yn ]T ∈ n is the system output vector, and u(t) ∈ m is the system input; and Ai , Bi , Ci and Di are, respectively, n × n, n × m, n × n and n × 1 matrices. We shall design a rule-based fuzzy controller, R i : If y1 is S1i , . . . , yn is Sn i , then u(t) = ri (t), i = 1, . . . , ,
(2)
to minimize the quadratic cost functional, ∞ J (u(·)) = [X T (t)LX(t) + uT (t)Su(t)] dt,
(3)
t0
where X T (t)LX(t) is state-trajectory penalties with L belonging to symmetric positive semi-definite n × n matrices, and uT (t)Su(t) denotes energy consumption; y1 , . . . , yn are the elements of output vector Y (t), S1i , . . . , Sn i are the input fuzzy terms in the ith control rule, and the plant input (i.e., control output) vector u(t) or ri (t) is in m space. Though both consequent parts (linear and affine type) are represented by linear state equations, there exists a constant singleton in the fuzzy rule consequence for the affine T-S fuzzy model. This constant term could be a bias for a biotechnological process [10], an offset in model prediction control of a chemical process [21,20], a system’s parameter (characteristics) [38], or an external disturbance. In our simulation, it respectively denotes the linearization residue for the four-pole differential-driving magnetic bearings system (case 1), the transformed system’s parameter in a nonlinear system (case 2), and a fuzzy singleton learned from a soft-computation network (case 3). ˙ Even the fuzzy subsystem is linear, the entire affine fuzzy system X(t) = ri=1 hi (X(t))(Ai X(t) + Bi u(t) + Di ) is highly nonlinear. We cannot transform it to a linear type by shift-operation directly. Theorem 1 (Regulating control). For an affine T-S fuzzy system in Eq. (1) and a fuzzy controller in Eq. (2), if Ai is T ¯ ) > 0, nonsingular, (Ai , Bi ) completely controllable (c.c.), (Ai , C) completely observable (c.o.) and ¯ −1 i i (L + Ai ∀i = 1, . . . , r, then (1) the local fuzzy regulating law is ri∗ (t) = −S −1 BiT ¯ i X ∗ (t) + r¯is ,
i = 1, . . . , r,
∗ where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ) and X¯ is = A−1 i Di ; their “blending’’ global fuzzy controller, u (t) = ∗ ∗ (X (t))ri (t), minimizes J (u(·)) in (3), where ∞ −1 T T b¯ s = − e[Ai −Bi S Bi ¯ i ] · d · LX¯ s , i
i
0
(4) r
i=1
hi
(5)
and ¯ i is the unique symmetric positive semi-definite solution of the Riccati equation Ki Ai + Ati Ki − Ki Bi S −1 Bit Ki + L = 0;
(6)
(2) the entire feedback fuzzy system is stable, X˙ ∗ (t) =
r i=1
hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) + Bi r¯is + Di ].
(7)
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Proof. (1) From the essence of the dynamic programming formalism, we know min J (u(·)) = min
u[t0 ,∞)
u[t,∞)
∞ t
(XlT LXl
+ uTl Sul ) dl
+ min
t
u[t0 ,t] t 0
(XlT LXl
+ uTl Sul ) dl
,
(8)
where lower index is used to denote time-dependence for notation simplification, i.e., Xl for X(l). A quadratic optimization problem is hence turned into a successively on-going dynamic problem with regards to the state resulting from the previous decision, i.e., the initial state (at time t) X0t = Xt∗ . Moreover, according to the signal flow of a fuzzy inference system in Fig. 1 with y1 denotes x˙1 , the overall behavior of the fuzzy system at any time l is X˙ l =
r
hi (Xl )(Ai Xl + Bi ul + Di ),
l ∈ [t, ∞),
(9)
i=1
with ul = i=1 wi (Yl )ril and X0t = Xt∗ ∈ n ; hi (Xl ) and wi (Yl ) denote, respectively, the normalized firing strength of the ith rule of the fuzzy model and of the ith fuzzy control rule; i.e., hi (Xl ) = i / ri=1 i with i = nj=1 Tj i (Xl ), where Tj i (Xl ) is the membership function of fuzzy term Tj i , and wi (Yl ) = i / i=1 i with i = lj =1 Sj i (Yl ), where Sj i (Yl ) is the membership function of fuzzy term Sj i . Therefore, we shall focus on successively finding the continuous global decision (global fuzzy controller) u∗t for minimizing Jt (ut ) =
∞
t
(XlT LXl + uTl Sul ) dl,
t ∈ [t0 , ∞),
(10)
and estimating Xt∗+ with regards to the initial state Xt∗ , where t + denotes the time instant slightly later time t; and then with the new initial state, Xt∗+ , resolving u∗t + to minimize Jt + (ut + ). At any time instant t, the local decision (local fuzzy control law) stems from minimizing Jt (ut ) in Eq. (10) with regards to the fuzzy subsystem, X˙ l = Ai Xl + Bi ul + Di ,
l ∈ [t, ∞), i = 1, . . . , r;
(11)
the global decision results from minimizing Jt (ut ) with regard to the entire fuzzy system in Eq. (9). For clarity, since u∗t is only a variable to be solved regardless of the aforementioned local optimization problem or the global optimization issue, we can use ri∗t to denote the local decision of the ith fuzzy subsystem. (2) Now, let l (Xl , ul ) and il (Xl , ril ), i = 1, . . . , r, denote, respectively, the entire energy and local energy at any ∞ ∞ time instant l, l ∈ [t, ∞). Then, Jt (ut ) = t l (Xl , ul ) dl and Jt (rit ) = t il (Xl , ril ) dl. At any time instant, the energy of the entire fuzzy system is some kind of (nonlinear) combination of fuzzy subsystems’ energy; we can also describe the global energy in terms of rule-based local energy. We note that this combination is the function of system states only and has nothing to do with system inputs. In other words, whether the system behavior is nonlinear to system inputs or not, the input term should not be included as a fuzzy precondition in physical reality even if it is reasonable in mathematical concept terms. Besides, this nonlinear combination is not necessary to be the same type as fuzzily blending subsystems into an entire system. Therefore, we use h (X(t)) to denote that (a) the nonlinear combination is only state dependent; (b) the energy relationship between the entire system and subsystems could be fully different from the behavior relationship, denoted by the normalized membership function, h(X(t)). We then write l (Xl , ul ) = ri=1 hi (Xl ) il (Xl , ril ). At the time instant t with initial condition Xt∗ , let ri∗t denote the local decision to minimize Jt (rit ) for all i = 1 . . . , r, i.e., ∞ * it (Xt∗ , rit ) * *Jt (rit ) = (X , r ) dl = l il il ∗ = 0, ∗ *rit r ∗ *rit t *rit r r it
it
2 * Jt (rit ) *ri2 t
ri∗t
2 * it (Xt∗ , rit ) = *ri2 t
> 0. ri∗t
it
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y1
... Layer 6
... Layer 5
a11
a21
b11 u
X
b21
X
u
a31
b31
X u ...
Layer 4
R2
R1
...
R3
Layer 3
... Layer 2
... Layer 1
X1
X2
Fig. 1. Neural-fuzzy inference network for T-S fuzzy system.
Then, their corresponding global decision u˘ t = ri=1 hi (Xt∗ )rit |ri∗ can satisfy t ∞ *Jt (ut ) * = l (Xl , ul ) dl *ut u˘ t *ut t u˘ t ∞ r * = hi (Xl ) il (Xl , ril ) dl *ut t i=1 u˘ t r ∗, r ) (X * i i = hi (Xt∗ ) t t t *ut i=1 u˘ t r ∗ ∗ * it (Xt , rit ) *rit hi (Xt ) · = 0, = *ut u˘ t *rit i=1
2 * Jt (ut ) *u2t
= u˘ t
r i=1
* hi (Xt∗ )
2
it (Xt∗ , rit ) *ri2t
·
*rit *ut
2 r 2 ∗ ∗ * it (Xt , rit ) * rit hi (Xt ) · + > 0, *rit *u2t u˘ i=1 u˘ t
t
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i.e., u˘ t = u∗t . So, at any time instant t, if we can find ri∗t to minimize Jt (rit ) then their composed global decision u∗t can be the global minimizer of the total cost Jt (ut ). According to the above analysis, we know that, at any time instant, solving the quadratic optimal control problem is to find only one corresponding optimal solution of the fuzzy controller for each rule of the fuzzy model. ˆ (3) We further assume Ai , i = 1, . . . , r, is nonsingular and let X(t) = X(t) + X¯ is , where X¯ is = A−1 i Di . Then the local affine-type quadratic problem is further reformulated as follows: given the fuzzy subsystem, ˙ˆ = A Xˆ + B r , X i il l i l
l ∈ [t, ∞), i = 1, . . . , r
(12)
with the initial state resulting from the previous decision, i.e., Xˆ 0t = Xˆ t∗ , (a) find the local decision at time instant t, ri∗t , for minimizing the cost functional, ∞ ((Xˆ lT − X¯ is )L(Xˆ l − X¯ is ) + ritl Sril ) dl; (13) Jt (rit ) = t
(b) obtain the global decision u∗t at time instant t for minimizing the entire cost functional Jt (ut ) in Eq. (10) by r ˆ∗ ∗ ˆ∗ fuzzily blending each local decision, i.e., u∗t = i=1 hi (Xt )rit . Notice that the next-decision initial state Xt + is ∗ ∗ ˙ˆ = r h (Xˆ ∗ )(A Xˆ ∗ + B u∗ ). There exists a one˙ˆ = r h (Xˆ ∗ )(A Xˆ ∗ + B r ∗ ) instead of from X from X i t i il i t i t t t t t i=1 i i=1 i to-one relationship between each fuzzy subsystem and the corresponding fuzzy control rule. Since the local fuzzy system (i.e., fuzzy subsystem) is linear, its quadratic optimization problem is the same as the general linear quadratic tracking issue. Therefore, we obtain the regulating law ri∗ (t) = −S −1 BiT [¯ i (X ∗ (t) + X¯ is ) + bis (t)], fuzzy subsystem X˙ ∗ (t) = (Ai − Bi S −1 BiT ¯ i )(X ∗ (t) + X¯ is ) − Bi S −1 Bit bis (t) with bis (t) satisfies b˙is (t) = −(Ai − Bi S −1 BiT ¯ i )t bis (t) + LX¯ is ,
(14)
0 . Therefore, we where bis (∞) = 0n×1 . Further, since (Ai , Bi ) is c.c. and (Ai , C) is c.o., (Ai − Bi S −1 BiT ¯ i ) ∈ C− s s obtain bi (t) = b¯i in Eq. (5). Thus, the fuzzy control design scheme for an affine T-S fuzzy system in Theorem 1 can easily be derived. (4) For the purpose of stability analysis, we regard X¯ is as an artificial target. Let U¯ iart = Bi r¯is + Di , an artificial-target associated constant input. Then the stability of the resultant feedback fuzzy system in Eq. (7) concurs with the zero-input system,
X˙ ∗ (t) =
r
hi (X ∗ (t))[Ai − Bi S −1 BiT ¯ i ]X ∗ (t).
(15)
i=1
Defining Lyapunov function V (X) = Xt P X, where P is a symmetric positive matrix. Via Eq. (6), we have Ai − T ¯ ), and hence, Bi S −1 BiT ¯ i = −¯ −1 i i (L + Ai V˙ (X) = X˙ t P X + X t P X˙
r r
T −1 T T t −1 T = hi (X(t))X (Ai − Bi S Bi ¯ i ) PX + X P hi (X(t))(Ai − Bi S Bi ¯ i )X i=1 r
=−
i=1 T hi (X(t)){X T [(L + ATi ¯ i )T ¯ −1 ¯ −1 ¯ i )]X} i P + P i (L + Ai
i=1 r
= −2
T hi (X(t))[X t P ¯ −1 ¯ i )X]. i (L + Ai
(16)
i=1
Further choosing P = I > 0, we obtain V˙ = −2
r
T hi (X(t))[X T ¯ −1 ¯ i )X] < 0 i (L + Ai
(17)
i=1 T ¯ ) > 0, since h (X(t)) is always a positive number. for ¯ −1 i i i (L + Ai
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As we know, if A > 0 and B is Hermitian then AB has the same number of positive, negative and zero eigenvalues as B. T¯ ) > Therefore, for ¯ i and also ¯ −1 i i being symmetric positive definite (Hermitian with positive eigenvalues), if (L+Ai T ¯ ) > 0, which ensure the stability of ¯ −1 0, then (L + ATi ¯ i )T > 0; further, (L + ATi ¯ i )T ¯ −1 i i > 0 and also i (L + Ai the closed-loop system. So we have a more simple sufficient criterion. Corollary. If (L + ATi ¯ i ) > 0, then the closed-loop system in Eq. (7) is stable. In the remainder section, we further propose some criteria to ensure the exponential stability of the generated system in Eq. (7). The converse theorem is adopted in the following theorems. Before further derivation, we emphasize here that T-S fuzzy system is basically a locally linearized system. Therefore linearizing this fuzzy system is different from generalizing the differential operation to the whole nonlinear system. As we know, linearizing a nonlinear system with respect to a point X0 is to capture the approximate linear behavior at X0 neighborhood. A T-S fuzzy system can be described or realized as a fuzzy inference network in Fig. 1, where output y1 denotes x˙1 . Based on this inference structure, linearizing a T-S fuzzy system with respect to X0 can be figured out to root in the Fact X0 and developed ˙ upward to get output X(t). A fuzzy system is just like a successive separation-and-aggregation interaction; at any time instant each individual (subsystem) starts from the same point (state), travels away from each other, meets and fuzzily blends at incoming instant into the final target (output). We further use X¯ is = A−1 i Di to denote an artificial s art ¯ ¯ target, and then Ui = Bi ri + Di is an artificial-target associated constant input. Then the stability of the resultant feedback fuzzy system in Eq. (7) concurs with the zero-input system in Eq. (15). Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with its corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )(Ai − Bi S −1 Bit ¯ i )X(t).
(18)
i=1
Based on this inference, some criteria are proposed in Theorems 2–4 to ensure the exponential stability of the systems in Eq. (18) with regard to all X0 ∈ n . Therefore, the stability of generated system in Eq. (7) is ensured. Theorem 2. If each feedback fuzzy subsystem satisfies max (Aci + ATci ) < 0 where Aci = Ai − Bi S −1 BiT ¯ i , then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Theorem 3. If (Ai , Bi ) is c.c., (Ai , C) is c.o. and Aci = Ai − Bi S −1 BiT ¯ i is a Hermitian matrix for each fuzzy subsystem, then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Theorem 4. If (Ai , Bi ) is c.c., (Ai , C) is c.o. and Aci = Ai − Bi S −1 BiT ¯ i , i = 1, . . . , r, are simultaneously similar to diagonalization, or if the more relax condition Aci is simultaneously similar to triangularization, then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Recently, we have successfully applied the previously proposed linear-type optimal fuzzy controller to magnetic suspension systems, four-pole and eight-pole active magnetic bearing systems, inverted pendulum systems, half-car active suspension systems, and even the Taiwan iTS-1 experimental car. To reinforce our previous stability theorem, some brand-new theorems are proposed in this work. We further adopt three different examples which are, respectively, satisfies Corollary, Theorem 1 and neither one of the above theorems to point out that there must exit more relax conditions than these. It is our cordial wish that those mathematical-solid-background researchers could propose more flexible conditions to guarantee the stability of the closed-loop fuzzy system.
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3. Affine-type fuzzy servo control design The optimal servo control design scheme is to control the fuzzy servo system in such a way as to push the output Y (t) close to follow an incoming signal Y d (t) without excessive control-energy consumption where Y d (t) belongs to a known class of signal, z˙ (t) = F z(t), Y d (t) = Ez(t),
(19)
with z(t0 ) = z0 , where z(k) ∈ h , F and E are matrices of h×h and n ×h, respectively. Therefore, our affine-type servo control problem can be described as: given an affine T-S fuzzy servo system and a fuzzy servo controller, respectively, in Eqs. (1) and (2) with X(t0 ) = X0 ∈ n , find the individual servo law, ri∗ (·), i = 1, . . . , , such that the composed servo controller u∗ (·) can minimize the quadratic cost functional, ∞ J (u(·)) = (20) [uT (t)Su(t) + X T (t)L1 X(t) + (Y (t) − Y d (t))T L2 (Y (t) − Y d (t))] dt t0
with L1 = [In − C T (CC T )−1 C]t L3 [In − C T (CC T )−1 C]; S, L2 and L3 are m × m, n × n and n × n nonnegative symmetric matrices; XT (t)L1 X(t) is the state-trajectory penalty to produce smooth response; uT (t)Su(t) is energy consumption; and the last term in J (u(·)) is related to error cost. The desired output Y d (t) is the incoming signal in Eq. (19). ˜ Define X(t) = [XT (t) zT (t)]T and rewrite the performance index in Eq. (20) as ∞ J (u(·)) = [uT (t)Su(t) + (X(t) − X d (t))t L(X(t) − X d (t))] dt, (21) t0
where L = L1 + C T L2 C and the desired trajectory X d (t) = C T [CC T ]−1 Y d (t). Then the servo problem can be further simplified as the following augmented regulating problem: Given an augmented affine T-S fuzzy regulating system, ⎤ ⎡ r ˙˜ ˜ ˜ ⎣A˜ i X(t) + B˜ i (22) hi (X(t)) wj (Y (t))rj (t) + D˜ i ⎦ X(t) = i=1
j =1
˜ ˜ 0 ) = X˜ 0 ∈ n+h and hi (X(t)) = hi (X(t)), t ∈ [t0 , ∞), find the individual regulating law, ri∗ (·), i = 1, . . . , , with X(t to minimize ⎡ ⎤ ∞ ˜ ⎣X˜ T (t)L˜ X(t) + wi (Y (t))wj (Y (t))riT (t)Srj (t)⎦ dt, (23) J (ri (·)) = t0
i=1 j =1
= where the parameters are B˜ i T [CC T ]−1 E −LC L L˜ = −E T [CC T ]−1 CL E T [CC T ]−1 CLC T [CC T ]−1 E .
Bi 0h×m
,
D˜ i
=
Di 0h×1
,
A˜ i
=
Ai 0n×h
0h×n F
and
Based on the regulating control theorem in Section 2, we can derive the servo control theorem as follows. Theorem 5 (Servo control). For an affine T-S fuzzy system in Eq. (1) and a fuzzy servo controller in Eq. (2), let the desired trajectory, Xd (t), come from Y d (t) = CXd (t), where Y d (t) is an incoming signal in Eq. (19). If Ai is T ¯ ) > 0, ∀i = 1, . . . , r, then nonsingular, (Ai , Bi ) is c.c., (Ai , C) is c.o. and ¯ −1 i i (L + Ai (1) the local fuzzy servo law is s ri∗ (t) = −S −1 BiT [¯ i X ∗ + ¯ 21 i z(t)] + r¯i , T
i = 1, . . . , r, (24) and their “blending’’ global fuzzy servo controller, u∗ (t) = ri=1 hi (X ∗ (t))ri∗ (t), minimizes J (u(·)) in (20), ¯s ¯ i is the unique solution of the Riccati where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ), X¯ is = A−1 i Di , bi satisfies Eq. (5),
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equation in Eq. (6) and ∞ T −1 T 21 eF · E T (CC T )−1 CL · e(Ai −Bi S Bi ¯ i ) · d; ¯ i = −
(25)
0
(2) the entire feedback fuzzy system is stable, X˙ ∗ (t) =
r
s hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) − Bi S −1 BiT ¯ 21 i z(t) + Bi r¯i + Di ]. T
(26)
i=1
Proof. (1) Based on Theorem 1 in Section 2, we have ri∗ (t) = −S −1 B˜ iT ˜ i (t, t1 )X˜ ∗ (t) + r¯is (t), ˙˜ ∗ (t) = [A˜ − B˜ S −1 B˜ T ˜ (t, t )]X˜ ∗ (t) + B˜ r¯ s (t) + D˜ , X 1 i i i i i i i ˙b˜ s (t) = −[A˜ − B˜ S −1 B˜ T ˜ (t, t )]T b˜ s (t) + L˜ X ˜¯ s , b˜ s (t ) = 0 i 1 i (n+h)×1 , i i i i i i 1 ¯s s s ˜¯ + b˜ s (t)), X ˜¯ = A˜ −1 D˜ = Xi and ˜ (t, t ) is the symmetric positive semiwhere r¯is (t) = −S −1 B˜ iT (˜ i (t, t1 )X i i 1 i i i i 0h×1 definite solution of the following Riccati equation: ˜ −K˙˜ i (t) = K˜ i (t)A˜ i + A˜ Ti K˜ i (t) − K˜ i (t)B˜i S −1 B˜ iT K˜ i (t) + L,
K˜ i (t1 ) = 0(n+h)×(n+h) .
˜ t In fact, ˜¯ i = limt1 →∞ (t, ˜ t1 ) a constant Here we replace b˜is (∞) and ˜¯ i by b˜is (t1 ) and (t, 1 ) for further derivation.
s 21T (t) K (t) K b (t) ˜ = X(t) for the infinite horizon problem. Now, let K˜ i (t) = K 21i (t) Ki22 (t) , b˜i (t) = b i (t) and X(t) z(t) . Considering i
2i
i
the infinite horizon problem, t1 = ∞, and via complicated matrix manipulation, we obtain ri∗ (t) in Eq. (24) and X ∗ (t) in Eq. (26), where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ); b¯is is in Eq. (5) as (Ai , Bi ) c.c. and (Ai , Ci ) c.o.; ¯ i is the unique solution of the Riccati equation in Eq. (6); b2i (t) and ¯ 21 i , respectively, satisfy b˙2i (t) = K21 (t)Bi S −1 Bit bis (t) + F t bi (t) − E T (CC T )−1 CLX¯ is , b2i (∞) = 0h×1 , −K˙ 21 (t) = F t K21 (t) + K21 (t)Ai − K21 (t)Bi S −1 BiT ¯ i − E T (CC T )−1 CL,
(27)
K21 (∞) = 0h×n .
(28)
(2) We now wish to show that the solution of Eq. (28) is
¯ 21 i
in Eq. (25). Rewriting Eq. (28) as
− K˙ 21 (t) = F t K21 (t) + K21 (t)Aci − L21 ,
(29)
where Aci = Ai − Bi S −1 BiT ¯ i and L21 = E T (CC T )−1 CL. Then we obtain K21 (t) = 1 (t, t0 )K21 (t0 ) T2 (t, t0 ) +
T t0
1 (t, )L21 T2 (t, ) d,
(30)
˙ ˙ where 1 (t, t0 ) and 2 (t, t0 ) are state transition matrices of X(t) = −F t X(t) and X(t) = −Atci X(t), respectively. Therefore, we know t1 K21 (t1 ) = 1 (t1 , t0 )K21 (t0 ) T2 (t1 , t0 ) + 1 (t1 , )L21 T2 (t1 , ) d, t 0t1 T K21 (t0 ) = 1 (t0 , t1 )K21 (t1 ) 2 (t0 , t1 ) − 1 (t0 , )L21 T2 (t0 , ) d. t0
Substituting K21 (t0 ) into Eq. (30), we have K21 (t) =
1 (t, t1 )K21 (t1 ) T2 (t, t1 ) −
t1 t
1 (t, )L21 T2 (t, ) d.
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Further, limt1 →∞ K21 (t1 ) = 0h×n . Using K¯ 21 (t) to denote K21 (t) with zero infinity, we obtain ∞ T T K¯ 21 (t) = − e−F (t−) · L21 · e−Aci (t−) d. t
2297
(31)
Hence, we have ¯ 21 i in Eq. (25) which satisfies Eq. (31) and also Eq. (28). T ¯s (3) Stability analysis: since S −1 BiT ¯ 21 i Z(t) in Eq. (26) is associated with the target only and Bi ri + Di in Eq. (26) can be regarded as an artificial-target constant as in Proof of Theorem 1, the proposed optimal closed-loop servo control system in Eq. (26) can be rewritten as X˙ ∗ (t) =
r
hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) − Bi r¯iext (t) + U¯ iart ]
(32)
i=1
¯s ¯ art with r¯iext (t) = S −1 BiT ¯ 21 i Z(t) and Ui = Bi ri + Di . Further, the stability of the nonlinear servo control fuzzy system in Eq. (32) is coincident with that of the zero-input fuzzy system in Eq. (15). This is demonstrated to be stable in Proof T ¯ ) > 0. of Theorem 1 once each fuzzy subsystem satisfies (Ai , Bi ) c.c., (Ai , C) c.o. and ¯ −1 i i (L + Ai T
4. Numerical simulation In this section, a four-pole active magnetic bearing system and a sinusoidal nonlinear system are adopted in Section 4.1 to illustrate the proposed affine-type fuzzy control scheme. And an affine TS-based nonlinear system in Section 4.2 is used as a servomechanism to examine the performance of the proposed affine-type fuzzy servo controller. 4.1. Regulating control A four-pole differential-driving magnetic bearings driven by sum of bias current and control current is considered [11,12]. Based on the assumptions of symmetric structure and rigid floating mass, we have the dynamic equation of rotor motion:
ib − i(t) 2 ib + i(t) 2 − , (33) mx(t) ¨ = G − x(t) G + x(t) where i(t) is control current, rotor mass m = 0.0126 (lb s2 /in), nominal air gap G = 0.02 in, force constants = 0.00186 (lb in2 /A2 ), sensitivity of air gap to shaft displacement = 0.974, and the bias current ib = 0.3 A [12]. Fig. 2(a) is the magnetic force with respect to control current i(t) and rotor displacement x(t). Fig. 2(b) is the magneticforce evolution with control current i(t) at operating points, x(t) = 0, 0.005, 0.01 in. We found that for operation x(t) = 0.01 in, the magnetic-force evolution is the parabolic function with an extreme point at around i(t) = −0.3 A; and for x(t) = 0.005 in, the magnetic-force evolution is the parabolic function with an extreme point at around i(t) = −0.6 A. Taylor’s expansion technique is used to linearize the magnetic bearing system in Eq. (33) with respect to operating points (x(t), i(t)) = (0, 0), (0.005, 0), (0.01, 0), (0.005, −0.3) and (0.01, −0.3). And the following affine T-S fuzzy system is obtained [12]: ˙ = A1 X(t) + B1 i(t) + D1 , R 1 : If x is ZE, then X(t) 2 ˙ R : If x is PM and i is ZE, then X(t) = A2 X(t) + B2 i(t) + D2 , 3 ˙ R : If x is PB and i is ZE, then X(t) = A3 X(t) + B3 i(t) + D3 , ˙ R 4 : If x is PM and i is NE, then X(t) = A4 X(t) + B4 i(t) + D4 , 5 ˙ R : If x is PB and i is NE, then X(t) = A5 X(t) + B5 i(t) + D5 , (34) T and the membership functions of the fuzzy terms are shown in Figs. 2(c) and (d); A = where X(t) ˙ 1 = [x(t) x(t)] 0 1 0 1 0 1 0 1 0 1 0 0 = = = = = = , A , A , A , A ; B , B , 2 3 4 1 2 5 6489.7 0 6750 0 9182.0 0 25022 0 3948 0 444.2 532 0 0 0 0 0 0 0 B3 = 944 , B4 = 287.3 , B5 = 201 ; D1 = 00 , D2 = −9.2 , D3 = −138.7 , D4 = −33 , D5 = −39.5 ; Y = CX with C = [1 0] for each rule.
30 20 10 0 -10 -20 -30 0.01 Ro 0.005 0 tor -0.005 dis tan -0.01 -1 ce (In ch )
ZE
-0.5
5
)
amp
ent (
Curr
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
x=5.0 x=0.005 x=0.01
-1
-0.5
0
0.5
1
Current (amp)
PM
0
1
0.5
0
Magnetic force (Ecs)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Magnetic force (Ecs)
2298
PB
10
x (mili-inch)
NE
ZE
-0.3
0 u (amp)
Fig. 2. (a) Magnetic-force evolution with respect to control current i(t) and rotor displacement x(t); (b) magnetic-force evolution with control current i(t) at x(t) = 0, 0.005, 0.01 in; (c) and (d) membership functions for x(t) and u(t).
Each fuzzy subsystem in Eq. (34) is c.c. and c.o. since rank[Bi Ai Bi ] = 2 and rank[C T Ai t C T ]T = 2 for i = 1, . . . , 5. Let the penalty matrices be set as L = I2 and S = 1. Therefore, based on Theorem 1 in Section 2, we have the unique S.S.R.E. solutions, 15.5786 0.0659 18.3768 0.0649 27.9757 0.0562 ¯ 1 = , ¯ 2 = , ¯ 3 = , 0.0659 0.0024 0.0649 0.0020 0.0562 0.0011 27.0919 0.1636 23.1968 0.1956 ¯ 4 = , ¯ 5 = , 0.1636 0.0040 0.1956 0.0059 and (L + ATi ¯ i ) > 0, for all i = 1, . . . , 5. Fig. 4 shows the evolution of the position, the velocity of the shaft rotor, and the proposed control current with initial conditions x(0) = [0.0082 0]T and i(0) = −0.15 A; x(0) = [0.006 0]T and i(0) = −0.33 A; x(0) = [0.004 0]T and i(0) = −0.53 A; x(0) = [0.002 0]T and i(0) = −0.73 A. Simulation results show that the dynamic system can be stabilized in around just 0.4 s. We now consider another nonlinear system [16], x˙1 (t) = −x3 (t) + x1 (t) cos x2 (t) + (1 + cos x2 (t))u(t), x˙2 (t) = x1 (t) + u(t), x˙3 (t) = x1 (t) + 3x2 (t) + sin x2 (t) with = 0.5. A diffeomorphic transform ⎡ ⎤ ⎡ ⎤ x1 (t) x3 (t) ⎣ x2 (t) ⎦ = ⎣ x (t) − x (t) − sin x (t) ⎦ 2 2 1 x3 (t) x2 (t)
(35)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
M1
M2
2299
M3
_ x3 -E3
-E2
-E1
E1 2π 5
E1
E2
E3
1π 2
3π 2
E2
E3
Fig. 3. Membership functions for affine fuzzy system in Eq. (37).
is used to get a constant input function. Hence, we can rewrite Eq. (35) as x˙1 (t) = x2 (t) + 4x3 (t) + ( + 1) sin x3 (t), x˙2 (t) = −x1 (t) − x2 (t) − x3 (t) − sin x3 (t), x˙3 (t) = x2 (t) + x3 (t) + sin x3 (t) + u(t).
(36)
Then the following affine T-S fuzzy system is derived to describe the dynamic behavior of the transformed system in Eq. (36) [16], ˙ R 1 : If x3 (t) is M1 , then X(t) = A1 X(t) + B1 u(t) + D1 , 2 ˙ R : If x3 (t) is M2 , then X(t) = A2 X(t) + B2 u(t) + D2 , ˙ R 3 : If x3 (t) is M3 , then X(t) = A3 X(t) + B3 u(t) + D3 ,
(37)
where X(t) = [x1 (t) x2 (t) x3 (t)]T . The membership functions of the fuzzy terms are shown in Fig. 3; ⎡
⎡ ⎤ ⎤ 2 2 ( + 1) 0 1 4 + ( + 1) ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ 2 2 ⎢ ⎢ ⎥ A1 = ⎢ −1 −1 −1 + ⎥ , A2 = ⎢ −1 −1 −1 − ⎥ ⎥, ⎣ ⎣ ⎦ ⎦ 2 2 0 1 1− 0 1 1+ ⎡ ⎤ 2 0 1 4 − ( + 1) ⎢ ⎥ ⎢ ⎥ 2 ⎥ A3 = ⎢ ⎢ −1 −1 −1 + ⎥ ; ⎣ ⎦ 2 0 1 1− ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2( + 1) 0 2( + 1) ⎦ , D2 = ⎣ 0 ⎦ , D3 = ⎣ −2 ⎦ ; 2 D1 = ⎣ −2 0 2 0
1 4−
⎡ ⎤ 0 B1 = B2 = B3 = ⎣ 0 ⎦ ; 1 Y = CX with C = [100] for each rule. 2 Since each fuzzy subsystem in Eq. (37) is c.c. and c.o. (rank[Bi Ai Bi A2i Bi ] = 3 and rank[C T Ai t C T Ai t C T ]T = 3 for i = 1, 2, 3). The penalty matrices are set to be L = I3 and S = 1. Hence we have Riccati
2300
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
0
0.008
-0.02
0.006
-0.04
x
xdot
0.01
0.004 0.002
-0.06 -0.08
0
-0.1 0
0.2
0.4 time
0.6
0.8
0
0.5 time
1
0 -0.2 -0.4 -0.6 -0.8
0
0.2
0.4
0.6
0.8
time Fig. 4. The rotor position x ∗ (t) and velocity x˙ ∗ (t) of four-pole AMB system in Eq. (33) and the proposed optimal control current i ∗ (t) with initial conditions x(0) = [0.00820]T and i(0) = −0.15 A; x(0) = [0.0060]T and i(0) = −0.33 A; x(0) = [0.0040]T and i(0) = −0.53 A; x(0) = [0.0020]T and i(0) = −0.73 A.
solutions,
⎡
1.1050 ¯ 1 = ⎣ 0.0443 0.9547 ⎡ 1.1050 ¯ 3 = ⎣ 0.0443 0.9547
⎤ 0.0443 0.9547 1.0441 1.0174 ⎦ , 1.0174 3.1090 ⎤ 0.0443 0.9547 1.0441 1.0174 ⎦ . 1.0174 3.1090
⎡
⎤ 0.8699 −0.0489 1.0477 ¯ 2 = ⎣ −0.0489 0.9507 0.9696 ⎦ , 1.0477 0.9696 4.5686
Fig. 5 is the state response and the proposed optimal controller with initial conditions x(0) = [−7 1 10]T , [−7 − 1 − 10]T and [−7 1 − 10]T . It shows that the proposed controller can stabilize the system in 5 s. However, neither one of the above theorems are satisfied. Hence, there must be more relax conditions than these. We hope mathematical-trained researchers can propose more flexible conditions in the future to guarantee the stability of the closed-loop fuzzy system. 4.2. Servo control Via an analytical or hybrid soft-computing technique, any nonlinear system can be approximated by an affine T-S fuzzy system. Here we consider the following affine-TS-based nonlinear system to demonstrate the performance of the proposed servo controller in Section 3. ˙ R 1 : If x(t) is F11 and x(t) ˙ is F21 , then X(t) = A1 X(t) + B1 u(t) + D1 , 2 1 2 ˙ ˙ is F2 , then X(t) = A2 X(t) + B2 u(t) + D2 , R : If x(t) is F1 and x(t) ˙ ˙ is F21 , then X(t) = A3 X(t) + B3 u(t) + D3 , R 3 : If x(t) is F12 and x(t) 4 2 2 ˙ ˙ is F2 , then X(t) = A4 X(t) + B4 u(t) + D4 , R : If x(t) is F1 and x(t)
(38)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
3
5
2
0
1
x1
x2
10
-5
0
-10
-1
-15
2301
-2 0
5
10 time
15
20
10
0
5
10 time
15
20
0
5
10 time
15
20
30
5
20 u
x3
0 10
-5 0
-10 -15
-10 0
5
10 time
15
20
Fig. 5. State response X∗ (t) in Eq. (36) and the proposed optimal controller u∗ (t) with initial conditions x(0) = [−7 1 10]T , [−7 − 1 − 10]T and [−7 1 − 10]T .
and the system output is Y (t) = CX(t) with C = [01] for each rule, where 1 1 1 0.3 x(t) ˙ , D2 = , D3 = , D4 = ; X(t) = ; D1 = 1 0.5 0 1 x(t) 0 −0.02 −0.225 −0.02 0 −1.5275 A1 = , A2 = , A3 = , 1 0 1 0 1 0 1 −0.225 −1.5275 , i = 1, . . . , 4. A4 = ; Bi = 0 1 0 (t) The membership functions are chosen as F 1 (x(t)) = 1− x2.25 , F 2 (x(t)) =
(t) = 1− x˙2.25 , F 2 (x(t)) ˙ = 2 5 x˙ 2 (t) d 2.25 . This affine TS system is used for following an incoming signal Y (t) in Eq. (19) with initial condition z0 = 10 1 1 − 30 5 1 −3 1 0 3 and parameters E = [3 1], F = −3 . Now, let the penalty matrices , or 1 −5 −0.2 5 −2 −0.2 5 − 15 − 9 0.0316 in Eq. (20) be L3 = I2 , L2 = I1 , and S = 0.001. The unique Riccati solutions are ¯ 1 = 0.0326 ¯2 = 0.0316 1.0311 , 0.0324 0.0316 0.0326 0.0301 0.0323 0.0301 −1 T ¯ 3 = 0.0301 1.0309 , ¯ 3 = 0.0301 1.0306 and ¯ i (L + Ai ¯ i ) > 0, ∀i = 1, . . . , r. Fig. 6 is the 0.0316 1.0311 , servo control performance. 2
1
1
x 2 (t) ˙ 2.25 , F21 (x(t))
2
5. Conclusions Though a linear T-S fuzzy system is the most popular fuzzy model and has been successfully applied in various fields, an affine T-S is still the preferred method for computation-intelligent modelling of mode-free or highly nonlinear complex model-based physical system. We propose an affine-type fuzzy controller to stabilize a nonlinear system and a fuzzy servo controller to track an incoming signal. A Lyapunov-based stability criterion is proposed to guarantee global stability of the closed-loop fuzzy system. Some extra criteria are proposed for the exponential-stability property. Three different systems are used to demonstrate the proposed regulator and servo controller can, respectively, achieve the
2302
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
25
20
output response
output response
30 F=[-3 1; -5 -0.2]
10 0 -10
15 10 5 0
0
5
10 time
15
20
0
100
6 F=[-1/30 0; 5 -0.2]
80
output response
output response
F=[-3 1; 5 -2]
20
60 40 20 0
5
10 time
15
20
x 106 F=[1/3 5; -1/5 -1/9]
4 2 0 -2 -4
0
50 time
100
0
50
100
time
Fig. 6. Output response (dashed line) of affine TS-based servomechanism in Eq. (38) for incoming signals (solid line) with initial condition z(0) = [5 10], parameters E = [3 1] and various F.
desired stabilizing and tracking effects in a very short time span. The three adopted systems satisfy Corollary, Theorem 1 and neither one of the above theorems which suggests that there must be more relaxed conditions than these criteria. We conclude that highly mathematically trained researchers can perhaps propose more flexible conditions to guarantee the stability of the closed-loop fuzzy system. We examined the influence of a modelling error on the offset term by comparing the performance of a system with and without the compensate term [38]. Better stabilization performance is achieved by when the compensate term is included in the system. We also found the corresponding performance to be predictable for the original nonlinear closed-loop system. Acknowledgement We gratefully acknowledge the editorial assistance of Marc Grenier, MEG International English Center, Tainan City, Taiwan. This research is supported by the Program NSC 95-2221-E-212-013. Appendix A. Proof of Theorem 2. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(39)
i=1
We know the real parts of eigenvalues of Q, e [ (Q)] (Q), where (Q) is any kinds of matrix measure. And, (·) is a convex function; i.e., [aM + (1 − a)N] a(M) + (1 − a)(N ),
∀a ∈ [0, 1], ∀M, N ∈ n×n .
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Let gi (X0 ) =
r hi (X0 ) , fi (X0 ) i=1,i = hi (X0 )
=
gi (X0 ) gi (X0 )
r
2303
and use hi , gi , fi to denote hi (X0 ), gi (X0 ), fi (X0 ) for notation
i=1,i =,
simplification, respectively. r r e hi Aci hi Aci i=1
i=1
⎛
(1 − h ) ⎝ ⎡
⎞
r
gi Aci ⎠ + h (Ac ),
i=1,i =
⎛
⎞
r
(1 − h ) ⎣(1 − g ) ⎝
⎤
fi Aci ⎠ + g (Ac )⎦ + h (Ac ),
i=1,i =,
h (Ac ) + (1 − h )g (Ac ) + (1 − h )(1 − g )f (Ac ) + · · · . T If we choose the induced 2-norm as our matrix measure, 2 (Q) = max Q+Q , we have 2 r
Ac + ATc Ac + ATc e h max hi Aci + (1 − h )g max 2 2 i=1 Ac + ATc + ··· < 0 +(1 − h )(1 − g )f max 2 for max (Aci + ATci ) < 0, i = 1, . . . , r.
(40)
Proof of Theorem 3. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(41)
i=1
Let eigenvalues are arranged in order with 1 [·] 2 [·] · · · n [·]. For Hermitian matrix Aci , i = 1, . . . , r, hi (X0 )Aci is also Hermitian and ⎡ ⎤ r
r hi (X0 )Aci = k ⎣ hj (X0 )Acj + h (X0 )Ac ⎦ k j =1,j =
i=1
⎡ k ⎣
r
k ⎣ ...
i=1
⎤ hj (X0 )Acj ⎦ + h (X0 ) n [Ac ]
j =1,j =
⎡
r
r
⎤ hj (X0 )Acj ⎦ + h (X0 ) n [Ac ] + h (X0 ) n [Ac ]
j =1,j =,
hj (X0 ) k [Acj ] +
r
hi (X0 ) n [Aci ],
∀k = 1, . . . , n.
(42)
i=1,i =j
e k
r i=1
! hi (X0 )Aci
hj (X0 )e { n [Acj ]} +
r
hi (X0 )e { k [Aci ]} < 0
i=1,i =j
o for the case (A , B ) is c.c. and (A , C) is c.o. since (Aci ) ∈ C− i i i
(43)
2304
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Proof of Theorem 4. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(44)
i=1
For Aci , i = 1, . . . , r, are simultaneously similar to diagonalization or to triangularization, we have the spectrum, r
r o
hi (X0 )Aci ⊆ hi (X0 ) (Aci ) ∈ C− , (45) i=1
i=1
as all (Ai , Bi ) are c.c. and (Ai , C) are c.o.
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[25] K. Tanaka, M. Iwazaki, H.O. Wang, Switching control of an R/C hovercraft: stabilization and smooth switching, IEEE Trans. Fuzzy Systems 31 (2001) 853–863. [26] K. Tanaka, T. Taniguchi, S. Hori, H.O. Wang, Structure-oriented design for a class of nonlinear systems, in: Proc. FUZZ-IEEE’01, Australia, 2001, pp. 696–699. [27] K. Tanaka, T. Taniguchi, H.O. Wang, Generalized Takagi–Sugeno fuzzy systems: rule reduction and robust control, in: Proc. FUZZ-IEEE’00, San Antonio, 2000, pp. 688–693. [28] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis, Wiley, New York, 2001. [29] T. Taniguchi, K. Tanaka, H. Ohtake, H.O. Wang, Model construction, rule reduction and robust compensation for generalized form of Takagi–Sugeno fuzzy system, IEEE Trans. Fuzzy Systems 9 (2001) 525–538. [30] C.S. Tseng, B.S. Chen, H∞ decentralized fuzzy model reference tracking control design for nonlinear interconnected systems, IEEE Trans. Fuzzy Systems 9 (2001) 795–809. [31] C.S. Tseng, B.S. Chen, H.J. Uang, Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model, IEEE Trans. Fuzzy Systems 9 (2001) 381–392. [32] H.J. Uang, G.S. Huang, A robust fuzzy model following observer-based control design for nonlinear system, IEEE Internat. Conf. on Control Application, Taiwan, 2004. [33] H.O. Wang, J. Li, D. Niemann, K. Tanaka, T-S fuzzy model with linear rule consequence and PDC controller: a universal framework for nonlinear control system, in: Proc. FUZZ-IEEE’00, 2000, pp. 549–554. [34] S.J. Wu, H.H. Chiang, H.H. Lin, T.T. Lee, Neural-network-based optimal fuzzy controller design for nonlinear systems, Fuzzy Set Systems 154 (2005) 182–207. [35] S.J. Wu, C.T. Lin, Optimal fuzzy controller design: local concept approach, IEEE Trans. Fuzzy Systems 8 (2000) 171–185. [36] S.J. Wu, C.T. Lin, Optimal fuzzy controller design in continuous fuzzy system: global concept approach, IEEE Trans. Fuzzy Systems 8 (2000) 713–729. [37] S.J. Wu, C.T. Lin, Discrete-time optimal fuzzy controller design: global concept approach, IEEE Trans. Fuzzy Systems 10 (2002) 21–38. [38] S.J. Wu, C.T. Wu, Affine-type neural-fuzzy gap control of electromagnetic suspension system, in: Proc. IECON-IEEE’06, 2006. [39] Y. Yang, G. Feng, J. Ren, A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems, IEEE Trans. Systems Man Cybernet.—Part A 34 (2004) 406–420. [40] J. Yen, L. Wang, C.W. Gillespie, Improving the interpretability of TSK fuzzy models by combining global learning and local learning, IEEE Trans. Fuzzy Systems 6 (1998) 530–537. [41] X.J. Zeng, M.G. Singh, Approximation theory of fuzzy systems—MIMO case, IEEE Trans. Fuzzy Systems 3 (1995) 219–235.
Affine TS-model-based fuzzy regulating/servo control design Shinq-Jen Wu∗ Department of Electrical Engineering, Da-Yeh University, Chang-Hwa,Taiwan, ROC Received 2 August 2006; received in revised form 14 May 2007; accepted 16 May 2007 Available online 24 May 2007
Abstract An affine T-S fuzzy system is derived naturally from linearizing a nonlinear system or from some data-driven identification techniques, for example, cluster-based algorithms or soft-computation learning structures. Little research is proposed for the intrinsic analysis of an affine-type fuzzy system. Further, the controllers to regulate or to achieve servo control of affine TS-based nonlinear systems are also few. Both affine-type fuzzy regulation and servo control design scheme are theoretically derived. A simple Lyapunovbased stability criterion and some extra conditions are proposed to guarantee the global stability of the generated closed-loop fuzzy systems. The exponential stability of the feedback fuzzy system is ensured under some conditions. The performance of the proposed fuzzy controllers and fuzzy servo controllers is examined by three case studies. Simulation results show that the proposed controllers can stabilize these affine fuzzy systems and the proposed servomechanism can adapt itself to various incoming signals in very short time spans. © 2007 Elsevier B.V. All rights reserved. Keywords: Linear T-S system; Affine T-S system; Completely controllable; Completely observable; Optimal control
1. Introduction The research in fuzzy modelling and fuzzy control has come of age [2,24,17,29]. There are two model-based approaches to theoretically construct a T-S fuzzy system of a nonlinear system. One is from local linear approximation, which generates linear consequence with a constant term included in each rule, called an affine T-S fuzzy system. Mollov and coauthors propose multi-step linearization along the predicted input–output trajectory to decrease the linearization error for long-range fuzzy model predictive control [20,19]. The other is via a sector nonlinearity concept [27,23,26] which, in general, results in constant-free linear consequence for each rule, called a linear T-S fuzzy system [33]. Both fuzzy systems are demonstrated to be universal approximations to any smooth nonlinear system [33,28,41]. However, it is impractical to theoretically convert a mathematical model into a T-S fuzzy model if the nonlinear system is too complex to describe. More and more researchers attempt to learn fuzzy modelling from input–output data [40,7]. Grisales and coauthors propose two cluster-based algorithms to construct the fuzzy subsystem for each output: Gustafson–Kessel and robust parallel-competitive-agglomerative algorithms
∗ Tel.: +886 4 8511888x2192; fax: +886 6 2512882.
E-mail addresses: [email protected], [email protected] (S. Wu). 0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.05.012
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[9,10]. We previously proposed two self-organization neural-fuzzy networks to learn affine and linear T-S fuzzy systems, respectively [34,38]. An affine T-S fuzzy model is much preferred over a linear type in providing one more adjustable parameter for soft-computation (hybrid neural fuzzy and evolution) learning [34,38]. The linear T-S fuzzy system is the most popular fuzzy model due to its further intrinsic analysis: the linear matrix inequality (LMI)-based fuzzy controller is to minimize the upper bound of the performance index [28]; structureoriented and switching fuzzy controllers are developed for more complicated systems [26,25,22]; the optimal fuzzy control technique is used to minimize the performance index from local-concept or global-concept approaches [35–37]; Yang and coauthors use an input-free T-S fuzzy system to approximate an uncertain nonlinear state function and then adopt hybrid sliding-mode, adaptive and backstepping control techniques to control a strick-feedback uncertainty nonlinear system [39]. Via fuzzy static output feedback, Lo and Lin reform a robust H∞ quadratic tracking regulating issue into bilinear matrix inequalities [18]. As for the tracking control problem, it is a common issue in real world but it is very difficult to solve. Little research is devoted to linear T-S fuzzy tracking control problems. Cuevas and Toledo deal with a chaotic synchronization problem to obtain two Lorenz’ attractors [8]. Uang and Hung address a model-following control issue [32]. Chen and coauthors reform the H∞ tracking problem into a LMI problem [31] and adopt this technique to derive the reference-tracking control design of interconnected systems [30]. They further use a T-S fuzzy model to describe a fuzzy stochastic moving-average model with control input, which is really a local linear system approximating a nonlinear stochastic system. Then they derive the minimum-variance and model-reference tracking controllers [6] where the fuzzy decentralized concept (one-to-one rule correspondence between fuzzy system and fuzzy controller) is introduced directly. However, no theoretical demonstration exists for such correspondence. For affine T-S fuzzy systems, few regulating controllers with no corresponding servo controller are proposed. Hsiao and coauthors propose a hybrid compensation controller [13]. Kim and coauthors synthesize the affine-type fuzzy controller via the convex optimization technique and recast it into an LMI problem [14,15]. They further specialize in an affine T-S fuzzy system with a constant input matrix and transform the regulating problem into bilinear matrix inequalities [16]. Bergsten and coauthors try to derive an affine-type observer; but the constant-term consequence in an affine T-S fuzzy system is just a trivial term for observer derivation and the simulation is in fact a typical lineartype formulation [1]. Based on a clustering technique, Grisales and coauthors identify the affine T-S fuzzy rule for each output, and then propose discrete static-gain (for zero reference) and integral-gain (for constant reference) biascompensation-based fuzzy controllers to control a FAMIMO biological benchmark, which is a continuous-flow aerated bio-reactor for waste-water treatment [10]. Nounou and the coauthor propose three fuzzy model predictive control algorithms [21]. Chang and coauthors propose LMI-based fuzzy controllers for continuous and discrete affine T-S fuzzy systems with time-delay or constraint [3–5]. Though both consequent parts (linear and affine type) are represented by linear state equations, there exists a constant singleton in the fuzzy rule consequence for an affine T-S fuzzy model. This constant term could be a bias for a biotechnological process [10], an offset in model prediction control of a chemical process [20,19,21], a system’s parameter (characteristics) [38], or an external disturbance. Even if the fuzzy subsystem is linear the entire affine fuzzy system is highly nonlinear, and we cannot transform it to a linear type by shift-operation directly. Chang and coauthors propose a PDE-based LMI approach by lumping both input- and constant-terms together. In our previous papers [35], we proposed a simple local-based control scheme for linear-type fuzzy system. However, the stability criterion is not serious enough; the case for matrix-multiplication uncertainty is missed (Ai Aj − Aj Ai is neither positive semi-definite nor negative semi-definite.). Here I propose four different criteria to guarantee the stability. Besides, readers always confused our local-concept approach with the PDC technique: in our approach, we denote that under energy consideration there exists ith-to-ith correspondence between system rules and control rules . However, the PDC technique only denotes that the controller shares the same fuzzy sets with the fuzzy system in the premise parts; the ith system rule can be correspondent to the jth control rule [28]. To clarify this point, we here provide a more completely theoretical derivation to reinforce our technique. (Global effect can be achieved by local-concept approach.) We shall develop the relative controlling and servo controlling techniques for affine TS-model-based nonlinear systems. The technical contributions of this paper can be described as follows. The regulating control and servo control for affine TS-based nonlinear systems is theoretically derived. Lyapunov-based criterion and some extra conditions to ensure the stability of generated closed-loop fuzzy regulating/servo systems are proposed. Several nonlinear systems are concerned to examine the performance of those regulating and servo controllers.
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2. Affine-type fuzzy controller design The following affine T-S fuzzy system is considered to describe a nonlinear physical system: ˙ R i : If x1 is T1i , . . . , xn is Tni , then X(t) = Ai X(t) + Bi u(t) + Di , i = 1, . . . , r, Y (t) = CX(t),
(1)
where R i denotes the ith rule of the fuzzy model; x1 , . . . , xn are system states; T1i , . . . , Tni are the input fuzzy terms in the ith rule; X(t) = [x1 , . . . , xn ]T ∈ n is the state vector, Y (t) = [y1 , . . . , yn ]T ∈ n is the system output vector, and u(t) ∈ m is the system input; and Ai , Bi , Ci and Di are, respectively, n × n, n × m, n × n and n × 1 matrices. We shall design a rule-based fuzzy controller, R i : If y1 is S1i , . . . , yn is Sn i , then u(t) = ri (t), i = 1, . . . , ,
(2)
to minimize the quadratic cost functional, ∞ J (u(·)) = [X T (t)LX(t) + uT (t)Su(t)] dt,
(3)
t0
where X T (t)LX(t) is state-trajectory penalties with L belonging to symmetric positive semi-definite n × n matrices, and uT (t)Su(t) denotes energy consumption; y1 , . . . , yn are the elements of output vector Y (t), S1i , . . . , Sn i are the input fuzzy terms in the ith control rule, and the plant input (i.e., control output) vector u(t) or ri (t) is in m space. Though both consequent parts (linear and affine type) are represented by linear state equations, there exists a constant singleton in the fuzzy rule consequence for the affine T-S fuzzy model. This constant term could be a bias for a biotechnological process [10], an offset in model prediction control of a chemical process [21,20], a system’s parameter (characteristics) [38], or an external disturbance. In our simulation, it respectively denotes the linearization residue for the four-pole differential-driving magnetic bearings system (case 1), the transformed system’s parameter in a nonlinear system (case 2), and a fuzzy singleton learned from a soft-computation network (case 3). ˙ Even the fuzzy subsystem is linear, the entire affine fuzzy system X(t) = ri=1 hi (X(t))(Ai X(t) + Bi u(t) + Di ) is highly nonlinear. We cannot transform it to a linear type by shift-operation directly. Theorem 1 (Regulating control). For an affine T-S fuzzy system in Eq. (1) and a fuzzy controller in Eq. (2), if Ai is T ¯ ) > 0, nonsingular, (Ai , Bi ) completely controllable (c.c.), (Ai , C) completely observable (c.o.) and ¯ −1 i i (L + Ai ∀i = 1, . . . , r, then (1) the local fuzzy regulating law is ri∗ (t) = −S −1 BiT ¯ i X ∗ (t) + r¯is ,
i = 1, . . . , r,
∗ where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ) and X¯ is = A−1 i Di ; their “blending’’ global fuzzy controller, u (t) = ∗ ∗ (X (t))ri (t), minimizes J (u(·)) in (3), where ∞ −1 T T b¯ s = − e[Ai −Bi S Bi ¯ i ] · d · LX¯ s , i
i
0
(4) r
i=1
hi
(5)
and ¯ i is the unique symmetric positive semi-definite solution of the Riccati equation Ki Ai + Ati Ki − Ki Bi S −1 Bit Ki + L = 0;
(6)
(2) the entire feedback fuzzy system is stable, X˙ ∗ (t) =
r i=1
hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) + Bi r¯is + Di ].
(7)
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Proof. (1) From the essence of the dynamic programming formalism, we know min J (u(·)) = min
u[t0 ,∞)
u[t,∞)
∞ t
(XlT LXl
+ uTl Sul ) dl
+ min
t
u[t0 ,t] t 0
(XlT LXl
+ uTl Sul ) dl
,
(8)
where lower index is used to denote time-dependence for notation simplification, i.e., Xl for X(l). A quadratic optimization problem is hence turned into a successively on-going dynamic problem with regards to the state resulting from the previous decision, i.e., the initial state (at time t) X0t = Xt∗ . Moreover, according to the signal flow of a fuzzy inference system in Fig. 1 with y1 denotes x˙1 , the overall behavior of the fuzzy system at any time l is X˙ l =
r
hi (Xl )(Ai Xl + Bi ul + Di ),
l ∈ [t, ∞),
(9)
i=1
with ul = i=1 wi (Yl )ril and X0t = Xt∗ ∈ n ; hi (Xl ) and wi (Yl ) denote, respectively, the normalized firing strength of the ith rule of the fuzzy model and of the ith fuzzy control rule; i.e., hi (Xl ) = i / ri=1 i with i = nj=1 Tj i (Xl ), where Tj i (Xl ) is the membership function of fuzzy term Tj i , and wi (Yl ) = i / i=1 i with i = lj =1 Sj i (Yl ), where Sj i (Yl ) is the membership function of fuzzy term Sj i . Therefore, we shall focus on successively finding the continuous global decision (global fuzzy controller) u∗t for minimizing Jt (ut ) =
∞
t
(XlT LXl + uTl Sul ) dl,
t ∈ [t0 , ∞),
(10)
and estimating Xt∗+ with regards to the initial state Xt∗ , where t + denotes the time instant slightly later time t; and then with the new initial state, Xt∗+ , resolving u∗t + to minimize Jt + (ut + ). At any time instant t, the local decision (local fuzzy control law) stems from minimizing Jt (ut ) in Eq. (10) with regards to the fuzzy subsystem, X˙ l = Ai Xl + Bi ul + Di ,
l ∈ [t, ∞), i = 1, . . . , r;
(11)
the global decision results from minimizing Jt (ut ) with regard to the entire fuzzy system in Eq. (9). For clarity, since u∗t is only a variable to be solved regardless of the aforementioned local optimization problem or the global optimization issue, we can use ri∗t to denote the local decision of the ith fuzzy subsystem. (2) Now, let l (Xl , ul ) and il (Xl , ril ), i = 1, . . . , r, denote, respectively, the entire energy and local energy at any ∞ ∞ time instant l, l ∈ [t, ∞). Then, Jt (ut ) = t l (Xl , ul ) dl and Jt (rit ) = t il (Xl , ril ) dl. At any time instant, the energy of the entire fuzzy system is some kind of (nonlinear) combination of fuzzy subsystems’ energy; we can also describe the global energy in terms of rule-based local energy. We note that this combination is the function of system states only and has nothing to do with system inputs. In other words, whether the system behavior is nonlinear to system inputs or not, the input term should not be included as a fuzzy precondition in physical reality even if it is reasonable in mathematical concept terms. Besides, this nonlinear combination is not necessary to be the same type as fuzzily blending subsystems into an entire system. Therefore, we use h (X(t)) to denote that (a) the nonlinear combination is only state dependent; (b) the energy relationship between the entire system and subsystems could be fully different from the behavior relationship, denoted by the normalized membership function, h(X(t)). We then write l (Xl , ul ) = ri=1 hi (Xl ) il (Xl , ril ). At the time instant t with initial condition Xt∗ , let ri∗t denote the local decision to minimize Jt (rit ) for all i = 1 . . . , r, i.e., ∞ * it (Xt∗ , rit ) * *Jt (rit ) = (X , r ) dl = l il il ∗ = 0, ∗ *rit r ∗ *rit t *rit r r it
it
2 * Jt (rit ) *ri2 t
ri∗t
2 * it (Xt∗ , rit ) = *ri2 t
> 0. ri∗t
it
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y1
... Layer 6
... Layer 5
a11
a21
b11 u
X
b21
X
u
a31
b31
X u ...
Layer 4
R2
R1
...
R3
Layer 3
... Layer 2
... Layer 1
X1
X2
Fig. 1. Neural-fuzzy inference network for T-S fuzzy system.
Then, their corresponding global decision u˘ t = ri=1 hi (Xt∗ )rit |ri∗ can satisfy t ∞ *Jt (ut ) * = l (Xl , ul ) dl *ut u˘ t *ut t u˘ t ∞ r * = hi (Xl ) il (Xl , ril ) dl *ut t i=1 u˘ t r ∗, r ) (X * i i = hi (Xt∗ ) t t t *ut i=1 u˘ t r ∗ ∗ * it (Xt , rit ) *rit hi (Xt ) · = 0, = *ut u˘ t *rit i=1
2 * Jt (ut ) *u2t
= u˘ t
r i=1
* hi (Xt∗ )
2
it (Xt∗ , rit ) *ri2t
·
*rit *ut
2 r 2 ∗ ∗ * it (Xt , rit ) * rit hi (Xt ) · + > 0, *rit *u2t u˘ i=1 u˘ t
t
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i.e., u˘ t = u∗t . So, at any time instant t, if we can find ri∗t to minimize Jt (rit ) then their composed global decision u∗t can be the global minimizer of the total cost Jt (ut ). According to the above analysis, we know that, at any time instant, solving the quadratic optimal control problem is to find only one corresponding optimal solution of the fuzzy controller for each rule of the fuzzy model. ˆ (3) We further assume Ai , i = 1, . . . , r, is nonsingular and let X(t) = X(t) + X¯ is , where X¯ is = A−1 i Di . Then the local affine-type quadratic problem is further reformulated as follows: given the fuzzy subsystem, ˙ˆ = A Xˆ + B r , X i il l i l
l ∈ [t, ∞), i = 1, . . . , r
(12)
with the initial state resulting from the previous decision, i.e., Xˆ 0t = Xˆ t∗ , (a) find the local decision at time instant t, ri∗t , for minimizing the cost functional, ∞ ((Xˆ lT − X¯ is )L(Xˆ l − X¯ is ) + ritl Sril ) dl; (13) Jt (rit ) = t
(b) obtain the global decision u∗t at time instant t for minimizing the entire cost functional Jt (ut ) in Eq. (10) by r ˆ∗ ∗ ˆ∗ fuzzily blending each local decision, i.e., u∗t = i=1 hi (Xt )rit . Notice that the next-decision initial state Xt + is ∗ ∗ ˙ˆ = r h (Xˆ ∗ )(A Xˆ ∗ + B u∗ ). There exists a one˙ˆ = r h (Xˆ ∗ )(A Xˆ ∗ + B r ∗ ) instead of from X from X i t i il i t i t t t t t i=1 i i=1 i to-one relationship between each fuzzy subsystem and the corresponding fuzzy control rule. Since the local fuzzy system (i.e., fuzzy subsystem) is linear, its quadratic optimization problem is the same as the general linear quadratic tracking issue. Therefore, we obtain the regulating law ri∗ (t) = −S −1 BiT [¯ i (X ∗ (t) + X¯ is ) + bis (t)], fuzzy subsystem X˙ ∗ (t) = (Ai − Bi S −1 BiT ¯ i )(X ∗ (t) + X¯ is ) − Bi S −1 Bit bis (t) with bis (t) satisfies b˙is (t) = −(Ai − Bi S −1 BiT ¯ i )t bis (t) + LX¯ is ,
(14)
0 . Therefore, we where bis (∞) = 0n×1 . Further, since (Ai , Bi ) is c.c. and (Ai , C) is c.o., (Ai − Bi S −1 BiT ¯ i ) ∈ C− s s obtain bi (t) = b¯i in Eq. (5). Thus, the fuzzy control design scheme for an affine T-S fuzzy system in Theorem 1 can easily be derived. (4) For the purpose of stability analysis, we regard X¯ is as an artificial target. Let U¯ iart = Bi r¯is + Di , an artificial-target associated constant input. Then the stability of the resultant feedback fuzzy system in Eq. (7) concurs with the zero-input system,
X˙ ∗ (t) =
r
hi (X ∗ (t))[Ai − Bi S −1 BiT ¯ i ]X ∗ (t).
(15)
i=1
Defining Lyapunov function V (X) = Xt P X, where P is a symmetric positive matrix. Via Eq. (6), we have Ai − T ¯ ), and hence, Bi S −1 BiT ¯ i = −¯ −1 i i (L + Ai V˙ (X) = X˙ t P X + X t P X˙
r r
T −1 T T t −1 T = hi (X(t))X (Ai − Bi S Bi ¯ i ) PX + X P hi (X(t))(Ai − Bi S Bi ¯ i )X i=1 r
=−
i=1 T hi (X(t)){X T [(L + ATi ¯ i )T ¯ −1 ¯ −1 ¯ i )]X} i P + P i (L + Ai
i=1 r
= −2
T hi (X(t))[X t P ¯ −1 ¯ i )X]. i (L + Ai
(16)
i=1
Further choosing P = I > 0, we obtain V˙ = −2
r
T hi (X(t))[X T ¯ −1 ¯ i )X] < 0 i (L + Ai
(17)
i=1 T ¯ ) > 0, since h (X(t)) is always a positive number. for ¯ −1 i i i (L + Ai
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As we know, if A > 0 and B is Hermitian then AB has the same number of positive, negative and zero eigenvalues as B. T¯ ) > Therefore, for ¯ i and also ¯ −1 i i being symmetric positive definite (Hermitian with positive eigenvalues), if (L+Ai T ¯ ) > 0, which ensure the stability of ¯ −1 0, then (L + ATi ¯ i )T > 0; further, (L + ATi ¯ i )T ¯ −1 i i > 0 and also i (L + Ai the closed-loop system. So we have a more simple sufficient criterion. Corollary. If (L + ATi ¯ i ) > 0, then the closed-loop system in Eq. (7) is stable. In the remainder section, we further propose some criteria to ensure the exponential stability of the generated system in Eq. (7). The converse theorem is adopted in the following theorems. Before further derivation, we emphasize here that T-S fuzzy system is basically a locally linearized system. Therefore linearizing this fuzzy system is different from generalizing the differential operation to the whole nonlinear system. As we know, linearizing a nonlinear system with respect to a point X0 is to capture the approximate linear behavior at X0 neighborhood. A T-S fuzzy system can be described or realized as a fuzzy inference network in Fig. 1, where output y1 denotes x˙1 . Based on this inference structure, linearizing a T-S fuzzy system with respect to X0 can be figured out to root in the Fact X0 and developed ˙ upward to get output X(t). A fuzzy system is just like a successive separation-and-aggregation interaction; at any time instant each individual (subsystem) starts from the same point (state), travels away from each other, meets and fuzzily blends at incoming instant into the final target (output). We further use X¯ is = A−1 i Di to denote an artificial s art ¯ ¯ target, and then Ui = Bi ri + Di is an artificial-target associated constant input. Then the stability of the resultant feedback fuzzy system in Eq. (7) concurs with the zero-input system in Eq. (15). Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with its corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )(Ai − Bi S −1 Bit ¯ i )X(t).
(18)
i=1
Based on this inference, some criteria are proposed in Theorems 2–4 to ensure the exponential stability of the systems in Eq. (18) with regard to all X0 ∈ n . Therefore, the stability of generated system in Eq. (7) is ensured. Theorem 2. If each feedback fuzzy subsystem satisfies max (Aci + ATci ) < 0 where Aci = Ai − Bi S −1 BiT ¯ i , then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Theorem 3. If (Ai , Bi ) is c.c., (Ai , C) is c.o. and Aci = Ai − Bi S −1 BiT ¯ i is a Hermitian matrix for each fuzzy subsystem, then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Theorem 4. If (Ai , Bi ) is c.c., (Ai , C) is c.o. and Aci = Ai − Bi S −1 BiT ¯ i , i = 1, . . . , r, are simultaneously similar to diagonalization, or if the more relax condition Aci is simultaneously similar to triangularization, then the closed-loop fuzzy systems in Eqs. (7) or (15) are exponentially stable. Proof. See the Appendix. Recently, we have successfully applied the previously proposed linear-type optimal fuzzy controller to magnetic suspension systems, four-pole and eight-pole active magnetic bearing systems, inverted pendulum systems, half-car active suspension systems, and even the Taiwan iTS-1 experimental car. To reinforce our previous stability theorem, some brand-new theorems are proposed in this work. We further adopt three different examples which are, respectively, satisfies Corollary, Theorem 1 and neither one of the above theorems to point out that there must exit more relax conditions than these. It is our cordial wish that those mathematical-solid-background researchers could propose more flexible conditions to guarantee the stability of the closed-loop fuzzy system.
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
2295
3. Affine-type fuzzy servo control design The optimal servo control design scheme is to control the fuzzy servo system in such a way as to push the output Y (t) close to follow an incoming signal Y d (t) without excessive control-energy consumption where Y d (t) belongs to a known class of signal, z˙ (t) = F z(t), Y d (t) = Ez(t),
(19)
with z(t0 ) = z0 , where z(k) ∈ h , F and E are matrices of h×h and n ×h, respectively. Therefore, our affine-type servo control problem can be described as: given an affine T-S fuzzy servo system and a fuzzy servo controller, respectively, in Eqs. (1) and (2) with X(t0 ) = X0 ∈ n , find the individual servo law, ri∗ (·), i = 1, . . . , , such that the composed servo controller u∗ (·) can minimize the quadratic cost functional, ∞ J (u(·)) = (20) [uT (t)Su(t) + X T (t)L1 X(t) + (Y (t) − Y d (t))T L2 (Y (t) − Y d (t))] dt t0
with L1 = [In − C T (CC T )−1 C]t L3 [In − C T (CC T )−1 C]; S, L2 and L3 are m × m, n × n and n × n nonnegative symmetric matrices; XT (t)L1 X(t) is the state-trajectory penalty to produce smooth response; uT (t)Su(t) is energy consumption; and the last term in J (u(·)) is related to error cost. The desired output Y d (t) is the incoming signal in Eq. (19). ˜ Define X(t) = [XT (t) zT (t)]T and rewrite the performance index in Eq. (20) as ∞ J (u(·)) = [uT (t)Su(t) + (X(t) − X d (t))t L(X(t) − X d (t))] dt, (21) t0
where L = L1 + C T L2 C and the desired trajectory X d (t) = C T [CC T ]−1 Y d (t). Then the servo problem can be further simplified as the following augmented regulating problem: Given an augmented affine T-S fuzzy regulating system, ⎤ ⎡ r ˙˜ ˜ ˜ ⎣A˜ i X(t) + B˜ i (22) hi (X(t)) wj (Y (t))rj (t) + D˜ i ⎦ X(t) = i=1
j =1
˜ ˜ 0 ) = X˜ 0 ∈ n+h and hi (X(t)) = hi (X(t)), t ∈ [t0 , ∞), find the individual regulating law, ri∗ (·), i = 1, . . . , , with X(t to minimize ⎡ ⎤ ∞ ˜ ⎣X˜ T (t)L˜ X(t) + wi (Y (t))wj (Y (t))riT (t)Srj (t)⎦ dt, (23) J (ri (·)) = t0
i=1 j =1
= where the parameters are B˜ i T [CC T ]−1 E −LC L L˜ = −E T [CC T ]−1 CL E T [CC T ]−1 CLC T [CC T ]−1 E .
Bi 0h×m
,
D˜ i
=
Di 0h×1
,
A˜ i
=
Ai 0n×h
0h×n F
and
Based on the regulating control theorem in Section 2, we can derive the servo control theorem as follows. Theorem 5 (Servo control). For an affine T-S fuzzy system in Eq. (1) and a fuzzy servo controller in Eq. (2), let the desired trajectory, Xd (t), come from Y d (t) = CXd (t), where Y d (t) is an incoming signal in Eq. (19). If Ai is T ¯ ) > 0, ∀i = 1, . . . , r, then nonsingular, (Ai , Bi ) is c.c., (Ai , C) is c.o. and ¯ −1 i i (L + Ai (1) the local fuzzy servo law is s ri∗ (t) = −S −1 BiT [¯ i X ∗ + ¯ 21 i z(t)] + r¯i , T
i = 1, . . . , r, (24) and their “blending’’ global fuzzy servo controller, u∗ (t) = ri=1 hi (X ∗ (t))ri∗ (t), minimizes J (u(·)) in (20), ¯s ¯ i is the unique solution of the Riccati where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ), X¯ is = A−1 i Di , bi satisfies Eq. (5),
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S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
equation in Eq. (6) and ∞ T −1 T 21 eF · E T (CC T )−1 CL · e(Ai −Bi S Bi ¯ i ) · d; ¯ i = −
(25)
0
(2) the entire feedback fuzzy system is stable, X˙ ∗ (t) =
r
s hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) − Bi S −1 BiT ¯ 21 i z(t) + Bi r¯i + Di ]. T
(26)
i=1
Proof. (1) Based on Theorem 1 in Section 2, we have ri∗ (t) = −S −1 B˜ iT ˜ i (t, t1 )X˜ ∗ (t) + r¯is (t), ˙˜ ∗ (t) = [A˜ − B˜ S −1 B˜ T ˜ (t, t )]X˜ ∗ (t) + B˜ r¯ s (t) + D˜ , X 1 i i i i i i i ˙b˜ s (t) = −[A˜ − B˜ S −1 B˜ T ˜ (t, t )]T b˜ s (t) + L˜ X ˜¯ s , b˜ s (t ) = 0 i 1 i (n+h)×1 , i i i i i i 1 ¯s s s ˜¯ + b˜ s (t)), X ˜¯ = A˜ −1 D˜ = Xi and ˜ (t, t ) is the symmetric positive semiwhere r¯is (t) = −S −1 B˜ iT (˜ i (t, t1 )X i i 1 i i i i 0h×1 definite solution of the following Riccati equation: ˜ −K˙˜ i (t) = K˜ i (t)A˜ i + A˜ Ti K˜ i (t) − K˜ i (t)B˜i S −1 B˜ iT K˜ i (t) + L,
K˜ i (t1 ) = 0(n+h)×(n+h) .
˜ t In fact, ˜¯ i = limt1 →∞ (t, ˜ t1 ) a constant Here we replace b˜is (∞) and ˜¯ i by b˜is (t1 ) and (t, 1 ) for further derivation.
s 21T (t) K (t) K b (t) ˜ = X(t) for the infinite horizon problem. Now, let K˜ i (t) = K 21i (t) Ki22 (t) , b˜i (t) = b i (t) and X(t) z(t) . Considering i
2i
i
the infinite horizon problem, t1 = ∞, and via complicated matrix manipulation, we obtain ri∗ (t) in Eq. (24) and X ∗ (t) in Eq. (26), where r¯is = −S −1 BiT (¯ i X¯ is + b¯is ); b¯is is in Eq. (5) as (Ai , Bi ) c.c. and (Ai , Ci ) c.o.; ¯ i is the unique solution of the Riccati equation in Eq. (6); b2i (t) and ¯ 21 i , respectively, satisfy b˙2i (t) = K21 (t)Bi S −1 Bit bis (t) + F t bi (t) − E T (CC T )−1 CLX¯ is , b2i (∞) = 0h×1 , −K˙ 21 (t) = F t K21 (t) + K21 (t)Ai − K21 (t)Bi S −1 BiT ¯ i − E T (CC T )−1 CL,
(27)
K21 (∞) = 0h×n .
(28)
(2) We now wish to show that the solution of Eq. (28) is
¯ 21 i
in Eq. (25). Rewriting Eq. (28) as
− K˙ 21 (t) = F t K21 (t) + K21 (t)Aci − L21 ,
(29)
where Aci = Ai − Bi S −1 BiT ¯ i and L21 = E T (CC T )−1 CL. Then we obtain K21 (t) = 1 (t, t0 )K21 (t0 ) T2 (t, t0 ) +
T t0
1 (t, )L21 T2 (t, ) d,
(30)
˙ ˙ where 1 (t, t0 ) and 2 (t, t0 ) are state transition matrices of X(t) = −F t X(t) and X(t) = −Atci X(t), respectively. Therefore, we know t1 K21 (t1 ) = 1 (t1 , t0 )K21 (t0 ) T2 (t1 , t0 ) + 1 (t1 , )L21 T2 (t1 , ) d, t 0t1 T K21 (t0 ) = 1 (t0 , t1 )K21 (t1 ) 2 (t0 , t1 ) − 1 (t0 , )L21 T2 (t0 , ) d. t0
Substituting K21 (t0 ) into Eq. (30), we have K21 (t) =
1 (t, t1 )K21 (t1 ) T2 (t, t1 ) −
t1 t
1 (t, )L21 T2 (t, ) d.
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Further, limt1 →∞ K21 (t1 ) = 0h×n . Using K¯ 21 (t) to denote K21 (t) with zero infinity, we obtain ∞ T T K¯ 21 (t) = − e−F (t−) · L21 · e−Aci (t−) d. t
2297
(31)
Hence, we have ¯ 21 i in Eq. (25) which satisfies Eq. (31) and also Eq. (28). T ¯s (3) Stability analysis: since S −1 BiT ¯ 21 i Z(t) in Eq. (26) is associated with the target only and Bi ri + Di in Eq. (26) can be regarded as an artificial-target constant as in Proof of Theorem 1, the proposed optimal closed-loop servo control system in Eq. (26) can be rewritten as X˙ ∗ (t) =
r
hi (X ∗ (t))[(Ai − Bi S −1 BiT ¯ i )X ∗ (t) − Bi r¯iext (t) + U¯ iart ]
(32)
i=1
¯s ¯ art with r¯iext (t) = S −1 BiT ¯ 21 i Z(t) and Ui = Bi ri + Di . Further, the stability of the nonlinear servo control fuzzy system in Eq. (32) is coincident with that of the zero-input fuzzy system in Eq. (15). This is demonstrated to be stable in Proof T ¯ ) > 0. of Theorem 1 once each fuzzy subsystem satisfies (Ai , Bi ) c.c., (Ai , C) c.o. and ¯ −1 i i (L + Ai T
4. Numerical simulation In this section, a four-pole active magnetic bearing system and a sinusoidal nonlinear system are adopted in Section 4.1 to illustrate the proposed affine-type fuzzy control scheme. And an affine TS-based nonlinear system in Section 4.2 is used as a servomechanism to examine the performance of the proposed affine-type fuzzy servo controller. 4.1. Regulating control A four-pole differential-driving magnetic bearings driven by sum of bias current and control current is considered [11,12]. Based on the assumptions of symmetric structure and rigid floating mass, we have the dynamic equation of rotor motion:
ib − i(t) 2 ib + i(t) 2 − , (33) mx(t) ¨ = G − x(t) G + x(t) where i(t) is control current, rotor mass m = 0.0126 (lb s2 /in), nominal air gap G = 0.02 in, force constants = 0.00186 (lb in2 /A2 ), sensitivity of air gap to shaft displacement = 0.974, and the bias current ib = 0.3 A [12]. Fig. 2(a) is the magnetic force with respect to control current i(t) and rotor displacement x(t). Fig. 2(b) is the magneticforce evolution with control current i(t) at operating points, x(t) = 0, 0.005, 0.01 in. We found that for operation x(t) = 0.01 in, the magnetic-force evolution is the parabolic function with an extreme point at around i(t) = −0.3 A; and for x(t) = 0.005 in, the magnetic-force evolution is the parabolic function with an extreme point at around i(t) = −0.6 A. Taylor’s expansion technique is used to linearize the magnetic bearing system in Eq. (33) with respect to operating points (x(t), i(t)) = (0, 0), (0.005, 0), (0.01, 0), (0.005, −0.3) and (0.01, −0.3). And the following affine T-S fuzzy system is obtained [12]: ˙ = A1 X(t) + B1 i(t) + D1 , R 1 : If x is ZE, then X(t) 2 ˙ R : If x is PM and i is ZE, then X(t) = A2 X(t) + B2 i(t) + D2 , 3 ˙ R : If x is PB and i is ZE, then X(t) = A3 X(t) + B3 i(t) + D3 , ˙ R 4 : If x is PM and i is NE, then X(t) = A4 X(t) + B4 i(t) + D4 , 5 ˙ R : If x is PB and i is NE, then X(t) = A5 X(t) + B5 i(t) + D5 , (34) T and the membership functions of the fuzzy terms are shown in Figs. 2(c) and (d); A = where X(t) ˙ 1 = [x(t) x(t)] 0 1 0 1 0 1 0 1 0 1 0 0 = = = = = = , A , A , A , A ; B , B , 2 3 4 1 2 5 6489.7 0 6750 0 9182.0 0 25022 0 3948 0 444.2 532 0 0 0 0 0 0 0 B3 = 944 , B4 = 287.3 , B5 = 201 ; D1 = 00 , D2 = −9.2 , D3 = −138.7 , D4 = −33 , D5 = −39.5 ; Y = CX with C = [1 0] for each rule.
30 20 10 0 -10 -20 -30 0.01 Ro 0.005 0 tor -0.005 dis tan -0.01 -1 ce (In ch )
ZE
-0.5
5
)
amp
ent (
Curr
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
x=5.0 x=0.005 x=0.01
-1
-0.5
0
0.5
1
Current (amp)
PM
0
1
0.5
0
Magnetic force (Ecs)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Magnetic force (Ecs)
2298
PB
10
x (mili-inch)
NE
ZE
-0.3
0 u (amp)
Fig. 2. (a) Magnetic-force evolution with respect to control current i(t) and rotor displacement x(t); (b) magnetic-force evolution with control current i(t) at x(t) = 0, 0.005, 0.01 in; (c) and (d) membership functions for x(t) and u(t).
Each fuzzy subsystem in Eq. (34) is c.c. and c.o. since rank[Bi Ai Bi ] = 2 and rank[C T Ai t C T ]T = 2 for i = 1, . . . , 5. Let the penalty matrices be set as L = I2 and S = 1. Therefore, based on Theorem 1 in Section 2, we have the unique S.S.R.E. solutions, 15.5786 0.0659 18.3768 0.0649 27.9757 0.0562 ¯ 1 = , ¯ 2 = , ¯ 3 = , 0.0659 0.0024 0.0649 0.0020 0.0562 0.0011 27.0919 0.1636 23.1968 0.1956 ¯ 4 = , ¯ 5 = , 0.1636 0.0040 0.1956 0.0059 and (L + ATi ¯ i ) > 0, for all i = 1, . . . , 5. Fig. 4 shows the evolution of the position, the velocity of the shaft rotor, and the proposed control current with initial conditions x(0) = [0.0082 0]T and i(0) = −0.15 A; x(0) = [0.006 0]T and i(0) = −0.33 A; x(0) = [0.004 0]T and i(0) = −0.53 A; x(0) = [0.002 0]T and i(0) = −0.73 A. Simulation results show that the dynamic system can be stabilized in around just 0.4 s. We now consider another nonlinear system [16], x˙1 (t) = −x3 (t) + x1 (t) cos x2 (t) + (1 + cos x2 (t))u(t), x˙2 (t) = x1 (t) + u(t), x˙3 (t) = x1 (t) + 3x2 (t) + sin x2 (t) with = 0.5. A diffeomorphic transform ⎡ ⎤ ⎡ ⎤ x1 (t) x3 (t) ⎣ x2 (t) ⎦ = ⎣ x (t) − x (t) − sin x (t) ⎦ 2 2 1 x3 (t) x2 (t)
(35)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
M1
M2
2299
M3
_ x3 -E3
-E2
-E1
E1 2π 5
E1
E2
E3
1π 2
3π 2
E2
E3
Fig. 3. Membership functions for affine fuzzy system in Eq. (37).
is used to get a constant input function. Hence, we can rewrite Eq. (35) as x˙1 (t) = x2 (t) + 4x3 (t) + ( + 1) sin x3 (t), x˙2 (t) = −x1 (t) − x2 (t) − x3 (t) − sin x3 (t), x˙3 (t) = x2 (t) + x3 (t) + sin x3 (t) + u(t).
(36)
Then the following affine T-S fuzzy system is derived to describe the dynamic behavior of the transformed system in Eq. (36) [16], ˙ R 1 : If x3 (t) is M1 , then X(t) = A1 X(t) + B1 u(t) + D1 , 2 ˙ R : If x3 (t) is M2 , then X(t) = A2 X(t) + B2 u(t) + D2 , ˙ R 3 : If x3 (t) is M3 , then X(t) = A3 X(t) + B3 u(t) + D3 ,
(37)
where X(t) = [x1 (t) x2 (t) x3 (t)]T . The membership functions of the fuzzy terms are shown in Fig. 3; ⎡
⎡ ⎤ ⎤ 2 2 ( + 1) 0 1 4 + ( + 1) ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ 2 2 ⎢ ⎢ ⎥ A1 = ⎢ −1 −1 −1 + ⎥ , A2 = ⎢ −1 −1 −1 − ⎥ ⎥, ⎣ ⎣ ⎦ ⎦ 2 2 0 1 1− 0 1 1+ ⎡ ⎤ 2 0 1 4 − ( + 1) ⎢ ⎥ ⎢ ⎥ 2 ⎥ A3 = ⎢ ⎢ −1 −1 −1 + ⎥ ; ⎣ ⎦ 2 0 1 1− ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2( + 1) 0 2( + 1) ⎦ , D2 = ⎣ 0 ⎦ , D3 = ⎣ −2 ⎦ ; 2 D1 = ⎣ −2 0 2 0
1 4−
⎡ ⎤ 0 B1 = B2 = B3 = ⎣ 0 ⎦ ; 1 Y = CX with C = [100] for each rule. 2 Since each fuzzy subsystem in Eq. (37) is c.c. and c.o. (rank[Bi Ai Bi A2i Bi ] = 3 and rank[C T Ai t C T Ai t C T ]T = 3 for i = 1, 2, 3). The penalty matrices are set to be L = I3 and S = 1. Hence we have Riccati
2300
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
0
0.008
-0.02
0.006
-0.04
x
xdot
0.01
0.004 0.002
-0.06 -0.08
0
-0.1 0
0.2
0.4 time
0.6
0.8
0
0.5 time
1
0 -0.2 -0.4 -0.6 -0.8
0
0.2
0.4
0.6
0.8
time Fig. 4. The rotor position x ∗ (t) and velocity x˙ ∗ (t) of four-pole AMB system in Eq. (33) and the proposed optimal control current i ∗ (t) with initial conditions x(0) = [0.00820]T and i(0) = −0.15 A; x(0) = [0.0060]T and i(0) = −0.33 A; x(0) = [0.0040]T and i(0) = −0.53 A; x(0) = [0.0020]T and i(0) = −0.73 A.
solutions,
⎡
1.1050 ¯ 1 = ⎣ 0.0443 0.9547 ⎡ 1.1050 ¯ 3 = ⎣ 0.0443 0.9547
⎤ 0.0443 0.9547 1.0441 1.0174 ⎦ , 1.0174 3.1090 ⎤ 0.0443 0.9547 1.0441 1.0174 ⎦ . 1.0174 3.1090
⎡
⎤ 0.8699 −0.0489 1.0477 ¯ 2 = ⎣ −0.0489 0.9507 0.9696 ⎦ , 1.0477 0.9696 4.5686
Fig. 5 is the state response and the proposed optimal controller with initial conditions x(0) = [−7 1 10]T , [−7 − 1 − 10]T and [−7 1 − 10]T . It shows that the proposed controller can stabilize the system in 5 s. However, neither one of the above theorems are satisfied. Hence, there must be more relax conditions than these. We hope mathematical-trained researchers can propose more flexible conditions in the future to guarantee the stability of the closed-loop fuzzy system. 4.2. Servo control Via an analytical or hybrid soft-computing technique, any nonlinear system can be approximated by an affine T-S fuzzy system. Here we consider the following affine-TS-based nonlinear system to demonstrate the performance of the proposed servo controller in Section 3. ˙ R 1 : If x(t) is F11 and x(t) ˙ is F21 , then X(t) = A1 X(t) + B1 u(t) + D1 , 2 1 2 ˙ ˙ is F2 , then X(t) = A2 X(t) + B2 u(t) + D2 , R : If x(t) is F1 and x(t) ˙ ˙ is F21 , then X(t) = A3 X(t) + B3 u(t) + D3 , R 3 : If x(t) is F12 and x(t) 4 2 2 ˙ ˙ is F2 , then X(t) = A4 X(t) + B4 u(t) + D4 , R : If x(t) is F1 and x(t)
(38)
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
3
5
2
0
1
x1
x2
10
-5
0
-10
-1
-15
2301
-2 0
5
10 time
15
20
10
0
5
10 time
15
20
0
5
10 time
15
20
30
5
20 u
x3
0 10
-5 0
-10 -15
-10 0
5
10 time
15
20
Fig. 5. State response X∗ (t) in Eq. (36) and the proposed optimal controller u∗ (t) with initial conditions x(0) = [−7 1 10]T , [−7 − 1 − 10]T and [−7 1 − 10]T .
and the system output is Y (t) = CX(t) with C = [01] for each rule, where 1 1 1 0.3 x(t) ˙ , D2 = , D3 = , D4 = ; X(t) = ; D1 = 1 0.5 0 1 x(t) 0 −0.02 −0.225 −0.02 0 −1.5275 A1 = , A2 = , A3 = , 1 0 1 0 1 0 1 −0.225 −1.5275 , i = 1, . . . , 4. A4 = ; Bi = 0 1 0 (t) The membership functions are chosen as F 1 (x(t)) = 1− x2.25 , F 2 (x(t)) =
(t) = 1− x˙2.25 , F 2 (x(t)) ˙ = 2 5 x˙ 2 (t) d 2.25 . This affine TS system is used for following an incoming signal Y (t) in Eq. (19) with initial condition z0 = 10 1 1 − 30 5 1 −3 1 0 3 and parameters E = [3 1], F = −3 . Now, let the penalty matrices , or 1 −5 −0.2 5 −2 −0.2 5 − 15 − 9 0.0316 in Eq. (20) be L3 = I2 , L2 = I1 , and S = 0.001. The unique Riccati solutions are ¯ 1 = 0.0326 ¯2 = 0.0316 1.0311 , 0.0324 0.0316 0.0326 0.0301 0.0323 0.0301 −1 T ¯ 3 = 0.0301 1.0309 , ¯ 3 = 0.0301 1.0306 and ¯ i (L + Ai ¯ i ) > 0, ∀i = 1, . . . , r. Fig. 6 is the 0.0316 1.0311 , servo control performance. 2
1
1
x 2 (t) ˙ 2.25 , F21 (x(t))
2
5. Conclusions Though a linear T-S fuzzy system is the most popular fuzzy model and has been successfully applied in various fields, an affine T-S is still the preferred method for computation-intelligent modelling of mode-free or highly nonlinear complex model-based physical system. We propose an affine-type fuzzy controller to stabilize a nonlinear system and a fuzzy servo controller to track an incoming signal. A Lyapunov-based stability criterion is proposed to guarantee global stability of the closed-loop fuzzy system. Some extra criteria are proposed for the exponential-stability property. Three different systems are used to demonstrate the proposed regulator and servo controller can, respectively, achieve the
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S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
25
20
output response
output response
30 F=[-3 1; -5 -0.2]
10 0 -10
15 10 5 0
0
5
10 time
15
20
0
100
6 F=[-1/30 0; 5 -0.2]
80
output response
output response
F=[-3 1; 5 -2]
20
60 40 20 0
5
10 time
15
20
x 106 F=[1/3 5; -1/5 -1/9]
4 2 0 -2 -4
0
50 time
100
0
50
100
time
Fig. 6. Output response (dashed line) of affine TS-based servomechanism in Eq. (38) for incoming signals (solid line) with initial condition z(0) = [5 10], parameters E = [3 1] and various F.
desired stabilizing and tracking effects in a very short time span. The three adopted systems satisfy Corollary, Theorem 1 and neither one of the above theorems which suggests that there must be more relaxed conditions than these criteria. We conclude that highly mathematically trained researchers can perhaps propose more flexible conditions to guarantee the stability of the closed-loop fuzzy system. We examined the influence of a modelling error on the offset term by comparing the performance of a system with and without the compensate term [38]. Better stabilization performance is achieved by when the compensate term is included in the system. We also found the corresponding performance to be predictable for the original nonlinear closed-loop system. Acknowledgement We gratefully acknowledge the editorial assistance of Marc Grenier, MEG International English Center, Tainan City, Taiwan. This research is supported by the Program NSC 95-2221-E-212-013. Appendix A. Proof of Theorem 2. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(39)
i=1
We know the real parts of eigenvalues of Q, e [ (Q)] (Q), where (Q) is any kinds of matrix measure. And, (·) is a convex function; i.e., [aM + (1 − a)N] a(M) + (1 − a)(N ),
∀a ∈ [0, 1], ∀M, N ∈ n×n .
S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Let gi (X0 ) =
r hi (X0 ) , fi (X0 ) i=1,i = hi (X0 )
=
gi (X0 ) gi (X0 )
r
2303
and use hi , gi , fi to denote hi (X0 ), gi (X0 ), fi (X0 ) for notation
i=1,i =,
simplification, respectively. r r e hi Aci hi Aci i=1
i=1
⎛
(1 − h ) ⎝ ⎡
⎞
r
gi Aci ⎠ + h (Ac ),
i=1,i =
⎛
⎞
r
(1 − h ) ⎣(1 − g ) ⎝
⎤
fi Aci ⎠ + g (Ac )⎦ + h (Ac ),
i=1,i =,
h (Ac ) + (1 − h )g (Ac ) + (1 − h )(1 − g )f (Ac ) + · · · . T If we choose the induced 2-norm as our matrix measure, 2 (Q) = max Q+Q , we have 2 r
Ac + ATc Ac + ATc e h max hi Aci + (1 − h )g max 2 2 i=1 Ac + ATc + ··· < 0 +(1 − h )(1 − g )f max 2 for max (Aci + ATci ) < 0, i = 1, . . . , r.
(40)
Proof of Theorem 3. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(41)
i=1
Let eigenvalues are arranged in order with 1 [·] 2 [·] · · · n [·]. For Hermitian matrix Aci , i = 1, . . . , r, hi (X0 )Aci is also Hermitian and ⎡ ⎤ r
r hi (X0 )Aci = k ⎣ hj (X0 )Acj + h (X0 )Ac ⎦ k j =1,j =
i=1
⎡ k ⎣
r
k ⎣ ...
i=1
⎤ hj (X0 )Acj ⎦ + h (X0 ) n [Ac ]
j =1,j =
⎡
r
r
⎤ hj (X0 )Acj ⎦ + h (X0 ) n [Ac ] + h (X0 ) n [Ac ]
j =1,j =,
hj (X0 ) k [Acj ] +
r
hi (X0 ) n [Aci ],
∀k = 1, . . . , n.
(42)
i=1,i =j
e k
r i=1
! hi (X0 )Aci
hj (X0 )e { n [Acj ]} +
r
hi (X0 )e { k [Aci ]} < 0
i=1,i =j
o for the case (A , B ) is c.c. and (A , C) is c.o. since (Aci ) ∈ C− i i i
(43)
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S. Wu / Fuzzy Sets and Systems 158 (2007) 2288 – 2305
Proof of Theorem 4. Via the converse theorem, we know the stability of the fuzzy system in Eq. (15) at X0 neighborhood concurs with that of the corresponding linearized fuzzy systems, ˙ X(t) =
r
hi (X0 )Aci X(t).
(44)
i=1
For Aci , i = 1, . . . , r, are simultaneously similar to diagonalization or to triangularization, we have the spectrum, r
r o
hi (X0 )Aci ⊆ hi (X0 ) (Aci ) ∈ C− , (45) i=1
i=1
as all (Ai , Bi ) are c.c. and (Ai , C) are c.o.
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