Intelligent Digital Redesign Approach - Semantic Scholar

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 5, OCTOBER 2005

701

Digitalizing a Fuzzy Observer-Based Output-Feedback Control: Intelligent Digital Redesign Approach Ho Jae Lee, Jin Bae Park, and Young Hoon Joo

Abstract—This paper concerns an intelligent digital redesign (IDR) technique for a Takagi–Sugeno fuzzy observer-based output-feedback control (FOBOFC) system. The term IDR involves converting an existing analog control into an equivalent digital counterpart in the sense of state-matching. The IDR problem is herein viewed as a minimization problem of the norm distances between nonlinearly interpolated linear operators to be matched. Its constructive condition with global rather than local state-matching is formulated in terms of bilinear matrix inequalities. The main features of the proposed method are that the state estimation error in the plant dynamics is considered in the IDR condition that plays a crucial role in the performance improvement; the stability property is preserved by the proposed IDR algorithm; the separation principle is shown when the premise variables are measurable; finally, the IDR condition for a more general FOBOFC—an estimated premise variable case is conducted. A numerical example is demonstrated to visualize the feasibility of the developed methodology. Index Terms—Fuzzy observer-based output-feedback control (FOBOFC), intelligent digital redesign (IDR), separation principle, Takagi–Sugeno (T–S) fuzzy system.

I. INTRODUCTION

D

IGITAL control of an engineering system often evolves an analog plant controlled by feeding sampled outputs back with analog-to-digital and digital-to-analog devices for interfacing in which continuous-time and discrete-time signals coexist. It makes traditional analysis tools for a homogeneous signal system unable to be directly used. To fully enjoy the advancement of the digital technology in control engineering as well as surmount the theoretical obstacles, various digital control techniques have been consistently pursued with tremendous effort by many researchers. Among these, yet another efficient approach is, so-called digital redesign (DR) [1], [8], [10], [15]–[17], [19]–[24], to convert a well-designed analog control into an equivalent digital one maintaining the property of the original analog control system in the sense of state-matching, by

Manuscript received February 20, 2004; revised December 6, 2004. H. J. Lee was with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea. He is now visiting the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204 USA (e-mail: [email protected]). J. B. Park is with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]). Y. H. Joo is with the School of Electronic and Information Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2005.856556

which the benefits of both the analog control and the advanced digital technology can be achieved. It is known that the DR was basically developed only for linear time-invariant (LTI) systems [1], [10], [16], [22]. For that reason, it has been strongly demanded to build some intelligent digital redesign (IDR) methodologies for complex nonlinear systems, in which the first attempt was initiated by Joo et al. [6]. They synergistically merged both the Takagi–Sugeno (T–S) fuzzy-modelbased control and the DR. It has been the impetus for several results we have obtained: Chang et al. extended the IDR to uncertain T–S fuzzy systems [5] and elaborated it [2]. Very recently, an IDR scheme was suggested ensuring the stabilizability as well as the global state-matching [7]. Note that all discussed works have concentrated on the state-feedback case. Judging from a practical engineering point of view, the output-feedback is appealing because all state variables of many industrial plants cannot be directly available through control process. There have been some investigations on DR focusing on LTI observer-based output-feedback control (OBOFC) from several disparate perspectives [8], [10], [21], [22], [23]. However, until now, no tractable IDR method for the T–S fuzzy OBOFC (FOBOFC) has been proposed. In establishing an IDR for FOBOFC, one may unintentionally try to use the currently accessible IDRs [5], [6] and the duality concept between a state-feedback control design and a Luenberger-type full-order observer design, to obtain the digital control gain matrices and the discrete observer gain matrices. Such a strategy for LTI system can be found in [21], [22], [24]. In the approach, however, although the state estimation error inevitably occurs, it was ignored in deriving the DR condition. Consequently, the redesigned digital control may not match the states, which is one of the main factors that degrade the performance of the IDR for FOBOFC. Clearly, to enhance the performance, the estimation error in the plant dynamics should be carefully deliberated in investigating the IDR condition. Moving to the state-matching issue, the previous IDRs were acquired under the local state-matching criterion of each sub-closed-loop system—the control loop closed with the th plant and the th control rule. Obviously, they do not guarantee the global statematching between the analog and digital control systems, which is also an underlying factor in performance deterioration. Moreover, preserving the stability has been assumed to be implicitly retained by closely matching the digitally controlled state with the analogously controlled stable state. Involving an explicit stability preserving condition into the IDR algorithm has not been treated. Thus far, the aforementioned remains yet to be theoretically challenging issues and thereby must be fully tackled.

1063-6706/$20.00 © 2005 IEEE

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Motivated by the previous observations, this paper aims at developing IDR for a T–S FOBOFC system. To alleviate the problems raised before, we propose an alternative way—numerical optimization-based IDR involving bilinear matrix inequalities (BMIs). Casting the IDR problem into such a format is highly desirable since it allows one to simultaneously reflect a variety of specifications in the design algorithm. The main contributions of this paper are twofold: i) a constructive IDR condition in the presence of an injective mapping from the output to the premise variables of the concerned T–S fuzzy system—a measured premise variable case; and ii) an IDR condition in the absence of any injective mapping from the output to the premise variables—an estimated premise variable case. Specifically, to achieve the former, the global state-matching problem in the IDR is viewed as a minimization problem of the norm distances between the nonlinearly interpolated linear operators to be matched. The matching condition of the state estimation error in the plant dynamics is incorporated. The derived condition guarantees, under the mild prerequisite of a global exponential analog FOBOFC, the global exponential stability (GES) of the digital FOBOFC system. The separation principle on IDR is explicitly verified. The IDR for the latter, a more general FOBOFC without any injective mapping from the output of the T–S fuzzy system to the premise variables is also investigated. The analysis of this case is complicated since the premise variables of a T–S fuzzy observer depend on the estimated state [11], [25]. The rest of this paper is organized as follows: The T–S FOBOFC system is briefly reviewed for analog and digital cases in Section II. New IDR methods are proposed for the concerned systems with measured and estimated premise variable cases in Section III. An example—the permanent magnet synchronous motor (PMSM) is provided in Section IV to show the effectiveness of the proposed method. Finally, Section V concludes the paper with some discussions.

One way to view a T–S fuzzy system is that it performs the nonlinearly interpolated linear mapping and so as to satisfy

with arbitrary small scalars and , which we call the universal approximation in the literature. that represent Suppose there exist triplets the local dynamic behavior of (1), such that the matrix polytope

contains the domain and the range , denotes a convex hull of the set , where and , and . Thus, one can and as find an adequate mapping at time instant with follows:

where

ranges over a matrix polytope

and

with . The central spirit of a T–S fuzzy inference system is to determine the coefficients in the convex combination of the given vertices by virtue of the available qualitative knowledge from domain experts, which is quantified by “IF–THEN” rule base. More precisely, the th rule of the T–S fuzzy system is formulated in the following form: IF

II. SYSTEM DESCRIPTION

is about

and

and

is about

THEN

A. Analog FOBOFC System Many physical systems are very complex in practice and have strong nonlinearities and uncertainties so that rigorous mathematical models can be difficult, if not impossible, to obtain [14]. Fortunately, certain class of nonlinear dynamical systems can be expressed in some forms of a linear mathematical model locally, or as an aggregation of a set of linear mathematical models. Consider a nonlinear dynamical system of the following form:

where

denotes the th fuzzy inference rule; , is the premise variable injectively mapped from , is the consistent fuzzy set of the th premise variable in the th fuzzy inference rule. Using the center-average defuzzification, product inference, and singleton fuzzifier, the global dynamics is inferred as

(2)

(1) where

is the state; is the control input; is the output. The subscript “ ” means the analog control, while the subscript “ ” will denote the digital control in the sequel. The vector field on some compact sets and , is assumed to be and piecewise ([26], Ch. 14), affine in is assumed and the output mapping . to be

where

LEE et al.: DIGITALIZING A FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL

and is the membership value of in , where the universe of discourse of is a compact set. In many real control problems, it is not always possible for all state variables in (2) to be measured. Under this circumstance, a possible remedy is to introduce a Luenberger-type full-order observer parameterized by the following rules: IF

is about

and

and

is about

THEN whose defuzzified output is represented by (3), as shown at the bottom of the page. Throughout this paper, a well-constructed analog FOBOFC warranting GES is assumed to be predesigned by cascading (3) over a control specified by the following rules: IF

is about is about

and

and

THEN whose defuzzified output is given by

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matching criterion at each sampling time instant. Thus, it is more convenient to express (6) as a discrete-time system for derivation of the IDR condition. There are several methods in discretizing an LTI system. Unfortunately, these approaches are not directly applicable to the discretization of the T–S fuzzy system since it is not LTI but implicitly time-varying [3]. Moreover, it is further strongly desired to maintain the polytopic structure of the discretized T–S fuzzy system for the construction of a digital fuzzy-model-based control. Thus we need a mathematical foundation for the discretization of T–S fuzzy systems. Assumption 1: Assume that the firing strength of the th rule is approximated by its , that is, for value at time . Consequently, the nonlinear matrix functions and can be approximated by constant matrices and , respectively, over any interval [2], [3], [27]. is chosen, AsIf a suitably small sampling period sumption 1 is reasonable and we have the following result. Proposition 1: The pointwise dynamical behavior of (6) can be efficiently approximated by

(4) Let the state estimation error be , then the augmented analog closed-loop T–S fuzzy system (2) with (3) and (4) is written as

(5) where

(7) where and Proof: The general solution is initial data

. to (6) with any

.

B. Digital FOBOFC System and its Discretization By sharing the premise parts of (2), a hybrid T–S fuzzy system closed by a digital FOBOFC is represented by (6) where is a piecewise-constant digital control to , where , be determined in a time interval is a nonpathological sampling period. and The (I)DR problem is usually set up in discrete-time setting, to find some relevant digital control satisfying the state-

where the state transition map satisfies and with the initial condition . Let the initial time be , then the exact solution to (6) can be written as evaluated at

where

and .

(3)

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The exact evaluation of is very difficult, if not impossible, since (6) exhibits nonlinear dynamical behavior. To get around with this difficulty, suppose that Assumption 1 is satand are approximately evaluated as isfied, then follows:

Remark 2: The discretized T–S fuzzy system (7) contains the . discretization error with order of To form a digital FOBOFC using only the sampled outputs, a discrete fuzzy-model-based observer rule is introduced by simply copying (7); see the equation at the bottom of the page, from , which is inferred as (8), to produce as shown at the bottom of the page. In this study, the intelligently redesigned digital control takes the following form: IF THEN for by

is

and

and

is

, and the overall control is given

(9)

and

during the sampling time interval. . Let the estimation error be Then the controlled system (7) closed by (8) and (9) is augmented to supply

(10) where . Corollary 1: The pointwise dynamical behavior of (5) can also be approximately discretized as (11) where where

and

. Remark 1: If is singular, lowing formula [22]:

can be computed by the folfor all

, where . Proof: The proof is straightforward from Proposition 1.

IF

is

and

and

is

THEN

(8)


Remark 3: It is noted that the block matrices that comprise are calculated as follows [9]:

and

for all pairs

.

III. MAIN RESULTS A. IDR of FOBOFC Systems Simply speaking, the goal of the IDR problem in our concerned system framework is to convert an existing analog exponential FOBOFC into an equivalent digital one in the sense of state-matching. To attain this, one may try to obtain the digital control gain matrices by using the previous IDR techniques [5], [6] and the discrete observer gain matrices by further applying the duality concept, respectively. Such a strategy for LTI systems can be found in [21], [22]. The (F)OBOFC, however, generally possesses the state estimation error in the plant dynamics [8]. Exploiting [6] and [5] is not suitable because the methods will employ the unperturbed sub-closed-loop differenfor derivation of the tial equation local state-matching condition, instead of the actual one for the global state-matching. As a result, the intelligently redesigned digital control cannot match the estimation error inside the plant dynamics. This is the reason why the existing methods are not directly applicable to the FOBOFC. Therefore, to well intelligently digitally redesign the FOBOFC, the global plant dynamics containing the estimation error should be utilized, and , and not only the state-matching between and or, but their estimation matching between equivalently, their estimation-error-matching between and should be deliberated. What is more, it is highly desired for the intelligently redesigned digital FOBOFC to preserve the stability property of the original analog FOBOFC system. To this end, we formulate the following IDR problem. Problem 1 (IDR for FOBOFC): Given the well-designed and for (4) and (3) that ensure the GES gain matrices and for (9), and for (8) such that the of (5), find followings are satisfied. 1) The state of the discrete representation (10) of the digitally controlled system (6) with (8) and (9) matches of the discrete representation (11) of the the state analogously controlled system (5) at every sampling time , as closely as possible. instance 2) The digitallycontrolledsystem(6)with (8)and(9)isGES. Consider the first objective in Problem 1. In parallel with (10) under the assumpand (11), to realize , it is desired to determine , and tion in such a way that the following matrix equality conditions: (12) (13) (14) hold for all pairs

.

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The second objective in Problem 1 can be handled in a discrete manner by virtue of the following proposition. Proposition 2: Suppose (10) is GES; then the zero equiliband of the hybrid rium points digital FOBOFC system (6), (8), and (9) are also GES. Proof: From the supposition of the GES of (10), any closedexponentiallyconvergesto as loopsolution ,foranyinitialdata .Next,itfollowsfrom that (6), (9), and Assumption 1 for

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Fig. 1. Comparison of the state trajectories of the controlled systems for sampling period duality concept (dashed).

where

is independent of . By the GES definition of (10): , for any initial data , we further proceed over the interval

where and . Therefore, we conclude that the trivial solutions to (6) and (8) with (9) are GES. Remark 4: The complexity of Problem 1 is threefold. 1) The equalities (12) and (13) may be solved for and if and , and and both are nonprovided singular. Similarly, (14) may be solved for and is nonsingular. However, these conthat ditions are not satisfied in general even in LTI case [8]. 2) It should be addressed that (12)–(14) are hardly solved in many cases of T–S fuzzy-model-based control, since , and should satisfy each variable different equality constraints, respectively, for the global matching [7]. 3) In addition, preserving the stability property in the IDR procedure should be critically secured. However, guaranteeing the GES of (10) weights the problem down. In order to cope with the difficulties, an alternative approach is applied by relaxing Problem 1 and searching the digital FOBOFC in a numerical manner. Our key idea is to find , and such a way that the norm distances between and , and , and and , respectively, are minimized by using a numerical optimization technique. It is observed that the norm distances minimization beand stands for the case of tween only, . Note that we are seeking arbitrary functions , so that such a minimized , distance is valid for any admissible value of which requires to examine whether it is guaranteed or not when . This is an infinitely constrained problem for

T

= 0:001 (s): analog (dotted), proposed (solid), and [6] plus the

which each constraint corresponds to a given point in . The following finitizes the dimensionality of . Lemma 1: Suppose in the sense of the induced two-norm measure for all triplets ; then the following holds:

Proof: Since the induced two-norm is a convex operator on a convex domain , we have

which establishes the claim. Hence, Problem 1 is restated as follows. Problem 2 ( -Suboptimal IDR for FOBOFC): Given the well-designed gain matrices and for (4) and (3) that and for (9), and for (8) ensures the GES of (5), find such that the following are satisfied. 1) Minimize and over , and subject to , and in the sense of the induced 2-norm measure, for all . 2) The discretized closed-loop system (10) is GES in the sense of Lyapunov. Now, we are in position to present the main results. Theorem 1 ( -Suboptimal IDR for FOBOFC): Given a fulfilling Assumption 1, if there exist sufficiently small


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Fig. 2. Time responses by the proposed method for T = 0:001 (s). (a) x (t) and x (t) (dashed), x (t) (solid), and x (kT ) (step). (b) x (t) and x (t) (dashed), x (t) (solid), and x (kT ) (step). (c) x (t) and x (t) (dashed), x (t) (solid), and x (kT ) (step). (d) u (t) (dashed) and u (t) (solid).

symmetric positive–definite matrices , symmetric , and matrices matrices with compatible dimensions, and possibly small scalars such that the following two minimization problem (MP) have solutions: Minimize subject to (15)

of the discrete representation (7) of (6) controlled then, by the intelligently redesigned digital FOBOFC (9) in cascade of the discrete representation with (8) closely matches (11) of the analog FOBOFC system (5), and (10) is GES in the denotes the sense of Lyapunov stability criterion, where means transposed element in symmetric positions and such that . all pairs Proof: First, consider the last constraint in the first objecwith a tive of Problem 2. Introducing a free matrix variable full-column rank of , we have

(16) (23) . Choosing such that , and from the definition of the induced two-norm, (23) holds if the following inequality is satisfied: where

(17) (18)

(24) (19) Minimize subject to

Using the Schur complement, applying the congruence transformation with , denoting , and finally abusing as yields (20). Next, the first constraint in the first objective of Problem 2 holds if and only if the following inequality is satisfied:

(20)

(21) (22)

which is equivalent to (15). We can again establish a similar argument to that from before with the second constraint of Problem 2, to obtain (16). The remaining constrains (17)–(19),

708


Fig. 3. x

Time responses by [6] plus the duality concept for T = 0:001 (s). (a) x (t) and x (t) (dashed), x (t) (solid), and x (kT ) (step). (b) x (t) and (t) (solid), and x (kT ) (step). (c) x (t) and x (t) (dashed), x (t) (solid), and x (kT ) (step). (d) u (t) (dashed) and u (t) (solid). x

(t) (dashed),


(21), and (22) directly follow from the standard Lyapunov GES theorem [12]. Their detailed derivations are shown in Appendix. This completes the proof of the theorem. Corollary 2 (Separation Principle on IDR): The separation principle holds for IDR of FOBOFC: The digital fuzzy-modelbased control can be intelligently redesigned to be GES, independent of the global exponential detectability of the intelligently redesigned discrete fuzzy-model-based observer, and the whole FOBOFC system is still GES. Proof: It is easy to see that MP 1 in Theorem 1 neither includes the MP 2 searching variables, nor MP 2 does vice , and for (9) and , versa. Hence and for (8) can be searched independently. The conclusion is immediate. Remark 5 (Sampling Rate Selection): It is noted that the mapping of a continuous-time system to its corre-

T

= 0:002 (s): Analog (dotted), proposed (solid), and [6] plus the

sponding discretized system can be one-to-one if a selected sampling period satisfy the sampling theorem [21]. If a sampling period that violates the sampling theorem is selected, then the satisfactory state-matching will not be such that achieved. Hence, it is suggested to choose , equally

to acquire an acceptable state-matching performance. and , and in Theorem 1 are When redundant and the IDR problem is reduced to a state-feedback case.


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Corollary 3 (IDR of Fuzzy State-Feedback Control): Given a sufficiently small fulfilling Assumption 1, if there exist sym, matrices and a possibly metric positive definite matrix such that the following MP is solved: small scalar

Remark 6: It is worth noticing that the number of matrix inequalities in Theorem 1 and Corollary 3 can be reduced by permuting indexes of which rules are fired simultaneously, for an example of (16) as follows:

Minimize subject to

as well as of (15) and (20). Remark 7: The matrix constrains regarding on , and for all have to be solved in IDR procedure.

such that , do not

B. Extension to a General FOBOFC: Estimated Premise Variable Case then, the intelligently redesigned digital state-feedback control is given through the relation

for all Proof: See [7].

, where

.

, rather When there exists an injective mapping from , to , the premise variables for (3) should than from , denoted by because all state varibe mapped from ables are not directly available. In this case, the firing strength , is different from that of (2), and in of (3), denoted by turn, the augmented closed-loop representation (5) is no longer valid. The analog closed-loop system is reaugmented in the form shown in (25) at the bottom of the page. According to Corollary 1, the pointwise dynamical behavior of (25) evaluated at

(25)

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Fig. 6. x


(t) (dashed),


with an initial time

is expressed

1)

as

(26) where the equation shown at the bottom of the next page holds, . On the other hand, the pointfor all wise dynamical behavior of the closed-loop digital FOBOFC system constituted by (7), (8), and (9) is likewise reconstructed as shown in (27) at the bottom of the next page, where the second equation shown at the bottom of the next page holds. Remark 8: There are the points to be specially considered.

2)

T

= 0:005 (s): Analog (dotted), proposed (solid), and [6] plus the

Note that contrary to (5), the estimation error dynamics in (25) and (27) are the function vectors not only in and but also in and . Conis neither longer zero matrix, nor sequently its discretized digital counterpart in (27). It signifies such block matrix operators should be considered in establishing the estimation-error-matching condition: should be minimized. Pursuing the GES of (27) via the intelligently redesigned digital FOBOFC may degrade the state-matching performance. For that reason, hereafter, we advocate to seek the local exponential stability (LES) rather than GES.


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Lemma 2: The term , that is

is uniformly in order of

(28)

Proof: Since the T–S fuzzy system is capable of univer[26, Ch. 14], it separates points sally approximating (1) on on . Thus, the existence of the injective mapping from to renders whenever , which implies , and

(27)

712


Fig. 9. x


(t) (dashed),

, from which (28) directly follows. Corollary 4: The LES of (27) on some compact set conassures that of (6) taining an equilibrium point with (8) and (9). Proof: It is readily proved from Proposition 2. The IDR for the estimated premise variable case is summarized in the next theorem. Theorem 2 (Estimated Premise Variable Case): Suppose , not from , there exists an injective mapping from . Given a sufficiently small fulfilling Assumption 1, if to , there exist symmetric positive–definite matrices , and matrices symmetric matrices with compatible dimensions, and possibly small such that the MP, shown in (29)–(38) at the bottom of the next page, has solutions, then, of the discrete representation (27) of (6) controlled by the intelligently redesigned digital FOBOFC (9) in cascade with of the discrete representation of (8) closely matches the analog FOBOFC system (26), and (27) is LES in the sense of Lyapunov stability criterion. Proof: The proof is along a similar line to the Proof of Theorem 1: (29), (30), (34), and (35) stand for the induced two-norm distance minimization between the partitioned block matrix operators in (26) and their digital counterparts in (27). The remaining matrix inequalities (31)–(33) and (36)–(38) directly follow from the standard Lyapunov stability theorem [12]. The detailed derivation is shown in Appendix, which completes the proof of the theorem. Remark 9: To reduce the number of the matrix inequalities in Theorem 2, the index permutation should be performed with respect to and in (29), (30), (34), and (38) because it is . generally true that

Remark 10: The above formulation leads to a BMI optimizaand , and and . Therefore, if tion problem in and are fixed, finding and becomes a linear matrix inequality (LMI) problem and vice versa. In what follows, we solve the problem in an iterative fashion by treating the BMI as a double LMI [13]. Notice that the double LMI problem is feasible only if the BMI problem is feasible. For brevity, we let MP denote MP 4 excluding (31) and a block-diagonal matrix whose nonzero entries (36), are the left-hand side of (31) and (36), and the maximum eigenvalue. Algorithm 1 depicts a summary of the double LMI approach to MP 4. Remark 11: MP 1 in Theorem 1 is also bilinear in and , hence the double LMI approach can be utilized to solve MP1. IV. SIMULATION STUDY: THE PMSM The example we consider is motivated by the control of PMSM [4]. Consider the dynamical behavior of the smooth-air-gap PMSM without the external load torque modeled as, based on the direct-quadrature axis

where are the direct and quadrature current components; is the motor angular velocity; and stand for the direct and the stator quadrature input voltage components; the direct and quadrature-axis winding resistance, stator inductors, the polar moment of inertia, the viscous damping coefficient, the the number of pole-pairs permanent-magnet flux, and [4].


In order to cast the PMSM model into a T–S fuzzy system under consideration, the nonlinear terms and should be expressed as convex combinations of the following form:

713

where an auxiliary assumption is that (39) yields

(39)

Minimize

. Solving

(40)

subject to

(29)

(30) (31) (32) (33)

(34)

(35) (36) (37)

(38)

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By using (40), a T–S fuzzy system of the PMSM is constructed as IF

is about

THEN

where , and

and to explicitly show the feasibility of the proposed method, we further assume the output matrices to be

From [11, Th. 7], the gain matrices for the analog FOBOFC (3) and (4) assuring GES of (5) are computed as follows:

which guarantee the LES of (25). We seek to intelligently digitally redesign the given analog FOBOFC. On the advice of Remark 5, the sampling period is set (s). Note that there exists no injective mapping to be to , thus the problem belongs to the estimated from premise variable case. Accordingly, Theorem 2 is applied to obtain

concept is also simulated in which one should notice that the redesigned digital gain is dependent only on each local dynamical behavior. The initial states are set to be and . The simulated trajectories and their time responses by the both methods are reported in Figs. 1–3. As one can immediately witness, the state trajectory by the proposed method is almost identical to that of the original analog control system and is well . However, the compared method inguided to deed heavily deteriorates the state-matching performance even for relatively small sampling period. In another set of simulaand (s). The contion runs, we take trolled state trajectories are depicted in Figs. 4 and 7. Figs. 5, 6, 8, and 9 draw the time responses and the applied controls. It is observed that the state-matching performances by the proposed method are somewhat degraded yet the state trajectories have a strong resemblance to the original one, whereas the others fail in control. V. CLOSING REMARKS In this paper, a new IDR has been investigated for a FOBOFC system. The developed technique, unlikely the existing methods, formulated the concerned IDR problem as constrained minimization problems and the associated conditions are spelled out by BMIs, so that a variety of design specifications are involved. To the authors’ best knowledge, this paper is the first attempt to develop an IDR technique for the FOBOFC system. Some interesting features are: i) the state estimation error in the plant dynamics is considered in the IDR condition, which is the main factor that improves the performance of the IDR significantly; ii) the exponential stability of the system closed by the intelligently redesigned digital FOBOFC is preserved by the proposed IDR algorithm; iii) it is shown that the separation principle on the IDR holds when the premise variables are measurable; iv) the IDR on a more general FOBOFC—the estimated premise variable case is explored. The simulation results on the PMSM convincingly demonstrated the advantage of the developed redesign method compared to the previous approach. It implies the potential of the proposed IDR method for reliable digital industrial applications. APPENDIX A. Proof of Theorem 1 First, we show the stabilizability via the digital control . To this end, it suffices to prove that, there exists a Lyapunov function such that

with . A possibly comparable approach combining the method in [6] and the duality

(41)


and (18) holds on entire for all triplets . Importing an auxiliary condition as (41) bilinear in

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Let or (19) renders

By Schur transforming, we arrive at (17). To show the detectability we assume a Lyapunov functional candidate where . Suppose (22) holds. The quantity then exponentially , the following decreases, if and only if for some holds:

then we can write

(42) By the Sylvester criterion, the right-hand side (RHS) of (42) is less than if and only if (43)

on for all pairs . Using Schur complement, , and taking the congruence transformation with produces (21). denoting Now, to prove the stability of (10), it only remains to show the existence of a Lyapunov function of the form , where and [18]. To this end, we prove the and . It can be verified that existence of satisfies on entire for some posto . The function satisfies itive constants . The rate of similar inequalities with constants increase of along the closed-loop trajectory of (10) is computed by

where is the eigenvalue with minimum absolute value of the matrix of the quadratic form in the RHS of (42). It is readily seen and small that one can always choose sufficiently large without violating (43). Hence we conclude that (10) is GES in the sense of Lyapunov, whenever (17)–(19), (21), and (22) are satisfied. B. Proof of Theorem 2 Proving (31)–(33) and (36)–(38) is along a similar line to that of (17)–(19) in Theorem 1, and from which, we know that there exists a Lyapunov functional satisfying and on the entire . Furthermore, Lemma 2 ensures that there exists a compact set for any such that whenever . To prove the stability , the rate of increase along the trajectory of (27) of (27) on is majorized by starting from

in which the RHS of the above constitutes the quadratic form in where

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From the Sylvester criterion, if

is small enough to satisfy

then

on for some constant , hence we conclude that the origin is an LES equilibrium point of (27). REFERENCES [1] W. Chang, J. B. Park, H. J. Lee, and Y. H. Joo, “LMI approach to digital redesign of linear time-invariant systems,” in Proc. Inst. Elect. Eng., Control Theory Appl., vol. 149, 2002, pp. 297–302. [2] W. Chang, J. B. Park, and Y. H. Joo, “GA-based intelligent digital redesign of fuzzy-model-based controllers,” IEEE Trans. Fuzzy Syst., vol. 11, no. 1, pp. 35–44, Feb. 2003. [3] Z. Li, J. B. Park, and Y. H. Joo, “Chaotifying continuous-time T–S fuzzy systems via discretization,” IEEE Trans. Circuits Syst. I, vol. 48, no. 10, pp. 1237–1243, Oct. 2001. [4] Z. Li, J. B. Park, Y. H. Joo, B. Zhang, and G. Chen, “Bifurcations and chaos in a permanemt-magent synchronous motor,” IEEE Trans. Circuits Syst. I, vol. 49, no. 3, pp. 383–387, Mar. 2002. [5] W. Chang, J. B. Park, Y. H. Joo, and G. Chen, “Design of sampleddata fuzzy-model-based control systems by using intelligent digital redesign,” IEEE Trans. Circuits Syst. I, vol. 49, no. 4, pp. 509–517, Apr. 2002. [6] Y. H. Joo, L. S. Shieh, and G. Chen, “Hybrid state-space fuzzy modelbased controller with dual-rate sampling for digital control of chaotic systems,” IEEE Trans. Fuzzy Syst., vol. 7, no. 4, pp. 394–408, Aug. 1999. [7] H. J. Lee, H. Kim, Y. H. Joo, W. Chang, and J. B. Park, “A new intelligent digital redesign: Global approach,” IEEE Trans. Fuzzy Syst., vol. 12, no. 2, pp. 274–284, Apr. 2004. [8] H. J. Lee, J. B. Park, and Y. H. Joo, “An effecient observer-based sampled-data control: Digital redesign approach,” IEEE Trans. Circuits Syst. I, vol. 50, no. 12, pp. 1595–1601, Dec. 2003. [9] T. Chen and B. Francis, Optimal Sampled-Data Control Systems. Berlin, Germany: Springer-Verlag, 1995. [10] J. W. Sunkel, L. S. Shieh, and J. L. Zhang, “Digital redesign of an optimal momentum management controller for the space station,” J. Guid. Control Dyna., vol. 14, no. 4, pp. 712–723, 1991. [11] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250–265, Apr. 1998. [12] E. Kim and H. Lee, “New approaches to relaxed quadratic stability conditions of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523–534, Oct. 2000. [13] K. Kiriakidis, “Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities,” IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 269–277, Apr. 2001. [14] G. Feng, “Approaches to quadratic stabilization of uncertain fuzzy dynamic systems,” IEEE Trans. Circuits Syst. I, vol. 48, no. 6, pp. 760–779, Jun. 2001. [15] S. M. Guo, L. S. Shieh, G. Chen, and C. F. Lin, “Effective chaotic orbit tracker: A prediction-based digital redesign approach,” IEEE Trans. Circuits Syst. I, vol. 47, no. 11, pp. 1557–1570, Nov. 2000. [16] L. S. Shieh, W. M. Wang, and M. K. A. Panicker, “Design of PAM and PWM digital controllers for cascaded analog systems,” ISA Trans., vol. 37, pp. 201–213, 1998. [17] J. Xu, G. Chen, and L. S. Shieh, “Digital redesign for controlling chaotic Chua’s circuit,” IEEE Trans. Aero. Electr., vol. 32, no. 8, pp. 1488–1499, Aug. 1996. [18] A. E. Golubev, A. P. Krishchenko, and S. B. Tkachev, “A separation principle for affine systems,” Diff. Equat., vol. 37, no. 11, pp. 1541–1548, 2001. [19] L. S. Shieh, W. M. Wang, J. Bain, and J. W. Sunkel, “Design of lifted dual-rate digital controller for X-38 vehicle,” J. Guid. Control Dyna., vol. 23, pp. 629–639, 2000. [20] C. A. Rabbath and N. Hori, “Reduced-order PIM methods for digital redesign,” Proc. Inst. Elect. Eng., Control Theory Appl., vol. 150, no. 4, pp. 335–346, 2003.

[21] S. M. Guo, L. S. Shieh, C. F. Lin, and J. Chandra, “State-space selftuning control for nonlinear stochastic and chaotic hybrid systems,” Int. J. Birfurcation Chaos, vol. 11, no. 4, pp. 1079–1113, 2001. [22] L. S. Shieh, W. M. Wang, and J. B. Zheng, “Robust control of sampled-data uncertain systems using digitally redesigned observer-based controllers,” Int. J. Control, vol. 66, no. 1, pp. 43–64, 1997. [23] L. S. Shieh, W. M. Wang, and J. S. H. Tsai, “Digital redesign of controller via bilinear approximation method for state-delayed systems,” Int. J. Control, vol. 70, no. 5, pp. 665–683, 1998. [24] L. S. Shieh, Y. J. Wang, and J. W. Sunkel, “Hybrid state-space self-tuning control of uncertain linear systems,” in Proc. Inst. Elect. Eng., Control Theory Appl., vol. 140, 1993, pp. 99–110. [25] C. S. Tseng, B. S. Chen, and H. J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381–392, Jun. 2001. [26] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001. [27] P. Apkarian, “On the discretization of LMI-synthesized linear parameter-varying controllers,” Automatica, vol. 33, no. 4, pp. 655–661, 1997.

H

Ho Jae Lee received the B.S., M.S., and Ph.D. degrees in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 1998, 2000, and 2004, respectively. He was a Lecturer in the Department of Electrical and Electronic Engineering, the Graduate School of Yonsei University, in 2004. He is now with the University of Houston, Houston, TX, as a Visiting Assistant Professor in the Department of Electrical and Computer Engineering. His current research interests include stability analysis in fuzzy control systems and hybrid dynamical systems, and digital redesign.

Jin Bae Park received the B.E. degree in electrical engineering from Yonsei University, Seoul, Korea, in 1977, and the M.S. and Ph.D. degrees in electrical engineering from Kansas State University, Manhattan, in 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, where he is currently a Professor. His research interests include robust control and filtering, nonlinear control, mobile robot, fuzzy logic control, neural networks, genetic algorithms, and Hadamard-transform spectroscopy. Dr. Park is serving as the Director for the Transactions of the Korean Institute of Electrical Engineers (KIEE) (1998–2003) and the Institute of Control, Automation, and Systems Engineers (1999–2003). He is currently an Editor for the International Journal of Control, Automation, and Systems.

Young Hoon Joo received the B.S., M.S., and Ph.D. degrees in electrical engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea as a Project Manager from 1986 to 1995. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a Visiting Professor in the Department of Electrical and Computer Engineering. He is currently an Associate Professor in the School of Electronic and Information Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent control, fuzzy modeling and control, genetic algorithms, intelligent robot, and nonlinear systems control. Dr. Joo is serving as Editor-in-Chief for the Journal of Fuzzy Logic and Intelligent Systems (KFIS) (2002–2004) and the Associate Editor for the Transactions of the Korean Institute of Electrical Engineers (KIEE) (2000–2004).

Intelligent Digital Redesign Approach - Semantic Scholar

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