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Fuzzy Control Approach for a Class of Markovian Jump Nonlinear Systems Natache S. D. Arrifano and Vilma A. Oliveira, Member, IEEE

Abstract—This paper addresses the problem of stabilizing a class of nonlinear systems subject to Markovian jump parameters using performance. The a robust stochastic fuzzy controller with class of jump nonlinear systems considered is described by a fuzzy model composed of two levels: A crisp level which represents the jumps and a fuzzy level which represents the system nonlinearities. Considering the approximation error between the fuzzy model and the jump nonlinear system as norm-bounded uncertainties, we develop a systematic technique based on a coupled Lyapunov function to obtain a robust stochastic fuzzy controller which guarantees the 2 gain of the closed-loop system in respect to external inputs to be equal to or less than a prescribed value. A simulation example on an industrial power plant operating in a cogeneration scheme is presented to illustrate the effectiveness of the proposed stabilizing controller in reducing oscillations as well as maintaining a desired operation condition in the presence of fluctuations in the local load. Index Terms—Coupled Lyapunov function, fuzzy system controller, Markovian jump systems, stabilizing models, controller.

I. INTRODUCTION HE class of systems with Markovian jump parameters can be used to model a variety of physical systems. This class of systems has two components in the state vector. The first one varies continuously and is referred to as the continuous state of the system. The second one varies discretely and is referred to as the mode assumed by the system. This class of systems can be used to represent complex real systems, which may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs, changing of subsystem interconnections, and abrupt environmental disturbances. Because of the difficulty inherent in the analysis of nonlinear dynamics, most attention has been devoted to the linear representation of the Markovian jump systems which was introduced by Krasovskii and Lidskii [1]. In this context, stochastic stability and stabilizability problems were addressed [2]–[6]. Lately, considerable attention has been paid to the robust control, robust stochastic stability and stabilizability of jump linear uncertain systems [7]–[11]. In general, the system uncertainties considered appear as norm-bounded sets, which facilitates the extension of the deterministic robust and optimal control techniques to the Markovian jump linear

T

Manuscript received May 17, 2004; revised September 15, 2005 and February 8, 2006. This work was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and the Conselho de Desenvolvimento Científico e Tecnológico (CNPq). The authors are with the Departamento de Engenharia Elétrica, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil (e-mail: [email protected]. usp.br, [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.877359

systems. However, a more realistic model should consider the nonlinearities of a real system, to the best of our knowledge, the problem of stochastic stability and stochastic stabilizability of nonlinear systems have not been drawn much attention yet. The control of a class of nonlinear systems with Markovian jump parameters was considered in [12]; wherein, the problem is formulated in terms of dynamic programming. Recently, control problem for a class of linear continuous-time the systems with Markovian jumping parameters and unknown nonlinearities was addressed in Boukas et al. [13]; wherein, the robust stochastic stabilizability and disturbance attenuation are considered. In this approach, the unknown nonlinearities are modeled as norm-bounded uncertainties satisfying the matching condition. A similar approach for discrete-time systems in terms of linear matrix inequality (LMI) is presented in [14]. However, if nonlinearity is not distinguished from uncertainty, the results obtained are in general conservative. The fuzzy-model-based control design techniques have been successful used to stabilize nonlinear systems [15]–[18]. In general, the nonlinear dynamics are represented by a Takagi–Sugeno (TS) fuzzy system which is described by fuzzy IF–THEN rules representing local input–output relations of the nonlinear system [19]. The basic idea of this approach is to decompose the input space into many subspaces, approximating the nonlinear system by a fuzzy blending of local linear systems associated to each subspace. In fact, it can be shown that TS fuzzy systems are universal approximators [20]. The fuzzy-model-based control design is based on the fuzzy system modeling via the so-called parallel distributed compensation (PDC) scheme. The idea of the PDC scheme is that a linear control is designed for each local linear system. The overall controller is again a fuzzy blending of all local linear controllers, which is nonlinear in general. The fuzzy-model-based control design techniques are, in general, based on a common positive definite matrix and the solution is given in terms of LMIs. If the LMIs set is not feasible, especially when more stringent performance bounds are specified, piecewise quadratic Lyapunov functions can be used [21], [22]. Recently, a fuzzy-model-based control technique for a class of Markovian jump nonlinear system was introduced in [23] and [24]. In this approach, the class of systems is represented by a fuzzy system model with two levels of structure: A crisp level which describes the jumps of the Markov process and a fuzzy level which describes the system nonlinearities. Using this fuzzy system, the control design is formulated in terms of a coupled-Lyapunov function and a stochastic stabilizability concept so that less restrictive conditions for stability are possible. In this paper, we include in the fuzzy control design the approximation error between the fuzzy model and the nonlinear

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system and the two-level structured fuzzy model as in [23], [24]. Actually, the effect of the approximation error can deteriorate the stability and control performance of nonlinear control systems [25]. Then, using a coupled Lyapunov function and stochastic stability, we develop a systematic technique for degain of signing a robust fuzzy control which guarantees the the closed-loop system in respect to bounded disturbances to be less or equal to a prescribed value. The remainder of the paper is organized as follows. Section II presents the formulation of problem using the fuzzy modeling with two levels of the structure. Section III establishes the main results on the robust stochastic fuzzy-model-based control design. We then, present the design and simulation results for an industrial power plant in cogeneration scheme. In Section V, we conclude with some remarks. and denote the -dimenThroughout this paper, sional Euclidean space and the set of all real matrices, respectively, the superscript “ ” denotes the transpose, means that is a positive–definite (negative–definite, positive–semidefinite, negative–semidefinite, respectively) , and dematrix of appropriate dimensions, note the minimum and the maximum eigenvalues, respectively, and denote the identity and zero matrices of appropriate didenotes, for instance, for , mensions, respectively, , denotes the expectancy operator with respect to some probability measure , stands for the space of square integrable vector functions over , refers to either the Euclidean vector the interval norm or the matrix norm which is the operator norm induced stands for the norm in by the standard vector norm, , denotes the norm in , and is a probability space. II. PROBLEM STATEMENT The class of Markovian jump nonlinear systems (MJNLS) considered in this paper is defined in the probability space and is described by the differential equation

. The matrix is called transition rate matrix. We assume that the Markov with process has initial distribution . be a mode indicator variable, a crisp set, and Let a fuzzy set. The MJNLS (1) and (2) is then represented by the Markovian jump fuzzy system (MJFS) as in [23], [24]. This fuzzy system is structured in an upper and lower levels deassociated to the jumps of the Markov fined by the crisp sets associated process determined by and by the fuzzy sets to the nonlinearities in the state vector . Thus, the th mode assumed by system (1) and (2) is given by

If is Then If is Then

, where ability rate between modes

, and , for

(2) is the prob; and

and

is (3)

where and are matrices of appropriate dimensions which describe local linear representations of the nonlinear is system in the vicinity of chosen operation points, the number of inference rules in each mode, and variables are components of the state vector . In the terminology of fuzzy systems, the IF-part of the MJFS is referred to as the premise part, the THEN-part is referred to as the conse” in the consequent part quent part. The term “ is referenced as the subsystem in mode associated to rule , and variables and in the premise part are known as premise variables. Usually, the premise variables may be functions of state variables, external disturbances, and/or time [26]. The overall MJFS is inferred by a fuzzy blending of subsystems, which are selected according to the mode assumed by the Markov process, that is (4)

(1) is the system state vector, is the where is a bounded external disturcontrol input vector, bance belonging to , and is a continuous-time Markovian process taking values in a finite space state denoted by which determines the mode the system is, with and are smooth nonlinear functions with respect to the first argument, and are the initial values of the state, and the mode at time , respectively. that deterThe evolution of the stochastic process mines the mode of the system at each time is assumed to be described by the probability

and

where is the mode indicator which yields when , for , and are normalized membership functions given by

(5)

with

the grade of membership of , in the fuzzy set . In addition, considering , , we have the fact that in (5) and . The universe of discourse for the MJFS is given by

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Following the PDC scheme, we consider a state feedback fuzzy controller which shares the same structure of the MJFS (3) in the premise part, that is

If

when Using the fact that we suppose there exist bounding matrices for each mode such that

, that is, and ,

is

,

(14)

Then If is Then

and

and

and

is (6)

are the local feedback gain vectors. Thus, where the overall fuzzy controller is given by

(15) for all trajectory , which can be described as [25] (16)

(7) and Considering that stituting (7) in (4), it results

,

,

, then sub(17)

(8) Now, using the fact that (9) and

, system (8) can be rewritten as

(10) and . Therefore, system (1) and (2) can be described by the uncertain fuzzy system (11), where and are the approximating errors defined in (12) and (13), respectively, as shown at the bottom of the page. where

, , , and and where are bounding matrices of appropriate structure and dimensions which can be chosen according to the system nonlinearity. denote the solution of the MJFS (11) with Let fuzzy control law (7) under the initial conditions and . is said to be stochasDefinition 1: The MJFS (11) with and admistically stabilizable if for all initial conditions sible uncertainties, there exists a fuzzy control law (7) satisfying

(18) , of appropriate dimension. for some matrix Definition 1 is similar to that of stochastic stabilizability of Markovian jump linear systems with unknow nonlinearities [13], [27]. Under Definition 1, stochastic stabilizability of the Markovian jump fuzzy system means that there exists a feedback fuzzy control law that asymptotically drives the state from any given initial condition to the origin in the mean square sense, which implies the asymptotic stability of the closed-loop system [28].

(11)

(12)

(13)

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The problem we address in this paper is the design of feed, , for the fuzzy back gain vectors control law (7) that stochastically stabilizes MJFS (11) and attenuates the effect of external disturbance . The attenuation of the external disturbance effect is formulated in terms of the norm. system Definition 2: Consider the MJFS (11). Let be a positive to is said to have the scalar, then the mapping from gain less than or equal to , if for all , initial mode , and fuzzy control law initial state , (7), the constraint

with , a constant matrix of appropriate dimension. The weak infinitesimal operator of (23) is given by

(19)

(24)

, holds for all admissible uncertainties, with a set of weighting matrices of appropriate dimensions which can be chosen to yield the desired performance. denote the system transfer matrix from to the regLet . Then, the norm of is ulated output

, . In Proof: Let mode at time be , that is, what follows, for simplicity of notation, denotes the solution of the MJFS (11) when under the initial condition with fuzzy control law (7). Take the candidate for the coupled Lyapunov function as

(23)

Using [29], it is possible to obtain

(25)

(20) and the problem of disHence, (19) implies control turbance attenuation corresponds to the suboptimal problem.

Substituting (11) in (25) and using the fact that , it results

when

III. ROBUST FUZZY CONTROL In this section, we present a sufficient condition for the robust stochastic stabilization of the MJFS (11) using a coupled Lyapunov function. In order to obtain a systematic fuzzy control design, we formulate a stabilizing control problem in the context of LMIs. is stochastically Proposition 1: The MJFS (11) with stabilizable with state feedback fuzzy control law (7) for all admissible uncertainties if there exists a set of positive–definite , of appropriate dimensions satisfying matrices

(26) Using follows:

, we can write (26) as

(21)

(27) Using the fact that (22) for all

.

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the weak infinitesimal operator given by (27) satisfies

Now, multiplying (21) by we have

and (22) by

,

(33) and (28) Now, considering (14) and (15) for and (17), respectively, we have

and

as in (16)

(29) and

(30) Using the fact that [30]

(34) Adding (33) to (34) and using (9) and for . obtain Now, defining

, we

inequality (30) can be written as (31) Thus, substituting (29) and (31) in (28), we obtain

(35) with

and , and substituting (35) in (32), we can write

(36) Therefore, for all

and

, we have

(37) where

(32)

is a positive scalar given by

(38)

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By the Dynkin’s formula [31], we have

(39) Then, substituting (37) in (39), we obtain (46)

(40) Using the Gronwall–Bellman Lemma [32] in (40), we have

, Proof: Let mode at time be , that is, and consider the closed-loop MJFS (11) with . Choosing the Lyapunov function candidate as in (23), substituting (11) in (25), and proceeding in a similar way as in the proof of Proposition 1, we obtain

(41) Integrating both sides of (41) and taking the limit as results

, it

(42) Considering the fact that in (42) , for all , the result follows by Definition 1. Remark 1: The result in Proposition 1 is similar to the result and found in [23] for the closed-loop MJFS (11) with for all , that is, inequalities (21) and (22) become (47)

(43)

Inequality (47) can be written as shown in (48) at the bottom of the next page. , and Let us now assume there exists a set of , satisfying (45) and (46). Multiplying both sides of (45) and both sides of (46) by , we obtain by

(44) Proposition 2: Consider the MJFS (11). Let and be the initial state vector and the initial mode, respectively. The closed-loop system with the fuzzy control law (7) has the performance (19) for all admissible uncertainties if there exists , of appropriate dimensions a set of matrices , satisfying and (49) and

(45)

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(50)

(48)

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Then, adding (49) to (50) and again using (9) and , we have

the mean square sense and that the control performance given by (19) is guaranteed. To obtain a systematic fuzzy control design in terms of LMIs, inequalities (45) and (46) are written as

(55) and

(51) Now, premultiplying (51) by can verify that

and post-multiplying by , we (56) Pre- and postmultiplying (55) and (56) by

, we obtain

(52) in (52) we have Note that, when disturbance input , which ensures the asymptotic stability of the closed-loop system according to Proposition 1. Integrating both sides of (52), we obtain

(57) and

(53) Finally, substituting (39) in (53), it yields

(54)

and the result follows by Definition 2. Remark 2: For bounded but non vanishing disturbance , that , using (52) we obtain whenever is, with , so that it can be shown, using a standard Lyapunov extension [32], that the solution of the closed-loop MJFS (11) is ultimately bounded in

(58) and . where , , and given in (59)–(62), reNow, defining spectively, as shown at the bottom of the next page, and using the Schur complements [33], (57) and (58) can be written as

(65)

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and

(66) Using Proposition 2 and LMIs (65) and (66), we formulate the following optimization problem to the stochastic fuzzy control performance. design with , and Problem 1: Find a set of matrices of appropriate dimensions to minimize a set of matrices subject to , (65), and (66) . If Problem 1 is feasible, we can obtain the state feedback gain , , . vectors as Remark 3: In the approach given, the fuzzy-model-based , , control design is based on the matrices . Thus, the control design based on LMIs conditions is related to the number of inference rules and to the modes assumed by the Markov process. The properties of the normalized membership functions can be explored in order to reduce the number of intersections among the fuzzy sets and thus producing more relaxed LMI conditions. Examples of fuzzy-model-based control design using relaxed LMI conditions are given in [15] and [17]. Remark 4: In order to consider the stochastic stabilization for the MJNLS in case the equilibrium point is not the origin, , one should perform a change of coordinates that is, to make the origin the equilibrium before designing the fuzzy control (7) using Problem 1. IV. SIMULATION EXAMPLE To illustrate the usefulness of the robust fuzzy control approach, the stabilization of an electrical power system in a co-

Fig. 1. Unifilar diagram of the single-machine infinite-bus power system.

generation scheme [34] as shown in Fig. 1 is considered. The power system is described by the differential equations (67) (68) (69) (70) where ; the power angle [rad]; the rotor speed [rad/s]; the synchronous the mechanical input power [p.u.]; machine speed [rad/s]; the transient EMF in the quadrature axis [p.u.]; the the quadrature equivalent EMF in the excitation coil [p.u.]; axis current [p.u.]; the direct axis current [p.u.]; the inertia the direct axis transient open circuit time constant [p.u.]; constant [s]; the regulator time constant [s]; the regulator the regulator reference voltage [p.u.]; the terminal gain;

(59) .. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

(60)

(61) (62) (63) (64)

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voltage [p.u.]; the power system stabilizer voltage [p.u.]; the external disturbance [p.u.]; the direct axis transient the direct axis reactance [p.u.]. reactance [p.u.]; and The reactive and active powers, and , respectively, are given by (71) (72) Suppose the generator supplies the equivalent load , with the resistance and the reactance, both in p.u. Hence, considering the synchronous-machine direct-quadrature axes modeling [35] with the power angle referenced to the infinite-bus, it is possible to obtain the currents and as follows: (73) (74) where

The numerical values of the physical parameters adopted are: , , , , , , , , , , , , and . In the nominal operating conditions of the power system, the power angle is kept in the range which assures a real power and a reactive power . The external disturbance is used to produce variations about in both the mechanical input power and the rotor speed of the generator during a period of time, due to possible torsional oscillations occuring in the system structure [35]. be the vector of state variables and Let be the input vector. System (67)–(70) presents the equiand . As menlibrium point tioned before, it is necessary to perform a change of coordinates to bring the equilibrium of the power system to the origin. For as the new system coordinate this purpose, we adopt and obtain

(75)

, , , , the infinite-bus voltage, and the equivalent transmission line reactance both in p.u. Under no load conditions, and , with , and . In this paper, the operating conditions of the power system according to the local load variations is modeled as a Markov , which chain with three different modes, that is, correspond to the possible combinations between loads and , with and the resistances in p.u., and and the reactances in p.u. as follows: ; • mode 1—only load 1, ; • mode 2—only load 2, . • mode 3—both loads 1 and 2, We assume that the system under nominal operating conditions is described by mode 3. However, during a period of time, the dynamics of the power system changes according to a Markov process with initial distribution

and transition rate matrix

, with where given in (76)–(79), respectively, as shown at the bottom of the , , next page, for . For , two local linear representacan be obtained around the linearization points tions • mode 1: and ; • mode 2: and ; and • mode 3: ; yielding the following matrices: , , for the power system using the procedure given in Appendix I. • mode 1:

• mode 2:

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• mode 3:

IF IF is THEN IF is THEN

THEN and

is

and

is

Thus, the overall fuzzy system model can be described as (81) The mode indicator membership functions , are crisp functions described before. In this example, the premise variable is determined by monitoring the load acti, vation instants. The normalized membership functions describe the range of the state variables and in each mode and are obtained from gaussian membership functions available in the Fuzzy Logic Toolbox of MATLAB (see Fig. 2). A suitable range for the state variables can be in the interval . determined by constraining Hence, the fuzzy system used to represent the power system (75) is given by IF IF is THEN IF is THEN IF IF is THEN IF is THEN

THEN and

is

and

is

Let the initial conditions be and . The following performance specifications are considered in the stabilization of the power system: 1) maintain the system operating in the equilibrium given by mode 3, and 2) improve the damping of the electromechanical oscillations of the power system. The robust fuzzy control deand between sign considers the approximating errors the nonlinear system and the fuzzy system model. In case of and has the following bounding the power system matrices. • mode 1:

• mode 2:

THEN and

is

and

is

(76)

(77)

(78) (79)

(80)

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Fig. 2. Adopted membership functions.

• mode 3:

A procedure to obtain the above bounding matrices can be found in Appendix II. The weighting matrices , are . Therechosen as , and , , , 2, we obfore, for tain the feedback gains for the stabilization of the SMIB power system (75) by solving the LMIs in Problem 1. Table I presents the control design results. With and as previously defined, the software provided in [36] is used to simulate transitions among operating modes which are given by the Markovian jump process . Fig. 3 shows the load variations during a period of operation of the power system. In Figs. 4 and 5, we compare the results obtained with the robust fuzzy control to the results obtained with a

TABLE I CONTROL DESIGN RESULTS

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Fig. 3. Load variations during a period of the operation of the power system.

classical power system stabilizer given by the transfer function

and

(84) (82) Each row of the matrix , washout filter time constant with gain and lead-lag controller parameters , tuned according to [37].

is given by the formula

, , (85)

V. CONCLUDING REMARKS In this paper, a fuzzy control design with performance for Markovian jump systems is developed. The feedback nonlinear system achieves a minimum gain for disturbance attenuation. The proposed approach is appropriated to practical control design for nonlinear systems with external disturbance and can be solved very efficiently by convex optimization techniques with the aid of the LMI Control Toolbox of MATLAB. The advantage of the fuzzy-model-based control design is that we can consider a more refined description of the parameter variations in the nonlinear system. Because of that, we can provide less conservative conditions for stability in the stochastic sense. Simulation results show the usefulness of the proposed approach to the stabilization of an electrical power system in cogeneration scheme, which is a complex real system subject to random abrupt variations in its dynamics. APPENDIX I LOCAL LINEAR REPRESENTATIONS FOR THE ELECTRICAL POWER SYSTEM IN COGENERATION SCHEME Consider system (75). Let mode at time be , that is, , and , be a linearization point translated to the origin. Using [16], we obtain matrix so that in the vicinity of we have

(83)

where , are the nonlinear functions in each row of as in (76)–(79) and is the gradient, a column vector of , evaluated at . We can use the function , available in the Symbolic Math Toolbox of MATLAB, in order to compute the gradient of the power system. ˙ linearization formula produces linear repThe Teixeira–Zak resentations instead of affine, usually obtained with the Taylor linearization formula. To verify this, consider the Taylor linearization formula

(86) The representation of the function given by

around

is thus

(87) Thus, whenever which occurs if is not the origin, this representation produces affine models instead ˙ of linear models, as mentioned. Hence, using the Teixeira–Zak linearization formula, we can obtain several local linear approximations , of a nonlinear system in any chosen linearization points and then built a fuzzy system representation.

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(94) (95) (96)

APPENDIX II APPROXIMATION ERROR FOR THE ELECTRICAL POWER SYSTEM IN COGENERATION SCHEME

can be written as [25]

Now, (92) with

Consider the approximation error between the nonlinear system (75) and the fuzzy system model (81)

(88)

where and , the matrices obtained in each in Section IV). mode using the linearization points (see Now, suppose that the mode at time is , that is, , . Assume there exist bounding matrices , such that

(93) where and . The procedure to obtain scalars (91) can be computationally solved by selecting a suitable interval for . For example, for the power system considered the angle variations must be about 20%, then a rasonable choice is to take in the interval and compute the interval of , , and using the equilibrium relations (94)–(96), as shown at the top of the page. , Hence, using MATLAB, we compute and follows.

for

in the adopted interval as

(89) , along the trajectory of the state vector , where , and is a bounding matrix of appropriate dimension. In what follows, we give a procedure to obtain . Let and be the th entry of and , respectively. Then, considering a sector description of [38], we define the constraint (90) where , , are positive scalars. Note that at the origin . From (90), maximum scalars for can be obtained as

(91) Thus, we can obtain the bounding matrices , , with , entries of matrix

for

, that is

(92) .

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Fig. 4. State variables: “

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” power system with robust fuzzy control and “. . .” power system with classical stabilizer.

When (89) is satisfied, we have the maximum value of in the interval considered. Then, we compute which results as follows. • mode 1:

, • mode 2:

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Fig. 5. Control input, active and reactive powers: “

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” power system with robust fuzzy control and “. . .” power system with classical stabilizer.

ACKNOWLEDGMENT The authors would like to thank Dr. R. A. Ramos and Dr. R. V. Oliveira for the co-generation power system model and the classical PSS results. • mode 3:

REFERENCES [1] N. N. Krasovskii and E. A. Lidskii, “Analytical design of controllers in systems with random attributes I, II, III,” Automat. Remote Control, vol. 22, pp. 1021–1025, 1961. [2] E. K. Boukas and Z. K. Liu, “Suboptimal design of regulators for jump linear system with time-multiplied quadratic cost,” IEEE Trans. Autom. Control, vol. 46, no. 1, pp. 131–136, Jan. 2001. [3] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “Continuous-time state-feedback control of Markovian jump linear systems via convex analysis,” Automatica, vol. 35, no. 2, pp. 259–268, 1999. [4] M. Mariton and P. Bertrand, “Output feedback for a class of linear systems with stochastic jumping parameters,” IEEE Trans. Autom. Control, vol. AC-30, no. 9, pp. 898–900, Sep. 1985. [5] ——, “Robust jump linear quadratic control: A mode stabilizing solution,” IEEE Trans. Autom. Control, vol. AC-30, no. 11, pp. 1145–1147, Nov. 1985. [6] D. D. Sworder, “Feedback control of a class of linear systems with jump parameters,” IEEE Trans. Autom. Control, vol. AC-14, no. 1, pp. 9–14, Jan. 1969.

H

Now, by inspecting

, we can write , with

obtain the bounding matrices as (see in Section IV).

in order to , for

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Natache S. D. Arrifano received the B.Eng. and M.S. degrees in electrical engineering from the Universidade Federal do Pará, Brazil, in 1998 and 2000, respectively, and the D.S. degree in electrical engineering from the Universidade de São Paulo, Brazil, in 2004. Her research interests include intelligent control applications.

Vilma A. Oliveira (M’94) received the B.Eng. degree in electronics from the Universidade do Estado do Rio de Janeiro, Brazil, in 1976, the M.Sc. degree in electrical engineering in the Universidade Federal do Rio de Janeiro, Brazil, in 1980, and the Ph.D. in electrical engineering from the University of Southampton, U.K., in 1989. She joined the Departamento de Engenharia Elétrica, Universidade de São Paulo, Brazil, in 1990, where she is currently a Professor. Her research interests include control design and applications.