LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 427 Theorem 5: [13] (sufficient condition) Fractional system (7) is t stable if matrix 0P , P MM , exists, such that 0 11 APPA T . Proof: See steps above. Also in [13]. 5.3 Validity of the stability condition Figure 3 presents stability domain D S of a fractional system characterized using theorem 5 according to fractional order and to Aspecarg . A comparison 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 3. Stability domain D S ( ) determined by criterion 2 according to the values of and . A simple explanation can be given. Systems (7) and (19) have strictly the same behavior. However, transformations given by relations (16) to (18) produce a matrix f A unstable modes thus created are compensated by zeros produced by matrix tB f such a situation, a method based on eigenvalue analysis of matrix f A can only produce pessimistic stability conditions. )2/(arg radinAspec D S ’ ' ' between Fig. 3 and Fig. 1 reveals that the entire stability domain is not identi- whose eigenvalues are outside the left-half complex plane. The thus leading to a stable response to nonzero initial conditions. Due to fied using theorem 5. It therefore leads to a sufficient but nonnecessary condition. 428 In order to analyze such a conservatism, let f be an arguments of an eigenvalue of matrix /1 A and be the one of system (6) state transition associates to f xx F 1 2,0,0 : . (20) Fig. 4. F as a function of and f , and deduced stable domain ( ). As high values of lead to detection of some instability within the fractional stability domain D S , 2 which is thus reduced to: D S , 2,1 2 14, 2 34,0 i ii . (21) A method leading to a necessary and sufficient condition for stability of fractional systems is therefore necessary. Moze, Sabatier, and Oustaloup matrix A of system (6). Line D in Fig. 4 represents the function F that ' decays towards, 0, the slope of D increases significantly such that according to : LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 429 6.1 Characterization of the entire stability domain In order to characterize the entire stability domain D S , it is necessary to define a function that associates every D S with ' belonging to a convex domain of whose characterization is performed through LMI in theorem 4. Such a function can be defined by: 2 1 2 1 ,0,0 : ' xx F , (22) Fig. 5. ' F as a function of and f , and deduced stability domain ( ). 6.2 Equivalent integer order system Using function 'F defined by (22), it is now possible to assess stability of a fractional system through stability analysis of an equivalent integer system whose state transition matrix is to be determined. Let j ea , where j is the complex variable and ,0 . As aj ln , (23) one can note that aFb argarg ' if 2 1 ab . (24) S ’ 6 Stability Theorem Based on a Geometric Analysis of the Stability Domain the complex plane. This convex domain may be the left-half complex plane ' which is represented by line (D ) in Fig. 5. 430 Thus, ; 2 arg, 2 arg 2 1 aiffa . Stability of system (6) can thus be deduced by applying theorem 4 to a fictive integer system with state transition matrix 2 1 A . Theorem 6: Fractional system (6) is t stable if and only if a positive definite matrix P exists such that 0 2 1 2 1 APPA T . Proof: See steps above. As 2 1 A is a complex matrix, theorem 6 needs to be slightly changed when implemented in a LMI solver. As any complex LMI can be turned into a real one [18], the following LMI is to be implemented: .0 ReReImIm ImImReRe 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 APPAAPPA APPAAPPA TT TT (25) 6.3 Validity of the method Figure 6 presents the stability domain D S determined using theorem 6, according to the values of and of . Fig. 6. Stability domain D S ( ) determined by theorem 6 according to the values of and . ¢ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 )2/(arg radAspec D S ’’ Moze, Sabatier, and Oustaloup '' '' LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 431 S here ( D S D S ). The criterion is therefore not only sufficient but also necessary for stability detection of fractional systems. However, LMI of theorem 6 is not linear in relation to matrix A, thus limiting its use in more specific control problems. 7.1 Problem definition This approach is based on the obvious fact that a fractional system is stable if and only if it is not unstable. Applied to system (6) it emerges that the eigenvalues of the matrix A lie in the stable domain if and only if they do not lie in the unstable one, which is, as previously mentioned, convex. 7.2 Characterization of the entire unstable domain u belongs to D u if and only if it belongs to both D u1 and D u2 defined by D u1 }0) 2 1exp(Re j , (26) and D u2 0) 2 1exp(Re j }. (27) Thus belongs to D u if and only if 0) 2 1exp(Re 0) 2 1exp(Re j j , (28) or if and only if 0) 2 1exp() 2 1exp( 0) 2 1exp() 2 1exp( * * jj jj , (29) which can be rewritten as: '' When compared with Fig. 1, the entire stability domain D is identified denote the unstable domain as depicted on Fig. 2a. It is obvious that Let D 7 Stability Criterion Based on Unstability Domain Characterization 432 0 0 ** ** rr rr , (30) where 2 cos 2 sin jr . Fractional system (6) is thus t stable if and only if D u , q 0,0: qqAI n , (31) or if and only if , q n , .0,0: 0 0 ** ** qqAI rr rr (32) As for some Aspec , Aspec * , and as D u1 and D u2 are symmetric in relation to the real axis of the complex plane, condition (32) becomes 11 , Aspec D u1 22 , Aspec D u2, (33) and fractional system (6) is t stable if and only if , q n , .0,0:0 ** qqAIrr (34) It is now possible to use the following lemma given in [20]. Lemma 1 [20]: There exists a vector 0 qp for some 0 * if and only if 0 ** qppq . ,0 rqp q n , 0,0:0 ** qqAIqppq , (35) or if and only if ,0 q q n , 0:0 **** qAIrqqrqq . (36) As Aqq , fractional system (6) is t stable if and only if 0 q , q n , 0 *** rAqqrAqq T . (37) Moze, Sabatier, and Oustaloup Applied to relation (32), fractional system (6) is thus t stable if and only if LMI CHARACTERIZATION OF FRACTIONAL SYSTEMS STABILITY 433 Theorem 7: Fractional system (6) is t stable if and only if there does not nn such that 0 2 sin 2 cos 2 cos 2 sin TT TT QAAQQAAQ QAAQQAAQ . Proof: See steps above. 8 Conclusion An analysis of an existing method and two new methods are presented in order to characterize stability of fractional systems through LMI tools. Matignon’s theorem developed for stability analysis of fractional systems is first presented. A new proof of its extension to systems whose fractional order verifies 21 is proposed. For such derivative orders, stability is granted if all the eigenvalues of its state transition matrix belong to a convex subset of the complex plane, called stability domain. A trivial LMI stability condition is thus presented. For fractional orders verifying 10 , stability domain is not a convex subset of the complex plane. Three stability conditions involving LMI are however proposed. system have strictly the same behavior, an explanation of the conservatism of the condition is presented. In order to overcome this problem, a third condition is proposed. It relies on the fact that instability domain is a convex subset of the complex plane when 10 . This work is a first step in fractional system stability analysis using LMI tools towards new conditions and applications. permission to publish this revised contribution of an ASME article. exist any nonnegative rank one matrix Q The first condition appears in [13] and appears after algebraic transfor- mations of the fractional system state-space representation. The obtained con- dition is only sufficient. Even if the derived system and the original fractional The second condition is new and relies on a geometric analysis of the sta- bility domain. The resulting LMI stability condition is sufficient and necessary but is not linear in relation to the state transition matrix of the fractional sys- tem state-space representation, which can limit its applicability. Thanks go to the American Society of Mechanical Engineers (ASME) for the Acknowledgment 434 Moze, Sabatier, and Oustaloup References 1. Podlubny I (1999) Fractional-order systems and PI λ D µ -Controllers, IEEE Trans. Automat. Control, 44(1):208–214. 2. Monje CA, Vinagre, BM, Chen YO, Feliu V, Lanusse P, Sabatier J (2004) Proposals for fractional PID tuning, First IFAC Workwhop on Fractional Derivative and its Application, FDA 04, Bordeaux, France, 2004. 3. Caponetto R, Fortuna L, Porto D (2004) A new tuning strategy for a non integer order PID controller, First IFAC Workwhop on Fractional Derivative and its Application, FDA 04, Bordeaux, France. 4. 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The method can be applied to suppress vibration in large flexible structures that have high modal density, even for relatively low frequencies. In this report, we formulate a feedback-type active wave control law, described as a transfer function including a Laplace transform with an s 1/2 or s 3/2 term. As an example, we present the fractional-order derivatives and integrals of structural responses in the vibration suppression of a thin, light cantilevered beam. 1 Introduction Flexible structures such as large-scale space structures (LSS) have a high active vibration suppression, vibration control approaches based on modal analysis must determine the limits of the spillover instability phenomenon. Hence it is necessary to establish a new control methodology that can be applied to flexible structures. Among such novel approaches, the active wave (absorption) control method has attracted attention. It is known that control laws derived from active wave control theory can be expressed using a transfer function including a non-integer order power of the Tsukuba, Ibaraki 305-8564, Japan; Tel: +81-29-861-7147, Fax: +81-29-861-7098, E-mail: m-kuroda@aist.go.jp Keywords vibration-mode density, even in the low-frequency domain. Therefore, to achieve © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Recently, active wave control theory has attracted great interest as a novel Fractional calculus, control, fractional-order transfer function, wave, flexible in Physics and Engineering, 435–448. structure. 435 436 variable s of the Laplace transform. However, there are difficulties implementing the transfer function due to the non-integer order power. In this report we present a formulation of a feedback-type active wave controller, designed to suppress vibration of a flexible cantilevered beam, described by a transfer function with s or ss and introducing a fractional-order derivative and integral. 2 Active Wave Control of a Flexible Structure Active wave control differs from conventional vibration control in the way it suppresses the vibration modes (standing waves) of a structure. The interaction of progressive and retrogressive waves creates a standing wave, each of which can be treated as a controlled object in the control method developed by von Flotow and Schafer [1]. Kuroda Fig. 1. Schematic diagram of the active wave control method. As an example, we consider the vibration control of a flexible cantilever (Fig. 1). A sensor and an actuator are placed near the middle of the beam. A disturbance is applied at the free end of the beam. The relationship between the progressive and retrogressive wave vectors generated by the disturbance on the cantilever can be described in matrix form using boundary conditions on the control point. Backward propagating waves are produced by the reflection of the progressive wave, but are also produced by the control input, allowing control of the back- ward propagating wave. The progressive wave vectors can also be controlled. However, we note that only one control force can control any one of the wave [...]... positions along the negative real axis, as shown in the following equation [2]: s 10 4 s 10 2 s 100 s 102 s 10 s 3 s 10 1 s 101 s 103 (4) 3 Fractional Calculus The transfer function can be defined in terms of fractional calculus, whereby the derivatives and integrals of a continuous function can be defined using nonintegers [5–8] The definition of a fractional derivative can be written as D q [ x(t... differentiation and integration Therefore, terms such as fractional- order differentiator or fractional derivative should be understood to imply both differentiator and integrator If implemented properly, fractional- order controllers will find their place in contributing to many real-world control systems It has to be borne in mind that a fractional- order controller is an infinite-dimensional linear filter,... link with a freely pivoting pin-joint The disk has a nominal mass of 54 g, and floats on the horizontal air table with minimal friction Since the mass of the link is small relative to that of the disk, and the pinned joint prevents generation of torque at 458 Feliu, Vinagre, and Monje the end of the link, the mechanical system behaves practically as an ideal, singledegree-of-freedom, undamped spring-mass... above continuous approximation by using the Tustin rule with pre-warping, the resulting controller being denoted RIDZ (z); (ii) direct discretization of the fractional operator by using continuous fraction expansion (CFE) of the Tustin discrete equivalent of the 460 Feliu, Vinagre, and Monje Laplace operator s, the resulting controller being denoted RTCFE(z); and (iii) direct discretization of the fractional. .. function in the form of a fractional expression of an integer power series Customarily, transfer functions including non-integer powers of s have been approximated by introducing a limitation in the frequency range Following the methods of MacMartin and others, the transfer function is substituted by an approximated 438 Kuroda transfer function with a finite number of poles and zeros located in the exponential... Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 449–462 © 2007 Springer Feliu, Vinagre, and Monje 450 Since fractional- order controllers have been used successfully in robust control problems, the present work considers a control scheme based on a controller of this type that compensates for undesired changes in the dynamics of... The fractional- order control (FOC) strategy of the outer loop, which is based on the operators of fractional calculus, is proposed in this paper FRACTIONAL CONTROL OF A FLEXIBLE MANIPULATOR 451 Fractional calculus generalizes the standard differential and integral operators by defining a single general fundamental operator (see [7–9]) There are two commonly used definitions for the generalized fractional. .. determines for a given the step response, or (b) the crossover frequency : (a) the speed of To select these parameters, one may work in the complex plane, the frequency domain or the time domain In the frequency domain, the selection can be regraded as choosing a fixed phase margin by selecting , and choosing a crossover frequency c, by selecting K for a given That is, 2 2 m , K c 43.75 (22) In our... FLEXIBLE STRUCTURES 5 447 Conclusions Active wave control including a 1/2-order or a 3/2-order derivative element can be formulated using fractional calculus In this paper we have reported the following: 1 Application of fractional calculus to vibration control 2 The basis for (a) calculating and (b) measuring the responses of 1/2-order and 3/2-order fractional derivatives of a cantilevered beam 3 Implementation... FLEXIBLE STRUCTURES 439 In this study we overcome the difficulties due to fractional derivative responses by constructing the responses at the actuation point from a linear combination of multiple signals at several sensing points, rather than from a signal from a single sensor In this method, special sensors with additional signal-conversion functions are not required and existing displacement and velocity . (4) 3 Fractional Calculus The transfer function can be defined in terms of fractional calculus, whereby the derivatives and integrals of a continuous function can be defined using non- integers. function in the form of a fractional expression of an integer power series. Customarily, transfer functions including non-integer powers of s have been approximated by introducing a limitation in. even in the low-frequency domain. Therefore, to achieve © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications National Institute