Observer-Based Control for Fractional-Order ... - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeBIn5.3
Observer-Based Control for Fractional-Order Continuous-time Systems Ibrahima N’doye, Michel Zasadzinski, Mohamed Darouach and Nour-Eddine Radhy
Abstract— In this paper, we study the asymptotic stabilization of fractional-order systems using an observer-based control law. The fractional-order systems under consideration are either linear or nonlinear and affine. A generalization of GronwallBellman which is proved in the appendix is used to derive the closed-loop asymptotic stability. Index Terms— Linear and nonlinear fractional-order systems, generalization of Gronwall-Bellman lemma, observerbased control.
I. I NTRODUCTION Fractional-order systems (i.e. systems containing fractional derivatives and/or integrals) have been studied by many authors in engineering sciences from an application point of view (see [1], [2] and references therein). Many systems can be described with the help of fractional derivatives. These systems are known to display fractional-order dynamics : electromagnetic systems [3], [4], dielectric polarization [5], viscoelastic systems [6], [7]. The anomalous diffusion phenomena in inhomogeneous media can be explained by non-integer derivative based equations of diffusion [8], [9]. Another example for an element with fractional order model is the fractance. The fractance is an electrical circuit with non-integer order impedance [10]. This element has properties lie between resistance and capacitance. The resistance-capacitance-inductance (RLC) interconnect model of a transmission line is a fractionalorder model [11]. Heat conduction can be more adequately modeled by fractional-order models than by their integer order counterparts [12]. In biology, it has been shown that the membranes of cells of biological organism have a fractionalorder electrical conductance [13]. In economics, it is known that some finance systems can display fractional order dynamics [14]. More recently, a lot of chaotic behaviors have been shown to be fractional-order systems [15], [16]. Then, controlling fractional-order systems becomes one of an active fields, especially controlling nonlinear fractionalorder systems. Also, fractional-order controller was used in [17] for flexible spacecraft attitude control. Ibrahima N’doye is with Nancy Universit´e, Centre de Recherche en Automatique de Nancy (CRAN UMR−7039, CNRS), IUT de Longwy, 186 rue de Lorraine 54400 Cosnes et Romain, France and with Universit´e Hassan II, Facult´e des Sciences Ain-Chock, Laboratoire Physique et Mat´eriaux Micro´electronique Automatique et Thermique BP: 5366 Maarif, Casablanca 20100, Morocco Michel Zasadzinski and Mohamed Darouach are with Nancy Universit´e, Centre de Recherche en Automatique de Nancy (CRAN UMR−7039, CNRS), IUT de Longwy, 186 rue de Lorraine 54400 Cosnes et Romain, France Nour-Eddine Radhy is with Universit´e Hassan II, Facult´e des Sciences Ain-Chock, Laboratoire Physique et Mat´eriaux Micro´electronique Automatique et Thermique BP: 5366 Maarif, Casablanca 20100, Morocco
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
The design of state estimators is one of the essential points in control theory whose solution, in the linear case, is the well known Luenberger’s observer. Many contributions to the extension of the Luenberger observer for nonlinear systems has been proposed in the literature (see[18], [19], [20], [21] and references therein). The observer-based control is usually applied when we do not have access to all the states of a system. Unlike in the linear case [22], [23], the separation principle does not work for nonlinear systems [24], [25]. The notion of detectability which has been extended to nonlinear systems in [26], [27], [28] allows to design nonlinear observer-based control using the separation principle. In this paper, we present new contributions to the observerbased stabilization of linear and nonlinear fractional-order systems using a generalization of the Gronwall-Bellman lemma. The proof of the “standard” Gronwall-Bellman lemma and its use in the nonlinear systems theory can be found in [29], [30], [31], and some useful generalizations of the Gronwall-Bellman lemma have been proposed in [32], [33], [34], [35]. The paper is organized as follows. In section II, we introduce the definition of fractional derivative in brief. Fractional calculus is the name of theory of integrations and derivatives of arbitrary order, which unifies and generalizes the notion of noninteger order differentiation and n-fold integration. So this unified operator is called as generalized differintegral. In section III, an asymptotic stabilization of linear fractional-order systems is proposed using an observer-based control law. In section IV, the proposed generalization of the Gronwall-Bellman lemma allows to design an observerbased control law for asymptotic stabilization of nonlinear affine fractional-order systems. The proposed generalization of the Gronwall-Bellman √ lemma is provenpin the appendix. Notations. kxk = xT x and kAk = λmax (ATA) are the Euclidean vector norm and the spectral matrix norm respectively where λmax (AT A) is the maximal eigenvalue of the symmetric matrix ATA. (f (·))i stands for the ith component of vector f (·) . II. P RELIMINARY DEFINITIONS A. Definition of the fractional derivative In this paper, the symbols for fractional differintegration have been standardized8as follows [36], [37] dα > > < d tα , α > 0 α a Dt = 1, α=0 > > : Rt (−α) dτ , α 0 is a real constant, then θ , γ 6 |arg(λi (A))| 6 π, i = 1 . . . n kEα,β (A)k 6 1 + kAk (11) where θ = max(C, kP kkP −1 kC), λi (A) denotes the ith eigenvalue of matrix A, P is a nonsingular coordinate C transformation giving the Jordan form of A, > 1 + kAk C0i , where C and C0i are given positive conmax 16i6n 1 + λi stants. (k)
where Eα,β =
k=0
(4) where s ∈ C denotes the Laplace operator. Upon considering all the initial conditions to be zero, (4) can be reduced to α d f (t) L = sα L(f (t)). (5) dtα Proper initialization is crucial in the solution and understanding of fractional differential functions. So, we adopted generally the notation for assuming causality of function or system (for 0 6 α < 1) α α α with a 6 (c = 0) < t c Dt f (t) = 0 dt f (t) + a d0 f (t), (6) th f (t) is a α derivative of f (t) starts at time a and where a dα t continues at time t. This means that Z t 1 d f (τ ) α d τ (7) a dt f (t) = d t Γ(1 − α) a (t − τ )α and where a dα 0 f (t) = Ψ(α, f, a, 0, t) is a initialization function. This means that Z 0 1 d f (τ ) α dτ a d0 f (t) = Ψ(α, f, a, 0, t) = d t Γ(1 − α) a (t − τ )α (8) B. Definition of the two-parameter Mittag-Leffler function The two-parameter Mittag-Leffler function is defined as follows [36], [39] ∞ X zk Eα,β (z) = , α > 0, β > 0 (9) Γ(αk + β) k=0
where Eα,1 (z) =
∞ X k=0
zk ≡ Eα (z) Γ(αk + 1)
is the one-parameter Mittag-Leffler function. The Laplace transform of the two-parameter Mittag-Leffler function is Z ∞
e−st tαk+β−1 Eα,β (atα ) d t =
0
(k)
sα−β k!
(sα −a)k+1
III. O BSERVER -BASED C ONTROL FOR F RACTIONAL -O RDER L INEAR S YSTEMS Consider the fractional-order linear system described by a differential equation of the following form α D x(t) = Ax(t) + Bu(t) 0 α > (gi (b x(t))−gi (x(t)))(Lb x(t))i > < D e(t) = (A−KC)e(t) + i=1
i=1
(26)
which can be rewritten as m X b Dα X(t) = AX(t) + ∆gi (X(t)) ([ 0 L ]X(t))i (27) i=1
with " X(t) =
#
e(t)
x b(t)
" ,
b= A
A − KC −KC
0
A + BL # " gi (b x(t))−gi (b x(t) − e(t)) . ∆gi (X(t)) = gi (b x(t))
# ,
(28)
We assume that the nonlinear fractional-order system (22) satisfies the following hypothesis. Assumption 3: 1) Assume that there exists a gain matrix K such that relation (11) in corollary 1 holds if matrix A is replaced by A − KC with β = 1. 2) Assume that there exists a gain matrix L such that relation (11) in corollary 1 holds if matrix A is replaced by A + BL with β = 1. 3) The functions ∆gi (X(t)) defined in (28) satisfy the following relation (for i = 1, . . . , m) q k∆gi (X(t))k 6 ρi kX(t)k (29) where q > 1 and ρi are given positives constants. m X In the sequel, we define ρ = ρi .
IV. O BSERVER -BASED C ONTROL FOR F RACTIONAL -O RDER N ONLINEAR A FFINE S YSTEMS
1) gi (0) = 0, 2) there exist an integer q > 1, such that kgi (x(t))k 6 µi kx(t)kq (23) where µi are given positives constants. The purpose in this section is to study the problem of asymptotical stabilization of the fractional-order nonlinear system (22) with a “Luenberger-type” fractional-order linear
m
X > > α > b(t) = (A+BL)b x(t) − KCe(t) + gi (b x(t))(Lb x(t))i : D x
which is equivalent to (18). Then, if the time t tends to infinity, kX(t)k converges to zero which implies the asymptotic stability of the zero solution.
Consider the nonlinear affine fractional-order system described by a differential equation of the following form m X α gi (x(t))ui (t)+Bu(t) D x(t) =Ax(t)+ i=1 0 0. a
Similarly, α < 1 and (t − τ ) 6 τ when τ ∈ [ 2t , t], we haveZ t (t − τ )α−1 1
dτ
b α q
b t τ ) A (1 + (1 +
A (t − τ )α ) 2 Z t (t − τ )α−1 1
6 dτ
b t α q (1 + kAk(t − τ )α ) 2 (1 + A (t − τ ) ) Z 2t η α−1 1
= (40)
b α q
b α d η, 0 (1 + A
(η) ) (1 + A
η ) substituting (t − τ ) by η. From (39) and (40), relation (37) can be written Z 2t 1 τ α−1
Φ(t) = 2
b α q
b α d τ 0 (1 + A
τ ) (1 + A
τ ) Z 2t τ α−1
(41) =2
b α q+1 d τ 0 (1 + A
τ ) which is to equivalent to Φ(t) =
2 1
t q .
b 1 −
b qα A 1 + A 2
(42)
Using (42), we have Φ(t) > 0 if t > 0 and the expression q 1 − qρkLkθq+1 kX0 k Φ(t) in relation (36) is minimal when t tends to infinity. Then, inequality (36) holds if the initial state X0 satisfies condition (31).
Since the previous inequality can be written as
From (34), the generalization of Gronwall-Bellman lemma given in lemma 3 yields the following inequality θ(t − τ ) k @ θkX ‚ 0‚ A ‚ ‚ d τ > 0, ∀ t > 0, 1−qρkLk θkX0 k ‚ b‚ α ‚ b‚
0 1 + ‚A 1 + ‚A ‚τ ‚ (t − τ )α
b α 1 +
A t (35) (43) kX(t)k 6 q1 we must check that the following inequality q+1 q 1 − qρθ kLkkX k Φ(t) 0 q 1 − qρkLkθq+1 kX0 k Φ(t) > 0, ∀ t > 0 (36) which is equivalent to inequality (30). holds, where Z t Then, if the time t tends to infinity, kX(t)k converges 1 (t − τ )α−1
Φ(t) = d τ. (37) to zero which implies the asymptotic stability of the zero
b α q
b 0 (1 + A
τ ) (1 + A
(t − τ )α ) solution. Z
t
0
1q
α−1
1935
WeBIn5.3 V. C ONCLUSION
holds, then
In this paper, observer-based control laws have been derived for asymptotic stabilization of fractional-order systems. The fractional-order systems are either linear or nonlinear and affine. In the linear case, the observer-based control satisfies the separation principle since the closed-loop stability results in the stability of both the state feedback and the observation error. In the nonlinear case, the separation principle does not work and the closed-loop stability is proven using a proposed generalized form of the GronwallBellman lemma which is given in the appendix.
r(t)
x(t) 6
t
Z 1 − (` − 1)
(r(s))
`−1
1 . `−1 f (s) d s
(50)
a
Proof: Relation (48) can be written as follows Z t f (s)(x(s))`−1 x(s) d s x(t) 6 r(t) + a
and from the classical Gronwall-Bellman lemma [33], [31], we obtain Z t f (s)(x(s))`−1 d s (51) x(t) 6 r(t) exp a
or equivalently t
„ Z (x(t))`−1 6 (r(t))`−1 exp (` − 1)
A PPENDIX
« f (s)(x(s))`−1 d s .
a
(52)
A PPENDIX : G ENERALIZATION OF G RONWALL -B ELLMAN LEMMA
Multiplying the above inequality by −(` − 1)f (t) gives
Lemma 2 (Gronwall-Bellman lemma): [31] (p 292) [30] (p 252) Let i) ii) iii) iv)
− (` − 1)f (t)(x(t))`−1 „ « Z t > −(`−1)(r(t))`−1 f (t) exp (` − 1) f (s)(x(s))`−1 d s
+
f , g and k, IR 7→ IR and locally integrable, g > 0, k > 0, g ∈ L∞ , gk is locally integrable on IR+ .
If u : IR+ 7→ IR satisfies Z t u(t) 6 f (t) + g(t) k(τ )u(τ ) d τ,
a
(53)
or equivalently t
„ Z − (` − 1)f (t)(x(t))`−1 exp −(` − 1)
« f (s)(x(s))`−1 d s
a
∀t > 0
> −(` − 1)(r(t))`−1 f (t)
(44)
(54)
0
then
Using the primitive of the exponential function, the above t
Z u(t) 6 f (t)+g(t)
„Z k(τ )f (τ ) exp
0
« inequality becomes t k(s)g(s) d s d τ, ∀ t > 0.
τ
(45)
+
Corollary 2: [33] Let k : IR 7→ IR, locally integrable on IR+ and k > 0 and c(t) be a positive, monotonic decreasing function. If u : IR+ 7→ IR+ satisfies Z t
u(t) 6 c(t) +
k(τ )u(τ ) d τ,
∀t > 0
(46)
0
then Z t u(t) 6 c(t) exp k(τ ) d τ ,
d dt
t
Z
`−1
exp −(` − 1)
f (s)(x(s))
ds
a
> −(` − 1)(r(t))`−1 f (t) (55) and integrating from a to t, we obtain Z t `−1 exp −(` − 1) f (s)(x(s)) ds a Z t > 1 − (` − 1) (r(s))`−1 f (s) d s. (56) a
∀ t > 0.
(47)
0
Lemma 3 (Generalization of Gronwall-Bellman lemma): Let i) a, b ∈ IR, 0 6 a < b, r(t) > 0 a positive, monotonic, decreasing function, an integer ` > 1, ii) f : [a, b] 7→ IR+ an integrable function such that, ∀ α, β ∈ [a, b], (0 6 α < β), we have Z β f (s) d s > 0 α
iii) x : [a, b] 7→ IR+ an essential bounded function such that Z t x(t) 6 r(t) + f (s)(x(s))` d s. (48)
Notice that the constant in the above integration is equal to 1 (this can be shown with t = a). If the inequality (49) holds, we have Z t exp (` − 1) f (s)(x(s))`−1 d s a
1
6
Z
(r(s))`−1 f (s) d s
1 − (` − 1) a
Inequalities (52) and (57) imply 1
(x(t))`−1 (r(t))−(`−1) 6
Z 1 − (` − 1)
t
(r(s))`−1 f (s) d s
a
or equivalently
a
If the following inequality Z t 1 − (` − 1) (r(s))`−1 f (s) d s > 0
. (57)
t
r(t)
x(t) 6 (49)
Z 1 − (` − 1)
(r(s)) a
a
1936
t `−1
1 . `−1 f (s) d s
(58)
WeBIn5.3 This completes the proof. R EFERENCES [1] I. Podlubny, Fractional Differential Equations. New York: Academic, 1999. [2] R. Hilfer, Applications of Fractional Calculus in Physics. Singapore: World Scientific Publishing, 2000. [3] O. Heaviside, Electromagnetic Theory. New York: Chelsea Publishing Company, 3rd ed., 1971. [4] N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism,” IEEE Trans. Antennas and Propagation, vol. 44, pp. 554–566, 1996. [5] H. Sun, A. Abdelwahad, and B. Onaral, “Linear approximation of transfer function with a pole of fractional order,” IEEE Trans. Aut. Contr., vol. 29, pp. 441–444, 1984. [6] R. Bagley and R. Calico, “Fractional order state equations for the control of viscoelastically damped structures,” J. Guidance, Contr. & Dynamics, vol. 14, pp. 304–311, 1991. [7] Y. Rossikhin and M. Shitikova, “Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system,” Acta Mechanica, vol. 120, pp. 109–125, 1997. [8] V. Arkhincheev, “Anomalous diffusion in inhomogeneous media: some exact results,” Modelling, Measurement and Control A, vol. 26, pp. 11– 29, 1993. [9] L. El Ghaoui and G. Scorletti, “Control of rational systems using linear-fractional representations and linear matrix inequalities,” Automatica, vol. 32, pp. 1273–1284, 1996. [10] A. Le M´ehaut´e and G. Crepy, “Introduction to transfer and motion in fractal media: The geometry of kinetics,” Solid State Ionics, vol. 9-10, pp. 311–322, 1983. [11] G. Chen and E. Friedman, “An RLC interconnect model based on Fourier analysis,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, vol. 24, pp. 170–183, 2005. [12] V. Jenson and G. Jeffreys, Mathematical Methods in Chemical Engineering. New York: Academic Press, 1977. [13] K. Cole, “Electric conductance of biological systems,” in Proc. Cold Spring Harbor Symp. Quantitative Biology, (New York, USA), pp. 107–116, 1933. [14] N. Laskin, “Fractional market dynamics,” Physica A:Statistical Mechanics and its Applications, vol. 287, pp. 482–492, 2000. [15] C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons & Fractals, vol. 22, pp. 549–554, 2004. [16] I. Petr´as˘, “A note on the fractional-order Chuas system,” Chaos, Solitons & Fractals, vol. 38, pp. 140–147, 2008. [17] S. Manabe, “A suggestion of fractional-order controller for flexible spacecraft attitude control,” Nonlinear Dynamics, vol. 29, pp. 251– 268, 2002. [18] J. Tsinias, “Observer design for nonlinear systems,” Syst. & Contr. Letters, vol. 13, pp. 135–142, 1989. [19] J. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems, applications to bioreactors,” IEEE Trans. Aut. Contr., vol. 37, pp. 875–880, 1992. [20] S. Raghavan and J. Hedrick, “Observer design for a class of nonlinear systems,” Int. J. Contr., vol. 59, pp. 515–528, 1994. [21] R. Rajamani, “Observer for Lipschitz nonlinear systems,” IEEE Trans. Aut. Contr., vol. 43, pp. 397–401, 1998. [22] H. Kwakernaak and R. Sivan, Linear Optimal Control. New York, USA: Wiley Interscience, 1972. [23] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, New Jersey: Prentice Hall, 1996. [24] W. Lin, “Input saturation and global stabilization of nonlinear systems via state and output feedback,” IEEE Trans. Aut. Contr., vol. 40, pp. 776–782, 1995. [25] Z. Zhang, J. Lu, J. Cao, and G. Lu, “H∞ observer-based compensator design for bilinear continuous-time systems,” in Proc. Cong. Intelligent Contr. and Aut., (Hangzhou, P.R. China), pp. 833–837, 2004. [26] M. Vidyasagar, “On the stabilization of nonlinear systems using state detection,” IEEE Trans. Aut. Contr., vol. 25, pp. 504–509, 1980. [27] E. Sontag and Y. Wang, “Output-to-state stability and detectability of nonlinear systems,” Syst. & Contr. Letters, vol. 29, pp. 279–290, 1997. [28] J. Hespanha and A. Morse, “Certainty equivalence implies detectability,” Syst. & Contr. Letters, vol. 36, pp. 1–13, 1999. [29] B. Pachpatte, “A note on Gronwall-Bellman inequality,” J. of Mathematical Analysis and Applications, vol. 44, pp. 758–762, 1973.
[30] C. Desoer and M. Vidyasagar, Feedback Systems Input-Output Properties. New York: Electrical Sciences. Academic Press, 1975. [31] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, New Jersey: Prentice Hall, 2nd ed., 1993. [32] B. Pachpatte, “On some integral inequalities similar to Bellman-Bihari inequalities,” J. of Mathematical Analysis and Applications, vol. 49, pp. 794–802, 1975. [33] B. Pachpatte, “On some generalizations of Bellmans lemma,” J. of Mathematical Analysis and Applications, vol. 51, pp. 141–150, 1975. [34] N. El Alami, Analyse et Commande Optimale des Syst`emes Bilin´eaires Distribu´es. Applications aux Proc´ed´es Energ´etiques. PhD thesis, Univesit´e de Perpignan, France, 1986. Doctorat d’Etat. [35] N. El Alami, “A generalization of Gronwall’s lemma,” in Conf´erence Internationale sur les Equations Diff´erentielles, (Marrakesh, Morocco), 1995. [36] S. Das, Functional Fractional Calculus for System Identification and Controls. Heidelberg: Springer, 2008. [37] K. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. New York: Academic Press, 1974. [38] I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, pp. 367–386, 2002. [39] A. Erd´elyi and H. Bateman, Higher Transcendental Functions, vol. 3, pp. 43–57. Hightstown, New Jersey: McGraw-Hill, 1955. [40] X. Wen, Z. Wu, and J. Lu, “Stability analysis of a class of nonlinear fractional-order systems,” IEEE Trans. Circ. Syst. II : Express Briefs, vol. 55, pp. 1178–1182, 2008.
1937
WeBIn5.3
Observer-Based Control for Fractional-Order Continuous-time Systems Ibrahima N’doye, Michel Zasadzinski, Mohamed Darouach and Nour-Eddine Radhy
Abstract— In this paper, we study the asymptotic stabilization of fractional-order systems using an observer-based control law. The fractional-order systems under consideration are either linear or nonlinear and affine. A generalization of GronwallBellman which is proved in the appendix is used to derive the closed-loop asymptotic stability. Index Terms— Linear and nonlinear fractional-order systems, generalization of Gronwall-Bellman lemma, observerbased control.
I. I NTRODUCTION Fractional-order systems (i.e. systems containing fractional derivatives and/or integrals) have been studied by many authors in engineering sciences from an application point of view (see [1], [2] and references therein). Many systems can be described with the help of fractional derivatives. These systems are known to display fractional-order dynamics : electromagnetic systems [3], [4], dielectric polarization [5], viscoelastic systems [6], [7]. The anomalous diffusion phenomena in inhomogeneous media can be explained by non-integer derivative based equations of diffusion [8], [9]. Another example for an element with fractional order model is the fractance. The fractance is an electrical circuit with non-integer order impedance [10]. This element has properties lie between resistance and capacitance. The resistance-capacitance-inductance (RLC) interconnect model of a transmission line is a fractionalorder model [11]. Heat conduction can be more adequately modeled by fractional-order models than by their integer order counterparts [12]. In biology, it has been shown that the membranes of cells of biological organism have a fractionalorder electrical conductance [13]. In economics, it is known that some finance systems can display fractional order dynamics [14]. More recently, a lot of chaotic behaviors have been shown to be fractional-order systems [15], [16]. Then, controlling fractional-order systems becomes one of an active fields, especially controlling nonlinear fractionalorder systems. Also, fractional-order controller was used in [17] for flexible spacecraft attitude control. Ibrahima N’doye is with Nancy Universit´e, Centre de Recherche en Automatique de Nancy (CRAN UMR−7039, CNRS), IUT de Longwy, 186 rue de Lorraine 54400 Cosnes et Romain, France and with Universit´e Hassan II, Facult´e des Sciences Ain-Chock, Laboratoire Physique et Mat´eriaux Micro´electronique Automatique et Thermique BP: 5366 Maarif, Casablanca 20100, Morocco Michel Zasadzinski and Mohamed Darouach are with Nancy Universit´e, Centre de Recherche en Automatique de Nancy (CRAN UMR−7039, CNRS), IUT de Longwy, 186 rue de Lorraine 54400 Cosnes et Romain, France Nour-Eddine Radhy is with Universit´e Hassan II, Facult´e des Sciences Ain-Chock, Laboratoire Physique et Mat´eriaux Micro´electronique Automatique et Thermique BP: 5366 Maarif, Casablanca 20100, Morocco
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
The design of state estimators is one of the essential points in control theory whose solution, in the linear case, is the well known Luenberger’s observer. Many contributions to the extension of the Luenberger observer for nonlinear systems has been proposed in the literature (see[18], [19], [20], [21] and references therein). The observer-based control is usually applied when we do not have access to all the states of a system. Unlike in the linear case [22], [23], the separation principle does not work for nonlinear systems [24], [25]. The notion of detectability which has been extended to nonlinear systems in [26], [27], [28] allows to design nonlinear observer-based control using the separation principle. In this paper, we present new contributions to the observerbased stabilization of linear and nonlinear fractional-order systems using a generalization of the Gronwall-Bellman lemma. The proof of the “standard” Gronwall-Bellman lemma and its use in the nonlinear systems theory can be found in [29], [30], [31], and some useful generalizations of the Gronwall-Bellman lemma have been proposed in [32], [33], [34], [35]. The paper is organized as follows. In section II, we introduce the definition of fractional derivative in brief. Fractional calculus is the name of theory of integrations and derivatives of arbitrary order, which unifies and generalizes the notion of noninteger order differentiation and n-fold integration. So this unified operator is called as generalized differintegral. In section III, an asymptotic stabilization of linear fractional-order systems is proposed using an observer-based control law. In section IV, the proposed generalization of the Gronwall-Bellman lemma allows to design an observerbased control law for asymptotic stabilization of nonlinear affine fractional-order systems. The proposed generalization of the Gronwall-Bellman √ lemma is provenpin the appendix. Notations. kxk = xT x and kAk = λmax (ATA) are the Euclidean vector norm and the spectral matrix norm respectively where λmax (AT A) is the maximal eigenvalue of the symmetric matrix ATA. (f (·))i stands for the ith component of vector f (·) . II. P RELIMINARY DEFINITIONS A. Definition of the fractional derivative In this paper, the symbols for fractional differintegration have been standardized8as follows [36], [37] dα > > < d tα , α > 0 α a Dt = 1, α=0 > > : Rt (−α) dτ , α 0 is a real constant, then θ , γ 6 |arg(λi (A))| 6 π, i = 1 . . . n kEα,β (A)k 6 1 + kAk (11) where θ = max(C, kP kkP −1 kC), λi (A) denotes the ith eigenvalue of matrix A, P is a nonsingular coordinate C transformation giving the Jordan form of A, > 1 + kAk C0i , where C and C0i are given positive conmax 16i6n 1 + λi stants. (k)
where Eα,β =
k=0
(4) where s ∈ C denotes the Laplace operator. Upon considering all the initial conditions to be zero, (4) can be reduced to α d f (t) L = sα L(f (t)). (5) dtα Proper initialization is crucial in the solution and understanding of fractional differential functions. So, we adopted generally the notation for assuming causality of function or system (for 0 6 α < 1) α α α with a 6 (c = 0) < t c Dt f (t) = 0 dt f (t) + a d0 f (t), (6) th f (t) is a α derivative of f (t) starts at time a and where a dα t continues at time t. This means that Z t 1 d f (τ ) α d τ (7) a dt f (t) = d t Γ(1 − α) a (t − τ )α and where a dα 0 f (t) = Ψ(α, f, a, 0, t) is a initialization function. This means that Z 0 1 d f (τ ) α dτ a d0 f (t) = Ψ(α, f, a, 0, t) = d t Γ(1 − α) a (t − τ )α (8) B. Definition of the two-parameter Mittag-Leffler function The two-parameter Mittag-Leffler function is defined as follows [36], [39] ∞ X zk Eα,β (z) = , α > 0, β > 0 (9) Γ(αk + β) k=0
where Eα,1 (z) =
∞ X k=0
zk ≡ Eα (z) Γ(αk + 1)
is the one-parameter Mittag-Leffler function. The Laplace transform of the two-parameter Mittag-Leffler function is Z ∞
e−st tαk+β−1 Eα,β (atα ) d t =
0
(k)
sα−β k!
(sα −a)k+1
III. O BSERVER -BASED C ONTROL FOR F RACTIONAL -O RDER L INEAR S YSTEMS Consider the fractional-order linear system described by a differential equation of the following form α D x(t) = Ax(t) + Bu(t) 0 α > (gi (b x(t))−gi (x(t)))(Lb x(t))i > < D e(t) = (A−KC)e(t) + i=1
i=1
(26)
which can be rewritten as m X b Dα X(t) = AX(t) + ∆gi (X(t)) ([ 0 L ]X(t))i (27) i=1
with " X(t) =
#
e(t)
x b(t)
" ,
b= A
A − KC −KC
0
A + BL # " gi (b x(t))−gi (b x(t) − e(t)) . ∆gi (X(t)) = gi (b x(t))
# ,
(28)
We assume that the nonlinear fractional-order system (22) satisfies the following hypothesis. Assumption 3: 1) Assume that there exists a gain matrix K such that relation (11) in corollary 1 holds if matrix A is replaced by A − KC with β = 1. 2) Assume that there exists a gain matrix L such that relation (11) in corollary 1 holds if matrix A is replaced by A + BL with β = 1. 3) The functions ∆gi (X(t)) defined in (28) satisfy the following relation (for i = 1, . . . , m) q k∆gi (X(t))k 6 ρi kX(t)k (29) where q > 1 and ρi are given positives constants. m X In the sequel, we define ρ = ρi .
IV. O BSERVER -BASED C ONTROL FOR F RACTIONAL -O RDER N ONLINEAR A FFINE S YSTEMS
1) gi (0) = 0, 2) there exist an integer q > 1, such that kgi (x(t))k 6 µi kx(t)kq (23) where µi are given positives constants. The purpose in this section is to study the problem of asymptotical stabilization of the fractional-order nonlinear system (22) with a “Luenberger-type” fractional-order linear
m
X > > α > b(t) = (A+BL)b x(t) − KCe(t) + gi (b x(t))(Lb x(t))i : D x
which is equivalent to (18). Then, if the time t tends to infinity, kX(t)k converges to zero which implies the asymptotic stability of the zero solution.
Consider the nonlinear affine fractional-order system described by a differential equation of the following form m X α gi (x(t))ui (t)+Bu(t) D x(t) =Ax(t)+ i=1 0 0. a
Similarly, α < 1 and (t − τ ) 6 τ when τ ∈ [ 2t , t], we haveZ t (t − τ )α−1 1
dτ
b α q
b t τ ) A (1 + (1 +
A (t − τ )α ) 2 Z t (t − τ )α−1 1
6 dτ
b t α q (1 + kAk(t − τ )α ) 2 (1 + A (t − τ ) ) Z 2t η α−1 1
= (40)
b α q
b α d η, 0 (1 + A
(η) ) (1 + A
η ) substituting (t − τ ) by η. From (39) and (40), relation (37) can be written Z 2t 1 τ α−1
Φ(t) = 2
b α q
b α d τ 0 (1 + A
τ ) (1 + A
τ ) Z 2t τ α−1
(41) =2
b α q+1 d τ 0 (1 + A
τ ) which is to equivalent to Φ(t) =
2 1
t q .
b 1 −
b qα A 1 + A 2
(42)
Using (42), we have Φ(t) > 0 if t > 0 and the expression q 1 − qρkLkθq+1 kX0 k Φ(t) in relation (36) is minimal when t tends to infinity. Then, inequality (36) holds if the initial state X0 satisfies condition (31).
Since the previous inequality can be written as
From (34), the generalization of Gronwall-Bellman lemma given in lemma 3 yields the following inequality θ(t − τ ) k @ θkX ‚ 0‚ A ‚ ‚ d τ > 0, ∀ t > 0, 1−qρkLk θkX0 k ‚ b‚ α ‚ b‚
0 1 + ‚A 1 + ‚A ‚τ ‚ (t − τ )α
b α 1 +
A t (35) (43) kX(t)k 6 q1 we must check that the following inequality q+1 q 1 − qρθ kLkkX k Φ(t) 0 q 1 − qρkLkθq+1 kX0 k Φ(t) > 0, ∀ t > 0 (36) which is equivalent to inequality (30). holds, where Z t Then, if the time t tends to infinity, kX(t)k converges 1 (t − τ )α−1
Φ(t) = d τ. (37) to zero which implies the asymptotic stability of the zero
b α q
b 0 (1 + A
τ ) (1 + A
(t − τ )α ) solution. Z
t
0
1q
α−1
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WeBIn5.3 V. C ONCLUSION
holds, then
In this paper, observer-based control laws have been derived for asymptotic stabilization of fractional-order systems. The fractional-order systems are either linear or nonlinear and affine. In the linear case, the observer-based control satisfies the separation principle since the closed-loop stability results in the stability of both the state feedback and the observation error. In the nonlinear case, the separation principle does not work and the closed-loop stability is proven using a proposed generalized form of the GronwallBellman lemma which is given in the appendix.
r(t)
x(t) 6
t
Z 1 − (` − 1)
(r(s))
`−1
1 . `−1 f (s) d s
(50)
a
Proof: Relation (48) can be written as follows Z t f (s)(x(s))`−1 x(s) d s x(t) 6 r(t) + a
and from the classical Gronwall-Bellman lemma [33], [31], we obtain Z t f (s)(x(s))`−1 d s (51) x(t) 6 r(t) exp a
or equivalently t
„ Z (x(t))`−1 6 (r(t))`−1 exp (` − 1)
A PPENDIX
« f (s)(x(s))`−1 d s .
a
(52)
A PPENDIX : G ENERALIZATION OF G RONWALL -B ELLMAN LEMMA
Multiplying the above inequality by −(` − 1)f (t) gives
Lemma 2 (Gronwall-Bellman lemma): [31] (p 292) [30] (p 252) Let i) ii) iii) iv)
− (` − 1)f (t)(x(t))`−1 „ « Z t > −(`−1)(r(t))`−1 f (t) exp (` − 1) f (s)(x(s))`−1 d s
+
f , g and k, IR 7→ IR and locally integrable, g > 0, k > 0, g ∈ L∞ , gk is locally integrable on IR+ .
If u : IR+ 7→ IR satisfies Z t u(t) 6 f (t) + g(t) k(τ )u(τ ) d τ,
a
(53)
or equivalently t
„ Z − (` − 1)f (t)(x(t))`−1 exp −(` − 1)
« f (s)(x(s))`−1 d s
a
∀t > 0
> −(` − 1)(r(t))`−1 f (t)
(44)
(54)
0
then
Using the primitive of the exponential function, the above t
Z u(t) 6 f (t)+g(t)
„Z k(τ )f (τ ) exp
0
« inequality becomes t k(s)g(s) d s d τ, ∀ t > 0.
τ
(45)
+
Corollary 2: [33] Let k : IR 7→ IR, locally integrable on IR+ and k > 0 and c(t) be a positive, monotonic decreasing function. If u : IR+ 7→ IR+ satisfies Z t
u(t) 6 c(t) +
k(τ )u(τ ) d τ,
∀t > 0
(46)
0
then Z t u(t) 6 c(t) exp k(τ ) d τ ,
d dt
t
Z
`−1
exp −(` − 1)
f (s)(x(s))
ds
a
> −(` − 1)(r(t))`−1 f (t) (55) and integrating from a to t, we obtain Z t `−1 exp −(` − 1) f (s)(x(s)) ds a Z t > 1 − (` − 1) (r(s))`−1 f (s) d s. (56) a
∀ t > 0.
(47)
0
Lemma 3 (Generalization of Gronwall-Bellman lemma): Let i) a, b ∈ IR, 0 6 a < b, r(t) > 0 a positive, monotonic, decreasing function, an integer ` > 1, ii) f : [a, b] 7→ IR+ an integrable function such that, ∀ α, β ∈ [a, b], (0 6 α < β), we have Z β f (s) d s > 0 α
iii) x : [a, b] 7→ IR+ an essential bounded function such that Z t x(t) 6 r(t) + f (s)(x(s))` d s. (48)
Notice that the constant in the above integration is equal to 1 (this can be shown with t = a). If the inequality (49) holds, we have Z t exp (` − 1) f (s)(x(s))`−1 d s a
1
6
Z
(r(s))`−1 f (s) d s
1 − (` − 1) a
Inequalities (52) and (57) imply 1
(x(t))`−1 (r(t))−(`−1) 6
Z 1 − (` − 1)
t
(r(s))`−1 f (s) d s
a
or equivalently
a
If the following inequality Z t 1 − (` − 1) (r(s))`−1 f (s) d s > 0
. (57)
t
r(t)
x(t) 6 (49)
Z 1 − (` − 1)
(r(s)) a
a
1936
t `−1
1 . `−1 f (s) d s
(58)
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