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Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems
Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
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Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems Manuel A. Duarte-Mermoud a,b, Norelys Aguila-Camacho a,b,⇑, Javier A. Gallegos a,b, Rafael Castro-Linares c a
Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile Advanced Mining Technology Center, University of Chile, Av. Tupper 2007, Santiago, Chile c Department of Electrical Engineering, CINVESTAV, Av. IPN 2508, México DF, Mexico b
a r t i c l e
i n f o
Article history: Received 18 August 2014 Accepted 13 October 2014 Available online xxxx Keywords: Fractional calculus Uniform stability of fractional order systems Fractional extension of Lyapunov direct method General quadratic Lyapunov functions Fractional adaptive systems
a b s t r a c t This paper presents two new lemmas related to the Caputo fractional derivatives, when a 2 ð0; 1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Fractional calculus relates with the calculus using integrals and derivatives of orders that may be real or complex. It has become very popular in recent years due to its demonstrated applications in many fields of science and engineering [1]. Fractional operators can be found in the identification field, when modeling systems using fractional differential equations, such as the diffusion process found in batteries [2], some heat transfer process [3], the effect of the frequency in induction machines [4], amongst others. Fractional operators can be found in the control field as well; e.g. Fractional Order PID controllers [5]; fractional high gain output feedback controllers [6,7] and Fractional Order Model Reference Adaptive Controllers (FOMRAC) [8–14], amongst others. The stability of these systems have to be proved using techniques specially developed for fractional order systems (FOS), either for the stability of the system itself, or for the stability of the controlled system. One of the available techniques to prove the stability of FOS is the fractional-order extension of Lyapunov direct method, proposed by Li et al. [15]. This method allows concluding asymptotic stability and Mittag–Leffler stability for FOS. However, ⇑ Corresponding author at: Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile. Tel.: +56 2 29784920; fax: +56 2 26720162. E-mail addresses: [email protected] (M.A. Duarte-Mermoud), [email protected] (N. Aguila-Camacho), [email protected] (J.A. Gallegos), [email protected] (R. Castro-Linares). http://dx.doi.org/10.1016/j.cnsns.2014.10.008 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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it does not address the frequent case when the fractional derivative of the Lyapunov function is only negative semidefinite, and conclusions about stability or uniform stability can be drawn. The work reported in [16] can be seen as a complement of the method proposed in [15], since it allows proving stability in the Lyapunov sense when the fractional derivative of the Lyapunov function is only negative semidefinite, although the case of uniform Lyapunov stability is not addressed in [16]. Nevertheless, the use of those methods is often a hard task, since finding Lyapunov functions is more complex in the fractional order case than in the integer order case. Recently, a new lemma for the Caputo fractional derivative of a quadratic function has been presented in [17]. This result allows the use of classic quadratic Lyapunov functions in many stability analysis of FOS. However, in some cases, those simple quadratic functions are not useful, and more general quadratic Lyapunov functions must be used instead. Furthermore, in many cases the use of the trace of matrices inside Lyapunov functions can be useful in proving the stability of FOS, but there is no well established results in these cases. This paper presents a new lemma related to the Caputo fractional derivatives, when a 2 ð0; 1, for general quadratic functions, which allows using general quadratic Lyapunov functions in the stability analysis of many FOS. Besides, another lemma related to the Caputo fractional derivative of the trace of a product of matrices is presented, which allows using the trace inside Lyapunov functions, for stability analysis of FOS. The paper is organized as follows; Section 2 presents some basic concepts about fractional calculus and stability of FOS, as well as some well known results that are used in this paper. Section 3 introduces the new lemmas for Caputo fractional derivatives of special functions, and also a theorem related to the fractional extension of Lyapunov direct method, for proving uniform stability in the Lyapunov sense. Section 4 presents the usefulness of the proposed lemmas and the theorem for the stability analysis of two FOMRAC schemes. Finally, Section 5 presents the conclusions of the work. 2. Preliminary concepts In this section, some basic definitions related to fractional calculus and some concepts and techniques related to the stability analysis of FOS are presented. 2.1. Fractional calculus In fractional calculus, the traditional definitions of the integral and derivative of a function are generalized from integer orders to real or complex orders. In the time domain, the fractional order derivative and fractional order integral operators are defined by a convolution operation. Several definitions exist regarding the fractional derivative of order a > 0, but the Caputo definition in (1) is used in most of the engineering applications, since this definition incorporates initial conditions for xðÞ and its integer order derivatives, i.e., initial conditions that are physically appealing in the traditional way. Definition 1 (Caputo Fractional Derivative [1]). The Caputo fractional derivative of order a 2 Rþ on the half axis Rþ is defined as follows C a t0 Dt xðt Þ
¼
1 Cðn aÞ
Z
t
t0
xðnÞ ðsÞ ðt sÞanþ1
ds;
t > t0
ð1Þ
with n ¼ min fk 2 N=k > ag; a > 0. A new property for Caputo fractional derivatives has been presented in [17], and is introduced here for completeness in Lemma 1. Lemma 1 ([17]). Let xðtÞ 2 Rn be a differentiable vector. Then, for any time instant t P t 0
1 C a T D x ðtÞxðtÞ 6 xT ðt ÞCt0 Dat xðt Þ; 2 t0 t
8a 2 ð0; 1Þ
ð2Þ
2.2. Stability of fractional order systems Regarding the stability analysis of FOS, the fractional-order extension of the Lyapunov direct method [15] is one of the available methodologies in the literature, and is stated in Theorem 1. Using the Caputo derivative, a FOS can be defined in a general form as C a t0 Dt xðt Þ
¼ f ðx; t Þ
ð3Þ
In this study we will consider a 2 ð0; 1Þ. Definition 2. A continuous function c : ½0; tÞ ! ½0; 1Þ is said to belong to class-K if it is strictly increasing and cð0Þ ¼ 0 [15].
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Theorem 1 (Fractional-order extension of Lyapunov direct method [15]). Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a Lyapunov function V ðxðt Þ; t Þ and class-K functions ci ði ¼ 1; 2; 3Þ satisfying
c1 ðkxkÞ 6 V ðxðtÞ; tÞ 6 c2 ðkxkÞ
ð4Þ
C b t 0 Dt V ðxðt Þ; t Þ
ð5Þ
6 c3 ðkxkÞ
where b 2 ð0; 1Þ. Then the origin of the system (3) is asymptotically stable. As can be seen, Theorem 1 can not be used when the fractional derivative of the Lyapunov function is only negative semidefinite. Another result that can be used in the stability analysis of fractional differential systems is the fractional comparison principle, which is introduced in Lemma 2. Lemma 2 (Fractional comparison principle [15]). Let
C b t 0 Dt xðt Þ
P Ct0 Dbt yðt Þ, b 2 ð0; 1Þ, and xðt0 Þ ¼ yðt 0 Þ. Then xðt Þ P yðtÞ.
2.3. Miscellaneous concepts In what follows, some well known mathematical results and definitions are presented, which will be used in the proofs of the main results of the paper. Definition 3 (Positive Definite Matrix [18]). An n n matrix P is positive definite if and only if
xT Px > 0;
8x – 0; x 2 Cn
ð6Þ
Definition 4 (Negative Definite Matrix [19]). An n n matrix P is negative definite if and only if
xT Px < 0;
8x – 0; x 2 Cn
ð7Þ
Theorem 2 (Diagonalization of a real symmetric matrix [19]). Let P 2 Rnn be a real symmetric matrix. Then it may be transformed into a diagonal form by means of an orthogonal transformation, that is to say, there is an orthogonal matrix B 2 Rnn and a diagonal matrix K 2 Rnn such that
P ¼ BKBT
ð8Þ
Lemma 3 (Relationship between positive definite functions and class-K functions [24]). A function V ðx; tÞ is locally (or globally) positive definite if and only if there exists a class-K function c1 such that V ð0; t Þ ¼ 0 and
V ðx; t Þ P c1 ðkxkÞ
ð9Þ
8t P t 0 and 8x belonging to the local space (or the whole space). A function V ðx; tÞ is locally (or globally) decrescent if and only if there exists a class-K function c2 such that V ð0; tÞ ¼ 0 and
V ðx; t Þ 6 c2 ðkxkÞ
ð10Þ
8t P t 0 and 8x belonging to the local space (or the whole space). 3. New lemmas related to the Caputo fractional derivative This section presents two new lemmas. The first one allows the use of general quadratic Lyapunov functions in the stability analysis of many FOS. The second lemma allows using the trace of a matrix into Lyapunov functions, also in order to prove stability of FOS. Besides, a theorem is presented, which allows proving uniform stability of FOS using Lyapunov functions. First of all, let us refer to the fractional extension of Lyapunov direct method. In the work reported in [15], sufficient conditions for proving the asymptotic stability of FOS were stated, using class-K functions. In the work reported in [16], sufficient conditions for proving Lyapunov stability are given. However, the corresponding sufficient conditions for proving Lyapunov uniform stability were not addressed in these works. There are many works proposing the analysis of uniform stability for fractional order systems using Lyapunov functions. It can be mentioned the case for fractional systems with time delay [20], for q-fractional systems [21], for discrete fractional systems [22], for neutral fractional systems [23] etc. However, we could not find results for systems of the form in (3) using Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Lyapunov functions and analyzing its Caputo fractional derivative. Since the conditions for proving uniform stability for these kind of systems using Lyapunov functions are available in many common situations, we present them in Theorem 3, and we also use it in Section 4 of this paper. Theorem 3 (Lyapunov Stability and Uniform stability of fractional order systems). Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a continuous Lyapunov function V ðxðtÞ; t Þ and a scalar class-K function c1 ð:Þ such that, 8x – 0
c1 ðkxðtÞkÞ 6 V ðxðtÞ; tÞ
ð11Þ
C b t0 Dt V ðxðt Þ; t Þ
ð12Þ
and
6 0;
with b 2 ð0; 1
then the origin of the system (3) is Lyapunov stable. If, furthermore, there is a scalar class-K function c2 ð:Þ satisfying
V ðxðt Þ; t Þ 6 c2 ðkxðt ÞkÞ
ð13Þ
then the origin of the system (3) is Lyapunov uniformly stable. Proof. The proof of this theorem follows the same outline than in the proof for the case of integer order systems in [24], with some modifications to treat the fractional case. The first part of the proof is not a new result, since it was done in [16]. Nevertheless, we put it here in order to present a more complete result and to show a more detailed proof. Let us prove Lyapunov stability first. To establish Lyapunov stability, we must prove that if (11) and (12) are satisfied, then for any e > 0, there exists a dðe; t0 Þ such that
kxðt0 Þk < d
)
kxðtÞk < e;
8t P t 0
ð14Þ
Given that condition (12) holds, we can use Lemma 2 (Fractional Comparison Principle) obtaining that
V ðxðt Þ; t Þ 6 V ðxðt0 Þ; t0 Þ;
8t P t 0
ð15Þ
Using expressions (11) and (15) it can be written that
c1 ðkxkÞ 6 V ðxðtÞ; tÞ 6 V ðxðt0 Þ; t0 Þ; 8t P t0
ð16Þ
Since V is continuous with respect to x and V ð0; t0 Þ ¼ 0 for being a Lyapunov function, then we can find d such that
kxðt0 Þk < d
)
V ðxðt 0 Þ; t 0 Þ < g ¼ c1 ðeÞ
ð17Þ
This means that if kxðt 0 Þk < d, then from expression (16) and (17) it holds that
c1 ðkxðtÞkÞ < c1 ðeÞ; 8t P t0
ð18Þ
Since c1 ð:Þ is nondecreasing for being a class-K function, expression (18) implies that
kxðtÞk < e;
8t P t 0
ð19Þ
Therefore, we have shown that expression (14) holds, and consequently the origin of the system (3) is Lyapunov stable if conditions (11) and (12) are fulfilled. Let us state now the proof of Lyapunov uniform stability, which is the contribution of this theorem. To establish Lyapunov uniform stability, we must prove that if (11)–(13) are satisfied, then for any e > 0, there exists a dðeÞ such that
kxðt0 Þk < d
)
kxðtÞk < e;
8t P t 0
ð20Þ
From expressions (11) and (13) it follows that
c1 ðkxðtÞkÞ 6 V ðxðtÞ; tÞ 6 c2 ðkxðtÞkÞ; 8t P t0
ð21Þ
Now for any e > 0, we can find dðeÞ > 0 such that c2 ðdÞ < c1 ðeÞ. Let the initial condition xðt 0 Þ be chosen such that kxðt0 Þk < d. Then using the result in expression (15) it can be written
c1 ðeÞ > c2 ðdÞ P V ðxðt0 Þ; t0 Þ P V ðxðtÞ; tÞ P c1 ðkxðtÞkÞ
ð22Þ
Given that c1 ð:Þ is non increasing, this implies that
8t P t0 ;
kxðt Þk < e
and the condition for uniformly stability holds, because in this case d is independent of t0 .
ð23Þ h
Remark 1. Given the relationship between class-K functions and positive definite functions and decrescent functions in Lemma 3, Theorem 3 can be presented as well in the following form. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
M.A. Duarte-Mermoud et al. / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
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Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a continuous function V ðxðtÞ; tÞ such that j
V ðxðtÞ; tÞ is positive definite. b C with b 2 ð0; 1Þ, is negative semidefinite. then the origin of system (3) is Lyapunov stable. Furthermore, if V ðxðtÞ; t Þ is decrescent, then the origin of system (3) is Lyapunov uniformly stable.
j t Dt V ðxðt Þ; t Þ; 0 j
In what follows, two new lemmas are presented, where some interesting relationships for the Caputo fractional derivative of special functions are derived. Those relationships will be useful for proving the uniform stability of FOS, using Theorem 3. Lemma 4. Let xðt Þ 2 Rn be a vector of differentiable functions. Then, for any time instant t P t0 , the following relationship holds
1 C a T T C a t 0 D x ðt ÞPxðt Þ 6 x ðt ÞP t 0 D xðt Þ; 2
8a 2 ð0; 1; 8t P t 0
ð24Þ
where P 2 Rnn is a constant, square, symmetric and positive definite matrix. Proof. Since matrix P is symmetric, using Theorem 2, there exist an orthogonal matrix B 2 Rnn and a diagonal matrix K 2 Rnn such that we can write T 1 T 1 1 T x ðt ÞPxðtÞ ¼ xT ðt ÞBKBT xðtÞ ¼ B xðt Þ K BT xðt Þ ð25Þ 2 2 2 Let us use an auxiliary variable yðtÞ defined as
yðt Þ ¼ BT xðtÞ
ð26Þ
Then from (25) it can be written
1 T 1 x ðt ÞPxðtÞ ¼ yT ðtÞKyðt Þ 2 2
ð27Þ
Since matrix K is diagonal, we can write n 1 T 1X y ðtÞKyðt Þ ¼ kii y2i ðt Þ 2 2 i¼1
ð28Þ
Then n n X 1 C aX 1 C a T 1 kii y2i ðt Þ ¼ kii Ct0 Da y2i ðt Þ 8a 2 ð0; 1Þ; 8t P t0 t 0 D y ðt ÞKyðt Þ ¼ t0 D 2 2 2 i¼1 i¼1
ð29Þ
Applying Lemma 1 to Eq. (29), and since kii > 0, results n X 1 C a T kii yi ðtÞCt0 Da yi ðt Þ; t 0 D y ðt ÞKyðt Þ 6 2 i¼1
8a 2 ð0; 1Þ; 8t P t 0
ð30Þ
Since kii are the diagonal elements of matrix K, then it can be written that n X kii yi ðtÞCt0 Da yi ðt Þ ¼ yT ðt ÞK Ct0 Da yðt Þ
ð31Þ
i¼1
and from (30) we can state that
1 C a T D y ðtÞKyðt Þ 6 yT ðt ÞK Ct0 Da yðt Þ; 2 t0
8a 2 ð0; 1Þ; 8t P t 0
ð32Þ
Then replacing expression (26) in (32) we can write that
T T 1 C a T B xðtÞ K BT xðt Þ 6 BT xðtÞ K Ct0 Da BT xðt Þ ; t0 D 2
8a 2 ð0; 1Þ; 8t P t 0
ð33Þ
Rearranging and using BKBT ¼ P in (33) results
1 C a T D x ðt ÞPxðtÞ 6 xT ðt ÞP Ct0 Da xðt Þ; 2 t0
8a 2 ð0; 1Þ; 8t P t0
ð34Þ
and the proof is complete for the case a 2 ð0; 1Þ. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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The case a ¼ 1, corresponds to the well known chain rule for integer order derivatives, where d ¼ xT ðtÞP dt xðtÞ. Therefore the proof is now complete. h
1 d 2 dt
T x ðtÞPxðt Þ
Regarding the trace of the product of two matrices, the following lemma allows concluding about its Caputo fractional derivative, when the product has a particular form. Specifically, when it is the product of a rectangular time varying matrix and its transpose. Lemma 5. Let Aðt Þ 2 Rmn be a time varying differentiable matrix. Then, for any time instant t P t 0 , the following relationship holds C a t0 D
h n oi n o tr AT ðtÞAðt Þ 6 2tr AT ðtÞCt0 Da Aðt Þ ;
8a 2 ð0; 1; 8t P t0
ð35Þ
Proof. From the matrix trace definition and the matrix product definition it directly follows that n n X m n o X X tr AT A ¼ AT A ¼ ATij Aji ii
i¼1
ð36Þ
i¼1 j¼1
Since ATij ¼ Aji , then expression (36) can be written as n X m n o X tr AT A ¼ A2ij
ð37Þ
i¼1 j¼1
Let us write expression (37) for the case of matrix Aðt Þ in the form
n o X tr AT ðtÞAðt Þ ¼ a2ij ðtÞ
ð38Þ
i;j
where aij ðt Þ are the elements of the matrix AðtÞ. Applying the Caputo fractional derivative to expression (38) we get C a t0 D
" # h n oi X T C a 2 tr A ðtÞAðt Þ ¼ t0 D aij ðtÞ
ð39Þ
i;j
Since the Caputo fractional derivative has the property of linearity, expression (39) can be written as C a t0 D
" # h n oi i X X ah T C C a 2 2 tr A ðtÞAðt Þ ¼ t0 D aij ðtÞ ¼ t 0 D aij ðt Þ i;j
ð40Þ
i;j
Applying Lemma 1 to expression (40), the following inequality arises C a t0 D
h n oi X tr AT ðtÞAðt Þ 6 2 aij ðtÞCt0 Da aij ðt Þ
ð41Þ
i;j
Using the fact that
n o X aij ðt ÞCt0 Da aij ðt Þ ¼ tr AT ðt ÞCt0 Da Aðt Þ
ð42Þ
i;j
expression (41) can be written in the form C a t0 D
h n oi n o tr AT ðtÞAðt Þ 6 2tr AT ðtÞCt0 Da Aðt Þ
and the proof is complete for the case a 2 ð0; 1Þ. case oi a ¼ 1 n corresponds to the well known result h The n o d d tr AT ðt ÞAðtÞ ¼ 2tr AT ðt Þ dt AðtÞ . Therefore the proof is now complete. dt
ð43Þ
for h
integer
order
derivatives,
where
4. Using Theorem 3 and Lemmas 4 and 5 in the stability analysis of fractional order systems One of the most used Lyapunov function candidate to prove the stability and convergence of integer systems is the quadratic function xT ðt Þxðt Þ. However, in the fractional case, it is no straightforward the use of those functions, as was stated in [17]. Moreover, sometimes those simple quadratic functions are not enough to conclude about the stability of FOS, and general quadratic Lyapunov functions xT ðt ÞPxðt Þ are required instead. Besides, in many cases the use of the trace of a product of two matrices (in the form of Lemma 5) is required when dealing with multivariable adaptive systems or when the state Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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vector of the systems is accessible. In what follows, two applications of Lemmas 4 and 5, and Theorem 3 are presented, to prove uniform stability of FOMRAC schemes. 4.1. Uniform stability of a FOMRAC scheme for plant with state accessible Let us consider a linear and time-invariant plant to be controlled, given by C a t 0 D t xp ð t Þ
¼ Ap xp ðtÞ þ Bp u
ð44Þ
nn
np
where Ap 2 R is an unknown constant matrix and Bp 2 R is a known constant matrix. The pair Ap ; Bp is controllable, the n-dimensional state xp ðt Þ is assumed to be accessible and u 2 Rp is the control input to be defined. Let a reference model be given by C a t 0 D t xm ð t Þ
¼ Am xm ðt Þ þ Bm r
ð45Þ
where Am 2 Rnn is a known Hurwitz constant matrix, Bm 2 Rnp is a known constant matrix and r 2 Rp is a uniformly bounded and piecewise-continuous reference input. It is assumed that xm ðt Þ, for all t P t 0 , represents the desired trajectory for xp ðtÞ. The aim here is to control the plant so that all the signals remain bounded and ideally lim kxp ðtÞ xm ðtÞk ¼ 0. t!1 Let us choose the control input as
uðtÞ ¼ HðtÞxp ðt Þ þ Q r ðtÞ where Hðt Þ 2 R
pn
ð46Þ
pp
is a matrix consisting of adjustable parameters and Q 2 R
is defined such that
Bp Q ¼ Bm
ð47Þ
pn
It is further assumed that a constant matrix H 2 R
exists such that
Ap þ Bp H ¼ Am
ð48Þ n
Defining the control error as eðt Þ ¼ xp ðt Þ xm ðt Þ 2 R , the fractional differential equation describing the evolution of the error can be expressed as C a t 0 Dt eðt Þ
¼ Am eðtÞ þ Bp UðtÞxp ðt Þ
ð49Þ
where Uðt Þ ¼ HðtÞ H 2 Rpn is the parameter error matrix. The aim now is to define an adaptive law for adjusting Hðt Þ in such a way that all the signals in the adaptive scheme remain bounded and ideally lim keðt Þk ¼ 0. Let us see what can be assured by using fractional extension of Lyapunov direct t!1 method. Let us propose the following Lyapunov function candidate
n o V ðe; UÞ ¼ eT ðt ÞPeðtÞ þ tr UT ðt ÞUðt Þ
ð50Þ
where P 2 Rnn is a constant symmetric positive-definite matrix and tr denotes the trace of a matrix. As can be seen, this Lyapunov function candidate is positive definite and decrescent. Calculating the Caputo fractional derivative of expression (50) with respect to time we get C a t 0 Dt V ðe;
h n
UÞ ¼ Ct0 Dat eT ðt ÞPeðtÞ þ Ct0 Dat tr UT ðt ÞUðtÞ
oi
ð51Þ
Applying Lemmas 4 and 5 to the right hand side of Eq. (51) results C a t 0 Dt V ðe;
n
UÞ 6 2eT ðt ÞP Ct0 Dat eðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ
o
ð52Þ
Substituting expression (49) in (52) and using properties of the transpose yields C a t 0 Dt V ðe;
h
i
n
UÞ 6 eT ðtÞ ATm P þ PAm eðtÞ þ þ2eT ðt ÞPBm Uðt Þxp ðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ
o
ð53Þ
Since Am is Hurwitz, then a symmetric positive definite matrix Q 0 exists such that
ATm P þ PAm ¼ Q 0
ð54Þ
and expression (53) can be written as C a t 0 Dt V ðe;
n
UÞ 6 eT ðt ÞQ 0 eðt Þ þ 2eT ðtÞPBm UðtÞxp ðt Þ þ 2tr UT ðt ÞCt0 Dat Uðt Þ
o
ð55Þ
Since Uðt Þ is unknown, it can’t be used in the adaptive law we are looking for. But we can choose the adaptive law such that
n o 2eT ðt ÞPBm Uðt Þxp ðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ ¼ 0
ð56Þ
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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The first term of the left hand side of expression (56) can be rewritten as
T T 2eT ðtÞPBm Uðt Þxp ðt Þ ¼ 2 eT ðtÞPBm UðtÞxp ðt Þ ¼ 2xTp ðt Þ eT ðt ÞPBm Uðt Þ ¼ 2xTp ðt ÞUT ðtÞBTm Peðt Þ Now using the property for the trace tr yxT ¼ xT y;
2xTp ðt Þ
T
U
ðtÞBTm Peðt Þ
n
T
¼ 2tr U
ðtÞBTm Peðt ÞxTp ðtÞ
o
ð57Þ
for any x; y 2 Rn , we can write that
ð58Þ
From (57) and (58) we can finally write
n o 2eT ðtÞPBm Uðt Þxp ðt Þ ¼ 2tr UT ðtÞBTm Peðt ÞxTp ðtÞ
ð59Þ
According to expression (59) and (56), it is straightforward to conclude that choosing the adaptive law as C a t 0 Dt
Uðt Þ ¼ Ct0 Dat HðtÞ ¼ BTm PeðtÞxTp ðt Þ
ð60Þ
then expression (56) holds. Thus, the Caputo fractional derivative with respect to time of the Lyapunov function (55), using the adaptive law (60), results C a t0 Dt V ðe;
UÞ 6 eT ðtÞQ 0 eðt Þ
ð61Þ
As can be seen, the Caputo fractional derivative with respect to time of the Lyapunov function is negative semidefinite. Using Theorem 3, together with the fact that V ðe; UÞ is positive definite and decrescent, we can conclude that the origin of the system (49) and (60) is Lyapunov uniformly stable. Remark 2. The fact that the matrix Am of the reference model is Hurwitz allows using expression (54) and concluding about the stability of the scheme. However, if the matrix Am has eigenvalues with positive real parts such that their arguments are greater than a2p (which is also a stable reference model from fractional viewpoint), the expression (54) does not holds, and consequently the stability can’t be proved using this methodology and the Lyapunov function (50). For comments in the control error convergence, see Remark 3. 4.2. Stability of a FOMRAC scheme for plants with state not accessible In the case of plants where the whole state is not accessible, but only the plant output is measured, more interesting problems arise when designing a FOMRAC. In that cases, some extra considerations must be done. Let us consider a fractional order single-input single-output (SISO) plant, with input–output pair uð:Þ; yp ð:Þ , defined by n o T T the triplet Ap ; bp ; hp ; Ap 2 Rnn ; bp ; hp 2 Rn . The plant is non-minimum phase and the sign of the high frequency gain of the plant kp is known. Let us consider a fractional order (SISO) reference model, asymptotically sable, with input–output pair fr ð:Þ; ym ð:Þg, n o T T defined by the triplet Am ; bm ; hm , where Am 2 Rnn ; bm ; hm 2 Rn ; r 2 R is a uniformly bounded and piecewise-continuous function of time and the high frequency gain of the reference model km is assumed to be positive. For this kind of problem, auxiliary signals must be constructed in order
to compute
a control signal u, to achieve that all the plant parameters and signals remain bounded, and ideally that lim yp ðtÞ ym ðt Þ ¼ 0. t!1 For more details of these type of solution see [25] for the equivalent integer order case. We omit here the design details of the FOMRAC, since we want to present the stability analysis of the scheme, in order to prove the usefulness of the results presented in this paper (Lemma 4 and Theorem 3 in this case). The differential equation describing the plant together with the controller for this problem can be represented as C a t0 Dt xðt Þ
¼ Amn xðt Þ þ bmn /T ðtÞxðt Þ þ k r ðtÞ ;
T
yp ¼ hmn xðtÞ
ð62Þ
where xðtÞ 2 R3n2 ; Amn 2 Rð3n2Þð3n2Þ ; bmn 2 R3n2 ; hmn 2 R3n2 ; /ðt Þ ¼ hðt Þ h 2 R2n corresponds to the parameter error, hðt Þ 2 R2n is the parameter vector to be adaptively adjusted, h 2 R2n is the unknown ideal controller parameter vector, k ¼ kkmp and xðtÞ 2 R2n corresponds to auxiliary signals of the scheme, obtained by filtering the plant input and output. When hðtÞ h , it follows that the reference model can be described by the ð3n 2Þth order fractional differential equation C a t0 Dt xmn ðt Þ
¼ Amn xmn ðt Þ þ bmn k r ðtÞ;
T
ym ¼ hmn xmn ðtÞ
ð63Þ
3n2
where xmn ðt Þ 2 R . Therefore, the error equation for the overall system may be expressed as: C a t 0 Dt eðt Þ T
¼ Amn eðtÞ þ bmn /T ðt Þxðt Þ
e1 ¼ hmn eðt Þ
ð64Þ
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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9
where eðt Þ ¼ xðtÞ xmn ðt Þ is not accessible and e1 ðtÞ ¼ yp ðt Þ ym ðtÞ 2 R corresponds to the output error. Moreover, in this configuration T
kp hmn ½sI Amn 1 bmn
ð65Þ
is a SPR function. In order to find the adaptive laws for the FOMRAC and proving the stability of the scheme, let us propose the following Lyapunov function candidate, which is positive definite and decrescent
1 V ðe; /Þ ¼ eT ðtÞPeðtÞ þ
/T ðt Þ/ðt Þ kp
ð66Þ
where P 2 Rð3n2Þð3n2Þ is a constant symmetric and positive definite matrix. Taking the fractional derivative to expression (66) and applying Lemma 4 and Lemma 1, it follows that C a t 0 Dt V ðe; /Þ
2 6 2eT ðt ÞP Ct0 Dat eðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp
ð67Þ
Using expression (64) in (67) we obtain C a t 0 Dt V ðe; /Þ
2 6 2eT ðt ÞP Amn eðt Þ þ bmn /T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 ¼ 2eT ðtÞPAmn eðtÞ þ 2eT ðtÞPbmn /T ðtÞxðt Þ þ
/T ðtÞCt0 Dat /ðtÞ kp h i 2 ¼ eT ðt Þ ATmn P þ PAmn eðt Þ þ 2eT ðtÞPbmn /T ðt Þxðt Þ þ
/T ðtÞCt0 Dat /ðtÞ kp
ð68Þ
Since the transfer function (65) is SPR, then using the Meyer–Kalman–Yakubovich (MKY) Lemma [25], we can assure that given a matrix Q ¼ Q T > 0, there exists a P ¼ PT > 0 such that
ATmn P þ PAmn ¼ Q
ð69Þ
Pbmn ¼ hmn kp Using this fact in (68) it follows that C a t 0 Dt V ðe; /Þ
2 6 eT ðtÞQeðtÞ þ 2eT ðt Þhmn kp /T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 T ¼ eT ðt ÞQeðt Þ þ 2kp hmn eðtÞ/T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 ¼ eT ðt ÞQeðt Þ þ 2kp e1 ðt Þ/T ðtÞxðt Þ þ
/T ðt ÞCt0 Dat /ðt Þ kp
ð70Þ
If we choose the adaptive law as C a t 0 Dt /ðt Þ
¼ Ct0 Dat hðt Þ ¼ sgn kp e1 ðt ÞxðtÞ
ð71Þ
then the Caputo fractional derivative of the Lyapunov function turns out to be C a t 0 Dt V ðe; /Þ
6 eT ðtÞQeðtÞ þ 2kp e1 ðt Þ/T ðt Þxðt Þ 2
sgn kp T
/ ðtÞe1 ðt Þxðt Þ ¼ eT ðt ÞQeðt Þ
kp
ð72Þ
Therefore, the Caputo fractional derivative of the Lyapunov function is negative semidefinite. Thus, using Theorem 3 together with the fact that V ðe; /Þ is positive definite and decrescent, we can conclude that the origin of the system (64) and (71) is uniformly stable in the Lyapunov sense and all the signals remain bounded. Remark 3. The two analysis presented so far allow proving uniform stability in the Lyapunov sense for FOMRAC schemes. However, no conclusions about the convergence of the errors to zero can be drawn from this analysis. In the integer order case, Barbalat’s Lemma is used to prove it, but this analysis cannot be directly used in the fractional order case. Thus, additional tools need to be developed, in order to prove the errors convergence in the fractional order case. Currently, efforts are being made to develop these tools.
5. Conclusions Two new lemmas for the Caputo fractional derivatives of general quadratic forms and for the trace of the product of a rectangular matrix and its transpose, when a 2 ð0; 1, have been presented in this paper. Both results allow using the fractional extension of Lyapunov direct method for the stability analysis of fractional order systems. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Also, a theorem providing conditions for uniform stability in the Lyapunov sense of FOS has been presented in this paper. The theorem can be seen as a complement of the fractional extension of Lyapunov direct method proposed in [15] and the method for proving Lyapunov stability presented in [16]. The usefulness of the two lemmas and the theorem was stated in the paper through the analysis of two FOMRAC schemes, where the Lyapunov uniform stability was proved, using Lyapunov functions containing general quadratic forms and the trace of the product of a matrix and its transpose. Acknowledgments This work has been supported by CONICYT-Chile, under the grants FB009 ‘‘Centro de Tecnología para la Minería’’, FONDECYT 1120453, ‘‘Improvements of Adaptive Systems Performance by using Fractional Order Observers and Particle Swarm Optimization’’; and CONICYT-Chile under the grants ‘‘CONICYT-PCHA/National PhD scholarship program, 2013-21130004’’. References [1] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. Elsevier; 2006. [2] Sabatier J, Aoun M, Oustaloup A, Gréegoire G, Ragot F, Roy P. Fractional system identification for lead acid battery state of charge estimation. Signal Process 2006;86:2645–57. [3] Gabano J-D, Poinot T. Fractional modelling and identification of thermal systems. Signal Process 2011;91:531–41. [4] Lin J, Poinot T, Trigeassou JC, Kabbaj H, Faucher J, Modélisation et identification d’ordre non entire d’une machine asynchrone. In: Conférence Internationale Francophone d’Automatique; 2000. [5] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Eng Pract 2009;17:1380–7. [6] Ladaci S, Loiseau JJ, Charef A. Fractional order adaptive high-gain controllers for a class of linear systems. Commun Nonlinear Sci Numer Simul 2008;13:707–14. [7] El Figuigui O, Elalami N, Application of fractional adaptive High-Gain controller to a LEO (Low Earth Orbit) satellite. In: International conference on computers industrial engineering; 2009. p. 1850–56. [8] YaLi H, RuiKun G, Application of fractional-order model reference adaptive control on industry boiler burning system. In: International conference on intelligent computation technology and automation (ICICTA) 2010, vol. 1. p. 750–53. [9] Duarte-Mermoud MA, Aguila-Camacho N, Fractional order adaptive control of simple systems. In: Narendra K, editor. Fifteenth yale workshop on adaptive and learning systems; 2011. p. 57–62. [10] Duarte-Mermoud MA, Aguila-Camacho N, Some useful results in fractional adaptive control. In: Narendra K, editor. Sixteenth yale workshop on adaptive and learning systems; 2013. p. 51–6. [11] Aguila-Camacho N, Duarte-Mermoud MA. Fractional adaptive control for an automatic voltage regulator. ISA Trans 2013;52:807–15. [12] Ladaci S, Charef A. On fractional adaptive control. Nonlinear Dyn 2006;43:365–78. [13] Vinagre BM, Petrás I, Podlubny I, Chen Y. Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn 2002;29:269–79. [14] Suárez JI, Vinagre BM, Chen Y. A fractional adaptation scheme for lateral control of an AGV. J Vib Control 2008;14:1499–511. [15] Li Y, Chen Y, Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability. Comput Math Appl 2010;59:1810–21. [16] Zhang F, Li Ch, Chen YQ. Asymptotical stability of nonlinear differential systems with Caputo derivative. Int J Differ Equ 2011;2011. [17] Aguila-Camacho N, Duarte-Mermoud MA, Gallegos J. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 2014;19:2951–7. [18] Horn RA, Johnson ChR. Matrix analysis. Cambridge University Press; 1995. [19] Bellman R. Introduction to matrix analysis. Society for Industrial and Applied Mathematics (SIAM); 1997. [20] Wang Y, Li T. Stability analysis of fractional-order nonlinear systems with delay. Math Prob Eng 2014;2014. [21] Jarad F, Abdeljawad T, Baleanu D. Stability of q-fractional non-autonomous systems. Nonlinear Anal Real World Appl 2013;14:780–4. [22] Jarad F, Abdeljawad T, Baleanu D, Bicen K. On the stability of some discrete non autonomous systems. Abstr Appl Anal 2012;2012:780–4. [23] Liu K, Jiang W. Uniform stability of fractional neutral systems: a Lyapunov–Krasovskii functional approach. Adv Difference Equ 2013;2013. [24] Slotine JJ, Li W. Applied nonlinear control. Prentice Hall; 1991. [25] Narendra KS, Annaswamy AM. Stable adaptive systems. Dover Publications Inc.; 2005.
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Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems Manuel A. Duarte-Mermoud a,b, Norelys Aguila-Camacho a,b,⇑, Javier A. Gallegos a,b, Rafael Castro-Linares c a
Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile Advanced Mining Technology Center, University of Chile, Av. Tupper 2007, Santiago, Chile c Department of Electrical Engineering, CINVESTAV, Av. IPN 2508, México DF, Mexico b
a r t i c l e
i n f o
Article history: Received 18 August 2014 Accepted 13 October 2014 Available online xxxx Keywords: Fractional calculus Uniform stability of fractional order systems Fractional extension of Lyapunov direct method General quadratic Lyapunov functions Fractional adaptive systems
a b s t r a c t This paper presents two new lemmas related to the Caputo fractional derivatives, when a 2 ð0; 1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Fractional calculus relates with the calculus using integrals and derivatives of orders that may be real or complex. It has become very popular in recent years due to its demonstrated applications in many fields of science and engineering [1]. Fractional operators can be found in the identification field, when modeling systems using fractional differential equations, such as the diffusion process found in batteries [2], some heat transfer process [3], the effect of the frequency in induction machines [4], amongst others. Fractional operators can be found in the control field as well; e.g. Fractional Order PID controllers [5]; fractional high gain output feedback controllers [6,7] and Fractional Order Model Reference Adaptive Controllers (FOMRAC) [8–14], amongst others. The stability of these systems have to be proved using techniques specially developed for fractional order systems (FOS), either for the stability of the system itself, or for the stability of the controlled system. One of the available techniques to prove the stability of FOS is the fractional-order extension of Lyapunov direct method, proposed by Li et al. [15]. This method allows concluding asymptotic stability and Mittag–Leffler stability for FOS. However, ⇑ Corresponding author at: Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile. Tel.: +56 2 29784920; fax: +56 2 26720162. E-mail addresses: [email protected] (M.A. Duarte-Mermoud), [email protected] (N. Aguila-Camacho), [email protected] (J.A. Gallegos), [email protected] (R. Castro-Linares). http://dx.doi.org/10.1016/j.cnsns.2014.10.008 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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it does not address the frequent case when the fractional derivative of the Lyapunov function is only negative semidefinite, and conclusions about stability or uniform stability can be drawn. The work reported in [16] can be seen as a complement of the method proposed in [15], since it allows proving stability in the Lyapunov sense when the fractional derivative of the Lyapunov function is only negative semidefinite, although the case of uniform Lyapunov stability is not addressed in [16]. Nevertheless, the use of those methods is often a hard task, since finding Lyapunov functions is more complex in the fractional order case than in the integer order case. Recently, a new lemma for the Caputo fractional derivative of a quadratic function has been presented in [17]. This result allows the use of classic quadratic Lyapunov functions in many stability analysis of FOS. However, in some cases, those simple quadratic functions are not useful, and more general quadratic Lyapunov functions must be used instead. Furthermore, in many cases the use of the trace of matrices inside Lyapunov functions can be useful in proving the stability of FOS, but there is no well established results in these cases. This paper presents a new lemma related to the Caputo fractional derivatives, when a 2 ð0; 1, for general quadratic functions, which allows using general quadratic Lyapunov functions in the stability analysis of many FOS. Besides, another lemma related to the Caputo fractional derivative of the trace of a product of matrices is presented, which allows using the trace inside Lyapunov functions, for stability analysis of FOS. The paper is organized as follows; Section 2 presents some basic concepts about fractional calculus and stability of FOS, as well as some well known results that are used in this paper. Section 3 introduces the new lemmas for Caputo fractional derivatives of special functions, and also a theorem related to the fractional extension of Lyapunov direct method, for proving uniform stability in the Lyapunov sense. Section 4 presents the usefulness of the proposed lemmas and the theorem for the stability analysis of two FOMRAC schemes. Finally, Section 5 presents the conclusions of the work. 2. Preliminary concepts In this section, some basic definitions related to fractional calculus and some concepts and techniques related to the stability analysis of FOS are presented. 2.1. Fractional calculus In fractional calculus, the traditional definitions of the integral and derivative of a function are generalized from integer orders to real or complex orders. In the time domain, the fractional order derivative and fractional order integral operators are defined by a convolution operation. Several definitions exist regarding the fractional derivative of order a > 0, but the Caputo definition in (1) is used in most of the engineering applications, since this definition incorporates initial conditions for xðÞ and its integer order derivatives, i.e., initial conditions that are physically appealing in the traditional way. Definition 1 (Caputo Fractional Derivative [1]). The Caputo fractional derivative of order a 2 Rþ on the half axis Rþ is defined as follows C a t0 Dt xðt Þ
¼
1 Cðn aÞ
Z
t
t0
xðnÞ ðsÞ ðt sÞanþ1
ds;
t > t0
ð1Þ
with n ¼ min fk 2 N=k > ag; a > 0. A new property for Caputo fractional derivatives has been presented in [17], and is introduced here for completeness in Lemma 1. Lemma 1 ([17]). Let xðtÞ 2 Rn be a differentiable vector. Then, for any time instant t P t 0
1 C a T D x ðtÞxðtÞ 6 xT ðt ÞCt0 Dat xðt Þ; 2 t0 t
8a 2 ð0; 1Þ
ð2Þ
2.2. Stability of fractional order systems Regarding the stability analysis of FOS, the fractional-order extension of the Lyapunov direct method [15] is one of the available methodologies in the literature, and is stated in Theorem 1. Using the Caputo derivative, a FOS can be defined in a general form as C a t0 Dt xðt Þ
¼ f ðx; t Þ
ð3Þ
In this study we will consider a 2 ð0; 1Þ. Definition 2. A continuous function c : ½0; tÞ ! ½0; 1Þ is said to belong to class-K if it is strictly increasing and cð0Þ ¼ 0 [15].
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Theorem 1 (Fractional-order extension of Lyapunov direct method [15]). Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a Lyapunov function V ðxðt Þ; t Þ and class-K functions ci ði ¼ 1; 2; 3Þ satisfying
c1 ðkxkÞ 6 V ðxðtÞ; tÞ 6 c2 ðkxkÞ
ð4Þ
C b t 0 Dt V ðxðt Þ; t Þ
ð5Þ
6 c3 ðkxkÞ
where b 2 ð0; 1Þ. Then the origin of the system (3) is asymptotically stable. As can be seen, Theorem 1 can not be used when the fractional derivative of the Lyapunov function is only negative semidefinite. Another result that can be used in the stability analysis of fractional differential systems is the fractional comparison principle, which is introduced in Lemma 2. Lemma 2 (Fractional comparison principle [15]). Let
C b t 0 Dt xðt Þ
P Ct0 Dbt yðt Þ, b 2 ð0; 1Þ, and xðt0 Þ ¼ yðt 0 Þ. Then xðt Þ P yðtÞ.
2.3. Miscellaneous concepts In what follows, some well known mathematical results and definitions are presented, which will be used in the proofs of the main results of the paper. Definition 3 (Positive Definite Matrix [18]). An n n matrix P is positive definite if and only if
xT Px > 0;
8x – 0; x 2 Cn
ð6Þ
Definition 4 (Negative Definite Matrix [19]). An n n matrix P is negative definite if and only if
xT Px < 0;
8x – 0; x 2 Cn
ð7Þ
Theorem 2 (Diagonalization of a real symmetric matrix [19]). Let P 2 Rnn be a real symmetric matrix. Then it may be transformed into a diagonal form by means of an orthogonal transformation, that is to say, there is an orthogonal matrix B 2 Rnn and a diagonal matrix K 2 Rnn such that
P ¼ BKBT
ð8Þ
Lemma 3 (Relationship between positive definite functions and class-K functions [24]). A function V ðx; tÞ is locally (or globally) positive definite if and only if there exists a class-K function c1 such that V ð0; t Þ ¼ 0 and
V ðx; t Þ P c1 ðkxkÞ
ð9Þ
8t P t 0 and 8x belonging to the local space (or the whole space). A function V ðx; tÞ is locally (or globally) decrescent if and only if there exists a class-K function c2 such that V ð0; tÞ ¼ 0 and
V ðx; t Þ 6 c2 ðkxkÞ
ð10Þ
8t P t 0 and 8x belonging to the local space (or the whole space). 3. New lemmas related to the Caputo fractional derivative This section presents two new lemmas. The first one allows the use of general quadratic Lyapunov functions in the stability analysis of many FOS. The second lemma allows using the trace of a matrix into Lyapunov functions, also in order to prove stability of FOS. Besides, a theorem is presented, which allows proving uniform stability of FOS using Lyapunov functions. First of all, let us refer to the fractional extension of Lyapunov direct method. In the work reported in [15], sufficient conditions for proving the asymptotic stability of FOS were stated, using class-K functions. In the work reported in [16], sufficient conditions for proving Lyapunov stability are given. However, the corresponding sufficient conditions for proving Lyapunov uniform stability were not addressed in these works. There are many works proposing the analysis of uniform stability for fractional order systems using Lyapunov functions. It can be mentioned the case for fractional systems with time delay [20], for q-fractional systems [21], for discrete fractional systems [22], for neutral fractional systems [23] etc. However, we could not find results for systems of the form in (3) using Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Lyapunov functions and analyzing its Caputo fractional derivative. Since the conditions for proving uniform stability for these kind of systems using Lyapunov functions are available in many common situations, we present them in Theorem 3, and we also use it in Section 4 of this paper. Theorem 3 (Lyapunov Stability and Uniform stability of fractional order systems). Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a continuous Lyapunov function V ðxðtÞ; t Þ and a scalar class-K function c1 ð:Þ such that, 8x – 0
c1 ðkxðtÞkÞ 6 V ðxðtÞ; tÞ
ð11Þ
C b t0 Dt V ðxðt Þ; t Þ
ð12Þ
and
6 0;
with b 2 ð0; 1
then the origin of the system (3) is Lyapunov stable. If, furthermore, there is a scalar class-K function c2 ð:Þ satisfying
V ðxðt Þ; t Þ 6 c2 ðkxðt ÞkÞ
ð13Þ
then the origin of the system (3) is Lyapunov uniformly stable. Proof. The proof of this theorem follows the same outline than in the proof for the case of integer order systems in [24], with some modifications to treat the fractional case. The first part of the proof is not a new result, since it was done in [16]. Nevertheless, we put it here in order to present a more complete result and to show a more detailed proof. Let us prove Lyapunov stability first. To establish Lyapunov stability, we must prove that if (11) and (12) are satisfied, then for any e > 0, there exists a dðe; t0 Þ such that
kxðt0 Þk < d
)
kxðtÞk < e;
8t P t 0
ð14Þ
Given that condition (12) holds, we can use Lemma 2 (Fractional Comparison Principle) obtaining that
V ðxðt Þ; t Þ 6 V ðxðt0 Þ; t0 Þ;
8t P t 0
ð15Þ
Using expressions (11) and (15) it can be written that
c1 ðkxkÞ 6 V ðxðtÞ; tÞ 6 V ðxðt0 Þ; t0 Þ; 8t P t0
ð16Þ
Since V is continuous with respect to x and V ð0; t0 Þ ¼ 0 for being a Lyapunov function, then we can find d such that
kxðt0 Þk < d
)
V ðxðt 0 Þ; t 0 Þ < g ¼ c1 ðeÞ
ð17Þ
This means that if kxðt 0 Þk < d, then from expression (16) and (17) it holds that
c1 ðkxðtÞkÞ < c1 ðeÞ; 8t P t0
ð18Þ
Since c1 ð:Þ is nondecreasing for being a class-K function, expression (18) implies that
kxðtÞk < e;
8t P t 0
ð19Þ
Therefore, we have shown that expression (14) holds, and consequently the origin of the system (3) is Lyapunov stable if conditions (11) and (12) are fulfilled. Let us state now the proof of Lyapunov uniform stability, which is the contribution of this theorem. To establish Lyapunov uniform stability, we must prove that if (11)–(13) are satisfied, then for any e > 0, there exists a dðeÞ such that
kxðt0 Þk < d
)
kxðtÞk < e;
8t P t 0
ð20Þ
From expressions (11) and (13) it follows that
c1 ðkxðtÞkÞ 6 V ðxðtÞ; tÞ 6 c2 ðkxðtÞkÞ; 8t P t0
ð21Þ
Now for any e > 0, we can find dðeÞ > 0 such that c2 ðdÞ < c1 ðeÞ. Let the initial condition xðt 0 Þ be chosen such that kxðt0 Þk < d. Then using the result in expression (15) it can be written
c1 ðeÞ > c2 ðdÞ P V ðxðt0 Þ; t0 Þ P V ðxðtÞ; tÞ P c1 ðkxðtÞkÞ
ð22Þ
Given that c1 ð:Þ is non increasing, this implies that
8t P t0 ;
kxðt Þk < e
and the condition for uniformly stability holds, because in this case d is independent of t0 .
ð23Þ h
Remark 1. Given the relationship between class-K functions and positive definite functions and decrescent functions in Lemma 3, Theorem 3 can be presented as well in the following form. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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5
Let x ¼ 0 be an equilibrium point for the non autonomous fractional-order system (3). Let us assume that there exists a continuous function V ðxðtÞ; tÞ such that j
V ðxðtÞ; tÞ is positive definite. b C with b 2 ð0; 1Þ, is negative semidefinite. then the origin of system (3) is Lyapunov stable. Furthermore, if V ðxðtÞ; t Þ is decrescent, then the origin of system (3) is Lyapunov uniformly stable.
j t Dt V ðxðt Þ; t Þ; 0 j
In what follows, two new lemmas are presented, where some interesting relationships for the Caputo fractional derivative of special functions are derived. Those relationships will be useful for proving the uniform stability of FOS, using Theorem 3. Lemma 4. Let xðt Þ 2 Rn be a vector of differentiable functions. Then, for any time instant t P t0 , the following relationship holds
1 C a T T C a t 0 D x ðt ÞPxðt Þ 6 x ðt ÞP t 0 D xðt Þ; 2
8a 2 ð0; 1; 8t P t 0
ð24Þ
where P 2 Rnn is a constant, square, symmetric and positive definite matrix. Proof. Since matrix P is symmetric, using Theorem 2, there exist an orthogonal matrix B 2 Rnn and a diagonal matrix K 2 Rnn such that we can write T 1 T 1 1 T x ðt ÞPxðtÞ ¼ xT ðt ÞBKBT xðtÞ ¼ B xðt Þ K BT xðt Þ ð25Þ 2 2 2 Let us use an auxiliary variable yðtÞ defined as
yðt Þ ¼ BT xðtÞ
ð26Þ
Then from (25) it can be written
1 T 1 x ðt ÞPxðtÞ ¼ yT ðtÞKyðt Þ 2 2
ð27Þ
Since matrix K is diagonal, we can write n 1 T 1X y ðtÞKyðt Þ ¼ kii y2i ðt Þ 2 2 i¼1
ð28Þ
Then n n X 1 C aX 1 C a T 1 kii y2i ðt Þ ¼ kii Ct0 Da y2i ðt Þ 8a 2 ð0; 1Þ; 8t P t0 t 0 D y ðt ÞKyðt Þ ¼ t0 D 2 2 2 i¼1 i¼1
ð29Þ
Applying Lemma 1 to Eq. (29), and since kii > 0, results n X 1 C a T kii yi ðtÞCt0 Da yi ðt Þ; t 0 D y ðt ÞKyðt Þ 6 2 i¼1
8a 2 ð0; 1Þ; 8t P t 0
ð30Þ
Since kii are the diagonal elements of matrix K, then it can be written that n X kii yi ðtÞCt0 Da yi ðt Þ ¼ yT ðt ÞK Ct0 Da yðt Þ
ð31Þ
i¼1
and from (30) we can state that
1 C a T D y ðtÞKyðt Þ 6 yT ðt ÞK Ct0 Da yðt Þ; 2 t0
8a 2 ð0; 1Þ; 8t P t 0
ð32Þ
Then replacing expression (26) in (32) we can write that
T T 1 C a T B xðtÞ K BT xðt Þ 6 BT xðtÞ K Ct0 Da BT xðt Þ ; t0 D 2
8a 2 ð0; 1Þ; 8t P t 0
ð33Þ
Rearranging and using BKBT ¼ P in (33) results
1 C a T D x ðt ÞPxðtÞ 6 xT ðt ÞP Ct0 Da xðt Þ; 2 t0
8a 2 ð0; 1Þ; 8t P t0
ð34Þ
and the proof is complete for the case a 2 ð0; 1Þ. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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The case a ¼ 1, corresponds to the well known chain rule for integer order derivatives, where d ¼ xT ðtÞP dt xðtÞ. Therefore the proof is now complete. h
1 d 2 dt
T x ðtÞPxðt Þ
Regarding the trace of the product of two matrices, the following lemma allows concluding about its Caputo fractional derivative, when the product has a particular form. Specifically, when it is the product of a rectangular time varying matrix and its transpose. Lemma 5. Let Aðt Þ 2 Rmn be a time varying differentiable matrix. Then, for any time instant t P t 0 , the following relationship holds C a t0 D
h n oi n o tr AT ðtÞAðt Þ 6 2tr AT ðtÞCt0 Da Aðt Þ ;
8a 2 ð0; 1; 8t P t0
ð35Þ
Proof. From the matrix trace definition and the matrix product definition it directly follows that n n X m n o X X tr AT A ¼ AT A ¼ ATij Aji ii
i¼1
ð36Þ
i¼1 j¼1
Since ATij ¼ Aji , then expression (36) can be written as n X m n o X tr AT A ¼ A2ij
ð37Þ
i¼1 j¼1
Let us write expression (37) for the case of matrix Aðt Þ in the form
n o X tr AT ðtÞAðt Þ ¼ a2ij ðtÞ
ð38Þ
i;j
where aij ðt Þ are the elements of the matrix AðtÞ. Applying the Caputo fractional derivative to expression (38) we get C a t0 D
" # h n oi X T C a 2 tr A ðtÞAðt Þ ¼ t0 D aij ðtÞ
ð39Þ
i;j
Since the Caputo fractional derivative has the property of linearity, expression (39) can be written as C a t0 D
" # h n oi i X X ah T C C a 2 2 tr A ðtÞAðt Þ ¼ t0 D aij ðtÞ ¼ t 0 D aij ðt Þ i;j
ð40Þ
i;j
Applying Lemma 1 to expression (40), the following inequality arises C a t0 D
h n oi X tr AT ðtÞAðt Þ 6 2 aij ðtÞCt0 Da aij ðt Þ
ð41Þ
i;j
Using the fact that
n o X aij ðt ÞCt0 Da aij ðt Þ ¼ tr AT ðt ÞCt0 Da Aðt Þ
ð42Þ
i;j
expression (41) can be written in the form C a t0 D
h n oi n o tr AT ðtÞAðt Þ 6 2tr AT ðtÞCt0 Da Aðt Þ
and the proof is complete for the case a 2 ð0; 1Þ. case oi a ¼ 1 n corresponds to the well known result h The n o d d tr AT ðt ÞAðtÞ ¼ 2tr AT ðt Þ dt AðtÞ . Therefore the proof is now complete. dt
ð43Þ
for h
integer
order
derivatives,
where
4. Using Theorem 3 and Lemmas 4 and 5 in the stability analysis of fractional order systems One of the most used Lyapunov function candidate to prove the stability and convergence of integer systems is the quadratic function xT ðt Þxðt Þ. However, in the fractional case, it is no straightforward the use of those functions, as was stated in [17]. Moreover, sometimes those simple quadratic functions are not enough to conclude about the stability of FOS, and general quadratic Lyapunov functions xT ðt ÞPxðt Þ are required instead. Besides, in many cases the use of the trace of a product of two matrices (in the form of Lemma 5) is required when dealing with multivariable adaptive systems or when the state Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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vector of the systems is accessible. In what follows, two applications of Lemmas 4 and 5, and Theorem 3 are presented, to prove uniform stability of FOMRAC schemes. 4.1. Uniform stability of a FOMRAC scheme for plant with state accessible Let us consider a linear and time-invariant plant to be controlled, given by C a t 0 D t xp ð t Þ
¼ Ap xp ðtÞ þ Bp u
ð44Þ
nn
np
where Ap 2 R is an unknown constant matrix and Bp 2 R is a known constant matrix. The pair Ap ; Bp is controllable, the n-dimensional state xp ðt Þ is assumed to be accessible and u 2 Rp is the control input to be defined. Let a reference model be given by C a t 0 D t xm ð t Þ
¼ Am xm ðt Þ þ Bm r
ð45Þ
where Am 2 Rnn is a known Hurwitz constant matrix, Bm 2 Rnp is a known constant matrix and r 2 Rp is a uniformly bounded and piecewise-continuous reference input. It is assumed that xm ðt Þ, for all t P t 0 , represents the desired trajectory for xp ðtÞ. The aim here is to control the plant so that all the signals remain bounded and ideally lim kxp ðtÞ xm ðtÞk ¼ 0. t!1 Let us choose the control input as
uðtÞ ¼ HðtÞxp ðt Þ þ Q r ðtÞ where Hðt Þ 2 R
pn
ð46Þ
pp
is a matrix consisting of adjustable parameters and Q 2 R
is defined such that
Bp Q ¼ Bm
ð47Þ
pn
It is further assumed that a constant matrix H 2 R
exists such that
Ap þ Bp H ¼ Am
ð48Þ n
Defining the control error as eðt Þ ¼ xp ðt Þ xm ðt Þ 2 R , the fractional differential equation describing the evolution of the error can be expressed as C a t 0 Dt eðt Þ
¼ Am eðtÞ þ Bp UðtÞxp ðt Þ
ð49Þ
where Uðt Þ ¼ HðtÞ H 2 Rpn is the parameter error matrix. The aim now is to define an adaptive law for adjusting Hðt Þ in such a way that all the signals in the adaptive scheme remain bounded and ideally lim keðt Þk ¼ 0. Let us see what can be assured by using fractional extension of Lyapunov direct t!1 method. Let us propose the following Lyapunov function candidate
n o V ðe; UÞ ¼ eT ðt ÞPeðtÞ þ tr UT ðt ÞUðt Þ
ð50Þ
where P 2 Rnn is a constant symmetric positive-definite matrix and tr denotes the trace of a matrix. As can be seen, this Lyapunov function candidate is positive definite and decrescent. Calculating the Caputo fractional derivative of expression (50) with respect to time we get C a t 0 Dt V ðe;
h n
UÞ ¼ Ct0 Dat eT ðt ÞPeðtÞ þ Ct0 Dat tr UT ðt ÞUðtÞ
oi
ð51Þ
Applying Lemmas 4 and 5 to the right hand side of Eq. (51) results C a t 0 Dt V ðe;
n
UÞ 6 2eT ðt ÞP Ct0 Dat eðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ
o
ð52Þ
Substituting expression (49) in (52) and using properties of the transpose yields C a t 0 Dt V ðe;
h
i
n
UÞ 6 eT ðtÞ ATm P þ PAm eðtÞ þ þ2eT ðt ÞPBm Uðt Þxp ðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ
o
ð53Þ
Since Am is Hurwitz, then a symmetric positive definite matrix Q 0 exists such that
ATm P þ PAm ¼ Q 0
ð54Þ
and expression (53) can be written as C a t 0 Dt V ðe;
n
UÞ 6 eT ðt ÞQ 0 eðt Þ þ 2eT ðtÞPBm UðtÞxp ðt Þ þ 2tr UT ðt ÞCt0 Dat Uðt Þ
o
ð55Þ
Since Uðt Þ is unknown, it can’t be used in the adaptive law we are looking for. But we can choose the adaptive law such that
n o 2eT ðt ÞPBm Uðt Þxp ðtÞ þ 2tr UT ðtÞCt0 Dat UðtÞ ¼ 0
ð56Þ
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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The first term of the left hand side of expression (56) can be rewritten as
T T 2eT ðtÞPBm Uðt Þxp ðt Þ ¼ 2 eT ðtÞPBm UðtÞxp ðt Þ ¼ 2xTp ðt Þ eT ðt ÞPBm Uðt Þ ¼ 2xTp ðt ÞUT ðtÞBTm Peðt Þ Now using the property for the trace tr yxT ¼ xT y;
2xTp ðt Þ
T
U
ðtÞBTm Peðt Þ
n
T
¼ 2tr U
ðtÞBTm Peðt ÞxTp ðtÞ
o
ð57Þ
for any x; y 2 Rn , we can write that
ð58Þ
From (57) and (58) we can finally write
n o 2eT ðtÞPBm Uðt Þxp ðt Þ ¼ 2tr UT ðtÞBTm Peðt ÞxTp ðtÞ
ð59Þ
According to expression (59) and (56), it is straightforward to conclude that choosing the adaptive law as C a t 0 Dt
Uðt Þ ¼ Ct0 Dat HðtÞ ¼ BTm PeðtÞxTp ðt Þ
ð60Þ
then expression (56) holds. Thus, the Caputo fractional derivative with respect to time of the Lyapunov function (55), using the adaptive law (60), results C a t0 Dt V ðe;
UÞ 6 eT ðtÞQ 0 eðt Þ
ð61Þ
As can be seen, the Caputo fractional derivative with respect to time of the Lyapunov function is negative semidefinite. Using Theorem 3, together with the fact that V ðe; UÞ is positive definite and decrescent, we can conclude that the origin of the system (49) and (60) is Lyapunov uniformly stable. Remark 2. The fact that the matrix Am of the reference model is Hurwitz allows using expression (54) and concluding about the stability of the scheme. However, if the matrix Am has eigenvalues with positive real parts such that their arguments are greater than a2p (which is also a stable reference model from fractional viewpoint), the expression (54) does not holds, and consequently the stability can’t be proved using this methodology and the Lyapunov function (50). For comments in the control error convergence, see Remark 3. 4.2. Stability of a FOMRAC scheme for plants with state not accessible In the case of plants where the whole state is not accessible, but only the plant output is measured, more interesting problems arise when designing a FOMRAC. In that cases, some extra considerations must be done. Let us consider a fractional order single-input single-output (SISO) plant, with input–output pair uð:Þ; yp ð:Þ , defined by n o T T the triplet Ap ; bp ; hp ; Ap 2 Rnn ; bp ; hp 2 Rn . The plant is non-minimum phase and the sign of the high frequency gain of the plant kp is known. Let us consider a fractional order (SISO) reference model, asymptotically sable, with input–output pair fr ð:Þ; ym ð:Þg, n o T T defined by the triplet Am ; bm ; hm , where Am 2 Rnn ; bm ; hm 2 Rn ; r 2 R is a uniformly bounded and piecewise-continuous function of time and the high frequency gain of the reference model km is assumed to be positive. For this kind of problem, auxiliary signals must be constructed in order
to compute
a control signal u, to achieve that all the plant parameters and signals remain bounded, and ideally that lim yp ðtÞ ym ðt Þ ¼ 0. t!1 For more details of these type of solution see [25] for the equivalent integer order case. We omit here the design details of the FOMRAC, since we want to present the stability analysis of the scheme, in order to prove the usefulness of the results presented in this paper (Lemma 4 and Theorem 3 in this case). The differential equation describing the plant together with the controller for this problem can be represented as C a t0 Dt xðt Þ
¼ Amn xðt Þ þ bmn /T ðtÞxðt Þ þ k r ðtÞ ;
T
yp ¼ hmn xðtÞ
ð62Þ
where xðtÞ 2 R3n2 ; Amn 2 Rð3n2Þð3n2Þ ; bmn 2 R3n2 ; hmn 2 R3n2 ; /ðt Þ ¼ hðt Þ h 2 R2n corresponds to the parameter error, hðt Þ 2 R2n is the parameter vector to be adaptively adjusted, h 2 R2n is the unknown ideal controller parameter vector, k ¼ kkmp and xðtÞ 2 R2n corresponds to auxiliary signals of the scheme, obtained by filtering the plant input and output. When hðtÞ h , it follows that the reference model can be described by the ð3n 2Þth order fractional differential equation C a t0 Dt xmn ðt Þ
¼ Amn xmn ðt Þ þ bmn k r ðtÞ;
T
ym ¼ hmn xmn ðtÞ
ð63Þ
3n2
where xmn ðt Þ 2 R . Therefore, the error equation for the overall system may be expressed as: C a t 0 Dt eðt Þ T
¼ Amn eðtÞ þ bmn /T ðt Þxðt Þ
e1 ¼ hmn eðt Þ
ð64Þ
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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where eðt Þ ¼ xðtÞ xmn ðt Þ is not accessible and e1 ðtÞ ¼ yp ðt Þ ym ðtÞ 2 R corresponds to the output error. Moreover, in this configuration T
kp hmn ½sI Amn 1 bmn
ð65Þ
is a SPR function. In order to find the adaptive laws for the FOMRAC and proving the stability of the scheme, let us propose the following Lyapunov function candidate, which is positive definite and decrescent
1 V ðe; /Þ ¼ eT ðtÞPeðtÞ þ
/T ðt Þ/ðt Þ kp
ð66Þ
where P 2 Rð3n2Þð3n2Þ is a constant symmetric and positive definite matrix. Taking the fractional derivative to expression (66) and applying Lemma 4 and Lemma 1, it follows that C a t 0 Dt V ðe; /Þ
2 6 2eT ðt ÞP Ct0 Dat eðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp
ð67Þ
Using expression (64) in (67) we obtain C a t 0 Dt V ðe; /Þ
2 6 2eT ðt ÞP Amn eðt Þ þ bmn /T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 ¼ 2eT ðtÞPAmn eðtÞ þ 2eT ðtÞPbmn /T ðtÞxðt Þ þ
/T ðtÞCt0 Dat /ðtÞ kp h i 2 ¼ eT ðt Þ ATmn P þ PAmn eðt Þ þ 2eT ðtÞPbmn /T ðt Þxðt Þ þ
/T ðtÞCt0 Dat /ðtÞ kp
ð68Þ
Since the transfer function (65) is SPR, then using the Meyer–Kalman–Yakubovich (MKY) Lemma [25], we can assure that given a matrix Q ¼ Q T > 0, there exists a P ¼ PT > 0 such that
ATmn P þ PAmn ¼ Q
ð69Þ
Pbmn ¼ hmn kp Using this fact in (68) it follows that C a t 0 Dt V ðe; /Þ
2 6 eT ðtÞQeðtÞ þ 2eT ðt Þhmn kp /T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 T ¼ eT ðt ÞQeðt Þ þ 2kp hmn eðtÞ/T ðt ÞxðtÞ þ
/T ðt ÞCt0 Dat /ðt Þ kp 2 ¼ eT ðt ÞQeðt Þ þ 2kp e1 ðt Þ/T ðtÞxðt Þ þ
/T ðt ÞCt0 Dat /ðt Þ kp
ð70Þ
If we choose the adaptive law as C a t 0 Dt /ðt Þ
¼ Ct0 Dat hðt Þ ¼ sgn kp e1 ðt ÞxðtÞ
ð71Þ
then the Caputo fractional derivative of the Lyapunov function turns out to be C a t 0 Dt V ðe; /Þ
6 eT ðtÞQeðtÞ þ 2kp e1 ðt Þ/T ðt Þxðt Þ 2
sgn kp T
/ ðtÞe1 ðt Þxðt Þ ¼ eT ðt ÞQeðt Þ
kp
ð72Þ
Therefore, the Caputo fractional derivative of the Lyapunov function is negative semidefinite. Thus, using Theorem 3 together with the fact that V ðe; /Þ is positive definite and decrescent, we can conclude that the origin of the system (64) and (71) is uniformly stable in the Lyapunov sense and all the signals remain bounded. Remark 3. The two analysis presented so far allow proving uniform stability in the Lyapunov sense for FOMRAC schemes. However, no conclusions about the convergence of the errors to zero can be drawn from this analysis. In the integer order case, Barbalat’s Lemma is used to prove it, but this analysis cannot be directly used in the fractional order case. Thus, additional tools need to be developed, in order to prove the errors convergence in the fractional order case. Currently, efforts are being made to develop these tools.
5. Conclusions Two new lemmas for the Caputo fractional derivatives of general quadratic forms and for the trace of the product of a rectangular matrix and its transpose, when a 2 ð0; 1, have been presented in this paper. Both results allow using the fractional extension of Lyapunov direct method for the stability analysis of fractional order systems. Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008
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Also, a theorem providing conditions for uniform stability in the Lyapunov sense of FOS has been presented in this paper. The theorem can be seen as a complement of the fractional extension of Lyapunov direct method proposed in [15] and the method for proving Lyapunov stability presented in [16]. The usefulness of the two lemmas and the theorem was stated in the paper through the analysis of two FOMRAC schemes, where the Lyapunov uniform stability was proved, using Lyapunov functions containing general quadratic forms and the trace of the product of a matrix and its transpose. Acknowledgments This work has been supported by CONICYT-Chile, under the grants FB009 ‘‘Centro de Tecnología para la Minería’’, FONDECYT 1120453, ‘‘Improvements of Adaptive Systems Performance by using Fractional Order Observers and Particle Swarm Optimization’’; and CONICYT-Chile under the grants ‘‘CONICYT-PCHA/National PhD scholarship program, 2013-21130004’’. References [1] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. Elsevier; 2006. [2] Sabatier J, Aoun M, Oustaloup A, Gréegoire G, Ragot F, Roy P. Fractional system identification for lead acid battery state of charge estimation. Signal Process 2006;86:2645–57. [3] Gabano J-D, Poinot T. Fractional modelling and identification of thermal systems. Signal Process 2011;91:531–41. [4] Lin J, Poinot T, Trigeassou JC, Kabbaj H, Faucher J, Modélisation et identification d’ordre non entire d’une machine asynchrone. In: Conférence Internationale Francophone d’Automatique; 2000. [5] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Eng Pract 2009;17:1380–7. [6] Ladaci S, Loiseau JJ, Charef A. Fractional order adaptive high-gain controllers for a class of linear systems. Commun Nonlinear Sci Numer Simul 2008;13:707–14. [7] El Figuigui O, Elalami N, Application of fractional adaptive High-Gain controller to a LEO (Low Earth Orbit) satellite. In: International conference on computers industrial engineering; 2009. p. 1850–56. [8] YaLi H, RuiKun G, Application of fractional-order model reference adaptive control on industry boiler burning system. In: International conference on intelligent computation technology and automation (ICICTA) 2010, vol. 1. p. 750–53. [9] Duarte-Mermoud MA, Aguila-Camacho N, Fractional order adaptive control of simple systems. In: Narendra K, editor. Fifteenth yale workshop on adaptive and learning systems; 2011. p. 57–62. [10] Duarte-Mermoud MA, Aguila-Camacho N, Some useful results in fractional adaptive control. In: Narendra K, editor. Sixteenth yale workshop on adaptive and learning systems; 2013. p. 51–6. [11] Aguila-Camacho N, Duarte-Mermoud MA. Fractional adaptive control for an automatic voltage regulator. ISA Trans 2013;52:807–15. [12] Ladaci S, Charef A. On fractional adaptive control. Nonlinear Dyn 2006;43:365–78. [13] Vinagre BM, Petrás I, Podlubny I, Chen Y. Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn 2002;29:269–79. [14] Suárez JI, Vinagre BM, Chen Y. A fractional adaptation scheme for lateral control of an AGV. J Vib Control 2008;14:1499–511. [15] Li Y, Chen Y, Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability. Comput Math Appl 2010;59:1810–21. [16] Zhang F, Li Ch, Chen YQ. Asymptotical stability of nonlinear differential systems with Caputo derivative. Int J Differ Equ 2011;2011. [17] Aguila-Camacho N, Duarte-Mermoud MA, Gallegos J. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 2014;19:2951–7. [18] Horn RA, Johnson ChR. Matrix analysis. Cambridge University Press; 1995. [19] Bellman R. Introduction to matrix analysis. Society for Industrial and Applied Mathematics (SIAM); 1997. [20] Wang Y, Li T. Stability analysis of fractional-order nonlinear systems with delay. Math Prob Eng 2014;2014. [21] Jarad F, Abdeljawad T, Baleanu D. Stability of q-fractional non-autonomous systems. Nonlinear Anal Real World Appl 2013;14:780–4. [22] Jarad F, Abdeljawad T, Baleanu D, Bicen K. On the stability of some discrete non autonomous systems. Abstr Appl Anal 2012;2012:780–4. [23] Liu K, Jiang W. Uniform stability of fractional neutral systems: a Lyapunov–Krasovskii functional approach. Adv Difference Equ 2013;2013. [24] Slotine JJ, Li W. Applied nonlinear control. Prentice Hall; 1991. [25] Narendra KS, Annaswamy AM. Stable adaptive systems. Dover Publications Inc.; 2005.
Please cite this article in press as: Duarte-Mermoud MA et al. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.10.008