A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay

A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay

Recommended by

Prof. T Parisini

Journal Pre-proof

A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay V.N. Phat, P. Niamsup, M.V. Thuan PII: DOI: Reference:

S0947-3580(19)30459-5 https://doi.org/10.1016/j.ejcon.2020.02.005 EJCON 420

To appear in:

European Journal of Control

Received date: Revised date: Accepted date:

9 October 2019 13 February 2020 19 February 2020

Please cite this article as: V.N. Phat, P. Niamsup, M.V. Thuan, A new design method for observerbased control of nonlinear fractional-order systems with time-variable delay, European Journal of Control (2020), doi: https://doi.org/10.1016/j.ejcon.2020.02.005

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 European Control Association. Published by Elsevier Ltd. All rights reserved.

A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay V.N. Phata , P. Niamsupb , M.V. Thuanc a

ICRTM, Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

b Department c Department

of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

of Mathematics and Informatics, Thai Nguyen University of Sciences, Thai Nguyen, Vietnam

Abstract In this paper, an LMI-based design is proposed for observer control problem of nonlinear fractionalorder systems subject to time-variable delay, where the delay function is non-differentiable, but continuous and bounded. Our novel technique is based on a new lemma concerning Caputo derivative estimation of quadratic functions. In this proposed approach, delay-dependent sufficient conditions in terms of linear matrix inequalities are obtained for the design state feedback controller and observer gains. A simulation-based example is given to illustrate the effectiveness of the theoretical result. Keywords: Fractional calculus, Observer control, Nonlinear perturbation, Stabilization, Razumikhin theorem, Time-variable delay, Linear matrix inequalities 1. Introduction The study of fractional-order systems (FOSs), which is one of the recently focused research topics, has drawn attention of researchers in view of applications, such as heat conduction electronic and abnormal diffusion [1-9]. The main reason for successful applications of fractional calculus is that these new fractional-order models are more accurate than integer-order models. It is worth noting that fractional-order derivatives are nonlocal and have weakly singular kernels, the stability analysis of FOSs is more complex than that of integer-order systems. Specially, the FOSs with delays can characterize a class of chaotic behaviors [11-12]. Generally, the time-delay variables included in the dynamics of FOSs are due to transportation of material, energy, or information. Moreover, the presence of time-delays, also called dead-time or after-effect, can cause the plant instability. In fact, problems of stability and observer control for FOS with delays have attracted the attention of many researchers, see, e.g. [13-20], also [21-24] for related topics concerning the design of state estimators. Note that LMI-based design method presents sufficient conditions for designing observer controllers in terms of linear matrix inequalities (LMIs), which can be numerically solved by convex programming problem. For linear FOSs, sufficient conditions for designing full-order observer controllers have been proposed in [25-29] by Lyapunov function method and LMI technique. Email address: Corresponding author:

[email protected] (V.N. Phat)

Preprint submitted to European Journal of Control

Compared with the study of linear FOSs, the stability and control of nonlinear FOSs is more difficult because of the complexity of their modeling and control structure. The observer-based control design for nonlinear FOSs, using Laplace transform, Mittag-Leffler function and BellmanGronwall approach was proposed in [30, 31]. In particular, the problem of non-fragile observerbased robust control for uncertain FOSs was solved in [32]. Note that all the above works concerned only the case of no delays. Very recently, nonlinear FOSs with constant delay have been investigated in [33,34]. In practice, we are not only interested in observer control problem of FOSs with constant delay, but also in the problem of the systems with time-variable delay [18, 27, 35]. Moreover, since the unknown time-variable delay in the full-order observer system can make it un-realizable to evaluate the observer error states. To the best of our knowledge, the observer control problem for FOSs with time-variable delays has not been fully investigated in the literature. In this setting, by using full-order memory state observer the authors of [36- 39] proposed sufficient conditions for observer-based control. However, there are some gaps in the proof of their main results, since the observer structure is not practical if the time-variable delay appeared in the full-order observer system such that we can not evaluate the unknown time-variable delay in the observer. To overcome these gaps the authors of [40] used memoryless state observers to construct state feedback and observer controllers, but for integer-order systems with time-variable delays. Motivated by the above discussions, this present paper focuses on observer-based control analysis of nonlinear FOSs with time-variable delays. Our main contributions are reflected in the following aspects: (i) The consideration of bounded and continuous time-variable delay, but not necessarily differentiable. (ii) Providing a new lemma on Caputo derivative of quadratic function xT (t)Px(t), where x(t) is only continuous. (iii) Providing a novel LMI design method for state feedback controller and observer gains. (iv) Applying our theoretical results to observer-based control of linear uncertain FOSs subject to time-variable delays. This paper is outlined as follows. Preliminaries, problem statement and some auxiliary lemmas are given in the Section 2. The third section presents main result on feedback controller and observer design, application to linear uncertain FOSs and a numerical example. Notations. Throughout this paper we note by Rn the n−dimensional linear vector space with the Euclidean norm k.k; by xT y or hx, yi the scalar product of x, y; by Rn×m the space of n × m matrices; by λmax (A) and λmin (A) the maximal and the minimal eigenvalue of A, respectively. A = [ai j ]i, j=1,n denotes the matrix A of ai, j elements. The norm of a real matrix A is defined by p kAk = λmax (AT A). C[0, T ] denotes the space of continuous functions on [0, T ]. L p [0, T ] denotes the space of all p− integrable functions on [0, T ]. The segment of the trajectory x(t) is denoted by xt = {x(t + s) : s ∈ [−τ, 0]} with its norm kxt k = sup kx(t + s)k. A matrix P is positive definite

s∈[−τ,0] T (P > 0) if x Px > 0, ∀x 6= 0; P > Q means P − Q > 0. The symmetric term in a matrix is denoted by

∗. H α [0, T ], α ∈ (0, 1), denotes the standard Holder space of continuous functions x(t) on [0, T ] such that |x(t) − x(s)| |x|H α := max |x(t)| + sup < ∞; (t − s)α t∈[0,T ] 0≤s
and H0α [0, T ], α ∈ (0, 1) denotes the subset of H α [0, T ], consisting of functions x(t) such that |x(t) − x(s)| → 0, as ε → 0. α 0≤s
2. Preliminaries and auxiliary lemmas In this section, we present some basic concepts of fractional calculus introduced in [1, 41]. For 0 < α < 1, f ∈ C[0, T ] the Riemann-Liouville integral J α f (t), and the Riemann-Liouville derivative DαR f (t) are defined respectively by Z

t 1 f (τ) dτ, J f (t) = Γ(α) 0 (t − τ)1−α d DαR f (t) = J 1−α f (t), dt α

R∞

where Γ(s) := e−t t s−1 dt, s > 0 is the Gamma function. The Caputo fractional derivative DCα f (t) 0

is defined via the Riemann-Liouville derivatives as DCα f (t) = DαR [ f (t) − f (0)]. Lemma 1. ([1, 6]) For any continuous and integrable functions x(t), y(t) on every finite interval [0,t], λ1 > 0, λ2 > 0, we have the following relation DCα [λ1 x(t) + λ2 y(t)] = λ1 DCα x(t) + λ2 DCα y(t). Lemma 2. ([41], Theorem 5.2, p. 475) For 0 < α < 1, T > 0, and v(t) ∈ C[0, T ], the following conditions are equivalent: (i)

The fractional derivative DCα v ∈ C[0, T ] exists; v(t) − v(0) = γ, exists and t→0 tα

(ii) A finite limit lim

Zt v(t) − v(s) ds → 0 when ξ → 1− ; sup 1+α (t − s) 0
(iii)

Rt

v(t) has the structure v(t)−v(0) = γt α +v0 , v0 ∈ H0α [0, T ], and (t −s)−α−1 (v(t)−v(s))ds =: 0

w(t) converges for every t ∈ (0, T ] defining a function w ∈ C(0, T ] which has a finite limit lim w(t) := w(0).

t→0

For v(t) ∈ C[0, T ] with DCα v ∈ C[0, T ], we have (DCα v)(0) = Γ(α + 1)γ and (DCα v)(t) =

v(t) − v(0) α + α Γ(1 − α)t Γ(1 − α) 3

Zt 0

v(t) − v(s) ds, (t − s)1+α

0 < t ≤ T.

Now, we present several technical lemmas for deriving the main results in this paper. First, inspired by the result obtained in [42] we introduce the following lemma on the Caputo derivative of some quadratic functions. Lemma 3. Let z(t) be a continuous function, z(t) = z(0) + J α v(t), v ∈ C[0, T ], P = PT > 0, |z|2P := zT Pz. Then, the function DCα (z(t)T Pz(t)) is continuous and (i)

D Pv(0) E |z(t)|2P − |z(0)|2P = 2 z(0), . t→0 tα Γ(α + 1) lim

(ii)

D E DCα (|z(t)|2P )(0) = 2 z(0), Pv(0) .

(iii) DCα (|z(t)|2P ) = (iv)

|z(t)|2P − |z(0)|2P α + α t Γ(1 − α) Γ(1 − α)

Zt 0

DCα [z(t)T Pz(t)] ≤ 2z(t)T PDCα z(t),

|z(t)|2P − |z(s)|2P ds. (t − s)α+1 t ≥ 0.

Proof. Using Lemma 2, function DCα z is continuous, we get Rt

(t − s)α−1 (v(s) − v(0))ds z(t) − z(0) v(0) J α v(t) J α v(0) 0 − = α − = α α α t Γ(α + 1) t t t Γ(α) Rt

α−1 ds (t − s) 1 0 ≤ sup v(s) − v(0) = sup v(s) − v(0) → 0, α Γ(α) t Γ(α + 1) s∈[0,t] s∈[0,t]

as t → 0, and hence

z(t) − z(0) v(0) = , α t→0 t Γ(α + 1) |z(t)|2P − |z(0)|2P lim t→0 t αD E D E z(t) − z(0), Pz(t) + z(0), P[z(t) − z(0)] = lim t→0 tα D z(t) − z(0) E D z(t) − z(0) E = lim , Pz(t) + z(0), P t→0 tα tα = 2(z(0), Pγ),

γ := lim

4

which shows (i). Moreover, Using some simple calculations we get Z t |z(t)|2P − |z(s)|2P ξt

(t − s)α+1

ds =

Z t hz(t) − z(s), 2Pz(t)i

ds (t − s)α+1 Z t hz(t) − z(s), P[z(t) − z(s)]i − ds (t − s)α+1 ξt :=I1 (t, ξ ) − I2 (t, ξ ), ξt

we derive from Lemma 2 that D Zt E (t − τ)−α−1 (z(t) − z(τ))dτ, 2Pz(t) |I1 (t, ξ )| = ξt Zt

≤

ξt

(t − τ)−α−1 (z(t) − z(τ))dτ 2|Pz(t)|

Zt −α−1 ≤ sup (t − τ) (z(t) − z(τ))dτ 2 sup |Pz(t)|, 0
t∈[0,T ]

ξt

which gives I1 (t, ξ ) → 0 when ξ → 1− . We also have

z = z(0) + γt −α + z0 , z0 ∈ H0α [0, T ], t ∈ (0, T ], which implies z(t) − z(s) t α − τ α z (t) − z (s) 0 0 ≤ γ + α α α (t − s) (t − s) (t − s) (t − s)αcα−1 z0 (t) − z0 (s) + ≤γ , (t − s)α (t − s)α ≤γα[1/ξ − 1]1−α + =h(ξ ),

z (t) − z (s) 0 0 α (t − s) 0≤s
for ξ t ≤ s < t ≤ T, ξ ∈ (0, 1], c ∈ (s,t). Then we have |I2 (t, ξ )| = ≤ =

Zt

hz(t) − z(s), P[z(t) − z(s)]i ds (t − s)α+1

ξt

(t − s)2α dskPkh2 (ξ ) = (t − s)α+1

ξt Zt

(t − tξ )α α

kPkh(ξ )2 ≤ 5

Zt

(t − s)α−1 ds kPkh2 (ξ )

ξt α T (1 − ξ )α

α

kPkh2 (ξ ).

Note that function h(ξ ) is independent on s,t and goes to 0 when ξ → 1− because of |t − s| ≤ |t − tξ | ≤ T (1 − ξ ), c ≥ s ≥ ξ t, and x0 ∈ H0α [0, T ], hence limξ →1− |I2 (t, ξ )| → 0. Therefore, we obtain that Zt sup (t − s)−α−1 (|z(t)|2P − |z(s)|2P )ds → 0, when ξ → 1− , 0
ξt

which shows the existence of DCα |z(t)|2P ∈ C[0, T ] due to conditions (i) and (ii) of Lemma 2. On the other hand, since |z(t)|2P , DCα |z(t)|2P ∈ C[0, T ], applying Lemma 2 again we obtain the desired conditions (ii) and (iii). To prove (iv) we use the derived conditions (ii) -(iii) to get the following z(t) − z(0) v(0) = v(0), = Γ(α + 1) α t→0 t Γ(α + 1) Zt  z(t) − z(0) 1 α z(t) − z(s)  α (DC z)(t) = + ds . Γ(1 − α) tα Γ(1 − α) (t − s)1+α

(DCα z)(0) =Γ(α + 1) lim

0

Combining the above equalities and the formula (iii) of DCα (|z(t)|2P ), gives for t = 0 : DCα (|z(t)|2P ) − 2hz(t), PDCα z(t)i =DCα (|z(t)|2P ) − 2hz(t), Pv(t)i, =2hz(0), Pv(0)i − 2hz(0), Pv(0)i = 0, and for t ∈ (0, T ] : DCα (|z(t)|2P ) − 2hz(t), PDCα x(t)i hz(t), Pz(t)i − hx(0), Px(0)i α = + α t Γ(1 − α) Γ(1 − α) D − 2 z(t), P =−

z(t) − z(0) 1 [ +α Γ(1 − α) tα

α |z(t) − z(0)|2P − α t Γ(1 − α) Γ(1 − α)

Zt 0

Zt 0

Zt 0

hz(t), Pz(t)i − hz(s), Pz(s)i ds (t − s)α+1

z(t) − z(s) E ds] (t − s)1+α

|z(t) − z(τ)|2P dτ ≤ 0. (t − τ)α+1

The proof is completed. Lemma 4. (Schur complement lemma [43]) Given constant matrices X,Y, Z with appropriate dimensions satisfying Y = Y T > 0, X = X T , we have   X ZT T −1 < 0. X + Z Y Z < 0 ⇐⇒ Z −Y

6

3. Observer-based control design In this section, we study the design problem of observer-based control for following nonlinear FOSs with time-variable delay  α  DC x(t) = Ax(t) + Ad x(t − h(t)) + f (t, x(t), x(t − h(t))) + Bu(t), t ≥ 0, (1) y(t) = Cx(t) + g(t, x(t)), t ≥ 0,   x(t) = φ (t), t ∈ [−h, 0]

where DCα is the Caputo derivative, α ∈ (0, 1), x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, y(t) ∈ R p is the measure output; A, Ad ∈ Rn×n , B ∈ Rn×m are given constant matrices; φ (t) is the continuous initial condition Assumption 1. The function h(t) is continuous and satisfies 0 < h(t) ≤ h,

t ≥ 0.

(2)

Assumption 2. Nonlinear functions f (t, x, y) : R+ × Rn × Rn → Rn and g(t, x) : R+ × Rn → R p , f (t, 0, 0) = 0, g(t, 0) = 0 satisfy k f (t, x1 , y1 ) − f (t, x2 , y2 )k2 ≤(x1 − x2 )T E1T E1 (x1 − x2 ) + (y1 − y2 )T E2T E2 (y1 − y2 ), kg(t, x1 ) − g(t, x2 )k2 ≤(x1 − x2 )T GT G(x1 − x2 ),

(3)

for all x1 , x2 , y1 , y2 ∈ Rn , where G, Ei , i = 1, 2 are given constant matrices of appropriate dimensions. Proposition 1. ([9]) Assume that the initial function ϕ(t) ∈ C([−h, 0], Rn ), then the FOSs (1) under Assumption 1 and Assumption 2 has a unique solution. Now, we consider the following memoryless full-order observer system for system (1): ( DCα x(t) ˆ = Ax(t) ˆ + Bu(t) + L[y(t) − (Cx(t) ˆ + g(t, x(t))], ˆ x(t) ˆ = 0, t ∈ [−h, 0]

(4)

and the controller u(t) = K x(t), ˆ where x(t) ˆ ∈ Rn is the observer, and K and L are the feedback and observer gain control matrices with appropriate dimensions. Defining error state e(t) = x(t) − x(t), ˆ we consider the following error closed-loop system  α  DC e(t) = (A − LC)e(t) + Ad x(t − h(t)) − L[g(t, x(t)) − g(t, x(t) − e(t))] (5) + f (t, x(t), x(t − h(t))),   α DC x(t) = (A + BK)x(t) + Ad x(t − h(t)) − BKz(t) + f (t, x(t), x(t − h(t))).

The objective is to design the feedback controller K and the observer gain L such that that the system (5) is asymptotically stable.

The proof of our main result is based on the following fractional stability theorem for FOSs with delays. 7

Proposition 2. (Fractional Razumikhin theorem [9]) Assume that u, v, w : R+ → R+ are continuous and nondecreasing, u(0) = v(0) = w(0) = 0, v(.) is strictly increasing. If there exist a number q > 1 and a continuous function V (.) : R+ × Rn → R+ such that: (i) u(kxk) ≤ V (t, x) ≤ v(kxk),t ≥ 0, x ∈ Rn and

(ii) DCα V (t, x(t)) ≤ −w(kx(t)k) if V (t + s, x(t + s)) < qV (t, x(t)), ∀s ∈ [−h, 0],t ≥ 0,

then the zero solution of functional fractional-order system DCα x(t) = f (t, xt ) is asymptotically stable. We are in position to present sufficient conditions for designing observer controllers for systems (1). Note that the error closed-loop system (5) can be described in the vector form: DCα z(t) = A z(t) + Ad z(t − h(t)) + F (t, z(t)),

(6)

where z(t) = [eT (t), xT (t)]T , and       A − LC 0 −Lg(t) ˆ + fˆ(t) 0 Ad A = , Ad = , F (t, v) = , −BK A + BK 0 Ad fˆ(t) g(t) ˆ =g(t, x(t)) − g(t, x(t) − e(t)),

f (t) = f (t, x(t), x(t − h(t))).

Let us denote   Z 0 P= , 0 P−1

Ψ11 = ZA + AT Z − XC −CT X T + GT G + I + Z,

Ξ11 = AP + PAT + BY +Y T BT + I + P.

Theorem 5. Under the assumptions 1-2, the observer error closed-loop system (5) is asymptotically stable if there exist symmetric positive definite matrices P, Q, Z, matrices X,Y such that the following LMIs hold: hQ < Z,  −P PE2T  ∗ −1I 2 ∗ ∗  Ψ11 0  ∗ −hQ   ∗ ∗   ∗ ∗ ∗ ∗  Ξ11 PE1T  ∗ −1I  2  ∗ ∗ ∗ ∗



(1 + h)P  < 0, 0 −(1 + h)I  ZAd X Z 0 0 0  −I 0 0   < 0, ∗ −I 0  ∗ ∗ −I  BY Ad 0 0   < 0. I − 2P 0  ∗ −hI

Moreover, the feedback control matrix is K = Y P−1 , the observer gain matrix is L = Z −1 X. 8

(7a) (7b)

(7c)

(7d)

Proof. Consider the Lyapunov function V (t, z(t)) for system (6) given by V (t, z(t)) = zT (t)Pz(t) = eT (t)Ze(t) + xT (t)P−1 x(t). It is easy to verify that λmin (P)kz(t)k2 ≤ V (t, z(t)) ≤ λmax (P)kz(t)k2 .

(8)

Note that V (.) is convex, differentiable function on Rn and V (0) = 0. Therefore, using Lemma 1 and Lemma 3 (iv), the fractional-order of α derivative of V (.) is estimated as follows. DCα V (t, z(t)) ≤ 2eT (t)Z[(A − LC)e(t) + Ad x(t − h(t)) − Lg(t) ˆ + fˆ(t)]

+ 2xT (t)P−1 [(A + BK)x(t) + Ad x(t − h(t)) − BKe(t) + fˆ(t)]

= eT (t)[ZA + AT Z − ZLC −CT LT Z T ]e(t) + 2eT (t)ZAd x(t − h(t)) − 2eT (t)ZLg(t) ˆ + 2eT (t)Z fˆ(t)

(9)

+ xT (t)[P−1 A + AT P−1 + P−1 BK + K T BT P−1 ]x(t)

+ 2xT (t)P−1 Ad x(t − h(t)) − 2xT (t)P−1 BKe(t) + 2xT (t)P−1 fˆ(t). Using condition (3) and Cauchy matrix inequality for the following estimations 2eT (t)ZAd x(t − h(t)) ≤ eT (t)ZAd ATd Ze(t) + xT (t − h(t))x(t − h(t)),

2 −2eT (t)ZLg(t) ˆ ≤ eT (t)ZLLT Ze(t) + kg(t)k ˆ

≤ eT (t)ZLLT Ze(t) + eT (t)GT Ge(t), 2eT (t)Z fˆ(t) ≤ eT (t)ZZe(t) + k fˆ(t)k2

≤ eT (t)ZZe(t) + xT (t)E1T E1 x(t) + xT (t − h(t))E2T E2 x(t − h(t)),

− 2xT (t)P−1 BKe(t) ≤ xT (t)P−1 BKK T BT P−1 x(t) + eT (t)e(t), 2xT (t)P−1 fˆ(t) ≤ xT (t)P−1 P−1 x(t) + k fˆ(t)k2

≤ xT (t)P−1 P−1 x(t) + xT (t)E1T E1 x(t) + xT (t − h(t))E2T E2 x(t − h(t)) = xT (t)[P−1 P−1 + E1T E1 ]x(t) + xT (t − h(t))E2T E2 x(t − h(t)),

we obtain that DCα V (t, z(t)) ≤ η T (t)M η(t) + ξ T (t)N ξ (t) + heT (t − h(t))Qe(t − h(t)) + xT (t − h(t))[2E2T E2 + (1 + h)I]x(t − h(t)),

where 

   e(t) x(t) η(t) = , ξ (t) = , e(t − h(t)) x(t − h(t))     M11 0 N11 P−1 Ad M= ,N = , 0 −hQ ∗ −hI

M11 = ZA + AT Z − ZLC −CT LT Z T + ZAd ATd Z + ZLLT Z + GT G + ZZ + I, 9

(10)

N11 = P−1 A + AT P−1 + P−1 BK + K T BT P−1 + P−1 BKK T BT P−1 + P−1 P−1 + 2E1T E1 . By condition (7a), we have heT (t − h(t))Qe(t − h(t)) ≤ eT (t − h(t))Ze(t − h(t)). From Schur complement lemma, Lemma 4, (7b) is equivalent to the following condition 2PE2 E2T P + (1 + h)PP < P.

(11)

Now, pre- and post-multiplying both sides of the above inequality with P−1 , we get that the condition (11) is equivalent to the following 2E2T E2 + (1 + h)I < P−1 .

(12)

From (12), we obtain xT (t − h(t))[2E2T E2 + (1 + h)I]x(t − h(t)) < xT (t − h(t))P−1 x(t − h(t)). Moreover, note that V (t, z(t)) = zT (t)Pz(t), employing the fractional Razumikhin theorem (Proposition 2), for arbitrary real number ε > 0 such that q = 1 + ε > 1 we have V (t + s, z(t + s)) < (1 + ε)V (t, z(t)), s ∈ [−h, 0]. Therefore, taking the inequality (10) into account we have DCα V (t, z(t)) <η T (t)M η(t) + ξ T (t)N ξ (t) + (1 + ε)eT (t)Ze(t) + (1 + ε)xT (t)P−1 x(t).

(13)

Since ε > 0 is an arbitrary parameter and the functions DCα V (.), M , N do not dependent on ε, we can let ε → 0+ such that the inequality (13) becomes DCα V (t, z(t)) ≤ η T (t)M η(t) + ξ T (t)N ξ (t),

(14)

where   M 11 0 M= , 0 −hQ

  N 11 P−1 Ad N = ∗ −hI

M 11 = M11 + Z, N 11 = N11 + P−1 .

Letting L = Z −1 X and using Schur complement lemma, Lemma 4, the condition M < 0 is equivalent to the condition (7a). Now, using Schur complement lemma the condition N < 0 is equivalent to the following   Ω11 P−1 BK P−1 A1 Ω= ∗ −I 0  < 0, ∗ ∗ −hI

where

Ω11 = P−1 A + AT P−1 + P−1 BK + K T BT P−1 + P−1 P−1 + 2E1T E1 + P−1 . 10

Letting K = Y P−1 , pre- and post-multiplying both sides of Ω with diag{P, P, I}, using the following inequality −P2 ≤ I − 2P,

and using Schur complement lemma again, we get that the condition Ω < 0 is equivalent to the condition (7d). From conditions (7c), (7d) and (14), there is a positive number λ > 0 such that DCα V (t, z(t)) ≤ −λ kz(t)k2 . Therefore, based on the fractional Razumikhin stability theorem, Proposition 2, the closed-loop system (5) is asymptotically stable. The proof of the theorem is completed. Remark 1. Note that Lemma 1 in [42] on estimation of Caputo derivative of quadratic function kx(t)k2 , where x(t) is differentiable, can not be applied to FOSs with delay, since the solution x(t) of FOSs with delay is not always differentiable. Therefore, our proposed Lemma 3 (iv) for quadratic function xT (t)Px(t), where x(t) is only continuous, is applicable to FOSs with delay (1). Remark 2. It is well-known that additional unknowns and free-weighting matrices are introduced to make the flexibility to solve the resulting LMI conditions such that too many unknowns and free-weighting matrices employed in the existing methods used in papers [29, 31, 35-40] make the system analysis complicated and significantly increases the computational demand. Theorem 5 gives sufficient conditions in terms of LMIs, which consist of less free-weighting matrices: LMIs (7a), (7b) have no free-weighting matrices, LMIs (7c), (7d) has one free-weighting matrix. We include this section with an application of Theorem 5 to observer-based control problem for linear FOSs with time-variable delay. Consider the following linear uncertain FOSs  α   DC x(t) = [A + Ea Fa (t)Ha ]x(t) + [Ad + Ed Fd (t)Hd ]x(t − h(t)) + Bu(t), (15) y(t) = [C + Ec Fc (t)Hc ]x(t),   x(t) = φ (t), t ∈ [−h, 0],

where α ∈ (0, 1), x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control vector, y(t) ∈ R p is the measure output vector; A, Ad , B, Ea , Ed , Ec , Ha , Hd , Hc are given matrices of appropriate dimensions; The initial function φ (.) is continuous; The delay function h(t) is known, continuous function and satisfies condition (2); Fa (t), Fd (t), Fc (t) are unknown real matrix functions satisfying FaT (t)Fa (t) ≤ I, FdT (t)Fd (t) ≤ I, FcT (t)Fc (t) ≤ I. We rewrite (16) in the form  α   DC x(t) = Ax(t) + Ad x(t − h(t)) + f (t, x(t), x(t − h(t))) + Bu(t), y(t) = Cx(t) + g(t, x(t)),   x(t) = φ (t), t ∈ [−h, 0],

where

f (t, x(t), x(t − h(t))) = Ea Fa (t)Ha x(t) + Ed Fd (t)Hd x(t − h(t)), 11

(16)

(17)

g(t, x(t)) = Ec Fc (t)Hc x(t). Therefore, using simple calculation we see that the nonlinear functions f (.), g(.) satisfy condition (3) with √ √ E1 = 2kEa kkHa kI, E2 = 2kEd kkHd kI, G = kEc kkHc kI.

For system (15), we consider the following full-order observer system: ( ˆ DCα x(t) ˆ = Ax(t) ˆ + Bu(t) + L[y(t) − (Cx(t) ˆ + g(t, x(t)))], x(t) ˆ = 0, t ∈ [−h, 0],

(18)

and the control law u(t) = K x(t) ˆ in which x(t) ˆ ∈ Rn is the observer state vector, and K and L are the feedback control and observer gain matrix with appropriate dimensions, respectively. Define error vector z(t) = x(t) − x(t), ˆ which denotes difference between the real state and the estimated state. Then we have the following closed-loop system ( DCα z(t) = (A − LC)z(t) + Ad x(t − h(t)) − L[g(t, x(t)) − g(t, x(t) − z(t))] + f (t, x(t), x(t − h(t))), DCα x(t) = (A + BK)x(t) + Ad x(t − h(t)) − BKz(t) + f (t, x(t), x(t − h(t))). (19) From Theorem 5, we derive the following result. Corollary 6. The closed-loop system (19) is asymptotically stable if there exist symmetric positive definite matrices P, Q, Z, matrices X,Y with appropriate dimensions such that the following LMIs hold: hQ < Z,  −P PE2T  ∗ −1I 2 ∗ ∗  Ψ11 0  ∗ −hQ   ∗ ∗   ∗ ∗ ∗ ∗  Ξ11 PE1T  ∗ −1I  2  ∗ ∗ ∗ ∗

where E1 =



(1 + h)P  < 0, 0 −(1 + h)I  ZAd X Z 0 0 0  −I 0 0   < 0, ∗ −I 0  ∗ ∗ −I  BY Ad 0 0   < 0. I − 2P 0  ∗ −hI

√ √ 2kEa k.kHa kI, E2 = 2kEd k.kHd kI, G = kEc k.kHc kI

Ψ11 = ZA + AT Z − XC −CT X T + GT G + I + Z,

Ξ11 = AP + PAT + BY +Y T BT + I + P.

Moreover, the feedback control matrix is K = Y P−1 , the observer gain matrix is L = Z −1 X. 12

(20a) (20b)

(20c)

(20d)

Remark 3. Extending the results of [21, 22, 25, 26] to linear uncertain FOSs, Corollary 6 provides sufficient LMI conditions for the observer-based control problem, which can be easily solved by using LMI Toolbox from MATLAB [44]. Moreover, the methods used in [18, 27, 35, 38, 39] are not applicable to fractional system (1) with differentiable time-variable delay h(t). Example 1. Consider system (1), where α ∈ (0, 1), h(t) = 0.3| sint| ,         −2 0 0.5 0 1 A= , Ad = ,B= ,C = 0 5 , 0.5 −4 −0.4 1 2 and nonlinear functions f (.), g(.) satisfy condition (3) with       E1 = 0.2 0 , E2 = 0 0 , G = 0.2 0 .

Note that the delay function h(t) is continuous and non-differentiable so that the observer control design methods proposed in [18, 27, 35, 38, 39] are not applicable to this system. By using LMI 0.6 u(t)=K(x(t)−e(t)) 0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

0.1

0.2

0.3

0.4

0.5 Time(sec)

0.6

0.7

0.8

0.9

1

Figure 1: Response of control input signal u(t) = K x(t) ˆ

Toolbox from MATLAB, the LMI conditions of Theorem 5 are satisfied with     0.7112 −0.0193 3.5711 0.1141 P= ,Q= , −0.0193 0.7172 0.1141 8.1699       1.6384 −0.1170 0.0189 Z= , X= , Y = −0.0560 −0.1651 . 0.2263 3.8403 2.7367

Therefore, the resulting state feedback and observer control gains for the system (1) are defined

by   K = Y P−1 = −0.0851 −0.2325 ,   0.0622 −1 L=Z X = . 0.7089 13

5 4 3 2 1 0 −1 −2 −3 e1(t)

−4 −5

e2(t) 0

0.1

0.2

0.3

0.4

0.5 Time(sec)

0.6

0.7

0.8

0.9

1

Figure 2: Trajectories of the error state e1 (t), e2 (t) for α = 0.42

5 4 3 2 1 0 −1 −2 −3 x1(t)

−4 −5

x2(t) 0

0.1

0.2

0.3

0.4

0.5 Time(sec)

0.6

0.7

0.8

0.9

1

Figure 3: Trajectories of x1 (t), x2 (t) for α = 0.42 of the closed-loop system

Figure 1 shows the trajectories of control input u(t) = K x(t) ˆ = K(x(t) − e(t)), Figure 2 shows the trajectories of error states e1 (t), e2 (t), Figure 3 shows the trajectories x1 (t), x2 (t) of the closed-loop system with α = 0.42, φ1 (t) = e1 (t) = 5, φ2 (t) = e2 (t) = −5, t ∈ [−0.3, 0]. 4. Conclusion In this paper, based on the fractional Razumikhin approach and LMI technique we have studied the observer-based control problem of nonlinear FOSs with time-variable delay. The delay function is non-differentiable, but continuous and bounded. We have proposed a new lemma on Caputo derivative estimation of some quadratic functions. Based on the approach, delay-dependent 14

conditions in terms of LMIs are established to design the feedback and observer controllers. The obtained result is applied to observer-based control of linear uncertain FOSs with time-variable delay. It is notable that the considered fractional-order delay system (1) is time-invariant, our future work is to extent the proposed results to fractional-order delay systems with time-variable coefficients A(t), Ad (t), B(t). The second proposal for future work is the study of observer-based control problem for FOSs with unbounded time-variable delay. Acknowledgments The research of V.N. Phat is supported by the Vietnam Academy of Science and Technology under grant NVCC01.14/20-20. The authors would like to thank the Editors, the Associate Editor and anonymous reviewers for their valuable comments and suggestions, which allow us to improve the presentation and quality of this paper. References [1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1998. [2] G B. Bandyopadhyay, S. Kamal, Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, Springer, Berlin, 2015. [3] C.F. Lorenzo , T.T. Hartley, Initialization of fractional differential equations: Theory and application, Proceedings of the ASME 2007 International Design Engineering Technical Conferences, Las Vegas, USA, 2007, 1-7. [4] T.T. Hartley, C.F. Lorenzo. Dynamics and control of initialized fractional-order systems. Nonlinear Dynamics, 29(2002), 201-233. [5] M. Ortigueira, J. Machado, Special issue on fractional signal processing and applications, Signal Processing, 11(2003), 2285-2480. [6] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York, 2006. [7] M.S. Tavazoei, M. Haeri, Stabilization of unstable fixed points of chaotic fractional order systems by a state fractional PI controller, European Journal of Control , 14(2008), 247-257. [8] J. Shen, J. Lam, (2014). Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica, 50(2014), 547-551. [9] A. Coronel-Escamilla, F. Torres, J.F. Gomez-Aguilar, R. F. Escobar-Jimenez, G.V. GuerreroRamirez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning, Multibody System Dynamics, 43(3) (2018), 257- 277. [10] Y. Wen, X.F. Zhou, Z. Zhang, S. Liu, Lyapunov method for nonlinear fractional differential systems with delay, Nonlinear Dynamics, 82(2015), 1015-1025. 15

[11] F.F. Wang, D.Y. Chen, X.G. Zhang, Y. Wu, The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay, Applied Mathematics Letters, 53(2016), 45–51. [12] C.J. Zuniga-Aguilar, A. Coronel-Escamilla, J.F. Gomez-Aguilar, V.M. Alvarado-Martinez, H.M. Romero-Ugalde, New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks, European Physical Journal Plus, 133(75) (2018), 1-16. [13] G.M. Lazarevic, Stability and stabilization of fractional-order time delay systems, Scientific Technical Review, 61(2011), 31-44. [14] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers & Mathematics with Applications, 59(2010), 1810-1821. [15] X.F. Zhang, L. Liu, G. Feng, Y.Z. Wang, Asymptotical stabilization of fractional-order linear systems in triangular form. Automatica, 49(2013), 3315-3321. [16] J.G. Lu, Y.Q. Chen, Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties, Fractional Calculus and Applied Analysis, 16(2013), 142-157. [17] N.A. Camacho, M.A. Duarte-Mermoud, M.E. Orchard, Fractional order controllers for throughput and product quality control in a grinding mill circuit, European Journal of Control, 2019, https://doi.org/10.1016/j.ejcon.2019.08.002. [18] Y. Boukal, M. Darouach, M. Zasadzinski, N.E. Radhy, H∞ observer-based-controller for fractional-order time-varying delay systems, in: Proceedings of the 25th Mediterranean Conference on Control and Automation, Valletta, Malta, 2017, 1287-129. [19] S. Liu, R. Yang, X.F. Zhou, W. Jiang, X. Li, X.W. Zhao, Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems, Communications in Nonlinear Science & Numerical Simulation, 73(2019), 351-362. [20] C.J. Zuniga-Aguilar, J.F. Gomez-Aguilar, R.F. Escobar-Jimenez, H.M. Romero-Ugalde, A novel method to solve variable-order fractional delay differential equations based in Lagrange interpolations, Chaos, Solitons & Fractals, 126 (2019), 266-282. [21] J.H. Park, On the design of observer-based controller of linear neutral delay-differential systems. Applied Mathematics and Computation, 150(2004), 195-202. [22] J.D. Chen, Robust H∞ output dynamic observer-based control design of uncertain neutral systems, Journal of Optimization Theory and Applications, 132(2007), 193 - 211. [23] H. Trinh, T. Fernando, Functional observers for dynamical systems, Springer, Berlin, 2012. [24] Y.H. Lan, H.B. Gu, C.X. Chen, Y Zhou, Y.P. Luo, An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks, Neurocomputing, 136(2014), 235 - 242. 16

[25] S. Ibrir, M. Bettayeb, New sufficient conditions for observer-based control of fractional-order uncertain systems, Automatica, 59(2015), 216-223. [26] B. Li, X. Zhang, Observer-based robust control of (0 < α < 1) fractional-order linear uncertain control systems, IET Control Theory and Application, 10(2016), 1724 -1731. [27] A. Mohammadzadeh, S. Ghaemi, Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay, Nonlinear Dynamics, 93(2018), 1809-1821. [28] A.S. Ammour, S. Djennoune, W. Aggoune, M. Bettayeb, Stabilization of fractional-order system with state and input delay, Asian Journal of Control, 18(2016), 1- 9. [29] T. Liu, F. Wang, W. Lu, X. Wang, Global stabilization for a class of nonlinear fractional-order systems, International Journal of Modeling, Simulation and Scientific Computing, 10(2019) , 1-10. [30] C. Li, J. Wang, J. Lu, Observer-based robust stabilization of a class of nonlinear fractionalorder uncertain systems: an linear matrix inequalities approach, IET Control Theory and Application, 6(2012), 2757-2764. [31] J. Qiu, Y. Ji, Observer-based robust controller design for nonlinear fractional-order uncertain systems via LMI, Mathematical Problems in Engineering, 2017, Article ID 8217126: 1-15. [32] Y. Lan, Y. Zhou, Non-fragile observer-based robust control for a class of fractional-order nonlinear systems, Systems & Control Letters, 62(2013), 1143-1150. [33] Y. Tan, M. Xiong, D. Du, S. Fei, Observer-based robust control for fractional-order nonlinear uncertain systems with input saturation and measurement quantization, Nonlinear Analysis: Hybrid Systems, 34 (2019), 45–57. [34] M. Parvizian, K. Khandani, V.J. Majd, An H∞ non-fragile observer-based adaptive sliding mode controller design for uncertain fractional-order nonlinear systems with time delay and input nonlinearity, Asian Journal of Control, 2019, https://doi.org/10.1002/asjc.2209. [35] Z. Yang, J. Zhang, Stability analysis of fractional-order bidirectional associative memory neural networks with mixed time-varying delays, Complexity, ID 2363707, 2019, 1-21, https://doi.org/10.1155/2019/2363707. [36] G. Chen, Z. Xiang, H.R. Karimi, Observer-based robust H∞ control for switched stochastic systems with time-varying delay, Abstract and Applied Analysis, ID 320703 (2013), 1-12. [37] P. Liu, Observer-based control for time-varying delay systems with delay-dependence, Universal Journal Electrical and Electronic Engineering, 1(2013), 31-40. [38] K. W. Yu, C.H. Lien, Delay-dependent conditions for guaranteed cost observer-based control of uncertain neutral systems with time-varying delays, IMA Journal of Mathematical Control and Information, 24(2007), 383 - 394.

17

[39] H. Liu, Y. Shena, X. Zhao, Delay-dependent observer-based H∞ finite-time control for switched systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 6(2012), 885 - 896. [40] M.V. Thuan, V.N. Phat, H. Trinh, Observer-based controller design of linear time-delay systems with an interval time- varying delay, International Journal of Applied Mathematics and Computer Science, 22(2012), 921-927. [41] G. Vainikko, Which functions are fractionally differentiable, Zeitschrift fur Analysis und ihre Anwendungen, 35(2016), 465-487. [42] N.A. Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems, Communications in Nonlinear Science & Numerical Simulation, 19(2014), 2951-2957. [43] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [44] P. Gahinet, A. Nemirovskii, A.J. Laub, M. Chilali, LMI control toolbox for use with MATLAB. The MathWorks Inc, Natick, 1995.

18