Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampled-data control

Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampled-data control

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Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampled-data control Quanxin Zhu, S. Vimal Kumar, R. Raja, Fathalla Rihan PII: DOI: Reference:

S0016-0032(19)30606-4 https://doi.org/10.1016/j.jfranklin.2019.08.032 FI 4116

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

14 November 2018 21 May 2019 25 August 2019

Please cite this article as: Quanxin Zhu, S. Vimal Kumar, R. Raja, Fathalla Rihan, Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampleddata control, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.08.032

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1

Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampled-data control Quanxin Zhu 1 2

1,2

, S.Vimal Kumar 3 , R.Raja 4 , Fathalla Rihan

5

MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China 3 4

Department of Mathematics, RVS Technical Campus-Coimbatore, Coimbatore - 641402, India. Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi - 630 004, India. 5

Department of Mathematical Sciences, UAE University, Al Ain 15551 Abu Dhabi, UAE

Abstract This paper addresses the issue of reliable feedback control of an uncertain aircraft flight control systems with disturbances via non-fragile sampled-data control approach. In particular, the parameter uncertainties are assumed to be randomly occurring which is described by the Bernoulli distributed sequences. By constructing a suitable Lyapunov-Krasovskii functional together with Wirtinger-based inequality, a new set of sufficient conditions in terms of linear matrix inequalities is obtained to ensure the asymptotic stability and extended dissipativity of the aircraft flight control systems not only when all actuators are operational, but also in case of some actuator failures. Finally, simulation results are conducted to validate the effectiveness of the proposed control design technique.

Keywords: Extended dissipativity, Fault-tolerant control, Non-fragile sampled-data control, Randomly occurring uncertainties. I. I NTRODUCTION In the last few years advanced control techniques capable to deal with the nonlinear systems have gained growing popularity within the aerospace community. Especially, the flight control system can still be with unexpected interaction, which increases the difficulty of guaranteeing the aircraft’s safety and reliability. In addition, the time delay and parametric uncertainties are unavoidable in real-time systems such as flight control systems, networked control systems and dynamics models, and also it is a main source of instability and poor performance of systems dynamics [4], [6], [7]. So the effects of those factors on the control problem have been addressed in the literature, for instance see [21], [23]. Generally speaking, the fault-tolerant controller (FTC) or reliable controller design is one of the most considerable problem in the control domain and it is one of the most significant demand particularly for practical safety-critical systems and flight control systems [5], [12], [15]–[17]. The FTC design approach can be broadly categorized into two groups: the passive approaches [7], [22] and the active approaches [4], [24], [25]. However, these controllers are designed under nominal flight conditions (no faults or no failures) and no external disturbances from the environment This work was jointly supported by the National Natural Science Foundation of China (61773217), Hunan Provincial Science and Technology Project Foundation (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101) and the Construct Program of the Key Discipline in Hunan Province. Corresponding author is Quanxin Zhu with e-mail: [email protected]

2

and typically cannot adapt themselves to adverse (or unaccounted) situations such as failure of the actuators and/or sensors, structural damage or exogenous excitations such as wind gust. Moreover, the sampled-data control technology has gained intensive attention in the field of engineering for its reliable and stable performance in comparison with control approaches. Sampled-data control scheme is mainly classified into three categories. The first is a discrete time approach, which was used to analyze the stability of the aperiodically sampled-data control systems. However, it is only suitable for nominal system and it cannot deal with uncertainties in the system matrices. The second is an impulsive approach, which can be used to investigate the systems which have uncertain and bounded sampling intervals. The third is concerned with the input delay approach, where the system was modeled as a continuous-time system with input delay [2], [26]. The input delay approach has been used extensively since it can be easily applied to Lyapunov stability analysis and all kinds of samples are adapted in this approach. Besides, due to the complexity of the aircraft flight control systems with the stochastic uncertainty and external disturbance, it is difficult to solve the aircraft flight control problem by using fixed control gain approach. Actually, system uncertainties may induce the gain fluctuation. Therefore, it is reasonable to consider the non-fragile controller to solve the problem aircraft flight control systems [1], [3], [10], [11], [20]. On another research front, the dissipative theory provides a framework for the design and analysis of control systems under the input-output description based on energy-related considerations. More specifically, dissipativity describes the systems whose inner energy will never exceed the amount of outer energy, that is to say, dissipative systems will be only able to consume energy instead of producing energy. So we can utilize dissipative performance to estimate the ability of control system to dissipate the energy generated by exogenous disturbances. Moreover, in this paper, the extended dissipative concept [6], [8], [9], [14], [18], [19] is adopted to deal with the external disturbance input for the aircraft flight control systems. The extended dissipativity encompasses the H∞ performance, L2 − L∞ performance, passivity and dissipativity by adjusting weighting matrices in a new performance index. Thus, it is reasonable and significant to consider the unified framework for the stabilization of aircraft flight control systems. Prompted by aforementioned discussions, this work conducts the analysis and non-fragile sampleddata control problem for aircraft flight control systems with randomly occurring uncertainties subject to nonlinear actuator fault. The novelty and contribution of this paper are summarized as follows: •

•

• •

Different from most published research results, in this paper we describe the stability behaviors of control systems and extended dissipative performance is taken to ensure the ability of closed-loop system to reject the outside disturbance. Moreover, the physical limitations, like actuator fault and parameter uncertainty are also considered in this paper. A novel fault-tolerant control scheme is presented for the aircraft flight control systems which guaranteeing the stochastic stability and extended dissipativity of the closed-loop system in the presence of both linear and nonlinear actuator faults. More precisely, the fault control input for the actuator is implemented to the reliable controller design and also the parametric uncertainties are taken as random manner that are practically more significant. The desired state feedback non-fragile reliable controller for the uncertain system can be constructed in terms of a new set of linear matrix inequalities (LMIs), which can be easily solved by using MATLAB LMI toolbox. II. P ROBLEM FORMULATION

Consider a standard aircraft flight control model from [21], [23] of the form: x(t) ˙ = Ax(t) + Bu(t) + Dw(t),

(1)

3

h iT is the longitudinal motion state vector, α ˆ , qb and θ represent the angle of attack, where x(t) = α ˆ qb θ the body-axis angular rates and the pitch angle, respectively; u(t) is the elevator deflection; w(t) is the external disturbance; D = [1 0 0]T shows how the disturbance adds to the system; A and B are system matrices given by     Z δz Zαˆ 1 −g sin(µ∗ /V∗ )     (2) A = Mαˆ Mq 0  , B = Mδz  , 0 0 1 0 where the parameters Zαˆ , Mαˆ , Mq , Zδz and Mδz are the force and moment dimensional derivatives, the subscript δz denotes the equivalent elevator deflection; µ∗ and V∗ represent the flight-path angle and the velocity on the equilibrium point, respectively. Furthermore, we consider the linearized flight control system (1) with non-linear actuator fault control input in the form x(t) ˙ = Ax(t) + BuF (t) + Dw(t), z(t) = Cx(t),

(3)

where x(t) ∈ Rn is the state vector; w(t) is the external disturbance which belongs to L2 [0, ∞); z(t) ∈ Rp is the control output; A, B , D and C are given matrices of appropriate dimensions; uF (t) is the signal from the actuator that has failed. In the field of aerospace engineering it is essential to maintain robustness, stability and required specifications of system with existence of component failures. Fault-tolerant control techniques are extensively employed to tackle component failures. In this paper, we adopt the following generalized actuator fault model [15]: uF (t) = G1 u(t) + Ψ(u(t)),

(4)

where 0 < G1 = diag{r1 } ≤ I and the vector function Ψ(u(t)) = [Ψ1 (u(t)) Ψ2 (u(t)) Ψ3 (u(t))] represents the nonlinear fault term of the true control input u(t) and is assumed to satisfy the following bound condition: ΨT (u(t))Ψ(u(t)) ≤ uT (t)G2 u(t),

(5)

where G2 = diag{m1 }. By considering the fault model (4), the dynamics of the flight control system (3) with actuator faults can be described as  x(t) ˙ = Ax(t) + B [G1 u(t) + Ψ(u(t))] + Dw(t), (6) z(t) = Cx(t). We consider the sampled-data control input. It may be represented as delayed control as follows: u(t) = ud (tk ) = ud (t − (t − tk )) = ud (t − h(t)), tk ≤ t ≤ tk+1 , h(t) = t − tk ,

(7)

where ud is a discrete-time control signal and the time varing delay 0 ≤ h(t) = t − tk is piecewise linear with ˙ the derivative h(t) = 1 for t 6= tk . tk is the sampling instant satisfying 0 < t1 < t2 < · · · < tk < · · · . Sampling ¯ for all tk . interval hk = tk+1 − tk may vary but it is bounded. Thus, we assume h(t) ≤ tk+1 − tk = hk ≤ h Then, the natural choice of a controller given by u(t) = Kx(tk ). Now, we represent a piecewise control law as a continuous-time control with a time varying piecewise continuous delay h(t) = t − tk as given in (7). Thus we look for a non-fragile sampled-data feedback controller of the form u(t) = (K + ∆K(t))x(t − h(t)).

(8)

where K is the nominal control gain to be determined and the real-valued matrix ∆K(t) denotes the controller

4

gain fluctuation. Moreover, the gain fluctuation is assumed to be represented the following norm bounded structure ∆K(t) = EF (t)H, F T (t)F (t) ≤ I, where E and H are known constant matrices. Using the controller (8), the system (6) can be written as  x(t) ˙ = Ax(t) + BG1 (K + ∆K(t))x(t − h(t)) + B(Ψ(u(t))) + Dw(t), (9) z(t) = Cx(t). It is known that control systems are often disturbed by parametric uncertainties, which may cause undesirable dynamic behaviors or poor performance. Moreover, the existence of parametric uncertainties are mostly in random manner. In order to make an effective design, the differences between the simplified model and the real plant are taken as the model uncertainties. Therefore, the closed-loop system of (9) with random parametric uncertainties can be modeled as follows:  x(t) ˙ = (A + α(t)∆A(t))x(t) + BG1 (K + ∆K(t))x(t − h(t)) + B(Ψ(u(t))) + Dw(t), (10) z(t) = Cx(t),

where ∆A(t) represent the parametric uncertainties. Without loss of generality, these parametric uncertainties are assumed to satisfy ∆A(t) = M F (t)N, where M and N are known constant matrices with appropriate dimensions and the uncertain matrix F (t) satisfies F T (t)F (t) ≤ I, ∀ t ∈ [0, ∞). The random variable α(t) is Bernoulli distributed white sequences taking values on either 0 or 1 with (11)

P rob{α(t) = 1} = E[α(t)] = α ¯,

where α ¯ ∈ [0, 1] are known constants and mutually independent. E [·] stands for the mathematical expectation operator with respect to the given probability. Remark 2.1: It is noted that in the system model (10), the random variable α(t) is introduced to characterize the phenomenon of the randomly occurring uncertainties (ROUs) and it is more suitable for reflecting parameter variations of a random nature. Such ROUs are due to a variety of reasons such as network-induced random failures and repair of components, sudden environmental disturbances, etc. Very recently, in [16], the concept of ROUs was introduced to model the randomly occurring parameter uncertainties for flight control system. Remark 2.2: It should be noted that if we choose Ψ(u(t)) = 0, then the controller (4) is deduced to the nominal fault-tolerant controller. The proposed fault-tolerant controller (4) in this paper is more general than the existing fault-tolerant controllers. In order to obtain the main results, the following lemma and definition are needed. Lemma 2.3: [13] For any constant matrix M > 0, the following inequality holds for all continuously differentiable function x(t) in [a, b] → Rn : (b − a)

Z

b

Z

a

b

T

x (s)M x(s)ds ≥

Z bZ

Z

a

b

T Z b  x(s)ds M x(s)ds + 3ΦT M Φ a

s 2 x(u)duds. (b − a) a a a Assumption 2.4: [27] Given matrices S1 , S2 , S3 and S4 satisfy the following conditions:

where Φ =

x(s)ds −

1) S1 = S1T ≤ 0, S3 = S3T > 0 and S4 = S4T ≥ 0, 2) (kS1 k + kS2 k) · kS4 k = 0.

5

Definition 2.5: Let x(t) be the state trajectory of system (10). Then system (10) is said to be robustly stochastically stable if  Z ∞ 2 (12) E kx(t)k dt|x0 < ∞ 0

Rn .

for any initial system state x0 ∈ Remark 2.6: From Definition 2.5 we can see that if the augmented system (10) is robustly stochastically stable, then the system (10) with α(t)∆A(t) = 0 is asymptotically stable. Therefore, the asymptotically stability of the original system is a necessary condition in this paper. Definition 2.7: [27] For given matrices S1 , S2 , S3 and S4 satisfying Assumption 2.4, system (10) is said to be extended dissipative if there exists a scalar η such that the following inequality holds for any tf ≥ 0 and all w(t) ∈ L2 [0, ∞): ) (Z tf

E

0

J(t)dt − sup z T (t)S4 z(t) 0≤t≤tf

≥ η,

(13)

where J(t) = z T (t)S1 z(t) + 2z T (t)S2 w(t) + wT (t)S3 w(t). III. M AIN R ESULTS In this section, we will first establish a criterion to implement the robust reliable non-fragile for aircraft flight control systems with time-varying delay and disturbances via extended dissipativity and sampled-data approach for closed-loop system (9). Then, we establish the same for the stochastic uncertain closed-loop system (10). ¯ and matrices G1 , G2 , Si (i = Theorem 3.1: Consider the closed-loop control systems (9). For given scalars κ, µ ρ, η , h 1, 2, 3, 4) satisfying Assumption 2.4, the control system (9) is said to be asymptotically stable and extended dissipative, if there exist symmetric matrices P > 0, Q > 0, Ri > 0, (i = 1, 2) and positive scalars k , (k = 1, 2) such that the following LMIs hold:  h i  ˆ1 ˆ2 ˆ3 ˆ4 ˆ5 ˆ6 ˆ7 ˆ Σ Σ Σ Σ Σ Σ Σ Σ 11×11    ∗ − µ1 X 0 0 0 0 0 0      −1  ∗ ∗ −κG2 0 0 0 0 0      ∗ ∗ ∗ − I 0 0 0 0 1 ˆ   < 0, Γ = (14)   ∗ ∗ ∗ ∗ −1 I 0 0 0     ∗ ∗ ∗ ∗ ∗ −2 I 0 0      ∗ ∗ ∗ ∗ ∗ ∗ −2 I 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I ¯ 2 − µX < 0, hR

(15)

ρP − C T S4 C > 0,

(16)

¯R ˆ 1,1 = AX +XAT + Q+ ˆ h ˆ 1 − ¯4 R ˆ , Σ ˆ 1,2 = BG1 Y + ¯4 R ˆ ,Σ ˆ 1,4 = B, Σ ˆ 1,10 = ¯62 R ˆ2, Σ ˆ 1,11 = D−XC T S2 , where Σ h 2 h 2 h ˆ 2,2 = − ¯8 R ˆ , Σ ˆ 2,3 = ¯4 R ˆ , Σ ˆ 2,9 = ¯62 R ˆ2, Σ ˆ 2,10 = − ¯62 R ˆ2, Σ ˆ 3,3 = −Q− ˆ ¯4 R ˆ , Σ ˆ 3,9 = − ¯62 R ˆ2, Σ ˆ 4,4 = −κ−1 I, Σ h 2 h 2 h h h 2 h ˆ 5,5 = − ¯4 R ˆ , Σ ˆ 5,6 = ¯62 R ˆ1, Σ ˆ 6,6 = − ¯123 R ˆ1, Σ ˆ 7,7 = − ¯4 R ˆ , Σ ˆ 7,8 = ¯62 R ˆ1, Σ ˆ 8,8 = − ¯123 R ˆ1, Σ ˆ 9,9 = − ¯123 R ˆ2, Σ h 1 h h 1 h h T  h T h · · 0} D · · 0} ˆ 10,10 = − ¯123 R ˆ2, Σ ˆ 11,11 = −S3 , Σ ˆ 1 = AX BG1 Y 0 B 0| ·{z ˆ 2 = 0 Y 0| ·{z ˆ3 = Σ , Σ , Σ h 6 times 9 times

6

√ T T T     T T ¯ 1E T 0 · · · 0 0 HX 0 · · · 0 0 · · · 0  hBG · · 0} 2 ˆ4 = ˆ 5 = | {z } ˆ6 = 0 H X 0 | {z } | {z } | ·{z , Σ , Σ , , Σ 10 times 9 times 10 times 9 times  √ T C −S1 X |0 ·{z · · 0} ˆ Σ7 = [ . Moreover, if the obtained LMIs are feasible, then the feedback controller gain matrix 10 times can be constructed as K = Y X −1 . Proof. Let us consider the following Lyapunov-Krasovskii functional (LKF) candidate for the system (9):

 1 E T G1T B T

V (x(t)) =

2 X

(17)

Vi (x(t)),

i=1

where

Z t V1 (x(t)) =xT (t)P x(t) + xT (v)Qx(v)dv, ¯ t−h Z t Z t Z t Z t T V2 (x(t)) = x (v)R1 x(v)dvds + x˙ T (v)R2 x(v)dvds. ˙ ¯ t−h

¯ t−h

s

s

Calculating the time derivatives of Vi (x(t)) (i = 1, 2) along the trajectories of (9), we can obtain ¯ ¯ V˙ 1 (x(t)) =2xT (t)P x(t) ˙ + xT (t)Qx(t) − xT (t − h)Qx(t − h)

=2xT (t)P [Ax(t) + BG1 (K + ∆K(t))x(t − h(t)) + B(Ψ(u(t))) + Dw(t)] ¯ ¯ + xT (t)Qx(t) − xT (t − h)Qx(t − h),

¯ 1 x(t) − V˙ 2 (x(t)) =xT (t)hR

Z

t

¯ t−h

¯ 2 x(t) xT (s)R1 x(s)ds + x˙ T (t)hR ˙ −

Next, the first and second integral terms of (19) can split as follows: Z t Z t−h(t) Z T T − x (s)R1 x(s)ds = − x (s)R1 x(s)ds − −

¯ t−h Z t

¯ t−h

x˙ T (s)R2 x(s)ds ˙ =−

¯ t−h Z t−h(t) ¯ t−h

x˙ T (s)R2 x(s)ds ˙ −

t

t−h(t) Z t

Z

t

¯ t−h

x˙ T (s)R2 x(s)ds. ˙

−

t−h(t)

¯ t−h

1 x˙ (s)R2 x(s)ds ˙ ≤ −¯ h T

3 −¯ h

"Z

t−h(t)

¯ t−h

"Z

t−h(t)

¯ t−h

x(s)ds ˙

#T

R2

"Z

(20)

x˙ T (s)R2 x(s)ds. ˙

(21)

t−h(t)

t−h(t)

x(s)ds ˙

¯ t−h

2 x(s)ds ˙ − ¯ v5 (t) h

(19)

xT (s)R1 x(s)ds,

By applying Lemma 2.3 to each of the integral terms in (20)-(21), we can get  T  Z t−h(t) 1 T 3 2 T − x (s)R1 x(s)ds ≤ − ¯ v1 (t)R1 v1 (t) − ¯ v1 (t) − ¯ v2 (t) R1 v1 (t) − ¯ h h h t−h  T  Z t 1 T 3 2 T − x (s)R1 x(s)ds ≤ − ¯ v3 (t)R1 v3 (t) − ¯ v3 (t) − ¯ v4 (t) R1 v3 (t) − h h h t−h(t) Z

(18)

#T

R2

"Z

#

t−h(t) ¯ t−h

 2 ¯ v2 (t) , h  2 ¯ v4 (t) , h

# 2 x(s)ds ˙ − ¯ v5 (t) , h

(22) (23)

(24)

7

Z

"Z

t

1 − x˙ T (s)R2 x(s)ds ˙ ≤ −¯ h t−h(t)

#T

t

x(s)ds ˙

t−h(t)

"Z

R2

"Z

t

x(s)ds ˙

t−h(t)

#

#T "Z # t 2 2 x(s)ds ˙ − ¯ v6 (t) R2 x(s)ds ˙ − ¯ v6 (t) , h h t−h(t) t−h(t)

3 −¯ h

t

(25)

where v1 (t) = v4 (t) =

Z

t−h(t)

¯ t−h Z t

t−h(t)

x(s)ds, v2 (t) =

Z

s

Z

t−h(t) Z s

¯ t−h

x(u)duds, v5 (t) =

t−h(t)

x(u)duds, v3 (t) =

¯ t−h Z t−h(t) Z s ¯ t−h

Z

t

x(s)ds,

t−h(t)

¯ t−h

x(u)duds, ˙ v6 (t) =

Z

t t−h(t)

Z

s

x(u)duds. ˙

t−h(t)

Additionally, from equation (5), for any positive scalar κ, we can have   κ−1 uT (t)G2 u(t) − ΨT (u(t))Ψ(u(t)) ≥ 0.

(26)

Combining the equations (18)-(26), we can get

V˙ (x(t)) ≤ ζ T (t)Γ0 ζ(t),

(27)

h i ¯ ψ T (u(t)) v T (t) v T (t)v T (t) v T (t) v T (t) v T (t) wT (t) , ζ T (t) = xT (t) xT (t − h(t)) xT (t − h) 1 2 3 4 5 6   T T T T ¯ 2 x(t) Γ0 = [Σ]11×11 + 2x (t)P Ax(t − h(t)) + x˙ (t)hR ˙ + e2 (t) ∆K (t)G2 ∆K(t) e2 (t),  T  T 0 I4 |0 ·{z · · 0} 0 ∆K T (t)G1T B T 0| ·{z · · 0} e2 (t) = , A= . 9 times 9 times

By invoking the Schur complement, Γ can be equivalently written as  h i Σ 11×11   ∗    ∗  Γ = ∗   ∗    ∗ ∗

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

− µ13 P −1 ∗ ∗ ∗ ∗ ∗

0 −κG2−1 ∗ ∗ ∗ ∗

0 0 −1 I ∗ ∗ ∗

0 0 0 −1 I ∗ ∗

0 0 0 0 −2 I ∗

0 0 0 0 0 −2 I



      ,     

(28)

¯ 1 − ¯4 R2 , Σ1,2 = P BG1 K + ¯4 R2 , Σ1,4 = P B, Σ1,10 = ¯62 R2 , Σ1,11 = P D, where Σ1,1 = P A + AT P + Q + hR h h h Σ2,2 = − h¯8 R2 , Σ2,3 = h¯4 R2 , Σ2,9 = h¯62 R2 , Σ2,10 = − h¯62 R2 , Σ3,3 = −Q− h¯4 R2 , Σ3,9 = − h¯62 R2 , Σ4,4 = −κ−1 I, Σ5,5 = − h¯4 R1 , Σ5,6 = h¯62 R1 , Σ6,6 = − h¯123 R1 , Σ7,7 = − h¯4 R1 , Σ7,8 = h¯62 R1 , Σ8,8 = − h¯123 R1 , Σ9,9 = − h¯123 R2 , Σ10,10 = − h¯123 R2 , Σ11,11 = −I,  T  T  T A BG1 K 0 B |0 ·{z · · 0} D 0 K 0| ·{z · · 0} 1 E T G1T B T P 0| ·{z · · 0} Σ1 = , Σ2 = , Σ3 = , 6 times 9 times 10 times √  T  T   T ¯ 1E 0 H 0 · · 0} 0 · · · 0 2 hBG 0 H T P 0| ·{z · · 0} | ·{z Σ4 = , Σ5 = | {z } , Σ6 = . Hence, we have Γ < 0. 9 times 10 times 9 times Next, we are proceeding to consider the extended dissipative condition for the considered system. Noted that if Γ < 0 and we write V˙ (x(t)) − J(t) ≤ 0.

By integrating on both sides of the above inequality from 0 to t(t ≥ 0), we can obtain

8

Z

t 0

J(s)ds ≥ V (x(t)) − V (x(0)) ≥ xT (t)P x(t) + η.

(29)

The following lines are concentrated to demonstrate the inequality (13) is true, thus, two cases are needed, i.e, kS4 k = 0 and kS4 k = 6 0. First, if kS4 k = 0, then (29) implies for any tf ≥ 0 that Z t J(s)ds ≥ xT (tf )P x(tf ) + η ≥ η, (30) 0

this signifies Definition 2.7 to be true. If kS4 k = 6 0 as mentioned in Assumption 2.4, we can conclude that the matrices S1 = 0, S2 = 0, S3 > 0, thus, for any tf ≥ t ≥ 0, we have Z tf Z t J(s)ds ≥ J(s)ds ≥ xT (t)P x(t) + η. (31) 0

0

Thus, there exists a scalar 0 < ρ < 1 such that Z tf J(s)ds ≥ η + ρxT (t)P x(t).

(32)

0

Therefore, according to (16), we have

z T (t)S4 z(t) = xT (t)C T S4 Cx(t) ≤ ρxT (t)P x(t).

(33)

It is clear that, for any t ≥ 0, tf ≥ 0 with tf ≥ t Z

0

tf

J(s)ds ≥ z T (t)S4 z(t) + η.

(34)

Thus, the inequality (13) holds for any tf ≥ 0. Combining (28)-(34), we can get V˙ (x(t)) ≤ ζ T (t)

i h  ¯ Γ 17×17 ζ(t) + ζ T (t)ΓT1 Γ1 ζ(t),

(35)

√ ¯ 1,11 = P D − C T S2 , Γ ¯ 11,11 = −S3 , Γ1 = [C T −S1 0 · · · 0 ] and other terms of Γ ¯ are as in Γ. where Γ | {z } 16 times Since the matrix in (35) is nonlinear, it is not possible to solve it in a MATLAB LMI toolbox. But by using ˆ= congruence transformation technique, we can convert it into LMI. For that purpose, let us define X = P −1 , Q ˆ 1 = XR1 X and R ˆ 2 = XR2 X . Further, pre-and post-multiplying the right-hand side of inequality (35) XQX, R ¯ 2 < µX and Y = KX in this theorem. by diag{X, X, X, I, X, X, X, X, X, X, I, I, I, I, I, I, I} and letting hR Then, we can easily obtain the LMI (14). Hence, the closed-loop system (9) is asymptotically stable and extended dissipative. This completes the proof.  Now, we further extend the results obtained in Theorem 3.1 to the stochastic uncertain closed-loop system (10). Theorem 3.2: The closed-loop control systems with randomly occurring uncertainties (10) is robustly stochastically stable and satisfies extended dissipative performance η > 0, if there exist symmetric matrices P > 0, Q > 0, ¯ and matrices Ri > 0, (i = 1, 2) and positive scalars k (k = 1, 2, 3, 4) and for given positive scalars κ, µ, π, ρ, h G1 , G2 , Si (i = 1, 2, 3, 4) satisfying Assumption 2.4, such that the following LMI together with (15) and (16) are holds:

9

h i  ˆ Φ1 Φ2 Φ3 Φ4 Γ   18×18   ∗ −3 I 0 0 0    ˆ = Ω < 0,  ∗ ∗ −3 I 0 0      ∗ ∗ ∗ −4 I 0   ∗ ∗ ∗ ∗ −4 I

(36)

T T T T    √ · · 0} α ¯ N X 0| ·{z · · 0} 4 M T 0| ·{z · · 0} 0| ·{z · · 0} α ¯ h1 N X 0| ·{z 0 ·{z · · 0} | , Φ2 = , Φ3 = , Φ4 = Φ1 = 17 times 17 times 17 times 10 times 7 times and other parameters are defined as in Theorem 3.1. Moreover, the feedback controller gain matrix can be constructed as K = Y X −1 . Proof. Let us consider the LKF candidate (17) for the system (10)  3 M T

V (x(t)) = V1 (x(t)) + V2 (x(t))

where

Z t V1 (x(t)) =xT (t)P x(t) + xT (v)Qx(v)dv, ¯ t−h Z t Z t Z t Z t T V2 (x(t)) = x (v)R1 x(v)dvds + x˙ T (v)R2 x(v)dvds ˙ ¯ t−h

¯ t−h

s

s

and define the infinitesimal operator L of V (x(t)) as follows:

1 {E {V (x(t + ∆))|x(t)} − V (x(t))} . ∆→0 ∆

LV (x(t)) = lim

By taking the infinitesimal operator L of the LKF (17) along the system (10) and applying the mathematical expectation on both sides, we can obtain ¯ ¯ E{LV˙ 1 (x(t))} = E{2xT (t)P x(t) ˙ + xT (t)Qx(t) − xT (t − h)Qx(t − h)}

= E{2xT (t)P [(A + α(t)∆A(t))x(t) + BG1 (K + ∆K(t))x(t − h(t)) + B(Ψ(u(t))) + Dw(t)]

¯ ¯ + xT (t)Qx(t) − xT (t − h)Qx(t − h)}, Z t Z ¯ 2 x(t) ¯ 1 x(t) − E{L{V˙ 2 (x(t))} = E{xT (t)hR xT (s)R1 x(s)ds + x˙ T (t)hR ˙ − ¯ t−h

t

¯ t−h

x˙ T (s)R2 x(s)ds}. ˙

Then using the same techniques as in the proof of Theorem 3.1, we get E{V˙ (x(t))} < 0, hence the closed-loop system (10) is robustly stochastically stable. Next, we are proceeding to consider the extended dissipative condition for the considered system. Noted that if Γ < 0 and we write E{LV˙ (x(t))} − J(t) ≤ 0.

Then using the same techniques as in the proof of Theorem 3.1, we get h  i ¯0 E{LV˙ (x(t))} ≤ ζ T (t) Γ ζ(t) + ζ T (t)ΓT1 Γ1 ζ(t) 17×17 ¯ 1 − ¯4 R2 , ¯ 1,1 = P (A + α where Σ ¯ ∆A(t)) + (A + α ¯ ∆A(t))T P + Q + hR h

(37)

10

 ¯ ∆A(t) BG1 K 0 B ¯1 = A + α Σ

T · · 0} D |0 ·{z and other parameters are defined as in Theorem 3.1. 6 times Using Schur complement formula, and applying pre-and post-multiplying the right-hand side of inequality (37) ¯ 2 < µX and Y = KX , we have by diag{X, X, X, I, X, X, X, X, X, X, I, I, I, I, I, I, I}, and defining hR ˆ = Γ ˆ + Φ1 ∆(t)Φ2 + [Φ1 ∆(t)Φ2 ]T + Φ3 ∆(t)Φ4 + [Φ3 ∆(t)Φ4 ]T . Ω

(38)

Applying Lemma 2.6 in [12] and Schur complement to Eq. (38) results in ˆ = Γ ˆ + 3 Φ1 ΦT1 + −1 ΦT2 Φ2 + 4 Φ3 ΦT3 + −1 ΦT4 Φ4 . Ω 3 4

where Φ1 , Φ2 , Φ3 and Φ4 are defined in (36). Thus, the LMI conditions (36) hold. Hence, by Definition 2.5 and 2.7, the uncertain closed-loop system (10) is robustly stochastically stable with extended dissipative. This completes the proof.  ¯ and matrices G1 , G2 , Si (i = Corollary 3.3: Consider the closed-loop control systems (9). For given scalars κ, µ, ρ, η, h 1, 2, 3, 4) satisfying Assumption 2.4, the system (9) is extended dissipative, if there exist symmetric matrices P > 0, Q > 0, Ri > 0, (i = 1, 2) with ∆K(t) = 0 such that the following LMI together with (15) and (16) are satisfied: h i  ˆ1 ˆ2 ˆ Σ Σ Σ   11×11 ˆ =   Π (39) ∗ −µX 0 <0  ∗ ∗ −κG2−1

and other parameters are defined as in Theorem 3.1. Proof: By following the similar steps as in Theorem 3.1 with some modifications, we can obtain the desired result. So, the proof is omitted here. Corollary 3.4: The closed-loop control systems with randomly occurring uncertainties (10) is stochastically stable and extended dissipative, if there exists symmetric matrices P > 0, Q > 0, Ri > 0, j > 0 (i = 1, 2, j = ¯ ρ, η and matrices G1 , G2 , Si (i = 1, 2, 3, 4) satisfying 1, 2, 3, 4) with ∆K(t) = 0 and for given positive scalars κ, µ, h, Assumption 2.4, such that the following LMI together with (15) and (16) are satisfied: h i  ˆ Φ1 Φ2 Φ3 Φ4 Π   14×14   ∗ − I 0 0 0   3    <0 ∗ ∗ − I 0 0 3     ∗ ∗ ∗ − I 0 4   ∗ ∗ ∗ ∗ −4 I

(40)

and other parameters are defined as in Theorem 3.2. Proof: The proof of this corollary is similar to that of Theorem 3.2, and so we omit it here. Remark 3.5: In theorem 3.1, LKF which considers the time-varying delay h(t) is used to derive the main results. In general, the sampling time-varying delay values in the existing literature are not directly involved in LKF ˙ because the time derivative h(t) = 1, which results in that the output term involving time-varying delay based on R t−h(t) T Rt the term t−h¯ x (s)R1 x(s)ds, t−h(t) xT (s)R1 x(s)ds will vanish. But in this article, we deal this term by using Wirtinger-based integral, which is different from the published works [12], [16]. Remark 3.6: By taking the gain fluctuation into account, based on the proposed state feedback controller design strategy (8), Theorems 3.1 and 3.2 present the robust stabilization of aircraft flight control systems with and without

11

stochastic uncertainty, respectively, where the optimal attenuation levels could be obtained by solving the constrained optimization problem. Our results are more general than some existing ones, for instance, in [21], [23] the authors considered flight control systems, where the stochastic uncertainty was not considered. IV. N UMERICAL E XAMPLE In this section, the simulation results are given to demonstrate the usefulness and applicability of the proposed reliable non-fragile controller for flight control system model as in [23]. Consider the aircraft flight control system model (10), which is borrowed from [23] as follows:  x(t) ˙ = (A + α(t)∆A(t))x(t) + BG1 (K + ∆K(t))x(t − h(t)) + B(Ψ(u(t))) + Dw(t), z(t) = Cx(t), where

      −0.5427 1 0 −0.113 1 h i       A =  −1.069 −0.4134 0 , B = −3.259 , D = 0 , C = 0.01 0 0 . 0 1 0 0 0

Then,using the non-fragile sampled-data controller (8), with perturbed matrices E = [−2 1 2], F (t) = 0.1 sin(−t),  2 1 −3   H = 0 2 −3 and the non-linear reliable feedback controller (4), the fault parameters are chosen as G1 = 0.7

1 2 1 and G2 = 0.01. For this uncertain case, it is also assumed that M = 0.5I , F (t) = 0.1 sin(−t) and N = 0.2I . The designing parameters are taken to be µ = 160, α ¯ = 0.1, κ = 2, η = 0.01, and ρ = 0.7 and solving the LMIs in Theorem ¯ = 1.1078. Moreover, the state feedback control gain matrix is 3.2, we can get the maximum allowable bound h obtained as h i K = 0.6180 2.1423 1.1590 .

With the obtained control gain matrix, the initial condition x(0) = [1 − 3 10.2] and the disturbance w(t) = 0.1 sin(t). From Fig. 1(a) and 1(b), we see that the simulation result for the closed-loop system is stochastically stable under the proposed extended-dissipative based non-fragile controller even in the presence of nonlinear actuator fault and control performance of uncertain closed-loop flight control systems (10), respectively. Further, it is noted from Fig. 2a that the open-loop system is unstable. The state responses of the disturbance input are presented in Fig. 2b. The computation results are listed in TABLE I. TABLE I: Allowable upper bounds Methods Theorem 3.2 Theorem 3.2 of [12] Theorem 3.1 of [16]

Upper Bound 1.1078 0.7298 1.0463

Control K = [ 0.6180 K = [ -0.4649 K = [ 0.0341

Gain Matrix 2.1423 1.1590 ] 0.3634 0.1473 ] 0.0433 0.0327 ]

¯ Remark 4.1: It should be pointed out that the Table I, we can easily see that the maximum allowed value of h obtained by Theorem 3.2 is far greater than the values in [12] and [16]. From the simulation results, we can conclude that the proposed controller effectively overcomes the presence of nonlinear actuator faults, stochastic uncertainty and gain variations. Therefore, the proposed control design is

12

15

1.5

x1 (t) x2 (t) x3 (t)

uF (t)

1

Control Response

State Responses

10 5 0

0.5 0

−0.5

−5

−10 0

−1

10

20

30

40

−1.5 0

50

10

Time (s)

20

30

40

50

Time (s)

(a) State Responses

(b) Control

12

0.2

10

x1 (t)

0.15

8

x2 (t)

6

x3 (t)

4 2

0.05 0

−0.05

0

−0.1

−2

−0.15

−4 0

w(t)

0.1

Disturbance

State Responses

Fig. 1: Output responses of uncertain system (10) according to Theorem 3.2

10

20

30

40

50

−0.2 0

Time (s)

(a) State responses

10

20

30

40

50

Time (s)

(b) Disturbance

Fig. 2: Openloop and disturbance of uncertain system (10) according to Theorem 3.2

very much suitable for real-time situations which shows the usefulness and applicability of the obtained theoretical results. V. C ONCLUSION In this paper, the robust analysis of non-fragile controller for delayed aircraft flight control systems via extended dissipative-based sampled-data approach has been discussed. The Lyapunov stability theory and Writinger-based inequality have been utilized to solve the proposed control problem. Moreover, sufficient conditions have been obtained in terms of LMIs to ensure the robustly extended dissipative and stochastic stability of the aircraft flight control systems for all admissible randomly occurring uncertainties and time delays. Also, attenuate the effect of disturbance input on the controlled output to a prescribed extended dissipative index level η > 0. The obtained theoretical results of all the fault scenarios have been validated through numerical simulations. The finite-time stabilization condition for aircraft flight control systems with actuator saturation and equivalent-input-disturbance via repetitive control is an untreated issue which will be topic of our future works. VI. ACKNOWLEDGMENT This work was jointly supported by the National Natural Science Foundation of China (61773217), RUSAPhase 2.0 Grant No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, UGC-SAP (DRS-I)

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Grant No.F.510/8/DRS-I/2016(SAP-I) and DST (FIST-level I) 657876570 Grant No.SR/FIST/MS-I/2018/17, Hunan Provincial Science and Technology Project Foundation (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101) and the Construct Program of the Key Discipline in Hunan Province. R EFERENCES [1] M. Fang and J. H. Park, Non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation, Applied Mathematics and Computation, 219 (2013) 8009-8017. [2] H. Gao, W. Sun and P. Shi, Robust sampled data H∞ control for vehicle suspension systems, IEEE Transactions on Control Systems Technology, 18 (2010) 238-245. [3] M. Kchaou, A. EI. Hajjaji and A. Toumi, Non-fragile H∞ output feedback control design for continuous-time fuzzy systems, ISA Transactions, 54 (2015) 3-14. [4] K.S. Kim, K.J. Lee, and Y.D. Kim, Reconfigurable Flight Control System Design Using Direct Adaptive Method, Journal of Guidance, Control, and Dynamics, 26 (2003) 543-550. [5] S. K. Kommuri, M. Defoort, H. R. Karimi and K. C. Veluvolu, A robust observer-based sensor fault-tolerant control for PMSM in electric vehicles, IEEE Transactions on Industrial Electronics, 63 (2016) 7671-7681. [6] T. H. Lee, M. J. Park, J. H. Park, O. M. Kwon and S. M. Lee, Extended dissipative analysis for neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 25 (2014) 1936-1941. [7] F. Liao, J. L. Wang, and G. H. Yang, Reliable Robust Flight Tracking Control: An LMI Approach, IEEE Transactions on Control Systems Technology, 10 (2002) 76-89. [8] R. Manivannan, R. Samidurai, Jinde Cao, Ahmed Alsaedi, Non-Fragile Extended Dissipativity Control Design for Generalized Neural Networks with Interval Time-Delay Signals, Asian Journal of Control, 2018. DOI:10.1002/asjc.1752 [9] R. Manivannan, R. Samidurai, Jinde Cao, Ahmed Alsaedi, Fuad E. Alsaadi, Design of extended dissipativity state estimation for generalized neural networks with mixed time-varying delay signals, Information Sciences, 424 (2018) 175-203. [10] M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee and E. J. Cha, Synchronization of discrete-time complex dynamical networks with interval time-varying delays via non-fragile controller with randomly occurring perturbation, Journal of the Franklin Institute, 351 (2014) 4850-4871. [11] R. Sakthivel, D. Aravindh, P. Selvaraj, S. Vimal Kumar and S. Marshal Anthoni, Vibration control of structural systems via robust non-fragile sampled-data control scheme, Journal of the Franklin Institute, 3 (2017) 1265-1284. [12] R. Sakthivel, S. Vimal Kumar, D. Aravindh and P. Selvaraj, Reliable dissipative sampled-data control for uncertain systems with nonlinear fault input, Journal of Computational Nonlinear Dynamics, 11 (2015) 1-9. [13] A. Seuret and F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica, 49 (2013) 2860-2866. [14] H. Shen, Y. Zhu, L. Zhang and Ju H. Park, Extended dissipative state estimation for Markov jump neural networks with unreliable links, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017) 346-358. [15] S. Vimal Kumar, R. Raja, S. Marshal Anthoni, Jinde Cao and Zhengwen Tu, Robust finite-time non-fragile sampled-data control for T-S fuzzy flexible spacecraft model with stochastic actuator faults, Applied Mathematics and Computation, 321 (2018) 483-497. [16] S. Vimal Kumar, S. Marshal Anthoni and R. Raja, Dissipative analysis for aircraft flight control systems with randomly occurring uncertainties via non-fragile sampled-data control, Mathematics and Computers in Simulation, 155 (2019) 217-226. [17] Y. Wei, J. Qiu and H. R. Karimi, Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults, IEEE Transactions on Circuits and Systems I: Regular Papers, 64 (2017) 170-181. [18] Wenqian Xie, Hong Zhu, Shouming Zhong, Dian Zhang, Kaibo Shi and Jun Cheng, Extended dissipative estimator design for uncertain switched delayed neural networks via a novel triple integral inequality, Applied Mathematics and Computation, 335 (2018) 82-102. [19] Wenqian Xie, Hong Zhu, Shouming Zhong, Jun Cheng, Kaibo Shi and Jun Cheng, Extended dissipative resilient estimator design for discrete-time switched neural networks with unreliable links, Nonlinear Analysis: Hybrid Systems, 32 (2019) 19-36. [20] Z. G. Wu, J. H. Park, H. Su and J. Chu, Non-fragile synchronization control for complex networks with missing data, International Journal of Control, 86 (2013) 555-566. [21] Y. Xiang and Z. Youmin, Design of passive fault-tolerant flight controller against actuator failure, Chinese Journal of Aeronautics, 28 (2015) 180-190. [22] G. H. Yang, J. L. Wang, and Y. C. Soh, Reliable H∞ Controller Design for Linear Systems, Automatica, 37 (2001) 717-725. [23] Z. Yingxin, W. Qing, D. Chaoyang, and J. Yifan, H∞ output tracking control for flight control systems with time-varying delay, Chinese Journal of Aeronautics, 26 (2013) 1251-1258.

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[24] Y. M. Zhang and J. Jiang, Integrated Active Fault-Tolerant Control Using IMM Approach, IEEE Transactions on Aerospace and Electronic Systems, 37 (2001) 1221-1235. [25] Y. M. Zhang and J. Jiang, Integrated Design of Reconfigurable Fault Tolerant Control Systems, Journal of Guidance, Control, and Dynamics, 24 (2001) 133-136. [26] Y. Zhang and Y. Tian, Consensus of data-sampled multi-agent systems with random communication delay and packet loss, IEEE Transactions on Automatic Control, 55 (2010) 939-943. [27] B. Zhang, W. Zheng and S. Xu, Filtering of Markovian jump delay systems based on a new performance index, IEEE Transactions on Circuits and Systems I: Regular Papers, 60 (2013) 1250-1263.