Robust Finite-Time H-Infinity Control with Transients for Dynamic

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 2838749, 17 pages https://doi.org/10.1155/2018/2838749

Research Article Robust Finite-Time H-Infinity Control with Transients for Dynamic Positioning Ship Subject to Input Delay Xiaogong Lin, Kun Liang , Heng Li, Yuzhao Jiao , and Jun Nie College of Automation, Harbin Engineering University, Harbin, China Correspondence should be addressed to Kun Liang; [email protected] Received 1 January 2018; Revised 4 April 2018; Accepted 16 May 2018; Published 26 June 2018 Academic Editor: Sabri Arik Copyright © 2018 Xiaogong Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents the problem of robust finite-time 𝐻∞ control with transients for ocean surface vessels equipped with dynamic positioning (DP) system in presence of input delay. The main objective of this work is to design a finite-time 𝐻∞ state feedback controller, which ensures that all states of ship do not exceed a given threshold over a fixed time interval, with better robustness and transient performance subject to time-varying disturbance. Based on a novel augmented Lyapunov-Krasovskii-like function (LKLF) with triple integral terms and a method combining the Wirtinger inequality and reciprocally convex approach, a less conservative result is derived. In particular, an 𝐻∞ performance index with nonzero initial condition is introduced to attenuate the overconservatism caused by the assumption of zero initial condition and enhance the transient performance of ship subject to external disturbance. More precisely, the controller gain matrix for the DP system can be achieved by solving the linear matrix inequalities (LMIs), which can be easily facilitated by using some standard numerical packages. Finally, a numerical simulation for a ship is proposed to verify the effectiveness and less conservatism of the controller we designed.

1. Introductions With the increasing development in the ocean exploitation, dynamic positioning (DP) systems, regulating the horizontal position and heading of the vessel exclusively by means of active thrusters, have been developed for various marine and offshore applications such as drilling, salvage, pipelaying, and oil production [1]. To achieve expected trajectory tracking or positioning, various control strategies have been proposed, including robust adaptive control [2], sliding mode control [3], prescribed performance control [4], hybrid control [5], and neural network control [6]. However, in some applications, it is significant to maintain the vessel’s states under some bounds, particularly of which transient performance is emphasized, during a specific time interval, for example, when facing the matter of saturations or when the task of trajectory tracking should be fulfilled in a prescribed time interval. In [7], the proposed controller can maintain the bound of ship over an infinity time interval regardless of disturbance. However, it is inconvenient to analyze and to enhance the transient performance when the operation should be arrived in short time, such as rescuing

works. On the other hand, time delay is encountered in many dynamical systems and often leads to performance deterioration which engenders strongly growing interest in this topic in recent decades. In the DP system, the main kind of time delay is encountered in actuators [8], while another obvious kind of delay is the one produced between the sensors and the activation of the control mechanism [9]. The effect caused by time delay will be more significant in finite-time interval and it is the first motivation of this paper. The classical 𝐻∞ control theory [10] defines the control law, under the assumption of zero initial condition, providing the minimal value to the performance measure that is the worst-case norm of the regulated output over all exogenous signals and is applied to DP system successfully [11]. However, there exist some situations, when the initial state of ship is possibly nonzero, such as when saturation is activated or controller is switched to another one. It will cause an additional unknown disturbance [12] and the promising robustness is achieved at the expense of degraded nominal performance [13]. In [14], researchers introduced a performance measure that is the induced norm of the regulated output over all exogenous signals and initial states for finite

2 and infinite horizons. Unfortunately, to the best the authors’ knowledge, it has not been extended to the system with input delay. On another research frontier, various approaches, in the framework of finite-time boundedness for time-delay systems, have been developed to obtain the results with less conservatism [15, 16], usually indicated by the bound of state [17]. For DP system, engineers are willing to obtain control strategies which maintain the state of ship varying in a small region around the desired set-point or tracking path instead of the acceptable minimum state bound. Thus, the second motivate is to obtain a result which reduces the overconservatism caused by the assumption of zero initial condition and the loss information in the proof for time delay systems in a practical view. Based on the discussion aforementioned, the problem of robust finite-time 𝐻∞ control for DP system with input delay is studied. The main contribution of this paper lies in three aspects: Firstly, a finite-time 𝐻∞ controller with transients is designed for DP systems with input delay, by solving a couple of LMIs, which can guarantee the state of ship within a desired value over a fixed time interval in the presence of time-varying disturbances. Secondly, the concept of 𝐻∞ control under nonzero initial condition is introduced to time-delay system and its advantage on enhancing the transient performance will be shown in a numerical simulation compared to the 𝐻∞ with zero initial condition. In particular, the results are established in forms of LMI, which can be easily facilitated by using some standard numerical packages. Thirdly, a novel augmented LKLF with triple integral terms, which contains more information is constructed; meanwhile a method combining the Wirtinger inequality and reciprocally convex approach is applied to obtain a tight bound of the integral terms of quadratic functions which may lead to a less conservative result compared to previous works. Moreover, the practical significance of this method in engineering will be demonstrated later. Finally, a numerical simulation for a ship is proposed to verify the effectiveness and advantage of the controller we designed. The rest of this paper is organized as follows: In Section 2, the problem formulation of finite-time 𝐻∞ control is detailed for vessels while some definitions and lemmas are introduced as preparation. In Section 3, the method to design finite-time 𝐻∞ controller is proposed, and the proposed control schemes are simulated in Section 4. In Section 5, the conclusion is drawn. Notation. Throughout this paper, R𝑛 is the 𝑛-dimensional Euclidean vector space, and R𝑚×𝑛 denotes the set of all 𝑚 × 𝑛 real matrices. For symmetric matrices 𝑋 and 𝑌, 𝑋 > 𝑌 (respectively, 𝑋 ≥ 𝑌 ) means that 𝑋 − 𝑌 is positive definite (respectively, positive semidefinite). The superscript “𝑇” represents the transpose. The symmetric terms in a symmetric matrix are denoted by “∗”. Moreover, we use 𝜆 max (⋅)(𝜆 min (⋅) ) to denote the maximum (minimum) eigenvalue of a symmetric matrix.

2. Problem Formulation 2.1. Model of DP System. At first, DP system model with three-DOP under low speed can be described as [18]

Mathematical Problems in Engineering 𝜂̇ (𝑡) = 𝐽 (𝜓) 𝜐 (𝑡)

(1)

𝑀𝜐̇ (𝑡) + 𝐷𝜐 (𝑡) = 𝑢 (𝑡 − ℎ (𝑡)) + 𝜔 (𝑡)

(2)

where 𝜐 = (𝜇 V 𝑟) is a vector of velocities given in the body-fixed coordinate system and 𝜂 = (𝑋 𝑌 𝜓) is the position and orientation of the vessel with respect to an inertial reference coordinate system. 𝑢(𝑡) is a control vector of forces and moments provided by the propulsion systems. ℎ(𝑡) is a time-varying function that expresses the actuator delay ̇ < 𝜇 . 𝜔(𝑡) is the disturbance input and satisfies ℎ(𝑡) < ℎ ℎ(𝑡) ∞ of the system and satisfies the condition of ∫0 𝜔(𝑡)𝑇 𝜔(𝑡)𝑑𝑡 < 𝑑 . 𝐽(𝜓) is the transformation matrix between the inertial and body-fixed coordinate frames. The inertia matrix 𝑀 includes hydrodynamic added inertia, and 𝐷 is a strictly positive damping matrix due to linear wave drift damping and laminar friction. Then, the structures of the matrices 𝐽(𝜓), 𝑀, and 𝐷 can be explicitly given as follows: cos (𝜓) −sin (𝜓) 0 𝐽 (𝜓) = ( sin (𝜓) cos (𝜓) 0) 0

0

𝑚 − 𝑋𝑢̇

0

0

𝑚 − 𝑌V̇

0

𝑚𝑥𝑔 − 𝑌𝑟̇

𝑀=(

𝑋𝑢 0 𝐷 = −( 0

1 0 𝑚𝑥𝑔 − 𝑌𝑟̇)

(3)

𝐼𝑧 − 𝑁𝑟̇

0

𝑌V 𝑌𝑟 )

0 𝑁V 𝑁𝑟 where 𝑋𝑢 ,𝑌V ,𝑌𝑟 ,𝑁𝑟 ,𝑋𝑢̇ ,𝑌V̇ ,𝑌𝑟̇, and 𝑁𝑟̇ are the hydrodynamic parameters of the vessel. 𝑚 is the mass of the ship, and 𝐼𝑧 is the moment of inertia about the yaw rotation. 𝑥𝑔 is the vertical distance from coordinate origin to center of gravity in body-fixed frame. To simplify the model, some assumptions are introduced first. Assumption 1. All the parameters of state are available. Assumption 2. The roll and pitch angles are small enough; for DP system, it is a reasonable assumption. Under assumption 2, we can obtain the simplified equation in form of state-space [19] 𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝐵1 𝑢 (𝑡 − ℎ (𝑡)) + 𝐵2 𝜔 (𝑡)

(4)

𝑧 (𝑡) = 𝐶𝑥 (𝑡) + 𝐷𝑢 (𝑡 − ℎ (𝑡))

(5)

where 𝑥 (𝑡) = (𝜂 (𝑡) 𝜐 (𝑡)) 𝐴=(

0

𝐼

)

0 −𝑀−1 𝐷

Mathematical Problems in Engineering 𝐵1 = 𝐵2 = (

0 −1

𝑀

3 fact that system state reaches the equilibrium point of system in a finite time [24].

). (6)

Under assumption 1, we design the full-state feedback as 𝑢 (𝑡 − ℎ (𝑡)) = 𝐾𝑥 (𝑡 − ℎ (𝑡))

(7)

so the (1) and (2) can be rewritten as 𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝐵1 𝐾𝑥 (𝑡 − ℎ (𝑡)) + 𝐵2 𝜔 (𝑡)

(8)

𝑧 (𝑡) = 𝐶𝑥 (𝑡) + 𝐷𝐾𝑥 (𝑡 − ℎ (𝑡)) .

(9)

Remark 3. In the situation of DP motion, the motion in heave, roll, and pitch will be ignored since we focus on the motion on the surface of sea. Meanwhile, velocity of vessel is low enough, so the Coriolis-Centripetal matrix and nonlinearities in damping matrix can be neglected [18]. Remark 4. In measurement subsystem of DP system, various sensors are installed to obtain the state of the vessel motion accurately, including global position system for the vessel’ s position, gyrocompass for the vessel’ s heading, and attitude sensor for the pitch and roll. Meanwhile, data fusion technology is applied to DP system to obtain more accurate state information of vessel motion [18]. 2.2. Preliminaries. In the sequel, some definitions and lemmas are introduced to obtain the results. Definition 5 ((FTB), see [20]). Given a positive definite matrix 𝑅 and three positive constants 𝑐1 , 𝑐2 , 𝑇 with 𝑐1 < 𝑐2 , the time-delay system (4) with 𝑢(𝑡) = 0 is said to be finite-time boundedness with respect to (𝑐1 𝑐2 𝑑 𝑇 𝜏 𝑅), ̇ if sup−𝜏 0 and a differentiable signal 𝑥 in [−𝑎, −𝑏] → R𝑛 , the following inequality holds: −𝑏

−𝑎

−𝑏 1 ⋅ Π1 (𝑆1 , 𝑥 (−𝑏) , 𝑥 (−𝑎) , ∫ 𝑥𝑇 (𝑢) 𝑑𝑢) 𝑎 − 𝑏 −𝑎 𝑇

Π1 (𝑆1 , 𝛼, 𝛽, 𝛾)

=

sup

‖𝜔(𝑡)‖2,[0,𝑇] 2 +𝑥(0)𝑇 𝑆𝑥(0)=0̸

[

‖𝑧 (𝑡)‖2,[0,𝑇]

‖𝜔 (𝑡)‖2,[0,𝑇] 2 + 𝑥 (0)𝑇 𝑆𝑥 (0)

]

Lemma 10 (see [29]). For constant matrices 𝑆1 > 0, 𝑆2 > 0 and constant scalars 𝑎 > 𝑏, the following inequalities hold for all continuously function 𝑥 in [−𝑎, −𝑏] → R𝑛 : −𝑏

𝑡−𝑏

−𝑎

𝑡+𝜃

∫ ∫

𝑥̇𝑇 (𝑢) 𝑆2 𝑥̇ (𝑢) 𝑑𝑢𝑑𝜃

≥ Π2 (𝑆2 , 𝑥𝑇 (𝑡 − 𝑏) , −𝑏

𝑡−𝑏

−𝑎

𝑡+𝜃

⋅∫ ∫ −𝑏

𝑡+𝜃

−𝑎

𝑡−𝑎

∫ ∫

where 𝛾 is a prescribed scalar and 𝑆 is a weighting diagonal matrix that penalizes the effect caused by the initial state. Remark 7. It is necessary to distinguish between finite-time stability and finite-time attractiveness [21]. The first concept is to maintain system states within a given boundary in a specified time interval [22, 23], while the latter describes the

𝑡−𝑏 1 1 ∫ 𝑥𝑇 (𝑢) 𝑑𝑢, 𝑎 − 𝑏 𝑡−𝑎 (𝑎 − 𝑏)2

𝑥𝑇 (𝑢) 𝑑𝑢𝑑𝜃)

(13)

𝑇

𝑥̇ (𝑢) 𝑆3 𝑥̇ (𝑢) 𝑑𝑢𝑑𝜃

≥ Π3 (𝑆3 , 𝑥𝑇 (𝑡 − 𝑎) ,

(10)

0,∑𝑖 𝛼𝑖 =1}

∑ 𝑖

1 𝑓 (𝑡) = ∑𝑓𝑖 (𝑡) + max∑𝑔𝑖,𝑗 (𝑡) 𝑔𝑖,𝑗 (𝑡) 𝛼𝑖 𝑖 𝑖 𝑖=𝑗̸

subject to {𝑔𝑖,𝑗 : 𝑅𝑚 󳨃󳨀→ 𝑅, 𝑔𝑗,𝑖 (𝑡) = 𝑔𝑖,𝑗 (𝑡) , [

Lemma 12 (see [31]). For any matrix 𝑆4 > 0 and a vector function 𝑥 in [−𝑎, −𝑏] → R𝑛 , if the integrals concerned are well defined, then the following inequality holds: −𝑏

∫ 𝑥𝑇 (𝑠) 𝑆4 𝑥 (𝑠) 𝑑𝑠 −𝑎

−𝑏 −𝑏 1 ≥ (∫ 𝑥𝑇 (𝑠) 𝑑𝑠) 𝑆4 (∫ 𝑥 (𝑠) 𝑑𝑠) . 𝑎 − 𝑏 −𝑎 −𝑎

𝑓𝑖 (𝑡) 𝑔𝑖,𝑗 (𝑡) 𝑔𝑗,𝑖 (𝑡) 𝑓𝑗 (𝑡)

(15) ] ≥ 0} .

3𝑊 𝑍2

(

∗

)>0 3𝑊

1 𝑐1 𝜃2 + ℎ2 𝑐1 𝜃3 + 𝜌ℎ𝑐1 𝜃4 + 𝜌ℎ𝑐1 𝜃5 + 𝜌ℎ3 𝑐1 𝜃6 2 1 1 + 𝜌ℎ3 𝑐1 𝜃7 + 𝜌ℎ3 𝑐1 𝜃8 + 𝛽𝜃9 𝑑 < 𝑒−𝛼𝑇 𝑐2 𝜃1 6 3

(16)

where 2.3. Control Objective. Finally, the objective of this paper is to derive the control gain 𝐾 such that (1) the closed-loop system (8) with 𝑢(𝑡) = 0 is FTB; (2) under given nonzero initial condition, the closed 𝑇 system (8) and (9) guarantees that 𝐽 = ∫0 𝑧𝑇 (𝑠)𝑧(𝑠) − 𝛾2 𝜔𝑇 (𝑠)𝜔(𝑠) − 𝛾2 (1/𝑇)𝑥𝑇 (0)𝑆𝑥(0)𝑑𝑠 < 0 for all nonzero 𝜔(𝑡) and 𝑡 ∈ [0, 𝑇].

The designing of the robust finite-time 𝐻∞ controller for the DP system with input delay is divided into three steps. 3.1. FTB Analysis of DP System. Firstly, the result guaranteeing the FTB of DP system is established in this subsection. Theorem 13. Given five positive scalars 𝛼, 𝛽, 𝑐1 , 𝑐2 , 𝑇 and positive define matrix 𝑅, the finite-time boundedness problem of system (4) is solvable if there exist positive scalars 𝜃𝑙 (𝑙 = 1, 2, ⋅ ⋅ ⋅ , 8) and matrices 𝑃11 > 0, 𝑃22 > 0, 𝑊 > 0, 𝑄𝑖 > 0 (𝑖 = 1, 2), 𝑈𝑗 > 0 (𝑗 = 1, 2), 𝑀 > 0, 𝑍1 , 𝑍2 with appropriate dimensions, satisfying the following conditions:

𝑊 𝑍1

(

∗ 𝑊

)>0

1 + 𝐴𝑇 ℎ2 (𝑈1 + 𝑈2 + 2𝑊) 𝐴 − 4𝜌𝑊 − 3𝜌𝑈1 2 − 9𝜌𝑈2 , Ω22 = −𝜌𝑄2 − 4𝜌𝑊 − 6𝜌𝑈2 , 1 Ω33 = 𝐾𝑇𝐵1 𝑇 ℎ2 (2𝑊 + 𝑈1 + 𝑈2 ) 𝐵1 𝐾 − 8𝜌𝑊 2

3. Main Results

Ω = (Ω𝑖𝑗 )8×8 < 0

Ω11 = 𝐴𝑇 𝑃11 + 𝑃11 𝐴 − 𝛼𝑃11 + 𝑄1 + 𝑄2

(17) (18)

+ 𝜌𝑍1 𝑇 + 𝜌𝑍1 − 𝜌𝑍2 𝑇 − 𝜌𝑍2 − (1 − 𝜇) 𝑄2 , 1 Ω44 = 𝐵2 𝑇 ℎ2 (2𝑊 + 𝑈1 + 𝑈2 ) 𝐵2 − 𝛽𝑀, 2 Ω66 = −12𝜌𝑊, Ω77 = −12𝜌𝑊, Ω55 = 𝛼ℎ2 𝑃22 − 18𝜌𝑈2 − 6𝜌𝑈1 , Ω88 = −36𝜌𝑈1 − 36𝜌𝑈2 , Ω12 = 𝜌𝑍1 − 𝜌𝑍2 + 6𝜌𝑈2 , Ω15 = ℎ𝑃22 + 12𝜌𝑈2 , Ω16 = 6𝜌𝑊, Ω17 = 2𝜌𝑍2 ,

(19)

(20)

Mathematical Problems in Engineering

5

1 Ω13 = 𝑃11 𝐵1 𝐾 + 𝐴𝑇 ℎ2 (2𝑊 + 𝑈1 + 𝑈2 ) 𝐵1 𝐾 − 2𝜌𝑊 2 − 𝜌𝑍1 − 𝜌𝑍2 , 1 Ω14 = 𝑃11 𝐵2 + 𝐴𝑇 ℎ2 (2𝑊 + 𝑈1 + 𝑈2 ) 𝐵2 , 2 Ω18 = −18𝜌𝑈2 + 6𝜌𝑈1 ,

̃𝑖 = 𝑅−1/2 𝑄𝑖 𝑅−1/2 𝑄

(𝑖 = 1, 2, 3) ,

̃ = 𝑅−1/2 𝑊𝑅−1/2 , 𝑊 (𝑗 = 1, 2) ,

0 < 𝜃1 𝐼 < 𝑃11 < 𝜃2 𝐼,

Ω25 = −ℎ𝑃22 − 6𝜌𝑈2 ,

0 < 𝑃22 < 𝜃3 𝐼,

Ω26 = 2𝜌𝑍2 𝑇 ,

0 < 𝑄1 < 𝜃4 𝐼,

Ω27 = 6𝜌𝑊,

0 < 𝑄2 < 𝜃5 𝐼,

Ω28 = 12𝜌𝑈2 , Ω34 = 𝐾 𝐵1

𝑃̃22 = 𝑅−1/2 𝑃22 𝑅−1/2 ,

̃𝑗 = 𝑅−1/2 𝑈𝑗 𝑅−1/2 𝑈

Ω23 = −𝜌𝑍1 𝑇 − 𝜌𝑍2 𝑇 − 2𝜌𝑊,

𝑇

𝑃̃11 = 𝑅−1/2 𝑃11 𝑅−1/2 ,

0 < 𝑊 < 𝜃6 𝐼,

𝑇1 2

2

ℎ (2𝑊 + 𝑈1 + 𝑈2 ) 𝐵2 ,

0 < 𝑈1 < 𝜃7 𝐼,

Ω36 = 6𝜌𝑊 + 2𝜌𝑍2 𝑇 ,

0 < 𝑈2 < 𝜃8 𝐼,

Ω37 = 6𝜌𝑊 + 2𝜌𝑍2 ,

0 < 𝑀 < 𝜃9 𝐼. (21)

Ω58 = 12𝜌𝑈1 + 24𝜌𝑈2 , Ω67 = −4𝜌𝑍2 ,

Proof. Consider the candidate augment LKLF as follows:

Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56 = Ω57 = Ω68 = Ω78 = 0, 𝜃1 = 𝜆 min (𝑃̃11 ) 𝜃2 = 𝜆 max (𝑃̃11 ) , 𝜃3 = 𝜆 max (𝑃̃22 ) , ̃1 ) , 𝜃4 = 𝜆 max (𝑄 ̃2 ) , 𝜃5 = 𝜆 max (𝑄 ̃ , 𝜃6 = 𝜆 max (𝑊) ̃1 ) , 𝜃7 = 𝜆 max (𝑈 ̃2 ) , 𝜃8 = 𝜆 max (𝑈 𝜃9 = 𝜆 max (𝑀) , 𝜌 = 𝑒𝛼ℎ ,

𝑉 (𝑡) = 𝑉1 (𝑡) + 𝑉2 (𝑡) + 𝑉3 (𝑡) + 𝑉4 (𝑡)

(22)

𝑉1 (𝑡) = 𝜀 (𝑡)𝑇 𝑃𝜀 (𝑡)

(23)

𝑉2 (𝑡) = ∫

𝑡

𝑡−ℎ

+∫

𝑒𝛼(𝑡−𝑠) 𝑥 (𝑠)𝑇 𝑄1 𝑥 (𝑠) 𝑑𝑠 (24)

𝑡

𝛼(𝑡−𝑠)

𝑒

𝑡−ℎ(𝑡)

0

𝑉3 (𝑡) = ℎ ∫ ∫

𝑡

−ℎ 𝑡+𝜃

0

0

0

𝑥 (𝑠) 𝑄2 𝑥 (𝑠) 𝑑𝑠

𝑒𝛼(𝑡−𝑠) 𝑥̇ (𝑠)𝑇 𝑊𝑥̇ (𝑠) 𝑑𝑠𝑑𝜃

𝑡

𝑉4 (𝑡) = ∫ ∫ ∫ −ℎ 𝜃

𝑇

𝑡+𝛽

𝜃

(25)

𝑒𝛼(𝑡−𝑠) 𝑥̇ (𝑠)𝑇 𝑈1 𝑥̇ (𝑠) 𝑑𝑠𝑑𝛽𝑑𝜃

+∫ ∫ ∫

𝑡

−ℎ −ℎ 𝑡+𝛽

(26) 𝛼(𝑡−𝑠)

𝑒

𝑇

𝑥̇ (𝑠) 𝑈2 𝑥̇ (𝑠) 𝑑𝑠𝑑𝛽𝑑𝜃

𝑡

where 𝜀(𝑡) = (𝑥(𝑡) ∫𝑡−ℎ 𝑥(𝑠)𝑑𝑠) 𝑃 = ( 𝑃∗11 𝑃022 ). To deal with the formulas conveniently, let us provide such definitions:

𝜉 (𝑡) = (𝑥 (𝑡) 𝑥 (𝑡 − ℎ) 𝑥 (𝑡 − ℎ (𝑡)) 𝜔 (𝑡) 0 ⋅ ⋅ ⋅ 0) . 0 ⋅ ⋅ ⋅ 0 1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑙𝑘 𝑇 = (⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑘−1

8−𝑘

𝑡 𝑡−ℎ(𝑡) 1 1 1 0 𝑡 1 𝑡 𝑥 (𝑠) 𝑑𝑠 𝑥 (𝑠) 𝑑𝑠 2 ∫ ∫ 𝑥 (𝑠) 𝑑𝑠𝑑𝜃) (27) ∫ 𝑥 (𝑠) 𝑑𝑠 ∫ ∫ ℎ 𝑡−ℎ ℎ (𝑡) 𝑡−ℎ(𝑡) ℎ − ℎ (𝑡) 𝑡−ℎ ℎ −ℎ 𝑡+𝜃

6

Mathematical Problems in Engineering

The time-derivative of 𝑉(𝑡) along system (4) can be bounded as

Λ1 = Λ2 = (

𝑉1̇ (𝑡) = 𝛼𝑉1 (𝑡) + 𝜀 ̇ (𝑡)𝑇 𝑃𝜀 (𝑡) + 𝜀 (𝑡)𝑇 𝑃𝜀 ̇ (𝑡) − 𝛼𝑉1 (𝑡) (28) 𝑉2̇ (𝑡)

𝑇

𝑉3̇ (𝑡) ≤ 𝛼𝑉3 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑊𝑥̇ (𝑡) − 𝑒𝛼ℎ 𝜛3 𝑇 Λ 3 𝜛3

𝑇

− (1 − ℎ̇ (𝑡)) 𝑒𝛼ℎ(𝑡) 𝑥 (𝑡 − ℎ (𝑡))𝑇 𝑄2 𝑥 (𝑡 − ℎ (𝑡))

where 𝜛3 𝑇 = ( 𝜛𝜛1 𝑇 ), Λ 3 = ( ∗∗ 2

1 𝑉4̇ (𝑡) ≤ 𝛼𝑉4 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑈1 𝑥̇ (𝑡) 2

− (1 − 𝜇) 𝑥 (𝑡 − ℎ (𝑡))𝑇 𝑄2 𝑥 (𝑡 − ℎ (𝑡))

1 + ℎ2 𝑥̇ (𝑡)𝑇 𝑈2 𝑥̇ (𝑡) − 𝑒𝛼ℎ 𝜛4 𝑇 Λ 4 𝜛4 2

𝑉3̇ (𝑡) = 𝛼𝑉3 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑊2 𝑥̇ (𝑡)

(30)

𝜛4 𝑇

1 = 𝛼𝑉4 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑈1 𝑥̇ (𝑡) 2 −𝑒

∫ ∫

𝑡

0

𝑡+𝛽

−ℎ 𝑡−ℎ

𝑥̇ (𝑠) 𝑈1 𝑥̇ (𝑠) 𝑑𝑠𝑑𝛽

6𝑈1 −12𝑈1

(31) Λ4 = (

𝑥̇ (𝑠)𝑇 𝑈2 𝑥̇ (𝑠) 𝑑𝑠𝑑𝛽.

𝑉3̇ (𝑡) = 𝛼𝑉3 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑊𝑥̇ (𝑡) − 𝑒𝛼ℎ ℎ ∫

𝑡−ℎ(𝑡) 𝑡−ℎ(𝑡)

− 𝑒𝛼ℎ ℎ ∫

𝑡−ℎ

𝑥̇ (𝑠)𝑇 𝑊𝑥̇ (𝑠) 𝑑𝑠

∗

∗

∗

0

6𝑈2 −12𝑈2 ∗

).

36𝑈2

𝑉 (𝑡) < 𝑒𝛼𝑡 𝑉 (0) + 𝛽 ∫ 𝑒𝛼(𝑡−𝑠) 𝜔𝑇 (𝑠) 𝑀𝜔 (𝑠) 𝑑𝑠 0

where 𝑇 𝑇

(𝑙1 − 𝑙3 ) 𝜛1 𝑇 = 𝜉𝑇 (𝑡) ( ) 𝑇 𝑇 (𝑙1 + 𝑙3 𝑇 − 2𝑙6 𝑇 ) 𝑇

(𝑙3 𝑇 − 𝑙2 𝑇 )

(37)

𝑡

(32)

ℎ ℎ − 𝑒𝛼ℎ 𝜛1 𝑇 Λ 1 𝜛1 − 𝑒𝛼ℎ 𝜛 𝑇Λ 𝜛 ℎ (𝑡) ℎ − ℎ (𝑡) 2 2 2

𝑇

∗

0

Assuming that Ω < 0 and integrating the left part of inequality (37),

≤ 𝛼𝑉3 (𝑡) + ℎ2 𝑥̇ (𝑡)𝑇 𝑊𝑥̇ (𝑡)

𝑇

36𝑈1

0

𝑉̇ − 𝛼𝑉 − 𝛽𝜔𝑇 (𝑡) 𝑀𝜔 (𝑡) < 𝜉𝑇 (𝑡) Ω𝜉 (𝑡) .

𝑥̇ (𝑠)𝑇 𝑊𝑥̇ (𝑠) 𝑑𝑠

𝑇

∗

0

(36)

Combining (28), (29), (34), and (35) with the definition of 𝜌 = 𝑒𝛼ℎ , we can obtain

Invoking Lemma 9, we can obtain

𝑡

) ) ) )

𝑇 1 ( 𝑙2 𝑇 − 𝑙5 𝑇 + 𝑙8 𝑇 ) ) ( 2

1 + ℎ2 𝑥̇ (𝑡)𝑇 𝑈2 𝑥̇ (𝑡) 2 − 𝑒𝛼ℎ ∫ ∫

𝑇 1 ( 𝑙1 𝑇 − 𝑙8 𝑇 ) 2 𝑇 (𝑙2 𝑇 − 𝑙5 𝑇 )

( ( = 𝜉𝑇 (𝑡) ( (

𝑇

−ℎ 𝑡+𝛽

𝑇

𝑇

(𝑙1 𝑇 − 𝑙2 𝑇 )

𝑥̇ (𝑠)𝑇 𝑊2 𝑥̇ (𝑠) 𝑑𝑠

0

(35)

where

𝑉4̇ (𝑡)

𝛼ℎ

0 0 𝑍2 ). 3𝑊

ity 𝑇

− 𝑒 𝑥 (𝑡 − ℎ) 𝑄1 𝑥 (𝑡 − ℎ) + 𝑥 (𝑡) 𝑄2 𝑥 (𝑡)

𝑡−ℎ

0 0 3𝑊 ∗

(34)

By applying Lemma 10 to (31), we can obtain the inequal-

𝑇

𝑡

𝑊 𝑍1 𝑊 ∗ ∗ ∗

𝑇

(29)

≤ 𝛼𝑉2 (𝑡) + 𝑥 (𝑡)𝑇 𝑄1 𝑥 (𝑡)

− 𝑒𝛼ℎ ℎ ∫

). ∗ 3𝑊 (33)

− 𝑒 𝑥 (𝑡 − ℎ) 𝑄1 𝑥 (𝑡 − ℎ) + 𝑥 (𝑡) 𝑄2 𝑥 (𝑡)

𝛼ℎ

0

In order to obtain a tighter bound of integral term, Lemma 11 is applied to (32) as follows:

= 𝛼𝑉2 (𝑡) + 𝑥 (𝑡)𝑇 𝑄1 𝑥 (𝑡) 𝛼ℎ

𝑊

𝑇

𝜛2 = 𝜉 (𝑡) ( ) (𝑙3 𝑇 + 𝑙2 𝑇 − 2𝑙7 𝑇 )

𝑇

𝑥 (𝑡) 𝑃11 𝑥 (𝑡) = 𝑥

𝑇

(𝑡) 𝑅1/2 𝑃̃11 𝑅1/2 𝑥 (𝑡)

> 𝜆 min (𝑃̃11 ) 𝑥𝑇 (𝑡) 𝑅𝑥 (𝑡) .

̃ 11 = 𝑃11 𝐴𝑇 + 𝐴𝑃11 − 𝛼𝑃11 + 𝑄1 + 𝑄2 − 4𝜌𝑊 Π − 3𝜌𝑈1 − 9𝜌𝑈2 ,

(40)

̃ 44 = −𝛾2 𝐼, Π ̃ 22 = −𝜌𝑄2 − 4𝜌𝑊 − 6𝜌𝑈2 , Π

Therefore, conditions (17) to (20) can guarantee the FTB of system (4). This completes the proof. Remark 14. Among the existing approaches, there are two threads: one is to construct a novel LKLF that involves more information of delay; the other is to find a tighter estimation of upper bound for cross terms coming from the derivative of the LKLF. In this paper, these two techniques are applied to obtain the result. In addition, the less conservatism in practical engineering will be shown in the simulations which are always ignored in most literature sources.

̃ 33 = −8𝜌𝑊 + 𝜌𝑍1 𝑇 + 𝜌𝑍1 − 𝜌𝑍2 𝑇 − 𝜌𝑍2 Π − (1 − 𝜇) 𝑄2 , ̃ 13 = 𝐵1 𝑌 − 2𝜌𝑊 − 𝜌𝑍1 − 𝜌𝑍2 , Π ̃ 14 = 𝐵2 , Π ̃ 66 = −12𝜌𝑊, Π ̃ 77 = −12𝜌𝑊, Π

3.2. Controller Design. In this subsection, we focus on the problem of finite-time 𝐻∞ state feedback designing based on Theorem 13, that is, designing a state feedback controller in the form of (7) such that the resulting DP system satisfies the control objective proposed in Section 1.

̃ 55 = 𝛼ℎ2 𝑃22 − 18𝜌𝑈2 − 6𝜌𝑈1 , Π

Theorem 15. For given positive 𝛼, 𝑐1 , 𝑐2 , 𝑇, 𝛾 and matrices 𝑅 > 0, 𝑆 ≥ 0, if there exist positive scalars 𝜃𝑙 (𝑙 = 1, 2, ⋅ ⋅ ⋅ , 8) and matrices 𝑃11 > 0, 𝑃22 > 0, 𝑊 > 0, 𝑄𝑖 > 0 (𝑖 = 1, 2), 𝑈𝑗 > 0 (𝑗 = 1, 2), 𝑌, 𝑍1 , 𝑍2 with appropriate dimensions, satisfying the following conditions:

̃ 15 = ℎ𝑃22 + 12𝜌𝑈2 , Π

̃ = (Π ̃ 𝑖𝑗 ) 0 ∗ 𝑊 3𝑊 𝑍2

(

∗

3𝑊

)>0

(45)

(43)

̃ 88 = −36𝜌𝑈1 − 36𝜌𝑈2 , Π ̃ 12 = 𝜌𝑍1 − 𝜌𝑍2 + 6𝜌𝑈2 , Π

̃ 16 = 6𝜌𝑊, Π ̃ 17 = 2𝜌𝑍2 , Π ̃ 25 = −ℎ𝑃22 − 6𝜌𝑈2 , Π ̃ 18 = −18𝜌𝑈2 + 6𝜌𝑈1 , Π ̃ 26 = 2𝜌𝑍2 𝑇 , Π ̃ 23 = −𝜌𝑍1 𝑇 − 𝜌𝑍2 𝑇 − 2𝜌𝑊, Π ̃ 27 = 6𝜌𝑊, Π

(44)

̃ 28 = 12𝜌𝑈2 , Π

(46)

8

Mathematical Problems in Engineering ̃ 36 = 6𝜌𝑊 + 2𝜌𝑍2 𝑇 , Π

Σ99 = −𝛾2 𝑑,

̃ 37 = 6𝜌𝑊 + 2𝜌𝑍2 , Π

Σ12 = √𝑐1 𝐻2 ,

̃ 58 = 12𝜌𝑈1 + 24𝜌𝑈2 , Π

Σ13 = √ℎ2 𝑐1 𝐻3 ,

̃ 67 = −4𝜌𝑍2 , Π

Σ14 = √𝜌ℎ𝑐1 𝐻4 ,

̃ 19 = 1 ℎ𝑃11 𝐴𝑇 , Π 2

Σ15 = √𝜌ℎ𝑐1 𝐻5 ,

̃ 39 = 1 ℎ𝑌𝑇 𝐵1 𝑇 , Π 2

1 Σ16 = √ 𝜌ℎ3 𝑐1 𝐻6 , 2

̃ 49 = 1 ℎ𝐵2 𝑇 , Π 2

1 Σ17 = √ 𝜌ℎ3 𝑐1 𝐻7 , 6

̃ 1,10 = 1 ℎ𝑃11 𝐴𝑇 , Π 2

1 Σ18 = √ 𝜌ℎ3 𝑐1 𝐻8 , 3

̃ 3,10 = 1 ℎ𝑌𝑇 𝐵1 𝑇 , Π 2

Σ19 = 𝑃11 ,

̃ 4,10 = 1 ℎ𝐵2 𝑇 , Π 2

(48) others

̃ 1,11 = ℎ𝑃11 𝐴𝑇 , Π

Σ𝑖𝑗 = 0

̃ 3,11 = ℎ𝑌𝑇 𝐵1 𝑇 , Π

then a state feedback 𝐻∞ controller in form of (7) exists, such that (1) the closed-loop system (8) with 𝑢(𝑡) = 0 is FTB; (2) under given nonzero initial condition, the closed 𝑇 system (8) and (9) guarantees that 𝐽 = ∫0 𝑧𝑇 (𝑠)𝑧(𝑠) − 𝛾2 𝜔𝑇 (𝑠)𝜔(𝑠) − 𝛾2 (1/𝑇)𝑥𝑇 (0)𝑆𝑥(0)𝑑𝑠 < 0 for all nonzero 𝜔(𝑡) and 𝑡 ∈ [0, 𝑇]. −1 The controller gain can be calculated by 𝐾 = 𝑌𝑃11 .

̃ 4,11 = ℎ𝐵2 𝑇 , Π ̃ 1,12 = 𝑃11 𝐶𝑇 , Π ̃ 3,12 = 𝑌𝑇 𝐷𝑇 , Π ̃ 99 = 1 𝑈1 − 𝑃11 , Π 2 ̃ 10,10 = 1 𝑈2 − 𝑃11 , Π 2

Proof. In preparation for designing, we set 𝑀 = 𝛾2 𝐼, 𝛽 = 1 for (37); then (37) and (38) can be rewritten as follows:

̃ 11,11 = 𝑊 − 2𝑃11 , Π ̃ 12,12 = −𝑒−𝛼𝑇 𝐼, Π

̂ (𝑡) 𝑉̇ − 𝛼𝑉 − 𝛾2 𝜔𝑇 (𝑡) 𝜔 (𝑡) < 𝜉𝑇 (𝑡) Ω𝜉 (47)

𝑡

0

̃ 𝑖𝑗 = 0, Π

Σ22 = −𝐻2 , Σ33 = −𝐻3 , Σ44 = −𝐻4 , Σ55 = −𝐻5 , Σ66 = −𝐻6 , Σ77 = −𝐻7 , Σ88 = −𝐻8 ,

(50)

𝑉 (𝑡) < 𝑒𝛼𝑡 𝑉 (0) + 𝛾2 ∫ 𝑒𝛼(𝑡−𝑠) 𝜔𝑇 (𝑠) 𝜔 (𝑠) 𝑑𝑠

others

Σ11 = −𝑒−𝛼𝑇 𝑐2 𝐻1 ,

(49)

𝛼𝑇