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 Β© 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 π»β 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 π»β 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 π»β 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 vesselβs 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 π»β 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 authorsβ 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 π»β control for DP system with input delay is studied. The main contribution of this paper lies in three aspects: Firstly, a finite-time π»β 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 π»β 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 π»β 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 π»β control is detailed for vessels while some definitions and lemmas are introduced as preparation. In Section 3, the method to design finite-time π»β controller is proposed, and the proposed control schemes are simulated in Section 4. In Section 5, the conclusion is drawn. Notation. Throughout this paper, Rπ is the π-dimensional Euclidean vector space, and RπΓπ denotes the set of all π Γ π real matrices. For symmetric matrices π and π, π > π (respectively, π β₯ π ) means that π β π is positive definite (respectively, positive semidefinite). The superscript βπβ represents the transpose. The symmetric terms in a symmetric matrix are denoted by βββ. Moreover, we use π max (β
)(π min (β
) ) 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 πΜ (π‘) = π½ (π) π (π‘)
(1)
ππΜ (π‘) + π·π (π‘) = π’ (π‘ β β (π‘)) + π (π‘)
(2)
where π = (π V π) is a vector of velocities given in the body-fixed coordinate system and π = (π π π) is the position and orientation of the vessel with respect to an inertial reference coordinate system. π’(π‘) is a control vector of forces and moments provided by the propulsion systems. β(π‘) is a time-varying function that expresses the actuator delay Μ < π . π(π‘) is the disturbance input and satisfies β(π‘) < β β(π‘) β of the system and satisfies the condition of β«0 π(π‘)π π(π‘)ππ‘ < π . π½(π) is the transformation matrix between the inertial and body-fixed coordinate frames. The inertia matrix π includes hydrodynamic added inertia, and π· is a strictly positive damping matrix due to linear wave drift damping and laminar friction. Then, the structures of the matrices π½(π), π, and π· can be explicitly given as follows: cos (π) βsin (π) 0 π½ (π) = ( sin (π) cos (π) 0) 0
0
π β ππ’Μ
0
0
π β πVΜ
0
ππ₯π β ππΜ
π=(
ππ’ 0 π· = β( 0
1 0 ππ₯π β ππΜ)
(3)
πΌπ§ β ππΜ
0
πV ππ )
0 πV ππ where ππ’ ,πV ,ππ ,ππ ,ππ’Μ ,πVΜ ,ππΜ, and ππΜ are the hydrodynamic parameters of the vessel. π is the mass of the ship, and πΌπ§ is the moment of inertia about the yaw rotation. π₯π 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] π₯Μ (π‘) = π΄π₯ (π‘) + π΅1 π’ (π‘ β β (π‘)) + π΅2 π (π‘)
(4)
π§ (π‘) = πΆπ₯ (π‘) + π·π’ (π‘ β β (π‘))
(5)
where π₯ (π‘) = (π (π‘) π (π‘)) π΄=(
0
πΌ
)
0 βπβ1 π·
Mathematical Problems in Engineering π΅1 = π΅2 = (
0 β1
π
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 π’ (π‘ β β (π‘)) = πΎπ₯ (π‘ β β (π‘))
(7)
so the (1) and (2) can be rewritten as π₯Μ (π‘) = π΄π₯ (π‘) + π΅1 πΎπ₯ (π‘ β β (π‘)) + π΅2 π (π‘)
(8)
π§ (π‘) = πΆπ₯ (π‘) + π·πΎπ₯ (π‘ β β (π‘)) .
(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 vesselβ s position, gyrocompass for the vesselβ 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 π
and three positive constants π1 , π2 , π with π1 < π2 , the time-delay system (4) with π’(π‘) = 0 is said to be finite-time boundedness with respect to (π1 π2 π π π π
), Μ if supβπ 0 and a differentiable signal π₯ in [βπ, βπ] β Rπ , the following inequality holds: βπ
βπ
βπ 1 β
Ξ 1 (π1 , π₯ (βπ) , π₯ (βπ) , β« π₯π (π’) ππ’) π β π βπ π
Ξ 1 (π1 , πΌ, π½, πΎ)
=
sup
βπ(π‘)β2,[0,π] 2 +π₯(0)π ππ₯(0)=0ΜΈ
[
βπ§ (π‘)β2,[0,π]
βπ (π‘)β2,[0,π] 2 + π₯ (0)π ππ₯ (0)
]
Lemma 10 (see [29]). For constant matrices π1 > 0, π2 > 0 and constant scalars π > π, the following inequalities hold for all continuously function π₯ in [βπ, βπ] β Rπ : βπ
π‘βπ
βπ
π‘+π
β« β«
π₯Μπ (π’) π2 π₯Μ (π’) ππ’ππ
β₯ Ξ 2 (π2 , π₯π (π‘ β π) , βπ
π‘βπ
βπ
π‘+π
β
β« β« βπ
π‘+π
βπ
π‘βπ
β« β«
where πΎ is a prescribed scalar and π 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
π‘βπ 1 1 β« π₯π (π’) ππ’, π β π π‘βπ (π β π)2
π₯π (π’) ππ’ππ)
(13)
π
π₯Μ (π’) π3 π₯Μ (π’) ππ’ππ
β₯ Ξ 3 (π3 , π₯π (π‘ β π) ,
(10)
0,βπ πΌπ =1}
β π
1 π (π‘) = βππ (π‘) + maxβππ,π (π‘) ππ,π (π‘) πΌπ π π π=πΜΈ
subject to {ππ,π : π
π σ³¨σ³¨β π
, ππ,π (π‘) = ππ,π (π‘) , [
Lemma 12 (see [31]). For any matrix π4 > 0 and a vector function π₯ in [βπ, βπ] β Rπ , if the integrals concerned are well defined, then the following inequality holds: βπ
β« π₯π (π ) π4 π₯ (π ) ππ βπ
βπ βπ 1 β₯ (β« π₯π (π ) ππ ) π4 (β« π₯ (π ) ππ ) . π β π βπ βπ
ππ (π‘) ππ,π (π‘) ππ,π (π‘) ππ (π‘)
(15) ] β₯ 0} .
3π π2
(
β
)>0 3π
1 π1 π2 + β2 π1 π3 + πβπ1 π4 + πβπ1 π5 + πβ3 π1 π6 2 1 1 + πβ3 π1 π7 + πβ3 π1 π8 + π½π9 π < πβπΌπ π2 π1 6 3
(16)
where 2.3. Control Objective. Finally, the objective of this paper is to derive the control gain πΎ such that (1) the closed-loop system (8) with π’(π‘) = 0 is FTB; (2) under given nonzero initial condition, the closed π system (8) and (9) guarantees that π½ = β«0 π§π (π )π§(π ) β πΎ2 ππ (π )π(π ) β πΎ2 (1/π)π₯π (0)ππ₯(0)ππ < 0 for all nonzero π(π‘) and π‘ β [0, π].
The designing of the robust finite-time π»β 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 πΌ, π½, π1 , π2 , π and positive define matrix π
, the finite-time boundedness problem of system (4) is solvable if there exist positive scalars ππ (π = 1, 2, β
β
β
, 8) and matrices π11 > 0, π22 > 0, π > 0, ππ > 0 (π = 1, 2), ππ > 0 (π = 1, 2), π > 0, π1 , π2 with appropriate dimensions, satisfying the following conditions:
π π1
(
β π
)>0
1 + π΄π β2 (π1 + π2 + 2π) π΄ β 4ππ β 3ππ1 2 β 9ππ2 , Ξ©22 = βππ2 β 4ππ β 6ππ2 , 1 Ξ©33 = πΎππ΅1 π β2 (2π + π1 + π2 ) π΅1 πΎ β 8ππ 2
3. Main Results
Ξ© = (Ξ©ππ )8Γ8 < 0
Ξ©11 = π΄π π11 + π11 π΄ β πΌπ11 + π1 + π2
(17) (18)
+ ππ1 π + ππ1 β ππ2 π β ππ2 β (1 β π) π2 , 1 Ξ©44 = π΅2 π β2 (2π + π1 + π2 ) π΅2 β π½π, 2 Ξ©66 = β12ππ, Ξ©77 = β12ππ, Ξ©55 = πΌβ2 π22 β 18ππ2 β 6ππ1 , Ξ©88 = β36ππ1 β 36ππ2 , Ξ©12 = ππ1 β ππ2 + 6ππ2 , Ξ©15 = βπ22 + 12ππ2 , Ξ©16 = 6ππ, Ξ©17 = 2ππ2 ,
(19)
(20)
Mathematical Problems in Engineering
5
1 Ξ©13 = π11 π΅1 πΎ + π΄π β2 (2π + π1 + π2 ) π΅1 πΎ β 2ππ 2 β ππ1 β ππ2 , 1 Ξ©14 = π11 π΅2 + π΄π β2 (2π + π1 + π2 ) π΅2 , 2 Ξ©18 = β18ππ2 + 6ππ1 ,
Μπ = π
β1/2 ππ π
β1/2 π
(π = 1, 2, 3) ,
Μ = π
β1/2 ππ
β1/2 , π (π = 1, 2) ,
0 < π1 πΌ < π11 < π2 πΌ,
Ξ©25 = ββπ22 β 6ππ2 ,
0 < π22 < π3 πΌ,
Ξ©26 = 2ππ2 π ,
0 < π1 < π4 πΌ,
Ξ©27 = 6ππ,
0 < π2 < π5 πΌ,
Ξ©28 = 12ππ2 , Ξ©34 = πΎ π΅1
πΜ22 = π
β1/2 π22 π
β1/2 ,
Μπ = π
β1/2 ππ π
β1/2 π
Ξ©23 = βππ1 π β ππ2 π β 2ππ,
π
πΜ11 = π
β1/2 π11 π
β1/2 ,
0 < π < π6 πΌ,
π1 2
2
β (2π + π1 + π2 ) π΅2 ,
0 < π1 < π7 πΌ,
Ξ©36 = 6ππ + 2ππ2 π ,
0 < π2 < π8 πΌ,
Ξ©37 = 6ππ + 2ππ2 ,
0 < π < π9 πΌ. (21)
Ξ©58 = 12ππ1 + 24ππ2 , Ξ©67 = β4ππ2 ,
Proof. Consider the candidate augment LKLF as follows:
Ξ©24 = Ξ©35 = Ξ©38 = Ξ©45 = Ξ©46 = Ξ©47 = Ξ©48 = Ξ©56 = Ξ©57 = Ξ©68 = Ξ©78 = 0, π1 = π min (πΜ11 ) π2 = π max (πΜ11 ) , π3 = π max (πΜ22 ) , Μ1 ) , π4 = π max (π Μ2 ) , π5 = π max (π Μ , π6 = π max (π) Μ1 ) , π7 = π max (π Μ2 ) , π8 = π max (π π9 = π max (π) , π = ππΌβ ,
π (π‘) = π1 (π‘) + π2 (π‘) + π3 (π‘) + π4 (π‘)
(22)
π1 (π‘) = π (π‘)π ππ (π‘)
(23)
π2 (π‘) = β«
π‘
π‘ββ
+β«
ππΌ(π‘βπ ) π₯ (π )π π1 π₯ (π ) ππ (24)
π‘
πΌ(π‘βπ )
π
π‘ββ(π‘)
0
π3 (π‘) = β β« β«
π‘
ββ π‘+π
0
0
0
π₯ (π ) π2 π₯ (π ) ππ
ππΌ(π‘βπ ) π₯Μ (π )π ππ₯Μ (π ) ππ ππ
π‘
π4 (π‘) = β« β« β« ββ π
π
π‘+π½
π
(25)
ππΌ(π‘βπ ) π₯Μ (π )π π1 π₯Μ (π ) ππ ππ½ππ
+β« β« β«
π‘
ββ ββ π‘+π½
(26) πΌ(π‘βπ )
π
π
π₯Μ (π ) π2 π₯Μ (π ) ππ ππ½ππ
π‘
where π(π‘) = (π₯(π‘) β«π‘ββ π₯(π )ππ ) π = ( πβ11 π022 ). To deal with the formulas conveniently, let us provide such definitions:
π (π‘) = (π₯ (π‘) π₯ (π‘ β β) π₯ (π‘ β β (π‘)) π (π‘) 0 β
β
β
0) . 0 β
β
β
0 1 βββββββββ ππ π = (βββββββββ πβ1
8βπ
π‘ π‘ββ(π‘) 1 1 1 0 π‘ 1 π‘ π₯ (π ) ππ π₯ (π ) ππ 2 β« β« π₯ (π ) ππ ππ) (27) β« π₯ (π ) ππ β« β« β π‘ββ β (π‘) π‘ββ(π‘) β β β (π‘) π‘ββ β ββ π‘+π
6
Mathematical Problems in Engineering
The time-derivative of π(π‘) along system (4) can be bounded as
Ξ1 = Ξ2 = (
π1Μ (π‘) = πΌπ1 (π‘) + π Μ (π‘)π ππ (π‘) + π (π‘)π ππ Μ (π‘) β πΌπ1 (π‘) (28) π2Μ (π‘)
π
π3Μ (π‘) β€ πΌπ3 (π‘) + β2 π₯Μ (π‘)π ππ₯Μ (π‘) β ππΌβ π3 π Ξ 3 π3
π
β (1 β βΜ (π‘)) ππΌβ(π‘) π₯ (π‘ β β (π‘))π π2 π₯ (π‘ β β (π‘))
where π3 π = ( ππ1 π ), Ξ 3 = ( ββ 2
1 π4Μ (π‘) β€ πΌπ4 (π‘) + β2 π₯Μ (π‘)π π1 π₯Μ (π‘) 2
β (1 β π) π₯ (π‘ β β (π‘))π π2 π₯ (π‘ β β (π‘))
1 + β2 π₯Μ (π‘)π π2 π₯Μ (π‘) β ππΌβ π4 π Ξ 4 π4 2
π3Μ (π‘) = πΌπ3 (π‘) + β2 π₯Μ (π‘)π π2 π₯Μ (π‘)
(30)
π4 π
1 = πΌπ4 (π‘) + β2 π₯Μ (π‘)π π1 π₯Μ (π‘) 2 βπ
β« β«
π‘
0
π‘+π½
ββ π‘ββ
π₯Μ (π ) π1 π₯Μ (π ) ππ ππ½
6π1 β12π1
(31) Ξ4 = (
π₯Μ (π )π π2 π₯Μ (π ) ππ ππ½.
π3Μ (π‘) = πΌπ3 (π‘) + β2 π₯Μ (π‘)π ππ₯Μ (π‘) β ππΌβ β β«
π‘ββ(π‘) π‘ββ(π‘)
β ππΌβ β β«
π‘ββ
π₯Μ (π )π ππ₯Μ (π ) ππ
β
β
β
0
6π2 β12π2 β
).
36π2
π (π‘) < ππΌπ‘ π (0) + π½ β« ππΌ(π‘βπ ) ππ (π ) ππ (π ) ππ 0
where π π
(π1 β π3 ) π1 π = ππ (π‘) ( ) π π (π1 + π3 π β 2π6 π ) π
(π3 π β π2 π )
(37)
π‘
(32)
β β β ππΌβ π1 π Ξ 1 π1 β ππΌβ π πΞ π β (π‘) β β β (π‘) 2 2 2
π
β
0
Assuming that Ξ© < 0 and integrating the left part of inequality (37),
β€ πΌπ3 (π‘) + β2 π₯Μ (π‘)π ππ₯Μ (π‘)
π
36π1
0
πΜ β πΌπ β π½ππ (π‘) ππ (π‘) < ππ (π‘) Ξ©π (π‘) .
π₯Μ (π )π ππ₯Μ (π ) ππ
π
β
0
(36)
Combining (28), (29), (34), and (35) with the definition of π = ππΌβ , we can obtain
Invoking Lemma 9, we can obtain
π‘
) ) ) )
π 1 ( π2 π β π5 π + π8 π ) ) ( 2
1 + β2 π₯Μ (π‘)π π2 π₯Μ (π‘) 2 β ππΌβ β« β«
π 1 ( π1 π β π8 π ) 2 π (π2 π β π5 π )
( ( = ππ (π‘) ( (
π
ββ π‘+π½
π
π
(π1 π β π2 π )
π₯Μ (π )π π2 π₯Μ (π ) ππ
0
(35)
where
π4Μ (π‘)
πΌβ
0 0 π2 ). 3π
ity π
β π π₯ (π‘ β β) π1 π₯ (π‘ β β) + π₯ (π‘) π2 π₯ (π‘)
π‘ββ
0 0 3π β
(34)
By applying Lemma 10 to (31), we can obtain the inequal-
π
π‘
π π1 π β β β
π
(29)
β€ πΌπ2 (π‘) + π₯ (π‘)π π1 π₯ (π‘)
β ππΌβ β β«
). β 3π (33)
β π π₯ (π‘ β β) π1 π₯ (π‘ β β) + π₯ (π‘) π2 π₯ (π‘)
πΌβ
0
In order to obtain a tighter bound of integral term, Lemma 11 is applied to (32) as follows:
= πΌπ2 (π‘) + π₯ (π‘)π π1 π₯ (π‘) πΌβ
π
π
π2 = π (π‘) ( ) (π3 π + π2 π β 2π7 π )
π
π₯ (π‘) π11 π₯ (π‘) = π₯
π
(π‘) π
1/2 πΜ11 π
1/2 π₯ (π‘)
> π min (πΜ11 ) π₯π (π‘) π
π₯ (π‘) .
Μ 11 = π11 π΄π + π΄π11 β πΌπ11 + π1 + π2 β 4ππ Ξ β 3ππ1 β 9ππ2 ,
(40)
Μ 44 = βπΎ2 πΌ, Ξ Μ 22 = βππ2 β 4ππ β 6ππ2 , Ξ
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.
Μ 33 = β8ππ + ππ1 π + ππ1 β ππ2 π β ππ2 Ξ β (1 β π) π2 , Μ 13 = π΅1 π β 2ππ β ππ1 β ππ2 , Ξ Μ 14 = π΅2 , Ξ Μ 66 = β12ππ, Ξ Μ 77 = β12ππ, Ξ
3.2. Controller Design. In this subsection, we focus on the problem of finite-time π»β 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.
Μ 55 = πΌβ2 π22 β 18ππ2 β 6ππ1 , Ξ
Theorem 15. For given positive πΌ, π1 , π2 , π, πΎ and matrices π
> 0, π β₯ 0, if there exist positive scalars ππ (π = 1, 2, β
β
β
, 8) and matrices π11 > 0, π22 > 0, π > 0, ππ > 0 (π = 1, 2), ππ > 0 (π = 1, 2), π, π1 , π2 with appropriate dimensions, satisfying the following conditions:
Μ 15 = βπ22 + 12ππ2 , Ξ
Μ = (Ξ Μ ππ ) 0 β π 3π π2
(
β
3π
)>0
(45)
(43)
Μ 88 = β36ππ1 β 36ππ2 , Ξ Μ 12 = ππ1 β ππ2 + 6ππ2 , Ξ
Μ 16 = 6ππ, Ξ Μ 17 = 2ππ2 , Ξ Μ 25 = ββπ22 β 6ππ2 , Ξ Μ 18 = β18ππ2 + 6ππ1 , Ξ Μ 26 = 2ππ2 π , Ξ Μ 23 = βππ1 π β ππ2 π β 2ππ, Ξ Μ 27 = 6ππ, Ξ
(44)
Μ 28 = 12ππ2 , Ξ
(46)
8
Mathematical Problems in Engineering Μ 36 = 6ππ + 2ππ2 π , Ξ
Ξ£99 = βπΎ2 π,
Μ 37 = 6ππ + 2ππ2 , Ξ
Ξ£12 = βπ1 π»2 ,
Μ 58 = 12ππ1 + 24ππ2 , Ξ
Ξ£13 = ββ2 π1 π»3 ,
Μ 67 = β4ππ2 , Ξ
Ξ£14 = βπβπ1 π»4 ,
Μ 19 = 1 βπ11 π΄π , Ξ 2
Ξ£15 = βπβπ1 π»5 ,
Μ 39 = 1 βππ π΅1 π , Ξ 2
1 Ξ£16 = β πβ3 π1 π»6 , 2
Μ 49 = 1 βπ΅2 π , Ξ 2
1 Ξ£17 = β πβ3 π1 π»7 , 6
Μ 1,10 = 1 βπ11 π΄π , Ξ 2
1 Ξ£18 = β πβ3 π1 π»8 , 3
Μ 3,10 = 1 βππ π΅1 π , Ξ 2
Ξ£19 = π11 ,
Μ 4,10 = 1 βπ΅2 π , Ξ 2
(48) others
Μ 1,11 = βπ11 π΄π , Ξ
Ξ£ππ = 0
Μ 3,11 = βππ π΅1 π , Ξ
then a state feedback π»β controller in form of (7) exists, such that (1) the closed-loop system (8) with π’(π‘) = 0 is FTB; (2) under given nonzero initial condition, the closed π system (8) and (9) guarantees that π½ = β«0 π§π (π )π§(π ) β πΎ2 ππ (π )π(π ) β πΎ2 (1/π)π₯π (0)ππ₯(0)ππ < 0 for all nonzero π(π‘) and π‘ β [0, π]. β1 The controller gain can be calculated by πΎ = ππ11 .
Μ 4,11 = βπ΅2 π , Ξ Μ 1,12 = π11 πΆπ , Ξ Μ 3,12 = ππ π·π , Ξ Μ 99 = 1 π1 β π11 , Ξ 2 Μ 10,10 = 1 π2 β π11 , Ξ 2
Proof. In preparation for designing, we set π = πΎ2 πΌ, π½ = 1 for (37); then (37) and (38) can be rewritten as follows:
Μ 11,11 = π β 2π11 , Ξ Μ 12,12 = βπβπΌπ πΌ, Ξ
Μ (π‘) πΜ β πΌπ β πΎ2 ππ (π‘) π (π‘) < ππ (π‘) Ξ©π (47)
π‘
0
Μ ππ = 0, Ξ
Ξ£22 = βπ»2 , Ξ£33 = βπ»3 , Ξ£44 = βπ»4 , Ξ£55 = βπ»5 , Ξ£66 = βπ»6 , Ξ£77 = βπ»7 , Ξ£88 = βπ»8 ,
(50)
π (π‘) < ππΌπ‘ π (0) + πΎ2 β« ππΌ(π‘βπ ) ππ (π ) π (π ) ππ
others
Ξ£11 = βπβπΌπ π2 π»1 ,
(49)
πΌπ