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Robust delay-dependent guaranteed cost controller design for uncertain neutral systems
Applied Mathematics and Computation 215 (2009) 2936–2949
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Robust delay-dependent guaranteed cost controller design for uncertain neutral systems M.N. Alpaslan Parlakçı Istanbul Bilgi University, Department of Computer Science, Kurtulusß Deresi Cad. No. 47, Dolapdere, 34440 Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Neutral systems Descriptor representation Uncertainty Parameter perturbations Quadratic cost Linear matrix inequality
a b s t r a c t In this paper, a novel robust delay-dependent guaranteed cost controller is introduced for a class of uncertain nonlinear neutral systems with both norm-bounded uncertainties and nonlinear parameter perturbations. A neutral memory state-feedback control law is chosen such that a quadratic cost function is minimized. On the basis of a descriptor type model transformation, an augmented descriptor form Lyapunov–Krasovskii functional is proposed in order to derive a linear matrix inequality (LMI) condition of the synthesis of a robust stabilizing guaranteed cost controller for uncertain nonlinear neutral systems. The descriptor approach is applied for both the delay-differential equation and the neutral difference operator allowing to introduce several additional free weighting matrices in order to obtain further relaxation in the synthesis process. The proposed method of introducing some relaxation into design problem also leads to develop LMI conditions which can be easily solved using interior point algorithms. Two numerical examples have been introduced to show the application of the theoretical results which indicate that the proposed approach improves the guaranteed cost performance. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Time-delay systems is a significantly fruitful field of research for several decades. The main reason for this is that time-delay naturally arises in many physical and dynamical systems leading to bring about many theoretical and practical problems which need to be resolved. Among some of them, one can notice that a system which is subject to time-delay may exhibit a poor performance, it may get into an unstable behavior or it may not endeavour with external disturbances or even it may excite undesired modes which cause to get far away from meeting the required specifications. Concerning linear time-invariant (LTI) control systems, time-delay may exist in the state of the system which constitutes a broad class of systems so called retarded time-delay systems. Another important class of delay-differential systems is referred to as time-delay systems of neutral type in which the time-delay may be available in both position and velocity of the system. For a theoretical and practical consideration of neutral systems, one can refer to [1–3]. The stability analysis or stabilization synthesis for time-delay systems of neutral type have generally been conducted with respect to two different aspects of interest in the community of time-delay systems. The analysis and synthesis conditions have been developed delay-independently or delay-dependently. For practical time-delay systems involving a delay of finite size, it is reasonable that the need of a condition being applicable for any size of delay changing from 0 to infinity may lead to conservativeness. Hence, in general, delay-independent results are usually considered to be more conservative than their counterparts of delay-dependent type. This also indicates that the stability or stabilization range of time-delay may be used as an evaluation criterion among different theoretical results in terms of conservativeness. Moreover, the noise rejection capability defined in
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.040
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
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terms of H1 norm bound and controller performance measured in terms of a quadratic cost bound can as well be listed as the other two aspects of objectives in the analysis and design problems for time-delay systems. In addition to delay phenomenon, one can notice that the systems under consideration may be also subject to uncertainties of norm-bounded type and/or nonlinear parameter perturbations which need to be handled. In recent years, a number of different guaranteed cost controller design investigations have been taken into consideration for neutral systems with or without uncertainties and/or nonlinear parameter perturbations. For example see [4–19] and the references therein. The common feature of the researches listed in [4–19] is that the proposed methods have employed linear matrix inequality (LMI) techniques. Concerning the effectiveness of a stability condition or the performance of a controller synthesis in terms of the measures of delay conservatism, noise rejection and/or quadratic control cost, it is well known that [20–22] the utilization of some free weighting parameters introduce a considerable rate of relaxation into the results in particular for systems under the existence of uncertainties. However, the existing work on the subject has not taken into account this useful phenomenon because of the fact that the involvement of such slack variables destroy the nature of the LMI form of the stability and/or synthesis conditions. Therefore, the motivation of the present research can be stated as developing a novel technique that allows to develop improved LMI form of delaydependent guaranteed cost controller synthesis through the use of free weighting matrices. The key feature of the proposed methodology is that the inclusion of some new slack parameters do not devastate the LMI nature of the results but more than that the theoretical conditions of a stabilizing controller synthesis become considerably relaxed yielding a better performance. A robust memoryless state-feedback LMI controller design is investigated in [4] for uncertain time-delay systems with norm-bounded uncertainty. Sufficient LMI conditions have been derived in [5] for the existence of a guaranteed cost statefeedback controller of memoryless type. The design problem of guaranteed cost control of uncertain neutral systems with norm-bounded uncertainties is taken into consideration in [6] by employing an LMI technique. Nian and Feng [7] have considered uncertain neutral systems with time-varying delay and have developed some sufficient LMI conditions for the synthesis of a guaranteed cost memoryless state-feedback controller. Employing a memory type of state-feedback controller, Kwon et al. [8] have studied the design of guaranteed cost controller for uncertain time-delay systems by using an LMI approach. A guaranteed cost memoryless state-feedback controller is introduced in [9] for the stabilization of a class of neutral differential systems with parametric uncertainties. Park [10] has proposed a solution to the optimal guaranteed cost control problem by utilizing state-feedback control laws for a class of time-delay systems of neutral type. Both a memory state-feedback based guaranteed cost control and an observer-based output feedback guaranteed cost control problems have been addressed for neutral systems in [11] via LMI methods. For a class of nominal neutral systems, a guaranteed cost state-feedback controller is developed on the basis of an LMI technique in [12]. Introducing a retarded type of integral memory state-feedback controller, Park and Kwon [13] have obtained some LMI criteria to design a guaranteed cost controller for neutral systems. An observer-based guaranteed cost controller synthesis is given in [14] by proposing an LMI feasibility condition. Chen and Liu [15] have investigated the design problem of fuzzy guaranteed cost controller for nonlinear fuzzy neutral systems in terms of the feasibility of some linear matrix inequalities. Utilizing a descriptor form of a system representation and a recent result on bounding of cross terms, a guaranteed cost memoryless state-feedback control of uncertain neutral systems has been studied in [16] on the basis of an LMI methodology. A memory state-feedback control law and a quadratic cost function based on a delayed state approach have been introduced in [17] for the guaranteed cost control design problem of uncertain neutral systems through the feasibility condition of a linear matrix inequality. Utilizing a neutral model transformation and Lyapunov function method, Kwon and Park [18] have developed a guaranteed cost controller for uncertain large scale time-delay systems with delayed feedback. Both delay-independent and delay-dependent guaranteed cost stabilization criteria have been derived in [19] for a class of uncertain neutral systems in terms of an LMI approach. In this paper, the design problem of a robust delay-dependent guaranteed cost control of a class of uncertain nonlinear systems with both norm-bounded uncertainties and nonlinear parameter perturbations has been studied on the basis of a descriptor form of system representation along with an augmented descriptor type of Lyapunov–Krasovskii functional which has been recently proposed by Parlakçı [22]. The proposed approach allows to introduce free weighting matrices embedded in the Lyapunov–Krasovskii functional which has a strong potential effect of providing relaxation for the theoretical results without preventing to develop a linear matrix inequality methodology. A memory state-feedback controller in the form of neutral structure is introduced to provide integral and delayed state-feedback for a better control performance. The use of descriptor system representation in the quadratic Lyapunov stability analysis has been applied to both delay-differential equation and neutral operator equation which lead to be able to include additional slack variables when compared to conventional descriptor approach. A sufficient robust delay-dependent guaranteed cost controller synthesis criterion is derived in terms of the feasibility condition of a considerably relaxed linear matrix inequality which can be easily solved using interior point algorithms. Moreover, a convex optimization problem with LMI constraints is formulated to design the optimal guaranteed cost delayed neutral state-feedback controller which minimizes the upper bound of the guaranteed cost function for the closed-loop uncertain nonlinear neutral system. In order to demonstrate a practical application of the proposed guaranteed cost controller synthesis, two numerical examples are presented from the literature. The numerical results show that the novel augmented descriptor LMI approach improves the guaranteed cost performance in comparison to the existing guaranteed cost controllers in the literature.
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2. Problem statement Let us consider a class of uncertain nonlinear neutral systems with norm-bounded uncertainties and nonlinear parameter perturbations
_ dÞ þ ½B þ DBðtÞuðtÞ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ _ xðtÞ ¼ ½A þ DAðtÞxðtÞ þ ½Ah þ DAh ðtÞxðt hÞ þ ½Ad þ DAd ðtÞxðt _ dÞ; tÞ; þ hðxðt xðtÞ ¼ UðtÞ;
ð1Þ
_ ðtÞ 8t 2 ½s; 0; _ xðtÞ ¼U
s > 0;
n
ð2Þ m
where xðtÞ 2 R is the state vector of the system, uðtÞ 2 R is the control input, A; Ah ; Ad ; B are known real constant system matrices all with appropriate dimensions, DAðtÞ; DAh ðtÞ; DAd ðtÞ; DBðtÞ are unknown real time-varying matrix functions with appropriate dimensions representing time-varying parametric uncertainties which are assumed to be of the following form
½ DAðtÞ DAh ðtÞ DAd ðtÞ DBðtÞ ¼ DFðtÞ½ Ea
Eh
Ed
Eb ;
ð3Þ
where D; Ea ; Eh ; Ed ; Eb are known constant matrices with appropriate dimensions, and FðtÞ is an unknown real time-varying matrix satisfying
F T ðtÞFðtÞ 6 I
ð4Þ
_ dÞ; tÞ represent nonlinear parameter perturbations satisfying and f ðxðtÞ; tÞ; gðxðt hÞ; tÞ; hðxðt
f ð0; tÞ ¼ gð0; tÞ ¼ hð0; tÞ ¼ 0
ð5Þ
and UðÞ is a vector valued initial condition function. Note that the parameter uncertainties of DAðtÞ; DAh ðtÞ; DAd ðtÞ; DBðtÞ are said to be admissible if both (3) and (4) hold. Moreover, it is assumed that the nonlinear parameter perturbations satisfy the following bounding inequalities
kf ðxðtÞ; tÞk 6 akxðtÞk 8t > 0;
ð6Þ
kgðxðt hÞ; tÞk 6 bkxðt hÞk 8t > 0;
ð7Þ
_ dÞk 8t > 0; _ dÞ; tÞk 6 ckxðt khðxðt
ð8Þ
where a; b, and c are known positive scalars. The discrete, h, and neutral, d, delays are scalar constants such that s ¼ maxðh; dÞ. A difference operator can be defined as r : Ið½s; 0; Rn Þ ! Rn such that
rðxt Þ ¼ xðtÞ þ Ah
Z
t
xðsÞ ds Ad xðt dÞ:
ð9Þ
th
Definition 1 ([1,2]). The difference operator r is said to be stable if the zero solution of the homogeneous difference equation rxt ¼ 0; t P 0; x0 ¼ w 2 f/ 2 Cð½s; 0Þ : r/ ¼ 0g is uniformly asymptotically stable. The stability of the difference operator r is necessary for the stability of system (1) with uðtÞ 0. Assumption 1 [13]. Given positive scalars h, d, and any constant matrices Ah ; Ad 2 Rnn , the operator rðxt Þ defined as in (9) is asymptotically stable if there exist a symmetric and positive definite matrix, K0 and positive scalars q1 ; q2 such that
q1 þ q2 < 1; 2 4
ATd K0 Ad q1 K0
3
T
hAd K0 Ah 2
q2 K0 þ h
ATh
K0 A h
5 < 0:
ð10Þ
A quadratic cost function associated with the uncertain nonlinear neutral system (1) can be defined as
J¼
Z
1
½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞ dt;
ð11Þ
0
where S1 , and S2 are given constant positive definite gain matrices. A linear memory type neutral state-feedback control law can be chosen in a similar manner with Park [13] as follows:
uðtÞ ¼ K rðxt Þ;
ð12Þ
where K 2 Rmn denotes the feedback gain to be determined appropriately. The objective of the present work can be summarized in the following sequel. The objective of the present research can be stated such that given the positive scalars h, and d, the aim is to design a memory neutral form state-feedback controller (12), the closed-loop system is guaranteed to be asymptotically stable and the quadratic cost function (11) has an upper bound.
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Definition 2 [13]. For the uncertain nonlinear neutral system (1) and the quadratic cost function in (11), if there exists a control law u ðtÞ and a positive J such that for all admissible uncertainties and delays, the system is asymptotically stable and the closed-loop value of the cost function (11) satisfies J 6 J , then J is said to be a guaranteed cost and u ðtÞ is said to be a guaranteed cost control law of system (1) and cost function (11).
3. Main results Let us assume that system (1) is not subject to any norm-bounded uncertainty, then one can obtain the closed-loop nonlinear delay-differential equation in view of the neutral state-feedback control law (12) as follows:
_ dÞ; tÞ: _ _ dÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt xðtÞ ¼ AxðtÞ þ Ah xðt hÞ þ Ad xðt
ð13Þ
Differentiating the neutral difference operator, rðxt Þ along the state trajectory of system (13) gives
_ ðxt Þ ¼ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt _ dÞ; tÞ; r
ð14Þ
where A0 ¼ A þ Ah . Applying a descriptor form of representation [20] for (14) gives
_ ðxt Þ ¼ yðtÞ; r
ð15Þ
_ dÞ; tÞ; yðtÞ ¼ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
where yðtÞ is the fast varying descriptor variable. The following theorem summarizes the synthesis of a robust guaranteed cost controller. Theorem 1. Given the positive scalars, h > 0; d > 0, and q1 ; q2 such that q1 þ q2 < 1, if there exist symmetric and positive definite matrices K0 ; X 1 ; Q ; R; S, Z, and matrices X i , Y, i ¼ 2; . . . ; 6 and positive scalars k, ej , j ¼ 1; 2; 3, ek , k ¼ 4; 5; 6 with appropriate dimensions satisfying the following linear matrix inequalities:
2 6 6 6 6 6 4
T
q1 K0
ATd K0
khEd Eh 2
K0
6 6 6 6 6 6 6 6 6 6 6 6 R¼6 6 6 6 6 6 6 6 6 6 6 6 6 4
3
7 0 7 7 7 < 0; K0 D 7 5
T
q2 K0 þ kh ETh Eh hAh K0
2
0
ð16Þ
kI
Rð1; 1Þ hRð1; 2Þ Rð1; 2Þ Rð1; 2Þ Rð1; 5Þ aRð1; 2Þ Rð1; 2Þ Rð1; 8Þ Q
0
Rð1; 9Þ
Rð1; 10Þ Rð1; 11Þ
0
0
0
0
0
0
0
0
R
0
0
0
0
0
0
0
0
S1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
e1 I
0
0
0
0
0
S1 2
7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; ð17Þ 7 7 7 7 7 7 7 7 7 7 7 5
S
0
0
0
0
e2 I
0
0
0
Z þ e6 DDT
0
0 0 e0 I
e3 I
3
where
2 6 6 6 6 6 6 6 6 6 Rð1; 1Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
X 2 þ X T2
X 3 þ Y T BT X T2 þ X T4 AT0
X 1 þ X T4
0
0
0
0
0
0
X 3 X T3 þ A0 X 5 þ X T5 AT0 þ e4 DDT
A0 X 6 þ X T5
0
0
e1 I
0
e2 I
0
0
0
0
0
Q
0
0
0
0
0
R
0
0
0
0 e1 I
0 S
0 0
X 6 þ X T6 þ e5 DDT Ah Q Ad R
0 0 e2 I
Z
0
0
3
7 e3 I 7 7 7 0 7 7 0 7 7 7 0 7 7; 0 7 7 7 0 7 7 0 7 7 7 0 5 e3 I
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3 3 2 2 3 2 3 2 T3 T hX 4 Y T BT þ X T4 AT 0 0 Y 7 6 6 T7 607 607 607 X T5 AT 7 6 6 hX 5 7 7 7 7 6 6 6 7 6 6 T7 T T 607 607 607 7 6 6 hX 6 7 X A 7 7 7 6 6 6 6 7 7 6 6 607 607 607 7 6 6 0 7 0 6 7 6 7 6 7 7 7 6 6 607 607 607 7 7 6 0 7 7 6 7 Rð1; 2Þ ¼ 6 7; Rð1; 10Þ ¼ 6 6 0 7; Rð1; 5Þ ¼ 6 6 0 7; 6 0 7; Rð1; 8Þ ¼ 6 0 7; Rð1; 9Þ ¼ 6 7 6 6 0 7 e1 I 7 7 6 7 6 6 7 7 6 6 607 6 bS 7 607 7 6 6 0 7 SATh 6 7 6 7 6 7 7 7 6 6 607 607 607 7 6 6 0 7 e I 7 7 6 7 6 6 2 7 7 6 6 4 cZ 5 405 405 T 5 4 4 0 5 ZAd 0 0 0 e3 I 0 2
2 6 6 6 6 6 6 6 6 6 Rð1; 11Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
Y T ETb þ X T4 ET0
0
Y T ETb þ X T4 ETa
X T5 ET0
0
X T5 ETa
X T6 ET0
0
X T6 ETa
0
0
0
QETh RETd
0
0
0
0
0
SETh
0
0
0
0 0
0 0
ZETd 0
0
3 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
2 and
e4 I
6 ¼4 0 0
e0
0
0
e5 I 0
3
7 0 5; e6 I
and (*) denotes the symmetricity, then a robust guaranteed cost neutral state-feedback controller with K ¼ YX 1 robustly asymptotically stabilizes system (1) via ensuring an upper bound for the cost function computed as
J ¼ rT ðUð0ÞÞX 1 rðUð0ÞÞ þ h Z
þ
Z
0
ðs þ hÞUT ðsÞQ 1 UðsÞ ds þ
h 0
Z
0
UT ðsÞR1 UðsÞ ds þ
d
Z
0
UT ðsÞS1 UðsÞ ds
h
U_ T ðsÞZ 1 U_ ðsÞ ds:
ð18Þ
d
Proof. In order to justify the linear matrix inequality condition given in (16) for the stability of uncertain neutral difference operator, one can reconsider (10) and apply Schur complement [23] to obtain
2 6
W¼4
q1 K0
0
q2 K0
ATd K0
3
7 T hAh K0 5 < 0;
ð19Þ
K0
then replacing Ah ; Ad with Ah þ DAh ðtÞ; Ad þ DAd ðtÞ, respectively, in (19) gives
W þ DWðtÞ þ DWT ðtÞ < 0;
ð20Þ
where
DWðtÞ ¼ PT0 FðtÞH0 with P0 ¼ ½ 0
0
DT K0 ; H0 ¼ ½ Ed
hEh
0 . Applying norm-bounding inequality yields
W þ DWðtÞ þ DWT ðtÞ < W þ k1 PT0 P0 þ kHT0 H0 < 0;
ð21Þ
which then by Schur complement [23] verifies the condition in (16). Let us choose an augmented descriptor type of a candidate Lyapunov–Krasovskii functional using the approach of [22] as follows:
VðxðtÞ; tÞ ¼
5 X
V i ðxðtÞ; tÞ;
ð22Þ
i¼1
Rt Rt Rt where V 1 ðxðtÞ; tÞ ¼ gT ðtÞEP gðtÞ; V 2 ðxðtÞ; tÞ ¼ h th ðs t þ hÞxT ðsÞQxðsÞ ds; V 3 ðxðtÞ; tÞ ¼ td xT ðsÞRxðsÞ ds; V 4 ðxðtÞ; tÞ ¼ th R t _ ds with gðtÞ ¼ ½ rT ðxt Þ yT ðtÞ xT ðtÞ T , and xT ðsÞSxðsÞ ds; V 5 ðxðtÞ; tÞ ¼ td x_ T ðsÞZ xðsÞ
2
3 I 0 0 6 7 E ¼ 4 0 0 0 5; 0 0 0
2
P1 6 P ¼ 4 P2 P4
0 P3 P5
3 0 7 0 5; P6
P T1 ¼ P1 > 0;
Q T ¼ Q > 0;
RT ¼ R > 0;
ST ¼ S > 0;
ZT ¼ Z > 0
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all with appropriate dimensions. Taking the time-derivative of VðxðtÞ; tÞ along the state trajectory of system (1), (12) and (15) gives
_ VðxðtÞ; tÞ ¼
5 X
V_ i ðxðtÞ; tÞ:
ð23Þ
i¼1
One can compute V_ 1 ðxðtÞ; tÞ in the following form
d d V_ 1 ðxðtÞ; tÞ ¼ 2 gT ðtÞEP gðtÞ ¼ 2gT ðtÞPT ET gðtÞ: dt dt
ð24Þ
It follows from (15) that one can write
_ dÞ; tÞ: 0 ¼ yðtÞ þ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
ð25Þ
Utilizing the neutral difference operator equation in (9) allows to get
0 ¼ rðxt Þ þ xðtÞ þ Ah
Z
t
xðsÞds Ad xðt dÞ:
ð26Þ
th
In view of (25) and (26), one can compute the following
3 2 3 2 yðtÞ yðtÞ d _ dÞ; tÞ 5 tÞ þ gðxðt hÞ; tÞ þ hðxðt E gðtÞ ¼ 4 0 5 ¼ 4 yðtÞ þ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; Rt dt 0 rðxt Þ þ xðtÞ þ Ah th xðsÞds Ad xðt dÞ Z t _ dÞ; tÞ xðsÞds þ C2 xðt dÞ þ C3 ½f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt ¼ AgðtÞ þ C1 T
ð27Þ
th
2
0 where A ¼ 4 BK I
I I 0
2 3 0 0 0 0 A0 5; C1 ¼ 4 0 5; C2 ¼ 4 0 5; C3 ¼ 4 I 5. Substituting (27) into (24) gives Ah Ad 0 I 3
2
3
2
Z V_ 1 ðxðtÞ; tÞ ¼ 2gT ðtÞPT AgðtÞ þ C1
3
t
_ dÞ; tÞ xðsÞds þ C2 xðt dÞ þ C3 ½f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
th
¼ vT ðtÞX1 vðtÞ;
ð28Þ
where
vðtÞ ¼ gT ðtÞ 2 6 6 6 6 6 6 6 6 X1 ¼ 6 6 6 6 6 6 6 6 4
R t
T xðsÞds th
T _ dÞ; tÞ xT ðt dÞ f T ðxðtÞ; tÞ xT ðt hÞ g T ðxðt hÞ; tÞ x_ T ðt dÞ h ðxðt
0 P T C3
0 P T C3
0
0
;
3
P T A þ AT P
P T C1
P T C2
P T C3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7 0 7 7 7 0 7 7 7 0 7 7: 7 0 7 7 7 0 7 7 7 0 5
0
0
T
Calculating V_ 2 ðxðtÞ; tÞ yields 2 V_ 2 ðxðtÞ; tÞ ¼ h xT ðtÞQxðtÞ h
Z
t
xT ðsÞQxðsÞds:
ð29Þ
th
One can reexpress xðtÞ in the following manner
xðtÞ ¼ C4 vðtÞ; where C4 ¼ ½ K1
h
Z
t
th
0
ð30Þ 0
0
0
0
0
Z xT ðsÞQxðsÞds 6
t
0 ; K1 ¼ ½ 0
T Z xðsÞds Q
th
0
I . It follows from Jensen integral inequality Gu et al. [3] that
t
xðsÞds
ð31Þ
th
Substituting (30), (31) into (29) gives 2 V_ 2 ðxðtÞ; tÞ 6 vT ðtÞðX2 þ h CT4 Q C4 ÞvðtÞ;
ð32Þ
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where
2
0
6 6 6 6 6 6 6 X2 ¼ 6 6 6 6 6 6 4
0
0 0 0 0 0 0
3
0 0 0 0 0 07 7 7 0 0 0 0 0 07 7 0 0 0 0 07 7 7 0 0 0 07 7 0 0 07 7 7 0 05
Q
0
Moreover, the time-derivative of V 3 ðxðtÞ; tÞ is computed as follows:
V_ 3 ðxðtÞ; tÞ ¼ xT ðtÞRxðtÞ xT ðt dÞRxðt dÞ ¼ vT ðtÞðX3 þ CT4 RC4 ÞvðtÞ;
ð33Þ
where
2
0 0
6 6 6 6 6 6 X3 ¼ 6 6 6 6 6 6 6 4
0
0
3
0 0 0 0 0
0 0 0 0 07 7 7 R 0 0 0 0 0 7 7 0 0 0 0 07 7 7: 0 0 0 07 7 0 0 07 7 7 0 05 0 0
Computing the time-derivative of V 4 ðxðtÞ; tÞ yields
V_ 4 ðxðtÞ; tÞ ¼ xT ðtÞSxðtÞ xT ðt hÞSxðt hÞ ¼ vT ðtÞðX4 þ CT4 SC4 ÞvðtÞ;
ð34Þ
where
2
0 0 0 0
6 6 6 6 6 6 6 X4 ¼ 6 6 6 6 6 6 4
0
0 0 0
3
0 0 07 7 7 0 0 0 0 0 07 7 0 0 0 0 07 7 7: S 0 0 0 7 7 0 0 07 7 7 0 05
0 0 0
0
0
Finally, one can compute V_ 5 ðxðtÞ; tÞ as follows:
_ x_ T ðt dÞZ xðt _ dÞ: V_ 5 ðxðtÞ; tÞ ¼ x_ T ðtÞZ xðtÞ
ð35Þ
_ Let us reexpress xðtÞ in the following form as
_ xðtÞ ¼ C5 vðtÞ; where C5 ¼ ½ K2
0
ð36Þ 0
I
Ah
I
Ad
I ; K2 ¼ ½ BK
V_ 5 ðxðtÞ; tÞ ¼ vT ðtÞðX5 þ CT5 Z C5 ÞvðtÞ; where
2
0 0 0 0 0 0
6 6 6 6 6 6 6 X5 ¼ 6 6 6 6 6 6 4
3
0
0
0 0 0 0 0
0
0 0 0 0
0
0 0 0
0
0 0
0
0
0
Z
07 7 7 07 7 07 7 7: 07 7 07 7 7 05 0
0
A . Substituting (36) into (35) gives
ð37Þ
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
2943
It follows from (6)–(8) that one can write the following inequality
_ e1 ½f T ðxðtÞ; tÞf ðxðtÞ; tÞ þ a2 xT ðtÞxðtÞ þ e2 ½g T ðxðt hÞ; tÞgðxðt hÞ; tÞ þ b2 xT ðt hÞxðt hÞ þ e3 ½hT ðxðt 2 _T _ _ dÞ; tÞhðxðt dÞ; tÞ þ c x ðt dÞxðt dÞ ¼ vT ðtÞðX6 þ e1 a2 CT4 C4 þ e2 b2 CT6 C6 þ e3 c2 CT7 C7 ÞvðtÞ P 0;
ð38Þ
where
2
0 0 0 0 6 0 0 0 6 6 0 6 0 6 6 e1 I X6 ¼ 6 6 6 6 6 6 4
0 0 0 0 0 0 0 0 0 0 e2 I
3 0 0 0 0 7 7 7 0 0 7 7 0 0 7 7 0 0 7 7 7 0 0 7 7 0 0 5 e3 I
and C8 ¼ ½ K3 0 0 0 0 0 0 0 ; K3 ¼ ½ K 0 0 . Therefore, substituting (28), (32), (33), (34), (37) into (23) and adding (38) to the resulting expression leads to compute the following inequality as follows:
_ VðxðtÞ; tÞ þ xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞ 6 vT ðtÞX0 vðtÞ; P6
2 T 4 ðh Q þ R T u ðtÞS2 uðtÞ
where X0 ¼ i¼1 Xi þ C _ VðxðtÞ; tÞ þ xT ðtÞS1 xðtÞ þ
ð39Þ T 5Z
2
T 6
T 7
þ S1 þ S þ e1 a2 IÞC4 þ C C5 þ e2 b C C6 þ e3 c2 C C7 þ C < 0, one needs to satisfy
T 8 S2
C8 . In order to guarantee
X0 < 0
ð40Þ
then applying Schur complement [23] to (40) allows to get
X < 0;
ð41Þ
where
2 6 6 6 6 6 6 6 6 6 X¼6 6 6 6 6 6 6 6 6 4
Xð1; 1Þ
hKT1
Q 1
KT1 KT1 0 0 R1 0 S1 1
KT3 aKT1 KT1 0 0 0 0 0 0 0 0 0 1 0 0 S2 0 e1 1 I S1
Xð1; 8Þ Xð1; 9Þ Xð1; 10Þ 0 0 0
0 0 0
0 0 0
0 0
0 0
0 0
0 e1 2 I
0 0
0 0
Z 1
0 e1 3 I
3 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
with
2
P T A þ AT P P T C 1 P T C 2 P T C 3 6 Q 0 0 6 6 6 R 0 6 6 e1 I Xð1; 1Þ ¼ 6 6 6 6 6 6 4 2 T3 2 3 0 K2 6 0 7 607 7 6 6 7 7 6 6 7 6 0 7 607 7 6 6 7 6 I 7 6 7 7; Xð1; 10Þ ¼ 6 0 7: Xð1; 9Þ ¼ 6 6 AT 7 607 6 h7 6 7 7 6 6 7 6 I 7 607 6 T7 6 7 4A 5 4 cI 5 d 0 I
0 P T C3 0 0 0 0 0 0 S 0 e2 I
0 0 0 0 0 0 Z
3 P T C3 0 7 7 7 0 7 7 0 7 7; 0 7 7 7 0 7 7 0 5 e3 I
3 0 607 6 7 6 7 607 6 7 607 7 Xð1; 8Þ ¼ 6 6 bI 7; 6 7 6 7 607 6 7 405 2
0
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M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
Then preand postmultiplying X<0 in (41) with MT and M, respectively, where M ¼ diagfX; Q ; R; e1 ; S; e2 ; Z; e3 ; I; I; I; I; I; I; I; I; Ig; X ¼ P1 ; Q ¼ Q 1 ; R ¼ R1 ; S ¼ S1 ; Z ¼ Z 1 ; ei ¼ e1 i ; i ¼ 1; 2; 3, and choosing K ¼ YX 1 give the following linear matrix inequality
R < 0;
ð42Þ
where
2 6 6 6 6 6 6 6 6 6 R¼6 6 6 6 6 6 6 6 6 6 4
Rð1; 1Þ hRð1; 2Þ Rð1; 2Þ Rð1; 2Þ Rð1; 5Þ aRð1; 2Þ Rð1; 2Þ Rð1; 8Þ Rð1; 9Þ Rð1; 10Þ Q
0
0
0
0
0
0
0
0
R
0
0
0
0
0
0
0
S1 1
0
0
0
0
0
0
S1 2
0 e1 I
0 0
0 0
0 0
0 0
S
0
0 e2 I
0
0
0
Z
0 e3 I
3 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
and
2 6 6 6 6 6 6 6 6 6 Rð1; 1Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
X 2 þ X T2
X 3 þ Y T BT X T2 þ X T4 AT0
X 1 þ X T4
0
0
0
0
0
0
X 3 X T3 þ A0 X 5 þ X T5 AT0
A0 X 6 þ X T5
0
0
e1 I
0
e2 I
0
X 6 þ X T6
Ah Q
Ad R
0
0
0
0
Q
0
0
0
0
0
R
0
0
0
0
0
0
S
0
0 e1 I
0
0 e2 I
Z
0
3
7 e3 I 7 7 7 0 7 7 0 7 7 7 0 7 7: 0 7 7 7 0 7 7 0 7 7 7 0 5 e3 I
Now, one can replace A; Ah ; Ad ; B with A þ DAðtÞ; Ah þ DAh ðtÞ; Ad þ DAd ðtÞ; B þ DBðtÞ, respectively, in (42) to get
R þ DRðtÞ þ DRT ðtÞ < 0;
ð43Þ
where
2
DRðtÞ ¼ PT FðtÞH; 2 6
H¼4
6
0 DT
P ¼ 40
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
DT
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
Eb Y þ E0 X 4
E0 X 5
E0 X 6
0
0
0
0
Eh Q
Eb Y þ E a X 4
Ea X 5
Ea X 6
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0
0
0
0
0
Ed R 0
0
0
0
0
0 Eh S 0 Ed Z
T
0
3T
7 05 ;
0
0 0 0 0 0 0 0 0 0 0
3T
7 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0
with E0 ¼ Ea þ Eh . Applying a norm-bounding inequality allows to rewrite (43) as follows: T R þ DRðtÞ þ DRT ðtÞ 6 R þ e0 PT P þ e1 0 H H < 0:
ð44Þ
Then, by Schur complement [23], one obtains the linear matrix inequality given in (17). This completes the proof. h Next, one can show that by the neutral state-feedback controller (12), the guaranteed cost function in (11) has an upper bound. For this purpose, one can notice that (39) and (42) imply
_ VðxðtÞ; tÞ 6 xT ðtÞS1 xðtÞ uT ðtÞS2 uðtÞ:
ð45Þ
Integrating both sides of (45) from 0 to T > 0 gives
VðxðTÞ; TÞ Vðxð0Þ; 0Þ 6
Z
T
½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞdt:
ð46Þ
0
As the system is ensured to be robustly asymptotically stable once the conditions in (16) and (17) are satisfied, for RT limT!1 VðxðTÞ; TÞ ! 0, one obtains limT!1 0 ½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞdt ¼ J 6 J where J is as defined in Theorem 1. One
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
2945
can notice that Theorem 1 describes how to synthesize a robust guaranteed cost neutral state-feedback controller. In order to introduce the design of an optimal robust guaranteed cost neutral state-feedback controller that minimizes the upper bound of the quadratic cost function (11), the following theorem is presented. Theorem 2 [13]For the uncertain nonlinear neutral system (1) and the associated quadratic cost function (11), consider the optimization problem defined as follows:
(
lþ
min
4 X
X;Q ;R;S;Z;Y;T i ;i¼1;...;4;ej ;j¼1;2;3;e0
)
traceðTi Þ
i¼1
subject to ð16Þ and ð17Þ; and " " " " "
l rT ðUð0ÞÞ T 1 T 2 T 3 T 4
#
X 1 # W T1 6 0; Q # W T2 6 0; R # W T3 6 0; S # W T4 60 Z
6 0;
ð47Þ ð48Þ ð49Þ ð50Þ ð51Þ
such that if a feasible solution set of X; Q ; R; S; Z; Y; T i ; i ¼ 1; . . . ; 4; ej ; j ¼ 1; 2; 3; e0 can be achieved then the neutral state-feedback control law (12) is said to be an optimal robust guaranteed cost control law ensuring that the quadratic cost function upper R0 bound given in (18) for uncertain nonlinear neutral system (1) is minimized, where h h ðs þ hÞUðsÞUT ðsÞds ¼ W 1 W T1 ; R0 R0 R0 T T T T T T _ ðsÞds ¼ W 4 W . UðsÞU ðsÞds ¼ W 2 W 2 ; h UðsÞU ðsÞds ¼ W 3 W 3 ; d U_ ðsÞU 4 d Proof. It follows from (47)–(51) and Schur complement [23] that one can obtain the equivalent inequalities of T 1 rT ðUð0ÞÞX 1 W 1 6 T 1 , W T2 R1 W 2 6 T 2 , W T3 S1 W 3 6 T 3 , W T4 Z 1 W 4 6 T 4 , respectively. Moreover, note that 1 rðUð0ÞÞ 6 l, W 1 Q utilizing the trace operator allows to compute the following expressions
h
Z
0
Z 0 Z 0 ðs þ hÞUT ðsÞQ 1 UðsÞds ¼ trace h ðs þ hÞUT ðsÞQ 1 UðsÞds ¼ trace h ðs þ hÞUT ðsÞUðsÞdsQ 1
h
h
h
¼ trace½W 1 W T1 Q 1 ¼ trace½W T1 Q 1 W 1 ; Z
0
Z
UT ðsÞR1 UðsÞds ¼ trace
d
Z
0
0
UT ðsÞR1 UðsÞds ¼ trace
Z
d
Z
0
UT ðsÞS1 UðsÞds ¼ trace
h
Z
0
d
Z
d
Z
0
d
0
UT ðsÞS1 UðsÞds ¼ trace
h
U_ T ðsÞZ 1 U_ ðsÞds ¼ trace
0
h
U_ T ðsÞZ 1 U_ ðsÞds ¼ trace
Z
0
d
UT ðsÞUðsÞdsR1 ¼ trace½W 2 W T2 R1 ¼ trace½W T2 R1 W 2 ;
UT ðsÞUðsÞdsS1 ¼ trace½W 3 W T3 S1 ¼ trace½W T3 S1 W 3 ;
U_ T ðsÞU_ ðsÞdsZ 1 ¼ trace½W 4 W T4 Z 1 ¼ trace½W T4 Z 1 W 4 :
Therefore, in view of (18) one can easily deduce that the following inequality
J 6 l þ
4 X
traceðTi Þ
ð52Þ
i¼1
is always satisfied. This shows that minimizing the right-hand part of Eq. 52 is equivalent to the minimization of the upper bound of the quadratic cost function (11) which is given in (18). Hence, the proof is completed. h Remark 1. It can be argued that the neutral state-feedback control is difficult to implement in physical systems as it requires the use of exact information on the delay terms. It is clear that an approximation will cause the obtained results to be questionable. Moreover, if the system has time-varying delays, then the proposed approach seems to be difficult to extend and implement. Therefore, we need to underline that the developed method gives improved results only for a class of neutral time-delay systems with constant and known discrete and neutral delays.
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4. Numerical examples In this section, two numerical examples are presented to demonstrate the application of Theorems 1 and 2. Example 1. An uncertain linear neutral system example is considered as follows:
_ dÞ þ ½B þ DFðtÞEb uðtÞ; _ xðtÞ ¼ ½A þ DFðtÞEa xðtÞ þ ½Ah þ DFðtÞEh xðt hÞ þ ½Ad þ DFðtÞEd xðt 0 0:5 0 0:2 0:2 ; Ah ¼ ; Ad ¼ 0 1 0:2 0:5 0 and the initial condition is defined as where A ¼
/ðtÞ ¼
et
et
ð53Þ
0 0:1 with the delay bounds being given as h ¼ 0:5; d ¼ 0:2 ;B ¼ 0:5 0:2
8t 2 ½0:5; 0:
1 0 ; S2 ¼ 0:2. If it is assumed that D ¼ 0, the system 0 1 rðUð0ÞÞ; W 1 ; W 2 yields in (52) reduces to the example studied in Park [12]. Calculating 0:8394 0:1516 0:1516 0:2871 0:2871 ; W1 ¼ ; W2 ¼ . Choosing q1 ¼ 0:1; q2 ¼ 0:2, the condirðUð0ÞÞ ¼ 0:5608 0:1516 0:1516 0:2871 0:2871 9:0812 1:3109 . A feation of (16) for the stability of the nominal neutral difference equation is satisfied with K0 ¼ 1:3109 11:7136 sible solution set has been obtained for satisfying (17) appropriately to get The gain matrices of the associated cost function are selected as S1 ¼
2
0:2989
0:1982
0:1982
0:1328
X1 ¼ 4 2 X4 ¼ 4
0:0088
3
2
5;
X2 ¼ 4
3
2
5;
X5 ¼ 4
2:3366
1:1936
3
2
5 10 ;
X3 ¼ 4
3
2
5;
X6 ¼ 4
4
1:1935 2:3366
0:0817
0:1657 0:1214
0:0800
0:1666 0:1215 2 T1 ¼ 4
Y ¼ ½ 0:0997 0:4124 ;
0:0097
0:0939
0:0939
0:0939
0:0939
2:3366 1:1935 1:1936
2:3365
0:7273 0:0284
3 5 104 ;
3 5;
0:0267 0:4748
3
2
5;
T2 ¼ 4
0:0512
0:0511
0:0511
0:0512
3
2
5;
Q ¼4
5:4325 1:6851 1:6851 0:9629
3 5;
0:7119 , and l ¼ 2:3707. Then computing the cost function upper bound gives J ¼ 2:6607 and the neutral 1:9663 gain matrix is obtained as K ¼ ½ 167:6345 253:3164 . However, for the purpose of comparison one noachievable cost function bound obtained in Park [12] is 9.7773 with the feedback gain matrix of 3:3077 . Hence the proposed methodology is shown to be less conservative than that of [12]. Let us 0:5 ; Ea ¼ ½ 0:1 0:2 ; Eh ¼ ½ 0:3 0:1 ; Ed ¼ ½ 0:2 0:1 ; Eb ¼ 0:1. Then applicanow choose the uncertainty matrices as D ¼ 0:3 tion of Theorems 1 and 2 results in the following set of feasible solution for the design parameters as 3:8941 0:7119 state-feedback tices that the K ¼ ½ 1:8737
R¼
2
0:2286
X1 ¼ 4 2
0:2065
0:2065
0:2734
0:1547
0:1858
3
2
5;
X2 ¼ 4
3
2
5;
X5 ¼ 4 2
Y ¼ ½ 0:2027 2:1228 ;
R¼4 1:5373
1:5373
3
2
5 10 ; 3
0:0180 0:0593 0:6232
0:8389 0:7164
2:0874
1:0709
X3 ¼ 4
0:1172
0:1172
0:1172
T1 ¼ 4
3
2
5;
X6 ¼ 4
0:6231
0:1172
9:5422 1:0703 0:5279
0:5280 9:5798
X4 ¼ 4
2
9:5423
3
2
5;
T2 ¼ 4
3 5 103 ;
9:5791
0:3376
0:1812
0:1813
0:3123
3 5;
0:0518
0:0518
0:0518
0:0518
3
2
5;
Q ¼4
0:3827
0:0731
0:0731
0:2774
3 5;
e4 ¼ 0:1667; e5 ¼ 0:1098; e6 ¼ 3:7400 104 ; and l ¼ 3:5325:
1:5952
The upper bound of the cost function is calculated using the solutions of the required matrices yields J ¼ 3:8706 and the feedback gain of K ¼ ½ 24:8891 26:5678 . This result is also the indicative of the observation that the performance of
3 5;
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the proposed guaranteed cost controller is still quite better than that of the method presented in Park [12] even in the case of norm-bounded uncertainties. Example 2. Another example of uncertain nonlinear neutral system is considered as follows:
_ dÞ þ ½B þ DFðtÞEb uðtÞ þ f ðxðtÞ; tÞ _ xðtÞ ¼ ½A þ DFðtÞEa xðtÞ þ ½Ah þ DFðtÞEh xðt hÞ þ ½Ad þ DFðtÞEd xðt _ þ gðxðt hÞ; tÞ þ hðxðt dÞ; tÞ;
ð54Þ
where
A¼
1
1
1
1
Ah ¼
;
Ea ¼ ½ 0 0:2 ;
0
0:2
0:2
0
Eh ¼ ½ 1 0:3 ;
;
Ad ¼
0 0:1 0 0:1
Ed ¼ ½ 0:2 0:2 ;
;B ¼
1 ; 1
D¼
0 0:2
;
Eb ¼ 0:5
and the delay bounds are defined as h ¼ 1:0; d ¼ 2:0. The initial condition is given as follows:
UðtÞ ¼
1 et
8t 2 ½2:0; 0
with the cost function gain matrices being chosen as S1 ¼
"
rðUð0ÞÞ ¼ " W4 ¼
1:1129
#
1:1865
0
"
0:5975 0:3782
W1 ¼
;
0:3782 0:3752
0 ; S2 ¼ 1. One can compute rðUð0ÞÞ; W i ; i ¼ 1; . . . ; 4 as 1
0:5 0
#
"
;
W2 ¼
1:3356 0:4650
#
"
;
0:4650 0:5241
0:8817 0:4719
W3 ¼
#
0:4719 0:4579
;
#
0
0 0:7006
:
Let us assume that the system in (54) is not subject to any nonlinear parameter perturbations, i.e. f ðxðtÞ; tÞ ¼ 0, _ dÞ; tÞ ¼ 0 leading to the example considered in Xu et al. [6]. The stability of the neutral operator, gðxðt hÞ; tÞ ¼ 0, hðxðt rðxt Þ is ensured by choosing q1 ¼ q2 ¼ 0:3 and obtaining a feasible solution set for the condition given in (16) gives 1:4214 0:0108 ; k ¼ 0:1281. The linear matrix inequality (17) has given a feasible solution set of K0 ¼ 0:0108 1:3121 1:8276 0:9529 , X1 ¼ 0:9529 1:4210
2 X2 ¼ 4
437:6883
447:7330
3
2
5;
X3 ¼ 4
429:1885
426:7856 447:3215 2
0:4632
0:5332
0:5099
0:4502
X5 ¼ 4 2
3:1256
3:2367
3:2367
9:0033
Q ¼4
443:2937 434:9854
3
2
5;
X6 ¼ 4
3
2
5;
R¼4
3
2
5;
X4 ¼ 4
443:2540
0:8077
0:1552
0:1570
0:7105
0:6365
0:5264
0:5009
0:4470
3 5;
3 5;
73:7035
10:1748
10:1748
4:9701
Y ¼ ½ 1:2172 0:8982 ;
3
2
5;
T1 ¼ 4
0:3005 0:2075
3
2
5;
T2 ¼ 4
0:2075 0:1603
0:1529 0:1155
3 5;
0:1155 0:1104
e4 ¼ 5:4000; e5 ¼ 5:2079; e6 ¼ 1:9049 103 and l ¼ 4:0658 which indicate that the cost function upper bound can be computed as J ¼ 4:7377 with a neutral state-feedback gain matrix of K ¼ ½ 1:5309 1:6587 . However, the achievable cost function bound obtained in Xu et al. [6] is 50.0275 with the feedback gain matrix of K ¼ ½ 0:9943 0:9615 . This shows that the proposed methodology is capable of yielding less conservative cost function bounds implying that when the proposed controller is employed, a considerably less amount of control effort is required than that of the controller presented in Xu et al. [6]. Now let us assume that the numerical example of neutral system given in (54) involves nonlinear parameter perturbations defined such that T
f ðxðtÞ; tÞ ¼ ½ a1 cos tjx1 ðtÞj a2 sin tjx2 ðtÞj ;
jai j 6 0:3; T
jbi j 6 0:3;
T
jci j 6 0:3;
gðxðt hðtÞÞ; tÞ ¼ ½ b1 cos tjx1 ðt hÞj b2 sin tjx2 ðt hÞj ; _ dðtÞÞ; tÞ ¼ ½ c1 cos tjx_ 1 ðt dÞj c2 sin tjx_ 2 ðt dÞj ; hðxðt The
simulation work to 0:6560 0:3237 , X1 ¼ 0:3237 0:5468
satisfy
the
synthesis
condition
given
in
i ¼ 1; 2: (17)
yields
a
feasible
solution
set
of
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M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
X2 ¼
2:2927
4:1680
2:2926 4:1678 0:3095 0:0220 X3 ¼ 104 ; X 4 ¼ ; 4:1676 2:2926 0:1967 0:1279 0:1962 0:0110 X6 ¼ ; Y ¼ ½ 1:0444 0:4810 ; 0:0108 0:2791 35:9732 5:2477 2:1589 0:1474 R¼ ; S¼ ; 5:2477 2:3589 0:1474 2:8522 0:6322 0:4484 0:3228 0:2624 0:4597 0:2845 T1 ¼ ; T2 ¼ ; T3 ¼ ; 0:4484 0:3228 0:2624 0:2259 0:2845 0:1877 #
104 ;
4:1676 2:2928 0:1563 0:2269 X5 ¼ ; 0:2436 0:2915 0:9945 0:6536 Q¼ ; 0:6536 2:6037 5:8783 0:1885 Z¼ ; 0:1885 4:9904 " 0 7:8343 106 ; T4 ¼ 0 0:0985
e1 ¼ 0:2438;
e2 ¼ 0:2949;
e3 ¼ 0:5390;
e4 ¼ 0:7739; e5 ¼ 2:3444;
e6 ¼ 30:8330; and l ¼ 9:6704 and the upper bound of the cost function is computed as J ¼ 11:9199 with the neutral state-feedback gain matrix of K ¼ ½ 2:8622 2:5742 . Consequently, it can be seen that even in the case of the existence of nonlinear parameter perturbations, the proposed robust guaranteed cost neutral state-feedback controller performs much better than that given in Xu et al. [6] which is, however, applicable for only norm bounded uncertainty situation.
5. Conclusions This paper has investigated robust delay-dependent guaranteed cost controller design for uncertain neutral systems with nonlinear parameter perturbations. A memory type neutral form of state-feedback control law is introduced. On the basis of employing a descriptor representation, an augmented descriptor form of a candidate Lyapunov–Krasovskii functional is adopted to study the stability of the neutral system. In particular, with free weighting matrices and new descriptor system approach concerned in system model and operator, sufficient robust delay-dependent linear matrix inequality (LMI) conditions are derived such that the existence of a guaranteed cost neutral state-feedback controller is ensured once the specified LMIs are satisfied. Moreover, several additional LMI conditions are also presented in the context of a convex optimization algorithm in such a manner that the upper bound of a quadratic cost function is minimized. Two numerical examples of neutral systems which have been presented in the literature are taken into consideration to illustrate the application of the newly developed LMI guaranteed cost controller synthesis approach. Several numerical case studies have concluded that the proposed method of the present paper has been able to assure a less conservative control cost implying that the requirements can be achieved with considerably less amount of control effort in comparison to some of the existing approaches. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
V.B. Kolmanovskii, V.R. Nosov, Stability of functional differential equations, Mathematics in Science and Engineering, vol. 180, Academic Press, 1986. J. Hale, S.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, 2003. S.H. Esfehani, S.O.R. Moheimani, I.R. Petersen, LMI approach to suboptimal guaranteed cost control for uncertain time-delay systems, IEE Proceedings – Control Theory and Applications 145 (6) (1998) 491–498. L. Yu, J. Chu, An LMI approach to guaranteed cost control of linear uncertain time-delay systems, Automatica 35 (6) (1999) 1155–1159. S.Y. Xu, J. Lam, C.W. Yang, E.I. Verriest, An LMI approach to guaranteed cost control for uncertain linear neutral delay systems, International Journal of Robust and Nonlinear Control 13 (1) (2003) 35–53. X. Nian, J. Feng, Guaranteed cost control of a linear uncertain system with multiple time-varying delays: an LMI approach, IEE Proceedings – Control Theory and Applications 150 (1) (2003) 17–22. O. Kwon, S. Won, D. Yue, Delayed feedback guaranteed cost controller design for uncertain time-delay systems, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences E86A (9) (2003) 2413–2418. Ju H. Park, Guaranteed cost control for uncertain large scale systems with time-delays via delayed feedback, Chaos, Solitons & Fractals 27 (3) (2006) 800–812. Ju H. Park, Delay-dependent guaranteed cost stabilization criterion for neutral delay-differential systems:matrix inequality approach, Computers & Mathematics with Applications 47 (10–11) (2004) 1507–1515. B. Chen, J. Lam, S. Xu, New observer-based cost control for neutral delay systems, Proceedings of the Institution of Mechanical Engineers Part I – Journal of Systems and Control Engineering 218 (13) (2004) 227–235. J.H. Park, Delay-dependent criterion for guaranteed cost control of neutral delay systems, Journal of Optimization Theory and Applications 124 (2) (2005) 491–502. J.H. Park, O. Kwon, On guaranteed cost control of neutral systems by retarded integral state-feedback, Applied Mathematics and Computation 165 (2) (2005) 393–404. C.H. Lien, Guaranteed cost observer-based controls for a class of uncertain neutral time-delays, Journal of Optimization Theory and Applications 126 (1) (2005) 137–156. B. Chen, X.P. Liu, Fuzzy guaranteed cost control of nonlinear neutral delay systems with parameter uncertainty, Dynamics of Continuous Discrete and Impulsive Systems – Series B – Applications 12 (2005) 381–396. D.M. Zhang, L. Yu, Delay-dependent guaranteed cost control for generalized uncertain neural systems, Dynamics of Continuous Discrete and Impulsive Systems – Series A – Mathematical Analysis 13 (2006) 974–982. B. Chen, J. Lam, S.Y. Xu, Memory state-feedback guaranteed cost control for neutral delay systems, International Journal of Innovative Computing Information and Control 2 (2) (2006) 293–303. O.M. Kwon, Ju H. Park, Guaranteed cost control for uncertain large scale systems with time-delays via delayed feedback, Chaos, Solitons & Fractals 27 (3) (2006) 800–812.
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[19] C.H. Lien, Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach, Chaos, Solitons and Fractals 33 (3) (2007) 1017–1027. [20] E. Fridman, U. Shaked, An improved stabilization method for linear time-delay systems, IEEE Transactions on Automatic Control 47 (11) (2002) 1931– 1937. [21] Y. He, M. Wu, J.-H. Shi, G.-P. Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems and Control Letters 51 (2004) 57–65. [22] M.N.A. Parlakçı, Robust stability of uncertain time-varying state-delayed systems, IEE Proceedings – Control Theory and Applications 153 (4) (2006) 469–477. [23] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1994.
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Robust delay-dependent guaranteed cost controller design for uncertain neutral systems M.N. Alpaslan Parlakçı Istanbul Bilgi University, Department of Computer Science, Kurtulusß Deresi Cad. No. 47, Dolapdere, 34440 Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Neutral systems Descriptor representation Uncertainty Parameter perturbations Quadratic cost Linear matrix inequality
a b s t r a c t In this paper, a novel robust delay-dependent guaranteed cost controller is introduced for a class of uncertain nonlinear neutral systems with both norm-bounded uncertainties and nonlinear parameter perturbations. A neutral memory state-feedback control law is chosen such that a quadratic cost function is minimized. On the basis of a descriptor type model transformation, an augmented descriptor form Lyapunov–Krasovskii functional is proposed in order to derive a linear matrix inequality (LMI) condition of the synthesis of a robust stabilizing guaranteed cost controller for uncertain nonlinear neutral systems. The descriptor approach is applied for both the delay-differential equation and the neutral difference operator allowing to introduce several additional free weighting matrices in order to obtain further relaxation in the synthesis process. The proposed method of introducing some relaxation into design problem also leads to develop LMI conditions which can be easily solved using interior point algorithms. Two numerical examples have been introduced to show the application of the theoretical results which indicate that the proposed approach improves the guaranteed cost performance. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Time-delay systems is a significantly fruitful field of research for several decades. The main reason for this is that time-delay naturally arises in many physical and dynamical systems leading to bring about many theoretical and practical problems which need to be resolved. Among some of them, one can notice that a system which is subject to time-delay may exhibit a poor performance, it may get into an unstable behavior or it may not endeavour with external disturbances or even it may excite undesired modes which cause to get far away from meeting the required specifications. Concerning linear time-invariant (LTI) control systems, time-delay may exist in the state of the system which constitutes a broad class of systems so called retarded time-delay systems. Another important class of delay-differential systems is referred to as time-delay systems of neutral type in which the time-delay may be available in both position and velocity of the system. For a theoretical and practical consideration of neutral systems, one can refer to [1–3]. The stability analysis or stabilization synthesis for time-delay systems of neutral type have generally been conducted with respect to two different aspects of interest in the community of time-delay systems. The analysis and synthesis conditions have been developed delay-independently or delay-dependently. For practical time-delay systems involving a delay of finite size, it is reasonable that the need of a condition being applicable for any size of delay changing from 0 to infinity may lead to conservativeness. Hence, in general, delay-independent results are usually considered to be more conservative than their counterparts of delay-dependent type. This also indicates that the stability or stabilization range of time-delay may be used as an evaluation criterion among different theoretical results in terms of conservativeness. Moreover, the noise rejection capability defined in
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.040
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terms of H1 norm bound and controller performance measured in terms of a quadratic cost bound can as well be listed as the other two aspects of objectives in the analysis and design problems for time-delay systems. In addition to delay phenomenon, one can notice that the systems under consideration may be also subject to uncertainties of norm-bounded type and/or nonlinear parameter perturbations which need to be handled. In recent years, a number of different guaranteed cost controller design investigations have been taken into consideration for neutral systems with or without uncertainties and/or nonlinear parameter perturbations. For example see [4–19] and the references therein. The common feature of the researches listed in [4–19] is that the proposed methods have employed linear matrix inequality (LMI) techniques. Concerning the effectiveness of a stability condition or the performance of a controller synthesis in terms of the measures of delay conservatism, noise rejection and/or quadratic control cost, it is well known that [20–22] the utilization of some free weighting parameters introduce a considerable rate of relaxation into the results in particular for systems under the existence of uncertainties. However, the existing work on the subject has not taken into account this useful phenomenon because of the fact that the involvement of such slack variables destroy the nature of the LMI form of the stability and/or synthesis conditions. Therefore, the motivation of the present research can be stated as developing a novel technique that allows to develop improved LMI form of delaydependent guaranteed cost controller synthesis through the use of free weighting matrices. The key feature of the proposed methodology is that the inclusion of some new slack parameters do not devastate the LMI nature of the results but more than that the theoretical conditions of a stabilizing controller synthesis become considerably relaxed yielding a better performance. A robust memoryless state-feedback LMI controller design is investigated in [4] for uncertain time-delay systems with norm-bounded uncertainty. Sufficient LMI conditions have been derived in [5] for the existence of a guaranteed cost statefeedback controller of memoryless type. The design problem of guaranteed cost control of uncertain neutral systems with norm-bounded uncertainties is taken into consideration in [6] by employing an LMI technique. Nian and Feng [7] have considered uncertain neutral systems with time-varying delay and have developed some sufficient LMI conditions for the synthesis of a guaranteed cost memoryless state-feedback controller. Employing a memory type of state-feedback controller, Kwon et al. [8] have studied the design of guaranteed cost controller for uncertain time-delay systems by using an LMI approach. A guaranteed cost memoryless state-feedback controller is introduced in [9] for the stabilization of a class of neutral differential systems with parametric uncertainties. Park [10] has proposed a solution to the optimal guaranteed cost control problem by utilizing state-feedback control laws for a class of time-delay systems of neutral type. Both a memory state-feedback based guaranteed cost control and an observer-based output feedback guaranteed cost control problems have been addressed for neutral systems in [11] via LMI methods. For a class of nominal neutral systems, a guaranteed cost state-feedback controller is developed on the basis of an LMI technique in [12]. Introducing a retarded type of integral memory state-feedback controller, Park and Kwon [13] have obtained some LMI criteria to design a guaranteed cost controller for neutral systems. An observer-based guaranteed cost controller synthesis is given in [14] by proposing an LMI feasibility condition. Chen and Liu [15] have investigated the design problem of fuzzy guaranteed cost controller for nonlinear fuzzy neutral systems in terms of the feasibility of some linear matrix inequalities. Utilizing a descriptor form of a system representation and a recent result on bounding of cross terms, a guaranteed cost memoryless state-feedback control of uncertain neutral systems has been studied in [16] on the basis of an LMI methodology. A memory state-feedback control law and a quadratic cost function based on a delayed state approach have been introduced in [17] for the guaranteed cost control design problem of uncertain neutral systems through the feasibility condition of a linear matrix inequality. Utilizing a neutral model transformation and Lyapunov function method, Kwon and Park [18] have developed a guaranteed cost controller for uncertain large scale time-delay systems with delayed feedback. Both delay-independent and delay-dependent guaranteed cost stabilization criteria have been derived in [19] for a class of uncertain neutral systems in terms of an LMI approach. In this paper, the design problem of a robust delay-dependent guaranteed cost control of a class of uncertain nonlinear systems with both norm-bounded uncertainties and nonlinear parameter perturbations has been studied on the basis of a descriptor form of system representation along with an augmented descriptor type of Lyapunov–Krasovskii functional which has been recently proposed by Parlakçı [22]. The proposed approach allows to introduce free weighting matrices embedded in the Lyapunov–Krasovskii functional which has a strong potential effect of providing relaxation for the theoretical results without preventing to develop a linear matrix inequality methodology. A memory state-feedback controller in the form of neutral structure is introduced to provide integral and delayed state-feedback for a better control performance. The use of descriptor system representation in the quadratic Lyapunov stability analysis has been applied to both delay-differential equation and neutral operator equation which lead to be able to include additional slack variables when compared to conventional descriptor approach. A sufficient robust delay-dependent guaranteed cost controller synthesis criterion is derived in terms of the feasibility condition of a considerably relaxed linear matrix inequality which can be easily solved using interior point algorithms. Moreover, a convex optimization problem with LMI constraints is formulated to design the optimal guaranteed cost delayed neutral state-feedback controller which minimizes the upper bound of the guaranteed cost function for the closed-loop uncertain nonlinear neutral system. In order to demonstrate a practical application of the proposed guaranteed cost controller synthesis, two numerical examples are presented from the literature. The numerical results show that the novel augmented descriptor LMI approach improves the guaranteed cost performance in comparison to the existing guaranteed cost controllers in the literature.
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2. Problem statement Let us consider a class of uncertain nonlinear neutral systems with norm-bounded uncertainties and nonlinear parameter perturbations
_ dÞ þ ½B þ DBðtÞuðtÞ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ _ xðtÞ ¼ ½A þ DAðtÞxðtÞ þ ½Ah þ DAh ðtÞxðt hÞ þ ½Ad þ DAd ðtÞxðt _ dÞ; tÞ; þ hðxðt xðtÞ ¼ UðtÞ;
ð1Þ
_ ðtÞ 8t 2 ½s; 0; _ xðtÞ ¼U
s > 0;
n
ð2Þ m
where xðtÞ 2 R is the state vector of the system, uðtÞ 2 R is the control input, A; Ah ; Ad ; B are known real constant system matrices all with appropriate dimensions, DAðtÞ; DAh ðtÞ; DAd ðtÞ; DBðtÞ are unknown real time-varying matrix functions with appropriate dimensions representing time-varying parametric uncertainties which are assumed to be of the following form
½ DAðtÞ DAh ðtÞ DAd ðtÞ DBðtÞ ¼ DFðtÞ½ Ea
Eh
Ed
Eb ;
ð3Þ
where D; Ea ; Eh ; Ed ; Eb are known constant matrices with appropriate dimensions, and FðtÞ is an unknown real time-varying matrix satisfying
F T ðtÞFðtÞ 6 I
ð4Þ
_ dÞ; tÞ represent nonlinear parameter perturbations satisfying and f ðxðtÞ; tÞ; gðxðt hÞ; tÞ; hðxðt
f ð0; tÞ ¼ gð0; tÞ ¼ hð0; tÞ ¼ 0
ð5Þ
and UðÞ is a vector valued initial condition function. Note that the parameter uncertainties of DAðtÞ; DAh ðtÞ; DAd ðtÞ; DBðtÞ are said to be admissible if both (3) and (4) hold. Moreover, it is assumed that the nonlinear parameter perturbations satisfy the following bounding inequalities
kf ðxðtÞ; tÞk 6 akxðtÞk 8t > 0;
ð6Þ
kgðxðt hÞ; tÞk 6 bkxðt hÞk 8t > 0;
ð7Þ
_ dÞk 8t > 0; _ dÞ; tÞk 6 ckxðt khðxðt
ð8Þ
where a; b, and c are known positive scalars. The discrete, h, and neutral, d, delays are scalar constants such that s ¼ maxðh; dÞ. A difference operator can be defined as r : Ið½s; 0; Rn Þ ! Rn such that
rðxt Þ ¼ xðtÞ þ Ah
Z
t
xðsÞ ds Ad xðt dÞ:
ð9Þ
th
Definition 1 ([1,2]). The difference operator r is said to be stable if the zero solution of the homogeneous difference equation rxt ¼ 0; t P 0; x0 ¼ w 2 f/ 2 Cð½s; 0Þ : r/ ¼ 0g is uniformly asymptotically stable. The stability of the difference operator r is necessary for the stability of system (1) with uðtÞ 0. Assumption 1 [13]. Given positive scalars h, d, and any constant matrices Ah ; Ad 2 Rnn , the operator rðxt Þ defined as in (9) is asymptotically stable if there exist a symmetric and positive definite matrix, K0 and positive scalars q1 ; q2 such that
q1 þ q2 < 1; 2 4
ATd K0 Ad q1 K0
3
T
hAd K0 Ah 2
q2 K0 þ h
ATh
K0 A h
5 < 0:
ð10Þ
A quadratic cost function associated with the uncertain nonlinear neutral system (1) can be defined as
J¼
Z
1
½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞ dt;
ð11Þ
0
where S1 , and S2 are given constant positive definite gain matrices. A linear memory type neutral state-feedback control law can be chosen in a similar manner with Park [13] as follows:
uðtÞ ¼ K rðxt Þ;
ð12Þ
where K 2 Rmn denotes the feedback gain to be determined appropriately. The objective of the present work can be summarized in the following sequel. The objective of the present research can be stated such that given the positive scalars h, and d, the aim is to design a memory neutral form state-feedback controller (12), the closed-loop system is guaranteed to be asymptotically stable and the quadratic cost function (11) has an upper bound.
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Definition 2 [13]. For the uncertain nonlinear neutral system (1) and the quadratic cost function in (11), if there exists a control law u ðtÞ and a positive J such that for all admissible uncertainties and delays, the system is asymptotically stable and the closed-loop value of the cost function (11) satisfies J 6 J , then J is said to be a guaranteed cost and u ðtÞ is said to be a guaranteed cost control law of system (1) and cost function (11).
3. Main results Let us assume that system (1) is not subject to any norm-bounded uncertainty, then one can obtain the closed-loop nonlinear delay-differential equation in view of the neutral state-feedback control law (12) as follows:
_ dÞ; tÞ: _ _ dÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt xðtÞ ¼ AxðtÞ þ Ah xðt hÞ þ Ad xðt
ð13Þ
Differentiating the neutral difference operator, rðxt Þ along the state trajectory of system (13) gives
_ ðxt Þ ¼ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt _ dÞ; tÞ; r
ð14Þ
where A0 ¼ A þ Ah . Applying a descriptor form of representation [20] for (14) gives
_ ðxt Þ ¼ yðtÞ; r
ð15Þ
_ dÞ; tÞ; yðtÞ ¼ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
where yðtÞ is the fast varying descriptor variable. The following theorem summarizes the synthesis of a robust guaranteed cost controller. Theorem 1. Given the positive scalars, h > 0; d > 0, and q1 ; q2 such that q1 þ q2 < 1, if there exist symmetric and positive definite matrices K0 ; X 1 ; Q ; R; S, Z, and matrices X i , Y, i ¼ 2; . . . ; 6 and positive scalars k, ej , j ¼ 1; 2; 3, ek , k ¼ 4; 5; 6 with appropriate dimensions satisfying the following linear matrix inequalities:
2 6 6 6 6 6 4
T
q1 K0
ATd K0
khEd Eh 2
K0
6 6 6 6 6 6 6 6 6 6 6 6 R¼6 6 6 6 6 6 6 6 6 6 6 6 6 4
3
7 0 7 7 7 < 0; K0 D 7 5
T
q2 K0 þ kh ETh Eh hAh K0
2
0
ð16Þ
kI
Rð1; 1Þ hRð1; 2Þ Rð1; 2Þ Rð1; 2Þ Rð1; 5Þ aRð1; 2Þ Rð1; 2Þ Rð1; 8Þ Q
0
Rð1; 9Þ
Rð1; 10Þ Rð1; 11Þ
0
0
0
0
0
0
0
0
R
0
0
0
0
0
0
0
0
S1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
e1 I
0
0
0
0
0
S1 2
7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; ð17Þ 7 7 7 7 7 7 7 7 7 7 7 5
S
0
0
0
0
e2 I
0
0
0
Z þ e6 DDT
0
0 0 e0 I
e3 I
3
where
2 6 6 6 6 6 6 6 6 6 Rð1; 1Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
X 2 þ X T2
X 3 þ Y T BT X T2 þ X T4 AT0
X 1 þ X T4
0
0
0
0
0
0
X 3 X T3 þ A0 X 5 þ X T5 AT0 þ e4 DDT
A0 X 6 þ X T5
0
0
e1 I
0
e2 I
0
0
0
0
0
Q
0
0
0
0
0
R
0
0
0
0 e1 I
0 S
0 0
X 6 þ X T6 þ e5 DDT Ah Q Ad R
0 0 e2 I
Z
0
0
3
7 e3 I 7 7 7 0 7 7 0 7 7 7 0 7 7; 0 7 7 7 0 7 7 0 7 7 7 0 5 e3 I
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3 3 2 2 3 2 3 2 T3 T hX 4 Y T BT þ X T4 AT 0 0 Y 7 6 6 T7 607 607 607 X T5 AT 7 6 6 hX 5 7 7 7 7 6 6 6 7 6 6 T7 T T 607 607 607 7 6 6 hX 6 7 X A 7 7 7 6 6 6 6 7 7 6 6 607 607 607 7 6 6 0 7 0 6 7 6 7 6 7 7 7 6 6 607 607 607 7 7 6 0 7 7 6 7 Rð1; 2Þ ¼ 6 7; Rð1; 10Þ ¼ 6 6 0 7; Rð1; 5Þ ¼ 6 6 0 7; 6 0 7; Rð1; 8Þ ¼ 6 0 7; Rð1; 9Þ ¼ 6 7 6 6 0 7 e1 I 7 7 6 7 6 6 7 7 6 6 607 6 bS 7 607 7 6 6 0 7 SATh 6 7 6 7 6 7 7 7 6 6 607 607 607 7 6 6 0 7 e I 7 7 6 7 6 6 2 7 7 6 6 4 cZ 5 405 405 T 5 4 4 0 5 ZAd 0 0 0 e3 I 0 2
2 6 6 6 6 6 6 6 6 6 Rð1; 11Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
Y T ETb þ X T4 ET0
0
Y T ETb þ X T4 ETa
X T5 ET0
0
X T5 ETa
X T6 ET0
0
X T6 ETa
0
0
0
QETh RETd
0
0
0
0
0
SETh
0
0
0
0 0
0 0
ZETd 0
0
3 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
2 and
e4 I
6 ¼4 0 0
e0
0
0
e5 I 0
3
7 0 5; e6 I
and (*) denotes the symmetricity, then a robust guaranteed cost neutral state-feedback controller with K ¼ YX 1 robustly asymptotically stabilizes system (1) via ensuring an upper bound for the cost function computed as
J ¼ rT ðUð0ÞÞX 1 rðUð0ÞÞ þ h Z
þ
Z
0
ðs þ hÞUT ðsÞQ 1 UðsÞ ds þ
h 0
Z
0
UT ðsÞR1 UðsÞ ds þ
d
Z
0
UT ðsÞS1 UðsÞ ds
h
U_ T ðsÞZ 1 U_ ðsÞ ds:
ð18Þ
d
Proof. In order to justify the linear matrix inequality condition given in (16) for the stability of uncertain neutral difference operator, one can reconsider (10) and apply Schur complement [23] to obtain
2 6
W¼4
q1 K0
0
q2 K0
ATd K0
3
7 T hAh K0 5 < 0;
ð19Þ
K0
then replacing Ah ; Ad with Ah þ DAh ðtÞ; Ad þ DAd ðtÞ, respectively, in (19) gives
W þ DWðtÞ þ DWT ðtÞ < 0;
ð20Þ
where
DWðtÞ ¼ PT0 FðtÞH0 with P0 ¼ ½ 0
0
DT K0 ; H0 ¼ ½ Ed
hEh
0 . Applying norm-bounding inequality yields
W þ DWðtÞ þ DWT ðtÞ < W þ k1 PT0 P0 þ kHT0 H0 < 0;
ð21Þ
which then by Schur complement [23] verifies the condition in (16). Let us choose an augmented descriptor type of a candidate Lyapunov–Krasovskii functional using the approach of [22] as follows:
VðxðtÞ; tÞ ¼
5 X
V i ðxðtÞ; tÞ;
ð22Þ
i¼1
Rt Rt Rt where V 1 ðxðtÞ; tÞ ¼ gT ðtÞEP gðtÞ; V 2 ðxðtÞ; tÞ ¼ h th ðs t þ hÞxT ðsÞQxðsÞ ds; V 3 ðxðtÞ; tÞ ¼ td xT ðsÞRxðsÞ ds; V 4 ðxðtÞ; tÞ ¼ th R t _ ds with gðtÞ ¼ ½ rT ðxt Þ yT ðtÞ xT ðtÞ T , and xT ðsÞSxðsÞ ds; V 5 ðxðtÞ; tÞ ¼ td x_ T ðsÞZ xðsÞ
2
3 I 0 0 6 7 E ¼ 4 0 0 0 5; 0 0 0
2
P1 6 P ¼ 4 P2 P4
0 P3 P5
3 0 7 0 5; P6
P T1 ¼ P1 > 0;
Q T ¼ Q > 0;
RT ¼ R > 0;
ST ¼ S > 0;
ZT ¼ Z > 0
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all with appropriate dimensions. Taking the time-derivative of VðxðtÞ; tÞ along the state trajectory of system (1), (12) and (15) gives
_ VðxðtÞ; tÞ ¼
5 X
V_ i ðxðtÞ; tÞ:
ð23Þ
i¼1
One can compute V_ 1 ðxðtÞ; tÞ in the following form
d d V_ 1 ðxðtÞ; tÞ ¼ 2 gT ðtÞEP gðtÞ ¼ 2gT ðtÞPT ET gðtÞ: dt dt
ð24Þ
It follows from (15) that one can write
_ dÞ; tÞ: 0 ¼ yðtÞ þ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
ð25Þ
Utilizing the neutral difference operator equation in (9) allows to get
0 ¼ rðxt Þ þ xðtÞ þ Ah
Z
t
xðsÞds Ad xðt dÞ:
ð26Þ
th
In view of (25) and (26), one can compute the following
3 2 3 2 yðtÞ yðtÞ d _ dÞ; tÞ 5 tÞ þ gðxðt hÞ; tÞ þ hðxðt E gðtÞ ¼ 4 0 5 ¼ 4 yðtÞ þ A0 xðtÞ þ BK rðxt Þ þ f ðxðtÞ; Rt dt 0 rðxt Þ þ xðtÞ þ Ah th xðsÞds Ad xðt dÞ Z t _ dÞ; tÞ xðsÞds þ C2 xðt dÞ þ C3 ½f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt ¼ AgðtÞ þ C1 T
ð27Þ
th
2
0 where A ¼ 4 BK I
I I 0
2 3 0 0 0 0 A0 5; C1 ¼ 4 0 5; C2 ¼ 4 0 5; C3 ¼ 4 I 5. Substituting (27) into (24) gives Ah Ad 0 I 3
2
3
2
Z V_ 1 ðxðtÞ; tÞ ¼ 2gT ðtÞPT AgðtÞ þ C1
3
t
_ dÞ; tÞ xðsÞds þ C2 xðt dÞ þ C3 ½f ðxðtÞ; tÞ þ gðxðt hÞ; tÞ þ hðxðt
th
¼ vT ðtÞX1 vðtÞ;
ð28Þ
where
vðtÞ ¼ gT ðtÞ 2 6 6 6 6 6 6 6 6 X1 ¼ 6 6 6 6 6 6 6 6 4
R t
T xðsÞds th
T _ dÞ; tÞ xT ðt dÞ f T ðxðtÞ; tÞ xT ðt hÞ g T ðxðt hÞ; tÞ x_ T ðt dÞ h ðxðt
0 P T C3
0 P T C3
0
0
;
3
P T A þ AT P
P T C1
P T C2
P T C3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7 0 7 7 7 0 7 7 7 0 7 7: 7 0 7 7 7 0 7 7 7 0 5
0
0
T
Calculating V_ 2 ðxðtÞ; tÞ yields 2 V_ 2 ðxðtÞ; tÞ ¼ h xT ðtÞQxðtÞ h
Z
t
xT ðsÞQxðsÞds:
ð29Þ
th
One can reexpress xðtÞ in the following manner
xðtÞ ¼ C4 vðtÞ; where C4 ¼ ½ K1
h
Z
t
th
0
ð30Þ 0
0
0
0
0
Z xT ðsÞQxðsÞds 6
t
0 ; K1 ¼ ½ 0
T Z xðsÞds Q
th
0
I . It follows from Jensen integral inequality Gu et al. [3] that
t
xðsÞds
ð31Þ
th
Substituting (30), (31) into (29) gives 2 V_ 2 ðxðtÞ; tÞ 6 vT ðtÞðX2 þ h CT4 Q C4 ÞvðtÞ;
ð32Þ
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where
2
0
6 6 6 6 6 6 6 X2 ¼ 6 6 6 6 6 6 4
0
0 0 0 0 0 0
3
0 0 0 0 0 07 7 7 0 0 0 0 0 07 7 0 0 0 0 07 7 7 0 0 0 07 7 0 0 07 7 7 0 05
Q
0
Moreover, the time-derivative of V 3 ðxðtÞ; tÞ is computed as follows:
V_ 3 ðxðtÞ; tÞ ¼ xT ðtÞRxðtÞ xT ðt dÞRxðt dÞ ¼ vT ðtÞðX3 þ CT4 RC4 ÞvðtÞ;
ð33Þ
where
2
0 0
6 6 6 6 6 6 X3 ¼ 6 6 6 6 6 6 6 4
0
0
3
0 0 0 0 0
0 0 0 0 07 7 7 R 0 0 0 0 0 7 7 0 0 0 0 07 7 7: 0 0 0 07 7 0 0 07 7 7 0 05 0 0
Computing the time-derivative of V 4 ðxðtÞ; tÞ yields
V_ 4 ðxðtÞ; tÞ ¼ xT ðtÞSxðtÞ xT ðt hÞSxðt hÞ ¼ vT ðtÞðX4 þ CT4 SC4 ÞvðtÞ;
ð34Þ
where
2
0 0 0 0
6 6 6 6 6 6 6 X4 ¼ 6 6 6 6 6 6 4
0
0 0 0
3
0 0 07 7 7 0 0 0 0 0 07 7 0 0 0 0 07 7 7: S 0 0 0 7 7 0 0 07 7 7 0 05
0 0 0
0
0
Finally, one can compute V_ 5 ðxðtÞ; tÞ as follows:
_ x_ T ðt dÞZ xðt _ dÞ: V_ 5 ðxðtÞ; tÞ ¼ x_ T ðtÞZ xðtÞ
ð35Þ
_ Let us reexpress xðtÞ in the following form as
_ xðtÞ ¼ C5 vðtÞ; where C5 ¼ ½ K2
0
ð36Þ 0
I
Ah
I
Ad
I ; K2 ¼ ½ BK
V_ 5 ðxðtÞ; tÞ ¼ vT ðtÞðX5 þ CT5 Z C5 ÞvðtÞ; where
2
0 0 0 0 0 0
6 6 6 6 6 6 6 X5 ¼ 6 6 6 6 6 6 4
3
0
0
0 0 0 0 0
0
0 0 0 0
0
0 0 0
0
0 0
0
0
0
Z
07 7 7 07 7 07 7 7: 07 7 07 7 7 05 0
0
A . Substituting (36) into (35) gives
ð37Þ
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
2943
It follows from (6)–(8) that one can write the following inequality
_ e1 ½f T ðxðtÞ; tÞf ðxðtÞ; tÞ þ a2 xT ðtÞxðtÞ þ e2 ½g T ðxðt hÞ; tÞgðxðt hÞ; tÞ þ b2 xT ðt hÞxðt hÞ þ e3 ½hT ðxðt 2 _T _ _ dÞ; tÞhðxðt dÞ; tÞ þ c x ðt dÞxðt dÞ ¼ vT ðtÞðX6 þ e1 a2 CT4 C4 þ e2 b2 CT6 C6 þ e3 c2 CT7 C7 ÞvðtÞ P 0;
ð38Þ
where
2
0 0 0 0 6 0 0 0 6 6 0 6 0 6 6 e1 I X6 ¼ 6 6 6 6 6 6 4
0 0 0 0 0 0 0 0 0 0 e2 I
3 0 0 0 0 7 7 7 0 0 7 7 0 0 7 7 0 0 7 7 7 0 0 7 7 0 0 5 e3 I
and C8 ¼ ½ K3 0 0 0 0 0 0 0 ; K3 ¼ ½ K 0 0 . Therefore, substituting (28), (32), (33), (34), (37) into (23) and adding (38) to the resulting expression leads to compute the following inequality as follows:
_ VðxðtÞ; tÞ þ xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞ 6 vT ðtÞX0 vðtÞ; P6
2 T 4 ðh Q þ R T u ðtÞS2 uðtÞ
where X0 ¼ i¼1 Xi þ C _ VðxðtÞ; tÞ þ xT ðtÞS1 xðtÞ þ
ð39Þ T 5Z
2
T 6
T 7
þ S1 þ S þ e1 a2 IÞC4 þ C C5 þ e2 b C C6 þ e3 c2 C C7 þ C < 0, one needs to satisfy
T 8 S2
C8 . In order to guarantee
X0 < 0
ð40Þ
then applying Schur complement [23] to (40) allows to get
X < 0;
ð41Þ
where
2 6 6 6 6 6 6 6 6 6 X¼6 6 6 6 6 6 6 6 6 4
Xð1; 1Þ
hKT1
Q 1
KT1 KT1 0 0 R1 0 S1 1
KT3 aKT1 KT1 0 0 0 0 0 0 0 0 0 1 0 0 S2 0 e1 1 I S1
Xð1; 8Þ Xð1; 9Þ Xð1; 10Þ 0 0 0
0 0 0
0 0 0
0 0
0 0
0 0
0 e1 2 I
0 0
0 0
Z 1
0 e1 3 I
3 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
with
2
P T A þ AT P P T C 1 P T C 2 P T C 3 6 Q 0 0 6 6 6 R 0 6 6 e1 I Xð1; 1Þ ¼ 6 6 6 6 6 6 4 2 T3 2 3 0 K2 6 0 7 607 7 6 6 7 7 6 6 7 6 0 7 607 7 6 6 7 6 I 7 6 7 7; Xð1; 10Þ ¼ 6 0 7: Xð1; 9Þ ¼ 6 6 AT 7 607 6 h7 6 7 7 6 6 7 6 I 7 607 6 T7 6 7 4A 5 4 cI 5 d 0 I
0 P T C3 0 0 0 0 0 0 S 0 e2 I
0 0 0 0 0 0 Z
3 P T C3 0 7 7 7 0 7 7 0 7 7; 0 7 7 7 0 7 7 0 5 e3 I
3 0 607 6 7 6 7 607 6 7 607 7 Xð1; 8Þ ¼ 6 6 bI 7; 6 7 6 7 607 6 7 405 2
0
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M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
Then preand postmultiplying X<0 in (41) with MT and M, respectively, where M ¼ diagfX; Q ; R; e1 ; S; e2 ; Z; e3 ; I; I; I; I; I; I; I; I; Ig; X ¼ P1 ; Q ¼ Q 1 ; R ¼ R1 ; S ¼ S1 ; Z ¼ Z 1 ; ei ¼ e1 i ; i ¼ 1; 2; 3, and choosing K ¼ YX 1 give the following linear matrix inequality
R < 0;
ð42Þ
where
2 6 6 6 6 6 6 6 6 6 R¼6 6 6 6 6 6 6 6 6 6 4
Rð1; 1Þ hRð1; 2Þ Rð1; 2Þ Rð1; 2Þ Rð1; 5Þ aRð1; 2Þ Rð1; 2Þ Rð1; 8Þ Rð1; 9Þ Rð1; 10Þ Q
0
0
0
0
0
0
0
0
R
0
0
0
0
0
0
0
S1 1
0
0
0
0
0
0
S1 2
0 e1 I
0 0
0 0
0 0
0 0
S
0
0 e2 I
0
0
0
Z
0 e3 I
3 7 7 7 7 7 7 7 7 7 7 7; 7 7 7 7 7 7 7 7 5
and
2 6 6 6 6 6 6 6 6 6 Rð1; 1Þ ¼ 6 6 6 6 6 6 6 6 6 6 4
X 2 þ X T2
X 3 þ Y T BT X T2 þ X T4 AT0
X 1 þ X T4
0
0
0
0
0
0
X 3 X T3 þ A0 X 5 þ X T5 AT0
A0 X 6 þ X T5
0
0
e1 I
0
e2 I
0
X 6 þ X T6
Ah Q
Ad R
0
0
0
0
Q
0
0
0
0
0
R
0
0
0
0
0
0
S
0
0 e1 I
0
0 e2 I
Z
0
3
7 e3 I 7 7 7 0 7 7 0 7 7 7 0 7 7: 0 7 7 7 0 7 7 0 7 7 7 0 5 e3 I
Now, one can replace A; Ah ; Ad ; B with A þ DAðtÞ; Ah þ DAh ðtÞ; Ad þ DAd ðtÞ; B þ DBðtÞ, respectively, in (42) to get
R þ DRðtÞ þ DRT ðtÞ < 0;
ð43Þ
where
2
DRðtÞ ¼ PT FðtÞH; 2 6
H¼4
6
0 DT
P ¼ 40
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
DT
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
Eb Y þ E0 X 4
E0 X 5
E0 X 6
0
0
0
0
Eh Q
Eb Y þ E a X 4
Ea X 5
Ea X 6
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0
0
0
0
0
Ed R 0
0
0
0
0
0 Eh S 0 Ed Z
T
0
3T
7 05 ;
0
0 0 0 0 0 0 0 0 0 0
3T
7 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0
with E0 ¼ Ea þ Eh . Applying a norm-bounding inequality allows to rewrite (43) as follows: T R þ DRðtÞ þ DRT ðtÞ 6 R þ e0 PT P þ e1 0 H H < 0:
ð44Þ
Then, by Schur complement [23], one obtains the linear matrix inequality given in (17). This completes the proof. h Next, one can show that by the neutral state-feedback controller (12), the guaranteed cost function in (11) has an upper bound. For this purpose, one can notice that (39) and (42) imply
_ VðxðtÞ; tÞ 6 xT ðtÞS1 xðtÞ uT ðtÞS2 uðtÞ:
ð45Þ
Integrating both sides of (45) from 0 to T > 0 gives
VðxðTÞ; TÞ Vðxð0Þ; 0Þ 6
Z
T
½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞdt:
ð46Þ
0
As the system is ensured to be robustly asymptotically stable once the conditions in (16) and (17) are satisfied, for RT limT!1 VðxðTÞ; TÞ ! 0, one obtains limT!1 0 ½xT ðtÞS1 xðtÞ þ uT ðtÞS2 uðtÞdt ¼ J 6 J where J is as defined in Theorem 1. One
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
2945
can notice that Theorem 1 describes how to synthesize a robust guaranteed cost neutral state-feedback controller. In order to introduce the design of an optimal robust guaranteed cost neutral state-feedback controller that minimizes the upper bound of the quadratic cost function (11), the following theorem is presented. Theorem 2 [13]For the uncertain nonlinear neutral system (1) and the associated quadratic cost function (11), consider the optimization problem defined as follows:
(
lþ
min
4 X
X;Q ;R;S;Z;Y;T i ;i¼1;...;4;ej ;j¼1;2;3;e0
)
traceðTi Þ
i¼1
subject to ð16Þ and ð17Þ; and " " " " "
l rT ðUð0ÞÞ T 1 T 2 T 3 T 4
#
X 1 # W T1 6 0; Q # W T2 6 0; R # W T3 6 0; S # W T4 60 Z
6 0;
ð47Þ ð48Þ ð49Þ ð50Þ ð51Þ
such that if a feasible solution set of X; Q ; R; S; Z; Y; T i ; i ¼ 1; . . . ; 4; ej ; j ¼ 1; 2; 3; e0 can be achieved then the neutral state-feedback control law (12) is said to be an optimal robust guaranteed cost control law ensuring that the quadratic cost function upper R0 bound given in (18) for uncertain nonlinear neutral system (1) is minimized, where h h ðs þ hÞUðsÞUT ðsÞds ¼ W 1 W T1 ; R0 R0 R0 T T T T T T _ ðsÞds ¼ W 4 W . UðsÞU ðsÞds ¼ W 2 W 2 ; h UðsÞU ðsÞds ¼ W 3 W 3 ; d U_ ðsÞU 4 d Proof. It follows from (47)–(51) and Schur complement [23] that one can obtain the equivalent inequalities of T 1 rT ðUð0ÞÞX 1 W 1 6 T 1 , W T2 R1 W 2 6 T 2 , W T3 S1 W 3 6 T 3 , W T4 Z 1 W 4 6 T 4 , respectively. Moreover, note that 1 rðUð0ÞÞ 6 l, W 1 Q utilizing the trace operator allows to compute the following expressions
h
Z
0
Z 0 Z 0 ðs þ hÞUT ðsÞQ 1 UðsÞds ¼ trace h ðs þ hÞUT ðsÞQ 1 UðsÞds ¼ trace h ðs þ hÞUT ðsÞUðsÞdsQ 1
h
h
h
¼ trace½W 1 W T1 Q 1 ¼ trace½W T1 Q 1 W 1 ; Z
0
Z
UT ðsÞR1 UðsÞds ¼ trace
d
Z
0
0
UT ðsÞR1 UðsÞds ¼ trace
Z
d
Z
0
UT ðsÞS1 UðsÞds ¼ trace
h
Z
0
d
Z
d
Z
0
d
0
UT ðsÞS1 UðsÞds ¼ trace
h
U_ T ðsÞZ 1 U_ ðsÞds ¼ trace
0
h
U_ T ðsÞZ 1 U_ ðsÞds ¼ trace
Z
0
d
UT ðsÞUðsÞdsR1 ¼ trace½W 2 W T2 R1 ¼ trace½W T2 R1 W 2 ;
UT ðsÞUðsÞdsS1 ¼ trace½W 3 W T3 S1 ¼ trace½W T3 S1 W 3 ;
U_ T ðsÞU_ ðsÞdsZ 1 ¼ trace½W 4 W T4 Z 1 ¼ trace½W T4 Z 1 W 4 :
Therefore, in view of (18) one can easily deduce that the following inequality
J 6 l þ
4 X
traceðTi Þ
ð52Þ
i¼1
is always satisfied. This shows that minimizing the right-hand part of Eq. 52 is equivalent to the minimization of the upper bound of the quadratic cost function (11) which is given in (18). Hence, the proof is completed. h Remark 1. It can be argued that the neutral state-feedback control is difficult to implement in physical systems as it requires the use of exact information on the delay terms. It is clear that an approximation will cause the obtained results to be questionable. Moreover, if the system has time-varying delays, then the proposed approach seems to be difficult to extend and implement. Therefore, we need to underline that the developed method gives improved results only for a class of neutral time-delay systems with constant and known discrete and neutral delays.
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M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
4. Numerical examples In this section, two numerical examples are presented to demonstrate the application of Theorems 1 and 2. Example 1. An uncertain linear neutral system example is considered as follows:
_ dÞ þ ½B þ DFðtÞEb uðtÞ; _ xðtÞ ¼ ½A þ DFðtÞEa xðtÞ þ ½Ah þ DFðtÞEh xðt hÞ þ ½Ad þ DFðtÞEd xðt 0 0:5 0 0:2 0:2 ; Ah ¼ ; Ad ¼ 0 1 0:2 0:5 0 and the initial condition is defined as where A ¼
/ðtÞ ¼
et
et
ð53Þ
0 0:1 with the delay bounds being given as h ¼ 0:5; d ¼ 0:2 ;B ¼ 0:5 0:2
8t 2 ½0:5; 0:
1 0 ; S2 ¼ 0:2. If it is assumed that D ¼ 0, the system 0 1 rðUð0ÞÞ; W 1 ; W 2 yields in (52) reduces to the example studied in Park [12]. Calculating 0:8394 0:1516 0:1516 0:2871 0:2871 ; W1 ¼ ; W2 ¼ . Choosing q1 ¼ 0:1; q2 ¼ 0:2, the condirðUð0ÞÞ ¼ 0:5608 0:1516 0:1516 0:2871 0:2871 9:0812 1:3109 . A feation of (16) for the stability of the nominal neutral difference equation is satisfied with K0 ¼ 1:3109 11:7136 sible solution set has been obtained for satisfying (17) appropriately to get The gain matrices of the associated cost function are selected as S1 ¼
2
0:2989
0:1982
0:1982
0:1328
X1 ¼ 4 2 X4 ¼ 4
0:0088
3
2
5;
X2 ¼ 4
3
2
5;
X5 ¼ 4
2:3366
1:1936
3
2
5 10 ;
X3 ¼ 4
3
2
5;
X6 ¼ 4
4
1:1935 2:3366
0:0817
0:1657 0:1214
0:0800
0:1666 0:1215 2 T1 ¼ 4
Y ¼ ½ 0:0997 0:4124 ;
0:0097
0:0939
0:0939
0:0939
0:0939
2:3366 1:1935 1:1936
2:3365
0:7273 0:0284
3 5 104 ;
3 5;
0:0267 0:4748
3
2
5;
T2 ¼ 4
0:0512
0:0511
0:0511
0:0512
3
2
5;
Q ¼4
5:4325 1:6851 1:6851 0:9629
3 5;
0:7119 , and l ¼ 2:3707. Then computing the cost function upper bound gives J ¼ 2:6607 and the neutral 1:9663 gain matrix is obtained as K ¼ ½ 167:6345 253:3164 . However, for the purpose of comparison one noachievable cost function bound obtained in Park [12] is 9.7773 with the feedback gain matrix of 3:3077 . Hence the proposed methodology is shown to be less conservative than that of [12]. Let us 0:5 ; Ea ¼ ½ 0:1 0:2 ; Eh ¼ ½ 0:3 0:1 ; Ed ¼ ½ 0:2 0:1 ; Eb ¼ 0:1. Then applicanow choose the uncertainty matrices as D ¼ 0:3 tion of Theorems 1 and 2 results in the following set of feasible solution for the design parameters as 3:8941 0:7119 state-feedback tices that the K ¼ ½ 1:8737
R¼
2
0:2286
X1 ¼ 4 2
0:2065
0:2065
0:2734
0:1547
0:1858
3
2
5;
X2 ¼ 4
3
2
5;
X5 ¼ 4 2
Y ¼ ½ 0:2027 2:1228 ;
R¼4 1:5373
1:5373
3
2
5 10 ; 3
0:0180 0:0593 0:6232
0:8389 0:7164
2:0874
1:0709
X3 ¼ 4
0:1172
0:1172
0:1172
T1 ¼ 4
3
2
5;
X6 ¼ 4
0:6231
0:1172
9:5422 1:0703 0:5279
0:5280 9:5798
X4 ¼ 4
2
9:5423
3
2
5;
T2 ¼ 4
3 5 103 ;
9:5791
0:3376
0:1812
0:1813
0:3123
3 5;
0:0518
0:0518
0:0518
0:0518
3
2
5;
Q ¼4
0:3827
0:0731
0:0731
0:2774
3 5;
e4 ¼ 0:1667; e5 ¼ 0:1098; e6 ¼ 3:7400 104 ; and l ¼ 3:5325:
1:5952
The upper bound of the cost function is calculated using the solutions of the required matrices yields J ¼ 3:8706 and the feedback gain of K ¼ ½ 24:8891 26:5678 . This result is also the indicative of the observation that the performance of
3 5;
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M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
the proposed guaranteed cost controller is still quite better than that of the method presented in Park [12] even in the case of norm-bounded uncertainties. Example 2. Another example of uncertain nonlinear neutral system is considered as follows:
_ dÞ þ ½B þ DFðtÞEb uðtÞ þ f ðxðtÞ; tÞ _ xðtÞ ¼ ½A þ DFðtÞEa xðtÞ þ ½Ah þ DFðtÞEh xðt hÞ þ ½Ad þ DFðtÞEd xðt _ þ gðxðt hÞ; tÞ þ hðxðt dÞ; tÞ;
ð54Þ
where
A¼
1
1
1
1
Ah ¼
;
Ea ¼ ½ 0 0:2 ;
0
0:2
0:2
0
Eh ¼ ½ 1 0:3 ;
;
Ad ¼
0 0:1 0 0:1
Ed ¼ ½ 0:2 0:2 ;
;B ¼
1 ; 1
D¼
0 0:2
;
Eb ¼ 0:5
and the delay bounds are defined as h ¼ 1:0; d ¼ 2:0. The initial condition is given as follows:
UðtÞ ¼
1 et
8t 2 ½2:0; 0
with the cost function gain matrices being chosen as S1 ¼
"
rðUð0ÞÞ ¼ " W4 ¼
1:1129
#
1:1865
0
"
0:5975 0:3782
W1 ¼
;
0:3782 0:3752
0 ; S2 ¼ 1. One can compute rðUð0ÞÞ; W i ; i ¼ 1; . . . ; 4 as 1
0:5 0
#
"
;
W2 ¼
1:3356 0:4650
#
"
;
0:4650 0:5241
0:8817 0:4719
W3 ¼
#
0:4719 0:4579
;
#
0
0 0:7006
:
Let us assume that the system in (54) is not subject to any nonlinear parameter perturbations, i.e. f ðxðtÞ; tÞ ¼ 0, _ dÞ; tÞ ¼ 0 leading to the example considered in Xu et al. [6]. The stability of the neutral operator, gðxðt hÞ; tÞ ¼ 0, hðxðt rðxt Þ is ensured by choosing q1 ¼ q2 ¼ 0:3 and obtaining a feasible solution set for the condition given in (16) gives 1:4214 0:0108 ; k ¼ 0:1281. The linear matrix inequality (17) has given a feasible solution set of K0 ¼ 0:0108 1:3121 1:8276 0:9529 , X1 ¼ 0:9529 1:4210
2 X2 ¼ 4
437:6883
447:7330
3
2
5;
X3 ¼ 4
429:1885
426:7856 447:3215 2
0:4632
0:5332
0:5099
0:4502
X5 ¼ 4 2
3:1256
3:2367
3:2367
9:0033
Q ¼4
443:2937 434:9854
3
2
5;
X6 ¼ 4
3
2
5;
R¼4
3
2
5;
X4 ¼ 4
443:2540
0:8077
0:1552
0:1570
0:7105
0:6365
0:5264
0:5009
0:4470
3 5;
3 5;
73:7035
10:1748
10:1748
4:9701
Y ¼ ½ 1:2172 0:8982 ;
3
2
5;
T1 ¼ 4
0:3005 0:2075
3
2
5;
T2 ¼ 4
0:2075 0:1603
0:1529 0:1155
3 5;
0:1155 0:1104
e4 ¼ 5:4000; e5 ¼ 5:2079; e6 ¼ 1:9049 103 and l ¼ 4:0658 which indicate that the cost function upper bound can be computed as J ¼ 4:7377 with a neutral state-feedback gain matrix of K ¼ ½ 1:5309 1:6587 . However, the achievable cost function bound obtained in Xu et al. [6] is 50.0275 with the feedback gain matrix of K ¼ ½ 0:9943 0:9615 . This shows that the proposed methodology is capable of yielding less conservative cost function bounds implying that when the proposed controller is employed, a considerably less amount of control effort is required than that of the controller presented in Xu et al. [6]. Now let us assume that the numerical example of neutral system given in (54) involves nonlinear parameter perturbations defined such that T
f ðxðtÞ; tÞ ¼ ½ a1 cos tjx1 ðtÞj a2 sin tjx2 ðtÞj ;
jai j 6 0:3; T
jbi j 6 0:3;
T
jci j 6 0:3;
gðxðt hðtÞÞ; tÞ ¼ ½ b1 cos tjx1 ðt hÞj b2 sin tjx2 ðt hÞj ; _ dðtÞÞ; tÞ ¼ ½ c1 cos tjx_ 1 ðt dÞj c2 sin tjx_ 2 ðt dÞj ; hðxðt The
simulation work to 0:6560 0:3237 , X1 ¼ 0:3237 0:5468
satisfy
the
synthesis
condition
given
in
i ¼ 1; 2: (17)
yields
a
feasible
solution
set
of
2948
M.N. Alpaslan Parlakçı / Applied Mathematics and Computation 215 (2009) 2936–2949
X2 ¼
2:2927
4:1680
2:2926 4:1678 0:3095 0:0220 X3 ¼ 104 ; X 4 ¼ ; 4:1676 2:2926 0:1967 0:1279 0:1962 0:0110 X6 ¼ ; Y ¼ ½ 1:0444 0:4810 ; 0:0108 0:2791 35:9732 5:2477 2:1589 0:1474 R¼ ; S¼ ; 5:2477 2:3589 0:1474 2:8522 0:6322 0:4484 0:3228 0:2624 0:4597 0:2845 T1 ¼ ; T2 ¼ ; T3 ¼ ; 0:4484 0:3228 0:2624 0:2259 0:2845 0:1877 #
104 ;
4:1676 2:2928 0:1563 0:2269 X5 ¼ ; 0:2436 0:2915 0:9945 0:6536 Q¼ ; 0:6536 2:6037 5:8783 0:1885 Z¼ ; 0:1885 4:9904 " 0 7:8343 106 ; T4 ¼ 0 0:0985
e1 ¼ 0:2438;
e2 ¼ 0:2949;
e3 ¼ 0:5390;
e4 ¼ 0:7739; e5 ¼ 2:3444;
e6 ¼ 30:8330; and l ¼ 9:6704 and the upper bound of the cost function is computed as J ¼ 11:9199 with the neutral state-feedback gain matrix of K ¼ ½ 2:8622 2:5742 . Consequently, it can be seen that even in the case of the existence of nonlinear parameter perturbations, the proposed robust guaranteed cost neutral state-feedback controller performs much better than that given in Xu et al. [6] which is, however, applicable for only norm bounded uncertainty situation.
5. Conclusions This paper has investigated robust delay-dependent guaranteed cost controller design for uncertain neutral systems with nonlinear parameter perturbations. A memory type neutral form of state-feedback control law is introduced. On the basis of employing a descriptor representation, an augmented descriptor form of a candidate Lyapunov–Krasovskii functional is adopted to study the stability of the neutral system. In particular, with free weighting matrices and new descriptor system approach concerned in system model and operator, sufficient robust delay-dependent linear matrix inequality (LMI) conditions are derived such that the existence of a guaranteed cost neutral state-feedback controller is ensured once the specified LMIs are satisfied. Moreover, several additional LMI conditions are also presented in the context of a convex optimization algorithm in such a manner that the upper bound of a quadratic cost function is minimized. Two numerical examples of neutral systems which have been presented in the literature are taken into consideration to illustrate the application of the newly developed LMI guaranteed cost controller synthesis approach. Several numerical case studies have concluded that the proposed method of the present paper has been able to assure a less conservative control cost implying that the requirements can be achieved with considerably less amount of control effort in comparison to some of the existing approaches. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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