A delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay

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  • 1. A delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay Pin-Lin Liu n Department of Automation Engineering Institute of Mechatronoptic System, Chienkuo Technology University, Changhua 500, Taiwan, ROC a r t i c l e i n f o Article history: Received 5 April 2012 Received in revised form 16 June 2012 Accepted 3 July 2012 Available online 25 July 2012 Keywords: Lyapunov–Krasovskii’s functional Stability Time-varying delay Linear matrix inequalities (LMIs) Maximum allowable delay bound (MADB) a b s t r a c t This paper is concerned with delay-dependent robust stability for uncertain systems with time-varying delays. The proposed method employs a suitable Lyapunov–Krasovskii’s functional for new augmented system. Then, based on the Lyapunov method, a delay-dependent robust criterion is devised by taking the relationship between the terms in the Leibniz–Newton formula into account. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full considera- tion, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Numerical examples are included to show that the proposed method is effective and can provide less conservative results. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Time delay is one of the instability sources for dynamical systems, and is a common phenomenon in many industrial and engineering systems such as those in communication networks, manufacturing, and biology. Since system stability is an essential requirement in many applications, much effort has been made to investigate stability criteria for various time-delay systems during the last two decades. For details, see the works [1–24] and references therein. Recently, some researchers have paid attention to the issue concerning delay-dependent stability analysis [2–24]. The devel- opment of the technologies for delay-dependent stability analysis has been focusing on effective reduction of the conservation of the stability conditions. The main aim is to derive a maximum allowable delay bound (MADB) of the time-delay such that the time delays’ system is asymptotically stable for any delay size less than the MADB. Accordingly, the obtained MADB becomes a key performance index to measure the conservatism of a delay- dependent stability condition. On the other hand, many uncertain factors exist in practical systems. Uncertainty in a control system may be attributed to modeling errors, measurement errors, parameter variations and a linearization approximation [9]. Therefore, study on the robust stability of uncertain systems with time-varying delay becomes significantly important. The results obtained will be very useful to further research for robust stability of uncertain systems with time-varying delay [4,6,8,9–16,18–20,23,24] and references therein. The development of the technologies for delay-dependent stability analysis has been focusing on effective reduction of the conservation of the stability conditions. Two effective technolo- gies have been widely accepted: the bounding technology [4], and the model transformation technology [2]. The former is used to evaluate the bounds of some cross terms arising from the analysis of the delay-dependent problem; while the latter is employed to transform the original delay models into simpler ones so that the stability analysis becomes easier. However, it is also known that the bounding technology and the model transformation technique are the main source of conserva- tion. To further improve the performance of delay-dependent stability criteria, there is a need to avoid this source of conservation. Therefore, much effort has been devoted recently to the develop- ment of the free weighting matrices method, in which neither the bounding technology nor model transformation is employed [3,11,19–21]. Generally speaking, the free weighting matrices appear in two forms: the one with a null summing term added to the Lyapunov functional derivative [19–21], and the one with free matrices item added to the Lyapunov functional combined with the descriptor model transformation [3,11]. However, this approach introduced some slack variables apart from the matrix variables appearing in Lyapunov–Krasovskii functionals (LKFs). When the upper bound of delay derivative may be larger than or equal to 1, Zhu and Yang [23,24] used a delay decomposition approach, and new stability results were derived. Compared with [6], the stability results in [23,24] are simpler and less conservative. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.07.001 n Tel.: þ886 47 111155; fax: þ886 47 111129. E-mail address: lpl@cc.ctu.edu.tw ISA Transactions 51 (2012) 694–701
  • 2. Motivated by the above discussions, we propose new robust stability criteria for uncertain systems with time-varying delays. Based on the Lyapunov function method, a novel robust delay- dependent criterion, which is less conservative than delay- inde- pendent one when the size of delays is small, is established in terms of LMIs which can be solved efficiently by using the optimization algorithms. In order to derive less conservative results, a new integral inequality approach (IIA) which utilizes free weighting matrices is proposed. Also, the model transforma- tion technique, which leads to an additional dynamics, is not used in this work. Finally, five numerical examples are shown to support that our results are less conservative than those of the existing ones. 2. Problem statement Consider the following uncertain system with a time-varying state delay: _xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ ð1aÞ xðtÞ ¼ fðtÞ, tA½Àh,0Š, ð1bÞ where xðtÞARn is the state vector, A and B are constant matrices with appropriate dimensions, f(t) is a smooth vector-valued initial function, h(t) is a time varying delay in the state, and h is an upper bound on the delay h(t). The uncertainties are assumed to be the form: DAðtÞ DBðtÞ Â Ã ¼ DFðtÞ Ea Eb½ Š, ð2Þ where D, Ea and Eb are constant matrices with appropriate dimensions, and F(t) is an unknown, real, and possibly time- varying matrix with Lebesgue–measurable elements satisfying FT ðtÞFðtÞrI, 8t: ð3Þ As in [2], we consider two different cases for time varying delays: Case I. h(t) is a differentiable function, satisfying for all tZ0: 0rhðtÞrh and _hðtÞ      rhd: ð4Þ Case II. h(t) is not differentiable or the upper bound of the derivative of h(t) is unknown, and h(t) satisfies 0rhðtÞrh ð5Þ where h and hdare some positive constants. To do this, three fundamental lemmas are reviewed. First, Lemma 1 induces the integral inequality approach. Lemma 1. [9] For any positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, ð6aÞ the following integral inequality holds: À Z t tÀhðtÞ _xT ðsÞX33 _xðsÞ ds r Z t tÀhðtÞ xT ðtÞ xT ðtÀhðtÞÞ _xT ðsÞ h i X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 0 2 6 4 3 7 5 xðtÞ xðtÀhðtÞÞ _xðsÞ 2 6 4 3 7 5 ds: ð6bÞ Secondary, we introduce the following Schur complement which is essential in the proofs of our results. Lemma 2. [1] The following matrix inequality: QðxÞ SðxÞ ST ðxÞ RðxÞ " # o0, ð7aÞ where QðxÞ ¼ QT ðxÞ, RðxÞ ¼ RT ðxÞ and SðxÞ depend on affine on x, is equivalent to RðxÞo0, ð7bÞ QðxÞo0, ð7cÞ and QðxÞÀSðxÞRÀ1 ðxÞST ðxÞo0: ð7dÞ Lemma 3. [1] Given matrices Q ¼ QT , D, E, and R ¼ RT 40 of appropriate dimensions, Q þDFðtÞEþET FT ðtÞDT o0, ð8Þ for all F satisfying FT ðtÞFðtÞrH, if and only if there exists some e40 such that Q þeDDT þeÀ1 ET HEo0: ð9Þ The purpose of this paper is to find new stability criteria, which are less conservative than the existing results. Firstly, we consider the nominal from system (1): _xðtÞ ¼ AxðtÞþBxðtÀhðtÞÞ, tZ0: ð10Þ For the nominal system (10), we will give a stability condition by using a delay decomposition approach as follows: Theorem 1. In Case I, if 0rh(t)rah, for given scalars h(h40), a(0oao1) and hd, the system described by (10) with (4) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0,ði ¼ 1,2,3Þ, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0 such that O ¼ O11 O12 O13 O14 O15 OT 12 O22 O23 O24 O25 OT 13 OT 23 O33 O34 O35 OT 14 OT 24 OT 34 O44 O45 OT 15 OT 25 OT 35 OT 45 O55 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 o0, ð11Þ and R1ÀX33 Z0, R2ÀY33 Z0, R1 þð1ÀhdÞR3ÀZ33 Z0, ð12Þ where O11 ¼ AT PþPAþQ1 þQ3 þahZ11 þZ13 þZT 13, O12 ¼ PBþahZ12ÀZ13 þZT 23, O15 ¼ AT ½ahR1 þð1ÀaÞhR2 þahR3Š, O22 ¼ Àð1ÀhdÞQ3 þahX11 þX13 þXT 13 þahZ22ÀZ23ÀZT 23, O23 ¼ ahX12ÀX13 þXT 23,O25 ¼ BT ½ahR1 þð1ÀaÞhR2 þahR3Š, O33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT 23 þð1ÀaÞhY11 þY13 þYT 13, O34 ¼ ð1ÀaÞhY12ÀY13 þYT 23, O44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT 23, O55 ¼ À½ahR1 þð1ÀaÞhR2 þahR3Š, O13 ¼ O14 ¼ O24 ¼ O35 ¼ O45 ¼ 0: P.-L. Liu / ISA Transactions 51 (2012) 694–701 695
  • 3. Proof. In Case I, a Lyapunov functional can be constructed as VðtÞ ¼ V1ðtÞþV2ðtÞþV3ðtÞ, ð13Þ where V1ðtÞ ¼ xT ðtÞPxðtÞ, V2ðtÞ ¼ Z t tÀah xT ðsÞQ1xðsÞdsþ Z tÀah tÀh xT ðsÞQ2xðsÞds þ Z t tÀhðtÞ xT ðsÞQ3xðsÞds, V3ðtÞ ¼ Z 0 Àah Z t t þy _xT ðsÞR1 _xðsÞdsdyþ Z Àah Àh Z t t þy _xT ðsÞR2 _xðsÞdsdy þ Z 0 ÀhðtÞ Z t t þy _xT ðsÞR3 _xðsÞdsdy: Taking the time derivative of V(t) for tA(0,N) along the trajectory of (10), it yields that: _V ðtÞ ¼ _V 1ðtÞþ _V 2ðtÞþ _V 3ðtÞ, ð14Þ where _V 1ðtÞ ¼ _xT ðtÞPxðtÞþxT ðtÞP_xðtÞ ¼ xT ðtÞðAT PþPAÞxðtÞþxT ðtÞPBxðtÀhðtÞÞþxT ðtÀhðtÞÞBT PxðtÞ, ð15Þ _V 2ðtÞ ¼ xT ðtÞðQ1 þQ3ÞxðtÞÀxT ðtÀhðtÞÞð1À_hðtÞÞQ3xðtÀhðtÞÞ þxT ðtÀahÞðQ2ÀQ1ÞxðtÀahÞÀxT ðtÀhÞQ2xðtÀhÞ rxT ðtÞðQ1 þQ3ÞxðtÞÀxT ðtÀhðtÞÞð1ÀhdÞQ3xðtÀhðtÞÞ þxT ðtÀahÞðQ2ÀQ1ÞxðtÀahÞÀxT ðtÀhÞQ2xðtÀhÞ, ð16Þ and _V 3ðtÞ ¼ _xT ðtÞ½ahR1 þð1ÀaÞhR2 þhðtÞR3Š_xðtÞÀ Z t tÀah _xT ðsÞR1 _xðsÞds À Z tÀah tÀh _xT ðsÞR2 _xðsÞdsÀð1À _hðtÞÞ Z t tÀhðtÞ _xT ðsÞR3 _xðsÞds r _xT ðtÞ½ahR1 þð1ÀaÞhR2 þahR3Š_xðtÞÀ Z t tÀah _xT ðsÞR1 _xðsÞds À Z tÀah tÀh _xT ðsÞR2 _xðsÞdsÀð1ÀhdÞ Z t tÀhðtÞ _xT ðsÞR3 _xðsÞds: ð17Þ Now, we estimate the upper bound of the last three terms in inequality (17) as follows: À Z t tÀah _xT ðsÞR1 _xðsÞdsÀ Z tÀah tÀh _xT ðsÞR2 _xðsÞds Àð1ÀhdÞ Z t tÀhðtÞ _xT ðsÞR3 _xðsÞds ¼ À Z tÀhðtÞ tÀah _xT ðsÞR1 _xðsÞdsÀ Z tÀah tÀh _xT ðsÞR2 _xðsÞds À Z t tÀhðtÞ _xT ðsÞðR1 þð1ÀhdÞR3Þ_xðsÞds ¼ À Z tÀhðtÞ tÀah _xT ðsÞðR1ÀX33Þ_xðsÞdsÀ Z tÀah tÀh _xT ðsÞðR2ÀY33Þ_xðsÞds À Z t tÀhðtÞ _xT ðsÞðR1 þð1ÀhdÞR3ÀZ33Þ_xðsÞds À Z tÀhðtÞ tÀah _xT ðsÞX33 _xðsÞdsÀ Z tÀah tÀh _xT ðsÞY33 _xðsÞds À Z t tÀhðtÞ _xT ðsÞZ33 _xðsÞds:Þ ð18Þ From integral inequality approach of Lemma 1 [9], noticing that R1ÀX33 Z0, R2ÀY33 Z0, and R1 þð1ÀhdÞR3ÀZ33 Z0, it yields that: À Z tÀhðtÞ tÀah _xT ðsÞX33 _xðsÞdsr Z tÀhðtÞ tÀah xT ðtÀhðtÞÞ xT ðtÀahÞ _xT ðsÞ h i  X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 0 2 6 4 3 7 5 xðtÀhðtÞÞ xðtÀahÞ _xðsÞ 2 6 4 3 7 5ds rxT ðtÀhðtÞÞðahÀhðtÞÞX11xðtÀhðtÞÞ þxT ðtÀhðtÞÞðahÀhðtÞÞX12xðtÀahÞ þxT ðtÀhðtÞÞX13 Z t tÀhðtÞ _xðsÞdsþxT ðtÀahÞðahÀhðtÞÞXT 12xðtÀhðtÞÞ þxT ðtÀahÞðahÀhðtÞÞX22xðtÀahÞþxT ðtÀahÞX23 Z t tÀhðtÞ _xðsÞds þ Z tÀhðtÞ tÀah _xT ðsÞdsXT 13xðtÀhðtÞÞþ Z tÀhðtÞ tÀah _xT ðsÞdsXT 23xðtÀahÞ rxT ðtÀhðtÞÞahX11xðtÀhðtÞÞþxT ðtÀhðtÞÞahX12xðtÀahÞ þxT ðtÀhðtÞÞX13 Z t tÀhðtÞ _xðsÞdsþxT ðtÀahÞahXT 12xðtÀhðtÞÞ þxT ðtÀahÞahX22xðtÀahÞþxT ðtÀahÞX23 Z t tÀhðtÞ _xðsÞds þ Z tÀhðtÞ tÀah _xT ðsÞdsXT 13xðtÀhðtÞÞþ Z tÀhðtÞ tÀah _xT ðsÞdsXT 23xðtÀahÞ ¼ xT ðtÀhðtÞÞ½ahX11 þXT 13 þX13ŠxðtÀhðtÞÞ þxT ðtÀhðtÞÞ½ahX12ÀX13 þXT 23ŠxðtÀahÞ þxT ðtÀahÞ½ahXT 12ÀXT 13 þX23ŠxðtÀhðtÞÞ þxT ðtÀahÞ½ahX22ÀX23ÀXT 23ŠxðtÀahÞ: ð19Þ Similarly, we obtain À Z tÀah tÀh _xT ðsÞY33 _xðsÞdsrxT ðtÀahÞ½ð1ÀaÞhY11 þYT 13 þY13ŠxðtÀahÞ þxT ðtÀahÞ½ð1ÀaÞhY12ÀY13 þYT 23ŠxðtÀhÞ þxT ðtÀhÞ½ð1ÀaÞhYT 12ÀYT 13 þY23ŠxðtÀahÞ þxT ðtÀhÞ½ð1ÀaÞhY22ÀY23ÀYT 23ŠxðtÀhÞ, ð20Þ and À Z t tÀhðtÞ _xT ðsÞZ33 _xðsÞds rxT ðtÞ½hðtÞZ11 þZT 13 þZ13ŠxðtÞþxT ðtÞ½hðtÞZ12ÀZ13 þZT 23ŠxðtÀhðtÞÞ þxT ðtÀhðtÞÞ½hðtÞhZT 12ÀZT 13 þZ23ŠxðtÞ þxT ðtÀhðtÞÞ½hðtÞZ22ÀZ23ÀZT 23ŠxðtÀhðtÞÞ rxT ðtÞ½ahZ11 þZT 13 þZ13ŠxðtÞþxT ðtÞ½ahZ12ÀZ13 þZT 23ŠxðtÀhðtÞÞ þxT ðtÀhðtÞÞ½ahZT 12ÀZT 13 þZ23ŠxðtÞ þxT ðtÀhðtÞÞ½ahZ22ÀZ23ÀZT 23ŠxðtÀhðtÞÞ: ð21Þ The operator for the term _xT ðtÞ½ahR1 þð1ÀaÞhR2 þahR3Š_xðtÞ is as follows: _xT ðtÞ½ahR1 þð1ÀaÞhR2 þahR3Š_xðtÞ ¼ ½AxðtÞþBxðtÀhðtÞފT ½ahR1 þð1ÀaÞhR2 þahR3Š½AxðtÞþBxðtÀhðtÞފ ¼ xT ðtÞAT ½ahR1 þð1ÀaÞhR2 þahR3ŠAxðtÞ þxT ðtÞAT ½ahR1 þð1ÀaÞhR2 þahR3ŠBxðtÀhðtÞÞ þxT ðtÀhðtÞÞBT ½ahR1 þð1ÀaÞhR2 þahR3ŠAxðtÞ þxT ðtÀhðtÞÞBT ½ahR1 þð1ÀaÞhR2 þahR3ŠBxðtÀhðtÞÞ: ð22Þ Combining (13)–(22), it yields: _V ðtÞrxT ðtÞXxðtÞÀ Z tÀhðtÞ tÀah _xT ðsÞðR1ÀX33Þ_xðsÞds À Z tÀah tÀh _xT ðsÞðR2ÀY33Þ_xðsÞds P.-L. Liu / ISA Transactions 51 (2012) 694–701696
  • 4. À Z t tÀhðtÞ _xT ðsÞðR1 þð1ÀhdÞR3ÀZ33Þ_xðsÞds, ð23Þ where xT ðtÞ ¼ xT ðtÞ xT ðtÀhðtÞÞ xT ðtÀahÞ xT ðtÀhÞ h i and X ¼ X11 X12 X13 X14 XT 12 X22 X23 X24 XT 13 XT 23 X33 X34 XT 14 XT 24 XT 34 X44 2 6 6 6 6 4 3 7 7 7 7 5 with X11 ¼ AT PþPAþQ1 þQ3 þahZ11 þZ13 þZT 13 þAT ½ahR1 þð1ÀaÞhR2 þahR3ŠA, X12 ¼ PBþahZ12ÀZ13 þZT 23 þAT ½ahR1 þð1ÀaÞhR2 þahR3ŠB, X22 ¼ Àð1ÀhdÞQ3 þhaX11 þX13 þXT 13 þahZ22ÀZ23 ÀZT 23 þBT ½ahR1 þð1ÀaÞhR2 þahR3ŠB, X23 ¼ ahX12ÀX13 þXT 23, X33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT 23 þð1ÀaÞhY11 þY13 þYT 13, X34 ¼ ð1ÀaÞhY12ÀY13 þYT 23, X44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT 23, X13 ¼ X14 ¼ X24 ¼ 0: From nominal system (10) and the Schur complement of Lemma 2, it is easy to see that _V ðxtÞo0 holds if: R1ÀX33 Z0, R2ÀY33 Z0, R1 þð1ÀhdÞR3ÀZ33 Z0, and 0rh(t)rah. Theorem 2. In Case I, if ahrh(t)rh, for given scalars h(h40), a(0oao1) andhd, the system described by (10) with (4) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0, ði ¼ 1,2,3Þ, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5 Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0 such that O ¼ O11 O12 O13 O14 O15 O T 12 O22 O23 O24 O25 O T 13 O T 23 O33 O34 O35 O T 14 O T 24 O T 34 O44 O45 O T 15 O T 25 O T 35 O T 45 O55 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 o0, ð24Þ and R1 þð1ÀhdÞR3ÀX33 Z0, R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, ð25Þ where O11 ¼ AT PþPAþQ1 þQ3 þahX11 þX13 þXT 13,O12 ¼ PB, O13 ¼ ahX12ÀX13 þXT 23, O15 ¼ AT ½ahR1 þð1ÀaÞhR2 þhR3Š, O22 ¼ Àð1ÀhdÞQ3 þð1ÀaÞhY22ÀY23ÀYT 23 þð1ÀaÞhZ11 þZ13 þZT 13, O23 ¼ ð1ÀaÞhYT 12ÀYT 13 þY23, O24 ¼ ð1ÀaÞhZ12ÀZ13 þZT 23, O25 ¼ BT ½ahR1 þð1ÀaÞhR2 þhR3Š, O33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT 23 þð1ÀaÞhYT 11 þY13 þYT 13, O44 ¼ ÀQ2 þð1ÀaÞhZ22ÀZ23ÀZT 23, O55 ¼ À½ahR1 þð1ÀaÞhR2 þhR3Š, O14 ¼ O34 ¼ O35 ¼ O45 ¼ 0: Proof. If ahrh(t)rh, it gets: À Z t tÀah _xT ðsÞR1 _xðsÞdsÀ Z tÀah tÀh _xT ðsÞR2 _xðsÞds Àð1ÀhdÞ Z t tÀhðtÞ _xT ðsÞR3 _xðsÞds ¼ À Z t tÀah _xT ðsÞðR1 þð1ÀhdÞR3Þ_xðsÞds À Z tÀah tÀhðtÞ _xT ðsÞðR2 þð1ÀhdÞR3Þ_xðsÞds À Z tÀhðtÞ tÀh _xT ðsÞR2 _xðsÞds ¼ À Z t tÀah _xT ðsÞðR1 þð1ÀhdÞR3ÀX33Þ_xðsÞds À Z tÀah tÀhðtÞ _xT ðsÞðR2 þð1ÀhdÞR3ÀY33Þ_xðsÞds À Z tÀhðtÞ tÀh _xT ðsÞðR2ÀZ33Þ_xðsÞdsÀ Z t tÀah _xT ðsÞX33 _xðsÞds À Z tÀah tÀhðtÞ _xT ðsÞY33 _xðsÞdsÀ Z tÀhðtÞ tÀh _xT ðsÞZ33 _xðsÞds: ð26Þ From integral inequality matrix [9], noticing that R1 þð1ÀhdÞ R3ÀX33 Z0, R2ÀZ33 Z0, and R2 þð1ÀhdÞR3ÀY33 Z0, it yields: À Z t tÀah _xT ðsÞX33 _xðsÞdsrxT ðtÞ½ahX11 þXT 13 þX13ŠxðtÞ þxT ðtÞ½ahX12ÀX13 þXT 23ŠxðtÀahÞ þxT ðtÀahÞ½ahXT 12ÀXT 13 þX23ŠxðtÞ þxT ðtÀahÞ½ahX22ÀX23ÀXT 23ŠxðtÀahÞ, ð27Þ À Z tÀah tÀhðtÞ _xT ðsÞY33 _xðsÞdsrxT ðtÀahÞ½ð1ÀaÞhY11 þYT 13 þY13ŠxðtÀahÞ þxT ðtÀahÞ½ð1ÀaÞhY12ÀY13 þYT 23ŠxðtÀhðtÞÞ þxT ðtÀhðtÞÞ½ð1ÀaÞhYT 12ÀYT 13 þY23ŠxðtÀahÞ þxT ðtÀhðtÞÞ½ð1ÀaÞhY22ÀY23ÀYT 23ŠxðtÀhðtÞÞ, ð28Þ À Z tÀhðtÞ tÀh _xT ðsÞZ33 _xðsÞdsrxT ðtÀhðtÞÞ½ð1ÀaÞhZ11 þZT 13 þZ13ŠxðtÀhðtÞÞ þxT ðtÀhðtÞÞ½ð1ÀaÞhZ12ÀZ13 þZT 23ŠxðtÀhÞ þxT ðtÀhÞ½ð1ÀaÞhZT 12ÀZT 13 þZ23ŠxðtÀhðtÞÞ þxT ðtÀhÞ½ð1ÀaÞhZ22ÀZ23ÀZT 23ŠxðtÀhÞ: ð29Þ Combining (13)–(22) and (26)–(29), it yields: _V ðtÞrxT ðtÞXxðtÞÀ Z t tÀah _xT ðsÞ½R1 þð1ÀhdÞR3ÀX33Š_xðsÞds À Z tÀah tÀhðtÞ _xT ðsÞ½R2 þð1ÀhdÞR3ÀY33Š_xðsÞds À Z tÀhðtÞ tÀh _xT ðsÞðR2ÀZ33Þ_xðsÞds, ð30Þ where X ¼ X11 X12 X13 X14 X T 12 X22 X23 X24 X T 13 X T 23 X33 X34 X T 14 X T 24 X T 34 X44 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 and X11 ¼ AT PþPAþQ1 þQ3 þahX11 þX13 þXT 13 þAT ½ahR1 þð1ÀaÞhR2 þhR3ŠA, X12 ¼ PBþAT ½ahR1 þð1ÀaÞhR2 þhR3ŠB, X13 ¼ ahX12ÀX13 þXT 23, X22 ¼ Àð1ÀhdÞQ3 þð1ÀaÞhY22ÀY23ÀYT 23 þð1ÀaÞhZ11 þZ13 þZT 13 þBT ½ahR1 þð1ÀaÞhR2 þhR3ŠB, P.-L. Liu / ISA Transactions 51 (2012) 694–701 697
  • 5. X23 ¼ ð1ÀaÞhYT 12ÀYT 13 þY23, X24 ¼ ð1ÀaÞhZ12ÀZ13 þZT 23, X33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT 23 þð1ÀaÞhYT 11 þY13 þYT 13, X44 ¼ ÀQ2 þð1ÀaÞhZ22ÀZ23ÀZT 23, X14 ¼ X34 ¼ 0: From nominal system (10) and the Schur complement, it is easy to see that _V ðxtÞo0 holds if: R1 þð1ÀhdÞR3ÀX33 Z0, R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, and ahrh(t)rh. Theorem 3. In Case II, for given scalars h(h40), a(0oao1), the system described by (10) with (5) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0, ði ¼ 1,2Þ, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5 Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0 such that C ¼ C11 C12 C13 C14 C15 CT 12 C22 C23 C24 C25 CT 13 CT 23 C33 C34 C35 CT 14 CT 24 CT 34 C44 C45 CT 15 CT 25 CT 35 CT 45 C55 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 o0, ð31Þ and R1ÀX33 Z0, R2ÀY33 Z0, R1ÀZ33 Z0, ð32Þ where C11 ¼ AT PþPAþQ1 þhZ11 þZ13 þZT 13,C12 ¼ PBþhZ12ÀZ13 þZT 23, C15 ¼ AT ½ahR1 þð1ÀaÞhR2Š,C22 ¼ ahX11 þX13 þXT 13 þhZ22ÀZ23ÀZT 23, C23 ¼ ahX12ÀX13 þXT 23,C25 ¼ BT ½ahR1 þð1ÀaÞhR2Š, C33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT 23 þð1ÀaÞhY11 þY13 þYT 13, C34 ¼ ð1ÀaÞhY12ÀY13 þYT 23,C44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT 23, C55 ¼ À½ahR1 þð1ÀaÞhR2Š,C13 ¼ C14 ¼ C24 ¼ C35 ¼ C45 ¼ 0: Proof. In Case II, a Lyapunov functional can be chosen as (13) with Q3 ¼ R3 ¼ 0: Similar to the above analysis, one can get that _V ðtÞo0 holds if Co0. Thus, the proof is completed. Now, extending Theorems 1–3 to uncertain system (1) with time-varying delays yields the following Theorems. Theorem 4. In Case I, if 0rh(t)rah, for given scalars h(h40), a(0oao1) and hd, the uncertain system described by (1) with (4) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0,ði ¼ 1,2,3Þ, e40, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0, such that: Oe ¼ O11 þeET a Ea O12 þeET a Eb O13 O14 O15 PD OT 12 þeET bEa O22 þeET bEb O23 O24 O25 0 OT 13 OT 23 O33 O34 O35 0 OT 14 OT 24 OT 34 O44 O45 0 OT 15 OT 25 OT 35 OT 45 O55 O56 DT p 0 0 0 OT 56 ÀeI 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 o0, ð33Þ and R1ÀX33 Z0, R2ÀY33 Z0, R1 þð1ÀhaÞR3ÀZ33 Z0, ð34Þ where O56 ¼ DT ðahR1 þð1ÀaÞhR2 þahR3Þ and Oij,ði,j ¼ 1,2,:::,5; iojr5Þ are defined in (11). Proof. Replacing A and B in (11) with AþDFðtÞEa and BþDFðtÞEb, respectively, we apply Lemma 3 [1] for system (1) which is equivalent to the following condition: OþGdFðtÞGe þGT e FðtÞGT d o0, ð35Þ where Gd ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2 þahR3ÞD h i , and Ge ¼ Ea Eb 0 0 0  à : By Lemma 3 [1], a sufficient condition guaranteeing (11) for system (1) is that there exists a positive number e40 such that: OþeÀ1 GT dGd þeGT e Ge o0: ð36Þ Applying the Schur complement of Lemma 2 shows that (36) is equivalent to (33). This completes the proof. Theorem 5. In Case I, if ahrh(t)rh, for given scalars h(h40),a(0oao1) and hd, the uncertain system (1) with (4) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0,ði ¼ 1,2,3Þ, e40, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0 such that: Oe ¼ O11 þeET aEa O12 þeET aEb O13 O14 O15 PD O T 12 þeET bEa O22 þeET bEb O23 O24 O25 0 O T 13 O T 23 O33 O34 O35 0 O T 14 O T 24 O T 34 O44 O45 0 O T 15 O T 25 O T 35 O T 45 O55 O56 DT p 0 0 0 O T 56 ÀeI 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 o0, ð37Þ and R1 þð1ÀhdÞR3ÀX33 Z0, R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, ð38Þ where O56 ¼ DT ðahR1 þð1ÀaÞhR2 þhR3Þ and Oij,ði,j ¼ 1,2,:::,5; iojr5Þ are defined in (24). P.-L. Liu / ISA Transactions 51 (2012) 694–701698
  • 6. Proof. Replacing A and B in (24) with AþDFðtÞEa and BþDFðtÞEb, respectively, we apply Lemma 3 [1] for system (1) which is equivalent to the following condition: O þGdFðtÞGe þG T e FðtÞG T d o0, ð39Þ where Gd ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2 þhR3ÞD h i , and Ge ¼ Ea Eb 0 0 0  à : By Lemma 3 [1], a sufficient condition guaranteeing (24) for system (1) is that there exists a positive number e40 such that O þeÀ1 G T dGd þeG T e Ge o0: ð40Þ Applying the Schur complement shows that (40) is equivalent to (37). This completes the proof. Theorem 6. In Case II, for given scalars h(h40), a(0oao1), the uncertain system (1) with (5) is asymptotically stable if there exist matrices P ¼ PT 40, Qi ¼ QT i Z0, Ri ¼ RT i Z0,ði ¼ 1,2Þ, e40, and positive semi-definite matrices: X ¼ X11 X12 X13 XT 12 X22 X23 XT 13 XT 23 X33 2 6 4 3 7 5Z0, Y ¼ Y11 Y12 Y13 YT 12 Y22 Y23 YT 13 YT 23 Y33 2 6 4 3 7 5Z0, Z ¼ Z11 Z12 Z13 ZT 12 Z22 Z23 ZT 13 ZT 23 Z33 2 6 4 3 7 5Z0: such that C ¼ C11 þeET aEa C12 þeET a Eb C13 C14 C15 PD CT 12 þeET bEa C22 þeET bEb C23 C24 C25 0 CT 13 CT 23 C33 C34 C35 0 CT 14 CT 24 CT 34 C44 C45 0 CT 15 CT 25 CT 35 CT 45 C55 C56 DT P 0 0 0 CT 56 ÀeI 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 o0 ð41Þ R1ÀX33 Z0, R2ÀY33 Z0, R1ÀZ33 Z0 ð42Þ where C56 ¼ DT ½ahR1 þð1ÀaÞhR2Š and Cij,ði,j ¼ 1,2,:::,5; iojr5Þ are defined in (31). Proof. Replacing A and B in (31) with AþDFðtÞEa and BþDFðtÞEb, respectively, we apply Lemma 3 [1] for system (1) is equivalent to the following condition: CþG dFðtÞG e þG T e FðtÞG T d o0, ð43Þ where G d ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2ÞD h i , and G e ¼ Ea Eb 0 0 0  à : By Lemma 3 [1], a sufficient condition guaranteeing (31) for system (1) is that there exists a positive number e40 such that CþeÀ1G T d G d þeG T e G e o0: ð44Þ Applying the Schur complement of Lemma 2 shows that (44) is equivalent to (41). This completes the proof. Remark 1. In the proof of Theorems 1–6, the interval [tÀh,t] is divided into two subintervals [tÀh,tÀah] and [tÀah,t], the infor- mation of delayed state x(tÀah) can be taken into account. It is clear that the Lyapunov function defined in Theorems 1–6 are more general than the ones in [6,20], etc. Since the delay decom- position approach is introduced in time delay, it is clear that the stability results are based on the delay decomposition approach. When the positions of delay decomposition are varied, the stability results of proposed criteria are also different. In order to obtain the optimal delay decomposition sequence, we proposed an implementation based on optimization methods. The proposed stability conditions are much less conservative and are more general than some existing results. Remark 2. In the previous works except [2,6,7,11,13,14,16,18,19], the time delay term h(t) was usually estimated as h when estimating the upper bound of some cross term, this may lead to increasing conservatism inevitably. In Theorems 1–6, the value of the upper bound of some cross term is estimated to be more exactly than the previous methods since h(t) is confined to the subintervals 0rh(t)rah or ahrh(t)rh. So, such decomposition method may lead to reduction of conservatism. Remark 3. In the stability problem, maximum allowable delay bound (MADB) h which ensures that time-varying delay uncertain system (1) is asymptotically stable for any h can be determined by solving the following quasi-convex optimization problem when the other bound of time-varying delay h is known. Maximize h Subjectto ð33Þ ðð37Þ or ð41ÞÞ: ( ð45Þ Inequality (45) is a convex optimization problem and can be obtained efficiently using the MATLAB LMI Toolbox. Five examples will be presented in the following section to highlight the effectiveness of the proposed method. 3. Illustrative examples In this section, five examples are provided to illustrate the advantages of the proposed stability results. Example 1. Consider the uncertain system with time varying delay as follows: _xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð46Þ where A ¼ À1:2 0:1 À0:1 À1 ! , B ¼ À0:6 0:7 À1 À0:8 ! , D ¼ I, Ea ¼ Eb ¼ diag 0:1,0:1: Now, our problem is to estimate the bound of delay time h to keep the stability of system. Solution: when hd ¼ 0:5, using the stability criteria in Parlakci [12], Qian et al. [16] and Theorem 4 of this paper, the calculated maximum allowable delay bound (MADB) for the time delay are h¼1.0097 and h¼1.3618, and h¼2.2241, respectively. So the proposed method in this paper yields a less conservative result than that given in Parlakci [12] and Qian et al. [16]. Example 2. Consider the uncertain system with time varying delay as follows: _xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð47Þ where A ¼ À2 0 0 À1 ! ,B ¼ À1 0 À1 À1 ! , D ¼ I, Ea ¼ diag 1:6,0:05f g, Eb ¼ diag 0:1, 0:3f g: Solution: for hd ¼ 0:1ða ¼ 0:5Þ, by Theorem 4, we can obtain the maximum upper bound on the allowable size to be h¼2.2143. However, applying criteria in [22,15,18], the maximum value of h for the above system is 0.92, 1.1072 and 1.1075. We also apply Theorems 4 and 5 to calculate the maximum allowable value for different hd Table 1 shows the comparison of our results with those in [4,8,10,16,19,21]. This example demonstrates that our robust stability condition gives a less conservative result. Hence, it P.-L. Liu / ISA Transactions 51 (2012) 694–701 699
  • 7. is obvious that the results obtained from our simple method are less conservative than those obtained from the existing methods. Example 3. Consider the uncertain system with time varying delay as follows: _xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð48Þ where A ¼ À0:5 À2 1 À1 ! , B ¼ À0:5 À1 0 0:6 ! , Ea ¼ Eb ¼ diag 0:2,0:2f g, D ¼ I: Solution: for comparison, the Table 2 also lists the maximum allowable delay bound (MADB) h obtained from the criteria [2,6,7,11,13,14,16,18,19]. It is clear that Theorems 4 and 5 give much better results than those obtained by [2,6,7,11,13,14,16, 18,19]. It is illustrated that the proposed robust stability criteria are effective in comparison to earlier and newly published results existing in the literature. Example 4. Consider the uncertain system with time varying delay as follows: _xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð49Þ where A ¼ À0:6 À2:3 0:8 À1:2 ! , B ¼ À0:9 0:6 0:2 0:1 ! , D ¼ diag l,l È É ,Ea ¼ Eb ¼ I: Solution: when h is a constant, by the proposed method of Theorem 6 in this paper and those in [11,13,16,20], for different values of l,we can get the maximum allowable delay bound (MADB) h and the results are listed in Table 3, which indicate that our result is less conservative than those in [11,13,16,20]. When l¼0.3, Table 4 gives the comparisons of the maximum allowed delay h for various hd It can be seen that the robust stability condition in this paper is less conservative than the one in [6,23]. Example 5. Consider the system with time varying delay as follows: _xðtÞ ¼ AxðtÞþBxðtÀhðtÞÞ, ð50Þ where A ¼ 0 À0:12þ12r 1 À0:465Àr " # , B ¼ À0:1 À0:35 0 0:3 ! : Solution: we let r¼0.035 as [5] did. From Table 5, we could easily find that the results proposed in this note are better than those of [5,6,17]. The conclusion we draw is better than [5,6,17] when hd is small. 4. Conclusions In this paper, we have proposed new robust stability criteria for uncertain systems with time-varying delays. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay- dependent sufficient robust stability criteria are obtained in terms of linear matrix inequalities (LMIs), which can be easily solved by various optimization algorithms. Since the delay terms are con- cerned more exactly, less conservative results are presented. Moreover, the restriction on the change rate of time-varying delays is relaxed in the proposed criteria. The proposed criteria are computationally attractive, and it provides less conservative results than the existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results. References [1] Boyd S, Ghaoui LE, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. PA, Philadelphia: SIAM; 1994. [2] Fridman E, Shaked U. An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control 2002;47:1931–7. [3] Fridman E, Shaked U. A descriptor system approach to HN control of linear time-delay systems. IEEE Transactions on Automatic Control 2002;47:253–70. Table 1 Maximum allowable delay bound (MADB) h compared with different methods in Example 2. hd 0 0.2 0.4 0.6 0.8 Han [4] 1.03 0.82 0.61 0.40 0.18 Lien [8] 1.149 1.063 0.973 0.873 0.760 Wu et al. [19] 1.149 1.063 0.973 0.873 0.760 Yue and Han [21] 1.149 1.063 0.973 0.873 0.760 Park and Jeong [10] 1.149 1.099 1.077 1.070 1.068 Qian et al. [16] 1.248 1.121 1.081 1.072 1.069 Theorem 4 (a¼0.5) 2.2980 2.1312 1.9664 1.8074 1.6727 Theorem 5 (a¼0.5) 1.2270 1.1428 1.0543 0.9590 0.9181 Table 2 Maximum allowable delay bound (MADB) h for different hdin Example 3. hd 0.5 0.9 Z1 Fridman and Shaked [2] 0.1820 – – Jing et al. [7] and Wu et al. [19] 0.2433 0.2420 0.2420 Parlaski [11] 0.3067 0.2512 – He et al. [6] (Corollary 1) 0.3155 0.3155 0.3155 He et al.[6] (Corollary 4) 0.3420 0.3378 0.3356 Wang and Shen [18] (Theorem 2) 0.3155 0.3155 0.3155 Wang and Shen [18] (Theorem 1) 0.3497 0.3497 0.3497 Peng and Tian [14] 0.4243 0.4095 0.4080 Peng and Tian [13] 0.4760 0.4760 0.4760 Qian et al. [16] (Case I) 0.6151 0.6044 0.6006 Qian et al. [16] (Case II) 0.6097 0.6011 0.5970 Theorem 4 (a¼0.4) 0.7889 0.7889 0.7889 Theorem 5 (a¼0.9) 0.7072 0.7072 0.7072 Table 3 Maximum allowable delay bound (MADB) h for different l (hd ¼ 0) in Example 4. l 0.3 0.4 0.5 0.6 Xu and Lam [20] 0.9514 0.7950 0.6426 0.2087 Parlaski [11] 2.5618 0.9654 0.7229 0.2139 Peng and Tiam [13] 2.5618 0.9654 0.7229 0.2139 Qian et al. [16] 2.5688 1.0634 0.7677 0.5608 Theorem 6 (a¼0.3) 3.1713 2.6501 2.1419 0.6955 Table 4 Maximum allowable delay bound (MADB) h for different hd in Example 4. hd 0.8 0.9 1.0 He et al. [6] 0.6871 0.6871 0.6871 Zhu and Yang [23] (Corollary 2) 0.6946 0.6913 0.6896 Zhu and Yang [23] (Corollary 3) 1.5633 1.5632 1.5630 Theorem 4 (a¼0.3) 2.2899 2.2898 2.2898 Table 5 Calculation results for Example 5. hd 0 0.1 0.3 0.9 any hd [5] (Lemma 1) 0.89 0.87 0.84 0.84 0.84 [6] (Theorem 1) 0.89 0.88 0.87 0.87 0.87 Song and Wang [17] 1.67 1.45 0.89 0.84 0.84 Theorem 1 (a¼0.4) 2.2214 2.1545 1.8459 1.5699 1.3936 Theorem 2 (a¼0.4) 2.2412 2.2014 2.1749 2.1235 1.3936 P.-L. Liu / ISA Transactions 51 (2012) 694–701700
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