00 Stability radii of discrete-time stochastic systems with respect to

切换线性随机系统的指数稳定分析：多Lyapunov函数方法丛屾;蒋海峰;盛遵冰【摘要】考虑随机扰动对于切换系统稳定性的影响.我们推广多Lyapunov函数方法并利用噪声的统计性质建立几乎必然指数稳定的条件.进而,给出了子系统反馈控制器的设计条件以使闭环系统是几乎必然指数稳定的.最后,通过一个仿真算例验证了方法的有效性.【期刊名称】《黑龙江大学工程学报》【年(卷),期】2011(002)001【总页数】5页(P100-104)【关键词】切换随机系统;几乎必然指数稳定;多Lyapunov函数方法【作者】丛屾;蒋海峰;盛遵冰【作者单位】黑龙江大学,机电工程学院,哈尔滨,150080;南京理工大学,自动化学院,南京,210094;黑龙江大学,机电工程学院,哈尔滨,150080【正文语种】中文【中图分类】TP271.740 引言随着现代控制技术的发展,切换作为一种控制手段广泛存在于各种控制系统的数学模型中,从理论角度来说我们自然关心切换是如何影响系统动力学行为的。

因此,在过去的十几年间切换系统成为控制理论领域中研究的热点问题,发展出来的方法与成果形成了一个独立研究分支。

特别是与切换系统稳定问题相关的一系列研究成果丰富了动力系统理论体系并完善了Lyapunov稳定性方法[1]。

另一方面,在系统建模过程中引入噪声是反映各种振动现象及不确定因素的有效方式,因此我们将考虑具有状态时滞的切换随机系统并基于多Lyapunov泛函方法分析其在均方意义下的稳定性。

较之于确定系统,随机系统包含更为丰富的研究内容,但是建立与之相应的理论体系却也更为困难,这是因为我们所使用的很多概念与方法潜在地依赖于具体的微积分法则。

以切换系统为例,描述切换驱动的状态转移过程是准确分析系统动力学的基础[2-3];然而对于随机系统,除一维情形外,一般无法以闭合的形式刻划状态转移过程并以此分析其统计特性[4]。

在随机分析的理论体系中考虑切换现象时通常假定其演化过程满足一定的统计规律,即所谓的时齐Markov过程。

相依随机序列滑动平均的强偏差定理摘要本文主要研究任意相依连续型随机变量滑动平均的强偏差定理(即用不等式表示的强极限定理),全文共分四章.绪论部分,首先简明扼要地介绍了概率论极限理论背景以及发展现状.其次介绍了刘文创立的研究随机变量强偏差定理的基本思想与方法.最后引入了随机变量滑动平均,滑动似然比以及滑动相对熵概念.第二章.利用随机序列滑动似然比以及滑动相对熵概念作为任意随机序列联合分布与参考乘积分布(服从指数分布的独立随机变量序列)的偏差的随机性度量,并通过滑动相对熵给出了样本空间的一个子集.在此子集上得到了一类用不等式表示的强偏差定理,即小偏差定理.第三章.利用滑动似然比的概念和Laplace变换的方法,研究相依连续型非负随机变量序列滑动和的极限性质,得到了一类用不等式表示的强偏差定理,即小偏差定理.第四章.设{ξn,n≥1}是任意相依连续型随机变量序列,{B n,n≥1}是实直线上的Lebesgue可测集,1Bn (x)是B n的示性函数,本章研究{1Bn(ξn),n≥1}的极限性质,得到了一类用不等式表示的强偏差定理,并且得到关于{1Bn(ξn),n≥1}的强大数定律作为本节定理的推论.关键词:滑动平均;滑动似然比;滑动相对熵;指数分布;强偏差定理The Strong Deviation Theorems Concerning Moving Averageof Dependent Random VariablesAbstractThis dissertation makes a main study of strong deviation theorems of any con-tinuous random variable moving average(namely the strong limit theorem with the inequality).The article is divided into four chapters.Introduction:Wefirst briefly introduce probability theory limit theory background and development present situation.Secondly introduce the basic ideas and methods of small deviation theorems of random variable Liuwen founded.Finally this chapter introduces the random variable moving average、moving likelihood ratio and moving relative entropy concept.The second chapter:We use the concept of random sequence moving likelihood ratio and moving relative entropy to study the random measurement of the deviation of the random sequence joint distribution and the reference distribution.A subset of Sample space is constructed with the moving relative entropy and we get a class of strong deviation theorems represented with inequalities on this subset,that is the small deviation theorem.The third chapter is devoted to a study of the limit property of the moving sum of dependent continuous non-negative random variable sequences.With the concept of moving likelihood ratio and the method of Laplace transformation,we obtain a kind of strong deviation theorems represented with inequalities i.e.the Small Deviation Theorem In chapter4,we study the limit property of{1B(ξn),n≥1}for an arbitrary depen-ndent continuous random variable sequence{ξn,n≥1}where{B n,n≥1}is a sequence of Lebesgue sets of the real line and1Bis the characteristic of B n for each n∈ω.A classnof strong deviation theorem represented with inequalities is obtain.Moreover,we show that the strong law of large numbers of{1B(ξn),n≥1}is an immediate consequence ofnthe theorem in this section.安徽工业大学硕士毕业论文5Keywords:moving average;moving likelihood ratio;moving relative entropy;expo-nential distribution;strong deviation theorem文中部分缩写与符号说明µ, µ———概率测度a.s.———几乎必然µ−a.s.———依概率µ几乎必然L n(ω)———滑动似然比———滑动相对熵h µµ1(.)———示性函数log———自然对数R+———正实数R−———负实数↗———单调递增↘———单调递减目录第一章绪论 (1)第二章基于指数分布的随机序列滑动平均的强偏差定理 (5)第三章Laplace变换与随机变量滑动和的一类强偏差定理 (10)第四章连续型随机变量滑动平均的一类强偏差定理 (17)第五章总结与展望 (24)参考文献 (25)致谢 (28)附录 (29)安徽工业大学硕士毕业论文第一章绪论概率论是研究大量随机现象的统计规律性的数学学科。

LetterRelaxed Stability Criteria for Delayed Generalized Neural Networks via a Novel Reciprocally Convex CombinationYibo Wang, Changchun Hua, and PooGyeon ParkDear Editor,This letter examines the stability issue of generalized neural net-works (GNNs) with time-varying delay based on a novel recipro-cally convex combination (RCC). By considering a new matrix poly-nomial, the proposed novel reciprocally convex method leads to a tight bound for integral inequality combination and encompasses sev-eral existing approaches as special cases. The relaxed stability condi-tions with less conservatism are developed by employing the pro-posed reciprocally convex combination and the Lyapunov-Krasovskii (L-K) functional. Finally, several numerical examples are conducted to show the superiorities of the stability conditions.GNNs are composed of an enormous amount of connected units or nodes called artificial neurons, which are connected to simulate neu-rons in a biological brain. As we know, time-varying delays are an inevitable factor in GNNs because of the limited switching speed and communication bandwidth. Stability is a prerequisite for dynamic system analysis and application, but time delays may lead to unac-ceptable dynamic responses or instability [1], [2]. Thus, it is funda-mental to focus on the stability of the GNNs with time delay.(1/ϱ)ηT 1Z η1+1/(1−ϱ)ηT2Z η2ϱ2ϱ2The convex combination () is fre-quently encountered when evaluating the integral term in the L-K functional. RCC plays a key role in dealing with the combination in the stability analysis of delayed GNNs. In the last few years, a num-ber of improvements in RCC have been obtained [3]–[5]. In [3], an RCC was developed by introducing a high-order matrix polynomial.A -dependent RCC was proposed in [4] by introducing related terms to study the dissipativity issue of delayed GNNs. By employ-ing m -degree matrix-valued polynomial, a generalized RCC was established in [5] to investigate delayed neural networks. From the above study, it is clear that matrix-valued polynomials have been widely employed in RCC. These slack matrix variables are valid for reducing the RCC estimation errors. However, few studies devel-oped the non-affine RCC for the study of delayed GNNs. There is much room for development in the RCC to estimate the combination accurately.The second rote is constructing the L-K functional that contains more information about the neuron activation function and system state. A suitable L-K functional can build the connection among neu-ron activation function and system vectors of GNNs. Thus, an L-K functional that contains more information is conclusive for diminish-ing the conservatism of the conditions.This letter establishes an RCC by considering a new matrix poly-nomial to evaluate the convex combination. The proposed RCC includes some inequalities in [5]–[7] as exceptional cases and can be instantly dealt with by linear matrix inequality (LMI). Then, the sta-bility conditions with less conservatism are derived by exploiting the new RCC and the L-K functional with delay-product terms. Severalnumerical examples are implemented to verify the effectiveness of the stability conditions.S n (S n +)n ×n He {O}=O T +O C n m =m !(m −n )!n !Notation: denotes the set of symmetric (positive-def-inite) real matrices. . Symmetric terms are expressedby “*”. .Problem statement: Consider a category of delayed GNNs as fol-W i f (W 2x (t ))=col {f 1(W 21x (t )),...,f n (W 2n x (t ))}d neuron state, respectively. are the interconnection weight matri-ces. stands for the neu-ron activation function. A is a diagonal matrix consisting of positive12mi pi p diag {k p 1,...,k pn }K m =diag {k m 1,...,k mn }where , and are given constants. Let and .Main results: The RCC is a crucial lever in the study of GNNs with time-varying delay. Then, a novel RCC is achieved by introduc-ing a matrix polynomial.Z ∈S n +Vj ∈n ×n (j =0,...,m )W n (i =1,...,2m )Proposition 1: For a given m and , if there existProof: According to (4) and Theorem 1 in [8], the following condi-tion with ϱ holds:Corresponding authors: Changchun Hua and PooGyeon Park.Citation: Y. B. Wang, C. C. Hua, and P. Park, “Relaxed stability criteria for delayed generalized neural networks via a novel reciprocally convex combination,” IEEE/CAA J. Autom. Sinica , vol. 10, no. 7, pp. 1631–1633, Jul.2023.Y. B. Wang and C. C. Hua are with the School of Electrical Engineering,Yanshan University, Qinhuangdao 066004, China (e-mail: wyb13032317@P. Park is with the Department of Electrical Engineering, Pohang University of Science and Technology, Pohang 37673, South Korea (e-mail:Digital Object Identifier 10.1109/JAS.2022.106025The following condition is yielded by substituting into (6):diag {√1−ϱϱI ,√ϱ1−ϱI }By pre-multiplying and post-multiplying (7) by, we haveZ = 1−ϱϱZ O ∗ϱ1−ϱZ where .Z By adding to both sides of (8), we obtainW i V j Remark 1: Proposition 1 is achieved as a generalized reciprocally convex inequality with different orders by considering a new matrix-valued polynomial. Some necessary variable matrices and are integrated into the proposed RCC, which has positive consequences for reducing the conservatism of stability criteria. Furthermore,Proposition 1 includes the RCCs in [5]–[7] as special cases.V 0=∑m j =0¯M j V i =∑i j =1C j i (−1)j ¯M i +¯Y i ,(i =1,...,m )1) By consideringand , (5) in Proposition 1 is identical to Lemma 4 in [5].m =22) By considering , (5) decreases to1331120123−W 4ϱ2+(W 2+W 4)ϱ. Note that the inequality in [6] can be covered by (10).m =13) By considering , (5) decreases towhich are identical to the result in [7]. Before proceeding, some matrices and vectors are defined in [9].¯d µ<¯µK p =diag {k +1,...,k +n }K m =Theorem 1: For given , , , and diag {k −1,...,k −n}P 1∈S 9n +P 2∈S 9n +Q1∈S 6n +Q 2∈S 6n +R ∈S n +Z ∈S 2n+G 1∈S 6n +G 2∈S14n +G 3∈S 14n +X k ∈S 3n (k =1,2,3,4)X z ∈R 2n Y ι∈R 3n (ι=0,1,2)L 1∈S n +L 2∈S n +H j ∈S n +T j ∈S n +(j =1,2,3)N 1∈R 6n N 2∈14n N 14n , the system (1) is asymptotically stable if there exist matrices , , , , , ,, , , , ,, diagonal matrices , , , , and skew-symmetric matrices , , , such that the following inequalities hold:whereBy differentiating along with the trajectory of (1), we yieldW 1(t )=−d t −¯d ˙x (s )R ˙x (s )ds ,W 2(t )=−d t −¯d ψ3(s )Z ψ3(s )ds where .W 1(t )W 2(t )For the integral terms and , the following conditionshold by employing auxiliary function-based inequality and Proposi-tion 1:t 2t 10where .According to the constraints of the activation function (3), some common positive definite items are given as follows:0≤κ1(t ,H 1)+κ1(t −d t ,H 2)+κ1(t −¯d ,H 3)(18)℧[d t ,˙d t ]Θj [˙d t ]Θj [˙d t ]℧[d t ,˙d t ]where and are defined in Theorem 1. are the coef-ficients of the matrix-valued polynomial .℧[d t ,˙d t ]˙d t ℧[d t ,˙d t ]<0∀˙d t ∈[u ,¯u ]℧[d t ,u ]<0℧[d t ,¯u ]<0℧[d t ,u ]<0℧[d t ,¯u ]<0˙V (t )≤−ςx T (t )x (t )ς>0Note that is affine on , holds for if and only if and . Then LMIs (12) and (13) are derived by utilizing Lemma 1 in [8] to ensure and , respectively. Hence, if the convex optimization condi-tions (11)−(13) are feasible, holds for a suffi-ciently small . ■¯d t t −¯dt u ψT 3(s )Z ψ3(s )dsdu Remark 2: Some new integral terms of activation function and dou-ble integral term are consolidated into the proposed L-K functional to include more information about time delay and activation function. Thus, more cross information about neuron activation function is contained in the L-K functional.Numerical example: This section provides two extensive delayed GNNs to indicate the superiorities of the derived stability criteria.Example 1: Consider the GNNs (1) withu For different and , the maximum allowable bounds are evalu-ated via Theorem 1. The numerical results are summarized in Table 1with those of diverse approaches. The maximum allowable bound is one of the crucial indicators for evaluating stability criteria. Accord-ing to Table 1, Theorem 1 improves maximum allowable bounds by 18% more than the leading methods. It can be concluded that Theo-rem 1 produces less conservative stability criteria for delayed DNNs.¯uTable 1. Maximum Allowable Delay for Various Values of Example 1¯u =−u 0.450.50.55NDVs[10]16.36313.060−152.5n 2+23.5n [5] (m = 3)17.34214.44612.670191n 2+40n Theorem 123.29017.52414.448596.5n 2+64.5n:¯d¯u =−u ={0.1,0.5,0.9}¯dbounds are derived for by solving the opti-mization problem. Table 2 shows that the bounds based on Theo-rem 1 are much larger than those based on other present conditions.Then it can be concluded that the technique of Proposition 1 plays a key role in reducing conservatism. Furthermore, the number of deci-sion variables (NDVs) of different methods are recorded in Table 2.How diminishing the NDVs of the proposed methods is an essential research direction.Conclusion: This letter establishes a new stability criteria for GNNs with time-varying delay. An improved RCC is proposed for delayed GNNs by considering a new matrix-valued polynomial. The proposed RCC encompasses some existing results as exceptionalcases. Then, sufficient stability conditions are achieved by exploitingthe RCC and L-K functional. Finally, several numerical examples are conducted to reveal the superiorities of the stability scheme.Acknowledgments: This work was supported by the Basic Sci-ence Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, Information and Communications Technology (ICT), and Future Planning (2020R1A2C2005709), the National Natural Science Foundation of China (618255304), and the Key Project of Natural Science Foundation of Hebei Province (F2021203054).ReferencesY. Chen and G. Chen, “Stability analysis of systems with time-varyingdelay via a novel Lyapunov functional,” IEEE/CAA J. Autom Sinica ,vol. 6, no. 4, pp. 1068–1073, 2019.[1]X. Zhang, Q.-L. Han, X. Ge, and B. Zhang, “Delay-variation-dependentcriteria on extended dissipativity for discrete-time neural networks with time-varying delay,” IEEE Trans. Neural Networks and Learning Systems , pp. 1–10, 2021. DOI: 10.1109/TNNLS.2021.3105591[2]J. Chen, J. H. Park, and S. Xu, “Improvement on reciprocally convexcombination lemma and quadratic function negative-definiteness lem-ma,” J. Franklin Institute , vol. 359, no. 2, pp. 1347–1360, 2022.[3]G. Tan and Z. Wang, “α2-dependent reciprocally convex inequality forstability and dissipativity analysis of neural networks with time-varying delay,” Neurocomputing , vol. 463, pp. 292–297, 2021.[4]H. Lin, H. Zeng, X. Zhang, and W. Wang, “Stability analysis fordelayed neural networks via a generalized reciprocally convex inequa-lity,” IEEE Trans. Neural Networks and Learning Systems , pp. 1–9,2022. DOI: 10.1109/TNNLS.2022.3144032[5]H. B. Zeng, H. C. Lin, Y. He, K. L. Teo, and W. Wang, “Hierarchicalstability conditions for time-varying delay systems via an extended reciprocally convex quadratic inequality,” J. Franklin Institute , vol. 357,no. 14, pp. 9930–9941, 2020.[6]A. Seuret and F. Gouaisbaut, “Delay-dependent reciprocally convexcombination lemma,” 2016. [Online]. Available: http://hal.archives-ouvertes.fr/hal-01257670/.[7]X. Zhang, Q.-L. Han, and X. Ge, “Novel stability criteria for lineartime-delay systems using Lyapunov-Krasovskii functionals with a cubic polynomial on time-varying delay,” IEEE/CAA J. Autom. Sinica , vol. 8,no. 1, pp. 77–85, 2020.[8]Y. Wang, “Appendix: Introduction to the main symbols,” 2022.[Online]. Available: /10.13140/RG.2.2.14675.66083/1.[9]F. Long, C. Zhang, Y. He, Q. Wang, and M. Wu, “Stability analysis fordelayed neural networks via a novel negative-definiteness determination method,” IEEE Trans. Cybernetics , vol. 52, no. 6, pp. 5356–5366, 2022.[10]X. Zhang, Q.-L. Han, and J. Wang, “Admissible delay upper bounds forglobal asymptotic stability of neural networks with time-varying delays,” IEEE Trans. Neural Networks and Learning Systems , vol. 29,no. 11, pp. 5319–5329, 2018.[11]J. Chen, X. Zhang, J. H. Park, and S. Xu, “Improved stability criteria fordelayed neural networks using a quadratic function negative-defini-teness approach,” IEEE Trans. Neural Networks and Learning Systems ,vol. 33, no. 3, pp. 1348–1354, 2022.[12]¯u Table 2. Maximum Allowable Delay for Various Values of Example 2¯u =−u 0.10.50.9NDVs[11] 1.14540.5806−115n 2+22n [5] (m = 3) 1.16410.63960.5361191n 2+40n [12] (N = 2) 1.16600.6480−193n 2+38n Theorem 11.20540.72500.5864596.5n 2+64.5nWANG et al .: RELAXED STABILITY CRITERIA FOR DELAYED GNNs VIA A NOVEL RCC 1633。

时延离散网络系统的均方指数稳定控制姚合军【摘要】针对带有随机时延和数据包丢失的不确定离散网络系统,得到了系统的均方指数稳定控制器设计策略.通过把网络诱导时延和数据包丢失看作满足Bernoulli 分布的等价时延,并结合等价时延在不同区间上的概率取值,建立了更加切合实际的网络系统数学模型,并在此基础上,结合Lyapunov稳定性理论,对网络系统的均方指数稳定性进行分析,同时给出了输出反馈鲁棒控制器设计方法.仿真算例说明了该方法的有效性.【期刊名称】《科学技术与工程》【年(卷),期】2019(019)011【总页数】5页(P178-182)【关键词】网络控制系统;时延;指数均方稳定;随机;数据丢失【作者】姚合军【作者单位】安阳师范学院数学与统计学院,安阳455000【正文语种】中文【中图分类】TP273网络系统是指由通信网络连接传感器，执行器和控制器的控制系统[1]。

与传统的点对点控制系统相比较,网络控制系统具有连线少,高性能,低成本,易于维护等优点。

因此,近二十年来，对网络控制系统的研究成果不断出现，网络控制系统已经成为控制界研究的重要分支之一[2—6]。

由于通信信号是通过网络进行传输，因此不可避免地会在系统中存在时延，数据包丢失，网络拥塞等现象，从而使得对网络系统的研究难度不断加大[7—12]。

信息在传感器、执行器和控制器之间传输过程中,以及控制器中的计算过程中不可避免地会出现网络诱导时延。

众所周知时延的存在常常会使系统性能变差,甚至不稳定[13—16]。

另一个区别于传统点对点系统的是由于不确定性和外部干扰的影响,网络系统中常常会存在数据丢失现象。

为分析网络诱导时延和丢包对系统性能影响,高会军给出了一个网络控制系统镇定的新方法,并给出了使闭环系统稳定的控制器设计方法[17]。

Huang通过分析随机时延与数据包丢失对系统的影响，建立了随机网络控制系统的数学模型，并通过设计系统的状态观测器，给出了系统的动态输出反馈控制器设计方法[18]。

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt.J.Robust Nonlinear Control 2003;13:1035–1056(DOI:10.1002/rnc.753)Input-to-state stability with respect to inputsand their derivativesDavid Angeli 1,n ,y ,Eduardo D.Sontag 2,z and Yuan Wang 3,}1Dip.Sistemi e Informatica,Universit "adi Firenze,Via di Santa Marta 3,50139Firenze,Italy 2Department of Mathematics,Rutgers University,New Brunswick,NJ 08903,USA3Department of Mathematics,Florida Atlantic University,Boca Raton,FL 33431,USASUMMARYA new notion of input-to-state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized.An example shows that this notion of stability is indeed weaker than the usual iss .Applications to the study of global asymptotic stability of cascaded non-linear systems are discussed.Copyright #2003John Wiley &Sons,Ltd.KEY WORDS :input-to-state stability;Lyapunov methods;dissipative systems;cascaded systems1.INTRODUCTIONA central question in control theory is how to formulate,for general non-linear systems,notions of robustness and stability with respect to exogenous input disturbances.The linear case is by now very well understood,and,at least in a finite-dimensional set-up,most ‘reasonable’definitions of ‘input-to-state’or ‘input–output’stability (provided in this last case that additional reachability and observability assumptions are met)boil down to local asymptotic stability,viz.to the classical condition on the systems poles lying in the complex open left half-plane.However,for non-linear systems the range of possibilities is much broader,and the goal of coming up with an effective classification for many different behaviours that might be labeled as ‘stable’together with methods which would allow to establish relationships between such stability notions has attracted a substantial research effort within the past years.In this respect,input to state stability (iss )and integral iss ,as well as H 1theory,have proven to be powerful tools,used successfully in order to tackle problems both of robustness analysis and control synthesis,[1–5].Received 29October 2001Revised 6August 2002Published online 31March 2003Accepted 4September 2002Copyright #2003John Wiley &Sons,Ltd.y E-mail:angeli@dsi.unifi.it n Correspondence to:David Angeli,Dipartimento di Sistemi e Informatica,Universit "adi Firenze,Via di Santa Marta 3,50139Firenze,Italy.z E-mail:sontag@ }E-mail:ywang@Contract/grant sponsor:U.S.Air Force;contract/grant number:F-49620-01-0063.Contract/grant sponsor:NSF;contract/grant number:DMS-0072620In the iss -related literature,a ‘disturbance’is a locally essentially bounded measurable function.Such an extremely rich set of possible input perturbations is well suited to model Gaussian and random noises,as well as constant or periodic signals,slow parameters drifts,and so on.If,on one side,this makes the notion of iss extremely powerful,on the other it is known that iss might sometimes be too strong a requirement [6].In the output regulation literature [7],instead,the focus is on ‘deterministic’disturbances,i.e.signals that can be generated by a finite dimensional non-linear systems,(usually smooth),when the state is evolving in a neighbour-hood of a neutrally stable equilibrium position.This is an extremely interesting class of persistent disturbances for which,roughly speaking,the following is true:jj d jj 1small )jj ’djj 1and derivatives of arbitrary order are also small Under similar circumstances,for instance when cascading asymptotically stable systems,regarding the ‘forcing’system’s state as a disturbance typically yieldslim sup t !þ1j d ðt Þj ¼0)lim sup t !þ1j ’d ðt Þj ¼0Nevertheless,the classical definition of input-to-state stability completely disregards such additional information.Tracking of output references,see Reference [8],is another area where ‘derivative’knowledge is usually disregarded (the analysis is often performed only taking into account constant set-points),whereas such information could be exploited to get tighter estimates for the steady-state tracking error due to time-varying,smooth reference signals.Analogous situations arise when parameters variations are taken into account (i.e.in adaptive control)and we expect the system to have nice and stable behaviour for slow parameters drifts.The study of systems with slowly varying parameters has long been an interesting topic in the literature,see e.g.References [9,10].The analysis of such a system is usually carried out by first considering the systems corresponding to ‘frozen’parameters.If for all the frozen parameters,the corresponding frozen systems possess certain stability property uniformly,then it is reasonable to expect that the system with slowly varying parameters will possess the same property.See,for instance,Reference [10]for a result of this type.A more general question is how the magnitudes of the time derivatives of the time varying parameters affect the behaviour of the systems.The main contribution of this note is to show how,in the context of iss ,stability notions can be adjusted in order to take into account robustness with respect to disturbances and their time derivatives.The new notion of D k iss is defined through an iss -like estimate which involves the magnitudes of the inputs and their derivatives up to the k th order.We also propose several properties related to the D k iss notion.All these properties serve to formalize the idea of ‘stable’dependence upon the inputs and their time derivatives.They differ in the formulation of the decay estimates which make precise how the magnitudes of derivatives affect the system.We illustrate by means of several interesting examples how these properties differ from each other and from the well known iss property.One of our main objectives is to provide equivalent Lyapunov characterizations for these properties.Interestingly enough,one of our Lyapunov formulations already appeared in Reference [9],(see formula (5)in that reference).In this work,we provide a stability property that is equivalent to the existence of this type of Lyapunov functions.As a key step in establishing the Lyapunov formulations,we show how the D k iss property can be treated as a Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1036special case of the input–output-to-state stability property(for detailed discussions on this property,see Reference[11]).A second objective is to discuss some applications of the newly introduced notions to the analysis of cascaded non-linear systems.The well known result that cascading preserves the iss property is generalized to the D k iss property.The paper is organized as follows:Section2provides the basic definitions.Sections3–5 contain the Lyapunov characterizations of the D k iss property and some other related properties. Sections6and7are devoted to the study of cascaded systems.Sections8and9provide discussions on the relation between the newly introduced stability notions and the well established iss property.2.BASIC DEFINITIONSConsider non-linear systems of the following form:’xðtÞ¼fðxðtÞ;uðtÞÞð1Þwhere xðtÞ2R n and uðtÞ2R m for each t50:The function f:R nÂR m!R n is locally Lipschitz continuous.Thus,for any measurable,locally essentially bounded function uðtÞ:R!R m;and any initial condition x2R n;there exists a unique solution xðt;x;uÞof(1)satisfying the initialcondition xð0;x;uÞ¼x;defined on some maximal intervalðTÀx;u ;Tþx;uÞ:Recall that system(1)is input-to-state stable(iss for short)if there exist g2K1n and b2KL so that the following holds:j xðt;x;uÞj4bðj x j;tÞþgðjj u½0;tÞjj1Þð2Þfor all t50;all x2R n;and all input signals u;where for any interval I;u I denotes the restrictionof u to I;and where jj v jj denotes the usual L m1-norm(possibly infinite)of v:Usually one can thinkof u as an exogenous disturbance entering the system.Note that if(2)holds for any trajectory on any interval where the trajectory is defined,then the system is automatically forward complete. We denote by W k;1ðJÞ;for any integer k51and any interval J;the space of all functions u:J!R m for which theðkÀ1Þst derivative uðkÀ1Þexists and is locally Lipschitz.For k¼0;we define W0;1ðJÞas the set of locally essentially bounded u:J!R m:When J¼½0;þ1Þ;we omit J and write simply W k;1:(Since absolutely continuous functions have essentially bounded derivatives if and only if they are Lipschitz,the definition of W k;1ðJÞ;for positive k;amounts to asking that theðkÀ1Þst derivative uðkÀ1Þexists and is absolutely continuous,and hence,its derivative,that is,uðkÞ;is locally essentially bounded.Thus W k;1ðJÞis a standard Sobolev space, justifying our notation.)Definition2.1System(1)is said to be k th derivative input-to-state stableðD k iss)if there exist some KL-function b;and some K-functions g0;g1;...;g k such that,for every input u2W k;1;the*A function F:S!R is positive definite if FðxÞ>0;8x2S;x=0and Fð0Þ¼0:A function g:R50!R50is ofclass K if it is continuous,positive definite,and strictly increasing.It is of class K1if it is also unbounded.Finally, b:R50ÂR50!R50is of class KL if for eachfixed t50;bðÁ;tÞis of class K and for eachfixed s>0;bðs;tÞdecreases to0as t!1:An important fact concerning K1functions which will often be used in the following sections is the so-called‘weak triangular inequality’gðaþbÞ4gð2aÞþgð2bÞfor all a;b50:Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056INPUT-TO-STATE STABILITY1037following holds:j x ðt ;x ;u Þj 4b ðj x j ;t Þþg 0ðjj u jjÞþg 1ðjj ’ujjÞþÁÁÁþg k ðjj u ðk ÞjjÞð3Þfor all t 50:As with iss ,we remark that if estimate (3)was instead only required to hold on the maximal interval of definition of the solution x ðt ;x ;u Þ;then j x ðt ;x ;u Þj is uniformly bounded on any subinterval of the maximal interval.Hence,the solution must be globally defined if u 2W k ;1;and the same definition results.We say simply that the system is D iss when it is D 1iss and,of course,iss is the same as D k iss for k ¼0:It is also clear that a system is D k iss if and only if there exist some b 2KL and some g 2K such thatj x ðt ;x ;u Þj 4b ðj x j ;t Þþg ðjj u jj ½k Þð4Þfor all t 50;where jj u jj ½k ¼max 04i 4k fjj u ði Þjjg :Lemma 2.2System (1)is D k iss if and only if property (4)holds for all smooth input functions (with the same b ;g ).ProofOne implication is trivial.To prove the non-trivial implication,assume for some b 2KL and g 2K ;estimate (4)holds for all smooth input functions.By causality,one may replace jj u jj ½k by jj u ½0;t Þjj ½k in (4).Let u 2W k ;1:Fix T >0such that x ðt ;x ;u Þis defined on ½0;T :Note that u ðk Þis essentially bounded on ½0;T :It is a routine approximation fact (reviewed in Corollary A.2in the appendix)that there exists an equibounded sequence of C 1functions f u j g such that*u j !u pointwise on ½0;T ;and *lim sup j !1jjðu j Þ½0;T Þjj ½k 4jj u ½0;T Þjj ½k :Applying (4)to the trajectories with the input function u j ;and then taking the limits,we get j x ðt ;x ;u Þj 4b ðj x j ;0Þþg ðjj u jj ½k ½0;T þx ;u ÞÞð5ÞHence,T þx ;u ¼1;that is,x ðt ;x ;u Þis defined on ½0;1Þ:Thus T can be picked arbitrarily,and (5)holds for all t 50where jj u jj ½k ½0;T þx ;uÞbecomes by jj u jj ½k :&3.A LYAPUNOV CHARACTERIZATION OF D k issFix k 51:For system (1),consider the auxiliary system’x¼f ðx ;z 0Þ;’z 0¼z 1;...;’z k À1¼v ð6ÞLet#x ðt ;x ;Z ;v Þ:¼x ðt ;x ;Z ;v Þz ðt ;Z ;v Þ!Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1038INPUT-TO-STATE STABILITY1039 denote the trajectory of(6)with the initial state xð0Þ¼x;zð0Þ¼Z;(note that the z-component of the solution is independent of the choice of x:)Observe that,if property(4)is known to hold for all inputs in W k;1;then,for the trajectories ðxðt;x;z0;vÞ;zðt;x;z0;vÞÞof the auxiliary system,the following property holds:j xðt;x;Z;vÞj4bðj x j;tÞþ*g0ðjj z jj½0;t Þþ*g1ðjj v jjÞð7Þfor all measurable,locally essentially bounded inputs v:Given the fact that j zðtÞj4jj z jj½0;tÞis always true,we getj#xðt;x;Z;vÞj4bðjðx;ZÞj;tÞþ#g0ðjj z jj½0;t Þþ#g1ðjj v jjÞfor some#g0;#g12K:This shows that if(1)is D k iss;then(6)is input–output-to-state stable,i.e., ioss,with v as input and z¼ðz0;z1;...;z kÀ1Þas outputs(cf.Reference[11]).On the other hand,if the auxiliary system(6)is ioss,then there exist some b2KL and g0;g2K such thatj#xðt;x;Z;vÞj4bðj x jþj Z j;tÞþg0ðjj z jj½0;tÞÞþgðjj v jjÞfor all locally essentially bounded inputs v:Observe thatbðj x jþj Z j;tÞ4bð2j x j;tÞþbð2j Z j;0Þ4bð2j x j;tÞþbð2jj z jj½0;tÞ;0ÞIt follows thatj#xðt;x;Z;vÞj4%bðj x j;tÞþ%g0ðjj z jj½0;tÞÞþgðjj v jjÞholds for all locally essentially bounded v;where%bðs;tÞ¼bð2s;tÞ;and%g0ðsÞ¼bð2s;0Þþg0ðsÞ:In particular,j xðt;x;Z;vÞj4%bðj x j;tÞþ%g0ðjj z jj½0;tÞÞþgðjj v jjÞThis implies that for any u2W k;1;the trajectory of system(1)with initial state x satisfies the estimate:j xðt;x;uÞj4bðj x j;tÞþg1ðjj u jj½k Þwhere g1ðsÞ¼%g0ðsÞþgðsÞ:We have therefore proved the following result that underlies the proofs of Theorems1and2to be given later.Lemma3.1Let k51:System(1)is D k iss if and only the associated auxiliary system(6)is ioss with v as input and z¼ðz0;z1;...;z kÀ1Þas output.By the main result in Reference[11],System(6)is ioss if and only if it admits an ioss-Lyapunov function,that is,a smooth function V:R nÂR km!R50such thata;%a2K1;it holds that*for some%aðjðx;zÞjÞ4Vðx;zÞ4%aðjðx;zÞjÞ8ðx;zÞ%Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056*for some a ;r 2K 1;@V @x ðx ;z Þf ðx ;z Þþ@V @z 0ðx ;z Þz 1þÁÁÁþ@V @z k À1ðx ;z Þv 4Àa ðV ðx ;z ÞÞþr ðjðz ;v ÞjÞfor all x ;z and v :Interpreting z as the input and its derivatives for system (1),we get the following:Theorem 1Let k 51:System (1)is D k iss if and only if there exists a smooth function V :R n ÂR km !R 50such that*there exist some %a ;%a 2K 1such that for all ðx ;m ½k À1 Þ2R n ÂR km ;it holds that %a ðjðx ;m ½k À1 ÞjÞ4V ðx ;m ½k À1 Þ4%a ðjðx ;m ½k À1 ÞjÞð8Þ*there exist some a 2K 1;r 2K 1such that for all x 2R n and all m ½k 2R m ðk þ1Þwith m ½k ¼ðm 0;m 1;...;m k Þ;it holds that@V @x ðx ;m ½k À1 Þf ðx ;m 0Þþ@V @m 0ðx ;m ½k À1 Þm 1þ@V @m 1ðx ;m ½k À1 Þm 2þÁÁÁþ@V @m k À1ðx ;m ½k À1 Þm k 4Àa ðV ðx ;m ½k À1 ÞÞþr ðj m ½k jÞð9ÞRemark 3.2Note that inequality (8)implies that%a ðj x jÞ4V ðx ;m ½k À1 Þ4%a ðjðx ;m ½k À1 ÞjÞð10ÞSuppose a system (1)admits a Lyapunov function V satisfying (9)and (10).Then it can be seen that,along any trajectory x ðt Þwith u 2W k ;1as the input,(9)yieldsd d tV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞ4Àa ðV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞÞþr ðjj u jj ½k Þfor almost all t 50:From this it follows that for some b 2KL and g 2K ;it holds thatV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞ4b ðj V 0j ;t Þþg ðjj u jj ½k Þ8t 50where V 0¼V ðx ð0Þ;u ð0Þ;...;u ðk À1Þð0ÞÞ:Combining this with (10),one sees that system (1)is D k iss :Hence,an equivalent Lyapunov characterization of D k iss is the existence of a smooth function Vsatisfying (9)and (10)for some %a ;%a ;a 2K 1and some r 2K :4.ASYMPTOTIC GAINSClearly,if a system is D k iss ;then it is forward complete (for u 2W k ;1)and for some g 0;g 1;...;g k 2K it holds thatlim sup t !1j x ðt ;x ;u Þj 4g 0ðjj u jjÞþg 1ðjj ’ujjÞþÁÁÁþg k ðjj u ðk ÞjjÞ:ð11ÞWe say that a forward complete system satisfies the asymptotic gain (AG)property in u ;...;u ðk Þif,for some g 0;...;g k 2K ;(11)holds for all x 2R n and all u 2W k ;1:Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG 1040INPUT-TO-STATE STABILITY1041 By Lemma3.1,the system(1)is D k iss if and only if the associated auxiliary system(6)is ioss with v as input and z¼ðz0;z1;...;z kÀ1Þas output.Applying the main result in Reference[12] about asymptotic gains for the ioss property to the auxiliary system(6),one can prove the following:Theorem2For a forward complete system as in(1),the following are equivalent:1.it is D k iss;2.it satisfies the AG property in u;...;uðkÞ;and the corresponding zero-input system’x¼fðx;0Þis(neutrally)stable.5.RELATED NOTIONSIn this section,we consider two properties related to D iss:We focus specifically on D iss(rather than D k iss)as it seems to be the most relevant in applications.As a matter of fact,the authors were not able tofind any example of a D2iss system not being D iss and it is therefore an open question whether or not D k iss(k51)is equivalent to D iss:We say that system(1)is iss in’u if,for some b2KL and some g2K;the following estimate holds for all trajectories with inputs in W1;1:j xðt;x;uÞj4bðj x j;tÞþgðjj’u jjÞ8t50ð12ÞWe say that system(1)is iss in constant inputs if,for some b2KL and g2K;the following estimate holds for all trajectories corresponding to constant inputs u:j xðt;x;uÞj4bðj x j;tÞþgðjj u jjÞ8t50:ð13ÞIt is obvious that if a system is iss in’u;then it is gas uniformly in all constant inputs,that is, for some b2KL;the following holds for all trajectories with constant inputs:j xðt;x;uÞj4bðj x j;tÞ8t50Also note thatðiss in’uÞ)ðD issÞ:The converse is in general false.This can be seen through the following argument.Suppose D iss implies iss in’u:Then we would haveðissÞ)ðD issÞ)ðiss in’uÞand hence,ðissÞ)ðiss in’uÞ:But this is false,as one can see that the linear system’x¼Àxþu is iss but not iss in’u:It is also clear that,for any k50;ðD k issÞ)ðiss in constant uÞAgain,the converse implication is in general false as shown by examples in ing similar arguments as in the proof of Lemma3.1,we get the following:*System(1)is iss in’u if and only if the auxiliary system’x¼fðx;zÞ;’z¼vð14ÞCopyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056is state-independent-input-to-output stable,i.e.siios (see Reference [13])with v as inputs and x as outputs;and *System (1)is iss in constant inputs if and only if the auxiliary system’x ¼f ðx ;z Þ;’z ¼0ð15Þis output-to-state-stable,i.e.oss (see Reference [11])with z as outputs.Applying Theorem 1.2of Reference [14]in conjunction with Remark 4.1in Reference [14]to the siios property for system (14),we get the following:Proposition 5.1System (1)is iss in ’uif and only if there exists a smooth Lyapunov function V :R n ÂR m !R 50satisfying the following:*for some %a ;%a 2K 1;%a ðj x jÞ4V ðx ;m 0Þ4%a ðj x jÞ8x 2R n ;8m 02R m ð16Þ*for some w 2K 1and some continuous,positive definite function a ;V ðx ;m 0Þ5w ðj m 1jÞ)@V @x ðx ;m 0Þf ðx ;m 0Þþ@V @m 0ðx ;m 0Þm 14Àa ðV ðx ;m 0ÞÞð17Þfor all x 2R n and all m 0;m 12R m :Observe that if one restricts the set where the input functions take values to be a bounded set U (as in the case of Reference [9]),then the Lyapunov characterization in Proposition 5.1isequivalent to the existence of a smooth Lyapunov function V satisfying (16)for some %a ;%a 2K 1such that for some a 2K 1and s 2K ;@V @x ðx ;m 0Þf ðx ;m 0Þþ@V @m 0ðx ;m 0Þm 14Àa ðV ðx ;m 0ÞÞþs ðj m 1jÞfor all x 2R n ;all m 02U ;and all m 12R m :Such a Lyapunov estimate was used in [9]to analyze the asymptotic behaviour of systems with slowly varying parameters.Applying Theorem 2of Reference [11]to the oss property for system (15),we have the following:Proposition 5.2System (1)is iss with respect to constant inputs if and only if it admits a smooth Lyapunov function V :R n ÂR m !R 50such that*for some %a ;%a 2K 1;%a ðjðx ;m ÞjÞ4V ðx ;m Þ4%a ðj V ðx ;m ÞjÞ8x 2R n ;8m 2R m ð18Þ*for some a 2K 1;s 2K ;@V @xðx ;m Þf ðx ;m Þ4Àa ðV ðx ;m ÞÞþs ðj m jÞð19Þfor all x 2R n ;m 2R m :Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1042Remark5.3It may also be interesting to consider the D iss property with different indexes on different components of the inputs.For instance,for a system’x¼fðx;u;vÞð20Þwithðu;vÞas inputs,one may consider the property that for some b2KL;g u2K and g v2K; it holds thatj xðt;x;u;vÞj4bðj x j;tÞþg uðjj u jjÞþg vðjj v jjÞþg vðjj’v jjÞð21ÞOne can also get a Lyapunov characterization for such a property by using the same argument as in the proof of Theorem1with the ioss results.For instance,a system as in(20)satisfies property(21)if and only if there exists a smooth Lyapunov function V such that*for some%a;%a2K1;%aðjðx;n0ÞjÞ4Vðx;n0Þ4%aðjðx;n0ÞjÞ*for some a2K1;some r u;r v2K;it holds that@V @x ðx;n0Þfðx;m0;n0Þþ@V@n0ðx;n0Þn14ÀaðVðx;n0ÞÞþr uðj m0jÞþr vðj n0jÞþr vðj n1jÞfor all x;m0;n0and n1:6.APPLICATION OF D iss TO THE ANALYSIS OF CASCADE SYSTEMSAn interesting feature of iss,which makes it particularly useful in feedback design,is that the property is preserved under cascades,(see Reference[15]).Unfortunately,this is not the case for the weaker notion of integral iss,as remarked in Reference[1](but,see Reference[16]for related work).Interestingly,however,although D iss is also a weaker property than iss,it is preserved under cascades,as shown in this section.For a system’x¼fðx;v;uÞwithðv;uÞas inputs,we say that the system is D k iss in v and D l iss in u if there exist b2KL and g2K such that the following holds along any trajectory xðt;x;v;uÞwith initial state x;any input ðv;uÞfor which v2W k;1and u2W l;1:j xðt;x;v;uÞj4bðj x j;tÞþgðjj v jj½k Þþgðjj u jj½l Þ8t50Lemma6.1Consider a cascade system’x¼fðx;z;uÞ’z¼gðz;uÞð22Þwhere xðÁÞand zðÁÞevolve on R n1and R n2;respectively,the input u takes values in R m;and where f is locally Lipschitz and g is smooth.Let k50:Suppose that the z-subsystem is D k iss with u as input,and that the x-subsystem D kþ1iss in z and D k iss in u:Then the cascade system(22)is D k iss:Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056INPUT-TO-STATE STABILITY1043ProofBy assumption,there exist b z 2KL and g z 2K such that,along any trajectory z ðt Þof the z -subsystem with input u ;it holds thatj z ðt Þj 4b z ðj z ð0Þj ;t Þþg z ðjj u jj ½k Þ8t 50ð23Þand there exist some b x 2KL and g x 2K such that,for any trajectory x ðt ;v ;u Þof the system ’x¼f ðx ;v ;u Þ;j x ðt ;x ;v ;u Þj 4b x ðj x ð0Þj ;t Þþg x ðjj v jj ½k þ1 Þþg x ðjj u jj ½k Þ8t 50ð24ÞTo prove Lemma 6.1,we need to find a suitable estimate for the x -component of solutions of(22).For this purpose,we define by induction for 14i 4k þ1:g i ða ;b 0;b 1;...;b i À1Þ¼@g i À1@a g ða ;b 0ÞþX i À2j ¼0@g i À1@b j b j þ1where g 1ða ;b 0Þ¼g ða ;b 0Þ:It can be seen that g i ð0;0;...;0Þ¼0for all 04i 4k ;hence,there exists some s i 2K such thatg i ða ;b 0;...;b i À1Þ4s i ðj a jÞþs i ðj b ½i À1 jÞAgain,by induction,one can show that,along any trajectory z ðt Þof the z -subsystem of (22)with an input u 2W k ;1;it holds thatd i d tz ðt Þ¼g i ðZ ðt Þ;d ðt Þ;’d ðt Þ;...;d ði À1Þðt ÞÞfor all 14i 4k þ1:It then follows thatjj z ½k þ1 jj 4s ðjj z jjÞþs ðjj u ½k jjÞð25Þfor some s 2K :It then follows from (24)and (25)that,for some r 2K ;it holds that along any trajectory ðx ðt Þ;z ðt ÞÞof (22),j x ðt Þj 4b x ðj x ð0Þj ;t Þþr ðjj z jjÞþr ðjj u jj ½k Þ8t 50ð26ÞApplying a standard argument to (26)and (23)as in the proof of the result that a cascade of iss systems is again iss ,one can show that system (22)is D k iss :To be more precise,(26)implies that j x ðt Þj 4b x j x ðt =2Þj ;t 2þr ðjj z jj ½t =2;t ÞÞþr ðjj u jj ½k ½t =2;t ÞÞ8t 50ð27Þalong any trajectory of (22).Fix an input u and pick any trajectory ðx ðt Þ;z ðt ÞÞof (22)with the input u :Let x 1¼x ðt =2Þ:We also have j x 1j 4b x j x j ;t 2þr ðjj z jj ½0;t =2ÞÞþr ðjj u jj ½k ½0;t =2ÞÞ8t 50Hence,there exist some *b x ;*b z and some *r 2K (which depend only on b x ;r )such that b x j x 1j ;t 24*b x ðj x j ;t Þþ*b z ðjj z jj ½0;t =2Þ;t Þþ*r ðjj u jj ½k ½0;t =2ÞÞð28Þfor all t 50:By (23),j z ðt Þj 4b z ðj z ð0Þj ;0Þþg z ðjj u jj ½k Þfor all t50;hence,*b z ðjj z jj ½0;t =2Þ;t Þ4%b z ðj z ð0Þj ;t Þþ%g z ðjj u jj ½k Þ8t 50ð29ÞCopyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1044。