Delay-Dependent Robust Control for Uncertain
一类不确定Lur'e奇异系统的可靠鲁棒H∞跟踪控制
ACTA AUTOMATICA SINICA
August, 2008
Reliable Robust H ∞ Tracking Control for a Class of Uncertain Lur e Singular Systems
WANG Hui-Jiao1, 2 LIN Yue-Song2 XUE An-Ke2 PAN Hai-Peng1 LU Ren-Quan2
Control of singular systems has been extensively studied in the past years due to the fact that singular systems can better describe physical systems than regular ones. A great number of results based on the theory of regular systems have been extended to the area of singular systems, such as robust stability and robust stabilization problem[1−3] , robust H∞ control problem[4−6] , robust H∞ filtering problem[7] , guarantee cost problem[8] , etc. However, all the aforementioned results are under a full reliability assumption that all control components of the system are in good working condition. As is well known, failures of actuators or sensors, in practical engineering systems, often occur, which may lead to intolerable performance of the closed-loop system. Therefore, from a safety as well as performance point of view, it is required to design a reliable controller which can tolerate actuator or sensor failures and guarantee the stability and performance of resultant closed-loop system. Furthermore, there are lots of results about reliable control for regular systems[9−12] . For singular systems, [13] studied the reliable H∞ control problem with actuator failures and multiple time delays, but the reliable controller design method is based on a basic assumption that control component failures are modeled as outages, that is, when a failure occurs, the signal (in the case of sensors) or the control action (in the case of actuators) simply becomes zero. The outage model is the simplest case of control component failures. In this paper, a more practical and general failure model is adopted for sensor and actuator failures, which consists of a scaling factor with upper and lower bounds to the signal to be measured or to the control action, plus a disturbance. This model was introduced by [14]. For actuators, this general failure model could represent faults in the driving circuitry before the final control actuator. For sensors, it could represent faults in the signal conditioning circuitry at the sensors. To our knowledge, the reliable control problem of singular systems based on such failure model has not yet been fully investigated. On the other hand, designing a controller which can guarantee the output of controlled system tracking the reference signal is of theoretical and practical meaning. There are some results presented on the problem of reliable trackReceived June 29, 2007; in revised form February 23, 2008 Supported by National Natural Science Foundation of China (60434020, 60604003) 1. Institute of Automation, College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China 2. Institute of Information and Control, Hangzhou Dianzi University, Hangzhou 310018, P. R. China DOI: 10.3724/SP.J.1004.2008.00893
具有时变时滞的Lurie系统绝对稳定性的时滞相关条件
ACTA AUTOMATICA SINICA
May, 2005
Delay-dependent Conditions for Absolute Stability of Lurie Control Systems with Time-varying Delay1)
HE Yong WU Min
m
V (x t ) = x T (t )P x (t ) + 2
j =1
σj
t
λj
0
fj (σj )dσj +
t−d(t)
xT (s)Qx(s)ds +
0 −τ
t t+ θ
˙ T (s )Z x ˙ (s)dsdθ (5) x
where P > 0, Q 0, Z > 0, Λ = diag(λ1 , λ2 , · · · , λm ) 0 are to be determined. Calculating the derivative of V (xt ) along the solutions of system (1) yields d V (x t ) dt 2[xT (t)P + f T (σ )ΛC T ] · [Ax(t) + B x(t − d(t)) + Df (σ )]+ xT (t)Qx(t) − (1 − µ)xT (t − d(t))Qx(t − d(t)) −
t t−d(t)
˙ T (s )Z x ˙ (s)ds+ x
(6)
τ [Ax(t) + B x(t − d(t)) + Df (σ )]T Z [Ax(t) + B x(t − d(t)) + Df (σ )] Using the Leibniz-Newton formula, for any matrices N1 , N2 ∈ Rn×n , N3 ∈ Rm×n , 2[xT (t)N1 + xT (t − d(t))N2 + f T (σ )N3 ] · [x(t) − x(t − d(t)) − X11 T On the other hand, for any semi-positive definite matrix X = X12 T X13 holds according to (2). z T (t)(τ X )z (t) − where z (t) = [xT (t) xT (t − d(t)) d V (x t ) dt
不确定广义时滞系统的时滞依赖鲁棒H∞控制
During the past decade, many important and interesting results have been reported on the robust H∞ controller design for uncertain time-delay systems by various methods[1−2] . Recently, the robust H∞ control problem for uncertain singular time-delay systems has also been considered. These methods may be classified into two categories: delay-independent cases[3−5] and delay-dependent cases. Generally speaking, delay-independent cases are likely to be conservative, especially when the time delays are small. [6−9] investigated the delay-dependent H∞ control problem for singular time-delay systems and several sufficient conditions for the solvability of this problem were proposed by linear matrix inequality (LMI) approach. However, in order to make use of the results of [6−7], the considered system should be transformed into an augmented system, where the elements of system matrices are used. This not only makes the results inapplicable to uncertain singular time-delay systems but also renders the analysis and design procedure relatively intricate and unreliable. The delay-dependent robust H∞ control problem for uncertain singular time-delay systems was considered in [10−12], where some sufficient conditions for the existence of H∞ controller were proposed and the obtained design methods are formulated in terms of nonstrict LMIs whose solutions are difficult to calculate since equality constraints are often fragile and usually not met perfectly. To the best of our knowledge, the delay-dependent robust H∞ control problem for uncertain singular time-delay systems has not yet been fully investigated. Particularly, strict LMI-based condition has never been reported in the published works. This paper deals with the problem of delay-dependent robust H∞ control for uncertain singular time-delay systems. A delay-dependent sufficient condition is established, which guarantees the nominal system to be regular, impulse free, and stable with disturbance attenuation level γ . Based on this, the delay-dependent robust H∞ control problem is solved and a strict LMI-based design method of the desired controller is proposed. Two numerical examples are provided to demonstrate the effectiveness of the proposed results.
低频活动漂浮潜水船声探测系统（LFATS）说明书
LOW-FREQUENCY ACTIVE TOWED SONAR (LFATS)LFATS is a full-feature, long-range,low-frequency variable depth sonarDeveloped for active sonar operation against modern dieselelectric submarines, LFATS has demonstrated consistent detection performance in shallow and deep water. LFATS also provides a passive mode and includes a full set of passive tools and features.COMPACT SIZELFATS is a small, lightweight, air-transportable, ruggedized system designed specifically for easy installation on small vessels. CONFIGURABLELFATS can operate in a stand-alone configuration or be easily integrated into the ship’s combat system.TACTICAL BISTATIC AND MULTISTATIC CAPABILITYA robust infrastructure permits interoperability with the HELRAS helicopter dipping sonar and all key sonobuoys.HIGHLY MANEUVERABLEOwn-ship noise reduction processing algorithms, coupled with compact twin line receivers, enable short-scope towing for efficient maneuvering, fast deployment and unencumbered operation in shallow water.COMPACT WINCH AND HANDLING SYSTEMAn ultrastable structure assures safe, reliable operation in heavy seas and permits manual or console-controlled deployment, retrieval and depth-keeping. FULL 360° COVERAGEA dual parallel array configuration and advanced signal processing achieve instantaneous, unambiguous left/right target discrimination.SPACE-SAVING TRANSMITTERTOW-BODY CONFIGURATIONInnovative technology achievesomnidirectional, large aperture acousticperformance in a compact, sleek tow-body assembly.REVERBERATION SUPRESSIONThe unique transmitter design enablesforward, aft, port and starboarddirectional transmission. This capabilitydiverts energy concentration away fromshorelines and landmasses, minimizingreverb and optimizing target detection.SONAR PERFORMANCE PREDICTIONA key ingredient to mission planning,LFATS computes and displays systemdetection capability based on modeled ormeasured environmental data.Key Features>Wide-area search>Target detection, localization andclassification>T racking and attack>Embedded trainingSonar Processing>Active processing: State-of-the-art signal processing offers acomprehensive range of single- andmulti-pulse, FM and CW processingfor detection and tracking. Targetdetection, localization andclassification>P assive processing: LFATS featuresfull 100-to-2,000 Hz continuouswideband coverage. Broadband,DEMON and narrowband analyzers,torpedo alert and extendedtracking functions constitute asuite of passive tools to track andanalyze targets.>Playback mode: Playback isseamlessly integrated intopassive and active operation,enabling postanalysis of pre-recorded mission data and is a keycomponent to operator training.>Built-in test: Power-up, continuousbackground and operator-initiatedtest modes combine to boostsystem availability and accelerateoperational readiness.UNIQUE EXTENSION/RETRACTIONMECHANISM TRANSFORMS COMPACTTOW-BODY CONFIGURATION TO ALARGE-APERTURE MULTIDIRECTIONALTRANSMITTERDISPLAYS AND OPERATOR INTERFACES>State-of-the-art workstation-based operator machineinterface: Trackball, point-and-click control, pull-down menu function and parameter selection allows easy access to key information. >Displays: A strategic balance of multifunction displays,built on a modern OpenGL framework, offer flexible search, classification and geographic formats. Ground-stabilized, high-resolution color monitors capture details in the real-time processed sonar data. > B uilt-in operator aids: To simplify operation, LFATS provides recommended mode/parameter settings, automated range-of-day estimation and data history recall. >COTS hardware: LFATS incorporates a modular, expandable open architecture to accommodate future technology.L3Harrissellsht_LFATS© 2022 L3Harris Technologies, Inc. | 09/2022NON-EXPORT CONTROLLED - These item(s)/data have been reviewed in accordance with the InternationalTraffic in Arms Regulations (ITAR), 22 CFR part 120.33, and the Export Administration Regulations (EAR), 15 CFR 734(3)(b)(3), and may be released without export restrictions.L3Harris Technologies is an agile global aerospace and defense technology innovator, delivering end-to-endsolutions that meet customers’ mission-critical needs. The company provides advanced defense and commercial technologies across air, land, sea, space and cyber domains.t 818 367 0111 | f 818 364 2491 *******************WINCH AND HANDLINGSYSTEMSHIP ELECTRONICSTOWED SUBSYSTEMSONAR OPERATORCONSOLETRANSMIT POWERAMPLIFIER 1025 W. NASA Boulevard Melbourne, FL 32919SPECIFICATIONSOperating Modes Active, passive, test, playback, multi-staticSource Level 219 dB Omnidirectional, 222 dB Sector Steered Projector Elements 16 in 4 stavesTransmission Omnidirectional or by sector Operating Depth 15-to-300 m Survival Speed 30 knotsSize Winch & Handling Subsystem:180 in. x 138 in. x 84 in.(4.5 m x 3.5 m x 2.2 m)Sonar Operator Console:60 in. x 26 in. x 68 in.(1.52 m x 0.66 m x 1.73 m)Transmit Power Amplifier:42 in. x 28 in. x 68 in.(1.07 m x 0.71 m x 1.73 m)Weight Winch & Handling: 3,954 kg (8,717 lb.)Towed Subsystem: 678 kg (1,495 lb.)Ship Electronics: 928 kg (2,045 lb.)Platforms Frigates, corvettes, small patrol boats Receive ArrayConfiguration: Twin-lineNumber of channels: 48 per lineLength: 26.5 m (86.9 ft.)Array directivity: >18 dB @ 1,380 HzLFATS PROCESSINGActiveActive Band 1,200-to-1,00 HzProcessing CW, FM, wavetrain, multi-pulse matched filtering Pulse Lengths Range-dependent, .039 to 10 sec. max.FM Bandwidth 50, 100 and 300 HzTracking 20 auto and operator-initiated Displays PPI, bearing range, Doppler range, FM A-scan, geographic overlayRange Scale5, 10, 20, 40, and 80 kyd PassivePassive Band Continuous 100-to-2,000 HzProcessing Broadband, narrowband, ALI, DEMON and tracking Displays BTR, BFI, NALI, DEMON and LOFAR Tracking 20 auto and operator-initiatedCommonOwn-ship noise reduction, doppler nullification, directional audio。
Stability of Time-Delay Systems Equivalence between Lyapunov and Scaled Small-Gain Conditio
Stability of Time-Delay Systems:Equivalence between Lyapunov and Scaled Small-Gain ConditionsJianrong Zhang,Carl R.Knopse,and Panagiotis Tsiotras Abstract—It is demonstrated that many previously reported Lyapunov-based stability conditions for time-delay systems are equivalent to the ro-bust stability analysis of an uncertain comparison system free of delays via the use of the scaled small-gain lemma with constant scales.The novelty of this note stems from the fact that it unifies several existing stability results under the same framework.In addition,it offers insights on how new,less conservative results can be developed.Index Terms—Stability,time-delay systems.II.I NTRODUCTIONThe analysis of linear time-delay systems(LTDS)has attracted much interest in the literature over the half century,especially in the last decade.Two types of stability conditions,namely delay-inde-pendent and delay-dependent,have been studied[17].As the name implies,delay-independent results guarantee stability for arbitrarily large delays.Delay-dependent results take into account the maximum delay that can be tolerated by the system and,thus,are more useful in applications.One of the first stability analysis results was the polyno-mial criteria[8]–[10].An important result was later provided by[3], which gives necessary and sufficient conditions for efficient compu-tation of the delay margin for the linear systems with commensurate delays.This result only requires the computation of the eigenvalues and generalized eigenvalues of constant matrices.Unfortunately,it is not straightforward to extend this to many problems of interest, such as the stability of general(noncommensurate)delays systems, H1performance of LTDS with exogenous disturbances,robust stability of LTDS with dynamical uncertainties,and robust controller synthesis,etc.Recently,much effort has been devoted to developing frequency-domain and time-domain based techniques which may be extendable to such problems.The frequency-domain approaches include integral quadratic constraints[6],singular value tests[25], framework-based criteria[4],and other similar techniques.In[20], the traditional -framework was extended for time-delay systems to obtain a necessary and sufficient stability condition,which was then relaxed to a convex sufficient condition.Other recent stability analysis results have been developed in the time-domain,based on Lyapunov’s Second Method using either Lyapunov–Krasovskii functionals or Lyapunov–Razumikhin functions [26],[12],[13],[16],[22],[14],[17],[19].These results are formulated in terms of linear matrix inequalities(LMIs),and,hence,can be solved efficiently[1].While these results are often extendable to the systems with general multiple delays and/or dynamical uncertainties,they can be rather conservative and the corresponding Lyapunov functionals are complex.A formal procedure for constructing Lyapunov functionals for LTDS was proposed in[11],but a Lyapunov functional,in general, Manuscript received June10,1999;revised August10,2000.Recommended by Associate Editor J.Chen.This work was supported by the National Science Foundation under Grant DMI-9713488.J.Zhang and C.R.Knospe are with the Department of Mechanical and Aerospace Engineering,University of Virginia,Charlottesville,V A22904-4746 USA(e-mail:jz9n@;crk4y@).P.Tsiotras is with the School of Aerospace Engineering,Georgia Institute of Technology,Atlanta,GA30332-0150USA(e-mail:p.tsiotras@). Publisher Item Identifier S0018-9286(01)01015-7.does not provide direct information on how conservative the resultant condition may be in practice.In this note,we show that several existing Lyapunov-based results, both delay-independent and delay-dependent,are equivalent to the scaled small-gain condition for robust stability of a comparison system that is free of delay.This result provides a new frequency-domain in-terpretation to some common Lyapunov-based results in the literature. Via a numerical example,we investigate the potential conservatism of the stability conditions,and demonstrate that a major source of conservatism is the embedding of the delay uncertainties in unit disks that the comparison system employs.This source of conservatism is hidden in the Lyapunov-based framework but is quite apparent in the comparison system interpretation.These results also provide insight into how to reduce the conservatism of the stability tests.After a conference version of this note appeared in[28],we be-came aware of the results of[15]and[7]which are related to our approach.Unlike the model transformation class in[15],which con-tains distributed delays,the comparison system employed herein is a delay-free uncertain system stated in frequency domain and permits the immediate application of the standard frequency-domain techniques, such as the framework.The results in[7]are based on a special case of our comparison system,namely M=I n.Neither[15]nor[7]exam-ined the equivalence of existing Lyapunov-based criteria and the scaled small-gain conditions,which is the contribution of this note.The notation is conventional.Let n2m)be the set of all real(complex)n2mmatrices,[f1g,I n be n2n identity matrix,W T be the transpose of real matrix W,and RH1:=f H(s): H(s)2H1,H(s)is a real rational transfer matrix g.P>0indicates that P is a symmetric and positive definite matrix,and k1k1indicates the H1norm defined by k G k1:=sup!2n2n with respect to a block structure 3is defined by 3(M)=0if there is no323such that I0M3 is singular,and3(M)=[min f (3):det(I0M3)=0;323g]01 otherwise.We also define the set1r:=f diag[ 1I n,111, r I n and the closed norm-bounded set B1r:=f12 H1:k1k1 1;1(s)21r g.Finally,for linear time-invariant system P(s)and its input x(t),we define a signal P(s)[x](t)asP(s)[x](t):=L01[P(s)X(s)]where X(s)is the Laplace transform of x(t),and L01[1]is the inverse Laplace operator.III.C OMPARISON S YSTEMFor ease of exposition,we will examine the single-delay case. However,the Lyapunov stability conditions examined here may all be straightforwardly extended to the case of systems with multiple (noncommensurate)delays.Consider the linear time-delay system_x(t)=Ax(t)+A d x(t0 )(1) where A2n2n are constant matrices,and the delay is constant,unknown,but bounded by a known bound as0 . The following assumption is a necessary condition when investigating asymptotic stability of the system(1).Assumption1:The system(1)free of delay is asymptotically stable, that is,the matrix A:=A+A d is Hurwitz.Taking Laplace transforms of both sides,the system(1)can be ex-pressed in the s domain assX(s)=AX(s)+A d e0 s X(s):(2)0018–9286/01$10.00©2001IEEEFig.1.A system with uncertainty.The results of this note depend on the notion of robust stability of afeedback interconnection of a finite-dimensional,linear,time-invariant (FDLTI)system and an uncertain system with known uncertainty struc-ture.The following definition clarifies the type of robust stability used herein.More on this definition can be found in [32].Definition 1:Consider a linear,time-invariant (finite-dimensional)system G (s )interconnected with an uncertain block 1,as shown in Fig.1.The uncertain block 1belongs to a known,uncertainty structure set 121.Then,the system is said to be robustly stable if G (s )is internally stable and the interconnection is well posed and remains internally stable for all 121:To proceed with our analysis,we need the following preliminary results.Lemma 1:Let M2e 0s 01 s e 0s 01 s=A +MA d )X (s )+(I 0M )A d e0 s X (s )+d AX(s )+e 0 s1d A d X (s ):In view of the fact that k e 0 s k 1=1and k (e 0 s 01)=( s )k 1== 1,it follows from the above equation that (2)is a special case of the uncertain system (3)with 11=e 0 s I n ,and 12=(e 0 s 01)=( s )I n .Therefore,the robust stability of (3)guarantees that (1)is asymptotically stable for all 2[0; ].As shown in the next section,the comparison system (3)can be rewritten as an interconnection of an FDLTI system G (s )with a block 1,where 1=diag[11;12]2B 12.Hence,the analysis of the ro-bust stability of the system (3)may be performed via -analysis,since the small- theorem applies even to the case where the uncertainty is nonrational [23].Because the calculation of is NP-hard in general[2],its upper bound with D scales is typically used instead.In partic-ular,the interconnection in Fig.1is robustly stable if G (s )2RH1is internally stable andsup !2(j!)D 01n 2n;D i =D 3i >0g :The test (4),although a convex optimization problem,requires a fre-quency sweep.Alternatively,the analysis of robust stability may be performed without the frequency sweep by solving an LMI.The fol-lowing lemma states this result.Additional conservatism is introduced in this formulation,however,since it implies satisfaction of (4)with the same constant real scaling matrix used for all frequencies.Lemma 2[21](Scaled Small-Gain LMI)1:Consider the system in-terconnection shown in Fig.1where the plant G (s )is FDLTI and the uncertainty block is such that 12B 1r .Let (A;B;C;D )be a min-imal realization of G (s )withG (s )=:Then,the closed-loop system is robustly stable if there exist matricesX >0and Q =diag[Q 1,Q 2,111,Qr ]>0,Qi 22nXA d AXA d A dA T A T d X011XA T d A T dX012X>0(7)where =0 01[(A +A d )T X +X (A +A d )]0( 011+ 012)X .b)[13]There exist matrices P >0;P 1>0and P 2>0satisfyingH P A T P A T dAP 0 P 10 A dP0 P 2<0(8)1Thesmall gain theorem applies to the case where the uncertainty blockscontain infinite dimensional dynamic systems [32].where H=P(A+A d)T+(A+A d)P+ A d(P1+ P2)A T d.c)[19]There exist matrices X>0;U>0;V>0and Wsatisfying10W A d A T A T d V (W+X)0A T d W T0U A T d A T d V0V A d A V A d A d0V0000V<0(9)where 1=(A+A d)T X+X(A+A d)+W A d+A T d W T+U.The following proposition shows that all of above conditions areequivalent to the SSGS conditions for the special case of the compar-ison system(3).Proposition1:For the comparison system(3),if M=0,the SSGScondition is equivalent to the condition(6),2and,if M=I n,the SSGScondition is equivalent to the condition(8)and can also be reducedto the condition(7).Moreover,the delay-dependent condition(9)isequivalent to the SSGS condition for(3)with M as a free-matrix vari-able.Proof:First,let M=0,then the comparison system(3)becomessX(s)=AX(s)+11A d X(s)112B11which can be described as the following closed-loop system:_x=Ax+A d uy=xu=11[y](t):WithG(s)=t h e S S G S c o n d i t i o n b e c o m e s(6).N e x t,w e l e tM=I n a n d13=1112.Equation(3)then becomessX(s)=(A+A d)X(s)+12 A d AX(s)+13 A d A d X(s)(10)with diag[12;13]2B12.The last equation can be rewritten as theclosed-loop system_x=(A+A d)x+ A d u1+ A d u2y1=Axy2=A d xu1=12[y1](t)u2=13[y2](t):Then,by applying Lemma2withG(s)=AA dw e s e e t h a t t h e s y s t e m(1)i s a s y m p t o t i c a l l y s t a b l e f o r a n y c o n s t a n t,0 ,i f t h e r e e x i s tX>0and Q=diag[Q1;Q2]>0suchthatR XA d XA d A T Q1A T d Q2A T d X0Q1000A T d X00Q200Q1A000Q10Q2A d0000Q2<02Similar observations can also be found,for example,in[26]and[4].where R=(A+A d)T X+X(A+A d).Multiplying bydiag[X01;I;I; Q011; Q012]on both sides and using Schurcomplements,the above inequality is equivalenttoH XA d A XA d A dA T A T d X0Q10A T d A T d X00Q2<0X>0;Q1>0;Q2>0(12)where H=(A+A d)T X+X(A+A d)+Q1+Q2.Now,lettingQ1= 011X and Q2= 012X,where constants 1>0and2>0,(12)is reduced to(7).Finally,consider the general case of(3)and rewrite it as the fol-lowing:_x=(A+MA d)x+(I0M)A d u2+ Mu1y1=A d Ax+A d A d u2y2=xu1=12[y1](t)u2=11[y2](t):(13)Therefore,applying Lemma2withG(s)=w h e r e^A=A+MA d,^B=[ M(I0M)AA I]Fig.2.Delay margin versus K.(1)Nyquist Criterion.(2) upper bound withfrequency-dependent D scaling.(3)Condition of[19].(4)Condition of[13].(5)Condition of[16].(6)Condition of[25],[26]for K<Kc o n t r o l o f u n c e r t a i n l i n e a r13t h I F A C W o r l d C o n g r.,1996,p p.113–118.d e l a y s y s t e m s,”i n[14]S.-I.N i c u l e s c u,“O n t h e s t a b i l i t y a n d s t a b i lw i t h d e l a y e d s t a t e,”P h.D.d i s s e r t a t i o n,L a b o r aG r e n o b l e,I N P G,1996.[15]S.-I.Niculescu and J.Chen,“Frequency sweeping tests for asymptoticstability:A model transformation for multiple delays,”in Proc.38th IEEE Conf.Decision Control,1999,pp.4678–4683.[16]S.-I.Niculescu,o,J.-M.Dion,and L.Dugard,“Delay-depen-dent stability of linear systems with delayed state:An LMI approach,”in Proc.34th IEEE Conf.Decision Control,1995,pp.1495–1497. [17]S.-I.Niculescu,E.I.Verriest,L.Dugard,and J.-M.Dion,“Stability androbust stability of time-delay systems:A guided tour,”in Stability and Robust Control of Time Delay Systems.New York:Springer-Verlag, 1997,pp.1–71.[18] A.Packard and J.C.Doyle,“The complex structured singular value,”Automatica,vol.29,no.1,pp.77–109,1993.[19]P.Park,“A delay-dependent stability criterion for systems with uncer-tain time-invariant delays,”IEEE Trans.Automat.Contr.,vol.44,pp.876–877,Apr.1999.[20]G.Scorletti,“Robustness analysis with time-delays,”in Proc.36th IEEEConf.Decision Control,1997,pp.3824–3829.[21]R.E.Skelton,T.Iwasaki,and K.Grigoriadis,A Unified Algebraic Ap-proach to Linear Control Design.New York:Taylor&Francis,1998.[22] E.Tissir and A.Hmamed,“Further results on stability of_x(t)=Ax(t)+Bx(t0 ),”Automatica,vol.32,no.12,pp.1723–1726,1996.[23] A.L.Tits and M.K.H.Fan,“On the small- theorem,”Automatica,vol.31,no.8,pp.1199–1201,1995.[24]J.Tlusty,“Machine dynamics,”in Handbook of High Speed MachiningTechnology,R.I.King,Ed.New York:Chapman&Hall,1985,pp.48–153.[25] E.I.Verriest,M.K.H.Fan,and J.Kullstam,“Frequency domain robuststability criteria for linear delay systems,”in Proc.32nd IEEE Conf.Decision Control,1993,pp.3473–3478.[26] E.I.Verriest and A.F.Ivanov,“Robust stability of systems with delayedfeedback,”Circuits,Syst.Signal Processing,vol.13,pp.213–222,1994.[27]M.Vidyasagar,Nonlinear Systems Analysis,2nd ed.EnglewoodCliffs,NJ:Prentice-Hall,1993.[28]J.Zhang,C.R.Knospe,and P.Tsiotras,“A unified approach to time-delay system stability via scaled small gain,”in Proc.Amer.Control Conf.,1999,pp.307–308.[29],“Toward less conservative stability analysis of time-delay sys-tems,”in Proc.38th IEEE Conf.Decision Control,1999,pp.2017–2022.[30],“Stability of linear time-delay systems:A delay-dependent cri-terion with a tight conservatism bound,”in Proc.2000Amer.Control Conf.,2000.[31],“Asymptotic stability of linear systems with multiple time-in-variant state-delays,”in Proc.2nd IFAC Workshop Linear Time Delay Systems,to be published.[32]K.Zhou,J. C.Doyle,and K.Glover,Robust and Optimal Con-trol.Englewood-Cliffs,NJ:Prentice-Hall,1996.Bounded Stochastic Distributions Control forPseudo-ARMAX Stochastic SystemsHong Wang and Jian Hua ZhangAbstract—Following the recently developed algorithms for the control of the shape of the output probability density functions for general dy-namic stochastic systems[6]–[8],this note presents the modeling and con-trol algorithms for pseudo-ARMAX systems,where,different from all the existing ARMAX systems,the considered system is subjected to any arbi-trary bounded random input and the purpose of the control input design is to make the output probability density function of the system output as close as possible to a given distribution function.At first,the relationship between the input noise distribution and the output distribution is estab-lished.This is then followed by the description on the control algorithm de-sign.A simulated example is used to demonstrate the use of the algorithm and encouraging results have been obtained.IndexTerms—i=1v i(k)B i(y)y2[a;b](1)whereu k control input;(y;u)measured probability density function of the system output;V(k)=(v1;v2;...;v M)T,weight vector;B i(y)pre-specified basis functions for the approximation of(y;u)[2];A andB constant matrices.Although there are several advantages in using this type of model to de-sign the required control algorithm,it is difficult to link such a model structure to a physical system.In particular,the key assumption that the control input only affects the weights of the output probability density function is strict for some applications.As such,it would be ideal if aManuscript received March30,2000;revised July31,2000.Recommended by Associate Editor Q.Zhang.This work was supported in part by the U.K. EPSRC under Grant(GB/K97721),and in part by the Overseas Scholarship Committee of the P.R.China.H.Wang is with the Department of Paper Science,Affiliated Member of Con-trol Systems Centre,University of Manchester Institute of Science and Tech-nology,Manchester M601QD,U.K.H.Zhang is on leave from the Department of Power Engineering,North China University of Electrical Power,Beijing,P.R.China.Publisher Item Identifier S0018-9286(01)01014-5.0018–9286/01$10.00©2001IEEE。
Direct adaptive fuzzy control for nonlinear systems with time-varying delays-陈兵2010
Direct adaptive fuzzy control for nonlinear systems with time-varying delays qBing Chen a,*,Xiaoping Liu b ,Kefu Liu b ,Peng Shi c ,Chong Lin aaInstitute of Complexity Science,Qingdao University,Qingdao,Shandong 266071,PR China bFaculty of Engineering of Lakehead University,Thunder Bay,On,Canada cSchool of Technology,University of Glamorgan,Pontypridd,CF371DL,UKa r t i c l e i n f o Article history:Received 24November 2008Received in revised form 29October 2009Accepted 3November 2009Keywords:Nonlinear systems Adaptive fuzzy control Time-delayRazumikhin functional Output trackinga b s t r a c tThis paper focuses on the problem of direct adaptive fuzzy control for nonlinear strict-feed-back systems with time-varying delays.Based on the Razumikhin function approach,a novel adaptive fuzzy controller is designed.The proposed controller guarantees that the system output converges to a small neighborhood of the reference signal and all the signals in the closed-loop system remain bounded.Different from the existing adaptive fuzzy con-trol methodology,the fuzzy logic systems are used to model the desired but unknown con-trol signals rather than the unknown nonlinear functions in the systems.As a result,the proposed adaptive controller has a simpler form and requires fewer adaptation parameters.Ó2009Elsevier Inc.All rights reserved.1.IntroductionIn recent years,considerable attention has been paid to fuzzy logic control.Some fuzzy logic control techniques have been developed,for instance,see [3,15,16,23,35,36]and the references therein.Approximation-based adaptive fuzzy control or neural control has emerged as a popular and convenient tool in analysis and synthesis of complex and ill-defined systems,to which the application of conventional control techniques is not straightforward or feasible.The main idea of such a control methodology is to utilize fuzzy logic systems or neural networks to approximate the unknown nonlinearities in dynamic sys-tems and the adaptive backstepping technique to construct the controllers.Following such an idea,some adaptive neural control or fuzzy control schemes were proposed for nonlinear delay-free systems with strict-feedback structure.In [1,4,6,37],the adaptive neural tracking control was addressed for nonlinear delay-free systems in strict-feedback form.The corresponding adaptive fuzzy control schemes were proposed in [11,20,31–33,38,25].All the aforementioned studies do not deal with state time-delays,which often occur in many dynamic systems,such as rolling mill systems,biological systems,metallurgical processing systems,network systems,and so on.It is well known that the existence of delays usually becomes the source of instability and performance deterioration of systems.Stability analysis and control synthesis of Takagi–Sugeno (T–S)fuzzy delayed systems were proposed in [2,12,17,26]and [24],respectively.Especially,approximation-based adaptive control has also been addressed for nonlinear systems with time-delay.In [5],the problem of adaptive fuzzy tracking control was addressed for nonlinear strict-feedback systems with time-delays.The proposed tracking controller guarantees that all the closed-loop signals remain bounded,while the output converges to a neighborhood of the reference signal.By using a wavelet fuzzy network online approximation model,the stabilization0020-0255/$-see front matter Ó2009Elsevier Inc.All rights reserved.doi:10.1016/j.ins.2009.11.004qThis work was supported in part by the national science foundation of China (Nos.60674055and 60774047).*Corresponding author.E-mail address:chenbing1958@ (B.Chen).Information Sciences 180(2010)776–792Contents lists available at ScienceDirectInformation Sciencesjournal homepage:www.else v i e r.c o m /l o c a t e /i n sproblem was discussed for nonlinear strict-feedback systems with time-delay in [7].In [21],the adaptive H 1control was investigated via backstepping and fuzzy networks technique.In [8],the observer-based adaptive neural controller was de-signed for a special class of nonlinear time-delay systems.By using fuzzy logic systems to approximate the unknown non-linear functions,the adaptive fuzzy control scheme was proposed in [28]for nonlinear delayed systems with lower-triangular form.The main advantage of the results proposed in [28]is that the controllers contain fewer adaptation ually,there are two approaches for stability analysis of time-delay systems.The first one is based on the Lyapunov–Krasovskii functionals [5,7,28],and the second one on the Lyapunov-Razumikhin functionals [9,13,18,34,19]and [27].The Lyapunov–Krasovskii functional methods usually make stability analysis and controller design more complex than the Raz-umikhin functional approach.All the existing results on the approximation-based adaptive control are obtained by using the Lyapunov–Krasovskii functional approach.To the authors’knowledge,there have not been any reported results on the use of the Razumikhin functional approach for this purpose.In addition,all the existing adaptive fuzzy or neural controllers were proposed only for the nonlinear systems with constant delays.Therefore,in theory,the existing adaptive fuzzy controllers are not capable of dealing with systems with time-varying delays.Thus,it is of importance to develop a new adaptive fuzzy control approach for nonlinear systems with time-varying delays.The above observations motivate the present research,in which we investigate adaptive fuzzy control for nonlinear sys-tems with time-varying delays.By utilizing the Razumikhin functional approach and fuzzy backstepping technique,a novel direct adaptive fuzzy controller is proposed.The proposed adaptive fuzzy controller guarantees that all the signals of the closed-loop system are bounded,while the system output converges to a small neighborhood of the reference signal.Three main contributions are made in this paper.(1)A Razumikhin lemma-based adaptive fuzzy controller design procedure is first pared with the existing results,the proposed fuzzy controllers can handle systems with both constant and time-varying delays.(2)A novel direct adaptive fuzzy control approach is presented.Unlike the existing approximation-based adaptive control methods,the direct adaptive fuzzy control technique employs fuzzy logic systems to approximate the desired but unknown control input signals rather than the nonlinear functions in dynamic systems.As a result,the direct adaptive fuzzy controllers have a simpler form and fewer tuning parameters.Furthermore,it is more convenient to imple-ment the controllers in practice.(3)Different from the existing adaptive fuzzy control methods,for an n th order system the proposed control scheme contains only n adaption laws,no matter how many fuzzy rule bases are used.Therefore,the com-putation burden is reduced,and the algorithm is easily realized in real-time.In addition,the prior knowledge of the upper bound of the virtual control coefficients is not required.The rest of the paper is organized as follows.Section 2provides preliminaries and the formulation of the problem.Sec-tion 3develops an adaptive fuzzy controller design procedure based on the Razumikhin functional approach and backstep-ping technique.Section 4presents four examples to illustrate the effectiveness of the proposed controllers.These are followed by conclusions in Section 5.2.Problem formulation and preliminariesConsider the nonlinear time-delay system described by_x i ¼f i x i ðÞþg i x i ðÞx i þ1ðt Þþq i x i t Às i ðt ÞðÞðÞ;16i 6n À1;_x n ¼f n x n ðÞþg n x n ðÞu ðt Þþq n x n t Às n ðt ÞðÞðÞ;y ¼x 1;ð1Þwhere x i 2R ;u 2R and y 2R are the state variable,the control input,and the output of the system (1),respectively,s i ðt Þde-notes the time-varying delay in the i th equation, x i ðt Þ¼½x 1ðt Þ;x 2ðt Þ;...;x i ðt Þ T and x ðt Þ¼x n ðt Þ¼½x 1ðt Þ;x 2ðt Þ;...;x n ðt Þ T ;f i ð:Þ;g i ð:Þand q i ð:Þare unknown smooth nonlinear functions satisfying f i ð0Þ¼q i ð0Þ¼0.For t 2½Às i ;0 ;x i ðt Þ¼U i ðt Þ;i ¼1;2;...;n ,where the initial function,U i ðt Þ,is smooth and bounded.For a given reference signal y d ðt Þ,the control objective is to design a direct adaptive controller such that the system’s out-put converges to a small neighborhood of the reference signal and all the variables of the resulting closed-loop system re-main bounded.To this end,the following assumptions are imposed on the system (1).Assumption 1.The signs of g i ð:Þ,for 16i 6n ,are known and there exist positive constants g 0and g M such that for i ¼1;2;...;n ,0<g 06j g i ð:Þj 6g M <1:Apparently,Assumption 1means that functions g i ð:Þ,for 16i 6n ,are either strictly positive or strictly negative.With-out loss of generality,it is further assumed that g i ð:Þ>g 0>0.In addition,in Assumption 1g 0and g M can be unknown.Assumption 2.There exist class-k 1functions Q i ð:Þsuch that for 16i 6n ,q i x i t Às i ðt ÞðÞðÞj j 6Q i k x i t Às i ðt ÞðÞk ðÞ:Remark 1.Let Q ð:Þbe a class-k 1function and a i P 0for 16i 6n .Then Q ðP n i ¼1a i Þ6P ni ¼1Q ðna i Þholds.Moreover,if u ð:Þis a class-k 1function,then Q ðu ð:ÞÞis also a class-k 1function.B.Chen et al./Information Sciences 180(2010)776–792777Assumption 3.The reference signal y d ðt Þand its time derivatives up to the n th order are continuous and bounded.Further-more,it is assumed that there exists a constant d such that j y d ðt Þj 6d .In this paper,we adopt the singleton fuzzifier,prod-uct inference,and the center-defuzzifier to deduce the following fuzzy rules:R i :IF x 1is F i 1and ...and x nis F i nTHEN y is B i ði ¼1;2;...;N Þ;where x ¼½x 1;...;x n T 2R n and y 2R are the input and output of the fuzzy system,respectively,F j i and B i are fuzzy sets in R .The fuzzy inference engine performs a mapping from fuzzy sets in R n to a fuzzy set in R based on the IF–THEN rules in the fuzzy rule base and the compositional rule of inference.The fuzzifier maps a crisp point x ¼½x 1;...;x n T 2R n into a fuzzy set A x in R .The defuzzifier maps a fuzzy set in R to a crisp point in R .Since the strategy of singleton fuzzification,center-average defuzzification and product inference is used,the output of the fuzzy system can be formulated asy x ðÞ¼P Nj ¼1W jQ ni ¼1l F j i x i ðÞP N j ¼1Qni ¼1l F j ix i ðÞh i :ð2Þwhere W j is the point at which fuzzy membership functionl B j ðW j Þachieves its maximum value.It is assumed thatl B j ðW j Þ¼1.Let S j ðx Þ¼Q ni ¼1l Fj iðx i ÞP N j ¼1½Q n i ¼1l F j iðx i Þ ;S ðx Þ¼½S 1ðx Þ;S 2ðx Þ;...;S N ðx Þ T and W ¼½W 1;...;W N T .Then the fuzzy logic system(2)can be rewritten asy ðx Þ¼W T S ðx Þ:ð3ÞIt has been proved in [29]and [30]that if Gaussian functions are used as membership functions,the following lemma holds.Lemma 1.For any continuous function f ðx Þover a compact set X x &R q and any given constant e >0,there exists a fuzzy logicsystem W T S ðx Þsuch that:f ðx Þ¼W T S ðx Þþd ðx Þ;d ðx Þj j 6e ;where d ðx Þis the approximation error.Razumikhin Lemma [10].Suppose f :R ÂC !R n maps R ÂB (B is a bounded subset of C )into a bounded set of R n andconsider the retarded functional differential equation (RFDE)_x¼f ðt ;x t Þ.Suppose that u ðs Þ;v ðs Þand w ðs Þare continuous non-decreasing functions and u ðs Þ!1as s !1.If there are a continuous function V :R ÂR n !R ,a continuous nondecreasing function p :R þ!R þ;p ðs Þ>s for s >0and a constant b 0P 0such that:V t þs ;x t þs ðÞðÞ<p V t ;x t ðÞðÞðÞ;8s 2½Àd ;0 ;u k x k ðÞ6V t ;x t ðÞ6v k x k ðÞ;_Vt ;x t ðÞ6Àw k x k ðÞþb 0;then the solutions of RFDE (f )are uniformly ultimately bounded (UUB).In this case,it is said that the system is UUB stable.Ifb 0¼0,the system is said to be asymptotically stable.In [14],it has been pointed out that for simplicity,the function p ð:Þcan be chosen as p ðs Þ¼ps with p >1.Furthermore,if the Lyapunov function is quadratic,i.e .,V ðx Þ¼x T Px with P being a positive definite matrix,then the conditionV ðt þs;x ðt þs ÞÞ<p ðV ðt ;x ðt ÞÞÞ;8s 2½Àd ;0 can be replaced by the condition k x ðt þs Þk 6q k x ðt Þk ;8s 2½Àd ;0 ,where q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip k M =k m p >1;k m and k M are the minimum and maximum eigenvalues of P ,respectively.In general,a backstepping-based design procedure for an n th order nonlinear strict-feedback system contains n designsteps.At Step i ,the fuzzy network W T i S ðZ i Þwill be used to approximate the unknown function f i ðZ i Þfor i ¼1;2;...;n .Thus,the virtual control signals a i ðZ i Þand control law u are constructed as follows:a i Z i ðÞ¼À12a 2i z i ðt Þ^h i ðt ÞS T Z i ðÞS Z i ðÞ;16i 6n À1;ð4Þu ðt Þ¼À12a 2nz n ðt Þ^h n ðt ÞS T Z n ðÞS Z n ðÞ;ð5Þwhere S ð:Þis the basis function vector,^h i is the estimation of the unknown constant h i which will be specified at Step i ,a i and a n are positive design parameters,andz i ðt Þ¼x i ðt ÞÀa i À1Z i ðÞ;Z i ðt Þ¼ x T i ;^h T ; y T dih i T ;^h ¼^h 1;...;^h n h i T;Z ðt Þ¼z 1;...;z n ;~h T h i T;h ¼h 1;...;h n ½ T ;~h ¼h À^h ;with a 0¼y d and y di ¼½y d ;_y d ;...;y ði Þd T.The i th adaption law is defined by778 B.Chen et al./Information Sciences 180(2010)776–792_^h i ¼r i 2a iz 2i ðt ÞS T Z i ðÞS Z i ðÞÀr i ^h i ðt Þ;i ¼1;...;n ;ð6Þwith r i andr i being positive design parameters.Remark 2.From (6),it is easy to prove that for any given initial condition ^h ðt 0ÞP 0;^h ðt ÞP 0holds for t P t 0.Thus,in thisresearch,it is always assumed that ^h ðt 0ÞP 0.To develop a simple backstepping-based design procedure,we introduce some useful Lemmas,which will be used in thecontroller design procedure.Lemma 2[22].For any u 2R ande >0,the following holds.06j u j Àu tan hu e6d e ;d ¼0:2785:Lemma 3.Let x i ¼½x 1;...;x i T ; z i ðt Þ¼½z 1;...;z i T and ai ¼½y d ;a 1;...;a i À1 T .Then there exists a class-k 1function u ð:Þsuch that:k x i ðt Þk 6u ðk Z ðt ÞkÞþd ;16i 6n ;where d is a constant defined as in Assumption 3and u ðs Þ¼s ða 0þb 0s Þwith a 0and b 0being positive constants.Proof.According to the definition of a j ,there are positive constants c j such that:k x i k 6k z i k þk ai k 6k Z ðt Þk þX i À1j ¼1k a j k þj y d j 6k Z ðt Þk þXi À1j ¼1c j j Z ðt Þkk ^h k þd6k Z ðt Þk þX i À1j ¼1c j k Z ðt Þk k ~h k þk h k þd 6k Z ðt Þk þX n À1i ¼1k Z ðt Þk c j k Z ðt Þk þc j k h k ÀÁþd 6k Z ðt Þk a 0k Z ðt Þk þb 0ðÞþd ¼u k Z ðt Þk ðÞþd ;where u ðs Þ¼s ða 0s þb 0Þwith a 0¼P n À1j ¼1c j and b 0¼1þP n À1j ¼1c j k h k .Obviously,u ð:Þis a class-k 1function.h3.Control design and stability analysisIn this section,a backstepping-based adaptive fuzzy controller design procedure will be developed.For the purpose ofsimplicity,the time variable t is omitted from the corresponding functions and x ðt Às ðt ÞÞis denoted as x ðs Þ.Step 1.Let z 1¼x 1Ày d .It follows from the first differential equation of the system (1)that:_z 1¼f 1 x 1ðÞþg 1 x 1ðÞx 2þq 1 x 1s 1ðÞðÞÀ_yd :Consider a Lyapunov function candidate asV 1¼12z 21þg 02r 1~h 21;where ~h 1¼h 1À^h 1.Differentiating V 1yields:_V 1¼z 1f 1 x 1ðÞþg 1 x 1ðÞx 2À_y d ðÞþz 1q 1 x 1s 1ðÞðÞÀg 01~h 1_^h 1:ð7ÞBy Assumption 2and Lemma 3:z 1q 1ð x 1ðs 1ÞÞ6j z 1j Q 1ðu ðk Z ðs 1ÞkÞþd Þ:ð8ÞSince Q 1ð:Þis a class-k 1function,one has:Q 1ðu ðk Z ðs 1ÞkÞþd Þ6Q 1ð2u ðk Z ðs 1ÞkÞÞþQ 1ð2d Þ:ð9ÞNote that u ð:Þis also a class-k 1function,therefore,1ð:Þ¼Q 1ð2u ð:ÞÞis still a class-k 1function.Furthermore,by usingthe Razumikhin Lemma,there exists a constant q >1such that k Z ðs 1Þk 6q k Z ðt Þk .Consequently,it follows from (9)that:Q 1ðu ðk Z ðs 1ÞkÞþd Þ6Q 1ðq k Z ðt ÞkÞþQ 1ð2d Þ:ð10ÞDue to k Z ðt Þk 6k Z 1ðt Þk þP ni ¼2j z i jwith Z 1¼½z 1;~h T T ,(10)implies that:Q 1ðu ðk Z ðs 1ÞkÞþd Þ6X n i ¼2Q 1ðl 1j z i jÞþQ 1ðl 1k Z 1ðt ÞkÞþQ 1ð2d Þ;with l 1¼q ðn À1þ1Þ.Because 1ð:Þis a class-k 1function,it can be expressed in the form 1ðs Þ¼s /1ðs Þwith /1ð:Þbeing afunction.As a result,the above inequality can be rewritten asB.Chen et al./Information Sciences 180(2010)776–792779Q1ðuðk Zðs1ÞkÞþdÞ6X ni¼2l1j z i j/1ðl1j z i jÞþ1ðl1k1ðtÞkÞþQ1ð2dÞ:ð11ÞSubstituting(11)into(8)and using completion of squares gives:z1q1ð x1ðs1ÞÞ6X ni¼21l21z21þX ni¼21z2i/21ðl1j z i jÞþj z1jð1ðl1k1ðtÞkÞþQ1ð2dÞÞ6X ni¼21l21z21þX ni¼21z2i/21ðl1j z i jÞþz1F1tanhz1F11þdb1;ð12Þwhere Lemma2is used to deal with the term j z1jðQ1ðl1k Z1ðtÞkÞþQ1ð2dÞÞand F1¼Q1ðl1k Z1ðtÞkÞþQ1ð2dÞ.Define a function asf 1ðZ1Þ¼f1À_y dþX ni¼212l21z1þF1tanhz1F1b1;Then,the following inequality can be verified._V 16z1ð f1þg1ð x1Þx2ÞÀg1~h1_^h1þX ni¼21z2i/21ðl1j z i jÞþdb1:ð13ÞApparently,when x2is viewed as the intermediate input,a desired control signal for thefirst subsystem is^a1¼ÀgÀ11ð f1þðk1þ0:5g21Þz1Þ.With^a1,one has:z1ð f1þg1ð x1Þx2Þ¼Àðk1þ0:5g21Þz21þz1g1ðx2Àa1Þþz1g1ða1À^a1Þ:ð14ÞHowever,since^a1consists of the unknown functions f1and gÀ11,it cannot be implemented in practice.By Lemma1,a fuzzy logic system is thus used to approximate^a1such that for any given positive constant e1:^a1¼W T1SðZ1Þþd1ðZ1Þ;k d1k26e21;with d1ðZ1Þindicating the approximation error.Furthermore,it follows that:Àz1g1^a1¼Àz1g1W T1SðZ1Þk W1kk W1kÀ1þz1g1d1ðZ1Þ6g2a21z21gMgk W1k2S TðZ1ÞSðZ1Þþ12a21gMþ12g21z21þ12e21¼g2a21z21h1S TðZ1ÞSðZ1Þþ12a21gMþ12g21z21þ12e21;ð15Þwhere h1¼g Mk W1k2.On the other hand,it follows from Remark2and(4)that:z1g1a16À12a1z21g^h1S TðZ1ÞSðZ1Þ:ð16ÞCombining(14)with(15)and(16)gives:z1ð f1þg1ð x1Þ^a2Þ6Àk1z21þ12ða21gMþe21Þþz1g1z2þg2a1z21~h1S TðZ1ÞSðZ1Þ;ð17Þwith z2¼x2Àa1.Substituting(17)into(13)yields:_V 16Àk1z21þ12a21gMþe21ÀÁþdb1þz1g1z2þgr1~h1r12a21z21S TðZ1ÞSðZ1ÞÀ_^h1þX ni¼212z2i/21ðl1j z i jÞ:ð18ÞRemark3.Unlike the existing indirect adaptive fuzzy control approaches in[28]and the references therein,the ideal control signal,i.e.,^a1,isfirst constructed by using the unknown nonlinear functions and constants.Then,the virtual control a1is designed by using a fuzzy logic system to approximate the unknown control signal^a1directly rather than the unknownfunctions f1ð x1Þand g1ð x1Þ.Therefore,this method is called direct adaptive fuzzy control.Such an idea will be implementedat each step henceforth.As shown in(4)and(5),by using the direct adaptive method,the virtual control signals a i and the real control signal u have simpler form and contain fewer design parameters.In addition,differing from the existing adaptive fuzzy control methods for nonlinear delayed systems,the proposed control laws are suitable for the case of time-varying delays,because the Razumikhin functional is used for stability analysis and control synthesis.Step k.ð26k6nÀ1ÞFirst,it is assumed that after Step kÀ1for the following Lyapunov function:V kÀ1¼V kÀ2þ12z2kÀ1þg2r kÀ1~h2kÀ1;780 B.Chen et al./Information Sciences180(2010)776–792with V0¼0and~h kÀ1¼h kÀ1À^h kÀ1,^h kÀ1being the estimation of h kÀ1¼g Mg0k W kÀ1k2,the following inequality holds:_VkÀ16ÀX kÀ1i¼1k i z2iþX kÀ1i¼112ða2igMþe2iÞþX kÀ1i¼1db iþX kÀ1i¼1gr i~hir i2a2iz2iS TðZ iÞSðZ iÞÀ_^h iþX ni¼k12z2iX kÀ1j¼1ðkÀjÞ/2jðl j j z i jÞþX kÀ1i¼2z i x iÀX nj¼1o a iÀ1o^h i_^hi!þz kÀ1g kÀ1z k;ð19Þwhere z k¼x kÀa kÀ1.Then,consider a Lyapunov function candidate asV k¼V kÀ1þ12z2kþg2r k~h2k:Differentiating V k gives:_V k ¼_V kÀ1þz kðf kÀ_a kÀ1þg k x kþ1Þþz k q kð x kðs kÞÞÀg0rk~hk_^hk;ð20ÞwhereÀz k_a kÀ1¼Àz kX kÀ1j¼1o a kÀ1o x jðf jþg j x jþ1ÞÀz kX kÀ1j¼1o a kÀ1o x jqjð x jðs jÞÞÀz kX kÀ1j¼0o a kÀ1o yðjÞdyðjþ1ÞdÀz kX nj¼1o a kÀ1o^h j_^hj:ð21ÞFor the term z k qkð x kðs kÞÞ,repeating the procedure from(8)–(12)with Z1being replaced by Z k¼½z1;...;z k;~h T T and l1by l k¼qðnÀkþ1Þproduces:z k qk ð x kðs kÞÞ6X ni¼kþ1l2k2z2kþX ni¼kþ112z2i/2kðl k j z i jÞþj z k jðQ kðl k k Z kðtÞkÞþQ kð2dÞÞ:ð22ÞSimilarly,one also has the following inequality:Àz kX kÀ1j¼1o a kÀ1o x jqjð x jðs jÞÞ6X kÀ1j¼1X ni¼kþ1l2k2z2ko a kÀ1o x j!2þX kÀ1j¼1X ni¼kþ112z2i/2jðl k j z i jÞþj z k jX kÀ1j¼1o a kÀ1o x jðj l k k Z kðtÞkÀÁþQ jð2dÞÞ:ð23ÞNow,combining(22)and(23)gives:z k qk ð x kðs kÞÞÀz kX kÀ1j¼1o a kÀ1jqjxjðs jÞÀÁ6X ni¼kþ1l2k z2kþX kÀ1j¼1X ni¼kþ1l2k z2ko a kÀ1j!2þX kj¼1X ni¼kþ11z2i/2jl k z i j jðÞþj z k j F k6X ni¼kþ1l2k2z2kþX kÀ1j¼1X ni¼kþ1l2k2z2ko a kÀ1o x j!2þX kj¼1X ni¼kþ112z2i/2jðl k j z i jÞþz k F k tanhz k F kb kþdb k;where Lemma2is used to handle the term j z k j F k with F k¼Q kðl k k Z kðtÞkÞþQ kð2dÞþP kÀ1j¼1j o a kÀ1jjðQ jðl k k Z k kÞþQ jð2dÞÞ.Define:f k ðZ kÞ¼f kþz kÀ1g kÀ1Àz kX kÀ1j¼1o a kÀ1jðf jþg j x jþ1ÞÀz kX kÀ1j¼0o a kÀ1o yðjÞdyðjþ1ÞdþF k tanhz k F kkþ1X ni¼kþ1l2kz k þX kÀ1j¼1X ni¼kþ1l2k2z ko a kÀ1o x j!2þ12z kX kÀ1j¼1ðkÀjÞ/2jðl j j z k jÞÀx kðZ kÞ;where yðjÞd denotes the j th order derivative of yd.Substituting(19),(22)and(23)and f k into(20)produces:_V k 6ÀX kÀ1i¼1k i z2iþX kÀ1i¼112a2igMþe2iÀÁþX ki¼1db iþX kÀ1i¼1gr i~hir i2aiz2iS TðZ iÞSðZ iÞÀ_^h iþX ki¼2z i x iÀX nj¼1o a iÀ1o^h j_^hj!þX ni¼kþ112z2iX kj¼1ðkÀjþ1Þ/2jðl j j z i jÞþz kð f kþg k x kþ1Þ:ð24ÞAlternatively,viewing x kþ1as the virtual control input and taking the virtual control input signal as^a k¼ÀgÀ1kð f kþðk kþ0:5g2kÞz kÞ,we have:z kð f kþg k x kþ1Þ¼Àðk kþ0:5g2k Þz2kþz k g k z kþ1þz k g kða kÀ^a kÞ:By Lemma1,a fuzzy logic system is again utilized to approximate the unknown^a k such that for any given constant e k>0:^a k¼W Tk SðZ kÞþd kðZ kÞ;B.Chen et al./Information Sciences180(2010)776–792781。
Observer-based adaptive fuzzy backstepping dynamic surface control for a class of non-linear ....
Published in IET Control Theory and Applications Received on 29th October 2010 Revised on 10th February 2011 doi: 10.1049/iet-cta.2010.0632
1426 & The Institution of Engineering and Technology 2011
fuzzy or neural backstepping control approaches suffer from the problem of ‘explosion of complexity’. The ‘explosion of complexity’ is caused by repeated differentiations of some non-linear functions at each step within the conventional backstepping technique. As a result, the complexity of a controller drastically grows as the order of the system increases. Recently, the dynamic surface control (DSC) technique has been proposed to avoid this problem by introducing a first-order low-pass filter at each step of the conventional backstepping design procedure [17 – 20]. In [17], a robust control is studied for non-linear systems in strict feedback form. In [18], a simplified DSC algorithm is developed for non-linear systems in parametric strict feedback form. In [19, 20], adaptive DSC approaches are proposed for adaptive tracking control of a class of canonical-form SISO systems with and without time-delays, respectively. However, the above-mentioned DSC approaches have two limitations. One is that these approaches require that the controlled non-linear dynamics models be known exactly or the unknown non-linear functions can be linearly parameterised. If those kinds of knowledge are not available a priori, these adaptive backstepping controllers cannot be applied. The other is that they require that all the states be available for measurements. To cope with the problem of ‘explosion of complexity’ inherent in the existing adaptive fuzzy or neural backstepping control schemes, an adaptive neural backstepping control
压电执行器动态迟滞建模与LQG最优控制器设计
压电执行器动态迟滞建模与LQG最优控制器设计佚名【摘要】为提高空间天文望远镜稳像系统中压电快摆镜(Fast Steering Mirror,FSM)的动态性能,对压电执行器(Piezoelectric Actuator,PZT)动态迟滞补偿和控制进行研究.鉴于基于广义Play算子Prandtl-Ishlinskii(PI)模型的求逆复杂性和迟滞曲线的非对称性,构造一种基于广义Stop算子PI逆模型来补偿压电执行器迟滞非线性.采用Hammerstein模型对压电执行器动态迟滞特性进行建模,以广义PI模型和自回归遍历模型(Auto-regressive Exogenous Model,ARX)分别表征Hammerstein迟滞模型中的静态非线性和率相关性,并针对迟滞率相关模型不确定性问题,提出一种前馈补偿和线性二次型Gauss最优控制算法(Linear Quadratic Gaussian,LQG)相结合的复合控制策略.利用自适应差分进化算法(Adaptive Differential Evolution algorithm,ADE)辨识和整定模型及控制器参数.实验结果表明:该动态迟滞模型能够有效描述1～100 Hz频率范围内压电执行器迟滞曲线,拟合均方根误差为0.077 1μm(@1 Hz)～0.512 3 μm(@100 Hz),相对误差为0.31％(@1 Hz)～2.09％(@100 Hz);实时跟踪幅值为24.5 μm的变频目标位移,LQG控制算法的跟踪精度相比于直接前馈控制和PID控制分别提高48.6％和27.02％.【期刊名称】《光学精密工程》【年(卷),期】2018(026)011【总页数】10页(P2744-2753)【关键词】压电执行器;动态迟滞特性;Hammerstein模型;广义Stop算子;LQG最优控制【正文语种】中文【中图分类】TL3611 引言空间天文望远镜主要通过快摆镜(Fast Steering Mirror，FSM)机构补偿卫星平台不稳定引起的像移，以实现稳像从而提高探测精度[1]。
Robust Control
Robust ControlRobust control is a branch of control theory that deals with the design of control systems that can operate effectively in the presence of uncertainties and disturbances. These uncertainties can arise from various sources, such as modeling errors, parameter variations, and external disturbances. The goal of robust control is to ensure that the system remains stable and performs satisfactorily despite these uncertainties.One of the key challenges in robust control is to design controllers that can handle uncertainties without sacrificing performance. This requires a careful balance between robustness and performance. On the one hand, a controller that is too robust may be overly conservative and result in poor performance. On the other hand, a controller that is too aggressive may be sensitive to uncertainties and result in instability or poor performance.To address this challenge, various techniques have been developed for robust control. These include H-infinity control, mu-synthesis, and robust model predictive control. These techniques differ in their approach and assumptions, but they all aim to achieve robustness while maintaining performance.H-infinity control is a popular technique for robust control that is based on the concept of optimal control. The goal of H-infinity control is to minimize the worst-case performance of the system subject to certain constraints. This approach is particularly useful when the uncertainties are bounded and can be modeled as a norm-bounded perturbation.Mu-synthesis is another technique for robust control that is based on the concept of structured uncertainty. The idea behind mu-synthesis is to design a controller that is robust to uncertainties that have a known structure. This approach is particularly useful when the uncertainties are not bounded and cannot be modeled as a norm-bounded perturbation.Robust model predictive control is a relatively new technique for robust control that is based on the concept of predictive control. The idea behind robust model predictive control is to design a controller that can handle uncertainties in the future by optimizing a sequence of control actions over a finite time horizon. This approach is particularly useful when the uncertainties are time-varying and cannot be modeled as a fixed perturbation.In addition to these techniques, there are also various tools and software packages available for robust control design. These include MATLAB, Simulink, and Robust Control Toolbox. These tools can be used to simulate and analyze the performance of robust control systems under various scenarios and uncertainties.In conclusion, robust control is an important area of control theory that deals with the design of control systems that can operate effectively in the presence of uncertainties and disturbances. This requires a careful balance between robustness and performance, which can be achieved through various techniques and tools. By incorporating robust control into the design of control systems, we can ensure that these systems remain stable and perform satisfactorily under a wide range of conditions.。
一种新的基于自适应观测器中立时滞系统故障估计的设计方法
ACTA AUTOMATICA SINICA
January, 2009
Hale Waihona Puke A New Criterion of Fault Estimation for Neutral Delay Systems Using Adaptive Observer
ZHANG Ke1 JIANG Bin1 SHUMSKY Alexey2
Control engineers are faced with increasingly complex systems where dependability considerations are sometimes more important than performance. Sensor, actuator, or process failures may drastically change the system behavior, resulting in performance degradation or even instability. Therefore, fault detection and isolation (FDI) algorithms and their applications to a wide range of industrial and commercial processes have been the subjects of intensive investigation over the past two decades. In order to improve efficiency, the reliability can be achieved by fault tolerant control (FTC), which relies on early detection of faults. So FDI has become a popular topic and has received considerable attention[1−3] during the past two decades. In general, fault tolerance can be achieved in two ways: 1) passively, using feedback control laws that are robust with respect to possible system faults[4−5] , or 2) actively, using an FDI module and accommodation techniques. Active FTC is obtained by fault accommodation[6−7] , which controls the faulty system, or by system reconfiguration, which controls the healthy part of the system. Therefore, in fault accommodation, FDI module must detect and isolate the faults, as well as estimate them. FDI is the first step in fault accommodation to monitor the system and determine the location of the fault. Then, fault estimation is utilized to on-line determine the magnitude of the fault. Finally, using the obtained fault information, an additive controller can be designed to compensate the effect caused by the fault. On the other hand, it is well known that time-delay frequently occurs in many practical systems, such as manufacturing, telecommunication, and economic systems. There are two types of time delay systems: retarded and neutral. The retarded type contains delays only in its states, whereas the neutral type contains delays both in its states and in the derivatives of the states. Neutral type delay systems can be found in such places as population ecology, distributed networks containing lossless transmission lines, heat exchangers, and robots in contact with rigid environments. Therefore, in recent years, much attention has been drawn to the development of stability conditions of neutral delay systems[8−9] . Generally speaking, delayReceived October 31, 2007; in revised form April 14, 2008 Supported by National Natural Science Foundation of China (60574083, 60811120024), Graduate Innovation Research Foundation of Jiangsu Province (CX08B-090Z), and Doctoral Innovation Foundation of Nanjing University of Aeronautics and Astronautics (BCXJ08-03) 1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China 2. Institute for Marine Technology Problems, Far Eastern Branch of Russian Academy of Sciences, Vladivostok 690950, Russia DOI: 10.3724/SP.J.1004.2009.00085
Robust control for a class of uncertain nonlinear
to the best of our knowledge, little attention has been focussed on treating the robust H 1 control problem for uncertain systems with the perturbation of dynamic uncertainties. Robust H 1 control designs have been proposed for uncertain linear systems with norm-bounded uncertainties by Xie and Souza [15] and Xie et al. [16]. Shen and Tamura [17] presented a solution of a robust H 1 control problem for an affine nonlinear system with gain-bounded uncertainty. Xie and Su [18] designed a nonlinear static state-feedback controller such that the closed-loop system achieves a finite L2 -gain performance for a class of uncertain nonlinear systems. Lee and Chen [19] investigated the problem of robust H 1 output regulation for a class of feedback linearisable nonlinear systems under matched linearly parametric uncertainties. Pan and Bas ¸ ar [20] developed a systematic procedure to obtain robust adaptive controllers that achieve asymptotic tracking and H 1 disturbance attenuation for a class of parametric strictfeedback nonlinear SISO systems. However, the adaptive H 1 control schemes developed by Lee and Chen [19] and Pan and Bas ¸ar [20] can only be applied to simple classes of nonlinear SISO systems that involve constant parametric uncertainties. On the other hand, Battilotti and Lanari [21] solved the disturbance attenuation problem with global stability for rigid and elastic jointed robots but only in the presence of unknown constant parameters. Dynamic state-feedback controllers that combine the adaptive and H 1 control techniques were developed. We now address the problem of designing robust H 1 controls for a large class of uncertain nonlinear time-varying systems perturbed by parametric uncertainties, unmodelled matched and mismatched perturbations, and external disturbances. The nonlinear H 1 control technique combined with the adaptive control technique and the VSC technique is to construct a hybrid robust adaptive stabilisation controller such that a H 1 control is achieved. This control is in the sense that a prescribed H 1 performance from the external disturbance to the state variable can be guaranteed for the closed-loop system regardless of the parametric type of the uncertainties and unmodelled perturbations. In order to counteract the effect of unmodelled perturbations on the H 1 control a modified Hamilton-Jacobi-like inequality must be solved which incorporates the bounded values of the a class of uncertain nonlinear time-varying systems and its application
具有时变输入时滞的不确定T-S模糊系统的鲁棒稳定性
88
ACTA AUTOMATICA SINICA Then
r
Vol. 34
[13]. This paper is organized as follows. In Section 1, we give a general description of uncertain T-S fuzzy systems with time-varying input delay and review some results to be used in this paper. The sufficient conditions for robust stability of uncertain T-S fuzzy systems with time-varying input delay and time-invariant input delay are proposed in Section 2. In Section 3, a numeric example is given to show that the proposed results are effective and less conservative. Some conclusions are drawn in Section 4.
Input delays are often encountered in various practical control systems, such as chemical process, biological systems and economical systems. The existence of input delays may deteriorate the performance of the system and make it difficult to stabilize the resulting closed-loop system rather. As a result, the control design problem of input delayed systems has attracted considerable attention recently[1−5] . Since Takagi and Sugeno[6] proposed the Takagi-Sugeno fuzzy model in 1985, fuzzy model control has been extensively studied[7−13] because Takagi-Sugeno (T-S) fuzzy models can provide an effective representation of complex nonlinear systems in terms of fuzzy sets, and fuzzy reasoning has been applied to a set of linear input-output submodels[9] . Up to now, the stabilization problem of uncertain T-S fuzzy systems with time-varying input delay has not been fully investigated, and it remains a challenging problem. The approaches to solving the control systems with time delay can be classified into two classes: one is LyapunovKrasovskii functional method, and the other is LyapunovRazumikhin functional method. The stabilization problem is thus reduced to the problem of finding solutions to Lyapunov or Riccati equations, or solving linear matrix inequalities (LMIs). In [13], the Lyapunov-Razumikhin functional technique makes some improvements such as avoiding the constraint that the upper bound of the timederivative of the time delay must be less than one, however, it requires some other supplementary assumptions in the process of completing the proof of the theorem, therefore the results obtained in [13] are still too conservative. In this paper, by constructing an appropriate LyapunovKrasovskii functional, we provide sufficient conditions for robust stability of uncertain T-S fuzzy systems with timevarying input delay, without using any property of the timederivative of the input delay. The results obtained in the present paper are less conservative than that obtained in
Adaptive-fuzzy-tracking-control-for-stochastic-nonli_2015_Applied-Mathematic
/10.1016/j.amc.2014.12.104 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
J. Li, H. Yue / Applied Mathematics and Computation 256 (2015) 514–528
Apion 256 (2015) 514–528
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515
such as [12,18]. Thus, if not under the risk-sensitive cost criterion, how to design the stochastic tracking controller by using the quadratic Lyapunov is worth studying. On the other hand, fuzzy logic systems (FLS) and neural networks (NNs) have been proved to be very useful tools for solving the control problem of uncertain systems. However, compared with NNs, the main advantage of FLS is that it can combine some experience and knowledge from designers or experts [9]. These experience and knowledge can initiate the estimated parameters in order to make them close to their optimal values, which is very important to improve the control transient performance. Therefore, this paper will use the FLS to deal with the uncertain nonlinear systems. Another challenging problem in real engineering systems is that time delays are frequently encountered, and the existence of time delays usually becomes the source of instability and degrading performance of systems. Therefore, the stability analysis and controller synthesis of nonlinear systems with time-delay are important. With the development of adaptive control and backstepping designs in nonlinear systems [22], recently, for nonlinear systems with constant delays [25,26], several fuzzy (neural) adaptive control schemes have been reported by combining Lyapunov–Krasovskii functionals and the backstepping technique. However, for the nonlinear systems with unknown time-varying delays, how to design the adaptive fuzzy controller by using the backstepping technique and the appropriate Lyapunov–Krasovskii functionals is a challenging subject. Based on the above observation, in this paper, the problem of output tracking is revisited for stochastic strict-feedback nonlinear systems with unknown time-varying delays using fuzzy control. Under the approximately assumptions, quadratic functions are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms, while the Lyapunov–Krasovskii functionals are constructed to compensate the unknown time-varying delays terms. Then, the FLS are employed to approximate the unknown nonlinear functions. Finally, the adaptive backstepping approach is utilized to construct the fuzzy controller. The two main advantages of the scheme are that: (1) In this paper, quadratic functions are used as Lyapunov functions to analyze the stability of the systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. Thus, the long-standing obstacle for stochastic systems control by quadratic Lyapunov function is overcome. (2) The developed control scheme can handle stochastic nonlinear systems with time-varying delays. It can be proven that all the signals in the closed loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense. Simulation results are provided to show the effectiveness of the proposed approach. 2. Preliminaries 2.1. Notations (1) C ð½Àd; 0; Rn Þ denotes the space of continuous Rn valued functions on ½Àd; 0 endowed with the norm kÁk defined by n kf k ¼ supx2½Àd;0 jf ðxÞj for f 2 C ð½Àd; 0; Rn Þ, and C b F 0  ð½Àd; 0; R Þ denotes the family of all F 0 -measurable bounded C ð½Àd; 0; Rn Þ-valued random variables n ¼ nðhÞ : Àd 6 h 6 0. (2) C i denotes the set of all functions with continuous ith partial derivatives; C 2;1 ðRn  ½Àd; 1; Rþ Þ denotes the family of all nonnegative functions V ðx; t Þ on Rn  ½Àd; 1Þ, which are C 2 in x and C 1 in t; and C 2;1 denotes the family of all functions, which are C 2 in the first argument and C 1 in the second argument. (3) K denotes the set of all functions: Rþ ! Rþ , which are continuous, strictly increasing, and vanish at zero; K 1 denotes the set of all functions that are of class K and unbounded. 2.2. Stochastic Stability Consider an n-dimensional stochastic time-delay system
具有广泛不确定性的模糊系统的鲁棒控制英文
控制工程C ontrol Engi neering of China Sep 12008V ol .15,S 12008年9月第15卷增刊文章编号:167127848(2008)S 120066206 收稿日期22; 收修定稿日期22 作者简介L F (62),,D q ,j ,,,f zzy ;S B K (2),,2f 李凤江(62),男,黑龙江大庆人,工程师,主要研究方向为鲁棒控制、模糊控制等;苏宝库(2),男,教授。
Delay 2dependent R obust H ∞Contr ol of FuzzySystem w ith G eneral U ncer ta intyLI Feng 2jiang 1,G ONG Cheng 2,SU Bao 2ku 2,(1.3rd Oil 2extraction Plant ,Daqi ng Oil field Limited Liability C om pany ,Daqing 163453,Chi na ;2.S pace C ontrol and Inertial Technology Res earch C enter ,Harbi n Ins tit ute of Technol ogy ,Harbin 150001,China)Abstract :T he delay 2dependent robus t control problem of T 2S f u zzy time 2delay s ystem w ith a generalized class o f uncertainty is pres ented.Both n orm b ounded uncertainty and conv ex polyhedral un certainty are in 2cluded in the generalized un certainty.A delay 2dependen t stability condition is derived by Lyapun ov 2K ra 2sovskii functi onal meth od.Based on the s tability condition ,a delay 2dependen t robust H ∞con troller ,wh ich guaran tees n ot only th e robust asy mptotic s tability of the uncertain system ,but als o the prescribed H ∞atten 2uation level for all admiss ible un certainties.The numerical ex ample sh ows the effectiveness o f the prop osed method.K ey w or ds :robus t H ∞control ;delay 2dependent ;polyhedral uncertain ty;norm b ounded uncertainty;fuzzy sys temCLC number :T P 273 Document code :A具有广泛不确定性的模糊系统的鲁棒控制李凤江1,巩 诚2,苏宝库2(1.大庆油田有限责任公司采油三厂,黑龙江大庆 163453;2.哈尔滨工业大学空间控制与惯性技术研究中心,黑龙江哈尔滨 150001)摘 要:研究了一类具有广泛不确定性的T 2S 模糊时滞系统的时滞依靠鲁棒H ∞控制问题。
Robust guaranteed cost control for singular Markovian jump systems with time-varying delay_2012
Robust guaranteed cost control for singular Markovian jump systemswith time-varying delayYueying Wang,Quanbao Wang n,Pingfang Zhou,Dengping DuanSchool of Aeronautics and Astronautics,Shanghai Jiaotong University,Dongchuan Road No.800,MinHang District,Shanghai200240,PR Chinaa r t i c l e i n f oArticle history:Received28November2011Accepted19April2012Available online19May2012Keywords:Singular Markovian jump systemGuaranteed cost controlDelay-range-dependent stabilityLinear matrix inequality(LMI)a b s t r a c tThis paper is concerned with the guaranteed cost control for continuous-time singular Markovian jumpsystems with time-varying delay.Without using the free weighting matrices method,a delay-range-dependent condition is derived in terms of strict linear matrix inequality(LMI),which guarantees thatthe singular system is regular,impulse free and mean-square exponentially stable with an H Nperformance.Based on this,the existence condition of the guaranteed cost state feedback controlleris proposed.A numerical example is given to illustrate the effectiveness and less conservatism of theproposed design method.&2012ISA.Published by Elsevier Ltd.All rights reserved.1.IntroductionOver the past years,considerable attention has been devotedto Markovian jump systems because this class of systems is morepowerful and appropriate to model varieties of systems mainlythose with abrupt changes in their structure.For more details onthis subject,we refer the readers to[1–5]and the referencestherein.Very recently,some efforts have been addressed to thestudy of singular Markovian jump systems.The study for suchsystems is more complicated than that for regular ones,becausethe singular systems usually have three types of modes,namely,finite-dynamic modes,impulsive modes and non-dynamic modes,while the latter two do not appear in state-space systems.Theresults related to singular Markovian jump systems can be foundin[5–9],and the references therein.On the other hand,guaranteed cost control is one of theapproaches to robustly stabilize dynamical uncertain systems andits main idea is to design a control system which is not only stable butalso guarantees an adequate level of performance.A great number ofresults on guaranteed cost control for all kinds of systems have beenreported in the literature;see,for example,[10–14],and thereferences therein.Up to date,three results have been reported onguaranteed cost control for singular Markovian jump systems.Fordiscrete-time singular Markovian jump systems[15],investigated theproblem of guaranteed cost control for singular Markovian jumpsystems with indefinite quardratic cost.For continuous-time singularMarkovian jump systems,based on LMI,some sufficient conditionsfor optimal guaranteed cost control are developed.It should bepointed out that no time delay was considered in[16],which oftencause instability and poor performance of control systems[17].Performing free weighting matrices method,the delay-dependentguaranteed cost control problem was discussed in[18].However,thedelay-dependent conditions derived in[18]are formulated in termsof nonstrict LMIs whose solutions are difficult to calculate sinceequality constraints are often fragile and usually not met perfectly.Inaddition,the time delay of[18]is time-invariant and the informationof the lower bound of delay was not considered,which limit thescope of applications.To the best of our knowledge,the problem ofdelay-range-dependent guaranteed cost control for continuous-timesingular Markovian jump systems with time-varying delays has notbeen fully investigated,which motivates the present study.Recently,some researchers have pointed out that too many free-weightingmatrices introduced in the theoretical derivation sometimes cannotreduce the conservatism of the obtained results,on the contrary,theymake the system analysis and synthesis complicated[19].Therefore,how to overcome the aforementioned disadvantages of free-weight-ing matrices method is also an important research topic of this paper.In this paper,without resorting to the free-weighting matricesmethod,we proposed a delay-range-dependent sufficient conditionfor the mean-square exponential admissibility with an H N perfor-mance of the system in terms of strict LMI.Then,the guaranteed costcontroller is established to satisfy the desired performance.Finally,anumerical example is presented to illustrate the effectiveness and lessconservatism of the proposed design method.2.Problem formulation and preliminariesLet{r t,t Z0}be a continuous-time Markov process with a rightcontinuous trajectory taking values in afinite set S¼{1,2,y,s}Contents lists available at SciVerse ScienceDirectjournal homepage:/locate/isatransISA Transactions0019-0578/$-see front matter&2012ISA.Published by Elsevier Ltd.All rights reserved./10.1016/j.isatra.2012.04.005n Corresponding author.Tel.:þ862134206743;fax:þ862134206630.E-mail address:quanbaowang@(Q.Wang).ISA Transactions51(2012)559–565with transition probability matrix L ¼{p ij }given byP ½r t þD t ¼j 9r t ¼i ¼p ij D þo ðD Þif j a i1þp ij D þo ðD Þif j ¼i(where lim D -0o ðD Þ=D ¼0;p ij 40,j a i and p ii ¼ÀPj a i p ij for each i A S .Fix a probability space (X ,F ,P )and consider the singular Marko-vian jump systems subjected to input nonlinearity as follows:E _x ðt Þ¼A ðr t Þx ðt ÞþA d ðr t Þx ðt Àd ðt ÞÞþBu ðt ÞþB w ðr t Þw ðt Þz ðt Þ¼C ðr t Þx ðt ÞþC d ðr t Þx ðt Àd ðt ÞÞþD ðr t Þw ðt Þx ðt Þ¼j ðt Þ,8t A ½Àd 2,08><>:ð1Þwhere x (t )A R n is the state vector,z (t )A R s is the controlled output,w (t )A R p is the disturbance input which belongs to L 2½0,1Þ,E A R n Ân may be singular,we assume that rank E ¼r r n.f (u )is a continuous function vector,j ðt ÞA C n ,d 2is a compatible vector valued initial function.A (r t ),A d (r t ),B (r t ),B w (r t ),C (r t ),C d (r t )and D (r t )are real constant matrices of appropriate dimensions.The time delays d (t )is a time-varying continuous function that satisfies0r d 1r d ðt Þr d 2,0r _dðt Þr m o 1,8i A Sð2Þwhere d 1,d 2and m are the constants.In this paper,we consider a linear controller of form u ðt Þ¼K ðr t Þx ðt Þð3Þwhere K (r t )is the nominal controller gain.Applying the controller in (3)in system (1)yields the following closed-loop system:E _x ðt Þ¼r t Þx ðt ÞþA d ðr t Þx ðt Àd ðt ÞÞþB w ðr t Þw ðt Þz ðt Þ¼C ðr t Þx ðt ÞþC d ðr t Þx ðt Àd ðt ÞÞþD ðr t Þw ðt Þx ðt Þ¼j ðt Þ,8t A ½Àd 2,08><>:ð4Þwhere A ðr t Þ¼A ðr t ÞþB ðr t ÞK ðr t Þ.In association with system (1),we define the following guaranteed cost function as follows:J ¼Z 1ðx T ðt Þ^Q ðr t Þx ðt Þþu T ðt Þ^W ðr t Þu ðt ÞÞdt ð5Þwhere ^Qðr t Þ40and ^W ðr t Þ40are given constant.For notational simplicity,in the sequel,for each possible r t A i ,i A S ,a matrix M (r t )will be denoted by M i ;for example,ðr t Þis denoted by A i ,A d (r t )is denoted by A di ,and so on.Throughout this paper,we shall use the following concepts.Definition 1.[7](1)Singular Markovian jump system (4)with w (t )¼0is said to beregular and impulse free for any time delay d (t )satisfying (2),if the pairs ðE ,A i Þand ðE ,A i þA di Þare regular and impulse free for every i A S .(2)Singular Markovian jump system (4)with w (t )¼0is said to bemean-square exponentially stable,if there exist scalars a 40and b 40such that E f :x ðt Þ:2g r a e Àb t :j ðt Þ:2d 2,t 40.(3)Singular Markovian jump system (4)with w (t )¼0is said to bemean-square exponentially admissible,if it is regular,impulse free and mean-square exponentially stable.Definition 2.Given a scalar g 40,the singular Markovian jump system (4)is said to be mean-square exponentially admissible with an H N performance level g ,if it is mean-square exponen-tially admissible with w (t )¼0,and under zero condition,fornonzero w (t )A L 2[0,N ],it holds thatJ ðt Þ¼E Z 10½z T ðs Þz ðs ÞÀg 2w Tðs Þw ðs Þ ds &'o 03.Main resultsIn this section,we shall give a solution to the delay-range-dependent H N guaranteed cost control problem.Initially,the following theorem is proposed,which will play a key role in solving the problem.Theorem 1.For given scalars g 40,0r d 1o d 2and 0r m o 1,the singular Markovian jump system (4)is mean-square exponen-tially admissible with an H N performance level g for any time delay d (t )satisfying (2),if there exist symmetric positive-definite matrices Q k ,W k ,k ¼1,2,3,P i ,a nonsingular matrix Y i and matrices R ,S such that for each i A S ,X i 11X i 12E T W 3E0X i 15d 12d 2A Ti W 1d 12A Ti W 2d 1A Ti W 3C T in X i 22E T W 2E X i 240d 12d 2A T di W 1d 12A T di W 2d 1A T di W 3C T di n n X i 33E TW 2E 00n n n X i 4400000n n n n Àg 2Id 12d 2B T wi W 1d 12B T wi W 2d 1B T wi W 3D T i n n n n n ÀW 1000n n n n n n ÀW 200n n n n n n n ÀW 30nnnnnnnnÀI26666666666666666643777777777777777775o 0ð6Þwhere d 12¼d 2Àd 1X i 11¼A Ti ðP i E þR T Y i S T ÞþðE T P i þSY T i R ÞA iþX s j ¼1p ij E TP j E þX 3k ¼1Q k Àd 212E T W 1E ÀE T W 3EX i 12¼ðE T P i þSY T i R ÞA di þd 212E T W 1EX i 15¼ðE T P i þSY T i R ÞB wiX i 22¼Àð1Àm ÞQ 3Àðd 212þd 12d 2ÞE T W 1E À2E TW 2EX i 24¼d 12d 2E T W 1E þE T W 2E X i 33¼ÀQ 1ÀE T W 2E ÀE T W 3E X i 44¼ÀQ 2Àd 12d 2E T W 1E ÀE T W 2Ewhere R and S are the matrices satisfying RE ¼0and ES ¼0.Proof.Since rank E ¼r r n ,there exist nonsingular matrices G and H such thatGEH ¼I r 000 !ð7ÞDenoteGA i H ¼A i 11A i 12A i 21A i 22"#,P i ¼ðP i E þR TY i S TÞ,GÀTP i H ¼P i 11P i 12P i 21P i 22"#ð8ÞIt is easy to see that E T P i ¼P Ti E .From (7)and (8),we can deduce that P i 12¼0for each i A S .Pre-multiplying and post-multiplyingX i 11o 0by H T and H ,respectively,we have A T i 22P i 22þP Ti 22A i 22o 0,which implies A i 22is nonsingular and the pair ðE ,i Þis regular and impulse free for each i A S .On the other hand,pre-multiplying andY.Wang et al./ISA Transactions 51(2012)559–565560post-multiplyingX i11X i12n X i22"#o0byII!TandII!,respectively,we haveðiþA diÞTðP i EþR T Y i S TÞþðE T P iþSY Ti RÞðiþA diÞþX sj¼1p ij E T P j E o0This implies that the pairðE,iþA diÞis regular and impulse free for each i A S according to Theorem10.1of[20].Then,by Definition1,singular Markovian jump system(4)is mean-square regular and impulse free for any time delay d(t)satisfying(2).Next,we shall show the mean-square exponential stability of the system(4).Define a new process{(x t,r t),t Z0}by x t(s)¼x(tþs),À2d2r s r0,then{(x t,r t),t Z0}is a Markov process with initial state(j(Á),r0).Now,choose a stochastic Lyapunov function candidate for the system(4)with w(t)¼0,Vðx t,r t,tÞ¼x TðtÞE T Pðr tÞExðtÞþX2k¼1Z ttÀd kx TðaÞQ k xðaÞd aþZ ttÀdðtÞx TðaÞQ3xðaÞd aþd212d2Z0Àd2Z ttþb_x TðaÞE T W1E_xðaÞd a d bþd12ZÀd1Àd2Z ttþb_x TðaÞE T W2E_xðaÞd a d bþd1Z0Àd1Z ttþb_x TðaÞE T W3E_xðaÞd a d bð9ÞLet L be the weak infinitesimal generator of the random process{x t,r t}.Then,for each i A S,we haveL Vðx t,i,tÞr2x TðtÞE T P i E_xðtÞþx TðtÞX sj¼1p ij E T P j E2 435xðtÞþX3k¼1x TðtÞQ k xðtÞÀX2k¼1x TðtÀd kÞQ k xðtÀd kÞÀð1ÀmÞx TðtÀdðtÞÞQ3xðtÀdðtÞÞþ_x TðtÞE Tðd212d22W1þd212W2þd21W3ÞE_xðtÞÀd212d2Z ttÀd2_x TðaÞE T W1E_xðaÞd aÀd12Z tÀd1tÀd2_x TðaÞE T W2E_xðaÞd aÀd1Z ttÀd1_x TðaÞE T W3E_xðaÞd að10ÞAccording to Jensen integral inequality[21],the following equation is true:Àd212d2Z ttÀd2_x TðaÞE T W1E_xðaÞd a¼Àd212d2Z ttÀdðtÞ_x TðaÞE T W1E_xðaÞd aÀd212d2Z tÀdðtÞtÀd2_x TðaÞE T W1E_xðaÞd arÀd212dðtÞZ ttÀdðtÞ_x TðaÞE T W1E_xðaÞd aÀðd2ÀdðtÞÞd12d2Z tÀdðtÞtÀd2_x TðaÞE T W1E_xðaÞd arÀd212Z ttÀdðtÞ_x TðaÞE T d a!W1Z ttÀdðtÞE_xðaÞd a!Àd12d2Z tÀdðtÞtÀd2_x TðaÞE T d a"#W1Z tÀdðtÞtÀd2E_xðaÞd a"#¼xðtÞxðtÀdðtÞÞxðtÀd2Þ264375TÀd212E T W1E d212E T W1E0nÀðd212þd12d2ÞE T W1E d12d2E T W1En nÀd12d2E T W1E26643775ÂxðtÞxðtÀdðtÞÞxðtÀd2Þ264375Similarly,we also getÀd12Z tÀd1tÀd2_x TðaÞE T W2E_xðaÞd arxðtÀdðtÞÞxðtÀd1ÞxðtÀd2Þ264375TÀ2E T W2E E T W2E E T W2EnÀE T W2E0n nÀE T W2E264375T xðtÀdðtÞÞxðtÀd1ÞxðtÀd2Þ264375Àd1Z ttÀd1_x TðaÞE T W3E_xðaÞd a¼xðtÞxðtÀd1Þ"#TÀE T W3E E T W3EnÀE T W3E"#xðtÞxðtÀd1Þ"#Noting RE¼0and ES¼0,we can deducex TðtÞSY TiRE_xðtÞþ_x TðtÞE T R T Y i S T¼0ð11ÞThus,by Schur complement and combining with(11),we havefor each i A SL Vðx t,i,tÞr x TðtÞY i x TðtÞwhereY i¼X i11X i12E T W3E0n X i22E T W2E X i24n n X i33E T W2En n n X i44266664377775þA T iA Tdi266664377775ðd212d22W1þd212W2þd21W3ÞA T iA Tdi266664377775TxðtÞ¼½x TðtÞx TðtÀdðtÞÞx TðtÀd1Þx TðtÀd2Þ TFrom(6)and by Schur complement,there exists a scalar r40such that for each i A S,L Vðx t,i,tÞrÀr:xðtÞ:2Since A i22is non-singular for each i A S,we set G¼I r A i12AÀ1i220AÀ1i22"#G.It is easy to see thatGEH¼I r000!,GA i H¼^Ai11^Ai21I"#where^A i11¼A i11ÀA i12AÀ1i22A i21and^A i21¼AÀ1i22A i21.DefinediH¼A id1A id2A id3A id4"#,ÀT i H¼^Pi11^Pi12^Pi21^Pi22"#,zðtÞ¼z1ðtÞz2ðtÞ"#¼HÀ1xðtÞThen,from E T P i¼P TiE,we can deduce that^P i1140and^P i12¼0for each i A S.Y.Wang et al./ISA Transactions51(2012)559–565561The singular Markovian jump system (4)with w (t )¼0is equivalent to _z 1ðt Þ¼^A i 11z 1ðt ÞþA id 1z 1ðt Àd ðt ÞÞþA id 2z 2ðt Àd ðt ÞÞÀz 2ðt Þ¼^A i 21z 1ðt ÞþA id 3z 1ðt Àd ðt ÞÞþA id 4z 2ðt Àd ðt ÞÞz ðt Þ¼c ðt Þ¼H À1j ðt Þ,t A ½Àd 2,08>><>>:ð12ÞWe define a new function as V ðx t ,r t ,t Þ¼e e t V ðx t ,r t ,t Þwhere e 40.By Dynkin’s formula,we get that for each i A S ,E f V ðx t ,r t ,t Þg r E f V ðx 0,r 0,0ÞgþE Z t 0e e s½e V ðx t ,r t ,t ÞÀr :x ðs Þ:2 ds&'ð13ÞBy using the similar method of [22],it can be seen from (9),(12)and (13)that,if e is chosen small enough,a constant k 40can be found such thatmin i A Sf l min ð^P i 11Þg E f :z 1ðt Þ:2g r E f x T ðt ÞE T P i x ðt Þg r E f V ðx t ,r t ,t Þg r k e Àe t :j ðt Þ:2d 2Then,for any t 40E f :z 1ðt Þ:2g r Z e Àe t :j ðt Þ:2d 2where Z ¼ðmin i A S f l min ð^Pi 11ÞgÞÀ1k .Following a similar approach of [23,24],we can deduce that z 2(t )is mean-square exponential stable.Then,system (4)with w (t )¼0is mean-square exponential admissible.In the following,we establish the H N performance of system (4).Under zero initial condition,it is easy to see thatJ ðt Þ¼E Z 10½z T ðs Þz ðs ÞÀg 2w Tðs Þw ðs Þ ds&'r EZ 10½z T ðs Þz ðs ÞÀg 2w Tðs Þw ðs ÞþL V ðx s ,r s ,s Þ ds &'r E ZtxTðs ÞðY i þL T i L i Þx s Þds&'whereY i ¼X i 11X i 12E T W 3E 0ðE T P i þSY T i R ÞB win X i 22E T W 2EX i 240nn X i 33E TW 2En n n X i 44nnnnÀg 2I266666664377777775þA T i A T di 00B T wi 266666664377777775ðd 212d 22W 1þd 212W 2þd 21W 3ÞA T i A T di 00B T wi266666664377777775T x t Þ¼½x T ðt Þx T ðt Àd ðt ÞÞx T ðt Àd 1Þx T ðt Àd 2Þw T ðt Þ T L i ¼C iC diD iÂÃHence,by Schur complement,we can obtain from (6)thatfor all t 40,J (t )o 0.Therefore,by Definition 2,the singular Markovian jump system (4)is mean-square exponentially admis-sible with an H N performance level g for any time delay d (t )satisfying (2).This completes the proof.&Remark 1.Without resorting to the free-weighting matrices method,a delay-range-dependent condition for the mean-square exponential admissibility with an H N performance for singular Markovian jump system (4)is derived in terms of strict LMI,which is in contrast with [18],where the free-weighting matricesmethod and equality constraints are used.In addition,Therefore,our condition has more computational advantages and more real applications than [18].In the following,we shall focus on the design of non-fragile guaranteed cost controller,which will guarantee the closed-loop system (4)is mean-square exponentially admissible with an H N performance level g for any time delay d (t )satisfying (2)and at the same time assures that the cost function (5)is bounded.Theorem 2.For given scalars g 40,0r d 1o d 2and 0r m o 1,there exists a state feedback controller of the form (3)such that the closed-loop system (4)is mean-square exponentially admis-sible with an H N performance level g for any time delay d (t )satisfying (2)and moreover the cost function (5)is bounded if there exist symmetric positive-definite matrices Q k ,W k ,k ¼1,2,3,^Pi ,a nonsingular matrix ^Y i and matrices J i A R m Ân ,H i A R m Â(n Àr ),R ,S such that for each i A S ,X i 11X i 12X i 130B wi X i 16X i 17X i 18Z T i C T i Z T i ^Q i X i 111n X i 22X i 23X i 240X i 2600Z T i C T di 0n n X i 33X i 340000000n n n X i 440000000n n n n Àg 2IX i 5600D T i00n n n n n X i 660000n n n n n n X i 77000n n n n n n n X i 88000n n n n n n n n ÀI00n n n n n n n n n À^Qi0nnnnnnnnnnÀ^Wi 266666666666666666666664377777777777777777777775o 0ð14Þwhere d 12¼d 2Àd 1,X i 11¼A i Z i þZ T i A T i þB i ðJ i E T þH i R ÞþðEJ T i þR T H T i ÞB T i þp ii EZ i þp ii Z T i ETÀp ii E ^P i E T Àðd 212þ1ÞZ T iE T Àðd 212þ1ÞEZ i þd 212W 1þW 3X i 12¼A di Z i þd 212Z T i E Tþd 212EZ i ÀW 1X i 13¼EZ i þZ T i E TÀ3X i 16¼½d 12d 2Z T i A T i þd 12d 2ðEJ T i þR T H T i ÞB T i ,d 12Z T i A T iþd 12ðEJ T i þR T H T i ÞB T i ,d 1Z T i A T i þd 1ðEJ T i þR T H T i ÞB TiX i 17¼½Z T i ,Z T i ,Z TiX i 18¼ffiffiffiffiffiffiffip i 1p Z T i E R ,...,ffiffiffiffiffiffiffiffiffiffiffiffiffip i ði À1Þp Z T i E R ,ffiffiffiffiffiffiffiffiffiffiffiffiffiffip i ði þ1Þp Z T i E R ,...,ffiffiffiffiffiffip is p Z Ti E Rh iX i 22¼Àð1Àm ÞZ T i Àð1Àm ÞZ i þð1Àm ÞQ 3Àðd 212þd 12d 2þ2ÞZ T i ETÀðd 212þd 12d 2þ2ÞEZ i þðd 212þd 12d 2Þ1þ22X i 23¼EZ i þZ T i E T ÀW 2X i 24¼ðd 12d 2þ1ÞZ T i E T þðd 12d 2þ1ÞEZ i Àd 12d 2W 1ÀW 2X i 26¼½d 12d 2Z T i A T di ,d 12Z T i A T di ,d 1Z T i A T diX i 33¼ÀZ i ÀZ T i þ1À2EZ i À2Z T i E T þ2þ3X i 34¼EZ i þZ T i E T ÀW 2X i 44¼ÀZ i ÀZ T i þQ 2Àðd 12d 1þ1ÞEZ i Àðd 12d 1þ1ÞZ T i E T þW 1þW 2X i 56¼½d 12d 2B T wi ,d 12B T wi ,d 1B T wiY.Wang et al./ISA Transactions 51(2012)559–565562X i 66¼Àdiag ðW 1,W 2,W 3ÞX i 77¼Àdiag ðQ 3,Q 1,Q 2ÞX i 88¼Àdiag ðE T R ^P 1E R ,...,E T R ^P i À1E R ,E T R ^P i þ1E R ,...,E T R ^P s E R ÞX i 111¼ðEJ T i þR T H T i Þ^W iZ i ¼^Pi E T þS ^Y i R where R and S are the matrices satisfying RE ¼0and ES ¼0.E L A R n Âr and E R A R n Âr are of full column rank and satisfy E ¼E L E T R .The desired state feedback controller K i is given byK i ¼ðJ i E T þH i R ÞZ À1i ,i A S ð15ÞThe corresponding cost function in (5)satisfiesJ r J n ¼E x Tð0ÞE T P ðr 0ÞEx ð0ÞþX 2k ¼1Z 0Àd k x T ða ÞQ k x ða Þd a(þZ 00Àd ð0Þx T ða ÞQ 3x ða Þd a þZ00Àd 2ðd 2þa Þ_x T ða ÞE T W 1E _x ða Þd a þZÀd 1Àd 2ðd 2þa Þ_xT ða ÞE T W 2E _x ða Þd a )ð16ÞProof.Since RE ¼0and ES ¼0,thus we get that RE L ¼0and E T R S ¼and [E R S ]is nonsingular.ThenE R S ÂÃÀ1¼ðE T R E R ÞÀ1E T R ðS T S ÞÀ1ST "#,E R ðE T R E R ÞÀ1E T R þS ðS T S ÞÀ1S T ¼I n ð17ÞFrom (17),we have½E R ðE T R E R ÞÀ1ðE TL P i E L ÞÀ1E T L þS ðS T S ÞÀ1Y À1i ðRR T ÞÀ1R ÀS ðS T S ÞÀ1Y À1i ðRR T ÞÀ1RP i E L ðE T L P i E L ÞÀ1E TL ðP i E þR T Y i S T Þ¼I n which implies P i E þR T Y i S T is nonsingular and its inverse can beexpressed as ðP i E þR T Y i S T ÞÀ1¼E R ðE T R E R ÞÀ1ðE TL P i E L ÞÀ1E T L þS ðS T S ÞÀ1Y À1i ðRR T ÞÀ1R ÀS ðS T S ÞÀ1Y À1i ðRR T ÞÀ1RP i E L ðE T L P i E L ÞÀ1E TL ¼ðE T R E R ÞÀ1E T R ðS T S ÞÀ1ST"#T ðE T L P i E LÞÀ1Y i 12n 0"#ðE T R E R ÞÀ1E T R ðS T S ÞÀ1ST"#E TþS ðS T S ÞÀ1Y À1i ðRR T ÞÀ1R whereY i 12¼ÀðE T L P i E L ÞÀ1E TL P i R T ðRR T ÞÀ1Y ÀT i :Let^Pi ¼ðE T R E R ÞÀ1E TR ðS T S ÞÀ1S T"#T ðE T L P i E L ÞÀ1Y i 12n"#ðE T R E R ÞÀ1E TR ðS T S ÞÀ1S T"#,^Y i ¼ðS T S ÞÀ1Y À1iðRR T ÞÀ1Then,we get ðP i E þR T Y i S T ÞÀ1¼^P i E T þS ^Y i R ¼Z i and E T R ^P i E R ¼ðE T L P i E L ÞÀ1.Applying Schur complement to (14)givesX þZ T i ^Q i Z i þðEJ T i þR T H T i Þ^W i ðJ i E T þH i R Þo 0ð18ÞwhereX ¼X i 11X i 12X i 130B wi X i 16X i 17X i 18Z T i C T in X i 22X i 23X i 240X i 2600Z T i C T di n n X i 33X i 3400000n n n X i 4400000n n n n Àg 2IX i 5600D T i n n n n n X i 6600n n n n n n X i 770n n n n n n n X i 880nnnnnnnnÀI26666666666666666643777777777777777775Pre-multiplying and post-multiplying (6)by diag ðZ T ði Þ,Z T ði Þ,Z T ði Þ,Z T ði Þ,I ,W À11,W À12,W À13,I Þand its transpose and let J i¼K i ^Pi ,H i ¼K i S ^Yi ,we have where~X i 11¼A i Z i þZ T i A T i þB i ðJ i E T þH i R ÞþðEJ T i þR T H T i ÞB T iþp ii Z T i E R ðE T R ^P i E R ÞÀ1E TR Z i þXs j ¼1,j a ip ii Z T i E R ðE T R ^P j E R ÞÀ1E TR Z iþX 3k ¼1Z T i Q k Z i Àd 212Z T i E T W 1EZ i ÀZ T i E TW 3EZ i~X i 12¼A di Z i þd 212Z T iE T W 1EZ i ~X i 16¼d 12d 2Z T i A T i þd 12d 2ðEJ T i þR T H T i ÞB T i~X i 17¼d 12Z T i A T i þd 12ðEJ T i þR T H T i ÞB T i ~X i 18¼d 1Z T i A T i þd 1ðEJ T i þR T H T i ÞB T i~Xi 11~X i 12Z T i E TW 3EZ i 0B wi ~Xi 16~Xi 17~Xi 18Z T i C T in ~Xi 22Z T i E T W 2EZ i~Xi 240d 12d 2Z T i A Tdid 12Z T i A Tdid 1Z T i A TdiZ T i C T di n n ~Xi 33Z T i E TW 2EZ i00000n n n ~Xi 4400000n n n n Àg 2Id 12d 2B T wi B Tid 12B T wi B Tid 1B T wi B TiD T i n n n n n ÀW À1100n n n n n n ÀW À120n n n n n n n ÀW À130nnnnnnnn ÀI2666666666666666666437777777777777777775o 0ð19ÞY.Wang et al./ISA Transactions 51(2012)559–565563。
自动控制专业英语词汇
自动控制专业英语词汇（一）acceleration transducer 加速度传感器acceptance testing 验收测试accessibility 可及性accumulated error 累积误差AC-DC-AC frequency converter 交-直-交变频器AC (alternating current) electric drive 交流电子传动active attitude stabilization 主动姿态稳定actuator 驱动器，执行机构adaline 线性适应元adaptation layer 适应层adaptive telemeter system 适应遥测系统adjoint operator 伴随算子admissible error 容许误差aggregation matrix 集结矩阵AHP (analytic hierarchy process) 层次分析法amplifying element 放大环节analog-digital conversion 模数转换annunciator 信号器antenna pointing control 天线指向控制anti-integral windup 抗积分饱卷aperiodic decomposition 非周期分解a posteriori estimate 后验估计approximate reasoning 近似推理a priori estimate 先验估计articulated robot 关节型机器人assignment problem 配置问题，分配问题associative memory model 联想记忆模型associatron 联想机asymptotic stability 渐进稳定性attained pose drift 实际位姿漂移attitude acquisition 姿态捕获AOCS (attritude and orbit control system) 姿态轨道控制系统attitude angular velocity 姿态角速度attitude disturbance 姿态扰动attitude maneuver 姿态机动attractor 吸引子augment ability 可扩充性augmented system 增广系统automatic manual station 自动-手动操作器automaton 自动机autonomous system 自治系统backlash characteristics 间隙特性base coordinate system 基座坐标系Bayes classifier 贝叶斯分类器bearing alignment 方位对准bellows pressure gauge 波纹管压力表benefit-cost analysis 收益成本分析bilinear system 双线性系统biocybernetics 生物控制论biological feedback system 生物反馈系统black box testing approach 黑箱测试法blind search 盲目搜索block diagonalization 块对角化Boltzman machine 玻耳兹曼机bottom-up development 自下而上开发boundary value analysis 边界值分析brainstorming method 头脑风暴法breadth-first search 广度优先搜索butterfly valve 蝶阀CAE (computer aided engineering) 计算机辅助工程CAM (computer aided manufacturing) 计算机辅助制造Camflex valve 偏心旋转阀canonical state variable 规范化状态变量capacitive displacement transducer 电容式位移传感器capsule pressure gauge 膜盒压力表CARD 计算机辅助研究开发Cartesian robot 直角坐标型机器人cascade compensation 串联补偿catastrophe theory 突变论centrality 集中性chained aggregation 链式集结chaos 混沌characteristic locus 特征轨迹chemical propulsion 化学推进calrity 清晰性classical information pattern 经典信息模式classifier 分类器clinical control system 临床控制系统closed loop pole 闭环极点closed loop transfer function 闭环传递函数cluster analysis 聚类分析coarse-fine control 粗-精控制cobweb model 蛛网模型coefficient matrix 系数矩阵cognitive science 认知科学cognitron 认知机coherent system 单调关联系统combination decision 组合决策combinatorial explosion 组合爆炸combined pressure and vacuum gauge 压力真空表command pose 指令位姿companion matrix 相伴矩阵compartmental model 房室模型compatibility 相容性，兼容性compensating network 补偿网络compensation 补偿，矫正compliance 柔顺，顺应composite control 组合控制computable general equilibrium model 可计算一般均衡模型conditionally instability 条件不稳定性configuration 组态connectionism 连接机制connectivity 连接性conservative system 守恒系统consistency 一致性constraint condition 约束条件consumption function 消费函数context-free grammar 上下文无关语法continuous discrete event hybrid system simulation 连续离散事件混合系统仿真continuous duty 连续工作制control accuracy 控制精度control cabinet 控制柜controllability index 可控指数controllable canonical form 可控规范型[control] plant 控制对象,被控对象controlling instrument 控制仪表control moment gyro 控制力矩陀螺control panel 控制屏，控制盘control synchro 控制[式]自整角机control system synthesis 控制系统综合control time horizon 控制时程cooperative game 合作对策coordinability condition 可协调条件coordination strategy 协调策略coordinator 协调器corner frequency 转折频率costate variable 共态变量cost-effectiveness analysis 费用效益分析coupling of orbit and attitude 轨道和姿态耦合critical damping 临界阻尼critical stability 临界稳定性cross-over frequency 穿越频率，交越频率current source inverter 电流[源]型逆变器cut-off frequency 截止频率cybernetics 控制论cyclic remote control 循环遥控cylindrical robot 圆柱坐标型机器人damped oscillation 阻尼振荡damper 阻尼器damping ratio 阻尼比data acquisition 数据采集data encryption 数据加密data preprocessing 数据预处理data processor 数据处理器DC generator-motor set drive 直流发电机-电动机组传动D controller 微分控制器decentrality 分散性decentralized stochastic control 分散随机控制decision space 决策空间decision support system 决策支持系统decomposition-aggregation approach 分解集结法decoupling parameter 解耦参数deductive-inductive hybrid modeling method 演绎与归纳混合建模法delayed telemetry 延时遥测derivation tree 导出树derivative feedback 微分反馈describing function 描述函数desired value 希望值despinner 消旋体destination 目的站detector 检出器deterministic automaton 确定性自动机deviation 偏差deviation alarm 偏差报警器DFD 数据流图diagnostic model 诊断模型diagonally dominant matrix 对角主导矩阵diaphragm pressure gauge 膜片压力表difference equation model 差分方程模型differential dynamical system 微分动力学系统differential game 微分对策differential pressure level meter 差压液位计differential pressure transmitter 差压变送器differential transformer displacement transducer 差动变压器式位移传感器differentiation element 微分环节digital filer 数字滤波器digital signal processing 数字信号处理digitization 数字化digitizer 数字化仪dimension transducer 尺度传感器direct coordination 直接协调disaggregation 解裂discoordination 失协调discrete event dynamic system 离散事件动态系统discrete system simulation language 离散系统仿真语言discriminant function 判别函数displacement vibration amplitude transducer 位移振幅传感器dissipative structure 耗散结构distributed parameter control system 分布参数控制系统distrubance 扰动disturbance compensation 扰动补偿diversity 多样性divisibility 可分性domain knowledge 领域知识dominant pole 主导极点dose-response model 剂量反应模型dual modulation telemetering system 双重调制遥测系统dual principle 对偶原理dual spin stabilization 双自旋稳定duty ratio 负载比dynamic braking 能耗制动dynamic characteristics 动态特性dynamic deviation 动态偏差dynamic error coefficient 动态误差系数dynamic exactness 动它吻合性dynamic input-output model 动态投入产出模型econometric model 计量经济模型economic cybernetics 经济控制论economic effectiveness 经济效益economic evaluation 经济评价economic index 经济指数economic indicator 经济指标eddy current thickness meter 电涡流厚度计effectiveness 有效性effectiveness theory 效益理论elasticity of demand 需求弹性electric actuator 电动执行机构electric conductance levelmeter 电导液位计electric drive control gear 电动传动控制设备electric hydraulic converter 电-液转换器electric pneumatic converter 电-气转换器electrohydraulic servo vale 电液伺服阀electromagnetic flow transducer 电磁流量传感器electronic batching scale 电子配料秤electronic belt conveyor scale 电子皮带秤electronic hopper scale 电子料斗秤elevation 仰角emergency stop 异常停止empirical distribution 经验分布endogenous variable 内生变量equilibrium growth 均衡增长equilibrium point 平衡点equivalence partitioning 等价类划分ergonomics 工效学error 误差error-correction parsing 纠错剖析estimate 估计量estimation theory 估计理论evaluation technique 评价技术event chain 事件链evolutionary system 进化系统exogenous variable 外生变量expected characteristics 希望特性external disturbance 外扰fact base 事实failure diagnosis 故障诊断fast mode 快变模态feasibility study 可行性研究feasible coordination 可行协调feasible region 可行域feature detection 特征检测feature extraction 特征抽取feedback compensation 反馈补偿feedforward path 前馈通路field bus 现场总线finite automaton 有限自动机FIP (factory information protocol) 工厂信息协议first order predicate logic 一阶谓词逻辑fixed sequence manipulator 固定顺序机械手fixed set point control 定值控制FMS (flexible manufacturing system) 柔性制造系统flow sensor/transducer 流量传感器flow transmitter 流量变送器fluctuation 涨落forced oscillation 强迫振荡formal language theory 形式语言理论formal neuron 形式神经元forward path 正向通路forward reasoning 正向推理fractal 分形体，分维体frequency converter 变频器frequency domain model reduction method 频域模型降阶法frequency response 频域响应full order observer 全阶观测器functional decomposition 功能分解FES (functional electrical stimulation) 功能电刺激functional simularity 功能相似fuzzy logic 模糊逻辑game tree 对策树gate valve 闸阀general equilibrium theory 一般均衡理论generalized least squares estimation 广义最小二乘估计generation function 生成函数geomagnetic torque 地磁力矩geometric similarity 几何相似gimbaled wheel 框架轮global asymptotic stability 全局渐进稳定性global optimum 全局最优globe valve 球形阀goal coordination method 目标协调法grammatical inference 文法推断graphic search 图搜索gravity gradient torque 重力梯度力矩group technology 成组技术guidance system 制导系统gyro drift rate 陀螺漂移率gyrostat 陀螺体Hall displacement transducer 霍尔式位移传感器hardware-in-the-loop simulation 半实物仿真harmonious deviation 和谐偏差harmonious strategy 和谐策略heuristic inference 启发式推理hidden oscillation 隐蔽振荡hierarchical chart 层次结构图hierarchical planning 递阶规划hierarchical control 递阶控制homeostasis 内稳态homomorphic model 同态系统horizontal decomposition 横向分解hormonal control 内分泌控制hydraulic step motor 液压步进马达hypercycle theory 超循环理论I controller 积分控制器identifiability 可辨识性IDSS (intelligent decision support system) 智能决策支持系统image recognition 图像识别impulse 冲量impulse function 冲击函数，脉冲函数inching 点动incompatibility principle 不相容原理incremental motion control 增量运动控制index of merit 品质因数inductive force transducer 电感式位移传感器inductive modeling method 归纳建模法industrial automation 工业自动化inertial attitude sensor 惯性姿态敏感器inertial coordinate system 惯性坐标系inertial wheel 惯性轮inference engine 推理机infinite dimensional system 无穷维系统information acquisition 信息采集infrared gas analyzer 红外线气体分析器inherent nonlinearity 固有非线性inherent regulation 固有调节initial deviation 初始偏差initiator 发起站injection attitude 入轨姿势input-output model 投入产出模型instability 不稳定性instruction level language 指令级语言integral of absolute value of error criterion 绝对误差积分准则integral of squared error criterion 平方误差积分准则integral performance criterion 积分性能准则integration instrument 积算仪器integrity 整体性intelligent terminal 智能终端interacted system 互联系统，关联系统interactive prediction approach 互联预估法，关联预估法interconnection 互联intermittent duty 断续工作制internal disturbance 内扰ISM (interpretive structure modeling) 解释结构建模法invariant embedding principle 不变嵌入原理inventory theory 库伦论inverse Nyquist diagram 逆奈奎斯特图inverter 逆变器investment decision 投资决策isomorphic model 同构模型iterative coordination 迭代协调jet propulsion 喷气推进job-lot control 分批控制joint 关节Kalman-Bucy filer 卡尔曼-布西滤波器knowledge accomodation 知识顺应knowledge acquisition 知识获取knowledge assimilation 知识同化KBMS (knowledge base management system) 知识库管理系统knowledge representation 知识表达ladder diagram 梯形图lag-lead compensation 滞后超前补偿Lagrange duality 拉格朗日对偶性Laplace transform 拉普拉斯变换large scale system 大系统lateral inhibition network 侧抑制网络least cost input 最小成本投入least squares criterion 最小二乘准则level switch 物位开关libration damping 天平动阻尼limit cycle 极限环linearization technique 线性化方法linear motion electric drive 直线运动电气传动linear motion valve 直行程阀linear programming 线性规划LQR (linear quadratic regulator problem) 线性二次调节器问题load cell 称重传感器local asymptotic stability 局部渐近稳定性local optimum 局部最优log magnitude-phase diagram 对数幅相图long term memory 长期记忆lumped parameter model 集总参数模型Lyapunov theorem of asymptotic stability 李雅普诺夫渐近稳定性定理自动控制专业英语词汇（二）macro-economic system 宏观经济系统magnetic dumping 磁卸载magnetoelastic weighing cell 磁致弹性称重传感器magnitude-frequency characteristic 幅频特性magnitude margin 幅值裕度magnitude scale factor 幅值比例尺manipulator 机械手man-machine coordination 人机协调manual station 手动操作器MAP (manufacturing automation protocol) 制造自动化协议marginal effectiveness 边际效益Mason's gain formula 梅森增益公式master station 主站matching criterion 匹配准则maximum likelihood estimation 最大似然估计maximum overshoot 最大超调量maximum principle 极大值原理mean-square error criterion 均方误差准则mechanism model 机理模型meta-knowledge 元知识metallurgical automation 冶金自动化minimal realization 最小实现minimum phase system 最小相位系统minimum variance estimation 最小方差估计minor loop 副回路missile-target relative movement simulator 弹体-目标相对运动仿真器modal aggregation 模态集结modal transformation 模态变换MB (model base) 模型库model confidence 模型置信度model fidelity 模型逼真度model reference adaptive control system 模型参考适应控制系统model verification 模型验证modularization 模块化MEC (most economic control) 最经济控制motion space 可动空间MTBF (mean time between failures) 平均故障间隔时间MTTF (mean time to failures) 平均无故障时间multi-attributive utility function 多属性效用函数multicriteria 多重判据multilevel hierarchical structure 多级递阶结构multiloop control 多回路控制multi-objective decision 多目标决策multistate logic 多态逻辑multistratum hierarchical control 多段递阶控制multivariable control system 多变量控制系统myoelectric control 肌电控制Nash optimality 纳什最优性natural language generation 自然语言生成nearest-neighbor 最近邻necessity measure 必然性侧度negative feedback 负反馈neural assembly 神经集合neural network computer 神经网络计算机Nichols chart 尼科尔斯图noetic science 思维科学noncoherent system 非单调关联系统noncooperative game 非合作博弈nonequilibrium state 非平衡态nonlinear element 非线性环节nonmonotonic logic 非单调逻辑nonparametric training 非参数训练nonreversible electric drive 不可逆电气传动nonsingular perturbation 非奇异摄动non-stationary random process 非平稳随机过程nuclear radiation levelmeter 核辐射物位计nutation sensor 章动敏感器Nyquist stability criterion 奈奎斯特稳定判据objective function 目标函数observability index 可观测指数observable canonical form 可观测规范型on-line assistance 在线帮助on-off control 通断控制open loop pole 开环极点operational research model 运筹学模型optic fiber tachometer 光纤式转速表optimal trajectory 最优轨迹optimization technique 最优化技术orbital rendezvous 轨道交会orbit gyrocompass 轨道陀螺罗盘orbit perturbation 轨道摄动order parameter 序参数orientation control 定向控制originator 始发站oscillating period 振荡周期output prediction method 输出预估法oval wheel flowmeter 椭圆齿轮流量计overall design 总体设计overdamping 过阻尼overlapping decomposition 交叠分解Pade approximation 帕德近似Pareto optimality 帕雷托最优性passive attitude stabilization 被动姿态稳定path repeatability 路径可重复性pattern primitive 模式基元PR (pattern recognition) 模式识别P control 比例控制器peak time 峰值时间penalty function method 罚函数法perceptron 感知器periodic duty 周期工作制perturbation theory 摄动理论pessimistic value 悲观值phase locus 相轨迹phase trajectory 相轨迹phase lead 相位超前photoelectric tachometric transducer 光电式转速传感器phrase-structure grammar 短句结构文法physical symbol system 物理符号系统piezoelectric force transducer 压电式力传感器playback robot 示教再现式机器人PLC (programmable logic controller) 可编程序逻辑控制器plug braking 反接制动plug valve 旋塞阀pneumatic actuator 气动执行机构point-to-point control 点位控制polar robot 极坐标型机器人pole assignment 极点配置pole-zero cancellation 零极点相消polynomial input 多项式输入portfolio theory 投资搭配理论pose overshoot 位姿过调量position measuring instrument 位置测量仪posentiometric displacement transducer 电位器式位移传感器positive feedback 正反馈power system automation 电力系统自动化predicate logic 谓词逻辑pressure gauge with electric contact 电接点压力表pressure transmitter 压力变送器price coordination 价格协调primal coordination 主协调primary frequency zone 主频区PCA (principal component analysis) 主成分分析法principle of turnpike 大道原理priority 优先级process-oriented simulation 面向过程的仿真production budget 生产预算production rule 产生式规则profit forecast 利润预测PERT (program evaluation and review technique) 计划评审技术program set station 程序设定操作器proportional control 比例控制proportional plus derivative controller 比例微分控制器protocol engineering 协议工程prototype 原型pseudo random sequence 伪随机序列pseudo-rate-increment control 伪速率增量控制pulse duration 脉冲持续时间pulse frequency modulation control system 脉冲调频控制系统pulse width modulation control system 脉冲调宽控制系统PWM inverter 脉宽调制逆变器pushdown automaton 下推自动机QC (quality control) 质量管理quadratic performance index 二次型性能指标qualitative physical model 定性物理模型quantized noise 量化噪声quasilinear characteristics 准线性特性queuing theory 排队论radio frequency sensor 射频敏感器ramp function 斜坡函数random disturbance 随机扰动random process 随机过程rate integrating gyro 速率积分陀螺ratio station 比值操作器reachability 可达性reaction wheel control 反作用轮控制realizability 可实现性,能实现性real time telemetry 实时遥测receptive field 感受野rectangular robot 直角坐标型机器人rectifier 整流器recursive estimation 递推估计reduced order observer 降阶观测器redundant information 冗余信息reentry control 再入控制regenerative braking 回馈制动，再生制动regional planning model 区域规划模型regulating device 调节装载regulation 调节relational algebra 关系代数relay characteristic 继电器特性remote manipulator 遥控操作器remote regulating 遥调remote set point adjuster 远程设定点调整器rendezvous and docking 交会和对接reproducibility 再现性resistance thermometer sensor 热电阻resolution principle 归结原理resource allocation 资源分配response curve 响应曲线return difference matrix 回差矩阵return ratio matrix 回比矩阵reverberation 回响reversible electric drive 可逆电气传动revolute robot 关节型机器人revolution speed transducer 转速传感器rewriting rule 重写规则rigid spacecraft dynamics 刚性航天动力学risk decision 风险分析robotics 机器人学robot programming language 机器人编程语言robust control 鲁棒控制robustness 鲁棒性roll gap measuring instrument 辊缝测量仪root locus 根轨迹roots flowmeter 腰轮流量计rotameter 浮子流量计，转子流量计rotary eccentric plug valve 偏心旋转阀rotary motion valve 角行程阀rotating transformer 旋转变压器Routh approximation method 劳思近似判据routing problem 路径问题sampled-data control system 采样控制系统sampling control system 采样控制系统saturation characteristics 饱和特性scalar Lyapunov function 标量李雅普诺夫函数SCARA (selective compliance assembly robot arm) 平面关节型机器人scenario analysis method 情景分析法scene analysis 物景分析s-domain s域self-operated controller 自力式控制器self-organizing system 自组织系统self-reproducing system 自繁殖系统self-tuning control 自校正控制semantic network 语义网络semi-physical simulation 半实物仿真sensing element 敏感元件sensitivity analysis 灵敏度分析sensory control 感觉控制sequential decomposition 顺序分解sequential least squares estimation 序贯最小二乘估计servo control 伺服控制，随动控制servomotor 伺服马达settling time 过渡时间sextant 六分仪short term planning 短期计划short time horizon coordination 短时程协调signal detection and estimation 信号检测和估计signal reconstruction 信号重构similarity 相似性simulated interrupt 仿真中断simulation block diagram 仿真框图simulation experiment 仿真实验simulation velocity 仿真速度simulator 仿真器single axle table 单轴转台single degree of freedom gyro 单自由度陀螺single level process 单级过程single value nonlinearity 单值非线性singular attractor 奇异吸引子singular perturbation 奇异摄动sink 汇点slaved system 受役系统slower-than-real-time simulation 欠实时仿真slow subsystem 慢变子系统socio-cybernetics 社会控制论socioeconomic system 社会经济系统software psychology 软件心理学solar array pointing control 太阳帆板指向控制solenoid valve 电磁阀source 源点specific impulse 比冲speed control system 调速系统spin axis 自旋轴spinner 自旋体stability criterion 稳定性判据stability limit 稳定极限stabilization 镇定，稳定Stackelberg decision theory 施塔克尔贝格决策理论state equation model 状态方程模型state space description 状态空间描述static characteristics curve 静态特性曲线station accuracy 定点精度stationary random process 平稳随机过程statistical analysis 统计分析statistic pattern recognition 统计模式识别steady state deviation 稳态偏差steady state error coefficient 稳态误差系数step-by-step control 步进控制step function 阶跃函数stepwise refinement 逐步精化stochastic finite automaton 随机有限自动机strain gauge load cell 应变式称重传感器strategic function 策略函数strongly coupled system 强耦合系统subjective probability 主观频率suboptimality 次优性supervised training 监督学习supervisory computer control system 计算机监控系统sustained oscillation 自持振荡swirlmeter 旋进流量计switching point 切换点symbolic processing 符号处理synaptic plasticity 突触可塑性synergetics 协同学syntactic analysis 句法分析system assessment 系统评价systematology 系统学system homomorphism 系统同态system isomorphism 系统同构system engineering 系统工程tachometer 转速表target flow transmitter 靶式流量变送器task cycle 作业周期teaching programming 示教编程telemechanics 远动学telemetering system of frequency division type 频分遥测系统telemetry 遥测teleological system 目的系统teleology 目的论temperature transducer 温度传感器template base 模版库tensiometer 张力计texture 纹理theorem proving 定理证明therapy model 治疗模型thermocouple 热电偶thermometer 温度计thickness meter 厚度计three-axis attitude stabilization 三轴姿态稳定three state controller 三位控制器thrust vector control system 推力矢量控制系统thruster 推力器time constant 时间常数time-invariant system 定常系统，非时变系统time schedule controller 时序控制器time-sharing control 分时控制time-varying parameter 时变参数top-down testing 自上而下测试topological structure 拓扑结构TQC (total quality control) 全面质量管理tracking error 跟踪误差trade-off analysis 权衡分析transfer function matrix 传递函数矩阵transformation grammar 转换文法transient deviation 瞬态偏差transient process 过渡过程transition diagram 转移图transmissible pressure gauge 电远传压力表transmitter 变送器trend analysis 趋势分析triple modulation telemetering system 三重调制遥测系统turbine flowmeter 涡轮流量计Turing machine 图灵机two-time scale system 双时标系统ultrasonic levelmeter 超声物位计unadjustable speed electric drive 非调速电气传动unbiased estimation 无偏估计underdamping 欠阻尼uniformly asymptotic stability 一致渐近稳定性uninterrupted duty 不间断工作制，长期工作制unit circle 单位圆unit testing 单元测试unsupervised learing 非监督学习upper level problem 上级问题urban planning 城市规划utility function 效用函数value engineering 价值工程variable gain 可变增益，可变放大系数variable structure control system 变结构控制vector Lyapunov function 向量李雅普诺夫函数velocity error coefficient 速度误差系数velocity transducer 速度传感器vertical decomposition 纵向分解vibrating wire force transducer 振弦式力传感器vibrometer 振动计viscous damping 粘性阻尼voltage source inverter 电压源型逆变器vortex precession flowmeter 旋进流量计vortex shedding flowmeter 涡街流量计WB (way base) 方法库weighing cell 称重传感器weighting factor 权因子weighting method 加权法Whittaker-Shannon sampling theorem 惠特克-香农采样定理Wiener filtering 维纳滤波work station for computer aided design 计算机辅助设计工作站w-plane w平面zero-based budget 零基预算zero-input response 零输入响应zero-state response 零状态响应zero sum game model 零和对策模型z-transform z变换。
不确定欠驱动系统的高阶自适应Super-Twisting滑模控制
不确定欠驱动系统的高阶自适应Super-Twisting滑模控制杨兴明;高银平【期刊名称】《模式识别与人工智能》【年(卷),期】2014(000)009【摘要】为实现一类不确定欠驱动系统在未知干扰情况下的鲁棒控制，针对传统滑模控制中存在的抖振问题，提出一种基于二次型Lyapunov函数的二阶Super-Twisting自适应滑模控制策略。
首先，控制器的不连续项采用二阶Super-Twisting算法，将不连续控制作用在滑模量的二阶导数。
然后，针对滑模面受不确定干扰影响的情况，为调节参数设计一种自适应律方法，该方法不受传统二阶滑模控制中干扰项的一阶导数边界已知的条件限制，保证滑模面在有干扰情况下的收敛，削弱控制器输入的抖振现象。
最后，以两轮自平衡车为实验对象验证该方法，并与传统滑模及普通二阶滑模方法做仿真对比。
仿真结果表明文中所提的二阶自适应滑模控制方法在控制效果和降低抖振方面表现更优。
%To achieve good robustness against disturbances for a class of uncertain underactuated systems, a second-order adaptive sliding mode control method is proposed based on quadratic Lyapunov function to reduce the inherent chattering of conventional sliding mode control ( SMC ) . Firstly, a second-order super-twisting algorithm is used by the discontinuous part of controller, which acts on the second-order derivative of sliding mode variables. Secondly, as for the effects of unknown disturbances on sliding mode surface, an adaptive law is designed to adjust the parameters. This method eliminates the restriction of the first derivative of disturbancesboundary being known in the traditional second-order sliding mode control, which not only keeps convergence of sliding mode surface but also reduces chattering. Finally, a two-wheeled self-balancing cart is used to test the proposed approach. The simulation results showthat compared with conventional SMC and ordinary second-order SMC, the proposed method outperforms the above methods on effectiveness and reducing chattering.【总页数】7页(P808-814)【作者】杨兴明;高银平【作者单位】合肥工业大学计算机与信息学院合肥230009;合肥工业大学计算机与信息学院合肥230009; 科大讯飞股份有限公司合肥230088【正文语种】中文【中图分类】TP273【相关文献】1.一类欠驱动系统的自适应神经滑模控制 [J], 杨兴明;汤星2.一类非线性不确定系统的高阶积分自适应滑模控制 [J], 杜文正;张金星;谢政;吴鹏3.多输入多输出非线性不确定系统连续高阶滑模控制 [J], 刘向杰;韩耀振4.不确定高阶MIMO系统的指对数型终端滑模控制 [J], 傅佳烨;牛玉刚5.基于高阶微分滑模面的不确定飞机\r控制系统自适应滑模控制 [J], 武晓晶;吴学礼;邵士凯;刘慧贤因版权原因，仅展示原文概要，查看原文内容请购买。
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II. DEFINITIONS AND PROBLEM FORMULATION
Consider the following uncertain discrete stochastic T-S fuzzy system with both time-varying delays and input delays:
+Edi(k)(k − τ (k)) + B2i(k)u(k)
+Bd2i(k)u(k − τ (k))]ω(k),
ห้องสมุดไป่ตู้
(1)
where x(k) = φ(k), k = −τ2, −τ2 + 1, · · · , 0 is the initial conditions. μij is the fuzzy set and r is the number of IFTHEN rules; x(k) ∈ Rn is the state; u(k) ∈ Rm is the control input;s1(k), s2(k), . . . , sg(k) are the premise variables which do not depend on the input variables u(k). Throughout this
Yongmin Li, Shengyuan Xu, and Baoyong Zhang School of Automation
Nanjing University of Science and Technology Nanjing 210094, Jiangsu, P.R. China
ymlwww@ & syxu02@
Plant Rule i: IF s1(k) is μi1 and s2(k) is μi2 . . . and sg(k) is μig THEN
x(k + 1) = Ai(k)x(k) + Adi(k)x(k − τ (k)) + B1i(k)u(k)
+Bd1i(k)u(k − τ (k)) + [Ei(k)x(k)
Index Terms—-Delay-dependent control, discrete TakagiSugeno (T-S) fuzzy model, input delays, relaxed LMI, stochastic systems.
I. INTRODUCTION
T-S fuzzy model were first proposed by Takagi and Sugeno [12]. When parametric uncertainties appear in a T-S fuzzy system, the robust stability problem was addressed in [11], where the stability conditions were expressed in terms of LMIs. The robust H∞ fuzzy control problem for uncertain T-S fuzzy systems were proposed in [2] and [6] by using the algebraic Riccati inequality-based approach and the LMI-based approach, respectively.
Yuming Chu School of Science Huzhou Teachers College Huzhou 313000, Zhejiang, P.R. China chuyuming@
Abstract— This paper deals with the problem of robust control for uncertain discrete stochastic fuzzy systems with both state and input time-varying delays. The uncertainties are time-varying but norm-bounded. The purpose is to design a state feedback fuzzy controller such that the resulting closed-loop system is robustly stochastically stable. Based on Takagi-Sugeno (T-S) fuzzy model and relaxed LMI approach, a delay-dependent sufficient condition for the solvability of the problem is obtained by utilizing a proper Lyapunov functional.
As it is well-known, stochastic modeling has come to play an important role in many branches of science and engineering applications. Many fundamental results for deterministic systems have been extended to stochastic systems [15]. When parameter uncertainties and time-delay appear in stochastic systems, the problems of robust stochastic stability analysis and controller design have been studied in [10], where some useful robust stochastic stability conditions were proposed. The combination of stochastic system and T-S fuzzy model can be seen in [8], where the stochastic stability and fuzzy controller design were investigated. The delay-dependent stabilization
problem for stochastic fuzzy systems with constant time delays was addressed in [16]. A sliding mode fuzzy controller was designed in [4] to stabilize a stochastic T-S fuzzy system with unknown nonlinearities and constant time delays. It should be pointed out that the studies in [4], [8], [16] are concerned with the continuous-time case. To the best of our knowledge, so far, no results on the delay-dependent roust stabilization for discrete stochastic T-S fuzzy systems are available in the literature, which is still open and remains unsolved. This motivates the present study.
It has been shown that the existence of time delays is often one of the main causes of instability and poor performance of a control system [5]. The stability analysis and stabilization problems for T-S fuzzy delay systems were considered in [3], where state feedback fuzzy controllers and fuzzy observers were designed via the so-called parallel distributed compensation (PDC) scheme. The robust H∞ control problem for uncertain T-S fuzzy systems with time delays was investigated in [7]. The corresponding results for the discrete case can be found in [14].
In this paper, we are concerned with the delay-dependent control problem for uncertain discrete stochastic fuzzy systems with state and input delays. The parameter uncertainties are assumed to be time-varying norm-bounded, and the system delay is supposed to be time-varying but bounded. We aim at designing a state-feedback fuzzy controller such that the resulting closed-loop system is robustly stochastically stable. A delay-dependent sufficient condition for the solvability of the formulated problem is proposed by using the relaxed LMI approach [9]. The desired state feedback controller is constructed by solving certain LMIs, which can be easily implemented by using standard numerical algorithms [1].