Systems&Control Letters55(2006)879–
886
https://www.doczj.com/doc/f14817312.html,/locate/sysconle
Stability,robust stabilization and H∞control of singular-impulsive systems
via switching control?
Jing Yao a,Zhi-Hong Guan a,?,Guanrong Chen b,Daniel W.C.Ho c
a Department of Control Science and Engineering,Huazhong University of Science and Technology,Wuhan430074,PR China
b Department of Electroni
c Engineering,City University of Hong Kong,Kowloon,Hong Kong
c Department of Mathematics,City University of Hong Kong,Kowloon,Hong Kong
Received18August2004;received in revised form11December2005;accepted6April2006
Available online15June2006
Abstract
In this paper,stability,robust stabilization and H∞control of singular-impulsive systems are studied.Some new fundamental properties are derived for switched singular systems subject to impulse effects.Applying the Lyapunov function theory,several suf?cient conditions are established for exponential stability,robust stabilization and H∞control of the corresponding singular-impulsive closed-loop systems.Some numerical examples are given to demonstrate the effectiveness of the proposed control and stabilization methods.
?2006Elsevier B.V.All rights reserved.
Keywords:Singular-impulsive systems;H∞control;Switching systems;Robustness
1.Introduction
Many practical systems involve a mixture of continuous and discrete dynamics.Systems in which these two kinds of dy-namics coexist and interact are usually called hybrid systems. Hybrid models have been used extensively to describe systems in a wide range of applications,including robotics,automo-tive electronics,manufacturing,automated highway systems, air traf?c management systems,integrated circuit design,and multimedia.Switched systems can be considered as a class of hybrid dynamical systems consisting of a family of continuous-and/or discrete-time subsystems,and a rule that orchestrates the switching between them[3,24,28].In recent years,there is signi?cant growth of interest in stability analysis and design of various switched systems,as surveyed in[7,16].
On the other hand,singular systems have attracted particu-lar interest in the literature for their important applications in, e.g.,circuits[23],robotics[22],aircraft modelling[25],social,
?Supported by the National Natural Science Foundation of China under Grants60274004and60573005,and City University of Hong Kong Project no.9040871.
?Corresponding author.Tel.:+862787542145;fax:+862787542145.
E-mail address:zhguan@https://www.doczj.com/doc/f14817312.html,(Z.-H.Guan).
0167-6911/$-see front matter?2006Elsevier B.V.All rights reserved. doi:10.1016/j.sysconle.2006.05.002biological,and multisector economic systems[19],dynamics of thermal nuclear reactors,singular perturbation systems,and so on.Progress in dealing with singular systems can be found in the books[1,4,6].
Many singular systems exhibit impulsive and switching behaviors,which are characterized by abrupt changes and switches of states at certain instants;that is,the systems switch with impulse effects[2,10,13,18,27].Moreover,impulsive and switching phenomena can be found in the?elds of informa-tion science,electronics,automatic control systems,computer networking,arti?cial intelligence,robotics,and telecommuni-cations[10].Many sudden and sharp changes occur instanta-neously,in the form of impulses and switches,which cannot be well described by using pure continuous or pure discrete models.Therefore,it is important and,in fact,necessary to study hybrid impulsive and switching singular systems.
For the control point of view,control techniques based on switching between different controllers have been applied extensively in recent years,due to their advantages in,for in-stance,achieving stability,improving transient responses,and providing effective mechanisms to cope with highly complex systems[5,14,21,26,29].For the following categories of con-trol problems,one might want or need to consider switching control(of course,a combination of some of these is also
880J.Yao et al./Systems &Control Letters 55(2006)879–886
possible)[15]:(1)due to the nature of the problem itself,con-tinuous control is not suitable;(2)due to sensor and/or actuator limitations,continuous control cannot be implemented;(3)the model of a system is highly uncertain,for which a single con-tinuous control law cannot be found.The interests in switched systems have grown recently for the theoretical and practi-cal signi?cance [5,9,12,14,20,26].However,to our knowledge there are very few reports on impulsive and switching singular systems and their corresponding control problems.This moti-vates the present investigation of impulsive and switching sin-gular systems.
The organization of the paper is as follows.In Section 2,the concept of a singular-impulsive mode is described.In Section 3,the asymptotic stability of hybrid impulsive and switching singular systems is studied.Section 4extends these results to robust stabilization.Then,in Section 5,the theory and approach of H ∞control for a class of singular and impulsive systems un-der arbitrary switch are investigated.Three examples are given in Sections 3,4and 5,respectively,for different purposes.Finally,in Section 6,some conclusions are presented.2.Some fundamental theories of singular-impulsive systems and the problem formulation
Let R +=[0,+∞),J =[t 0,+∞)(t 0 0),and R n denote the n -dimensional Euclidean space.For
x =(x 1,...,x n ) ∈R n ,the norm of x is x :=( n i =1x 2i )1/2.Correspondingly,for
A =(a ij )n ×n ∈R n ×n , A := 1/2max (A T A).The identity matrix
of order n is denoted as I n (or simply I if no confusion arises).A linear singular-impulsive control system with impulses at ?xed instants is described by E ˙x =Ax +Bu ,t =t k ,
x =U k (t,x),t =t k ,
(2.1)
where t ∈J (t 0 0),x ∈R n is the state variable,u ∈R m is the control input,the matrix E ∈R n ×n may be singular,and it is assumed that rank (E)=r n ,A and B are known real constant matrices of appropriate dimensions.
A sequence {t k ,U k (t k ,x(t k ))}has the effect of suddenly changing the state of system (2.1)at the instants t k ,where t 1 t k =∞,(2.2) and t 1>t 0;that is, x |t k =:x(t +k )?x(t ? k )=U k (t k ,x(t k )), where x(t +k )=lim h →0+x(t k +h)and x(t ?k )=lim h →0+x(t k ?h). For simplicity,it is assumed that x(t ? k )=x(t k ).Furthermore,U k (t k ,x(t k ))can be chosen as c 2k x(t k )with c 2k being constants for k =1,2,.... Construct a switching controller u for system (2.1)as follows: u(t)= ∞ k =1 C 1k x(t)l k (t),(2.3) where C 1k is an n ×n constant matrix and l k (t)is given by l k (t)= 1,t k ?1 0,otherwise .(2.4) If one chooses the switching control gain matrices {C 1k }as C 1,C 2,...,C m ,that is,C 1k ∈{C 1,C 2,...,C m },then the closed-loop system of (2.1)under control (2.3)becomes ? ??E ˙x =(A +BC i k )x,t ∈(t k ?1,t k ], x =c 2k x(t k ),t =t k , x(t +0)=x 0,k =1,2,...,(2.5)where i k ∈{1,2,...,m }. System (2.5)is also called a hybrid impulsive and switch-ing singular system ,which can be rewritten in the following compact form:? ??E ˙x =A i k x,t ∈(t k ?1,t k ], x =c k x(t k ),t =t k , x(t +0)=x 0,k =1,2,...,(2.6)where t ∈J ,x ∈R n is the state variable,A i k =A +BC i k are n ×n matrices,c k are constants for k =1,2,...,i k ∈{1,2,...,m }is the switch index,and the sequence {t k }satis?es (2.2). It is obvious that system (2.6)has m different modes;that is,E ˙x =A i x, i =1,2,...,m , (2.7) switching in the interval J .It is assumed that the pair (E,A i )(i =1,2,...,m)are regular;that is,det (sE ?A i )=0for some complex number s . De?nition 2.1.System (2.6)is said to be exponentially stable if there exist constants a >0,b >0such that for t >t 0its state x(t)satis?es x(t) x(t 0) a e ?b(t ?t 0), t >t 0. (2.8) De?nition 2.2(Dolezai [8]).System (2.6)is said to be E-exponentially stable if there exist constants a >0,b >0such that Ex(t) Ex(t 0) a e ?b(t ?t 0), t >t 0. (2.9) The following lemma characterizes the relationship between the exponential stability and the E-exponential stability for the impulsive and switching singular system (2.6). Lemma 2.1.For system (2.6),its E-exponential stability is equivalent to its exponential stability . A proof of Lemma 2.1is similar to that in [11],so details are omitted. 3.Hybrid impulsive and switching singular system In this section,some asymptotic properties of the hybrid system (2.6)under the arbitrary switching are discussed. J.Yao et al./Systems&Control Letters55(2006)879–886881 In system(2.6),let (1+c k)2 k,k=1,2,...,(3.1) and assume that there exist constants0< < such that ln k?2 (t k?t k?1) 0,k=1,2,....(3.2) Theorem3.1.If there exists a constant invertible matrix P such that E P=P E 0,(3.3) A i P+P A i+2 E P<0,i=1,2,...,m,(3.4) then the trivial solution of system(2.6)is globally exponentially stable under arbitrary switching,where k is given by(3.1). Proof.For system(2.6),construct a Lyapunov function in the form of v(t)=x (t)E Px(t),(3.5) and let v(t):=v(x(t)),where P is an invertible constant matrix satisfying(3.3)and(3.4).For t∈(t k?1,t k],the total derivative of v(x(t))with respect to(2.6)is ˙v(x(t))|(2.6)=x A i k Px+x P A i k x=x (A i k P+P A i k )x which implies that v(t) v(t+k?1)e?2 (t?t k?1),t∈(t k?1,t k],k=1,2,.... (3.6) On the other hand,it follows from(2.6)and(3.5)that v(t+k)=x (t+k)E Px(t+k)=(1+c k)2x (t k)E Px(t k) =(1+c k)2v(t k) k v(t k),k=1,2,...,(3.7) where k are de?ned by(3.1). The following results come from(3.6)and(3.7).For t∈(t0,t1], v(t) v(t+0)e?2 (t?t0), which leads to v(t1) v(t+0)e?2 (t1?t0), and v(t+1) 1v(t1) 1v(t+0)e?2 (t1?t0). Similarly,for t∈(t1,t2], v(t) v(t+1)e?2 (t?t1) 1v(t1)e?2 (t?t1) 1v(t+0)e?2 (t?t0).In general,for t∈(t k,t k+1], v(t) v(t+0) 1··· k e?2 (t?t0) =v(t+0) 1··· k e?2 (t?t0)e?2( ? )(t?t0) v(t+0) 1··· k e?2 (t k?t0)e?2( ? )(t?t0) =v(t+0) 1e?2 (t1?t0)··· k e?2 (t k?t k?1)e?2( ? )(t?t0) v(t+0)e?2( ? )(t?t0),t∈(t k,t k+1], namely, v(t) v(t+0)e?2( ? )(t?t0),t>t0, where is given by(3.2).Since E P=P E 0and P is invertible,there exists a positive de?nite symmetric matrix Q such that E P=E QE.Thus,we obtain min(Q) Ex(t) 2 v(x(t)) v(t+0)e?2( ? )(t?t0) Ex(t+0) 2 max(Q)e?2( ? )(t?t0),t>t0, that is, Ex(t) Ex(t+0) max(Q) min(Q)e ?( ? )(t?t0),t>t0, implying by Lemma2.1that system(2.6)is E-exponentially stable under arbitrary switching;that is,system(2.6)is expo-nentially stable under arbitrary switching.This completes the proof. Remark3.1.Based on Theorem3.1,the suggested controller design procedure for systems(2.1)or(2.6)is summarized as follows. Step1:For system(2.6),c k is known,so one can get k(k= 1,2,...)by(3.1),and then go to the next step. Step2:From the impulsive time sequence{t k},one can pick constants and such that0< < and(3.2)holds,and then go to the next step. Step3:Design control gain matrices C i,i=1,2,...,m, substitute them into(3.4)with A i=A+BC i,and then solve the two inequalities(3.3)and(3.4):if there exists a solution of constant invertible matrix P,then one obtains the switching controller u(t)de?ned by(2.3)for system(2.1);otherwise,go back to step1. Example3.1.In early1940s,economists proposed a singu-lar system model to analyze some real problems,such as pro-duction structures,commodity prices,increasing rates,interest rates,etc.For example,the well-known fundamental dynamic Leontief model of economic systems[19]is a singular system. Since administrative effects are included,the discontinuous im-pulsive perturbations at instants t1,t2,...are often inevitable. In this case,the production state vector is no longer continuous and the corresponding Leontief system reduces to a singular-impulsive model.Because of the impulsive effects,the behav-iors of such economic systems in different time intervals are 882J.Yao et al./Systems &Control Letters 55(2006)879–886 00.10.20.30.40.50.60.7 -1012 3x 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1-0.50 0.5 x 2 00.10.20.3 0.40.50.60.7 -15 -10-50 5t x 3 Fig.1.System states of the controlled system (2.6). different.So,it is necessary to apply switching control strategies for this kind of discontinuous systems. Consider the linear singular and impulsive system ??????? ??100010000??˙x =??011?130000??x +??02 1011 ??u,t =t k x =?1.2x(t k ), t =t k . It is easy to see that (1+c k )2< k =1.Select =0.5, =1and t k ?t k ?1=0.06such that ln k ?2 (t k ?t k ?1)=?0.06<0.If one designs C 1= ?11.110?3 ?2.210 and C 2= ?100.36?5 ?1.52?0.5 , then by substituting the above-speci?ed matrices into Eqs.(3.3) and (3.4)and then solving them,one obtains P =??0.3637?0.96550 ?0.96550.196401.3107 0.2156 ??, which implies by Theorem 3.1that the solution of the controlled system (2.6)is globally exponentially stable under arbitrary switching. Simulation results are shown in Fig.1. 4.Robust stabilization of a hybrid impulsive and switching singular system Since uncertainties are frequently a source of instability and encountered in various engineering systems,the issue of ro-bust stabilization for uncertain singular-impulsive systems is considered. In the subsequent discussion,the following lemma will be needed. Lemma 4.1(Lin [17]).Let x be a vector and P ,D ,E and F be matrices with compatible dimensions ,satisfying F F I .The following inequality holds for any real constant >0:2x PDFE x (x PDD P x)+ 1 x E Ex .(4.1) Consider system (2.1)with parameter uncertainties described by E ˙x =(A + A)x +(B + B)u,t =t k x =c 2k x(t k ), t =t k , (4.2) where A and B are time-invariant matrices representing norm-bounded parameter uncertainties,which are assumed to be in the following form: A =MF ( )N, B =RH ( )S , (4.3) where M ,N ,R and S are known real constant matrices with appropriate dimensions.The uncertain matrices F ( )and H ( )satisfy F ( )F ( ) I, H ( )H ( ) I , (4.4) and ∈R p is the uncertain parameter vector. The switching controller (2.3)is applied to system (4.2),and the control gain matrices {C 1k }are chosen as C 1,...,C m ,that is C 1k ∈{C 1,...,C m },leading to the following closed-loop form: ? ??E ˙x =(A i k + A i k )x,t ∈(t k ?1,t k ], x =c k x(t k ),t =t k ,x(t +0)=x 0,k =1,2,..., (4.5) where A i k =A +BC i k , A i k = A + BC i k ,i k ∈{1,2,...,m },and c k =c 2k . For system (4.5),similar to (2.6),assume that assumptions (3.1)and (3.2)hold.Then we have the following conclusion.Theorem 4.1.If there exist two constants 1>0, 2>0and an invertible constant matrix P ,satisfying E P =P E 0, (4.6) A i P +P A i + 1P MM P + 1 1 N N + 2P RR P + 1 2 C i S SC i +2 E P <0,(4.7) with i =1,2,...,m ,then the trivial solution of system (4.5)is globally exponentially stable under arbitrary switching ,where k is given by (3.1). J.Yao et al./Systems &Control Letters 55(2006)879–886883 Proof.The derivative of the Lyapunov function (3.5)along the trajectory of (4.5)is ˙v(x(t))|(4.5)=x (A i k + A i k ) Px +x P (A i k + A i k )x =x (A i k P +P A i k )x +2x P MF ( )Nx +2x P RH ( )SC i k x x A i k P +P A i k + 1P MM P +11N N + 2P RR P +12C i k S SC i k x t ∈(t k ?1,t k ]. (4.8) The remaining part of the proof is similar to that for Theorem 3.1,so details are omitted. Example 4.1.The controlled design procedure here is similar to that in Remark 3.1and therefore is omitted. Consider the linear continuous uncertain singular and impul-sive system (4.2)with parameters given by E =??100010000 ??,A =??01.53?12.500 1 ??,B =? ?0.0700.10.49101.501? ?, c 2k =?0.8, M =??0.2 0.10.1 ??,N =[0.110], R =??100 ??, S =[00.010]. F ( )=H ( )=0.1sin ( ),where F ( ) = H ( ) 0.1.It is easy to verify that (1+c k )2< k =1.Select =0.5, =1and t k ?t k ?1=0.06such that ln k ?2 (t k ?t k ?1)=?0.06<0.If one designs C 1=???3.3924.5?24.5 3.38?36.811.80 1??,C 2=? ?0 0?19.6?3 ?29.6000.1 0? ?and 1= 2=1, then there exists an invertible matrix, P =???0.5584?0.03030?0.03030.270800.050600.2573 ??, satisfying (4.6)and (4.7),which implies by Theorem 4.1 that the solution of the controlled system (4.5)is globally E-exponentially stable under arbitrary switching. 00.10.20.30.40.50.60.7 012 3x 1 00.10.20.30.40.50.60.7 -1-0.5 0x 2 0.1 0.2 0.3 0.40.50.60.7 012 3t x 3 Fig.2.System states of the controlled system (4.5). Simulation results are shown in Fig.2. 5.H ∞control of a hybrid impulsive and switching singular system Consider the following linear uncertain singular-impulsive dynamic system:?????????E ˙x =Ax +B 1u +D 1w,t =t k ,Z =Gx +B 2u +D 2 w, x =c k x(t k ),t =t k ,x(t + 0)=x 0, k =1,2,...,(5.1) where w ∈R p is the disturbance input,Z ∈R q is the con-trolled output,B 1,B 2,G,D 1and D 2are known matrices of appropriate dimensions,and A and E are n ×n matrices with det (E)=0. For the disturbance signal w(·)∈R p ,de?ne w T = T 0 w(t) 2d t 1/2= T 0 w(t) w(t)d t 1/2 , (5.2) where T >0is an arbitrary constant.Then,w is said to belong to L 2[0,T ],if w T <∞.Throughout the paper,it is assumed that the disturbance input w ∈L 2[0,T ]. Now,one is in a position to consider the robust H ∞control problem for system (5.1).When the controller (2.3)is used,similar to the above discussion,that is C 1k ∈{C 1,...,C m },the closed-loop system of (5.1)becomes ?????????E ˙x =A i k x +D 1w,t ∈(t k ?1,t k ],Z =G i k x +D 2 w, x =c k x(t k ),t =t k ,x(t + 0)=x 0,k =1,2,..., (5.3) 884J.Yao et al./Systems&Control Letters55(2006)879–886 where A i k =A+B1C i k ,G i k =G+B2C i k ,and i k∈{1,2,...,m}. It is assumed[6]that the pair(E,A i)are regular;that is, det(sE?A i)=0for some complex number s. With the above preliminaries,the robust H∞control problem to be addressed here can be formulated as a problem of using a switching controller to achieve the following objectives: (i)The closed-loop system(5.3)is exponentially stable under arbitrary switching when w=0. (ii)Under the zero-initial condition,the controlled output Z satis?es Z T w T, for any nonzero w(·)∈L2[0,T],where >0is a pre-scribed scalar. The above conditions are also called robust H∞criteria for the closed-loop system(5.3). For system(5.3),similar to(2.6),assume that assumptions (3.1)and(3.2)hold. Theorem5.1.If there exist two nonnegative constantsεand ,such thatε+ D2 2< 2and an invertible constant matrix P,satisfying E P=P E 0,(5.4) i=A i P+P A i+G i G i+εI +εC i C i+( 2?ε? D2 2)?1 ×(D 1P+D 2G i) (D 1P+D 2G i) +2 E P<0,(5.5) with i=1,2,...,m,then the closed-loop impulsive system(5.3) satis?es the robust H∞criteria(i)and(ii). https://www.doczj.com/doc/f14817312.html,e the Lyapunov function(3.5)and de?ne (t)=˙v(t)+2 v(t)+ Z 2? 2 w 2 +ε( x 2+ u 2+ w 2),t∈(t k?1,t k],(5.6) which yields from(5.3)that (t)=(x A i k +w D 1)Px+x P (A i k x+D1w) +x G i k D2w+w D 2G i k x +x (2 E P+G i k G i k +εI+εC i k C i k )x ?( 2?ε)w w+w D 2D2w x (A i k P+P A i k +2 E P+G i k G i k +εI+εC i k C i k )x +x (P D1+G i k D2)w+w (D 1P+D 2G i k )x ?( 2?ε? D2 2)w w,t∈(t k?1,t k]. Subtract[( 2?ε? D2 2)?1(D 1P+D 2C ik)x?w] ( 2?ε? D2 2)[( 2?ε? D2 2)?1(D 1P+D 2C ik)x?w]from both sides of x i x.It follows that for all x and for all w∈L2[0,∞),there exists an i∈{1,2,...,m}such that (t)<0,t∈(t k?1,t k], namely, ˙v(t)+2 v(t)+ Z 2? 2 w 2 0,t∈(t k?1,t k],(5.7) For any given T∈(t k?1,t k],integrating(5.7)from0to T with x(0)=0gives T ˙v(t)d t+2 T v(t)d t+ Z 2T? 2 w 2T<0, where T ˙v(t)d t= t 1 ˙v(t)d t+ t 2 t1 ˙v(t)d t +···+ t k t k?1 ˙v(t)d t+ T t k ˙v(t)d t =v(t1)?v(0)+v(t2)?v(t+1)+···+v(t k) ?v(t+k?1)+v(T)?v(t+k) =v(t1)?v(t+1)+···+v(t k?1) ?v(t+k?1)+v(t k)?v(t+k)+v(T) =? k i=1 c i(2+c i)v(t i)+v(T) 0. Thus, T ˙v(t)d t+2 T v(t)d t 0, and Z 2T? 2 w 2T<0,(5.8) which immediately reduces to Z T< w T. When w=0,from(5.7)and(5.8),one has ˙v(t)+2 v(t)<0,t∈(t k?1,t k], The remaining part of the proof is similar that for Theorem3.1,so details are omitted. Example5.1.The robust H∞control design procedure here is given following that from stability analysis discussed in Remark3.1. J.Yao et al./Systems &Control Letters 55(2006)879–886885 0.2 0.4 0.6 0.8 1 1.2 1.4 -202 4x 1 0.2 0.4 0.6 0.8 1 1.2 1.4 -1-0.50 0.5x 2 0.2 0.4 0.6 0.8 1 1.2 1.4 -10 -50 5t x 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -202 4z 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -202 4z 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -202 4t z 3 Fig.3.States x and controlled output Z of the closed-loop system with w =0. Consider the linear uncertain singular-impulsive system ????????????????????????? ???????????????????????? ??100010000??˙x =??1.200.31.100.5000.2??x +???0.10 0.2411.50.8??u +??0?0.05000?0.01?0.0500??w,t =t k Z =???0.26?0.11?0.19?0.37?0.64?0.116?0.11?0.01?0.03??x +??2.640.66 3.780.731.170.78??u +??0.8000?0.40000.3??w x =?0.5x(t k ),t =t k .It is easy to verify that (1+c k )2< k =1.Select =0.5, =1and t k ?t k ?1=0.06such that ln k ?2 (t k ?t k ?1)=?0.06<0.If one designs C 1= 0?0.500.01?0.1?0.5 ,C 2= ?2?2.5?0.3 13?0.4 , ε=0.6 and =5.3852, then there exists an invertible matrix,P =? ??30.35560.37360 0.3736 ?30.4121 011.1547 6.0516 ? ?,satisfying (5.4)and (5.5),which implies by Theorem 5.1that the closed-loop impulsive system (5.3)satis?es the robust H ∞criteria (i)and (ii). 00.20.40.60.81 1.2 1.4 1.6 1.82 51015202530 t R a t i o o f t h e o u t p u t e n e r g y t o t h e d i s t u r b a n c e i n p u t e n e r g y Fig.4.The ratio of the output energy to the disturbance input energy. When w =[13sin (2 t ?1),23cos (2 t +2),23sin ( t ?1)] , the trajectories of states and the controlled output of the closed-loop system (5.3)are shown in Fig.3.The ratio of the output energy to the disturbance input energy,i.e., T 0Z(t) Z(t)d t/ T 0w(t) w(t)d t ,is depicted in Fig.4,which shows that the maximum value of the ratio is about 28.2965.So =√28.2965=5.3194,which is less than the prescribed value 5.3852. 6.Conclusions The issues of stability,robust stabilization and H ∞control of singular-impulsive systems using switching controllers have been studied and corresponding results have been presented. 886J.Yao et al./Systems&Control Letters55(2006)879–886 Some new fundamental properties have been derived for the switched singular systems with impulse effects.Based on the Lyapunov function theory,some new conditions for exponential stability,robust stabilization and H∞criteria of the resulting closed-loop systems have also been derived.Three examples have been shown to verify the effectiveness of the proposed control and stabilization methods.To the best of our knowledge, this paper offers the?rst approach with some fundamental re-sults about impulsive and switching singular systems and their corresponding control problems.More research efforts will be devoted to this important class of systems and related problems in our future research,targeting more real-world applications of this kinds of important control systems. References [1]J.D.Aplevich,Implicit Linear Systems,Springer,New York,1991. [2]D.D.Bainov,P.S.Simeonov,Stability Theory of Differential Equations with Impulse Effect:Theory and Applications,Ellis Horwood,Chichester, 1989. [3]M.S.Branicky,Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,IEEE Trans.Automat.Control43(4) (1998)475–482. [4]K.E.Brenam,S.L.Campbell,L.R.Petzold,Numerical Solution of Initial Value Problems in Differential-Algebraical Equations,Elsevier,New York,1989. [5]J.Daafouz,P.Riedinger,C.Iung,Stability analysis and control synthesis for switched systems:a switched Lyapunov function approach,IEEE Trans.Automat.Control47(11)(2002)1883–1887. [6]L.Dai,Singular Control Systems,Springer,New York,1989. [7]R.A.Decarlo,M.S.Branicky,S.Pettersson,B.Lennartson,Perspectives and results on the stability and stabilizability of hybrid systems,in: Proceedings of IEEE,2000,pp.1069–1082. 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[22]https://www.doczj.com/doc/f14817312.html,ls,A.A.Goldenberg,Force and position control of manipulators during constrained motion tasks,IEEE Trans.Robot.Automat.38(1989) 30–46. [23]R.W.Newcomb,B.Dziurla,Some circuits and systems applications of semistate theory,Circuits Systems Signal Process.8(1989)235–260. [24]S.Pettersson,B.Lennartson,LMI for stability and robustness of hybrid systems,in:Proceedings of the American Control Conference,1997, pp.1714–1718. [25]B.L.Stevens,F.L.Lewis,Aircraft Modeling,Dynamics and Control, Wiley,New York,1991. [26]Z.Sun,S.S.Ge,On dynamic output feedback stabilization of a class of switched linear systems,IEEE Trans.Circuits and Systems50(8) (2003)1111–1115. [27]J.A.K.Suykens,T.Yang,J.Vandewalle,L.O.Chua,Impulsive control and synchronization of chaos,in:G.Chen(Ed.),Controlling Chaos and Bifurcations in Engineering Systems,CRC Press,Boca Raton,FL,USA, 1999,pp.275–298. [28]H.Ye,A.N.Michel,L.Hou,Stability analysis of switched systems, Conference on Decision Control,Kobe,Japan,1996. [29]M.Zhang,T.J.Tarn,A hybrid switching control strategy for nonlinear and underactuated mechanical systems,IEEE Trans.Automat.Control 48(10)(2003)1777–1782. TECHNICAL RESOURCE Introduction Proteins comprise an extremely heterogeneous class of biological macromolecules. They are often unstable when not in their native environments, which can vary considerably among cell compartments and extracellular fluids. If certain buffer conditions are not maintained, extracted proteins may not function properly or remain soluble. Proteins can lose activity as a of the protein and the storage conditions used. Optimal conditions for storage are distinctive to each protein; nevertheless, it is possible to suggest some general guidelines for protein storage and stability. Common conditions for protein storage are summarized and compared in Table 1. Generally, there are tradeoffs associated with each method. For example, proteins stored in solution at 4°C can be dispensed conveniently as needed but require more diligence to prevent microbial or proteolytic degradation; such proteins may not be stable for more than a few days or weeks. By contrast, lyophilization allows for long-term storage of protein with very little threat of degradation, but the protein must be reconstituted before use and may be damaged by the lyophilization process. Table 1. Comparison of Protein Storage Conditions Characteristic Solution at 4°C Solution in 25-50% glycerol or ethylene Frozen at -20° to -80°C or in liquid nitrogen Lyophilized (usually also frozen) Typical shelf life 1 month 1 year Years Years Requires sterile conditions or addition of antibacterial agent Yes Usually No No Number of times a sample may be removed for use Many Many Once; repeated freeze-thaw cycles generally degrade proteins impractical to lyophilize a sample multiple times Protein stability and storage l t O w .d o c u -t r a c k .c C i c k o b u y N w w o m 冻干法 第八讲 稳定性模型 虽然动态过程的变化规律一般要用微分方程建立的动态模型来描述，但是对于某些实际问题，建模的主要目的并不是要寻求动态过程每个瞬时的性态，而是研究某种意义下稳定状态的特征，特别是当时间充分长以后动态过程的变化趋势。譬如在什么情况下描述过程的变量会越来越接近某些确定的数值，在什么情况下又会越来越远离这些数值而导致过程不稳定。为了分析这种稳定与不稳定的规律常常不需要求解微分方程，而可以利用微分方程稳定性理论，直接研究平衡状态的稳定性就行了。 引言：微分方程稳定性理论简介 定义1 称一个常微分方程（组）是自治的，如果方程（组） ? ???? ?????==),(),(),(1t x f t x f t x F dt dx N M （1） F 中的，即在)(),(x F t x F =中不含时间变量。 t 事实上，如果增补一个方程，一个非自治系统可以转化自治系统，就是说，如果定义 ， ??????=t x y ?? ????=1),()(t x F y G 且引入另一个变量，则方程（1）与下述方程 s )(y G ds dy = 是等价的。这就是说自治系统的概念是相对的。下面仅考虑自治系统，这样的系统也称为动力系统。 定义2 系统 )(x F dt dx = （2） n R 2=n 的相空间是以为坐标的空间),,(1n x x L ，特别，当时，称相空间为相平面。 空间n R 中的点集 },,1,)2()(|),,{(1n i t x x x x i i n L L ==满足 称为系统（2）的轨线，所有轨线在相空间中的分布图称为相图。 定义3 相空间中满足的点称为系统（2）的奇点（或平衡点）。 0)(0=x F 0x 奇点可以是孤立的，也可以是连续的点集。例如，系统 ???????+=+=dy cx dt t dy by ax dt t dx )() ( （3） 当时，有一个连续的奇点的集合。当0=?bc ad 0≠?bc ad 时，是这个系统的 唯一的奇点。下面仅考虑孤立奇点。为了知道何时有孤立奇点，给出下述定理： )0,0( ANDAs: Stability Testing of Drug Substances and Products U.S. Department of Health and Human Services Food and Drug Administration Center for Drug Evaluation and Research (CDER) June 2013 Generics ANDAs: Stability Testing of Drug Substances and Products Additional copies are available from: Office of Communications Division of Drug Information, WO51, Room 2201 Center for Drug Evaluation and Research Food and Drug Administration 10903 New Hampshire Ave., Silver Spring, MD 20993 Phone: 301-796-3400; Fax: 301-847-8714 druginfo@https://www.doczj.com/doc/f14817312.html, https://www.doczj.com/doc/f14817312.html,/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/default.htm U.S. Department of Health and Human Services Food and Drug Administration Center for Drug Evaluation and Research (CDER) June 2013 Generics 微分方程平衡点及其稳定性理论 这里简单介绍下面将要用到的有关内容： 一、 一阶方程的平衡点及稳定性 设有微分方程 ()dx f x dt = （1） 右端不显含自变量t ，代数方程 ()0f x = （2） 的实根0x x =称为方程（1）的平衡点（或奇点），它也是方程（1）的解（奇解） 如果从所有可能的初始条件出发，方程（1）的解()x t 都满足 0lim ()t x t x →∞ = （3） 则称平衡点0x 是稳定的（稳定性理论中称渐近稳定）；否则，称0x 是不稳定的（不渐近稳定）。 判断平衡点0x 是否稳定通常有两种方法，利用定义即（3）式称间接法，不求方程（1）的解()x t ，因而不利用（3）式的方法称直接法，下面介绍直接法。 将()f x 在0x 做泰勒展开，只取一次项，则方程（1）近似为： 0'()()dx f x x x dt =- （4） （4）称为（1）的近似线性方程。0x 也是（4）的平衡点。关于平衡点0x 的稳定性有如下的结论： 若0'()0f x <，则0x 是方程（1）、（4）的稳定的平衡点。 若0'()0f x >，则0x 不是方程（1）、（4）的稳定的平衡点 0x 对于方程（4）的稳定性很容易由定义（3）证明，因为（4）的一般解是 0'()0()f x t x t ce x =+ （5） 其中C 是由初始条件决定的常数。 二、 微分方程组的平衡点和稳定性 方程的一般形式可用两个一阶方程表示为 112212()(,)()(,)dx t f x x dt dx t g x x dt ?=????=?? （6） 右端不显含t ，代数方程组 1212 (,)0(,)0f x x g x x =??=? （7） 的实根0012 (,)x x 称为方程（6）的平衡点。记为00012(,)P x x 如果从所有可能的初始条件出发，方程（6）的解12(),()x t x t 都满足 101lim ()t x t x →∞= 202lim ()t x t x →∞ = （8） 则称平衡点00012(,)P x x 是稳定的（渐近稳定）；否则，称P 0是不稳定的（不渐 近稳定）。 为了用直接法讨论方法方程（6）的平衡点的稳定性，先看线性常系数方程 1111222122()()dx t a x b x dt dx t a x b x dt ?=+????=+?? （9） 系数矩阵记作 1122a b A a b ??=???? 并假定A 的行列式det 0A ≠ 于是原点0(0,0)P 是方程（9）的唯一平衡点，它的稳定性由的特征方程 det()0A I λ-= 的根λ（特征根）决定，上方程可以写成更加明确的形式： 2120()det p q p a b q A λλ?++=?=-+??=? （10） 将特征根记作12,λλ，则 ICH Stability Climatic Zones Countries of climatic zones I and II: Europe: EU, Belarus, Bulgaria, Estonia, Hungary, Latvia, Lithuania, Norway, Rumania, Russia, Switzerland, Ukraine America: USA, Argentina, Bolivia, Chile, Canada, Mexico, Peru, Uruguay Africa: Egypt, Algeria, Canary Islands, Libya, Morocco, Namibia, Rwanda, South Africa, Tunisia, Zambia, Zimbabwe Asia: Japan, Afghanistan, Armenia, Azerbaijan, China, Georgia, Iran, Israel, Kazakhstan, Kirghizia, Korea, Lebanon, Nepal, Syria, Tadzhikistan, Turkey, Turkmen, Uzbekistan, Australia: New Zealand. Countries of climatic zones III and IV: America: Barbados, Belize, Brazil, Costa Rica, Dominican Republic, Ecuador, El Salvador, Guatemala, Guyana, Haiti, Honduras, Jamaica, Columbia, Cuba, Nicaragua, Dutch Antilles, Panama, Paraguay, Puerto Rico, Venezuela. All these countries are assigned to CZ IV. Africa: Angola, Ethiopia, Benin, Botswana (111), Burkino Faso, Burundi, Djibouti, Ivory Coast, Gabon, Gambia, Ghana, Guinea, TINA-TI应用实例：运算放大器的稳定性分析 原创：TI美国应用工程经理：Tim Green 译注：TI中国大学计划黄争Frank Huang 负反馈电路在运算放大器的应用中起着非常重要的作用，它可以改善运放的许多特性，比如稳定增益，减小失真，扩展频带，阻抗变换等。但是任何事情都有两面性，同样地，负反馈的引入也有可能会使得运放电路不稳定。不稳定轻则可能带来时域上的过冲，而最坏情况就是振荡，即输出中产生预料之外的持续振幅和频率信号。当不期望的振荡发生时，通常会给电路带来许多负面影响：一个最明显的例子是，当恒压源通过运放缓冲后送到ADC的参考电压端，如果运放发生振荡，会给整个电路的测量结果带来完全不可靠的数据。 本章中主要分析了电压反馈型运算放大器不稳定的原因；给出了使用伯特图来分析运放稳定性的方法；最后结合TINA-TI SPICE仿真软件，通过一个实例介绍了分析和解决运算放大器稳定性问题的方法。关于TINA-TI与运放稳定性的更深入讨论可以参考TI公司线性产品应用经理Tim Green先生所撰写的《Operational Amplifier Stability》一文[1]。这里也感谢Tim Green先生对本文提供的大量原始资料和技术指导。 5.1 运算放大器为什么会不稳定？ 要分析和解决运放的稳定性问题，首先要清楚为什么运算放大器会不稳定。我们还是先从负反馈电路谈起，以同相放大器的方框图为例来推导反馈系统的一系列方程，如图5.1。同时为更形象地描述运算放大器中的负反馈，绘制一个与图5.1等效的同相放大器如图5.2，注意β等系数在两图中的对应关系。 图5.1 负反馈框图 1 Leader-to-Formation Stability Herbert G.Tanner George J.Pappas Vijay Kumar Abstract—We investigate the stability for robot formations from a different perspective,focusing on the dependence of group con-?guration to incoming input singals or disturbances.Our idea builds on the notion of input-to-state stability to de?ne leader-to-formation stability(LFS),as a means to analyze error propagation and performance characterization.Contrary to other notions of stability for interconnected systems,in leader-to-formation stabil-ity the focus is shifted from disturbance rejection to quantifying transient and steady state behaviour,thus being able to relax con-ditions,address a larger class of systems and provide insight to issues of performance improvement in relation to interconnection topology.The new concepts are implemented numerically in the case of a formation of mobile robots where(LFS)also indicates the formation structures that ensure the smallest errors during maneuvering. I.I NTRODUCTION Interconnected systems have lately received considerable at-tention,motivated by recent advances in computation and com-munication,which provide the enabling technology for appli-cations such as automated highway systems[1],cooperative robot reconnaissance[2],[3]and manipulation[4],[5],forma-tion?ight control[6],[7],satellite clustering[8]and control of groups of unmanned vehicles[9],[10],[6].Formation control is one aspect of the study of a patricular class of interconnected systems. One research thrust aims at network architectures and coor-dination methods as a means of generating a group behavior in a formation of vehicles.In behavior-based approaches[2], [11],[12]the group behavior emerges as a combination of group member behaviors,selected among a set of primitive ac-tions.Agent behavior has alternatively been designed so that the group members move as being particles in a rigid virtual structure[13].Similar ideas have been combined with potential ?eld-like controllers for multi agent control[14],[4],[15],and decentralized formation forming[16].The leader-follower ap-proach[17],[18],[19]distinguishes a designated group leader which the other agents follow either directly or indirectly. Another research direction focuses on the stability of the in-terconnected system.In[20]a distributed control scheme is designed for spatially interconnected systems that is shown to inherit the same topological structure with the target system. String stability[21],[22],[1]and mesh stability[23],the latter Herbert Tanner is with the Department of Electrical and Systems Engineer-ing,3401Walnut Street,Suite301C,University of Pennsylvania,Philadelphia, PA19104-6228(e-mail:tanner@https://www.doczj.com/doc/f14817312.html,) George Pappas is with the Department of Electrical and Systems Engineering, 200South33rd Street,University of Pennsylvania,Philadelphia,PA19104(e-mail:pappasg@https://www.doczj.com/doc/f14817312.html,) Vijay Kumar is with the Department of Mechanical Engineering and Ap-plied Mechanics,3401Walnut Street,Suite301C,University of Pennsylvania, Philadelphia,PA19104-6228,(e-mail:kumar@https://www.doczj.com/doc/f14817312.html,)being the generalization of the former in more than two dimen- sions,express the property of the system to attenuate distur-bances as they propagate through the interconnections.While earlier works[24],[21],[22],[25]have used the notion of string stability in the frequency domain,string stability of in-terconnected systems has recently been studied in a state space framework[1].In[1],suf?cient conditions for string stability were derived,requiring global Lipschitz continuity of vector ?elds and exponential stability of the unconnected subsystems. Weaker notions of string stability include string stability [1],which is the only type of stability that can be guaranteed for an interconnected system without a special structure.It is also known that in certain cases string stability is impossible [26],[25],[22]. The class of formations discussed in this paper generally do not fall under the class of string stable interconnected systems. On the other hand,it is generally the case that formations are stable at least locally,in the sense that local controllers can en-sure some boundedness of errors.A question then arises as to whether further stability analysis can be performed to address issues such as boundedness of propagating errors and perfor-mance characterization.This question motivates the introduc-tion of a different framework,that brings forward the depen-dence of the internal state of an interconnected system to in-coming input signals or disturbances,rather than ensuring con-vergence of signals to zero. In this paper we de?ne Leader-to-Formation Stability(LFS), a different notion of stability for leader-following formations, and we use it to characterize how the motion of the group lead-ers can affect the motion of the group.This notion is based on input-to-state stability[27]and its propagation properties through certain interconnections[28],[29].In previous work [30],[31],[32]we have been able to characterize the effect of the behavior of the group leaders to the rest of the group for cer-tain types of formation interconnections and obtain quantitative measures of the stability of the formation with respect to the motion of the leaders.In this paper we generalize these results and exploit LFS for analysis and design of robot formations. II.D EFINITIONS AND P RELIMINARY R EMARKS We consider formations that are based on leader-follower ar-chitectures.In that framework,a formation will be broadly de-?ned as a network of vehicles interconnected through their con-trollers,with the latter being designed so that the motion of the vehicles meet certain speci?cations.In the formations consid-ered,a number of vehicles are identi?ed as designated leaders in the sense that their motion is not constrained by the forma-tion speci?cations.Related work[6],[17],[18],[33],[34]has motivated the use of graphs to represent agent interaction.We Abstract: Ship plates are stiffened using different stiffeners. In this paper, a Y stiffener is considered and investigated. A Y stiffener–plate combination model is used to represent the stiffened panel. Our contribution includes characterizing the local instability (buckling) of Y stiffeners in stiffened panels under the action of uniaxial compressive loads. The mathematical derivations have been carried out to find the elastic buckling coefficient for the web of the T-part of the Y stiffener under suitable boundary conditions. Then, the critical value of the buckling stress has been calculated. Using the value of the critical stress and the assumption of uniform stress distribution, the buckling load is calculated for the Y stiffener–plate combination model. Using curve fitting of the analytically obtained results, approximate expressions for calculation of the elastic buckling coefficient of the T-part of the Y stiffener are obtained. These approximate expressions enable designers to calculate easily the elastic buckling coefficient from which the critical buckling stress of the T-part is obtained. 船板用不同的扶强材加固。在本文中，考虑和调查Y加劲肋。Y加劲板组合模型用于表示加劲板。我们的贡献包括在单轴压缩载荷的作用下，在加劲板中表征Y加劲肋的局部不稳定性（屈曲）。进行数学推导以在合适的边界条件下找到Y加劲肋的T形部分的腹板的弹性屈曲系数。然后，计算弯曲应力的临界值。使用临界应力的值和均匀应力分布的假设，计算Y加强板组合模型的屈曲载荷。使用解析获得的结果的曲线拟合，获得用于计算Y加强件的T部分的弹性屈曲系数的近似表达式。这些近似表达式使得设计者能够容易地计算得到T部分的临界屈曲应力的弹性屈曲系数。 INTRODUCTION Ship plates are stiffened using stiffeners that are steel sec- tions having various shapes. In continuous stiffened ship plating, a stiffener (minor member) with attached plating is idealized by plate–stiffener combination model whose length extends perpendicularly between two adjacent major support members (transverse bulkheads). Various shapes of stiffeners may be classified according to their complexity into two groups. The first group of stiffeners that may be termed con- ventional (or typical) has simple shapes, easy fabrication, easy maintenance, and relatively low cost. However, this group in some cases is not capable of giv- ing the required stiffening that is necessary to prevent fail- ure of ship plating. Among this group of stiffeners, we can mention flat bar, T section and angle section. Geometri- cal configurations of conventional (typical) plate–stiffener combination sections are shown in Figure 1.The second group of stiffeners that may be termed non- conventional has relatively complicated shapes. These non- conventional stiffeners give stronger support to ship plates. Among the non-conventional stiffeners we can men- tion, In use stability专题研究 Zhulikou431 内部培训 2012 中国 谨记 纸上得来终觉浅， 绝知此事要躬行！ ---陆游 1.本培训资料参考文献更新至20121101. 2.本专题资料主要针对药品in use stability研究。 3.任何宝贵建议，请联系zhulikou431@https://www.doczj.com/doc/f14817312.html,. 目录（contents） 第01章：概念解析 第02章：中国药典对药品使用期间稳定性要求 第03章：cde指导原则对药品使用期间稳定性要求第04章：EMA指南对药品使用期间稳定性要求 第05章：WHO对药品使用期间稳定性要求 第06章：cde电子刊物对药品使用期间稳定性要求第07章：其他法规文献阐述 In use stability专题研究 第01章：in use stability概念解析 In use stability，顾名思义，指的是药品使用期间稳定性研究项目。 对于如下使用条件的药品，需要考察in use stability 项目： ---药品使用前，需要重新配置或者稀释； ---药品标签声明，和其他药品混合仍然具有稳定性；---药品包装多次打开以后，药品需要继续保持质量稳定性。 In use stability专题研究 第02章：中国药典对药品使用期间稳定性要求中国药典2010版附录XIX C《原料药与药物制剂稳定性试验指导原则》也要求： 此外，有些药物制剂还应考察临用时配制和使用过程中的稳定性。 In use stability专题研究 第03章：cde指导原则对药品使用期间稳定性要求 中国cde2005年发布了《化学药物稳定性研究技术指导 原则》，其中对于in use stability，进行了明确规定： 稳定性试验要求在一定的温度、湿度、光照条件下进行，这些放置条件的设置应充分考虑到药品在贮存、运输及使用过程中可能遇到的环境因素。 对于需要溶解或者稀释后使用的药品，如注射用无菌粉末、溶液片剂等，还应考察临床使用条件下的稳定性。 l 9-Fluorenylmethyl carbamate, Fmoc amino, Fmoc amine, Fmoc amide: H 2O: pH < 1, 100°C pH = 1, RT pH = 4, RT pH = 9, RT pH = 12, RT pH > 12, 100°C Bases: LDA NEt 3, Py t-BuOK Others: DCC SOCl 2 Nucleophiles: RLi RMgX RCuLi Enolates NH 3, RNH 2 NaOCH 3 Electrophiles: RCOCl RCHO CH 3I Others: :CCl 2 Bu 3SnH Reduction: H 2 / Ni H 2 / Rh Zn / HCl Na / NH 3 LiAlH 4 NaBH 4 Oxidation: KMnO 4 OsO 4 CrO 3 / Py RCOOOH I 2, Br 2, Cl 2 MnO 2/CH 2Cl 2 l t -Butyl carbamate, Boc amine, Boc amino, Boc amide: H 2O: pH < 1, 100°C pH = 1, RT pH = 4, RT pH = 9, RT pH = 12, RT pH > 12, 100°C Bases: LDA NEt 3, Py t-BuOK Others: DCC SOCl 2 Nucleophiles: RLi RMgX RCuLi Enolates NH 3, RNH 2 NaOCH 3 Electrophiles: RCOCl RCHO CH 3I Others: :CCl 2 Bu 3SnH Reduction: H 2 / Ni H 2 / Rh Zn / HCl Na / NH 3 LiAlH 4 NaBH 4 Oxidation: KMnO 4 OsO 4 CrO 3 / Py RCOOOH I 2, Br 2, Cl 2 MnO 2 / CH 2Cl 2 l Benzyl carbamate, Cbz-NR 2 / Z-NR 2: H 2O: pH < 1, 100°C pH = 1, RT pH = 4, RT pH = 9, RT pH = 12, RT pH > 12, 100°C Bases: LDA NEt 3, Py t-BuOK Others: DCC SOCl 2 Nucleophiles: RLi RMgX RCuLi Enolates NH 3, RNH 2 NaOCH 3 Electrophiles: RCOCl RCHO CH 3I Others: :CCl 2 Bu 3SnH Reduction: H 2 / Ni H 2 / Rh Zn / HCl Na / NH 3 LiAlH 4 NaBH 4 Oxidation: KMnO 4 OsO 4 CrO 3 / Py RCOOOH I 2, Br 2, Cl 2 MnO 2 / CH 2Cl 2 l Acetamide, Ac-NR 2: H 2O: pH < 1, 100°C pH = 1, RT pH = 4, RT pH = 9, RT pH = 12, RT pH > 12, 100°C Bases: LDA NEt 3, Py t-BuOK Others: DCC SOCl 2 Nucleophiles: RLi RMgX RCuLi Enolates NH 3, RNH 2 NaOCH 3 Electrophiles: RCOCl RCHO CH 3I Others: :CCl 2 Bu 3SnH Reduction: H 2 / Ni H 2 / Rh Zn / HCl Na / NH 3 LiAlH 4 NaBH 4 Oxidation: KMnO 4 OsO 4 CrO 3 / Py RCOOOH I 2, Br 2, Cl 2 MnO 2 / CH 2Cl 2 l Trifluoroacetamide: H 2O: pH < 1, 100°C pH = 1, RT pH = 4, RT pH = 9, RT pH = 12, RT pH > 12, 100°C Bases: LDA NEt 3, Py t-BuOK Others: DCC SOCl 2 Nucleophiles: RLi RMgX RCuLi Enolates NH 3, RNH 2 NaOCH 3 Electrophiles: RCOCl RCHO CH 3I Others: :CCl 2 Bu 3SnH O N O O N O O N O N O N O F F F 汽车电器与电子设备 电子稳定控制系统Electronic Stability Program 重庆汽车学院 目录 1 概述 (3) 2实际意义 (4) 3 生产商 (4) 4各种相似系统 (4) 5 发展史 (5) 6 关键技术 (6) 7 国内现状 (8) 附一：国产30万以下标配ESP车型 (9) 附二：模拟ESP效果图 (10) 1 概述 Electronic Stability Program（以下简称为ESP），汉语翻译为电子稳定控制系统。它其实就是牵引力控制系统（Traction Control System）的升级版本，与其他牵引力控制系统比较，ESP不但控制驱动轮，而且可控制从动轮。如后轮驱动汽车常出现的转向过多情况，此时后轮失控而甩尾，ESP 便会刹慢外侧的前轮来稳定车子；在转向过少时，为了校正循迹方向，ESP则会刹慢内后轮，从而校正行驶方向。 ESP系统包含ABS（Anti-lock Brake System，防抱死刹车系统）及ASR（Acceleration Slip Regulation防侧滑系统），是这两种系统功能上的延伸。因此，ESP称得上是当前汽车防滑装置的最高级形式。ESP系统由控制单元及转向传感器（监测方向盘的转向角度）、车轮传感器（监测各个车轮的速度转动）、侧滑传感器（监测车体绕垂直轴线转动的状态）、横向加速度传感器（监测汽车转弯时的离心力）等组成。控制单元通过这些传感器的信号对车辆的运行状态进行判断，进而发出控制指令。有ESP与只有ABS及ASR的汽车，它们之间的差别在于ABS及ASR只能被动地作出反应，而ESP则能够探测和分蛋白保存方法(Protein_stability_and_storage_)
稳定性模型
Stability Testing of Drug Substances and Products(FDA)
数学建模平衡点稳定性
ICH Stability Climatic Zones
Operational Amplifier Stability
Leader-to-formation stability
stability 翻译
in use stability专题研究
常用氨基保护基的稳定性Stability-PGofAmine
Electronic Stability Program电子稳定控制系统
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