(i.0 where x~n-dimensional state vector, u--r-dimensional control matrix, A--nx,

Transcription

1 Applied Mathematics and Mechanics (English Edition, Vol.3, No.5, Oct. 1982) Published by HUST Press, Wuhan, Hubei. i APPLICATION OF THE THEORY OF BRANCHING OF SOLUTIONS OF NONLINEAR EQUATIONS IN THE ESTIMATION OF POLE PERTURBANCE OF LINEAR CLOSED-LOOP SYSTEMS* Zhang Rong-xiang(~) (Shandong Engineering Institute) Chen Zha0-kuan( St~ ) (Shandong Abstract university) (Received Nov. 16, 198t) The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic ABSTRACT index of detonation products equals to three. In behavior of the When reflection the state shock and in the input explosive matrices products, of a and multivariable applying the small linear parameter purtime-invariant system are perturbed, the problem of the esterbation method, an analytic, first-order approximate solution is obtained for the problem of flying timation of pole perturbance of closed-loop system is conplate driven by various high explosives with polytropic indices other than but nearly equal to three. sidered by making use of the theory of branching of solutions Final velocities for of nonlinear flying plate equations. obtained agree very well with numerical results by computers. Thus I. index) Problem for estimation Statement of the velocity of flying plate is established. The pole assignment of linear closed-loop systems is one of the important problems in control system design 1. which Introduction affects considerably the quality of re- gulation. However, there are always certain errors in the mathematical model materials constructed under for intense any impulsive practical loading, system. shock synthesis It implies of diamonds, that the and system explosive is welding to be and percladding turbed. of The metals. existence The method of of this estimation perturbation of flyor velocity will and certainly the way of have raising an it influence are questions on the poles Pre-assigned and thereby on the regulation quality of the system.thus Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach the estimation of solving of the pole problem perturbance of motion of flyor linear is to closed-loopsystem solve the following system is quite.an of equations imgoverning portant the problem flow field whether of detonation in theory products or behind practice. the flyor (Fig. I): given by The problem is put forward as follows: Assume that a multi-variable ap +u_~_xp linear + time-invariantunperturbed au au au 1 system is. dz =M~+~u (1.1') dt where x~n-dimensional state vector, u--r-dimensional control matrix, A--nx, as as state matrix and B~n control matrix, and in terms of n-poles to be assig- ned: where p, p, S, u are pressure, density, ~L, specific 1=, "", entropy ~, and particle velocity of detonation products (1.2) trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- * Communicated by Li Hao. meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave

2 720 Zhang Rong-xiang the state feedback matrix K : has been obtained. K=[k,,],. (1.3) becomes Now the system model is perturbed: matrix A becomes A+~AI and matrix B are as follows: B+~B~, here ~ is the magnitude of perturbation, matrices A, and B l Let A= [a,,]... B= [b.,]... A,=[a,,]... la,,l<la,,i (1.4) B,=[~,,]... I~,,l~lb,,I (l.s) thus the perturbed system is Abstract dx The one-dimensional problem dt = (A of +ea, the motion )x+ (B+~B,)u of a rigid flying plate under explosive attack ( has I. 6) an analytic solution only when the polytropic index of detonation products equals to three. In If the closed-loop for system (1.6) is still constructed by the feedback matrix behavior K of eq.(l.3) of the reflection (as is shock usaully in the the explosive case in products, practice), and applying then the small equation parameter of pur- the perturbed terbation method, closed-loop an analytic, system first-order is approximate solution is obtained for the problem of flying plate driven by various high explosives dx with polytropic indices other than but nearly equal to three. = (A+eA~)x+ (B+eB~)Kx ( 1.7) Final velocities of flying plate dt obtained agree very well with numerical results by computers. Thus and an analytic at this formula time with the two poles parameters of closed-loop of high explosive system (i.e. eq.(l.7) detonation must velocity have and been polytropic offindex) for estimation of the velocity of flying plate is established. set from the originally assigned poles in eq. (1.2). This offset of the pole is called pole perturbance of the closed-loop system. The present problem to be considered is: how to express the major part of pole Explosive perturbance driven of flying-plate the closed-loop technique system ffmds its for important some use perturbation in the study satisfying of behavior of conditions materials under (1.4) intense and impulsive (1.5) loading, at fixed shock e? synthesis And then: of diamonds, how to and estimate explosive the welding major and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions part of maximum pole perturbance for all perturbations in the form of (1.4) and (1.5)? Under In the this assumptions paper the of one-dimensional authors apply plane the detonation theory of and branching rigid flying plate, of solutions the normal for approach nonlinear of solving equations the problem to of solve motion these of flyor two~roblems. is to solve the following Furthermore, system of using equations the governing the flow field of detonation products behind the flyor (Fig. I): same theory the authors have solved the well-posed problem in the most econo- mical structure synthesis of the linear control system proposed by [I] (refer to [2] ) ap +u_~_xp + au II. Solution of the Problem au au 1 Let the equation of unperturbed closed-loop system be dx dt =(A+BK)x (2.1) If the coefficient matrix on the right-hand side of eq.(2.1) is written byf, then eq.(2.1) becomes where p, p, S, u are pressure, density, dx specific entropy and particle velocity of detonation products respectively, with the trajectory R d----i- of = reflected Fx shock of detonation wave D as a boundary and (2.2) the trajectory and again F of let flyor the as equation another boundary. of perturbed Both are closed-loop unknown; the position system of be R and the state parameters on it are governed by -~t the flow = [ A + field BK I + of e( central A, + B~K rarefaction ) ]x wave behind the detonation wave (2.3)

3 Estimation of Pole Perturbance of Linear Closed-Loop Systems 7211 If (.4,+B,K)=F,, then eq. (2.3) can be written as dx --d'{- =( F +eft)x (2.4) The poles of closed-loop system (2.2) are the roots of the following charac- teristic equation: det[ii--f]=a'+c.-,i'-t+'"+c,a+co=o (2.5) The poles of perturbed closed-loop system f2.4) are the roots of the following characteristic equation : where and d.,j are det[ll--( F +eft)= l'+c,_,(e)l'-' +.'.+C,( e )l +Co(e )=O (2.6) C.(e)=d,+d,,te+...+d...*'-', (r-~-o, 1, -'., n--l) (2.7') k-order homogeneous functions Abstract of perturbation elements a,, and ~,,. For e~-0, matrix (F+eF,) will reduce to F, so The one-dimensional problem of the motion of a rigid flying C.(O)=d.=C.. plate under explosive attack has an analytic Let l0 solution be a root only of when eq. the (2.5). polytropic In order index of to detonation derive the products expression equals to of three. the In general, major part a numerical of perturbance analysis is required. for the In root this of paper, eq. (2.6) however, corresponding by utilizing the to "weak" A0, we shock inbehavior of the reflection shock in the explosive products, and applying the small parameter purtroduce a lemma first. terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final tural velocities nl;mher. of flying If the plate left-hand obtained agree side very of eq. well with (2.6) numerical is transformed results by computers. and arranged Thus according to the descending order of (l--10), we have: index) for estimation of the velocity of flying plate is established. (~--~o)" +e._,(8)(~--~o)'-' +...+ej(e)(~--~o)k+".+e~(e)(~--ld+eo(e)=o (2.8) where Lemma 2.1: Let A~10 be a k-multiple root of eq.(2.5), k is a certain na- materials Then it under must intense have been impulsive that loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method ek:]tf:01, of estimation ek_l=0i. of..., flyor el=0 velocity, eo and the way of raising it are questions of common Proof: interest. Because A~ is a k-multiple root of eq. (2.5), expanding the left- Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal hand side of eq. (2.2) in terms of the descending order arrangement of (A--A0) approach of solving the problem of motion of flyor is to solve the following system of equations governing we can certainly the flow field obtain of detonation products behind the flyor (Fig. I): (;t--lo)" +C; _,(l--lo) ~ -'+ "" +C~ (1-- ~0)~= 0 (2. I0) and C'.~0 ap +u_~_xp + au For era0, eq. (2.8) will be reduced to eq. (2.10), and it gives, the following au au 1 ed O ) =ek=c'~ results ek_l(0)=e~_t=0,..., e,(o)=et=0, eo(o)=eo----0 and ek=c~ ~0 Lemma 2.1 Q.E.D. Let l-----a0 be a root of eq. (2.5), in order to seek for the solution of eq. where (2.6) p, in p, the S, u neighbourhood are pressure, density, of specific /=A0 for entropy the perturbation and particle velocity Satisfying of detonation conditions products (1.4) and (1.5), we may utilize Newtun's convex-polygon method which is used trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters to find on branching it are governed solutions by the flow of field nonlinear I of central equations. rarefaction wave The behind theoretical detonation basis wave of

4 122 Zhang Rong-xiang this method will be found in [3 ]. Therefore, first transforming eq. (2.6) in- to eq.(2.8), and assuming I--I0=~, we obtain: F(~,e)f~'+e.-,(e)~'-'+'"+ei(e)~+'"+e,(e)~+eo(e)=O (2.11) and then the problem is turned into one of how to find the branching solutions of eq.(2.11) near the origin. Now let us briefly introduce Newton's convex-polygon method as follows: Consider the solution of eq.(2.11), which can be expanded into the fol- lowing series: ~=~ +~,ea' +~,e~~ (2.12) where ~<~t<~m<..., ~0, or for simplicity reduce eq. (2.12) Abstract ~_~aea+u (2.13) where The 0=O(ee) one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In For seeking the possible value of ~ and ~a, substitute eq.(2.13) into eq. behavior (2.11), of and the then reflection collect shock together in the explosive the coefficients products, and applying of e, the which small have parameter the same purterbation order. method, Let each an analytic, of them first-order equal zero. approximate This solution may start is obtained from for the the terms problem of of the flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. lowest-order. However, for the time being, due to the uncertainty of ~, we Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an do analytic not know formula which with terms two parameters would be of considered. high explosive For (i.e. detonation determining velocity ~, and Newton polytropic gave index) a so-called for estimation convex-polygon of the velocity method: of flying plate first is established. of all let us find the lowest-order of e from the coefficient of each term on the left-hand side of eq.(2.11),for example, the lowest-order of e 1. in coefficient Introduction e~(e) corresponding to ~J is p~. Mark out the correspondingpoint (k, p~)in the rectangular coordinate system, materials and then under connect intense a impulsive part of loading, the points shock marked synthesis with of diamonds, broken-lines and explosive so that welding they and cladding compose of a metals. convex-polygon The method of and estimation all the of remaining flyor velocity marked and the way points of raising are it located are questions over it as shown in Fig.2.1. Under the assumptions of one-dimensional Thus the slope plane detonation of broken-line and rigid of flying each plate, descending the normal approach of solving the problem of section motion of of flyor this is to convex-polygon solve the following with system respect of equations to negoverning the flow field of detonation products behind the flyor (Fig. I): (O.p=) gative real axis can be taken as the value of in eq.(2.11). Corresponding to various slopes, ~ different ap +u_~_xp + branching au solutions can be obtained. [~ After au determining au 1 the value of ~, substitute eq. o " (2.13) into eq.(2.11) and put the coefficient of Fig. 2.1 the term, which has the lowest-order of e ', to equal zero, thus we can get the value of ~a 9 What we just mentioned above is Newton's convex-polygon method used in where finding p, p, the S, u branching are pressure, solutions density, specific of nonlinea~ entropy and equations. particle velocity of detonation products Making use of this method the expression of the major part of eigenvalue trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters perturbance on it are of governed matrix by the A+B/~ flow field can I of be central found rarefaction when perturbation wave behind the matrices detonation Al wave and D B~ and are by initial given. stage However, of motion of there flyor still also; the remains position of the F and problem the state how parameters to find of products the es-

5 Estimation of Pole Perturbance of Linear Closed-loop Systems 72~. timating expression for the major part of maximum eigenvalue perturbance of matrix A+BK for all perturbation satisfying conditions (1.4) and (1.5). Thereby we have to discuss the branching solutions of eq.(2.1 ) through Newton's approach. The discussion is made in several cases as follows. THEOREM 2;I: If I0 is a single root of eq.(2.5), and the coefficient of term e in e0(e )Of eq.(2.11) is not identified with zero, then ~ in eq.(2.13) is one, and ~e is a linear homogeneous function of a,, and ~,. For getting the maximum perturbance of ~, a., has to take the value of ~ or --a,,and ~, the value of b., or--b.,respectively, depending on the signs of coefficients in the expression of ~a 9 Proof: Because i=i0 is a single Abstract root of eq.(2.5), the free term of coefficient The e,(e)of one-dimensional term ~ problem in eq.(2.11) of the motion is certainly of a rigid flying not zero plate under (from explosive Lemma 2.1).Thereattack has an fore analytic Newton's solution convex-polygon only when the comprises polytropic index only of one detonation descending products section equals whose to three. slope In with general, respect a numerical to negative analysis is real required. axis In is this equal paper, to however, I as shown by utilizing in Fig.2.2. the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation Then method, eq.(2.13) an analytic, becomes first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to (~.14) three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus From Lemma 2.1 we know that the free term in r index) is zero, for estimation so the lowest-order of the velocity of of flying 9 in plate the is established. e- quation obtained by substituting eq. (2.14) into eq.(2.11) is one. Letting 1. its Introduction coefficient be equal to zero,we get: ~(l,o) " Explosive driven flying-plate technique ffmds its important use in the e,~a+co,l=o Fig. study 2.2 of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of ~ =-f0,, estimation --~--~:, of flyor velocity and the way of raising it are questions (2.,s) " ", ~ e I ~ C~ where Under Cl is the the assumptions coefficient of one-dimensional of term (I--20) plane detonation in eq. (2.10) and rigid and flying is plate, independent the normal of approach a., and of fl., solving. From the problem eq.(2.6) of motion it can of be flyor seen is to that solve the the coefficients following system of of e equations in governing the flow field of detonation products behind the flyor (Fig. I): polynomials Ch(e) are composed'of linear homogeneous functions of a,, and fl.,, so ~ is also a linear homogeneous function of a,, and fl,,. For la~,i<~.lo,,l and l~.,l~ib,,i, in order for ap ~, in +u_~_xp eq. (2.15) + au to reach its maximum, it is neces- sary to take a,, or --a,~ as a,, and au b,, au or--b,,as 1 ~, respectively depending on the signs of coefficients in the expression of ~ Note 2.1: Theorem 2.1 Q.E.D. In the following we shall take the expression of maximum which is reached by the major part ~ e~ of ~ as the estimating expression of maximum perturbance where p, p, S, u of are eigenvalue pressure, density, l0 specific entropy and particle velocity of detonation products respectively, THEOREM with 2.2: the trajectory If 10 is R a of k-multiple reflected shock root of detonation of eq.(2.5) wave and D as the a boundary coefficient and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraof meters term on e it in are governed e0(e) of by eq.(2.11) the flow field is not I of central identified rarefaction with wave zero, behind then the 6 detonation in eq.(2.13) wave D equals and by i initial and stage ~5 is of motion the k-th of flyor root also; of the linear position homogeneous of F and the state function parameters of a.,and of products ~,,

6 724 Zhang Rong-xiang (there are h-multiple roots)9 For getting the maximum of ~5, u,, has to take the value of a.s or--u,,, and ~,, the value of b,, or--b,,respectively depending on the signs of coefficients in the expression of ~n. Proof: Because A=10 is a h-multiple root of eq.(2.5), from Lemma 2.1 we know that the free term in the coefficientel(g) of ~L in eq.(2.11) is certainly not zero, while ali the free terms in el-,(~), ek_~(~),..., ex(e), e0(e) are zero. Thus the descending section of Newton's convex-polygon comprises only the sec- tion AB as shown in Fig.2.3, whose slope with respect to the negative real axis 1 1 is~, i.e., ~ in eq.(2.13) is ~. Then we have: ' (2.16) Abstract For e0=e I... el_j=0, el~0, the lowest-order of ~ in The one-dimensional problem the of the equation motion of obtained a rigid flying by substituting plate under explosive eq. (2.16) attack has inan analytic solution only when the polytropic index of detonation products equals to three. In.,ICO, 1) to (2.11) is one and its corresponding coefficient behavior of the reflection shock in equation the explosive is products, and applying the small parameter pur- O B(k.0) terbation method, an analytic, first-order approximate solution el~=-eo,, is obtained for the problem of flying plate driven Fig. by various 2.3 high explosives with polytropic indices other than but nearly equal to three. 9 ~e~= eo,t = eo,t (2.17) Final velocities of flying plate obtained agree very well with el numerical C'. results by computers. Thus SO an analytic ~a is the formula k-th with root two of parameters --e~ of (there high explosive are k-multiple (i.e. detonation roots) velocity where and polytropic e0,, is index) for estimation of the velocity of C~ flying plate is established. a linear homogeneous function of a,, and ~,,. For l=,,l~<]a,,l and l~,,l~<ib,,l, in order to enable ~8 to reach its maximum, a,, has to take the value of u,, or --u,, and ~,, the value of b,, or--b,, respectively depending on the signs of coef- ficients Explosive in the driven expression flying-plate of technique ~. ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and Theroem explosive welding 2.2 Q.E.D. and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions Theorems 2.1 and 2.2 give respectively the major part expression of the branching Under solution the assumptions of eq. of one-dimensional (2.11) and that plane of detonation maximum and perturbance rigid flying plate, when the I=I0 normal is a single approach root of solving or a the multiple problem root of motion of eq. of (2.5). flyor is to But solve both the following of them system are on of the equations assumpgoverning the flow field of detonation products behind the flyor (Fig. I): tion that ee,l is not identified with zero. Now if ee,1=ee,z="'=e0,.----0, e,,,+,~0, then how to find the solution? For this case the problem can still be solved in a similar way as described ap +u_~_xp above, + au yet for concrete conditions it must be analyzed concretely. For instance, au au when 1 I=I0 is a single root, e0,1=e0, e0,.=0, and e0,., and we have 6=r+1 The calculating rules of ~a are similar to those mentioned above. When I----I0 is a multiple root, for instance l=io is a triple root of eq. (2.5), moreover, ee,,=ee,z-----0, and e0,s~0, then the terms of e,(e) should be considered. where p, p, S, u If are pressure, el,,~0 density,, then specific the descending entropy and section particle velocity of Newton's of detonation convex-polyproducts gon trajectory of this F of case flyor as comprises another boundary. sections Both C are and unknown; C as shown the position in Fig.2.4. of R and the The state slope para- Of meters section it are AC governed with respect by the flow to field negative I of central real rarefaction axis is wave 2 and behind that the of detonation section wave CB is I/2. Corresponding to section ~C we get a branching solution of eq. (2.11)

7 Estimation of Pole Perturbance of Linear Closed-loop Systems 725 and corresponding to section CB we get two. If the estimating expression of maximum perturbance is required, then we must take the latter. On this we ought to write k ~=~se*'~+ v (2.1S),,,4(0.3) The lowest order of ~ in the equation obtained by substituting eq. (2.18) into eq. (2 11) is 8/2 and its coefficient is 9 9. ~=+,/-~ el~l o B(3.01 Fig. 2.4 where el,j is a linear homogeneous Abstract function of a.~ and ~,i- Therefore, for get- ting the maximum of ~, ~,, should still take the value of a,, or--a,,and ~,, the The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an value analytic of br, solution or --b,,respectively, only when the polytropic depending index on of the detonation signs of products the coefficients equals to three. in In,t "." general, eq.(2.19). a numerical ~ As can analysis be seen, is required. the problem In this for paper, this however, case by can utilizing still the be "weak" solved shock with behavior of the reflection shock in the explosive products, and applying the small parameter pur- Newton's convex-polygon method, and the problem of finding maximum perturbance terbation method, an analytic, first-order approximate solution is obtained for the problem of flying becomes plate driven a by linea various r programming high explosives one with with polytropic some simple indices other inequality than but constraints. nearly equal to three. Final velocities A more complicated of flying plate condition obtained agree is very that well if with et,,=0 numerical in the results above by computers. example,then Thus the solution is a little bit different. In this case the descending section index) for estimation of the velocity of flying plate is established. of Newton's convex-polygon comprises only section AB as shown in Fig.2.5. ItS slope with respect to negative 1. real Introduction axis is one, hence Explosive 8=! driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, A(O0s and explosive welding and and eq. (2.13) becomes cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions ~=~,~+v (29 o The lowest-order Under the assumptions of e in of the one-dimensional equation obtained plane detonation by and rigid flying plate, B(3.0) the normal approach of solving the problem of motion of flyor is to solve the following substituting eq. (2.20) into eq. (2.11) is 3 and its Fig. system 2.5 of equations governing the flow field of detonation products behind the flyor (Fig. I): coefficient equation is (2.21) ap +u_~_xp + au where e0,s is a cubic homogeneous function of a., and ~.,, ~,,2 a quadratic hoau au 1 mogeneous function of a., and ~,,, and ez,~ a linear homogeneous function of a., ~J 2 e,~, + e2,1~, +el,,~, +eo,,=o and ~,,. For [~,,I~[a,,[, and [~,,[~[b,,[, the problem of finding the values of a,, and ~,~ so that ~5 reaches its maximum is turned into one of finding the solution for the extreme values of implicit funciton defined by eq. (2.21).This can be dealt with the method concerned in numerical analysis, though the prowhere p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products cedures of solution finding are somewhat complex. In general, if there are several trajectory points F of flyor marked as another on a boundary. certain Both section are unknown; of Newton's the position convex-polygon,the of R and state paraabove-mentioned meters on it are governed problem by the of flow extreme field I value of central of rarefaction implicit wave function behind will the detonation occur. wave (2.19)

8 726 Zhang Rong-xiang... III. Calculation Example A third-order linear control system is given by.i.,l 11 _i ix, l I'l dt x= = x= + 0. x, I x, 0 Assuming that the poles of closed-loop system to be assigned are ~, 1 4, =I~= - ~, = --~- (3.1) (s.~) and in terms of these poles the following feedback matrix Khas K=[--2, Abstract 21 4_~_I]41 t6 ' ~z~ been designed: then The the one-dimensional vector equation problem of of closed-loop the motion of system a rigid flying is plate under explosive attack has an analytic solution only when the polytropic dx index of detonation products equals to three. In general, a numerical analysis is required. In dt this =(A+BK')x paper, however, by utilizing the "weak" shock (3.4) behavior Write it of in the its reflection component shock in form: if the explosive j1 products, jlxj and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. d Final velocities of flying plate 'dr obtained x2 = agree very 1 well l with -- 1 numerical x= results by computers. Thus (3.5) x, x, index) for estimation of the velocity of flying plate is established. Now matrix ~ is perturbed by 8.41 ~a' G~j (Z=,, (ZZ8 - ~2tj G=I G[. s materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding and matrix of metals. B is The perturbed method of by estimation of flyor velocity and the way of raising it are questions Then Under the coefficient the assumptions matrix of one-dimensional of the perturbed plane detonation closed-loop and rigid system flying plate, is the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field --1+e(u,i--2~,)--l~+et- of detonation products behind,,,-~,0 the 21R. flyor (Fig. ~2+e I): (a,,+~bi,) 41 (3.3) eb,=*ls,, p., /~,,]r, 1.8,,l<lb,,I (3.7) 1 -ke(az, --2,8u),-ke(uzz--~6fl,,) e (azs4-~21,sz,) (3.8) ap +u_~_xp + au and the characterisitic equation of unperturbed closed-loop system is Simplifying the above equation, we get where p, p, S, u are pressure, density, specific 1 entropy I' and particle velocity of detonation products trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- AS can be seen, the unperturbed closed-loop system has indeed the poles to be meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D assigned and by initial eq. stage (3.2), of motion where of flyor A,=I, also; the I-- position is a duple of F and root the state and parameters A,=--89 of a products single root. lae, l<la,,i (3.6) au au 1

9 we get (i) Estimation of Pole Perturbance of Linear Clo,sed-loop S,ystem.7217 The major part of perturbance of pole An=--~ 1 Expanding eq.(3.9) in terms of the descending order arrangement of(a+i), I ~ I' I = and similarly for det{ii--[a+ea,+(b+eb,)k]}, we get! " +e, (e) (e) (3.10) where et(e)=14-o(8) eo(e)----e~--o.2~a,,--o.sal=--als--o.125a=,--o.25a,,--o.sa=s+o, lo937s~=l "FO.21875asz-1"O.4375ans--O.125fltn--O.O625fl=~-t-O.O546875~s~J..I-O(e) (3.tl) Abstract (3~12) From eqs. (3.11) and (3.12) we know that for equation The one-dimensional problem of the motion of a rigid flying plate under explosive attack has ~'+e=(e)~'+e1(e)~+r =0 (3.13) an analytic solution only when the polytropic index of detonation products equals to three. In general, the descending a numerical section analysis of is required. Newton's In convex-polygon this paper, however, merely by utilizing comprises the "weak" the section shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- AB as shown in Fig.3. I, and its slope with respect to negative real ax~s is i, terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic.'. &=Le+v indices other than but nearly equal to three. (3.14) Final The lowest-order velocities of flying of plate e in obtained the equation agree very obtained well with by numerical sub- results by computers. Thus stituting eq.(3.14) into eq.(3.10) is one. Put its index) for estimation of the velocity of flying plate is established. coefficient equal to zero, we obtain A(O,I) ~6 =t6[o.25a,,+o.5ai=+a,,+o.125a=, a== +0.Sazs--O.lO9375=,,--O.21875a== --O.4375azn-FO.125~u+O.O625~u ~,t] (3.15) Fig. 3.1 materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common From interest. eq.(3.15) we can find the maximum of ~a - When ~6 reaches its positive maximum, Under there the assumptions must be of one-dimensional plane detonation and rigid flying plate, the normal t approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I): Gst~Op GjZ~-- 1, GSS~--lp ~lt~lp ~ts~op ~$t~o. The maximum of ~e is ap +u_~_xp + au (~)m=x=54.50 If e=0.00l, then the major part au of maximum au 1 y possible =0, perturbance of closed-loop pole -0.5 is ~ae~-_ I (2) The major part of perturbance of pole A,=I,=--~ Expanding eq.(3.9) in terms of the descending order arrangement of(anu~), we get I ' l I = where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products trajectory and similarly F of flyor for as 4et41I--[A+eA,+(B+sB,)K]}, another boundary. Both are unknown; we get the position of R and the state parameters on it are governed by + r the (e) flow field I +e; of central rarefaction wave behind the detonation wave (A+~)' (A+88 (e) (A+~)+e;(e)=O (3.111) 0 m ;g

10 728 Zhang Rong-xiang where e;(e) =l+o(e) (3.17) e; (e) =el -- O. 0635,rzl, at=--a,=-- O , -- O. 5625a** a=s as=..1.. O.,16875as= u,,3 -- O Bt,-- O. 1289/~== + O ,8=, ] -I"O(e) (3.18) From eqs.(3.17) and (3.18) we know that for equation ~''1" e; (e)~*'1"e; (e)~'1" eo (e) = 0 (3.19) the descending section of Newton's convex-polygon comprises only the section AB as shown in Fig.3.2, and its slope with respect to negative real axis is~, 1 Abstract.'. ~=~,e"=+v (3.20) The one-dimensional problem The of the lowest-order motion of a rigid of e flying the plate equation under explosive obtained attack by has an analytic solution only when the polytropic index of detonation products equals to three. In substituting eq. (3.20) into eq. (3.19) is one.put its behavior A(On) of the reflection shock in coefficient the explosive equal products, to and zero, applying we obtain the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying Q ~ =2[ O. 0625a, l + O. 75alf +a,, + O. O46875a=, plate driven by various a(2.0) high explosives with polytropic indices other than but nearly equal to three. Fig a=,+0.75 a=3--0, a== Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive --0. (i.e aa=--0. detonation 625a,-' ,B=1 velocity and polytropic index) for estimation of the velocity of flying plate is established ,6'=, ,8,, ]" (3.21) From eq. (3.21) we can find the maximum of ~8 9 there must be In the case of positive maximum materials under intense impulsive Or==----~ ], loading, a=s~l shock, synthesis ase.~-0, of (2':12 diamonds, ~---- ~ I p and explosive welding and cladding of metals. The method a,,=--l. of estimation B.=I, of flyor #=,=0, velocity,8,~=0. and the way of raising it are questions The maximum of ~, is Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion ILl=~x=~.212 of flyor is to solve the following system of equations governing Therefore, the flow for field the of pole detonation --l---of products closed-loop behind the system flyor (Fig. the I): estimating value of major 4 part of maximum possible perturbance is ~elz2=~o ap +u_~_xp (here + au we still take e=0.001). Therefore, when the pole au of closed-loop au 1 system is multiple, it is observed that its maximum perturbance is greater. where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave

11 Estimation of Pole Perturbance of Linear Closed-loop System 729 References I. Tu Xu-yen, The Practical Value of Controllability and Observability and the Problem of "Most Economical Structural Synthesis" of Control Systems, S~posium of the First National Congress on Control Theory and Its Application, May (1979). (in Chinese) 2. Zhang Rong-xiang and Chen Zhao-kuan, The Well-Posed Problem in MESSgnthesis Df Linear Control System, Acta Automatica Sinica, Volo7, No.4,October(1981). (in Chinese) 3. Vainberg, M. M. and Trenogin, V.A., Theorey of Branching of Solutions of Nonlinear Equations, Noordhoff International Publishing House, (1974). Abstract The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus index) for estimation of the velocity of flying plate is established. materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I): ap +u_~_xp + au au au 1 as as where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave