Karma sistemlerin tümleyen değişkenli modelleri
Complementarity modeling of hybrid system
- Tez No: 83127
- Danışmanlar: DOÇ. DR. KÜLMİZ ÇEVİK
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1999
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 73
Özet
ÖZET Bu tezde, tümleyen değişkenli sistemler ele alınmış olup bu tür sistemlerin genel hali olan hibrit sistemler hakkında da bilgi verilmektedir. Bu sistemler, sürekli dinamik sistemlerin ayrık olaylar ile etkileşmesi sonucu ortaya çıkan karma sistemler olarak düşünülebilir. Sistem, mod olarak bilinen her bir ayrık durumda birbirinden farklı diferansiyel ve cebirsel denklemler ile tanımlanmaktadır. Modlar arası geçiş, o moddaki eşitsizlik kısıtlarının ihlal edilmesiyle olmaktadır. Sistemleri tanımlamakta verilmiş olan formülasyon ayrık durum (mod) kavramı üzerine kurulmuştur. Bu formülasyon, modlann değişmesiyle sistemin, o moddaki denklemlerinin nasıl değişeceğini ifade etmeye uygundur. Bunlara bağlı olarak her bir mod için bazı uzay tanımları verilip, sistemin otonom olması tanımlanan bu uzaylar cinsinden ifade edilmiştir. Tümleyen değişkenli sistemler için önemli olan bir konu da, sistemin çözümünün varlığı ve tekliği konusudur. Bu çalışmada bu konu iyi-tanımlılık olarak ifade edilmiş ve çözümün var ve tek olması için gerek ve yeter koşullar verilmiştir. Temel bir başka problem de olay am dediğimiz, bir moddan başka bir moda geçilmesi gereken am yakalayabilmek ve bu andan sonra sistemin çözümü için hangi modun uygun olduğunu bulmaktır. Bu problemin çözümü olarak iki yöntem tanıtılmış ve bu işlem için bazı kurallar verilmiştir. Bu çalışmada, ayrıca çokça karşılaşılan parça-parça doğrusal fonksiyonların tümleyen değişkenli gösterilimine dönüştürülmesi anlatılmış ve dönüşümün olması için parça-parça doğrusal fonksiyonun sağlaması gereken koşullar verilmiştir. Anlatılmış olan bu adımların bir örnek üzerinde uygulaması yapılıp, bunlar ayrı ayrı gösterilmiştir.
Özet (Çeviri)
COMPLEMENTARITY MODELING OF HYBRID SYSTEM SUMMARY The idea of a system Jhat interacts with its environment plays a role in computer science as well as it does in the theory of dynamical systems; in the computer science terminology, one speaks in this context of reactive systems. Often a computer system will operate in an environment that is described by continuous rather than discrete variables, for instance in the control of chemical processes or of mechanical systems, and in these cases one deals with, a combined system that incorporates features both of finite automata and of continuous dynamical systems. In a mirroring development, there is a trend in control theory [3] moving away from the standard paradigm that is formulated entirely in terms of differential and/or difference equations, towards formulations that also allow the presence of discrete elements, often described in this context as switching logic. Actually this trend follows rather than precedes control engineering practice, since switching elements have already been used successfully although on an ad-hoc basis in many control applications. The incorporation of discrete elements in continuous dynamical systems has also attracted the interest of researchers in the area of classical dynamical systems as a means for obtaining properties such as robust stability. In this way, from the efforts of computer scientists, control theorists, and dynamists a new research field is emerging for which the name hybrid systems are now mostly used. Such hybrid systems arise for instance in rule-based control of continuous processes, in situations where continuous processes affect the operation of a discrete device (timing constraints, clock drift), and in general in any situation where logic and dynamics interact with each other. VI.Proposals for defining the general class of hybrid systems or languages for it have been made in computer science as well as in control theory and in the theory of dynamical systems. The development of such general definitions will not be the main concern. Rather than a top-down strategy we prefer to follow a bottom-up approach in which we work with only a preliminary, not fully detailed definition of what a hybrid system is, and study special classes for which a good intuition is available about how they should behave. Even the study of quite restricted subclasses will be worthwhile if these offer a strong basis for theoretical development, which may then serve as a guideline and a reference for other theories. In this work we consider such a class of hybrid systems, which we have named the linear complementarity systems after a term used in optimization theory. As we hope to demonstrate, there are indeed a number of nontrivial conclusions to draw from the study of this special class. It is easy to give real-world examples of complementarity systems. Electrical networks containing diodes, hydraulic systems containing one-way valves and mechanical systems with stops can all be interpreted as complementarity systems. An advantage of the fact that these systems occur in nature is that a strong intuition is available about their operation. A second advantage that will be employed below is the presence of the concept of energy, which allows making general statements about the trajectories of complementarity systems. We also like to point out that natural structures often serve as guidelines for artificial design; in this context one may refer for instance to simulated annealing, or to the use of passivity in adaptive and non-linear-control. Consider the following system of linear differential and algebraic equations and inequalities (written in the usual form, with implicit“and”):. Jc(0=A.x(t) + B.u(t) (1.a) y(t)=C.x(t) + D.u(t) (1.b) y(t)>0, u(t)>0. yT(t).u(t)=0. (1.c) Equations (l.a) and (l.b) constitute a linear system in state space form; the number of inputs is equal to the number of outputs. The relations (l.c) are called complementarity conditions. Because of the nonnegativity constraints, the vanishing of vuthe inner product yT(t).u(t) implies that actually for each index i at least one of the variables y;(t) and m(t) must be zero. The set of indices for which y;(t) = 0 (we shall call this the active index set) need not be constant in time, so that the system may switch from one“operating mode”to another. To define the dynamics of (1) completely, we will have to specify when these mode switches occur, what their effect will be on the state variables, and how a new mode will be selected. A proposal for answering these questions will be explained below. The specification of the complete dynamics of (1) defines a class of dynamical systems called linear complementarity systems. Let n denote the length of the vector x(t) in the equations (l.a,l.b) and let k denote the number of inputs and outputs. There are then 2k possible choices for the active index set. The equations of motion when the active index set is I are given by. x(/)=A.x(t) + B.u(t) y(t)=C.x(t) + D.u(t) (2) yi(t)=0,iel UiO^.igf where 1° denotes the index set that is complementary to I, that is, r={ie{l,..., k}| igl} We shall say that the above equations represent the system in mode I. An equivalent and somewhat more explicit form is given by the (generalised) state equations. x(t)=A.x(t)+ B^uiit) 0=C -x(t) + Dn.ui(t) (3) together with the output equations j/C(t) = C/c;x(t)+JD/C/.ui(t) (4) «/c(0=0 Here and below, the notationM^ where M is a matrix of size (mxk) and I is a subset of {1,..., k} denotes the submatrix of M formed by taking the columns of M whose indices are in I. The notation M^, denotes the submatrix obtained by taking the rows with indices in the index set I. V1UIn order to formulate an event rule, we first need to introduce some concepts taken from the geometric theory of linear systems. Denote by Vi the consistent subspace of mode I, i.e. the set of initial conditions xo for which there exist smooth functions x(-) and ui(.), with x(0) = xo, such that (3) is satisfied. The space Vi can be computed as the limit of the sequence defined by V°=Rn Vi+1={xeVi | 3u eRlrl s.t. A.x+^u(t) e V, CT x(t) + Dn.ui(t)=0} (5) I» There exists a linear mapping Fi such that (3) will be satisfied for xoeVi by taking ui(t) = Fi.x(t). The mapping Fi is uniquely determined, and more generally the function ui(-) that satisfies (3) for given XoeVi is uniquely determined, if the full- column-rank condition ~B_ ker Pn. (6) holds and moreover we have VmTi={0}, (7) where Ti is the subspace that can be computed as the limit of the following sequence T°={0} Ti+1={xeRn|3xer, u eR|x|s.t. x = Ax+Bm[.u, C/#.x+Dn.u} (8) As will be indicated below, the subspace Ti is best thought of as the jump space associated to mode I, that is, as the space along which fast motions will occur that take an inconsistent initial state instantaneously to a point in the consistent space Vi; note that under the condition (7) this projection is uniquely determined. The projection can be used to define a jump rule. However, there are 2k possible projections, corresponding to all possible subsets of {1,..., k}: which one of these to choose should be determined by a mode selection rule. For the formulation of a mode selection rule we have to relate in some way index sets to continuous states. Such a relation can be established on the basis of the so-called rational complementarity problem (RCP). The RCP is defined as follows. Let a rational vector q(s) of length k and a rational matrix M(s) of size (kxk) be given. The rational IXcomplementarity problem is to find a pair of rational vectors y(s) and u(s) (both of length k) such that y(s) = q(s)+M(s).u(s) (9) and moreover for all indices 1^ i < k we have either yi(s)=0 and Ui(s)>0 for all sufficiently large s, or Ui(s) = 0 and yi(s)>0 for all sufficiently large s. The vector q(s) and the matrix M(s) are called the data of the RCP and we write RCP(q(s),M(s)). We shall also consider an RCP whose data are a quadruple of constant matrices (A,B,C,D) (such as could be used to define (l.a, l.b)) and a constant vector xo, namely by setting q(s) = C(sl - A)“1*) and M(s) = C(sl-A)”1. B + D. We say that an index set I e{l,.., k} solves the RCP (9) if there exists a solution (y(s), u(s)) with yi(s) = 0 for i el and u,(s) = 0 for i gl. A linear complementarity system is now defined as follows. We assume that a quadruple (A,B,C,D) is given whose transfer matrix G(s)=C(sI-A)'1.B+D is totally invertible, i.e. for each index set I the (kxk) matrix Gn(s) is non-singular. Under this condition, the two subspaces Vi and Ti as defined above form for all I a direct sum decomposition of the state space Rn, so that the projection a long Ti onto Vi is well- defined. We denote this projection by Pi. After the jump x((f) is equal to Pi.x(O'). The solution is most easily written down in terms of its Laplace transform: X(s)=(sl - A^xo + (si- A)'1 £“ Ui(s) (10) Ui(s) = -Ghl (s). Q..(si- AyVxo (11) Where Gn(s)=C/.(sI-A)-1 B^+Du (12> Note that the notation is consistent in the sense that Gn(s) can also be viewed as the (LI) submatrix of the transfer matrix G(s) = C(sI-A)”1.B+D. It is shown in [10] that the transfer matrix Gn(s) associated to the system parameters in (3) is left invertible when (6) and (7) are satisfied. Since the transfer matrices Gn(s) that we consider are square, left invertibility is enough to imply invertibility, and so (by duality) we also have Vi+Ti=Rn. While going on in selected mod I, it is necessary to pass in to another discrete state because of breaking of inequalities. After the event time, which is the moment MÖbBrfiTJ KUTULUwhen inequalities are broken, which mode to continue is decided by solving mode selection problem. The solution of mode selecting problem is done by converting in to one of the following problem. Markov parameters; I'd ; i=0 H*= J., (13) C.A^.B; i=l,2,... and mode selection problem is in t domain and so as to have the equations y^C.xo+D.u1 y2=C.A.xo+C.B.u1+D.u2 y3=C.A2.x0+C.A.B.u1+C.B.u2+D.u3 (14) yi=C.Ai"1.xo+ Xffİ~J -^ 1 and (u1 u2... uk)>:0 or (y1 y2... y^O and (u1 u2... uk) = 0 Here yI+1 represent i. derivation of y If mode selection problem is wanted to be solved in s domain, the same problem may be defined as the solution of the problem of rational functions Y(s) and U(s), which helps to solve the equation; Y(s) = C. (s.I-A)-1.xo + [C.(sJ-A)-1.B+D].U(s) (15.a) and inequations; Vs > so Y(s) > 0 ; U(s) > 0 ; Y(s)TU(s)=0 (15.b) Moreover, some studies have been made on how the piecewise linear functions, which are frequently seen in modelling, analysing and designing nonlinear systems, are changed in to complementarity form. Here the main problem is that y=f(x) piecewise linear function y = Ax + B.u +e. (16-a) j =C.x + D.u+f (16.b) j>0 u>0 jTu(t)=0 (16x) XIcan have complementarity form. Supposing Rn space is divided by (n-1) dimentional planes, called hyperplanes, and Hi hyperplanes and Bi are defined as following Hi={xeRn | cti.x+3i=0} i=l,2,...,k B Ja/.w + #0 i el (17) (18) (19) Suppose the function in each area is defined as this; f(x)=Ai.x+bi, xeBi for the adjasent two areas diveded by Hi hyperplane to be difined as [Ai-Aiu{i} bı-bıW{i}]=bi.[ai ft] ^o) if bi continuity coefficent can be found, it can be said that y=f(x) piecewise linear function is transformed in to complementarity form. To come this, supposing there are k hyperplanes C=,D=Ik, A=A0, e=b0 B=[bi b2... bk] (21) is taken and complementarity form is obtained. In addition to those mentioned above, unlike continuous time dynamic systems, the matters such as event detection, mode selection, re-initialisation etc. Which are defined for this kind of systems, are also studied, and all these mentioned problems are practised on a sample. XII
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