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Control Techniques for Constrained Discrete Time Systems Control techniques for constrained discrete time systems Paolo d’Alessandro, Elena De Santis April 27, 2004 ii Contents Introduction vii 0.1 Notations terminology and recalls . vii iii iv CONTENTS Preface ??? v vi PREFACE Introduction Reading the book requires some mathematical background. This is proposed in form functional to the present purposes in [7] and consists in the …rst place of rudiments of topology, linear spaces, Hilbert spaces and convex analysis. The style of exposition therein is concise and therefore for most proofs and details it is useful to make reference to the classical books: [8], [11] and [12]. In addition to that, we need here some of the results contained in this latter book, especially some basic ideas underlying the conical approach and the main results of the dual conical methods for the solution of linear feasibility and optimality problems. Rather then duplicating the presentation in [7], the following section aims at establishing the notations used in the present book as well as providing a sort of check list for the reader of the concept and results that will be used in the sequel. It does not substitute reading the above references, unless one is already familiar with the required background. We do not exclude a few re…nements later on, wherever required. 0.1 Notations terminology and recalls We shall consider, unless otherwise stated, points and sets in the Euclidean real spaces Rn equipped with the usual product topology, even though many of the involved concept are valid for more general linear spaces. At times we refer to a generic linear space, especially when there is no additional e¤ort to achieve maximum generality. We assume familiarity with the theory of linear bases. n (i) (i) We assume the space equipped with the inner product (x; y) = i=1 x y , which makes it into a Hilbert space. The symbol x(i) denotes the i.th component of the vector x. The inner product induces a norm by: P x 2 = (x; x) k k Any other Haussdor¤ (distinct points have disjoint neighborhoods so that limits are unique) vector topology is equivalent to the one generated by this norm, because of …nite dimensionality of the space [8]. Again at times we shall refer to a generic Hilbert space. We shall leave to the context however the distinction between di¤erent inner product, and use for all of them the same symbol. vii viii INTRODUCTION Given an arbitrary integer m > 0, m vectors x1; ::; xm and m scalars 1; ::; m, m we call linear combination of x1; ::; xm the vector i=1 ixi: This is called a¢ ne m combination if i=1 = 1; conical combination if i 0 i; convex combina- tion if it is both a¢ ne and conical. P 8 P n Next we de…ne set operations. If A and B are non-void subsets of R and if and are scalars, then: n A + B = z : z R ; z = x + y; x A; y B f 2 2 2 g n A + B = z : z R ; z = x + y; x A; y B f 2 2 2 g Henceforth, in formulas similar to the above, we shall omit to specify which symbols denote scalars. This will be left to the context. Sometimes simpli…ed notations are useful: e.g. for a vector x Rn 2 x + A = x + A f g where x is the singleton of x. Thef lineg segment joining two points x and y of a linear space is the set z : z = x + y; 0; 0; + = 1 f g and is denoted by [x : y]. The vectors x and y are called extremal points of the segment; it may well be the case that x and y coincide and hence the segment reduces to x . Notice thatf g [x : y] = z : z = x + (1 )y; 0 1 f g A subset F of Rn is said to be a linear subspace of Rn if it is closed under linear combinations. This is equivalent to say x + y F whenever x; y for any reals and . We can also write this as: + 2 . The set 02 is f g a linear subspace of Rn, often called the trivial subspace. Two linear subspaces F and G of a linear space E are called complementary subspaces if any vector of the space can be uniquely expressed as the sum of a vector of F plus a vector of G. In formulas: z E !x F !y G; z = x + y 8 2 9 2 9 2 where ! means exists and is unique. It is easy to prove that two linear subspaces are complementary9 if and only if their intersection is 0 and their sum is the whole space. Given a subspace F , all linear subspaces Gf gsuch that F and G are complementary have the same dimension, which is called the codimension of F . A translate x+F of a linear subspace is said to be an a¢ ne subspace. A¢ ne subspaces are closed under a¢ ne combinations. >From the theory of linear topological spaces [8], we know that linear and a¢ ne subspaces of Rn are necessarily closed. A subset of a linear space is said to be a convex cone (for brevity cone) if it is closed under conical combinations. This is equivalent to say x + y 2 0.1. NOTATIONS TERMINOLOGY AND RECALLS ix whenever x; y , 0; 0 and + = 1. We can also write this as: + ,2 ; 0 : + = 1. Finally another equivalence: is convex if 8 x; y ; [x : y] 8 2 The whole space Rn is a set of all the above sorts. In other words it is a linear subspace (of itself), an a¢ ne subspace, a cone and a convex set. It is also both closed and open. The linear, a¢ ne, conical, convex hull (or extension) of a set is, respec- tively, the minimum linear, a¢ ne, conical, convex set that contains . They are respectively denoted by: ( ); ( ); o( ) and ( ). It is immediate to verifyL thatA theseC extensionsC always exist. In fact, for example, ( ) is readily seen to be the intersection of all linear subspaces that contain L. Such family of sets is non-void because the whole space is one of its members. A similar proposition holds with the obvious changes, for the other hulls, and is proved with exactly the same argument mutatis mutandis. This provides external descriptions of hulls, because they are obtained by intersecting a family of sets. The linear subspace ( ) is the set of all linear combinations built using points of . Similarly forL ( ); o( ) and ( ), substituting, respectively, a¢ ne, conical and convex combinationsA C to linearC combinations. These state- ments yield the internal descriptions of these sets as opposed to the external descriptions depicted above. Of course there is the problem of giving compact, that is minimal in some sense, descriptions as well. For example we know from the theory of linear bases that ( ) is also the linear extension of a maximal linearly independent set containedL in . For a¢ ne sets it is easy to follow a sim- ilar paradigm introducing the notion of a¢ ne independence. The same problem is much more substantial for convex sets and cones. Later on, for the …rst case we will recall a major result of topology, namely the Krein-Milman theorem and for the second we shall recall the solution of this problem for the special case of polyhedral cones. Because linear subspaces and a¢ ne subspaces of Rn are always closed, it follows that ( ) and ( ) are also the minimal closed subspace and the mini- mal closed a¢L ne subspaceA containing . In other words they are also the closed linear hull, denoted by ( ) and the closed a¢ ne hull, denoted by ( ), of . L A The situation is di¤erent for convex sets and cones that are not necessarily closed. Denote with o( ) and ( ) the minimal closed cone and, respec- tively, the minimal convexC set containingC ; which are respectively the inter- section of all closed cones containing and all convex sets containing . In general o( ) o( ) and ( ) ( ). But what is their internal descrip- tion? AgainC in view C of the factC that the C topology is a linear one a simpli…cation occurs. More precisely it is possible to determine these closed extension by the formulas: o( ) = ( o( )) C C x INTRODUCTION ( ) = ( ( )) C C The proofs readily follow directly from linearity of the underlying topology. Before passing on we recall two more noteworthy computation rules for con- vex extensions for arbitrary non-void sets A and B and scalars and : ( A + B) = (A) + (B) C C C (A B) = [x : y]; x (A)& y (B) C [ f[ 2 C 2 C g Linear subspaces are particular examples of cones and a cone may well con- tain a (non-trivial) linear subspace (all cones contain the subspace 0 ). Thus it is interesting to single out cones that do not contain linear subspaces.f g The key concept to this purpose is that of lineality space of a cone, which is, intuitively, the maximum linear subspace contained in the cone in question Given a cone , the lineality space of , denoted as lin(), is given by lin() = ( ) If the lineality space of is the singleton of the origin then is called a pointed\ cone. We shall see momentarily that if a cone is not pointed, it can be decomposed in the sum of a pointed cone plus its lineality space. A noteworthy example of cone is the recession cone of a convex set . The recession cone of is the set de…ned by: y : x + y ; 0; x .
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