An Algebraic Approach to Multidimensional Behaviors

An Algebraic Approach to Multidimensional Behaviors

Diego Napp Avelli

RIJKSUNIVERSITEIT GRONINGEN

An Algebraic Approach to Multidimensional Behaviors Proefschrift

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op vrijdag 18 januari 2008 om 16:15 uur

door

Diego Napp Avelli geboren op 2 november 1973 te Buenos Aires, Argentina

Promotores: Prof. dr. Harry L. Trentelman Prof. dr. M. van der Put Beoordelingscommissie: Prof. Jan C. Willems, Katholieke Universiteit Leuven, Belgium. Prof. S. Shankar, Chennai Mathematical Institute, India. Prof. Ulrich Oberst, Institut f¨ ur Mathematik, Innsbruck, Austria.

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Acknowlegments I would like to express my gratitude and profound respect to my two supervisors Harry Trentelman and Maris van der Put, for their reliable guidance and clarification of important issues encountered during the thesis. Thanks Harry for giving me the chance to do a PhD and kindly introduce me to the world of systems and control theory. Before our first achievements he had to correct me innumerable mathematical and English mistakes. I am very grateful to him. In the last period of my thesis I went to consult with Marius since I was stuck on a problem, and he selfless offered his help with my thesis. He was very very patient and generous with me. Le estoy eternamente agradecido. It was very memorable to meet Shiva Shankar and Mohamed Barakat. They both taught me many things but I especially appreciate their friendship, support and help. I give them my heartfelt thanks. Thanks U. Oberst, J.C. Willems, S. Shankar and M. Barakat for accepting the burden of being part of the reading committee. I am also thankful to P. Rocha, R.F. Curtain, P. Rapisarda, H. Gluesing-Luerssen, E. Zerz and H. Pillai for helping me in different ways at various moments throughout these last four years. Life would have been just too tedious without the daily contact with my friends Alessio, Minh, Sijbo, Tomas, Florian, Irene, Carmen, Mavi, Ricardo, Ele, Graci, Ivan, Kanat, Steve, Rosti, Aneesh, Alef, Olga, Muhammad, Mark, Martin, Harsh, Shaik... Last but not least I thanks Maria, Sebi and Torcho for being always there.

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Contents 1 Introduction to the behavioral approach 1.1 Motivating the behavioral approach . . . . . . 1.2 Dynamical systems . . . . . . . . . . . . . . . 1.2.1 Differential systems . . . . . . . . . . . 1.3 Latent variables and the fundamental principle 1.4 Control as interconnection . . . . . . . . . . . 1.5 The problems treated in the thesis . . . . . .

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1 1 4 6 8 10 15

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17 18 18 21 23 27 32

regular implementation of nD systems Implementability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Does Proposition 3.1.2 hold for nD systems? . . . . . . . . . . . . . . . . . The canonical controller and regular implementability . . . . . . . . . . . .

37 38 41 45

2 Behaviors versus modules 2.1 Some algebra . . . . . . . . . . . . . 2.1.1 Noetherian rings and modules 2.1.2 Homs and exact sequences . . 2.1.3 The duality . . . . . . . . . . 2.2 Dictionary . . . . . . . . . . . . . . . 2.3 Finite dimension and codimension . . 3 On 3.1 3.2 3.3

4 Regular implementability in 4.1 Introduction . . . . . . . . 4.2 Preliminaries . . . . . . . 4.3 Implementation . . . . . .

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the space of compactly supported functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 54

5 Interconnection and decomposition problems up to finite dimension 59 5.1 From torsion free to free with finite codimension . . . . . . . . . . . . . . . 61 5.2 Regular “almost” implementability . . . . . . . . . . . . . . . . . . . . . . 70 5.3 Autonomous-controllable decomposition with finite dimensional intersection 72 5.4 Decomposition of the controllable part . . . . . . . . . . . . . . . . . . . . 77 5.5 Complete generalization of interconnection and decomposition problems up to finite dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 iii

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CONTENTS

6 Algorithms for multidimensional spectral factorization and sum of squares 83 6.1 Quadratic Differential Forms and 2n-variable polynomial matrices . . . . . 84 6.2 Lifting n variable to 2n variable polynomial matrices . . . . . . . . . . . . 88 6.3 Spectral factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.4 Sum of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 Discrete behaviors 107 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 The minimal injective cogenerator for D . . . . . . . . . . . . . . . . . . . 108

Chapter 1 Introduction to the behavioral approach The aim of this first chapter is to describe an approach to modelling and control of systems: the behavioral approach. We will set out the basic principles that form the backbone of the approach, particularly in the context of distributed parameter systems described by partial differential equations. We hope to convince the reader that such an approach occurs naturally as a result of this way of regarding modelling. We shall see how latent variables arise and explain the importance of the problem of elimination. We end this chapter by reviewing the notion of interconnection which is the basis of control in the behavioral framework.

1.1

Motivating the behavioral approach

Traditionally, mathematicians and engineers are used to develop modelling and control of physical systems within the framework of input/output thinking. Such a framework is, of course, often perfectly suitable to describe the interaction between a system and its environment. It is indeed appealing since one can interpret it as establishing cause and effect, i.e. each variable is either causing the evolution (and hence this variable is an input) or producing an effect (due to the input, and hence this variable is called output). However, it can also not be denied that the input/output approach and the state-space paradigms have many important shortcomings. As an example, consider the usual procedure in modelling: a system is viewed as an interconnection of subsystems, and modelling consists of describing the individual subsystems and their interconnections laws. The result of such a modelling procedure will be a model containing high order differential equations which involves manifest variables (the variables we try to model) and latent variables (the variables describing the subsystems and their interconnections). Consequently, in order to obtain first order or transfer function models, some manipulation has to be performed or 1

2

Chapter 1. Introduction to the behavioral approach

physical insight into the structure of the system must be used. Such manipulations can become very awkward for complicated systems. It is simply much too restrictive to consider the input-output approach as the starting point in the description of open systems. Many physical systems do not exhibit a causal dependence between the variables describing them. And, most of all, interaction between systems does not always happen by feeding the output of a system into the input of another one as called for in the context of input/output models. A modelling and simulation package like Matlab’s Simulink is a typical product of the input/output thinking. The package is, in fact, entirely based on block diagrams, series, parallel and feedback interconnections. However, in Simulink is not convenient to handle even the most basic of examples. We will provide a simple example to clarify this fact.

Figure 1.1: Tank

Example 1.1.1 : Consider a tank (in figure 1.1) for which we want to describe the relationship between the flow (f), the pressure (p), and the height (h) of the outlet. We regard it as a system with one fluid variable, corresponding to the outlet, and the variable s = (p, f, h) taking its value in R3 . The dynamic evolution of this variable is best described by introducing the height of the fluid in the tank (Ht ) as internal variable. The continuity equation then yields d (1.1) St Ht = −f, where τ denotes time, dτ with the constant St , the surface of the tank, a parameter of the model. If we neglect the kinetic energy of the fluid in the outlet and assume the velocity profile in the tank to be flat, application of the principle of conservation of energy yields a relationship between the heights Ht and h and the pressure p: St2 d d2 ρ 2 (1 − 2 )( Ht ) + ρHt 2 Ht + ρg(Ht − h) = p. 2 S dτ dτ

(1.2)

1.1. Motivating the behavioral approach

3

Here ρ the density of the fluid in the tank, and S, the cross section of the outlet, are other parameters of the model. A logical way to choose variables is z = (Ht , dtd Ht ) as state variable, u = (p, h) as inputs and y = f as output.

Figure 1.2: Two tanks

Assume, now we want to connect two such tanks as shown in figure 1.2. Interconnection then calls for p1 = p2 , h1 = h2 and f1 = −f2 , in other words, for equating two inputs and two outputs, exactly the opposite of what is called for by classical input-output thinking. We believe that different representations are suitable for different purposes; the equations of a model can be manipulated and rearranged, and variables can be eliminated in terms of convenience for simulation, for analysis, for control, and so on. In [70,71], J.C. Willems has proposed a paradigm for the description of systems that covers the input-output and the state-space one, and overcomes their inadequacies. This framework relies on the idea that control systems are described by equations, but their properties of interest are most naturally expressed in terms of the set of all solutions to the equations. This is formalized by the relatively new notion of system behavior. The approach centers around the idea of a system as a set of trajectories that includes besides the variables whose evolution we try to model, some latent variables introduced in the modelling process. The advantage of taking collections of trajectories as a starting point is that it becomes possible to discuss properties of dynamical systems without reference to some specific representation. Hence, the approach does not assume any a priori decomposition into inputs and outputs. Moreover, it makes a sharp distinction between the system, i.e. the behavior, and its representation, the set of equations that describe it.

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This approach does not wish to neglect or forget the paradigmatic input/output approach; rather it offers new possibilities.

1.2

Dynamical systems

In this section, necessary mathematical background is given in order to start developing the approach to modelling systems discussed in the previous section. The starting point of our discussion is the idea that a system is identified with the collection of all trajectories consistent with the mathematical laws of the system. This idea will be the corner-stone of this thesis. Formally we define: Definition 1.2.1 A dynamical system is defined as a triple Σ = (A, q, B), where A is a set called signal space, q ∈ Z+ is the number of components, and B ⊂ Aq is called the behavior of the system. The elements of B are called trajectories and are denoted by ”w”. This definition is the main concept and the essence of our approach to open dynamical systems. Basically one defines a model as an exclusion law, allowing to select some feasible trajectories out of all possible ones. Then a behavior is just a collection of trajectories consistent with the laws describing the system. Thus, B = {w ∈ Aq | w is compatible with the laws of Σ }. In order to make this clear we give some historical examples. We denote the space of all infinitely often differentiable functions from Rn to K by C∞ (Rn , K), and all K-valued 0 distributions on Rn by D (Rn , K), where K = R or C. Example 1.2.2 : Kepler’s laws The possible motions of a planet around the sun can be described by Kepler’s laws. This defines a dynamical system with A = C∞ (R, R), q = 3 and B the set of all smooth maps w : R → R3 that satisfy Kepler’s laws, i.e. the paths w are ellipses in R3 with the sun in one of the focal points, the radius vector from the sun to the planet must sweep out equal areas in equal time, and the squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axes (the ”half-length” of the ellipse) of their orbits. Example 1.2.3 : Heat conduction Consider the phenomenon of heat conduction in a homogeneous medium. Here the relevant functions are w1 (x, y, z, t) and w2 (x, y, z, t). The first one is the absolute temperature at a point (x, y, z) and time t. The second measures the heat produced in each point (x, y, z) of the medium at time instant t. The relationship between w2 and w1 is given by the following partial differential equation: µ 2 ¶ ∂ w1 ∂ 2 w1 ∂ 2 w1 ∂w1 =κ + + + w2 , (1.3) ρε ∂t ∂x2 ∂y 2 ∂z 2

1.2. Dynamical systems

5

where the constants ρ, ε, and κ are respectively the medium density, its specific heat, and its thermal conductivity. Supposing that the medium is unbounded, the dynamical system is defined by the signal space A = C∞ (R4 , R), the number of signals q = 2 and the system behavior B = {(w1 , w2 ) : R3 × R −→ R2 | (1.3) is satisfied }. Example 1.2.4 : One-dimensional wave equation Consider the one-dimensional wave equation describing the displacement w(t, x) from equilibrium of an elastic string: 2 ∂ 2w 2∂ w − τ = 0, (1.4) ∂t2 ∂x2 where ρ and τ are physical constants related to the mass density and the elasticity of the string. The equation defines a dynamical system with signal space A = C∞ (R2 , R), number of signals q = 1, and behavior

ρ2

B = {w ∈ C∞ (R2 , R) | w satisfies (1.4)}, Example 1.2.5 : Vibrating plate Let w(t, x, y) be the displacement of an infinite vibrating plate in the position (x, y) at time t; then it can be shown that w satisfies the PDE ∂ 2w ∂ 4w ∂ 2w ∂ 4w ρ 2 + + 2 + = 0, (1.5) ∂t ∂x4 ∂x∂y ∂y 4 where ρ is a constant depending on the physical properties of the plate. Such equation defines a dynamical system with signal space A = C∞ (R3 , R), number of signals q = 1, and behavior B = {w ∈ C∞ (R3 , R) | w satisfies (1.5)}. Example 1.2.6 : Maxwell’s equations Maxwell’s equations describe the relation be~ : R × R3 → R3 , the magnetic field B ~ : R × R3 → R3 , the current tween the electrical field E 3 3 3 density ~j : R × R → R , and the charge density ρ : R × R → R in free space by: ~ − ρ = 0, ∇·E ²0 ~ ~ + ∂ B = 0, ∇×E ∂t ~ ~ ~ − ∂ E − j = 0, c2 ∇ × B ∂t ²0 ~ = 0, ∇·B (1.6) where the constants c and ²0 stand for the speed of light in vacuum and the electric ~ B, ~ ~j, ρ) ∈ C∞ (R × constant, respectively. Then q = 10, A = C∞ (R × R3 , R) and B = {(E, R3 , R3 × R3 × R3 × R) | (1.6) is satisfied }. Example 1.2.7 : Convolutional codes Let A = F[x] with F a finite field and H ∈ Aq×q , then C := {w ∈ Aq | Hw = 0} is a convolutional code. Convolutional codes are thus dynamical systems according to the abstract definition 1.2.1.

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1.2.1


Differential systems

The main object of interest when modelling a dynamical system is the behavior B, i.e. the set of all trajectories that comply with the laws of the system. This set may be described in many possible ways, for instance it can be described through graphs or equations. Even when it is represented by equations, this representation is highly non unique. Then it is conceptually misleading to identify the dynamical system with equations, graphs or any other representations. However systems described by differential equations play an outstanding role in physical and engineering applications. Therefore we will consider behaviors B consisting of all solutions w(x) ∈ Aq (where A is a space of (generalized) functions) of a finite set of partial differential equations of the form: f (x, w(x), . . . ,

∂k w(x), . . . ) = 0, ∂xk

where f is a function involving x = (x1 , x2 . . . , xn ),

∂k ∂xk

=

∂ k1 k ∂x1 1

(1.7) kn

∂ . . . ∂x kn , and assuming a n

finite number of such partial derivatives. These B’s are called differential behaviors. A very important issue is which type of solutions (i.e. the choice of A) we are looking for. Normally this depends on the particular application in mind. In the following chapter we will address this matter, and see in how far the behaviors depend on the choice of the signal space A. We call Σ = (A, q, B) a linear differential system (with constant coefficients), if B is the solution set of a system of linear partial differential equations with constant coefficient. More precisely, taking a subclass of (1.7) of the form: R(

∂ ∂ ∂ , ,..., )w = 0, ∂x1 ∂x2 ∂xn

(1.8)

where w ∈ Aq and R ∈ K•×q [ξ1 , ξ2 , . . . , ξn ] is a polynomial matrix in n indeterminates with q columns and an arbitrary number of rows, where K = R or C. Therefore R( ∂x∂ 1 , ∂x∂ 2 , . . . , ∂x∂n ) is a linear partial differential operator with constant coefficient and (1.8) is a system of ∂ ∂ partial differential equations. We often write ∂x = ( ∂x∂ 1 , ∂x∂ 2 , . . . , ∂x∂n ), B = ker(R( ∂x )) or even B = ker(R). This representation of B is called a kernel representation. Then we consider B = {w ∈ Aq | w satisfies (1.8)}. (1.9) 0

Definition 1.2.8 For every vector x0 = (x01 , . . . , x0n ) ∈ Kn , one defines a shift σ x : (Aq ) → 0 (Aq ) by the formula σ x w(x1 , . . . , xn ) = w(x1 + x0 1 , . . . , xn + x0 n ). Note that the subspace, B, defined by a linear differential systems with constant coefficients is shift-invariant under all shifts. From now on, throughout the rest of this thesis, we will just consider our systems (behaviors) to be linear differential systems with constant coefficients, also called linear nD systems (behaviors) or just nD behaviors. Often we will use the terminology multidimensional behavior. We will make little distinction between a

1.2. Dynamical systems

7

system and its behavior.

Example 1.2.9 : For the differential behavior given in example 1.2.6 (Maxwell’s equations) we have that B ⊂ A10 and for a kernel representation of B (i.e. in the form 1.9) ~ B, ~ ~j, ρ) ∈ R10 and one writes w = (E,   ξ1 ξ2 ξ3 0 0 0 0 0 0 − ²10  0 −ξ3 ξ2 −ξ4 0 0 0 0 0 0     ξ3  0 −ξ 0 −ξ 0 0 0 0 0 1 4    −ξ2 ξ1 0 0 0 −ξ4 0 0 0 0    R=  0 0 0 ξ ξ ξ 0 0 0 0 1 2 3   1 2 2  ξ4  0 0 0 −c ξ c ξ 0 0 0 3 2 ²   0 1 2 2  0 ξ4 0 c ξ3 0 −c ξ1 0 ²0 0 0  0 0 ξ4 −c2 ξ2 c2 ξ1 0 0 0 ²10 0 where R ∈ R8×10 [ξ1 , ξ2 , ξ3 , ξ4 ] and ξi acts as the partial derivative operator

∂ . ∂xi

Remark 1.2.10 The special case n = 1, describes ordinary differential equations. There is only one independent variable, normally considered to be the time. Remark 1.2.11 In the same way as we define linear differential systems with constant coefficients, we can proceed with linear difference systems with constant coefficients. One can distinguish two types: n

1. The discrete signal space is A = KN , and B ⊂ Aq . We consider the shift operator σi instead of the partial derivative operator ∂x∂ i , where the shift operator σi is defined as (σi w)(t1 , t2 , . . . , tn ) = w(t1 , t2 , . . . , ti−1 , ti + 1, ti+1 , . . . , tn ). Then the behavior can be represented as B = {w ∈ Aq | R(σ1 , σ2 , . . . , σn )w = 0} ⊂ Aq ,

(1.10)

where R ∈ K•×q [σ1 , σ2 , . . . , σn ] is a polynomial matrix in n indeterminates with q columns and an arbitrary number of rows. n

2. The discrete signal space is A = KZ , B ⊂ Aq and the behavior represented by B = {w ∈ Aq | R(σ1−1 , σ2−1 , . . . , σn−1 , σ1 , σ2 , . . . , σn )w = 0} ⊂ Aq ,

(1.11)

where R ∈ K•×q [σ1 , σ2 , . . . , σn , σ1−1 , σ2−1 , . . . , σn−1 ] is a Laurent polynomial matrix with q columns and an arbitrary number of rows, σi is the shift operator and σi−1 acts as the inverse shift, i.e. (σi−1 w)(t1 , t2 , . . . , tn ) = w(t1 , t2 , . . . , ti−1 , ti − 1, ti+1 , . . . , tn ).

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The first developments within the behavioral approach were due to Willems [70, 72] in the context of 1D systems (n = 1) and are partially extended to the 2D systems (n = 2) case by Rocha [56]. In the last fifteen years, the behavioral framework has introduced quite a significant change in traditional system modelling. Two of the principal reasons are that it does not require any particular representation such as input/state/output representation or an a priori distinction between inputs and outputs variables, and that it provides a general framework in which it is possible to analyze a vast number of system-theoretic concepts such as controllability, observability, poles and zeros, state-space models and many others. Although this is already true for 1D systems, it is probably in the context of multidimensional systems that the behavioral approach has proved to be most effective. This is due to the fact that higher dimensional systems do not admit the state space machinery that holds for 1D systems.

1.3

Latent variables and the fundamental principle

In modelling physical systems from first principles, one often needs to incorporate some auxiliary variables in the model to make the modelling easier. For example, in electrical circuits, we may be interested in the relationship between voltage and currents across certain nodes. However, in order to model this relationship we may need to model the whole circuit including voltages and currents across other nodes. As a result one may obtain a set of equations with additional variables, called latent variables. The latent variables are different from the ones we are interested in, which are called manifest variables. Definition 1.3.1 A dynamical system with latent variables is a quadruple ΣL = (A, A0 , q+ l, Bf ull ), where A is a set called manifest signal space, A0 is a set called latent variable signal space, q and l ∈ Z+ are the number of components of the manifest variable and latent variable respectively, and Bf ull ⊆ Aq ⊕ (A0 )l is called the full behavior of the system. We will assume A = A0 and then write just ΣL = (A, q + l, Bf ull ). As before we will consider full behaviors Bf ull consisting of all solutions of a finite set of linear partial differential equations with constant coefficients. These are exactly those which admit a kernel representation of the form: µ ¶ ¡ ¢ w q l ∂ ∂ Bf ull = {(w, `) ∈ A ⊕ A | R( ∂x ) M ( ∂x ) = 0}, (1.12) ` where M ∈ K•×q [ξ1 , ξ2 , . . . , ξn ], R ∈ K•×l [ξ1 , ξ2 , . . . , ξn ] are polynomial matrices in n indeterminates. For sake of brevity we write Bf ull = ker(R M ). Thus, the trajectories of a system with latent variables are pairs (w, `), with w a manifest variable trajectory and ` a latent variable trajectory. Example 1.3.2 : In example 1.2.9 (Maxwell equations) we consider the manifest variable ~ B, ~ ~j, ρ). Nevertheless, in case we are just interested in the evolution of to be w = (E,

1.3. Latent variables and the fundamental principle

9

~ ~j) as the manifest variable and the electrical and current field, we may define w = (E, ~ ρ). Then a kernel magnetic field and charge density as latent variables, i.e. ` = (B, representation of the full behavior (i.e. as in 1.12) can be written with       N =     

ξ1 ξ2 ξ3 0 −ξ3 ξ2 ξ3 0 −ξ1 −ξ2 ξ1 0 0 0 0 ξ4 0 0 0 ξ4 0 0 0 ξ4

0 0 0 0 0 1 ²0

0 0

0 0 0 0 0 0 1 ²0

0

0 0 0 0 0 0 0 1 ²0





          ,R =           

0 0 0 − ²10 −ξ4 0 0 0 0 −ξ4 0 0 0 0 −ξ4 0 ξ1 ξ2 ξ3 0 2 2 0 −c ξ3 c ξ2 0 2 2 c ξ3 0 −c ξ1 0 −c2 ξ2 c2 ξ1 0 0

      .     

(1.13)

Once we have distinguished between manifest and latent variables, one can obtain a dynamical system (in the sense of definition 1.2.1) that is induced by the latent variable system as follows. Definition 1.3.3 Let ΣL = (A, q + l, Bf ull ) be a dynamical system with latent variables. The manifest dynamical system induced by ΣL is the dynamical system Σ = (A, q, B) with B defined as B = pr1 (Bf ull ) = {w ∈ Aq | ∃` such that (w, `) ∈ Bf ull } Note that we have just projected the full behavior onto the manifest variables w. Now an important question arises, namely, whether the manifest system Σ induced by the linear shift-invariant differential system ΣL is again a linear shift-invariant differential system, i.e. is pr1 (Bf ull ) an nD behavior? We will show in the next chapter that the answer depends crucially on the signal space considered. In this thesis we are mainly interested in the case that B contains either smooth or distributional solutions, i.e. A = C∞ (Rn , K) or D0 (Rn , K). Hence, we address the above question for these signal spaces. First, the following definition is needed. Definition 1.3.4 Given a polynomial matrix M ∈ Kq×m [ξ1 , . . . , ξn ] then R ∈ Kg×q [ξ1 , . . . , ξn ] is called a minimal left annihilator (MLA) of M if RM = 0 and R0 M = 0 implies R0 = XR for some X ∈ K•×g [ξ1 , . . . , ξn ]. Theorem 1.3.5 Given w ∈ Aq and M ∈ Rq×` [ξ1 , . . . , ξn ], there exists ` ∈ Al such that ∂ ∂ )` if and only if R( ∂x )w = 0 where R is MLA of M . w = M ( ∂x This important theorem is sometimes called the Fundamental Principle and is due to Ehrenpreis (1970), Malgrange (1964) and Palamodov (1970). Let D denote the polynomial ring

10


in n variables, K[ξ1 , ξ2 , . . . , ξn ], consisting of polynomials with coefficients in K, where K is either R or C. The fundamental principle in effect says that the signal spaces considered (A = C∞ (Rn , K) and A = D0 (Rn , K)) are injective D-modules, i.e. Hom(−, A) is an exact functor. We want to underline the importance of the choice of these signal spaces, since for instance the space of locally integrable or compactly supported functions are not injective D-modules, so the fundamental principle does not hold. We will make this clear in the following chapter. Theorem 1.3.6 Let A = C∞ (Rn , K) or D0 (Rn , K). Let Σ = (A, q, pr1 (Bf ull )) be a man∂ ∂ ifest dynamical system induced by the latent variable representation R( ∂x )w = M ( ∂x )`. ∂ ∂ Then w ∈ pr1 (Bf ull ) if and only if N ( ∂x )R( ∂x )w = 0 with N a MLA of M . The previous theorem, called elimination theorem, is a consequence of theorem 1.3.5 and says that pr1 (Bf ull ) is indeed an nD behavior. Its importance lies in the fact that it makes it possible to eliminate the latent variables. That is, starting with a linear nD behavior with latent variables we can project it on the set of manifest variables, thus obtaining again a linear nD behavior, so a systems represented by linear constant coefficient partial differential equations. Finally, we give a corollary that renders an explicit kernel representation of the behavior of such a system. Corollary 1.3.7 Let B be the manifest behavior induced by the latent variable represen∂ ∂ ∂ ∂ tation R( ∂x )w = M ( ∂x )`. Then B = Ker(N ( ∂x )R( ∂x )) where N is a MLA of M . The behavioral approach, in particular kernel representations, supplies an effective framework for modelling, where latent variables come up naturally. The process of elimination is always possible provided A is an injective D-module (explained in the following chapter). Its importance from the point of view of modelling is crucial.

1.4

Control as interconnection

In this section we will define interconnection of dynamical systems. We have already explained that we do not need to choose any special structure, like input/output structure, in our dynamical systems. Instead, our systems interact with their environment via some of its variables. When we interconnect two dynamical systems, we obtain a new dynamical system whose trajectories satisfy the laws of both systems. We shall now define interconnection in a mathematically precise way. Consider the following two dynamical systems Σ1 and Σ2 , given by Σ1 = (A, q + p, B1 ) , Σ2 = (A, p + s, B2 ) ,

(1.14)

with common signal space A. We assume that the system variables of Σ1 and Σ2 are (w1 , w2 ) and (w2 , w3 ), respectively, where w1 , w2 and w3 take their values in Kq , Kp and Ks , respectively. In other words we assume B1 ⊂ Aq ⊕ Ap , and B2 ⊂ Ap ⊕ As . Note

1.4. Control as interconnection

11

that Σ1 and Σ2 have the variable w2 in common. The interconnection of Σ1 and Σ2 is now defined as the dynamical system Σ1 ∧w2 Σ2 = (A, q + p + s, B) ,

(1.15)

B = {(w1 , w2 , w3 ) | (w1 , w2 ) ∈ B1 , (w2 , w3 ) ∈ B2 } .

(1.16)

with behavior B is given by

The shared variable w2 is referred to as the interconnection variable. The system variable

w 1

SYSTEM

SYSTEM

1

2 w

w 1

w

3

2

SYSTEM

SYSTEM

1

2 w

w

3

2

Figure 1.3: Interconnection of systems. of the interconnection of Σ1 and Σ2 is (w1 , w2 , w3 ), see figure 1.3. Sometimes, after interconnection, the interconnection variable w2 is considered as a latent variable, and we are only interested in the behavior of the two remaining variables w1 and w3 . This dynamical system can be obtained by eliminating w2 , with behavior : Bw1 ,w3 = {(w1 , w3 ) | ∃ w2 such that (w1 , w2 , w3 ) ∈ B} .

(1.17)

We are now ready to introduce the notion of control in the behavioral setting. Given is a dynamical system whose behavior will be called the full plant behavior and which will be denoted by Pf ull , (the to be controlled system). We want to restrict it so that certain desired properties (the specifications) are satisfied. Such a restriction is carried out by interconnecting it with another system called a controller. The interconnected system is called the full controlled system or behavior. This idea is explained in more detail as follows.

12


We assume that the full plant behavior carries to be controlled variables w and control variables c The controller has only one type of variables, the control variables c, see figure 1.4. Before interconnecting our system to the controller, the variables w and c only obey the restrictions imposed by the behavior Pf ull , in other words © ª Pf ull = (w, c) ∈ Aq ⊕ Ak | (w, c) satisfies the full plant equations . (1.18) As before, let pr1 denote the projection of Aq ⊕ Ak onto Aq . The manifest plant behavior, denoted by (Pf ull )w , contains the to be controlled variable trajectories w and is defined as (Pfull )w = {w ∈ Aq | ∃ c such that (w, c) ∈ Pfull } = pr1 (Pfull ).

w

w

Plant

Plant

c

(1.19)

Controller

Controller

Figure 1.4: Interconnection of systems. Partial interconnection case.

Thus the manifest plant behavior is obtained by eliminating c from Pfull . Remember that (when A = C∞ (Rn , K) or D0 (Rn , K)) using the elimination theorem (Theorem 1.3.6) we may conclude that (Pfull )w is an nD behavior. As we already explained above, the controller has only one type of variables, c. We define the controller behavior C as © ª (1.20) C = c ∈ Ak | c satisfies the controller equations . After interconnecting the systems we obtain a new system in which c satisfies both the laws of the plant and the controller, i.e., c will satisfy both the equations of Pfull and C. This then determines a new behavior which appears as the result of interconnecting the

1.4. Control as interconnection

13

controller with the plant, and it is called the full controlled behavior. This interconnection behavior is given by © ª Kfull (C) = (w, c) ∈ Aq ⊕ Ak ) | (w, c) ∈ Pfull and c ∈ C = Pfull ∩ (Aq ⊕ C).

(1.21)

After interconnecting the full plant behavior and the controller behavior, the restrictions imposed on c by the controller will also restrict the variables w, and in this way one has the power to influence the behavior of the manifest variable w by appropriate choice of the controller. The basic problem of control is then to choose the controller in such a way that in the interconnected system the variables w have a desired behavior (see the lower part of figure 1.4). Since the interconnection only takes place through the variable c and not through the whole system variable (w, c), we call this kind of interconnection partial interconnection. through c As we mentioned before, after interconnection the control variable is often considered as a latent variable. We can get rid of this variable by applying the elimination theorem. This then defines the manifest control behavior, defined as follows: (Kfull (C))w = {w ∈ Aq ) | ∃ c such that (w, c) ∈ Kfull (C)} = pr1 (Kfull (C)).

(1.22)

The main idea in control problems is to find a controller behavior C in such a way that (Kfull (C))w equals a given desired behavior K. In that case, we say that C implements K by partial interconnection through c. If a given desired behavior K has the property that there exists a controller C such that K = (Kfull (C))w , then we call K implementable by partial interconnection (through c). We now discuss two other relevant frameworks for interconnection. The first one is a particular case of partial interconnection, and is called full interconnection. This is actually the simplest variant, and has the feature that in Pfull , w coincides with c, i.e. all variables are available for interconnection, and therefore no projection is necessary to obtain the manifest controlled behavior. The second is a more general type of interconnection, called general interconnection. In this case, apart from the interconnection variable c, C carries an additional variable v, and has the following form: © ª C = (c, v) ∈ Ak ⊕ Av ) | (c, v) satisfies the controller equations .

(1.23)

14


Interconnection of the full plant behavior and the controller then leads to the interconnected behavior given by © ª Kfull (C) = (w, c, v) ∈ Aw ⊕ Ak ⊕ Av ) | (w, c) ∈ Pfull and (c, v) ∈ C = (Pfull ⊕ Av ) ∩ (Aq ⊕ C).

(1.24) (1.25)

Again, often the interconnection variable c is interpreted as a latent variable, i.e. we are mainly interested in the behavior of the variables (w, v), and the manifest controlled behavior is given by: (Kfull (C))w,v = {(w, v) ∈ Aq ⊕ Av ) | ∃ c such that (w, c, v) ∈ Kfull (C)} = pr1,3 (Kfull (C)).

w

w

Plant

Plant

c

Controller

Controller

(1.26) (1.27)

v

v

Figure 1.5: Interconnection of systems. General case

So, in this case, Pfull is the same as before, but the controller has an extra variable v. One is then interested in the behavior of (w, v), where the variables w are restricted by the laws of the plant, v by the laws of C and c by both. See figure 1.5. For examples of similar partitions of variables, see [24]. Of particular interest is the kind of interconnection that is called regular interconnection, which was introduced by Willems in [72]. In a regular interconnection, the restrictions imposed on the plant by the controller are not redundant, i.e. the restrictions of the controller are independent of the restrictions already present in the plant. We will discuss this in more detail in Chapter 3 of this thesis.

1.5. The problems treated in the thesis

1.5

15

The problems treated in the thesis

In this final section of this Chapter 1, we give an overview of the problems treated in this thesis. Formal mathematical statements of the problems will be given later. In Chapter 3 we consider the signal space A to be either C∞ (Rn , K) or D0 (Rn , K) and study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. This refers to Section 1.4, and can be made explicit as follows. Let pr1 : Aq ⊕ Ap −→ Aq and pr2 : Aq ⊕ Ap −→ Ap be the projections. Given are B ⊂ Aq ⊕ Ap , and the behavior K ⊂ Aq ⊕ {0}. Find necessary and sufficient conditions on the pair (B, K) for the existence of a behavior C ⊂ {0} ⊕ Ap such that pr1 (B ∩ C) = K and B + C = Aq ⊕ Ap . We note that in this context the canonical controller is defined as pr2 (B ∩ K). In [6] such conditions were obtained in the context of 1D systems. In this chapter we show that the conditions obtained in [6] are no longer valid in general in the nD context. We also show that under additional assumptions, the conditions in [6] still remain relevant. We also reinvestigate the conditions for regular implementability by partial interconnection in terms of the canonical controller that were obtained in [57]. Using the algebra of the finitely generated modules over the ring K[ξ1 , . . . ξn ] (of differential operators with constant coefficients) we generalize a result on regular implementability from the 1D to the nD case. Finally, we study how, in the 1D context, the conditions from [6] and [57] are connected. This chapter is based on the contributions [36] and [63] (joint work with H. L. Trentelman). The signal space A used in Chapter 3 is an injective cogenerator over K[ξ1 , . . . , ξn ]. Chapter 4 extends these results to the case that the signal space A is no longer an injective cogenerator. For instance, the space A = D of compactly supported smooth functions on Rn is not an injective cogenerator. For this A, the one-to-one correspondence between behaviors and modules, explained in Chapter 2, fails. In fact, there exists a bijection between behaviors and special modules, the ones called Willems’ closed. The projections of nD behaviors need no longer be behaviors, however we characterize the smallest behavior containing such a projection. This leads to an adapted version of the problem of Chapter 3 because pr1 (B) and pr2 (B) may not be behaviors. This chapter is based on the article [33], (joint work with S. Shankar and H. L. Trentelman). In Chapter 5 the following problem is addressed: given a plant (i.e., a behavior), under what conditions does there exist a controller such that their interconnection is “almost” regular, and has finite codimension (to be defined below) with respect to a certain desired system. This can be made explicit as follows. Given the behaviors K ⊂ B ⊂ Aq , when does there exist a behavior C ⊂ Aq such that

16


dimK ((B ∩ C)/K) < ∞ and B + C ⊂ Aq has finite codimension on K (i.e., dimK (Aq /(B + C)) < ∞)? A constructive and complete solution of the problem is presented in the 2D case. This chapter is based on the contributions [37] and [32]. In Chapter 6, algorithms are developed for the problem of spectral factorization and sum of squares of polynomial matrices with n indeterminates. The algorithms reduced the problems either to a problem of factoring a constant matrix or to an eigenvalue problem. They are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. Remarkable relations with dissipativity theory in the behavioral framework (see also [35]) are shown. This chapter is based on the contributions [38] and [34](joint work with H.L.Trentelman). Chapter 7 is concerned with discrete behaviors (see Remark 1.2.11) and presents a concrete description of the minimal injective cogenerator for the difference operators K[σ1 , . . . , σn ]. This is close to a result of U. Oberst [40]. The technical basis for Chapters 3, 4 and 5 is developed in Chapter 2. Here one first recalls general notions of modules, injectivity, exact sequences, homomorphisms, etc. It gives deep results on free modules (Quillen-Suslin) and injective cogenerators (Malgrange, Palamodov and Oberst). The relation between finitely generated modules and behaviors is given in the form of a duality. A dictionary between terms of control theory and commutative algebra is given. Some new items are introduced, e.g. the sheaf-theoretic notion of a Hausdorff autonomous behavior. This notion is strongly related to hypoelliptic partial differential operators. For 2D systems some (apparently) new results on commutative algebra over polynomial rings in 2 variables are found. These are responsible for the new results in 2D systems, namely implementability and decomposition problems up to finite (co)dimension. M. Barakat has provided algorithms implementing these results using the MAPLE package “homalg” [2] developed by M. Barakat and D. Robertz.

Chapter 2 Behaviors versus modules In this chapter we provide an introduction to the combined use of commutative algebra and behavioral theory in the study of multidimensional linear differential systems with constant coefficients (nD systems). The first developments of this behavioral-algebraic duality theory are due to Malgrange (1964) in [30], Ehrenpreis (1970) in [17] and Palamodov (1970) in [42], and later on to Oberst (1990) in [39]. Its huge importance lies in the possibility to translate analytic properties of the systems into algebraic properties of the underlying modules. This enables the application of the powerful machine of commutative algebra in order to solve problems for nD systems (e.g by using Groebner bases [4] or Involutive bases through homalg [2]). In Chapter 1 the main ideas of the behavioral approach were explained. We now recall briefly the definition of the specific behaviors considered in this thesis. We refer to [59], [18] or [14] for the basic notions in commutative algebra such as rings, ideals and modules.

As before, D denotes the polynomial ring K[ξ1 , ξ2 , . . . , ξn ] consisting of the polynomials in n variables with coefficients in K where K is either R or C. Recall our choice of signal space A = C∞ (Rn , K) or D0 (Rn , K), and A is a D-module, where ξi acts as the partial derivative operator ∂x∂ i , i.e. for any polynomial d = d(ξ1 , . . . , ξn ) ∈ D the action of D on A is defined as ∂ ∂ d · w := d( ,..., )w. ∂x1 ∂xn Later on, we will omit the “·” and simply write dw. Also, if w = (w1 , . . . , wn ) ∈ Aq and d = (d1 , . . . , dn ) ∈ Dq then  w1 n  ..  X di wi . dw := (d1 , . . . , dn )  .  = i=1 wn 

17

18

Chapter 2. Behaviors versus modules

A behavior B ⊂ Aq is given by {w ∈ Aq | r1 w = · · · = rn w = 0} for some subset {r1 , · · · , rs } ⊂ Dq . One often writes Rw = 0 when R ∈ Ds×q is the matrix with rows r1 , . . . , rs . In Chapter 1 we already called such a representation a kernel representation of B and we write B = ker(R). Behaviors can have different representations. If M ∈ Dq×m , then B = {w ∈ Aq | there exists ` ∈ Am s.t. w = M `} is also a behavior. In this case M is called an image representation of B and we write B = im(M ).

2.1

Some algebra

In this section we introduce some definitions, for more details see [75], [18] and [77].

2.1.1

Noetherian rings and modules

R will denote any noetherian commutative ring. A subset Γ of an R-module is said to be P independent if for any finite sum pi wi , pi ∈ R, wi ∈ Γ we have that X pi wi = 0 =⇒ (pi = 0, ∀i). In other words the only way in which 0 can be obtained as a linear combination of elements of Γ is by choosing all the coefficients in R equal to zero. For an R-module M , a subset Γ is called a generating set for M if every element of M is a finite sum of elements of Γ multiplied by coefficients in R. The set of such R-linear combinations of is denoted by h{Γ}i. If M has a finite generating set, then M is called finitely generated. If M has an independent generating set then M is called free . If this set consists of n elements then M is isomorphic to Rn . A module M is called projective if it is a direct summand of a free module. Remark 2.1.1 Serre’s problem (1955), also called the Serre’s conjecture or the QuillenSuslin theorem, asserts that any finitely generated projective module over a polynomial ring D = k[x1 , . . . , xn ], where k is a field, is free. The following properties of an R-module M are equivalent : 1. M is noetherian. 2. M obeys the ascending chain condition, i.e. every strictly increasing set of submodules is finite. 3. Every collection of submodules of M has a maximal element. 4. Every submodule of M is finitely generated. 5. M is finitely generated.

2.1. Some algebra

19

Any finitely generated module over R can be written in the form Rq /M for some q and some submodule of M of Rq . A ring R is called principal ideal domain (abbreviated PID) if R has no zero divisors and every ideal is principal ( i.e. generated by only one element). Example 2.1.2 : D = K[ξ1 , ξ2 , . . . , ξn ] is noetherian for any n ∈ N and it is a PID when n = 1. Example 2.1.3 : Consider the D = K[ξ]-module A = C∞ (R, K). Each Ak := {f ∈ A | f is zero on [−1/k, 1/k]} is a submodule. Then A1 ( A2 ( A3 . . . is an infinite ascending sequence of submodules. Hence A is not a finitely generated D-module. Corollary 2.1.4 Let {di }i∈I be an infinite subset of Dq . If B = {w ∈ Aq | di w = 0 i ∈ I}, then there exists a finite subset of J ⊂ I such that B = {w ∈ Aq | di w = 0 i ∈ J}. Proof : Dq is finitely generated over D. This implies that every submodule of Dq is finitely generated (equivalence of properties (4) and (5)). Thus, the module M generated by {di }{i∈I} is finitely generated and therefore there exists a finite subset J ⊂ I such that M is generated by {di }i∈J . Hence, B = {w ∈ Aq | di w = 0 i ∈ I} = {w ∈ Aq | di w = 0 i ∈ J}. ¤ Let M be an R-module. An element m ∈ M, m 6= 0 is called a torsion element if there exists d ∈ R, d 6= 0 such that dm = 0. The set Mtors of all torsion elements is a submodule of M . If Mtors = 0, then M is called torsion-free , and if it is Mtors = M , then M is called a torsion module. Definition 2.1.5 A module A over R is called injective if for any pair of modules K ⊂ E, any homomorphism ` : K −→ A can be extended to a homomorphism L : E −→ A (i.e. L · i = `). ∀`

/A > ~ ~ ~ i ~~ ² ~~ ∃L

K E

Remark 2.1.6 Baer’s criterion for A to be an injective module is: Any homomorphism I −→ A, where I is an ideal of R, can be extended to a homomorphism R −→ A. Definition 2.1.7 Let M, E be R-modules. Then HomR (M, E) is the R-module consisting of all R-linear homomorphisms from M to E. We will omit an explicit reference to the ring R as there will be no ambiguity and write Hom(, ) instead of HomR (, ). Definition 2.1.8 An R-module A is called a cogenerator if for every finitely generated R-module K 6= 0 one has HomR (K, A) 6= 0.

20


Remark 2.1.9 Definition 2.1.8 does not coincide with the definition given in [28, Def. 19.5]. However, for injective modules, definition 2.1.8 is equivalent to [28, Def. 19.5] when the category of left R-modules, R M, is replaced by the category of finitely generated left R-modules, fRg M. Theorem 2.1.10 The D-module A = C∞ (Rn , K) is an injective cogenerator module. Proof : The injectivity of A = C∞ (Rn , K) and many other important signal spaces was first proven by Ehrenpreis (1961), but with mistakes in the proof. Malgrange (1962) and, above all, Palamodov (1963) gave correct expositions. The books by Ehrenpreis (see bibliography of [41]) and Palamodov [42] were first published in 1970. The injectivity part is due to Ehrenpreis [17], Malgrange (1964) [30] and Palamodov (1963) and the cogenerator part to Oberst (1990) [39]. In order to give a flavour of the proof (using the previous notation) we will consider the case when E = D and K = m any maximal ideal. For convenience we consider the case K = C. Using Hilbert’s Nullstellensatz one can see that m can be written as m = hξ1 − α1 , . . . , ξn − αn i for any α1 , . . . , αn ∈ C. Then ` ∈ Hom(m, A) is determined by `(ξ1 −α1 ) = f1 , . . . , `(ξn −αn ) = fn where f1 , . . . , fn ∈ A. Since (ξi −αi )(ξj −αj ) = (ξj −αj )(ξi −αi ) ∈ m then automatically ( ∂x∂ i −αi )fj = ( ∂x∂ j −αj )fi must hold for i, j = 1, . . . , n. Denote f = L(1). The condition L(ξi − αi ) = `(ξi − αi ) = (ξi − αi )L(1) = (ξi − αi )f = ( ∂x∂ i − αi )f implies that ( ∂x∂ i − αi )f = fi must be satisfied for i = 1, . . . , n. Thus the problem is to produce an f ∈ A with ( ∂x∂ i − αi )f = fi for all i. The first step is to note that there exists g ∈ A with ( ∂x∂ 1 − α1 )g = f1 (by variation of constants). Then write f = g + f˜ and replace (f1 , . . . , fn ) by (f1 , . . . , fn ) − (( ∂x∂ 1 − α1 )g, . . . , ( ∂x∂n − αn )g) = (0, f˜2 , . . . , fñ ). One has to find f˜ ∈ A such that ( ∂x∂ 1 − α1 )f˜ = 0 and ( ∂x∂ i − αi )f˜ = f˜i for i = 2, . . . , n, and still one has the relations ( ∂x∂ 1 − α1 )f˜i = 0 and ( ∂x∂ i − αi )f˜j = ( ∂x∂ j − αj )f˜i for 2 ≤ i, j ≤ n. Then f˜k = eα1 x1 gk where gk depends only on x2 , . . . , xn and the desired f˜ as f˜ = eα1 x1 g with g just depending on x2 , . . . , xn . Thus, we have the relations ( ∂x∂ i − αi )g˜j = ( ∂x∂ j − αj )g˜i for 2 ≤ i, j ≤ n and by induction on n a g exists such that ( ∂x∂ i − αi )g = gi for i = 2, . . . , n. It can be shown that for a complete proof of injectivity one has to replace the ideal m used above by any ideal I generated by n elements (see [15, Prop. 1.2.2]). For the cogenerator property, note that any finitely generated module over a polynomial ring can be written in the form Dq /N for some q and some submodule N ⊂ Dq . By the Noetherian property there exists a submodule N + maximal such that N ⊂ N + Dq . q + Then D /N has no proper submodules and therefore is isomorphic to D/m with m = hξ1 −α1 , . . . , ξn −αn i, α1 , . . . , αn ∈ C a maximal ideal. Then is sufficient to show that there exists an ` ∈ Hom(D[ξ1 , . . . , ξn ]/m, A), ` 6= 0. This is equivalent to the existence of an ∂f ∂f − α1 )f = 0, . . . , ( ∂x − αn )f = 0. Now f = ceα1 x1 . . . eαn xn ∈ A f = `(1) that satisfies ( ∂x n 1 satisfies this property. ¤

2.1. Some algebra

21

Remark 2.1.11 The previous theorem also holds if one replaces C∞ (Rn , K) by C∞ (Ω, K) where Ω ⊂ Rn is a non-empty connected open set. Example 2.1.12 : The D-module Cc := {w ∈ C∞ (Rn , K) | w has compact support} is neither an injective module nor a cogenerator. To see this we will follow the reasoning and notation as in the above proof. Take n = 1 and α = 0. Thus we have to find R x a function ∂ ∞ f ∈ Cc (R, K) such that ( ∂x )f = g for a given g ∈ A. The solution f (x) = 0 g(t)dt has, in general, no compact support hence the module is not injective. The module is not a cogenerator because the non zero solutions of ( ∂x∂ 1 − α1 )f = 0, . . . , ( ∂x∂n − αn )f = 0 do not have compact support. 0

Example 2.1.13 : The set of distributions, D (Rn , K), is an injective module and cogenerator. The proof is similar to the one for C∞ (Rn , K). n

n

Example 2.1.14 : The D-modules defined in remark 1.2.11 (i.e. A = RN and A = RZ ) are injective cogenerators (see [39]). For more examples see [60, 62].

2.1.2

Homs and exact sequences

Given the R-modules B, C, and E and the R-linear maps α : B −→ C, we define Hom(α, E) : Hom(C, E) −→ Hom(B, E) by

ϕ 7−→ ϕ ◦ α. dj

dj+1

Definition 2.1.15 A sequence · · · −→ Aj−1 −→ Aj −→ Aj+1 −→ · · · of R-modules and homomorphisms dj is called exact if for every (relevant) j one has ker dj+1 = im dj . β

α

For example the sequence 0 −→ A1 −→ A2 −→ A3 −→ 0 is exact if and only if α is injective, β surjective and ker β = im α. In other words, A1 can be identified with a submodule of A2 , and A3 with the module of A2 /A1 . Exact sequences provides an easy way to express algebraic and system-theoretic properties. Lemma 2.1.16 If the sequence

α

β

0 −→ B −→ C −→ D −→ 0

is exact, then so is

β˜

α ˜

Hom(B, A) ←− Hom(C, A) ←− Hom(D, A) ←− 0 where α ˜ = Hom(α, E) and β˜ = Hom(β, E). Moreover, A is injective if and only if α ˜

β˜

0 ←− Hom(B, A) ←− Hom(C, A) ←− Hom(D, A) ←− 0 is exact for any exact sequence 2.1.

(2.1) (2.2)

22


Lemma 2.1.17 Suppose that A is injective. Then the following properties are equivalent: 1. A is a cogenerator. 2. A complex β

α

0 −→ B −→ C −→ D −→ 0 is exact if β˜

α ˜

0 ←− Hom(B, A) ←− Hom(C, A) ←− Hom(D, A) ←− 0

(2.3)

is exact. Proof : (1) ⇒ (2): It is enough to show that ker(α) = 0, im(β) = D and im(α) =ker (β). First we proof ker(α) = 0. Construct the following exact sequence: i

α0

0 −→ ker(α) −→ B −→ im(α) −→ 0,

(2.4)

where α0 (b) = α(b), for all b ∈ B, and i is the identity map. Since A is injective one has that ˜i α˜0 0 ←− Hom(ker(α), A) ←− Hom(B, A) ←− Hom(im(α), A) ←− 0 (2.5) α0

α

i

is exact and therefore α˜0 is injective. Now write B −→ C as B −→ im(α) −→ C to obtain α˜0

˜i

α ˜

Hom(B, A) ←− Hom(im(α), A) ←− Hom(C, A). By (2.3) one has that Hom(B, A) ←− α˜0

Hom(C, A) is surjective and hence Hom(B, A) ←− Hom(im(α), A) is also surjective. This implies that Hom(B, A) ≈ Hom(im(α), A) and therefore Hom(ker(α), A) = 0. Finally, Hom(ker(α), A) = 0 implies ker(α) = 0 using the assumption that A is a cogenerator. Now we proof that im(β) = D. Construct the following exact sequence: i

0 −→ im(β) −→ D −→ D/im(β) −→ 0.

(2.6)

Since A is injective one has that ˜i

0 ←− Hom(im(β), A) ←− Hom(D, A) ←− Hom(D/im(β), A) ←− 0

(2.7)

0

β β i is exact and therefore ˜i is surjective. Now write C −→ D as C −→ im(β) −→ D (i.e. β˜0

˜i

i · β 0 = β) to obtain Hom(C, A) ←− Hom(im(β), A) ←− Hom(D, A). By (2.3) one has β˜

˜i

that Hom(C, A) ←− Hom(D, A) is injective and hence Hom(im(β), A) ←− Hom(D, A) is also injective. This implies that Hom(D/im(β), A) = 0 and using the assumption that A is a cogenerator one obtains that D/im(β) = 0, or equivalently D = im(β) .

2.1. Some algebra

23

Finally we show that im(α) =ker (β): Construct the following exact sequence: i

0 −→ im(α) −→ ker(β) −→ ker(β)/im(α) −→ 0.

(2.8)

Since A is injective one has that ˜i

0 ←− Hom(im(α), A) ←− Hom(ker(β), A) ←− Hom(ker(β)/im(α), A) ←− 0

(2.9)

is exact and therefore ˜i is surjective. Now we want to show that ˜i is also injective i.e. ker(˜i) = 0. Suppose ker(˜i) 6= 0, i.e. there exists ϕ ∈ Hom(ker(β), A), ϕ 6= 0 and ˜i(ϕ) = ϕ · i = 0. Hence, ϕ must satisfy ϕ(x) = 0 for all x ∈ im(α) and since ϕ 6= 0 there exists x0 ∈ ker(β) such that ϕ(x0 ) 6= 0. Consider the sequence i0

0 −→ ker(β) −→ C.

(2.10)

Since A is injective one has that i˜0

0 ←− Hom(ker(β), A) ←− Hom(C, A)

(2.11)

is exact. Hence there exists ϕ0 ∈ Hom(C, A) such that ˜i0 (ϕ0 ) = ϕ0 · i0 = ϕ and therefore ˜ which is a contradiction ϕ0 (x0 ) 6= 0. Therefore one has that ϕ0 ∈ ker(˜ α) but ϕ0 ∈ / im(β), ˜ which implies that ker(˜i) = 0. with ker(˜ α) = im(β) Hence, Hom(ker(β)/im(α), A) = 0 and since A is a cogenerator ker(β)/im(α) = 0 ⇒ ker(β) = im(α). (2) ⇒ (1): If Hom(C, A) = 0 then 0 ←− 0 ←− Hom(C, A) ←− 0 ←− 0 is trivially exact and one has that 0 ←− 0 ←− C ←− 0 ←− 0 must be exact, which implies that C = 0. ¤

2.1.3

The duality

In this subsection we give some important consequences for behaviors whose signal space is an injective cogenerator D-module, e.g. C∞ (Rn , K). Fix a positive interger q. For any behavior B ⊂ Aq we define the submodule B⊥ of Dq by B⊥ := {d ∈ Dq | d(ξ1 , . . . , ξn )w = 0 for all w ∈ B} ⊂ Dq and call it the orthogonal module. Conversely, for any submodule M of Dq we define the behavior M⊥ := {w ∈ Aq | d(ξ1 , . . . , ξn )w = 0 for all d ∈ M} ⊂ Aq .

24


Example 2.1.18 : Let q = 2, A = C∞ (R2 , K) and M = h(ξ1 , ξ2 ), (ξ2 , 0)i. The homomorphisms h : D2 −→ A are determined by the values h(1, 0) = f1 , h(0, 1) = f2 with f1 , f2 ∈ A. The condition h(m) = 0 for all m ∈ M, is equivalent to h(ξ1 , ξ2 ) =

∂ ∂ ∂ f1 + f2 = 0 and h(ξ2 , 0) = f1 = 0. ∂x1 ∂x2 ∂x2

(2.12)

Thus M⊥ = {(f1 , f2 ) ∈ A2 | (2.12) holds }. One can also describe M⊥ explicitly. Note that f1 (x1 , x2 ) = g1 (x1 ) for some g1 ∈ C∞ (R, K), since f1 depends only on x1 . Thus f2 = −x2 g˙ 1 (x1 ) + g2 (x2 ) and M⊥ = {(g1 (x1 ), −x2 g˙ 1 (x1 ) + g2 (x2 )) | g1 , g2 ∈ C∞ (R, K)}. We now provide a lemma which was an observation of Malgrange [30], and is the key for the development of the theory which establish the correspondence (duality) between B ⊂ Aq and Dq /B⊥ . Lemma 2.1.19 Let A be a D-module. For any submodule M ⊂ Dq these is an natural isomorphism: ∼ χM : Hom(Dq /M, A) −→ M⊥ ⊂ Aq . (2.13) Proof : Let {e1 , e2 , . . . , eq } be the standard basis of Dq and write {¯ e1 , e¯2 , . . . , e¯q } for their q images in D /M. χM is defined by, χM (`) = (`(¯ e1 ), `(¯ e2 ), . . . , `(¯ eq )) ∈ Aq , ` ∈ Hom(Dq /M, A). First note that (`(¯ e1 ), `(¯ e2 ), . . . , `(¯ eq )) lies in M⊥ . Indeed, if m = (m1 , m2 , . . . , mq ) ∈ M one has m ¯ = 0 and 0 = `(m) ¯ = m1 `(¯ e1 ) + m2 `(¯ e2 ) + . . . , +mq `(¯ eq ). χM is injective because {¯ e1 , e¯2 , . . . , e¯q } generate Dq /M. Let f = (f1 , f2 , . . . , fq ) ∈ M⊥ then there exists ` ∈ Hom(Dq /M, A) withχM (`) = f . Indeed, define L : Dq −→ Aq by L(ei ) = fi for all i. We have to show that L(M) = 0. Then it would follow that L induces the required map ` : Dq /M −→ Aq . Now for P m = (m1 , m2 , . . . , mq ) ∈ M one has L(m) = mi fi = 0 because f ∈ M⊥ . ¤ Let M be a finitely generated D-module, we use the notation D(M) := HomD (M, A) and M∗ := HomD (M, D). We will omit an explicit reference to the ring D as there will be no ambiguity and write Hom(, ) instead of HomD (, ). The relation between Dq /B⊥ and B establishes an association between an algebraic object on the one hand and trajectories of dynamical systems on the other. This is extraordinarily useful since we can express system theoretical properties of nD systems in terms of algebraic properties of finitely generated modules over polynomial rings in n variables. Remark 2.1.20 It is important to note that in the previous lemma neither the injectivity nor the cogenerator property of A are needed. We now see how the property of injectivity of A is related with the elimination problem, namely how differential equations that only involve the manifest variables can be obtained starting from the original equations involving both manifest and latent variables (see previous chapter).

2.1. Some algebra

25

Proposition 2.1.21 Let M ⊂ Dw1 ⊕ Dw2 and B = M⊥ ⊂ Aw1 ⊕ Aw2 . Then pr1 (B) is the behavior of M ∩ Dw1 . Proof : First note that b1 ∈ Aw1 belongs to pr1 (B) if and only if there exists b2 ∈ Aw2 such that d(b1 , b2 ) = 0 for all d ∈ M. In particular, d(b1 ) = 0 for all d ∈ M ∩ Dw1 and pr1 (B) ⊂ (M ∩ Dw1 )⊥ . The exact sequence 0 −→ Dw1 /M ∩ Dw1 −→ Dw1 +w2 /M −→ Dw2 /pr2 (M) −→ 0 yields an exact sequence 0 → Hom(Dw2 /pr2 (M), A) → Hom(Dw1 +w2 /M, A) = B → Hom(Dw1 /(Dw1 ∩M), A) → 0 Hence Hom(Dw1 /(Dw1 ∩ M), A) = (Dw1 ∩ M)⊥ ⊂ Aw1 ⇒ (Dw1 ∩ M)⊥ ⊂ pr1 (B) ⇒ (Dw1 ∩ M)⊥ = pr1 (B). ¤ Note that only the injective property of A is needed. For an analog result when A is not assumed to be injective see [62]. However, for an injective cogenerator signal space, A, the correspondence in Lemma 2.1.19 is much more stronger. Indeed, we have the following theorem: Theorem 2.1.22 Suppose A is any injective cogenerator. The maps B −→ B⊥ and M −→ M⊥ between the set of behaviors in Aq and the set of submodules of Dq are each other inverses (i.e. (B⊥ )⊥ = B and (M⊥ )⊥ = M). Moreover ⊥ has the following properies: ⊥ ⊥ ⊥ 1. ⊥ inverts inclusions (i.e. B1 ⊂ B2 ⇒ B⊥ 2 ⊂ B1 ; M1 ⊂ M2 ⇒ M2 ⊂ M1 ); ⊥ 2. (B1 ∩ B2 )⊥ = B⊥ 1 + B2 ; ⊥ 3. (M1 ∩ M2 )⊥ = M⊥ 1 + M2 ;

4. If B = ker(R) , R ∈ Dp×q then B⊥ is the submodule of Dq of all D-linear combinations of the rows of R and is denoted by hRi, imRT or Dp R. Proof : It is easy to see that B ⊂ B⊥⊥ . We first prove that B⊥⊥ = B and M⊥⊥ = M: It is easy to see that M ⊂ M⊥⊥ . Thus M⊥ ⊂ (M⊥ )⊥⊥ . We apply (1) to M ⊂ M⊥⊥ to obtain M⊥ ⊃ (M⊥⊥ )⊥ , and therefore M⊥ = M⊥⊥⊥ . Thus, B = M⊥ implies B = B⊥⊥ for any B. Consider M ⊂ N ⊆ Dq two D-modules. Then the exact sequence 0 −→ N /M −→ Dq /M −→ Dq /N −→ 0 yields the exact sequence 0 −→ Hom(D/N , A) = N ⊥ −→ Hom(Dq /M, A) = M⊥ −→ Hom(N /M, A) −→ 0

26


since A is injective. N 6= M implies Hom(N /M, A) 6= 0 using the cogenerator property of A and therefore N ⊥ ( M⊥ . In particular if N = M⊥⊥ one gets that (M⊥ )⊥ = M and therefore we obtain a bijection between modules and behaviors. (2) follows from (3). ψ

We now prove (3): Consider Dq −→ Dq /M1 ⊕ Dq /M2 , m 7−→ (m mod M1 , m mod M2 ) for m ∈ Dq . The kernel of ψ is M1 ∩ M2 . Thus he following sequence is exact: α

0 −→ Dq /M3 −→ Dq /M1 ⊕ Dq /M2 ,

(2.14)

with M3 := M1 ∩ M2 ⊂ Aq . If we apply Hom(−, A) to this sequence we obtain α ˜

q q ⊥ ⊥ 0 ← Hom(Dq /M3 , A) = M⊥ 3 ← Hom(D /M1 , A) ⊕ Hom(D /M2 , A) = M1 ⊕ M2 ⊥ ⊥ which is exact. Since α ˜ (`1 , `2 ) = `1 + `2 , one gets that M⊥ 3 = M1 + M2 . (4) is obvious.

¤

Remark 2.1.23 Note that for parts 1,2 and 3 of the previous theorem only injectivity is used. ⊥ Remark 2.1.24 By induction we have that (M1 ∩ M2 ∩ · · · ∩ Ms )⊥ = M⊥ 1 + M2 + · · · + M⊥ s , but in the following example we show that this does not hold for infinitely many Mi . See [61, Prop.3.1] for related results.

Example 2.1.25 : Take D = K[ξ], A = C∞ (R, R) and Mi = ξ i D = hξ i i for i = 1, 2, 3, . . . . One has that ∩i>0 Mi = 0 and thus (∩i>0 Mi )⊥ = A, however M⊥ i = {f ∈ d i ⊥ x ⊥ A | ( dx ) f = 0} and ∪i>0 Mi ( A since e ∈ / ∪i>0 Mi .

Remark 2.1.26 For general function-distributional spaces, i.e., for any choice of A, we have that B⊥⊥ = B since B = M⊥ for some D-module M and M⊥⊥⊥ = M⊥ . However, in general M⊥⊥ ) M, which means that a module of equations defining B fails to describe the ’dynamics’ of the behavior B. We will call M⊥⊥ the Willems closure of M. The problem is to determine the modules M satisfying M = M⊥⊥ . This problem is sometimes called the Willems closure problem, and is related to the cogenerator property of the signal space. In [60], S. Shankar addresses this question for signal spaces, A which are not injective and cogenerators e.g. the space of compactly supported functions or distributions, the Schwartz space or the space of temperate distributions.

2.2. Dictionary

27

Example 2.1.27 : Consider n = 1 i.e. D = K[ξ], A = K[x] and M ⊂ Dq . Then M⊥⊥ = {d ∈ Dq | ∃ p ∈ D with p(0) 6= 0 and pd ∈ M}.

(2.15)

Proof : Using the structure theorem for finitely generated modules over a principal ideal domain we write Dq /M = F ⊕ (⊕i D/Pi ) ⊕ (⊕j D/Pj ) where F is free, Pi = pni where pi is an irreducible polynomial, monic and pi 6= ξ, Pj = ξ m and n, m ≥ 1. Note that: (a) Hom(D, A) = A. (b) Hom(D/ξ N , A) = {h ∈ K[x] | (d/dx)N h = 0} = {h ∈ K[x] | degree(h) < N }. (c) Hom(D/pn , A) = 0 if p is irreducible and f 6= ξ. Hence (a) ∀q ∈ F (free), q 6= 0 ∃ ` : F → A with `(q) 6= 0. (b) ∀q ∈ D/(ξ)m , q 6= 0, ∃` : Dq /M → A ⇒ `(q) 6= 0. (c) ∀q ∈ D/pn , q 6= 0 ∀` : D/pn → A ⇒ `(q) = 0. From definition one has that M⊥⊥ /M = {z ∈ Dq /M | `(z) = 0 for every ` ∈ Hom(Dq /M, A)}. Conclusion: M⊥⊥ /M = ⊕(⊕i D/pni ), where pi ∈ D is an irreducible polynomio, monic, pi 6= ξ and n ≥ 1. This translates into (2.15). ¤ q Example 2.1.28 : Consider D = K[ξ], A = C∞ c (R, K) and M ⊂ D . Then

M⊥⊥ = {d ∈ Dq | ∃p ∈ D, p 6= 0, s.t. pd ∈ M}.

(2.16)

Proof : Similar to the above. Note that here Hom(K[ξ], A) 6= 0, Hom(K[ξ]/pn (ξ), A) = 0 if p monic irreducible and p 6= ξ, and Hom(K[ξ]/ξ N , A) = 0. ¤ In the main part of the thesis we will focus on solutions spaces in Aq where A is an injective cogenerator and thus M⊥⊥ = M for all M ⊂ Dq . If not specified, we always assume B to contain just smooth or distribution solutions. Remark 2.1.29 : Let N be any finitely generated D-module, then one can view Hom(N, A) as a behavior. Namely, give for N a presentation as N = Dq /M then Hom(N, A) = M ⊥ as submodule of Aq . Note that N has many presentations (and with different q’s). In particular Hom(N, A) can be viewed in many ways as a behavior. Maybe one has to call a behavior B ⊂ Aq , an embedded behavior and Hom(N, A) as a general behavior.

2.2

Dictionary

The tight relation between B and Dq /B⊥ is the key of the algebraic approach to control theory for nD systems. Hence, basic theoretical concepts of nD systems as controllability, strong controllability, autonomy, flatness or observability can be translated into properties of the module Dq /B⊥ . This is the content of this subsection.

28


Definition 2.2.1 Let B = ker(R) ⊂ Aq . Then the i-th component wi of w is called free or input if πi : B −→ A given by w 7−→ wi is surjective. The behavior is called autonomous if it has no free variables. Theorem 2.2.2 (see [46, 76, 77]) Given B = {w ∈ Aq | Rw = 0} ⊂ Aq with A any injective cogenerator. The following are equivalent: 1. B is autonomous; 2. M := Dq /B⊥ is torsion; 3. R has full column rank. A stronger form of autonomy for the case where A = C∞ (Rn , K) is defined below. We note that for this the choice of the A is essential. Definition 2.2.3 Let A = C∞ (Rn , K) and B ⊂ Aq be a behavior given by B = {w ∈ Aq | r1 w = · · · = rn w = 0} for some subset {r1 , · · · , rs } ⊂ Dq . One associates to B the sheaf shB of C-vector spaces on Rn given by shB(U ) = {f ∈ C∞ (U, K)q | r1 f = · · · = rn f = 0} for every open U ⊂ Rn . In particular B = shB(Rn ). B is called Hausdorff autonomous if for every open U 6= ∅, the restriction map shB(Rn ) −→ shB(U ) is injective. Thus, a behavior is a Hausdorff autonomous if every trajectory in the behavior is determined by its values on any open subset of Rn . The term Hausdorff was suggested by Shiva Shankar during his stay in Groningen in the spring of 2007. It comes from the observation that B induces the sheaf shB on Rn which, seen as an etale space (espace étalé), is Hausdorff if the restriction map B → B|U is injective, see [21]. The following definition was first given by Pillai and Shankar in [46]. Definition 2.2.4 A behavior B ⊂ Aq is called strongly autonomous if B has finite dimension over K. Theorem 2.2.5 Given a behavior B ⊂ Aq , then B strongly autonomous ⇒ B Hausdorff autonomous ⇒ B autonomous. Proof : First we proof strongly autonomous ⇒ B Hausdorff autonomous: Write N = Dq /B ⊥ , then B = Hom(N, A). Suppose that N has finite dimension. Then there exists an ideal I = (P1 (ξ1 ), . . . , Pn (ξn )), where each Pi is a non zero polynomial, such that I · N = 0 (see next subsection). Put AI = {f ∈ A| Pi (ξi )f = 0 for all i}. Clearly, AI is a finite dimensional vector space consisting of linear combinations of products of polynomials in x1 , . . . , xn and exponential functions of the type eλ1 x1 +···+λn xn . It follows that AqI → A|qU is injective for any open non empty U . Further, B ⊂ AqI follows from I · N = 0.

2.2. Dictionary

29

Now we prove Hausdorff autonomous ⇒ B autonomous in several steps. (1) Consider an exact sequence of finitely generated D-modules 0 → N3 → N2 . If the behavior of N2 is Hausdorff autonomous then the same holds for the behavior of N3 . Indeed, Hom(N3 , A) → Hom(N2 , A) → Hom(N2 , shA(U )) → Hom(N3 , shA(U )) and the maps are injective. (2) Suppose that N 6= 0 is torsion free, then the behavior of N is not Hausdorff autonomous. Proof: N ⊂ F , where F is a free module with basis e1 , . . . , es and where there exists a P ∈ D, P 6= 0 with P · F ⊂ N . Consider a homomorphism L : F → A such that L(e1 ), . . . L(es ) are general C ∞ functions with support in, say, some compact set S. Choose L an open U with U ∩S = ∅. Then the P ·L(e1 ), . . . , P ·L(es ) are non zero and ` : N ⊂ F → A ` is non zero. However N → A → shA(U ) is zero. (3) Consider the exact sequence 0 → Ntors → N → N 0 → 0 and N 0 has no torsion. If the behavior is Hausdorff automomous of N , then by (1) and (2), one has N = Ntors . ¤ Note that for n = 1 all three notions of autonomy are equivalent. Example 2.2.6 : Consider n = 2. Then B = Hom(C[ξ1 , ξ2 ]/(ξ2 ), A) is autonomous but not Hausdorff autonomous and B = Hom(C[ξ1 , ξ2 ]/(ξ12 + ξ22 ), A) is Hausdorff autonomous but not strongly autonomous. The first statement of Example 2.2.6 is obvious, the second statement follows from the fact that the solution of the Laplace equations are harmonic and they are real analytic. We recall the following. A real or complex valued function f defined on an open subset U of Rn is said to be real analytic (sometimes just called analytic) if f ∈ C∞ (U, K) is locally 1 at every point of U equal to its Taylor expansion. We note that the function 1+x 2 is a real analytic function on R however not globally equal to its Taylor expansion at 0. We will use the following property of a real analytic function f on Rn : If f is zero on some non empty open set, then f is identically zero (see [64] p.22). Definition 2.2.7 ( [64], p.22) Let Ω be an open set in Rn . A linear partial differential operator P in Ω is said to be (real) analytic-hypoelliptic if, given any open subset U of Ω and any distribution u in U , u is an analytic function in U if this is true of P u. If P ∈ D is (real) analytic-hypoelliptic, then B = Hom(D/P, A) ⊂ A is Hausdorff autonomous. Indeed, if u ∈ B, then P u = 0 and u is real analytic on Rn . Hence B → shB(U ) is injective for all open, non empty U . Theorem 2.2.8 (see [22, 64]) P ∈ D is real analytic hypoelliptic if and only if the homogeneous part Pm of highest degree has the property Pm (v) = 0 and v ∈ Rn implies v = 0.

30


Example 2.2.9 : For n = 2, P = 12 (ξ1 + iξ2 ) ∈ D. Then P is a differential analytic∂ ∂ ∂ hypoelliptic operator. Indeed, it is the Cauchy-Riemann operator ∂z = 12 ( ∂x + i ∂y ). The corresponding behavior is not strongly autonomous but Hausdorff autonomous.

Example 2.2.10 : Let B = Hom(C[x1 , x2 , x3 , x4 ]/(P, Q), A) with P ∈ C[x1 , x2 ] and Q ∈ C[x3 , x4 ] both real analytic hypoelliptic. Then B is Hausdorff autonomous because a solution f (x1 , x2 , x3 , x4 ) ∈ B is separately (real) analytic in the pair x1 , x2 and the pair x3 , x4 . Then f is analytic in x1 , x2 , x3 , x4 , (see [23]). An interesting special case is P = 21 ( ∂x∂ 1 + i ∂x∂ 2 ) and Q = 12 ( ∂x∂ 3 + i ∂x∂ 4 ). The corresponding behavior B consists of the holomorphic functions in the complex variables z1 = x1 + ix2 , z2 = x3 + ix4 . Open problems: It is unknown whether every Hausdorff behavior consists of real analytic functions. Except for the above examples we have no algebraic criterion for a D-module N equivalent to Hom(N, A) is Hausdorff autonomous. In particular, for n ≥ 2 we do not know for which ideals I ⊂ D the behavior Hom(D/I, A) is Hausdorff autonomous (or consists of real analytic functions). ¤ One of the most important definitions in system theory is the one of controllability. For state space systems, controllability is defined as the possibility of transferring the state from any initial value to any final value. In the behavioral approach controllability is defined independently of an a priori given input/state/output system structure, and actually coincide with the classical definitions when applied to the special case of state space systems. Definition 2.2.11 A behavior B is said to be controllable if for all w1 , w2 ∈ B and all sets U1 ,U2 ⊂ Rn with disjoint closure, there exist a w ∈ B such that w |U1 = w1 |U1 and w |U2 = w2 |U2 . Thus, for a behavior to be controllable, one must be able to consider the restrictions of two of its elements to open subsets and always find a third element which coincides with the first two on the sets where they have been restricted. Theorem 2.2.12 (see [46, 76, 77]) Let B = {w ∈ Aq | Rw = 0} ⊂ Aq . The following are equivalent: 1. B is controllable; 2. M := D/B⊥ is torsion free; 3. B admits an image representation i.e. there exists a polynomial matrix M ∈ Dq×m such that B = {w ∈ Aq | w = M ` with ` ∈ Am } = im(M ). Another central definition in systems theory is the one of observability. For this property one needs to split the variables of the system in two sets; the first set of variables is interpreted as the observed variables an the other is the set of ’to be deduced’ variables.

2.2. Dictionary

31

Definition 2.2.13 Let B = {w ∈ Aq | Rw = 0} ⊂ Aq with w = (w1 , w2 ) be a partition of w ∈ Aq .Then w2 it is said to be observable from w1 in B if given any two trajectories 0 0 00 00 0 00 0 00 (w1 , w2 ), (w1 , w2 ) ∈ B we have that w1 = w1 implies w2 = w2 . So, observability only becomes an intrinsic property of the behavior after a partition of the manifest variable w is given. It is clear that we can divide the set of variables in many ways, however when one is dealing with a latent variable representation of the behavior, it is natural to ask whether the latent variables are observable from the manifest variables. This is the case when B = {w ∈ Aq | w = M ` with ` ∈ Am } =im(M ), where w represents the manifest variables and ` the latent ones. Definition 2.2.14 We say that B = ker(R) has an observable image representation if there exists a polynomial matrix M ∈ Dq×m such that w ∈ B ⇔ ∃ a unique ` ∈ Am such that w = M `. For controllable 1D systems it can be shown that there always exists an observable image representation. This is not true for nD systems. Definition 2.2.15 [58] A behavior B is said to be strongly controllable if Aq /B⊥ is free. Strongly controllability as defined above has previously been called “rectifiability” and is equivalent to the concepts of free-controllability and flatness due to Fliess and coworkers (e.g [19]). Definition 2.2.16 A behavior B is said to be complementable if there exists another behavior B0 such that B ⊕ B0 = Aq . Definition 2.2.17 A polynomial matrix R ∈ Dg×q is called zero left prime (ZLP) if its gth order minors generate the polynomial ring D as an ideal. A polynomial matrix R ∈ Dg×q is called zero right prime if its transpose RT is zero left prime. Theorem 2.2.18 [50, 58, 78] Let B = {w ∈ Aq | Rw = 0} ⊂ Aq . The following are equivalent: 1. B has an observable image representation ; 2. B is complementable; 3. B is strongly controllable; 4. there exists a ZLP R such that B = ker(R). We end this section with a notion that will be used in the coming chapters. Definition 2.2.19 A behavior B = kerR is said to be regular if it has a full row rank kernel representation i.e. B⊥ is free.

32


For the case n = 1, all behaviors are regular. This is not longer true for n ≥ 2, take for x instance the 2D differential behavior B =ker( 1 ), consisting of all constant functions, x2 which cannot be described as the kernel of a single polynomial operator. Note that by Quillen-Suslin theorem Aq /B⊥ free implies B⊥ free, i.e. B strongly controllable implies B regular. Regular behaviors were also introduced and investigated in [41, Th. 7.39] as those whose module has projective dimension at most one. In dimension n ≤ 2 all controllable behaviors are regular according to [41, Th. 7.42].

2.3

Finite dimension and codimension

As mentioned before an important class of behaviors are whose which are finite-dimension as a vector space over a field K, see [20, 41] ( we also use the notation dimK (B) < ∞ to denote that B is finite-dimension over the field K). In the 1D case, all autonomous behaviors are finite-dimensional, which also means that the state space is finite dimensional. For multivariable behaviors this is, in general, not longer true, since it could have an infinite set of initial conditions. In this final section we introduce the notion of codimension and provide some results concerning finite dimensional nD and 2D behaviors. These results will be mainly used in Chapter 5. Example 2.3.1 : Let q = 1, with M = hξ12 , . . . , ξn2 i ⊂ D = C[ξ1 , . . . , ξn ]. Then M⊥ = {f ∈ A | ( ∂x∂ i )2 f = 0 for i = 1, . . . , n } and its basis is {1, x1 , . . . , xn , xi xj , . . . , x1 x2 · · · xn } which has 2n elements. In the other hand D/M has a C-basis {1, ξ1 , . . . , ξn , ξj ξj , . . . } = {ξ1α1 ξ2α2 · · · ξnαn with αi = {0, 1}, i = 1, . . . , n} that also has 2n elements. We will now introduce the notion of codimension . Codimension is a term used to indicate the difference between the dimensions of a certain set and one of its subsets. Definition 2.3.2 Let A ⊂ B be finitely generated D-modules. We say that A has finite codimension in B if the dimension of B/A as a vector space over K is finite i.e. dimK (B/A) < ∞. If this condition holds, then we write A ⊂ B.