### Paradigms and puzzles in the theory of dynamical systems

J.C. Willems
1991 IEEE Transactions on Automatic Control
Was ich hier geschrieben habe, macht im Einzelnen uberhaupt nicht die Anspruch auf Neuheit (Wiltgenstein, Tractatus). theory of dynamical systems. The basic ingredients form a triptych, with Abstract-We will outline the main features of a framework for a the behavior of a system in the center, and behavioral equations and latent variables as side panels. We will discuss a variety of representation and parametrization problems, in particular, questions related to input/output and state models.
more » ... is shown how the interconnection of systems fits into this framework, in particular, problems of feedback control. The final issue to which we pay attention is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. Now, a (deterministic) mathematical model for the phenomenon (viewed purely from the behavioral, the black box point of view) claims that certain outcomes are possible, while others are not. Hence a model recognizes a certain subset 23 of M. This subset will be called the behavior (of the model). Formally: Definition I.1: A mathematical model is a pair (U, 8) with U the universum-its elements are called outcomes-and 8 the behavior. Example: The fact that the absolute temperature is nonnegative may be viewed as modeling the phenomenon temperature. Thus U = R and \$3 = [ -273, CQ) (with the temperature expressed in 'C). Example: During the ice age, shortly after Prometheus stole fire from the Gods, man realized that H,O could appear, depending on the temperature, as water, steam, or ice. It took a while longer before this situation was captured in a mathematical model. The generally accepted model is as follows: Example: Economists believe that there exists a relation between the production P of a particular economic resource, the capital K invested in the necessary infrastructure, and the labor L expended towards its production. A typical model will look like: M = R: , and 23 = { ( P , K , LjER:I P = F ( K , Lj}, In applications, models are more often than not described by equations. Definition 1.2: Let QJ be a universum, E an abstract set, called the equating spaces, and j l , j z : equation representation The best way of looking at the behavioral equations f l ( u ) = j 2 ( u ) is as equilibrium conditions: the behavior 8 consists of those attributes for which two (sets of) quantities are in balance. A few remarks are in order. First, in many applications models will be described by behavioral inequalities: simply take in the aforementioned definition I to be an ordered space and consider the behavioral inequality f , ( u ) 5 j 2 ( u ) . Many ( x = f " x and its generalizations). Notwithstanding the fact that Most dynamical systems are indeed described by behavioral many areas Of physics (as Hamiltonian mechanics and quantum equations. These are often differential or difference equations, mechanics) fit this framework very nicely and notwithstanding sometimes integral equations. We will postpone the introduction the impressive phenomenological achievements of this theory (as of such descriptions until later. First, we want to introduce some chaos and strange attractors). this line of thought suffers from general properties of dynamical systems in our framework. two serious drawbacks. First, it considers a system as isolated These qualitative properties can be of various type: involving from its environment and, second, it assumes that a model is structure (as linearity), symmetry (as time invariance), related to already in minimal state form (that is, very roughly speaking, the memory (as the state), involving the interaction with the that it assumes that the differential equations describing the environment (as inputs and outputs), etc. system are first order) and explicit in x. Both conditions are Definition 11.2: A dynamical system X = (p, W, 8 ) is said rarely satisfied in real applications. In engineering, particularly to be linear if W is a vector space (over a field %-for the in control and signal processing, there has always been a tenpurposes of this paper, think of it as R) and 8 is a linear d e w to view systems as processors, Producing output signals subspace of W v (which is a vector space in the obvious way by from input signals. In many applications in control engineering pointwise addition and multiplication by a scalar). and signal processing, it will, indeed, be eminently clear what Thus, ]inear systems obey the superposition principle in its the inputs and the outputs are. However, there are also many simplest form: {wl(.), w2(.) E 8 ; applications where this input-output structure is not at all evip w,(. ) E 8 1. dent (an example at point is in the terminal behavior of an we can view time-invariance as a of electrical circuit). We will view a dynamical system in the we w i~ therefore first introduce logical context of Definition 1.1. The only distinguishing feature kt be a family ofdyna_micd systems, and @ = (sg, E 8) is that now the phenomenon produces outcomes that are funca transformation group on C. T h p @ is a group, and S, defines tions of time: the universum is a function space. This leads to for each E 8 a _bijection on x satisfying sgl ~ g2 = s" 0 sg2 . the following definition. We will call C E C @-symmetric if S, Z = B for all g E @ , (U, W, 8) with D & R the time axis, W the signal space, and @ consists of the one point 2 9 E W v the behavior. Thus, a dynamical system2 is defined by U, the time instants axis = z, Take = z , viewed as a group under addition. Example: Let C denote all dynamical systems with the time of interest W, the space in which the time signals which the L~~ ur: wz -, ~" 3 " denote the backward t-shift: ("1f)(~3 := system produces take on their values, and 8, a family of f ( t f + t ) . N~~ define, for E p, s,: 2 -, by W-valued time trajectories. The sets e and w define the setting, s,((z,w, 8)) := (8, w, , , t~) , and @ := (st, The the mathematization of the problem, while 8 formalizes the dynamical systems which are in this will be model C, time signals in 8 can in principle occur, are compati-in a trivial way to continuous time (u = w) (and-in a ble with the laws governing 2 , while those outside 8 cannot bit less trivial way--to arbitrary time sets). occur, are prohibited. Example: Let B denote all time-invariant dynamical systems We will use the terms "system" and "dynamical system" with time axis = R, consider the (trivial) group consisting of interchangeably. Our notion of a dynamical system hardly does two points: 8 = 11, g } , 1. ~~f i~~