Realization theory of discrete-time dynamical systems.
The objective of the book is to develop a realization theory for discrete-time non-linear systems in a general approach. The first author has constructed such a theory for non-linear continuous-time systems in [\textit{T. Matsuo}, Realization Theory of Continuous-Time Dynamical Systems, Lecture Notes in Control and Information Sciences, 32 (1981)], using as a starting point Kalman's algebraic theory for linear systems. The present monograph studies the realization theory for the discrete-time counterparts of the continuous-time General Dynamical Systems, Linear Representation Systems and Affine Dynamical Systems presented in the book quoted above and extends this theory to some special sub-classes called Pseudo Linear Systems, Almost Linear Systems and So-called Linear Systems (which include the category of Linear Systems). The monograph contains eight chapters and a suitable list of references. The chapters 3-10 have two parts: the first part presents the notions, the main results, the historical notes and the concluding remarks; the second part is an appendix which completes the list of definitions or theorems and contains the proofs of the mentioned results. The introductory Chapter 1 presents the aim and the structure of the book, as well as the inclusion diagram of different sub-classes of dynamical systems and automata. Chapter 2 collects the basic definitions necessary to the concept of input/output map. The free-monoid over an alphabet \(U\) is said to be a {concatenation monoid} \(\Omega\); the concatenation of two words \(\omega_{1}\) and \(\omega_{2}\) is denoted by \(\omega_{2}| \omega_{1}\) and \(| \omega| \) denotes the length of the word \(\omega\). For a set \(Y\), an { input/output relation} is defined as a map \(A^{\#}:\Omega\rightarrow F_{S}(\mathcal{N},Y):=\{\gamma:[0,n]\subset\mathbb{N}\rightarrow Y| \;\text{for some}\;n\in\mathbb{N}\}\). If \(A^{\#}(\omega_{2}| \omega_{1})(n)= A^{\#}(\omega_{3}| \omega_{1})(n),\;0\leq n\leq| \omega| \) for any \(\omega_{1},\omega_{2},\omega_{3}\in\Omega\) then the map \(A^{\#}\) is said to be an { input/output map with causality}. The function \(a:\Omega\rightarrow Y\) defined by \(a(\omega)=A^{\#}(\omega)(| \omega| )\) is said to be an { input response map}. Chapter 3 introduces the concept of General Dynamical System as a suitable model for the input/output relations; a { General Dynamical System} (GDS) is a collection \(\sigma=((X,f),x^{0},h)\) where \(X\) and \(Y\) are two sets (the state and the output set respectively), \(f\) is an operator \(f:\rightarrow F(X):=\{\;x:X\rightarrow X\}\), \(x^{0}\in X\) (\(x^{0}\) is the initial state of \(\sigma\)) and \(h:X\rightarrow Y\) is an operator; \(\sigma\) is described by the equations \[ \begin{aligned} x(t+1)=f(\omega(t+1))x(t) \tag{1} \\ x(0)=x^{0} \tag{2} \\ \gamma(t)=hx(t) \tag{3} \end{aligned} \] for any \(t\in\mathbb{N}\). Equation (1) is said to be a U-action and it defines the { behavior} of \(\sigma\), i.e. the input response map \(a_{\sigma}:\Omega\rightarrow Y;\omega\rightarrow h\phi_{f}(\omega)x^{0}\) where \(\phi_{f}(\omega)=f(\omega(| \omega| ))f(\omega(| \omega| -1))\cdots f(\omega(1))\) is the transition map of \(\sigma\). The notions of reachability and distinguishability are introduced and a GDS is said to be {canonical} if it is both reachable and distinguishable. The realization theorem contains the existence and the unicity of canonical realizations. The existence part gives for any input response two canonical GDS that realize it, one of them derived from Nerode equivalence (as in Kalman's algebraic theory). The uniqueness part shows that any two canonical realizations are isomorphic by a unique system isomorphism. Finite GDS (i.e. GDS with finite state \(X\)) are studied and it is shown that a complete Eilenberg automaton is such a GDS. Minimal control systems are introduced, as well as some extensions of minimal automata; a realization theory is constructed for them. The control theory is applied to derive a condition for a given black-box to be obtained by a single experiment. Chapter 4 is devoted to Linear Representation Systems (LRS), i.e. particular GDS (1) (2) (3) with \(X\) and \(Y\) linear spaces over a field \(K\) and \(f\) and \(h\) linear operators. A realization theory is provided, which is similar to the realization theory of GDS. The case of finite dimensional LRS is investigated and a criterion for the canonicity of these systems is given, as well as a representation theorem of equivalence classes. A criterion for the existence of finite dimensional LRS is obtained on the basis of the rank of a Hankel matrix and a realization procedure is derived. It is shown that Eilenberg's K-U-automata are finite dimensional LRS. The last part of this chapter contains the partial realization theory. The notion of natural partial realization is introduced and a necessary and sufficient condition for the existence and uniqueness of the minimal natural partial realization is given by a rank condition for a finite sized Hankel Matrix. An algorithm is provided to obtain a natural partial realization from a given partial input response map. Another sub-class of GDS is studied in Chapter 5: a GDS is said to be an { Affine Dynamical System} (ADS) if \(X\) and \(Y\) are affine spaces and \(f\) and \(h\) are affine maps. A similar theory to that described in Chapter 4 is developed in this framework. Chapter 6 deals with Pseudo Linear Systems (PLS), which are special cases of ADS. A similar theory is presented. The main theorem states that for any input/output map with causality and time-invariance there exist at least two canonical PLS which realize it and any two canonical PLS with the same behavior are isomorphic. A criterion for the behavior of an \(n\)-dimensional canonical PLS is obtained by using a rank condition of an input/output matrix. Moreover, a real time partial realization problem is discussed and a mathematical model is derived from on-line data by a single experiment. The goal of Chapter 7 is to examine the Almost Linear Systems (ALS) which form a sub-class of the PLS class. The main realization theorem is similar to that of PLS by adding the condition of affinity of the input/output map. The ALS contain the sub-class of linear systems with a non-zero initial state, named in the book So-called Linear Systems (SLS) and these systems are studied in Chapter 8. A new method to obtain an SLS realization directly from input/output data is given. The real-time partial realization theorem states that for any causal, time-invariant, affine input/output map obtained on real time (i.e. by a single experiment) there exists a controllable and observable (hence minimal) SLS which realize it and all minimal realizations which have the same partial behavior are isomorphic. An algorithm which provides a partial realization SLS is presented.
scientific article; zbMATH DE number 2002748