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This book comprehensively examines various significant aspects of linear time-invariant systems theory, both for continuous-time and discrete-time. Using a number of new mathematical methods it provides complete and exact proofs of all the systems theoretic and electrical engineering results, as well as important results and algorithms demonstrated with nontrivial computer examples. The book is intended for readers who have completed the first two years of a university mathematics course. All further mathematical results required are proven in the book.

Chapter 1. Introduction

Abstract
Our goal in writing this book was to derive systems theoretic and electrical engineering results, to prove them using only mathematics which is taught in the first 2 years of a mathematics course at university, and to provide algorithms for all important results. In this introductory chapter, we provide an overview of the covered topics. We describe the employed mathematical methods and compare our approach and methodology to those which are used in most books on systems theory.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 2. A Survey of the Book’s Content

Abstract
This chapter is a detailed comment on the content of the book and a self-contained survey over larger parts of linear time-invariant systems theory and electrical engineering, on the basis of mathematical knowledge of two university years. For simplicity, we restrict ourselves here to the continuous-time case over the complex field $${\mathbb {C}}$$. We present the most important methods and results of the book and refer to the chapters or theorems, where they are discussed as well as to corresponding results in the literature. This chapter contains no proof and does not assume any knowledge from other chapters.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 3. The Language and Fundamental Properties of Behaviors

Abstract
A technical system as, for example, a machine or an electrical or mechanical network, is typically modeled as a set of equations. The solution set of such a systemof equations is called a behavior,while the individual solutions are the trajectories or signals of the system. In this book, we consider time systems, where the trajectories are functions of a continuous or discrete-time variable which satisfy linear differential or difference equations with constant coefficients. We study behaviors bymeans of their equations or, more precisely, the module of their equations, which allows an algebraic characterization of behavior properties. This approach requires a sufficiently close relationship between the (analytical) signals on the one hand and the (algebraical) equationmodules on the other hand. If the signal space is an injective cogenerator over the ring of operators-a property that is satisfied in all standard cases-such a close relationship is ensured. In fact, an injective cogenerator signal module leads to a perfect duality between the category of systems or behaviors and the category of finitely generated modules with distinguished sets of generators. The language and the mathematical framework introduced here forms the fundament for the rest of the book.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 4. Observability, Autonomy, and Controllability of Behaviors

Abstract
In this chapter we study the three fundamental system properties observability, autonomy, and controllability. We characterize these algebraically and in terms of the behavior’s trajectories, and derive Kalman’s famous results on state space systems as special cases. Both controllability and autonomy of a behavior can be described algebraically via the torsion module of the system module.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 5. Applications of the Chinese Remainder Theorem

Abstract
It applies to coprime ideals in a not necessarily commutative ring.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 6. Input/Output Behaviors

Abstract
In this chapter, we discuss IO behaviors, i.e., behaviors whose trajectories are decomposed into an input and an output component. Most non-autonomous systems are of this type. Every IO behavior has a transfer matrix, which we derive algebraically by module-behavior duality and which we will use in Sect. 8.​2.​4 for defining the impulse response and the Laplace transform for continuous-time systems. This approach is different from the usual one, where the Laplace transform is introduced analytically and then used for defining the transfer matrix.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 7. Interconnections of Input/Output Behaviors

Abstract
Complex systems are often obtained by interconnecting more elementary input/output behaviors. For example, every state-space system with proper transfer matrix can be constructed by interconnecting elementary building blocks, namely, adders, multipliers, and integrators (continuous time) or delay elements (discrete time). Basic types of interconnections are the series or cascade connection, the parallel connection, and the feedback interconnection. The latter is fundamental for the construction of stabilizing compensators and for controller design in Chap. 10.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 8. The Transfer Matrix as Operator or Input/Output Map

Abstract
In the standard discrete-time and continuous-time cases and for relevant classes of signals, we interpret the transfer matrix of an input/output behavior as transfer operator which maps input signals to output signals. In the most important settings, this action is given by convolution. The transfer operator generalizes the gain matrix. We characterize external stability and, in particular, bounded input/bounded output (BIBO) stability of transfer operators. The derived theory is then applied to electrical and mechanical networks.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 9. Stability via Quotient Modules

Abstract
We use quotient modules and quotient behaviors for algebraically treating the stability of autonomous and of IO behaviors. The prototypical example for such a quotient is the quotient field (e.g., $${\mathbb {Q}}$$) of a commutative integral domain (e.g., $${\mathbb {Z}}$$). The technique for rings and modules is presented and proven in Sections 9.1 and 9.2 with an emphasis on principal ideal domains. In Sect. 9.3, we apply this method to autonomous behaviors and in Sect. 9.4 to IO behaviors. In Sect. 9.5, we introduce the ring of proper and stable rational functions which will be essential in Chap.  10 for constructing stabilizing compensators with proper transfer matrices.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 10. Compensators

Abstract
We treat the two topics stabilization and compensator design in this chapter. Stabilization of an unstable IO behavior (the plant) by output feedback means to construct a suitable IO behavior (the stabilizing compensator or stabilizing controller) such that the feedback or closed-loop behavior is well-posed and T-stable for a suitable monoid T of stable polynomials in the sense of Chap. 9. In Sect. 10.1, we characterize the existence of a stabilizing compensator and then parameterize all of them. Our main emphasis is on the case where the compensator as well as the feedback behavior is proper IO behaviors, whereas the plant is not assumed proper. Compensator design means to construct stabilizing compensators that realize desired tasks like tracking, disturbance rejection, or model matching; and this is one of the key tasks in systems theory. We discuss the existence and parametrization of such restricted compensators in detail in Sect. 10.2. In the standard situations, we also show that the constructed compensators are robust. This signifies that the compensator is valid not only for the given plant, but also for all nearby ones. This obviously necessitates to topologize the set of all plants.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 11. Observers

Abstract
Often certain components of a behavior are of interest, but cannot be measured directly. An observer is an input/output behavior that uses the known quantities in order to produce an estimate for the desired information. Typically, the estimate is required to approximate the desired signal asymptotically, a concept that we incorporate by using the stability theory from Chap. 9. The setting includes, but is not restricted to, classical state observers as introduced by Luenberger. We derive an algorithmic construction of all possible observers for a given observer problem.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 12. Canonical State Space Realizations of IO Systems via Gröbner Bases

Abstract
In Theorem 6.​3.​11, we presented Fliess’ unconstructive proof that every IO behavior over a univariate polynomial ring admits an observable state-space realization, which is unique up to similarity. In this chapter, we use the Gröbner basis theory to construct the canonical observability and observer realizations of a behavior. If the behavior is controllable, we also construct the canonical controllability and controller realizations. The canonical realizations depend only on the 30 given IO behavior and on a chosen term order in the Gröbner basis theory, but not on the particular matrices that define the behavior. If the transfer matrix of the IO behavior is proper, these realizations can be built with elementary building blocks. In particular, the proper compensators and observers from Chaps. 10 and 11 can be implemented as state-space behaviors in this fashion.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher

Chapter 13. Generalized Fractional Calculus and Behaviors

Abstract
We show in this chapter that the module-behavior duality can also be applied to fractional or symbolic calculus, to suitably defined fractional behaviors and to the constructive solution of general fractional differential systems. We generalize the standard fractional calculus considerably.
Ulrich Oberst, Martin Scheicher, Ingrid Scheicher