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We collect three instances where the theory of generalized functions may still make contributions to the study of signals and systems. In the first, a purely algebraic approach is presented for LTI-ODE's, in terms of two operators, D and T, respectively the differentiation operator and the multiplication-by-the-independent-variable operator. This formalism adds simplicity, a duality theory, and nicely generalizes to other classes of operator equations and their solutions. In the second part we extend the classical bilateral Laplace transform to include Bohl functions with support inℝ by invoking Sato's hyperfunctions. Finally, in the third case we use the Colombeau algebra to allow for products of generalized functions. This is important in the study of (smooth) nonlinear systems driven by impulsive inputs, and hybrid system theory.

Model continuity plays an important role in applications like system identification, adaptive control, and machine learning. This paper provides sufficient conditions under which input-output systems represented by locally convergent Chen-Fliess series are jointly continuous with respect to their generating series and as operators mapping a ball in an Lp-space to a ball in an Lq-space, where p and q are conjugate exponents. The starting point is to introduce a class of topological vector spaces known as Silva spaces to frame the problem and then to employ the concept of a direct limit to describe convergence. The proof of the main continuity result combines elements of proofs for other forms of continuity appearing in the literature to produce the desired conclusion.

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and static output feedback is applied. The first step is to reconsider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output feedback, and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener-Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant.

A new type of Chen-Fliess series is first introduced which depends on the input and delayed versions of the input. It is then shown how a class of analytic differential delay systems with a single delay and constant initial conditions has input-output maps representable in terms of this new functional series. As in the classical case, the coefficients can be computed by iterated Lie derivatives, but here the method is applied to an infinite dimensional embedding of the original state space system. Finally, the more technical issue of series convergence is addressed. Sufficient conditions are produced to guarantee convergence in both a local and global sense.

This paper extends earlier work on balanced realizations for smooth nonlinear systems to the discrete time setting. The main idea behind the global balanced structure, if it exists, is that the linearized system along a nominal solution is balanced in the usual linear time-varying sense. Hence the global nonlinear balanced system is defined as the one that closes the commutative diagram between linearization and balancing.

This paper addresses state-space realizations for nonlinear adjoint operators. In particular the relationship among nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are clarified. The characterization of controllability, observability and Hankel operators, and controllability and observability functions will be derived based on it. Furthermore a duality between the controllability and observability functions will be proven. The statespace realizations ofsuch operators provide new insights to nonlinear control systems theory.

In linear system theory, Gramian matrices help quantify the input-tostate and state-to-output interactions in a state space model. In this paper two different existing generalizations of this idea are examined for the case of a stable bilinear system. One method is based on using the L2norm to measure signal sizes and leads to the notion of the controllability and observability functions, which are referred to collectively as energy functions. The other method is motivated by introducing algebraic generalizations of the linear system Lyapunov equations, the solutions of which are called the algebraic Gramians. While these generalizations are distinct, in this paper some new relationships between the two approaches are presented.