Luis A. Duffaut Espinosa

Left inversion of analytic nonlinear SISO systems via formal power series methods

Given a single-input, single-output (SISO) system, F , and a function y in the range of F , the left inversion problem is to determine an input u such that y = F u . The goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where F is an analytic mapping in the sense that it has a convergent Chen-Fliess functional expansion, and y is a real analytic function. In particular, it will be shown that given a certain condition on the generating series c of F , a corresponding unique analytic u can always be determined via operations on formal power series. The condition on c turns out to be equivalent to having a well-defined relative degree when F has an input-affine analytic state space realization with finite dimension. But the method is applicable even when F does not have such a realization. The technique is demonstrated on four examples, including a continuous stirred chemical reactor.

A combinatorial Hopf algebra for nonlinear output feedback control systems

Abstract In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given.

Analytic left inversion of multivariable Lotka-Volterra models

There is great interest in managing populations of animal species that are vital food sources for humans. A classical population model is the Lotka-Volterra system, which can be viewed as a nonlinear input-output system when time-varying parameters are taken as inputs and the population levels are the outputs. If some of these inputs can be actuated, this sets up an open-loop control problem where a certain population profile as a function of time is desired, and the objective is to determine suitable system inputs to produce this profile. Mathematically, this is a left inversion problem. In this paper, the general left inversion problem is solved for multivariable input-output systems that can be represented in terms of Chen-Fliess series using concepts from combinatorial Hopf algebras. The method is then applied to a three species, two-input, two-output Lotka-Volterra system. The biological goal is to change the population dynamics of the top-level predator species in a food chain in order to prevent extinction.

Analytic left inversion of SISO Lotka-Volterra models

There is great interest in managing populations of animal species such as fish that are vital food sources for humans. A classical population model is the Lotka-Volterra system, which can be viewed as a nonlinear input-output system where time-varying parameters are taken as inputs and the population levels are the outputs. If some of these inputs can be actuated, this sets up an open-loop control problem where a certain population profile as a function of time is desired, and the objective is to determine suitable system inputs to produce this profile. Mathematically, this is a left inversion problem. In this paper, this inversion problem is solved analytically using known methods based on combinatorial Hopf algebras. The focus is on the simplest case, two species models and single-input, single-output (SISO) systems.

Towards a macromodel for Packetized Energy Management of resistive water heaters

This paper presents a state bin transition (macro)model for a large homogeneous population of thermostatically controlled loads (TCLs). The energy use of these TCLs is coordinated with a novel bottom-up asynchronous, anonymous, and randomizing control paradigm called Packetized Energy Management (PEM). A macro-model for a population of TCLs is developed and then augmented with a timer to capture the duration and consumption of energy packets and with exit-ON/OFF dynamics to ensure consumer quality of service. PEM permits a virtual power plant (VPP) operator to interact with TCLs through a packet request mechanism. The VPP regulates the proportion of accepted packet requests to allow tight tracking of balancing signals. The developed macro-model compares well with (agent-based) micro-simulations of TCLs under PEM and can be represented by a controlled Markov chain.

Discrete-time approximations of Fliess operators

A convenient way to represent a nonlinear input–output system in control theory is via a Chen–Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are asymptotically achievable for certain worst case inputs. The paper then focuses on the special case where the operators are rational, i.e., they have rational generating series, and thus are realizable in terms of bilinear ordinary differential state equations. In particular, it is shown that a discretization of the state equation via a kind of Euler approximation coincides exactly with the discrete-time Fliess operator approximator of the continuous-time rational operator.

Discrete-time approximations of fliess operators

A common way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated sums and to provide accurate error bounds for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error bounds are asymptotically achievable in certain worst case scenarios.

Fractional Fliess operators: Two approaches

A useful representation of an input-output map in a nonlinear control system is the Chen-Fliess functional series or Fliess operator. It can be viewed as a noncommutative generalization of a Taylor series, and its algebraic nature is especially well suited for a number of important applications. The objective of this paper is to describe a generalization of this class of operators, so called fractional Fliess operators. These are functional series whose coefficients have a certain fractional growth rate (Gevrey series) and whose iterated integrals are defined in terms of fractional integrals. The motivation for this idea is two-fold. First, fractional system theory is a well developed area with a variety of applications, so this concept is a natural generalization in its own right. But even in the classical case it has been observed that the cascade interconnection of two Fliess operators can result in a composite system that has a certain fractional nature. Hence, developing this generalization may also provide some insight into this issue.

Pre-Lie algebra characterization of SISO feedback invariants

Transformation groups have been used extensively in system theory since its inception. Recently, a feedback transformation group for a nonlinear input-output system characterized by a Chen-Fliess functional expansion was described by the authors. In particular, an algorithm was given to identify a class of feedback invariant series. Their relationship to series having well-defined relative degree was also developed. This paper is a continuation of that work, but with three innovations. First, newly developed algebraic tools from the field of pre-Lie algebras are applied to give more insight into the invariance theory. The role of relative degree in this paper is diminished in favor of systems with arbitrary generating series. Finally, dynamic output feedback, as described by the feedback product, is considered explicitly.

Integration of output tracking and trajectory generation via analytic left inversion

In this paper, an algorithm for the integration of output tracking and trajectory generation via analytic left inversion is provided. The first step is to identify an output path using a known trajectory generation algorithm, and then a spline approximation of the path is computed. The second step is to solve the output tracking problem by explicitly computing the left inverse of the input-output map of the system to render the Taylor series of the desired input for each polynomial section of the spline approximation. In practice, these infinite series must be truncated and executed over a finite interval of time, so the relationship between truncation, tracking error and execution time is characterized. A case study is presented involving the kinematics of a bi-steerable car.