Christopher Diego Cruz-Ancona

Generalized multi-synchronization viewed as a multi-agent leader-following consensus problem

The work presents the generalized multi-synchronization of strictly different chaotic systems problem as a multi-agent leader-following consensus problem with fixed and not strongly connected topology from a differential algebra point of view. This problem is solved by using the differential primitive element as a linear combination of state measurements and control inputs from which it is possible to construct a family of transformations to carry out the multi-agent systems to a Generalized Observability Canonical Form Multi-output (GOCFM). Moreover, a dynamic consensus protocol is designed such that the states of the followers asymptotically converge to the state of the leader.

Generalized multi-synchronization

In this paper, the problem of Generalized Multi-Synchronization (GMS) in master-slave topology is addressed. Within a differential algebraic framework this problem is interpreted as a leader-following consensus problem of Multi-Agent Systems (MAS). Here, a multi-agent system is treated as a network of interconnected systems with strictly different dynamics of same dimension, fixed and not strongly connected topology. Multi-agent system is carried out to a Multi-output Generalized observability Canonical Form (MGOCF) with a family of transformations obtained from an adequate selection of the differential primitive element as a linear combination of state measurements and control inputs. This allow us to explicitly give the synchronization algebraic manifold and design a dynamic consensus protocol able to asymptotically achieve consensus for all agents in the network. Finally a worked out example is provided to illustrate the methodology proposed.

Algebraic Observability for Nonlinear Systems

The idea of observability is equal in importance to the idea of controllability , one of the key concepts introduced five decades ago in systems theory.

Estimators for a class of commensurate fractional order systems with caputo derivative

In this paper we propose a couple of fractional order observers for nonlinear commensurate fractional order systems i.e. a reduced order observer and a fractional Luenberger observer in the algebraic and differential setting. We introduce the Fractional Algebraic Observability (FAO) property, like a measurement degree of fractional observability of states variables. Finally a comparison between two observers illustrates the effectiveness of the suggested approach, this is performed with two different numerical examples: a linear mechanical oscillator with integer and fractional order damping; and a nonlinear fractional order Duffing System.

Differential Algebra in Control Theory

This chapter presents some elementary definitions and ideas from differential algebra on which this book is based.

More General Nonlinear Systems Case

The synthesis of observers is translated to the stability of the error dynamics. The simplest case is given when the observation error dynamics is linear, a condition that implies exponential stability of the error, which is translated to an appropriate calculation of eigenvalues. On the other hand, some difficulties may arise in the extension of observers to the nonlinear case. There exist very special cases, but in general, estimation error dynamics is not linear and must be stable for the observer synthesis to become feasible. In the worst-case scenario, the study of stability is extremely difficult, considering the fact that it depends on parameters from the system and the observer that are mostly unknown.

A Separation Principle for Nonlinear Systems

Since the early 1990s, a variety of approaches have been proposed for the synthesis of observers and controllers for nonlinear systems. A considerable number of researchers have studied the stability and asymptotic output tracking problems from different perspectives. An appealing approach is based on differential-geometric methods that are summarized in Isidori’s outstanding book [24]. In isidori’s treatment, a clear connection is established with the concepts of the inverse system and the zero dynamics using the notion of relative degree or relative order and the associated normal canonical form for nonlinear systems [2, 24]. This was an interesting generalization to the problem of exactly linearizing a nonlinear control system by means of a static-state feedback, which was solved independently by Jakubczyk and Respondek [25] and by Hunt et al. [23]. We refer to the references [5, 24] for a survey on this topic and for some material on input–output linearization. Over the past three decades, Charlet et al. [4] have tried to weaken the aforementioned conditions by allowing dynamic state feedbacks. They were able to prove, among other things, that for single-input systems, dynamic and static feedback condition coincide. In [28], interesting results on output stabilization for observed nonlinear systems via dynamic output feedback were considered, allowing one to deal with singularities that can appear. On the other hand, very important contributions have been made by Fliess and coworkers [9–19] using techniques based on differential algebra. Fliess’s ideas have contributed to a revision and clarification of the deeply rooted state-space approach [18, 42]. This approach has succeeded in clearly establishing basic concepts such as controllability, observability, invertibility, model matching, realization, exact linearization , and decoupling. Within this viewpoint, canonical forms [7–19, 31–34, 38–45] for nonlinear controlled systems are allowed to explicitly exhibit time derivatives of the control input functions on the state and output equations. Elimination of these input derivatives from the state equations via control-dependent state coordinate transformations is possible in the case of linear systems.

Observers for a Class of Nonlinear Systems

In this chapter will give a synthesis of the proposed observers for a class of nonlinear systems . We recall that in general, the main goal of the synthesis of an observer is the complete reconstruction of the state of the system. In the linear case, it is easy to see how this is proposed: the observer is a copy of 5h3 original system with a correction term given by the output error. Eventually, if the outputs match, the trajectories of the original system and the observer system are the same.

Parametric Identification of Time-Varying Nonlinear Systems

In this chapter we will study the problem of identification of unknown time-varying and time-invariant parameters of nonlinear systems. The suggested approach employs very simple differential-algebraic tools. The general idea consists in converting the identification problem into an observation problem. In order to solve this problem, we extend the system via system immersion [3].Systems identification of continuous time-varying systems can be performed, among other techniques, with aid of adaptive observers or neural networks [1, 8]. In this chapter, a differential-algebraic technique is used for identification of a certain class of continuous time-varying nonlinear systems [2, 4, 6]. The problem is to identify a parameter θ possibly depending on time of a continuous-time system. The methodology proposed consists in the following: Define first a function η(x, θ) as an extra state of the system [9]. This function is defined in terms of the states and the unknown parameter of the system. The dynamics of this new state is not known (i.e., \(\dot{\eta }\left (x,\theta \right )\) is unknown). The original system is then converted to an extended system in which the dynamics of the extra state is not known. The original identification problem is converted to an observation problem, where the aim is to observe this extra state of the system.

Observer-Based Local Stabilization and Asymptotic Output Tracking

In recent years, a variety of approaches has been used for the study of observer synthesis and control algorithms. In particular, applications of modern estimation and control techniques have been widely reported. For example, Lynch and Ramirez [11] designed optimal controllers with a Kalman filter for state estimation of a continuous stirred tank reactor (CSTR) . In Alvarez, Suarez, and Sanchez [1], a semiglobal nonlinear controller is designed that solves the problem of output tracking with disturbance rejection applied to a CSTR. Huan, Chao, and Chen [8] used an adaptive control for these types of processes. Cebuhar and Constanza [4] implemented control strategies for input–affine models of chemical reactors. The former monitoring techniques assume a complete knowledge of the state vector at every time. That is not always possible, since measurement techniques of some process variables are mostly indirect. Due to this restriction, it is necessary to design and implement state estimators. In general, introduction of a state estimator within a control scheme solves the measurement problem.Here, the method of estimation and control law algorithms are based on differential-algebraic techniques. Given a dynamical system described by a set of ordinary differential equations, it has been shown in [6] that there is an associated generalized controller canonical form depending on the controller inputs and their derivatives. Also, assuming that the system’s output is the differential primitive element, a generalized observability canonical form can be given. The generalized observability canonical form is obtained through a coordinate transformation based on the output and the control input (differential primitive element ), which possibly depends on a finite number of their time derivatives. For example, applications of this approach to modeling physical systems are given in [13, 14, 15].