Generalized Observability Canonical Form Research Articles

Generalized synchronization of commensurate fractional-order chaotic systems: Applications in secure information transmission

In this work, a class of chaotic nonlinear fractional systems of commensurate order called Liouvillian systems is considered to solve the problem of generalized synchronization. To solve this problem, the master and the slave systems are expressed in the Fractional Generalized Observability Canonical Form (FGOCF), then a fractional-order dynamical control law is designed to achieve the generalized synchronization. The encryption of color images is presented as an application to the proposed synchronization method, the encryption algorithm allows to decrypt data without loss. The synchronization and its applications are then illustrated with numerical examples.

Dynamical distributed control and synchronization

It is well known that most of real-world phenomena are described by partial differential equations. Nevertheless, for control design purposes it is very common to approximate them with a set of ordinary differential equations, since conventional design methods, such as calculus of variations or differential geometry, turn out to be very complex for this class of systems. However, by doing this, valuable properties are lost. In this work, we present a dynamical distributed control for nonlinear partial differential equation systems and we focus on solving the Generalized Synchronization problem, since this topic has multiple applications in the disciplines of engineering, biology, physics, etc. For the design of the control, we utilize a differential algebraic approach. The key ingredient of our design method is to find a canonical form of the given systems by means of the so-called partial differential primitive element. This representation is known as Generalized Observability Canonical Form and allows us to design a dynamical distributed control in a natural way. Additionally, to avoid a functional analysis for the stability of the resultant synchronization error, we propose to utilize tools from semi-group and spectral theory of infinite dimensional systems in a Hilbert space. As a result, we present a design approach less complex and, therefore, more accessible than most common design methods. Besides, with the proposed stability analysis, we obtain an easy criterion to select the control gains; hence, we can solve the generalized synchronization problem of partial differential equation systems in a simple way. To validate the effectiveness of the proposed control, we present two examples of generalized synchronization for reaction-diffusion systems and show their respective numerical results.

A dynamical controller with fault-tolerance: Real-time experiments

We propose a dynamical controller that is based on the high-gain and the reduced-order observers, is fault-tolerant and is obtained by means of a multi-input multi-output generalized observability canonical form generated from a differential primitive element. The dynamical controller is able to linearize the tracking errors, achieving ultimate uniform boundedness with measurement noise. To accomplish this, a fault diagnosis is required, involving additive and multiplicative faults, which have to be reconstructed simultaneously and online. Some real-time results are presented to illustrate the effectiveness of the proposed methodology.

Generalized multi-synchronization: A leader-following consensus problem of multi-agent systems

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.

Tracking through singularities using sliding mode differentiators

In this work, an alternative solution to the tracking problem for a SISO nonlinear dynamical system exhibiting points of singularity is given. An inversion-based controller is synthesized using the Fliess generalized observability canonical form associated to the system. This form depends on the input and its derivatives. For this purpose, a robust exact differentiator is used for estimating the control derivatives signals with the aim of defining a control law depending on such control derivative estimates and on the system state variables. This control law is such that, when applied to the system, bounded tracking error near the singularities is guaranteed.

Fractional generalized synchronization in a class of nonlinear fractional order systems

Generalized synchronization in nonlinear fractional order systems occurs whether the states of one system by means of a functional mapping are identical to states of another. This mapping can be obtained if there exists a fractional differential primitive element whose elements are fractional derivatives which generate a differential transcendence basis. In this contribution we investigate the fractional generalized synchronization (FGS) problem for a class of strictly different nonlinear fractional order systems and we consider the master-slave synchronization scheme. As well as, of a natural manner we construct a fractional generalized observability canonical form, we introduce a fractional algebraic observability property, and we design a fractional dynamical controller able to achieve synchronization. These particular forms of FGS are illustrated with numerical results.

Non-linear observer design by reduced generalized observer canonical form

A non-linear canonical form observer design approach for multi-input multi-output systems , y = h ( x , u ), with linearizable error dynamics is proposed in the paper. The method involves a transformation of the original non-linear system in a reduced generalized observer canonical form (RGOCF) by a non-linear state transformation. The name reduced comes from the fact that this non-linear canonical form has a reduced dependency on derivatives of the inputs. Necessary and sufficient conditions for transformability are derived and a transformation algorithm based on those conditions is developed. The non-linear observer is firstly designed in the canonical coordinates of the RGOCF and then it is presented in the original and observability coordinates. A final example including simulation is given for illustration.

Fault Diagnosis for Nonlinear Uncertain Systems Using Robust/Sliding Mode Observers

The authors investigate the fault diagnosis problem for a class of nonlinear multiple-input multiple-output (MIMO) systems with uncertainty. Under some geometric conditions, the system is transformed into two different subsystems with uncertainty. The first one is in the generalized observer canonical form, which is not affected by actuator/sensor faults. Nonlinear robust observer or sliding mode observer can be constructed for this subsystem depending on different assumptions on the uncertainties. The second subsystem, whose states can be measured, is affected by the faults. The observation scheme is then used for actuator/sensor fault estimation with good accuracy. Finally, a flexible joint robotics example is given to illustrate the efficiency of the proposed methods.

FAULT DIAGNOSIS USING SLIDING MODE OBSERVER FOR NONLINEAR SYSTEMS

In this paper, the fault diagnosis problem for a class of nonlinear multiple-input multiple-output (MIMO) systems with uncertainty is investigated. Under some geometric conditions, the system is transformed into two different subsystems. One is in the generalized observer canonical form and is not affected by actuator faults, so a nonlinear sliding mode observer for this subsystem is constructed. The other whose states can be measured is affected by the faults. The observation scheme is then used for actuator fault diagnosis with good accuracy. Extension to sensor fault diagnosis is also made. Finally, a numerical example is used to illustrate the efficiency of the proposed method.

High-Gain Nonlinear Observer for the Fault Detection Problem: Application to a Bioreactor

We propose a high-gain nonlinear observer to solve the Fault Detection Problem in a biotechnological process, in addition, the parametric and state estimation is obtained as well. The construction of High-Gain Nolinear Observers which are based upon the Multioutput Generalized Observability Canonical Form (MGOCF) is the main ingredient in our approach. We define the failure mode and we introduce the concept of algebraically observable failure mode. The proposed observer together with the failure mode rule is the, so called, residual generator in the fault detection problem and its performance is shown through numerical simulations.

Asymptotic output tracking of a class of nonlinear systems by means of an observer

AbstractAn observer‐based controller strategy for the asymptotic output tracking of a class of nonlinear systems is proposed. The generalized observability canonical form (𝒢𝒪𝒞ℱ) introduced by Fliess, is the main ingredient in our approach. Our tracking controller actually achieves uniform asymptotic stability by linearizing the tracking error. Two examples are considered to illustrate the effectiveness of the suggested approach. Copyright © 2001 John Wiley & Sons, Ltd.

Observers, Adaptive Observers and Output Feedback Stabilizing Controllers for Uniformly Observable Bilinear Systems

Abstract An adaptive observer for uniformly observable bilinear systems is proposed in this paper. The adaptive observer is derived from the generalized observer canonical form proposed by Williamson, where the conventional input- and output-injection are replaced by the injection of the output and the derivatives of the input. Then, an output-feedback stabilizing controller, using the observer dependent on the derivatives of the input, is proposed. To avoid the differentiation of the input, integrators are placed at the input channel, which makes the direct design of the controller impossible. To overcome the problem, an indirect design scheme based on backstepping is proposed.

Observer based tracking of bilinear systems: A differential algebraic approach

Techniques of differential algebra are used to derive an observer-based controller for bilinear systems. The generalized observability canonical form (GOCF) and the generalized controller canonical form (GCCF) introduced by Fliess are the main ingredients in our approach. Our controller actually achieves both stability and tracking by linearizing the tracking error. The controller is then of error-driven type; the dynamics of the error is estimated by an observer in the closed loop.

Dynamical adaptive pulse-width-modulation control of DC-to-DC power converters: a backstepping approach

Adaptive regulation of pulse-width-modulation (PWM) controlled DC-to-DC power supplies is proposed using a suitable combination of dynamical input-output linearization and the ‘backstepping’ controller design method. A nominal parameter, input-dependent, state coordinate transformation of the average PWM converter models leads to a type of pure parameter feedback canonical form associated with the Fliess generalized observability canonical form of such average models. A back-stepping design procedure can then be immediately devised which leads to a dynamical adaptive regulation scheme for the generation of the stabilizing duty ratio function. The validity of the proposed approach, regarding control objectives and robustness with respect to unmodelled, yet unmatched, and bounded stochastic perturbation inputs, is tested through digital computer simulations.

Dynamical Adaptive Backstepping Control/Observer Design for Uncertain Nonlinear Systems

In this paper the design of observers for a class of nonlinear uncertain systems, transformable into the Generalized Observer Canonical Form, is addressed. A backstepping-like algorithm is used to design a dynamical adaptive control to ensure the output tracks a prescribed bounded reference signal, based on the estimation of the unmeasured states. The destabilizing effects of the observation errors are compensated via nonlinear damping terms.

Observers for a multi-input multi-output bilinear system class: A differential algebraic approach

An exponential observer is constructed for a multi-output observable Bilinear System Class (BSC) in the Diop-Fliess' Observability sense. A differential algebraic approach is proposed for the estimation of the state of a class of bilinear system. A result on Multi-output Translated Fliess' Generalized Observability Canonical Form (MTGOCF) is given.

Pulse width modulated control of the full bridge buck converter

Two approaches are examined for the design a pulse width modulated (PWM) feedback control scheme regulating a full bridge buck converter in both stabilization and AC signal tracking tasks. The first approach is based on a direct strategy for the specification of a static PWM feedback control which accomplishes asymptotic stabilization to a preselected constant operating point for the output capacitor voltage. The same approach is also examined for the asymptotic output tracking of AC reference signals. The second approach, or indirect approach, proposes a dynamical feedback regulation scheme based on input inductor current stabilization or tracking using reference signals obtained via partial inversion of the system dynamics. The indirect approach emphasizes the use of Fliess' generalized observability canonical form of the average converter model.

Dynamical discontinuous feedback strategies in the regulation of nonlinear chemical processes

In this article, a unified approach is proposed for the design of dynamical discontinuous feedback controllers leading to the chattering-free stabilization of nonlinear single-input single-output systems describing chemical processes. The adopted framework is that of a generalized state representation form of the given nonlinear plant. Use is made of the associated generalized observability canonical form of such representation. Unification of discontinuous feedback policies is achieved by zeroing of an input-dependent auxiliary output function using simple discontinuous feedback control paradigms of various kinds. The zeroing of such scaler stabilizing function induces asymptotically stable controlled dynamics on the given nonlinear minimum-phase plant. Pulse-frequency-modulation, pulse-width-modulation and sampled sliding mode control strategies are considered from this unified viewpoint. Examples are provided including simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A dynamical variable structure control strategy in asymptotic output tracking problems

A dynamical discontinuous feedback strategy of the sliding mode type is presented for asymptotic output tracking problems in nonlinear dynamical systems. N. Fliess' (1989, 1990) generalized observability canonical form (GOCF) is used in the derivation of the dynamical variable structure feedback controller. For a large class of nonlinear systems, a truly effective smoothing of the sliding mode controlled responses is possible, while substantially reducing the chattering for the control output. For this reason, the technique is especially suitable for the control of some mechanical and electromechanical systems. An example, including simulations, is provided. >

Dynamic compensator design in nonlinear aerospace systems

Based on differential algebraic results, dynamic controllers are proposed for the feedback regulation of typical aerospace systems. Fliess's generalized observability canonical form (GOCF) is used for specifying a dynamic compensator that smoothly regulates the plant dynamics. The synthesis approach used is also applicable to the design of nonlinear pulsewidth-modulation (PWM) controllers, as well as to sliding-mode control strategies. The three underlying nonlinear control techniques, explored with the aid of illustrative examples, are commonly encountered in aerospace control system design problems. Simulations are also included. >