Exponential tracking of general references and rejection of general disturbances for nonlinear control systems with applications to the blood glucose regulation system

In solving the problem of exponential tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected are generated by an exosystem such as a finite dimensional system with pure imaginary eigenvalues. The aim of this note is to show that this assumption can be removed. For any nonlinear control system subject to a general disturbance, it can be split into a linear exponentially-stable system and a dynamical regulator system. If the dynamical regulator system has a solution, then there exists a feedback and feedforward controller such that an output of the control system exponentially tracks a desired general reference. The result is applied to the blood glucose regulation system.

Exponential tracking of general references and rejection of general disturbances for nonlinear control systems with applications to the blood glucose regulation system

In solving the problem of exponential tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected are generated by an exosystem such as a finite dimensional system with pure imaginary eigenvalues. The aim of this note is to show that this assumption can be removed. For any nonlinear control system subject to a general disturbance, it can be split into a linear exponentially-stable system and a dynamical regulator system. If the dynamical regulator system has a solution, then there exists a feedback and feedforward controller such that an output of the control system exponentially tracks a desired general reference. The result is applied to the blood glucose regulation system.

Model of Dynamic Insulin-Glucose in Diabetic

In the present work a study of the dynamic insulin-glucose in a healthy person is made; the different types of diabetes are indicated as well as the symptoms that characterize each one of them. The model that simulates the insulin-glucose dynamics for a person with diabetes is presented, and for the critical case of a pair of pure imaginary eigenvalues of the matrix of the linear part of the system, the system is simplified, a qualitative study is done of the system of equations and conclusions are given on the future behavior of the disease.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

Global Dynamics of a Compressor Blade with Resonances

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

Application of Normal Forms to a Laminated Composite Piezoelectric Rectangular Plate with One-to-Three Internal Resonance

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate under the parametrically and externally excitations, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues.

Stabilizing and destabilizing perturbations of PT -symmetric indefinitely damped systems

Eigenvalues of a potential dynamical system with damping forces that are described by an indefinite real symmetric matrix can behave as those of a Hamiltonian system when gain and loss are in a perfect balance. This happens when the indefinitely damped system obeys parity–time ( ) symmetry. How do pure imaginary eigenvalues of a stable -symmetric indefinitely damped system behave when variation in the damping and potential forces destroys the symmetry? We establish that it is essentially the tangent cone to the stability domain at the exceptional point corresponding to the Whitney umbrella singularity on the stability boundary that manages transfer of instability between modes.

An Algebraic Method on the Eigenvalues and Stability of Delayed Reaction-Diffusion Systems

The eigenvalues and stability of the delayed reaction-diffusion systems are considered using the algebraic methods. Firstly, new algebraic criteria to determine the pure imaginary eigenvalues are derived by applying the matrix pencil and the linear operator methods. Secondly, a practical checkable criteria for the asymptotic stability are introduced.