Special Canonical Form Research Articles

EXISTENCE AND COMPUTATION OF INVARIANT ALGEBRAIC CURVES FOR PLANAR QUADRATIC DIFFERENTIAL SYSTEMS

<abstract> Some necessary conditions are given for the existence of invariant algebraic curves for planar quadratic differential systems in a special canonical form. An efficient algorithm is then designed for computations of invariant algebraic curves. From the algorithm, a quadratic differential system is found with two Hopf bifurcations as the parameter varies, each leading to an invariant algebraic limit cycle of degree 5. A family of degree 6 invariant algebraic limit cycles is also produced. To further demonstrate the capability of the algorithm, we provide a quadratic system with a family of degree 7 invariant algebraic curves enclosing one or two centers, and a system possessing a degree 16 irreducible invariant algebraic curve with a singular point of multiplicity 8 on the curve. </abstract>

Bounded control of multiple oscillators system by using state-dependent saturation functions

This paper is concerned with the global stabilisation problem of multiple oscillators system with bounded controls by constructing state-dependent saturation functions. First, two types of planar systems subject to input additive disturbances are respectively stabilised by bounded feedbacks whose formulas involve state-dependent saturation function. Next, these results are used to stabilise multiple oscillators system of arbitrary length with bounded control based on a special canonical form. Explicit conditions are provided to guarantee the stability of the closed-loop system. Compared with the existing results, the proposed controllers contain the so-called state-dependent saturation functions and thus can improve the control performance significantly. Moreover, the corresponding results for the discrete-time case are also established. Two numerical examples are given to show the effectiveness and superiority of the proposed approaches.

Three-dimensional isometric tensor networks

Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values. Here we introduce a method for the time-evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form. Our method involves a tetrahedral site-splitting which allows to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position. Using imaginary time-evolution to find an isometric tensor network representation of the ground state of the 3D transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results. We show that the obtained energy matches the exact groundstate result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions. This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.

Group classification and exact solutions of a higher-order Boussinesq equation

We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for a one-dimensional lattice model considering higher order effects, and later it was re-derived in the context of nonlinear nonlocal elasticity. In the sense of one-dimensional wave propagation in solids, the nonlinearity function of the equation is connected with the stress-strain relation of the physical model. Considering a general nonlinearity, we determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. We find that the maximal dimension of the symmetry algebra is four, which is realized when the nonlinearity assumes some special canonical form. After that we perform reductions to ordinary differential equations. In case of a quadratic nonlinearity, we provide several exact solutions, some of which are in terms of elliptic functions.

Global Stabilization of the Discrete-Time Integrators System by Bounded Controls

This paper is concerned with global stabilization of discrete-time multiple integrators with bounded controls by utilizing the energy function based approach. First, a discrete-time double integrators system subject to input additive disturbances is stabilized by a bounded feedback whose formula involves a linear function of the state and a saturation function only. Next, this result is used to stabilize a discrete-time multiple integrators of arbitrary length by bounded control with the aid of a special canonical form. Compared with the existing results, the proposed controllers require fewer saturation functions, which allow a better use of the control energy. Moreover, some free parameters that are introduced into these controllers can help improve the transient performance of the closed-loop systems significantly. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Bounded control of feedforward time‐delay systems with linearized systems consisting of chain of oscillators

SummaryFor the global stabilization of a family of feedforward nonlinear time‐delay systems whose linearized systems consist of a chain of identical oscillators, a saturated feedback control is established based on a special canonical form. The proposed control laws use not only the current states but also the delayed states for feedback, which helps maintain the decoupling property in the recursive design. Moreover, explicit conditions guaranteeing the stability of the closed‐loop system are provided. When the nonlinear terms vanish and even the oscillators are distinct, a modified saturated feedback control can still be established for the corresponding global stabilization problem. Two numerical examples are given to illustrate the effectiveness of the proposed approaches.

Global stabilization of discrete-time multiple integrators with bounded and delayed feedback

This paper investigates the global stabilization problem of discrete-time multiple integrators with bounded and delayed feedback. In the absence of input delay, two classes of nonlinear feedback laws are proposed. The first one consists of parallel connections of saturation functions and the other one consists of nested saturation functions. Some free parameters and the so-called state-dependent saturation functions are introduced into these two types of control laws which can help to improve the transient performance of the closed-loop system. In the presence of input delay, with the aid of a special canonical form, two types of nonlinear control laws are also proposed to achieve global stability of the closed-loop system. Two numerical examples are given to illustrate the effectiveness of the proposed methods.

Global stabilization of discrete‐time feedforward time‐delay systems by bounded controls

SummaryThis paper studies the problem of global stabilization of a family of discrete‐time feedforward time‐delay systems with bounded controls. Two classes of nonlinear control laws are established based on a special canonical form of the considered system. The proposed control laws use not only the current states but also the delayed states for feedback and, moreover, contain some free parameters. These advantages can help to improve the transient performance of the closed‐loop system significantly. A practical example is given for illustration.

On Bounded Control of A Class of Feedforward Nonlinear Time-Delay Systems

For the global stabilization of a family of feedforward nonlinear time-delay systems with linearized identical oscillators, a saturated feedback control is established based on a special canonical form for the considered system. The proposed control laws use not only the current states but also the delayed states for feedback, and, moreover, contain some free parameters. These advantages can help to improve the transient performance of the closed-loop system significantly. A numerical example is given to illustrate the effectiveness of the proposed approaches.

Global stabilization of feedforward nonlinear time-delay systems by bounded controls

The problem of global stabilization for a family of feedforward nonlinear time-delay systems by bounded controls is considered. Based on a special canonical form of the considered nonlinear system, two types of new nonlinear control laws are proposed to achieve global stabilization. The new special canonical form used in this paper contains not only time delay in the input but also time delays in the state, which leads to natural cancellation in the recursive design. Moreover, some free parameters are introduced into these controllers. These advantages can help to simplify the proof for the global stability of the closed-loop system and improve the transient performance of the closed-loop system significantly. A practical example is given to illustrate the effectiveness of the proposed approaches.

Global Stabilization of Multiple Oscillator Systems by Delayed and Bounded Feedback

The problem of global stabilization for the multiple oscillator systems by bounded and delayed controls is considered. Based on a special canonical form of the considered system, a nonlinear control law consisting of nested saturation functions is proposed to achieve global stabilization. The new special canonical form used in this brief contains not only time delay in the input but also time delays in its state, which leads to natural cancellation and helps to not only provide explicit conditions guaranteeing the global stability of the closed-loop systems but also improves the transient performance of the closed-loop system significantly. The proposed feedback law is also adopted to achieve restricted tracking. A numerical simulation is given to illustrate the effectiveness of the proposed method.

First-order systems of linear partial differential equations: normal forms, canonical systems, transform methods

Abstract In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial value problems in appropriate basic areas, and you can try to achieve a solution of these problems by means of transform methods.

Tree Matrices and a Matrix Reduction Algorithm of Belitskii

Inspired by the bimodule matrix problem technique and various classification problems in poset representation theory, finite groups and algebras, we study the action of Belitskii algorithm on a class of square n by n block matrices M with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M∞ with respect to the conjugation by elementary transformations defined by a class of matrices chosen in a subalgebra of the full matrix algebra $\mathbb{M}_n$(K). The algorithm can be successfully applied in the study of indecomposable linear representations of finite posets by a computer search using numeric and symbolic computation. We mainly study the case when the di-graph (quiver) associated to the output matrix M∞ of the algorithm is a disjoint union of trees. We show that exceptional representations of any finite poset are determined by tree matrices. This generalizes a theorem of C.M. Ringel proved for linear representations of di-graphs.

On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays

This paper deals with the positivity properties of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate constant point delays. The main idea behind the investigation is that its main body is performed based on the construction of the whole state-space trajectory solution without using as usual equivalence or similarity transformations on the matrix of dynamics in order to split the state-trajectory solution into two parts, one being typically associated with a nilpotent matrix. In that way, the whole state trajectory solution contains impulsive terms associated with the initial conditions and inputs. Some extensions concerning positivity aspects are given for a special canonical form which separates the dynamics associated with the nilpotent matrix obtained from an equivalence transformation on the singular matrix of the dynamic system.

Quantum entanglement via nilpotent polynomials

We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed.

Observer design for nonlinear systems using linear approximations

There exist several approaches to the design of observers for nonlinear systems, including the separation of the nonlinear system into a linear part and a nonlinear perturbation of the system with a bounded condition [2], the use of Lie derivatives and the inversion of the Jacobian of a coordinate transformation to obtain the gain of the nonlinear observer [10] and the use of a Lyapunov equation to design the observer for a nonlinear system represented in a special canonical form [8]. The design of observers for linear systems is better understood (see [11], [14]) since in nonlinear theory there is a necessity to use more complex mathematics. Therefore, there exist interest in the development of simpler and general methods to solve the problem of nonlinear state reconstruction. This paper deals with the design of observers for nonlinear systems by using a recent technique in which the nonlinear dynamical system is represented as the limit of a sequence of linear time-varying approximations that converge to the solution of the nonlinear system under a local Lipschitz condition.

ON A GENERALIZED LORENZ CANONICAL FORM OF CHAOTIC SYSTEMS

This paper shows that a large class of systems, introduced in [Čelikovský &amp; Vaněček, 1994; Vaněček &amp; Čelikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Čelikovský &amp; Vaněček, 1994; Vaněček &amp; Čelikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen &amp; Ueta, 1999; Ueta &amp; Chen, 2000].Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some other closely related topics are also studied and discussed in the paper.

Optimal algorithms for well-conditioned nonlinear systems of equations

We propose solving nonlinear systems of equations by function optimization and we give an optimal algorithm which relies on a special canonical form of gradient descent. The algorithm can be applied under certain assumptions on the function to be optimized, that is, an upper bound must exist for the norm of the Hessian, whereas the norm of the gradient must be lower bounded. Due to its intrinsic structure, the algorithm looks particularly appealing for a parallel implementation. As a particular case, more specific results are given for linear systems. We prove that reaching a solution with a degree of precision /spl epsiv/ takes /spl Theta/(n/sup 2/k/sup 2/ log /sup k///sub /spl epsiv//), k being the condition number of A and n the problem dimension. Related results hold for systems of quadratic equations for which an estimation for the requested bounds can be devised. Finally, we report numerical results in order to establish the actual computational burden of the proposed method and to assess its performances with respect to classical algorithms for solving linear and quadratic equations.

A Simplified Approach for Biased Model Reduction of Linear Systems in Special Canonical Form

A new method is proposed for the modelling of high-order linear continuous time systems represented in special canonical form. New algorithms are developed to generate low order biased models which retain both the initial time-moments and Markov parameters of the original high-order system to give better approximation in both transient and steady state parts of the time response. The procedure is computationally simple and direct and avoids the necessity of matrix inversions and multiplications of the original high order system matrices. The new method avoids the calculation of the time moments and Markov parameters of the original high order system beforehand unlike many of the available methods for the model reduction in canonical form. The proposed method is digital computer oriented and very useful for the analysis and design of high order systems in special canonical form. The method is illustrated through typical numerical examples.

Estimating regions of asymptotic stability with sliding for relay-control systems

The object of this paper is the estimation of stability boundaries and regions of asymptotic stability with sliding for a class of relay-control systems. The direct method of Lyapunov is used to obtain these estimates. A coordinate transformation which brings the system into a special canonical form is utilized to facilitate the stability analysis. The proposed approach to stability regions estimation is applied to a class of second-order systems and analytical expressions for stability regions are derived.