Special Canonical Forms Research Articles

Existence of flipped orthogonal conjugate symmetric Jordan canonical bases for real H-selfadjoint matrices

For real matrices selfadjoint in an indefinite inner product there are two special canonical Jordan forms, that is (i) flipped orthogonal (FO) and (ii) γ-conjugate symmetric (CS). These are the classical Jordan forms with certain additional properties induced by the fact that they are H-selfadjoint. In this paper, we prove that for any real H-selfadjoint matrix, there is a γ-FOCS Jordan form that is simultaneously flipped orthogonal and γ-conjugate symmetric.

Bounded controls for discrete-time linear systems subject to input time delay

This paper investigates the global stabilization of discrete-time linear systems with input time delay by bounded controls. Based on some special canonical forms containing time delays both in its input and state, two special discrete-time linear systems---multiple integrators and oscillators are first considered. The global stabilizing controllers are respectively established, and moreover, explicit conditions are established to guarantee the stability of the closed-loop systems. Subsequently, a concise design method is proposed for globally stabilizing general discrete-time linear system by combining the design methods for multiple integrators and oscillators. The designed controller is in the explicit form with explicit stability conditions being given, and thus is easier to use than the existing results. Finally, numerical simulations illustrate the effectiveness of the proposed approaches.

Global stabilization of linear systems subject to input saturation and time delays

This note is concerned with global stabilization of linear systems subject to input saturation and time delays. Based on the Luenberger canonical form, two new decoupling methods are proposed. For the decoupled system, according to some special canonical forms, we propose two control laws for systems with input time-delays and systems with input saturation and time-delays, and give explicit conditions to ensure the global stability of the closed-loop system. Two special canonical forms contain time delays in input and state vectors, which is essential in recursive design. In addition, for the system subject to input saturation and time-delay, we introduce some free parameters when designing the controller, which can improve the instantaneous performance of the closed-loop system. Finally, the proposed approach is applied on the multi-agent system to design global stabilizing controllers and the effectiveness of the proposed controllers are illustrated by numerical simulations.

Generalization of Bass --- Gura Formula for Linear Dynamic Systems with Vector Control

The compact analytic formula of calculating the feedback law (controller matrix) coefficients is developed for solving the synthesis problem of modal controller providing desired pole placement by means of the fully measured state vector in linear dynamic systems with vector control. This formula represents the generalization of the known Bass --- Gura formula, used for synthesizing modal controllers in systems with scalar control, to systems with vector control. The obtained solution is applicable to systems with state-space dimension divisible by the number of control inputs and the matrix composed of the linearly independent first block columns of the Kalman controllability matrix by a number corresponding to the quantity of the mentioned multiplicity is reversible. To use the mentioned formula, it's not required to additionally transfer the described systems of the indicated class to special canonical forms. This formula may be applied to solve both numeric and analytic problems of modal control in mentioned class, independently on a specific ratio of state-vector and control-vector dimensions as well as on existence and multiplicity of real-value poles and complex-conjugate pairs of poles in original and desirable spectrums of state matrix. The examples are considered that prove the possibility of applying the generalized block-matrix Bass --- Gura formula to calculate modal controllers for the described class of systems with vector control

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.

Global Stabilization of the Multiple Integrators System by Delayed and Bounded Controls

This note is concerned with global stabilization of multiple integrators by delayed and bounded controllers. Three types of new nonlinear control laws are proposed based on some special canonical forms and explicit conditions are proposed to guarantee the global stability of the closed-loop systems. Differently from the existing canonical forms, the new special canonical forms used in this note contain not only time delay in the input but also time delays in its state, which are essential in the recursive design since they lead to natural cancellation. Moreover, some free parameters are introduced into these controllers. These advantages can help to improve the transient performance of the closed-loop system significantly. Numerical simulations for the fourth order integrators illustrate that the proposed new controllers can indeed outperform the existing one.

A novel and computationally efficient algorithm for stability analysis of multi input-multi output process control systems

An efficient method based on the Faddeev-Leverrier algorithm combined with the Adomian decomposition method is devised to facilitate the stability analysis of multi-input multi-output control systems. In contrast to prior eigenvalue algorithms, our method affords all eigenvalues of the state matrix, either real or complex. Specifically, the calculation of the complex eigenvalues is made possible through special canonical forms, mainly involving square root operators, of the characteristic equation of the state matrix. Moreover, the proposed method does not require an initial guess, which is often a matter of concern since an inappropriate guess can cause failure in such available schemes. For the sake of illustration, a number of numerical examples, including chemical reaction processes, are also provided that demonstrate the efficiency of our new technique.

Eigenfrequencies of symmetric planar trusses via weighted graph symmetry and new canonical forms

PurposeThe purpose of this paper is to describe how the method recently developed for mass‐spring systems and frame structures is modified to include the free vibration of trusses.Design/methodology/approachHere, two methods are presented for calculating the eigenfrequencies of structures. The first approach is graph theoretical and uses graph symmetry. The graph models are decomposed into submodels and healing processes are employed such that the union of the eigenvalues of the healed submodels contain the eigenvalues of the entire model. The second method has an algebraic nature and uses special canonical forms. The present method is illustrated through three simple examples with odd and even number of bays.FindingsThe inter‐relation for the mechanical properties of elements is established using new weighted graphs, enabling easy calculation of the eigenvalues involved. Two methods are presented for calculating the eigenfrequencies of the truss structures.Originality/valueSymmetry is used for easy calculation of the eigenfrequencies of structures.

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.

Weakly elliptic systems with obstacle constraints Part II — An N × N model problem

Existence, uniqueness (even stability), and regularity are established for a special system of variational inequalities of obstacle type. The system is only considered to be elliptic in the weakest possible sense. The system includes a version of the biharmonic and polyharmonic obstacle problems. The main aspect of this problem and the approach here is its reduction via algebraic invarients to special canonical forms. One might view this work as the beginnings of a “group analysis” of variational inequalities.

A State and Parameter Identification Scheme for Linearly Parameterized Systems

This paper presents an adaptive algorithm to estimate states and unknown parameters simultaneously for nonlinear time invariant systems which depend affinely on the unknown parameters. The system output signals are filtered and reparameterized into a regression form from which the least squares error scheme is applied to identify the unknown parameters. The states are then estimated by an observer based on the estimated parameters. The major difference between this algorithm and existing adaptive observer algorithms is that the proposed algorithm does not require any special canonical forms or rank conditions. However, an output measurement condition is imposed. The stability and performance limit of this scheme are analyzed. Two examples are then presented to show the effectiveness of the proposed schemes.

Estimation of continuous-time MIMO linear dynamic models from sampled data by hybrid parametrization

The existing methods for estimating continuous-time MIMO linear dynamic models are mostly confined to the use of so-called indirect methods. However, such methods are possible only for those models that are parametrized in some special canonical forms, which usually do not embody the physical meaning of dynamic systems. Furthermore, the conversion from the discrete-time model to its corresponding continuous-time form is usually subject to limitations, and the results thus obtained are vague owing to the existence of stochastic noise. To circumvent the difficulties mentioned, a method that directly estimates the continuous-time MIMO linear dynamic model using hybrid parametrization is proposed. In this method, the continuous-time model can be parametrized according to the description of the physical system and then a discrete-time realization, which includes the parametrization of the stochastic noise, is formed as the basis for parameter estimation. This method is capable of estimating linear dynamic syst...

An Analysis of Least Squares Derived Techniques for Parallel Implementation in the Mimo-Arma Situation

This work presents recent results obtained in the field of recur sive parameter identification for discrete time MIMO systems utilizing Least Squares Derived techniques. The purpose is twofold: to extend existing techniques by considering a general ARMA model for the noise and to analyze in which situations the algorithms obtained can be partitioned to allow a parallel implementation. The algorithms considered are: Extended Matrix, Generalized Least Squares, Approximated Maximum Likelihood and a new partitioned algorithm obtained from the Extended Matrix technique. The main results show that in the ARMA situation the first and last algorithms lead directly to a parallel implementation. The other two algorithms require the utilization of special canonical forms which, in general, will imply an in crease in the number of parameters to be estimated. Nevertheless the more accurate approximation utilized by the Approximated Maximum Likelihood tech nique compensates for this fact.

A theory of nonenergic N‐ports

AbstractA circuit element is nonenergic if the instantaneous power flow into it is always zero. Well‐known examples include the ideal diode, transformer, gyrator and circulator. Most of the interesting nonenergic elements are nonlinear N‐ports with N ⩾ 2, and many of their properties are quite counterintuitive. For example, there exists a surprisingly large class of nonenergic multiport capacitors and inductors, all of which, it turns out, are nonlinear and reciprocal. Nonenergic linear N‐ports, on the other hand, are necessarily resistive and antireciprocal.In this paper, we present a rigorous fundamental theory of nonenergic N‐ports that results in a general canonical representation. Special canonical forms are developed for nonenergic resistors, capacitors and inductors, and numerous examples are given.