On Operator Fields in an Upper Triangular Toeplitz Form

Using the Lyapunov function method, this study investigates both state and output feedback stabilisation control design problems for fractional order non‐linear systems in the lower triangular form, and presents a number of new results. First, some new properties for Caputo fractional derivative are presented. Second, by introducing appropriate transformations of coordinates, the feedback stabiliser design problem is converted into the determination of finding some parameters, which can be obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, both state and output feedback stabilisers are explicitly designed to make the closed‐loop system asymptotically stable. The study of an illustrative example shows that the obtained results are effective in designing feedback stabilisers for fractional order non‐linear systems in the lower triangular form.

Matrisler bilgisayar bilimlerinde sıklıkla kullanılan veri yapılarıdır. Üçgensel matris gibi özel matris türleri için programlama dillerinde hazır yapılar bulunmamaktadır. Bir veri yapısı olarak üçgensel matris kullanmak gerekiyorsa programlama dilinden kaynaklanan bu eksiklik nedeniyle bellek ve zaman karmaşıklığı yönünden etkin olmayan yazılımlar kodlanmaktadır. Bu çalışmada üçgensel matris tipindeki yapıların bir boyutlu dizi olarak temsil edilmesinde kullanılan dönüşüm ve ters dönüşüm formülleri elde edilerek programların yer ve zaman karmaşıklığı açısından etkinliğinin arttırılması amaçlanmaktadır.

In this paper, adaptive neural network (NN) control is investigated for a class of multiinput and multioutput (MIMO) nonlinear systems with unknown bounded disturbances in discrete-time domain. The MIMO system under study consists of several subsystems with each subsystem in strict feedback form. The inputs of the MIMO system are in triangular form. First, through a coordinate transformation, the MIMO system is transformed into a sequential decrease cascade form (SDCF). Then, by using high-order neural networks (HONN) as emulators of the desired controls, an effective neural network control scheme with adaptation laws is developed. Through embedded backstepping, stability of the closed-loop system is proved based on Lyapunov synthesis. The output tracking errors are guaranteed to converge to a residue whose size is adjustable. Simulation results show the effectiveness of the proposed control scheme.

This paper discusses the transformation of the various matrixes in application of electromagnetic theory,which are coordinate matrix transformation,operator matrix transformation and field matrix transformation.It is proposed that unit vector,differential and operator in different coordinates are corresponding to invariable matrix transformation.Also,the transformation matrix which longitudinal fields convert into the transverse fields of the cylindrical waveguide is invariable.

Inverting linear combinations of identity and generalized Catalan matrices

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz matrices and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz contractions. We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact and strictly convex, in the spectral norm, with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact, convex and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline German upper T Subscript upper N subset-of-or-equal-to e x t German upper T Subscript upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:mi>ext</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _{\mathfrak {L}_{N}} \mathfrak {T}_{N} \subseteq \operatorname {ext}\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub>

Necessary and sufficient sonditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request.

The paper deals with the problem of analysis of the transient temperature field in the plate junction zone of a thin-walled branched structure of an anisotropic material. In the junction zone of the plates of such structures, great temperature oscillations take place, which have an influence on the analysis of the stress-strain state of the structure. The junction zone of the plates was modelled by a connective finite element with a quadrangular base. The thermal conductivity matrix of the connective finite element was obtained in the case of an anisotropic material. The coordinate transformation and temperature value transformation formulae were derived, which allowed composing a grid of the finite elements of the examined object both with the corner and middle side nodes of the connective finite element. This allows a more efficient modelling of joining a few plates with different normals to a horizontal surface. The derived formulae were checked by the solution of a problem of the temperature field of a cooled plate. DOI: http://dx.doi.org/10.5755/j01.mech.18.2.1562

A quasi-primitive n-port is described by an admittance matrix of the form Y(p)=p/sup -1/ A+B+pC in which A, B, C, are real, symmetric, semidefinite n/spl times/n matrices. Any m-port, m<n, derived by ignoring some chosen (m-n) ports of the n-port described by Y(/spl middot/) is a descendant of Y(/spl middot/). Any such descendant is described by its own admittance matrix Y'(/spl middot/) called then a descendant of Y(/spl middot/). This note presents two results, the first a canonical form into which any given quasi-primitive Y(/spl middot/) can be put by a transformation of coordinates. Specifically, given Y(/spl middot/), quasi-primitive and of size n/spl times/n, there exists a real, invertible, n/spl times/n matrix Q such that Q*Y(/spl middot/)Q is of block-diagonal form bldiag{Y/sub SP/(p), Y/sub ND/(p), Y/sub TI/(p)} in which the blocks are quasi-primitive and describe respectively an n/sub 1/-port, an n/sub 2/-port, a 2n/sub 3/-port, separate networks, each of which, if not of size zero, exhibits distinctive characteristic properties. The second result, based on this structure, is a property of a given m/spl times/m positive-real matrix-valued function Y'(/spl middot/) necessary that Y'(/spl middot/) be a descendant of some quasi-primitive matrix Y(/spl middot/).

The optimal (adaptive) linear combiner (beamformer) weights for a sensor array are expressed in terms of the inverse of the multi-channel (MC) covariance matrix. Also, minimum variance (Capon) spectral estimators of the sensor array also depend on the same inverse. Rather than form an estimate of the covariance matrix directly from the available data and inverting it, an alternative direct estimate of the inverse may be obtained by forming parametric MC linear prediction estimates and then expressing the inverse in terms of these parametric MC estimates. The resulting parametric estimate of the inverse is typically more accurate than inverting the estimate of the covariance matrix. This paper reveals the structure of the the inverse of the covariance matrix for the MC version of the covariance least squares linear prediction algorithm. The inverse structure involves products of triangular block MC Toeplitz matrices, which leads to fast computational solutions. An example of a fast MC minimum variance spectral estimator illustrates this exploitation.

Der Verfasser bringt zunachst die verschiedenen Formeln, die fur einfache Koordinatentransformationen, d. h. zur Umrechnung von zwei unveranderlich miteinander verbundenen Punkten ohne Wechsel des Referenzellipsoides, angewendet werden, in Erinnerung und empfiehlt dann die Formeln (4) und (5) mit den Verbesserungsausdrucken (9), wobei die Bezeichnungenr, s, t, q durch die Gleichungen (6) und (8) definiert sind.

The formulae and algorithms for Lagrange-power basis transformation and Lagrange–Newton transformation

Catalan matrix and related combinatorial identities

Displacement operator based decompositions of matrices using circulants or other group matrices