On the generalized Sylvester mapping and matrix equations

Analyzing the stability of many control systems required solving a couple of crisp Sylvester matrix equations (CSMEs) simultaneously. However, there are some situations in which the crisp Sylvester matrix equations are not well equipped to deal with the uncertainty problem during the stability analysis of control systems. This paper constructs analytical and numerical methods for solving a couple of trapezoidal fully fuzzy Sylvester matrix equations (CTrFFSMEs) to overcome the drawbacks of the existing crisp methods. In developing these new methods, fuzzy arithmetic multiplication is applied on the CTrFFSME to transform it into an equivalent system of four CSMEs. Then, the fuzzy solution is obtained analytically by the fuzzy matrix vectorization method and numerically by gradient and least square methods. The analytical method can obtain the exact solution; however, it is limited to small-sized systems while the numerical methods can approximate the solution for large dimensional systems up to 100 × 100 with a very small error bound for any initial value. In addition, the proposed methods are applied to other fuzzy systems such as Sylvester and Lyapunov matrix equations. The proposed methods are illustrated by solving numerical examples with different size systems.

An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this approach is that the matrix F is firstly transformed into triangular form by Schur decomposition and then unimodular transformation or singular value decomposition are employed. The results can be easily extended to second order case and high order case and can provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1, Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round‐off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least‐norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system, J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.

Relevance Switching voltage converters are widely used in industry, which makes the study of such devices relevant. As a rule, these systems are built with feedback on the output voltage in order to stabilize it, so at the design stage it is necessary to calculate the control system in order to ensure stability and the required dynamic characteristics. The use of an input filter as part of pulse voltage converters, on the one hand, can improve the quality of the converter input current, and on the other hand, can lead to a loss of its stability as an aggregated «input filter — converter» system. This requires the development of methods for analyzing the dynamics of such systems. The applied mathematical models and computational methods must ensure maximum calculation speed, take into account the dynamic nonlinearities of pulsed power conversion systems and the periodic nature of electromagnetic processes with a frequency equal to or a multiple of the pulse modulation frequency. In some cases, chaotic oscillations may occur. The listed features make it relevant to use the point mapping method, based on nonlinear difference equations, which makes it possible to construct stability regions of the desired dynamic regime with minimal computer time. Aim of research Construction of a nonlinear dynamic model of a push-pull zero-DC voltage converter in the form of a nonlinear discrete Poincaré map, construction of two-parameter maps of dynamic modes, frequency characteristics of the system and, on their basis, identification of the main patterns of system behavior under conditions of changes in its input voltage and load resistance in a wide range for different input parameters filter. Research methods To analyze the dynamics, the following were used: small-signal modeling, point mapping method, frequency methods of stability analysis, theory of linear automatic control systems, theory of nonlinear dynamic systems, numerical methods for solving systems of nonlinear equations. Results A nonlinear dynamic model of a push-pull DC-DC converter with an input filter in the form of a Poincaré map has been constructed, and the main patterns of loss of stability of the desired dynamic mode have been identified when the input voltage and load resistance change over a wide range. The specificity of using various methods in analyzing the stability of the system under consideration is demonstrated. It is shown that the use of the point mapping method and accompanying methods for analyzing the stability of fixed points of dynamic modes makes it possible to use a single criterion for the loss of stability of the desired dynamic mode in different areas of system parameters.

Fundamentals of the theory of non-linear pluse control systems

The magnetostatic field produced by an air coil that possesses one-dimensional periodicity can be expressed as a one-dimensional discrete convolution of two functions. The first function expresses the field produced by a single turn of the coil. The second is a shape function; it expresses the spatial position and strength of current of each turn of the coil. The discrete convolution of these two functions gives the magnetostatic field produced by the coil. This result can be interpreted with linear system theory. Under this interpretation, the first function, the response from a single turn, can be thought of as the impluse response function of a hypothetical linear system or "black box." The second function, the expression of position and current strength can be thought of as the input signal to the black box. The advantage of this approach is that it allows the application of highly developed methods of linear system theory to air-coil problems. This paper presents one application of linear system theory to an air-coil calculation, the use of the Fast Fourier Transform (FFT) in computing magnetostatic fields from air coils. A program is described which uses FFT convolution to perform this calculation.

The properties of the matrix sign function, a relatively new object of the matrix theory, were described. In the modern computational algebra, it underlies an efficient technology enabling one to resolve the topical problems of the control theory. The potentialities of this technology were demonstrated by the example of the spectral problems and the Lyapunov, Sylvester, and Riccati algebraic matrix equations. Practical criteria for stability of the linear dynamic system were proposed on the basis of the iterative procedure for computation of the matrix sign function. An example of stability analysis of the mathematical model of a high-order power system was presented. A generalized matrix sign function for the generalized bundles of square matrices was described. For the generalized bundles of rectangular matrices, the matrix sign function was defined relying on the algebra of linear relations.

I . In t roduc t ion . P r o b a b l y the f i r s t n o n t r i v i a l example of the u se of a l g e b r a and geome t ry in l i n e a r s y s t e m theory was the so lu t ion of a m a t h e m a t i c a l p r o b l e m posed in 1868 by J . C . Maxwel l [ 26 ] . He p r o p o s e d the p r o b l e m of f inding computab le a l ge b r a i c c r i t e r i a to d e t e r m i n e if a s y s t e m of l i nea r , cons t an t coef f ic ien t o r d i n a r y d i f f e r e n t i a l equa t ions i s a s y m p t o t i c a l l y s tab le . Th i s p r o b l e m had l a rge ly been so lved in 1856 by H e r m i t e [17] who r e l a t ed the computa t ion of the Cauchy index [9] to the s i gna tu r e of a q u a d r a t i c f o rm . Subsequent ly , the comple te so lu t ion was given by Routh [30] and Hurwitz [18]. While this early work in linear system theory used important algebro-geometric techniques much of the subsequent development did not. The work that was done during World War II and subsequently through the development of the space program relied heavily on the use of analysis for the solution of various problems in linear system theory. It has been only recently that algebra and geometry have again begun to play an important role in system theory. The purpose of this paper is to provide a brief review of some of the highlights of these results, to apply geometric techniques in the derivation and the analysis of the solutions of the linear quadratic optimization and Gaussian estimation problems, and to sketch how algebro-geometric techniques can be used to obtain solutions to problems that are related to the usual quadratic optimization and Gaussian estimation problems but are nontrivial generalizations of these problems. The n a t u r a l g e o m e t r i c se t t ing for the a l g e b r o g e o m e t r i c t echn iques that a re u sed for the so lu t ions of the e s t i m a t i o n and the op t imiza t i on p r o b l e m s wi l l be e m p h a s i z e d . A n u m b e r of r e a s o n s ex i s t for the use of a l g e b r o g e o m e t r i c t echn iques in s y s t e m theory . A la rge n u m b e r of ob jec t s in s y s t e m theory a re n a t u r a l l y a lge b r a i c or g e o m e t r i c . T r a n s f e r func t ions for l i ne a r , cons t an t coef f ic ien t s y s t e m s a re m a t r i c e s of r a t i o n a l func t ions which have been s tudied for m o r e than a cen -

For the further development we use the state-variable description of linear time-invariant systems. This is the standard form used in linear system theory. Introducing this notation brings some advantages. The state-variable description is the most general form and is equally applicable for scalar and for multivariable systems. Matrix calculus is employed instead of working with polynomials. The whole theory of linear systems is available.

This paper presents iterative learning control (ILC) based on two-dimensional (2D) system theory for singular linear continuous-time systems. It is shown that the singular 2D linear continuous-discrete Roesser's model can be applied to describe the ILC process of singular linear continuous-time systems. Necessary and sufficient conditions are given for impulse-free convergence of the proposed ILC rules. Eventually, two numerical examples are given in order to show the results investigated.

Modern engineering systems, such as intelligent transportation systems, smart grids, smart manufacturing systems, etc., often comprise a large number of interconnected components. The design requirements for the components of such systems are becoming increasingly complex. Consequently, the development of components usually requires specialized expertise and is, thus, handled by different (independent) teams. But independent teams seldom have the facilities to collaborate effectively, which can cause long and costly delays. This can be avoided by adopting a method for specifying design requirements that is inherently modular, i.e., that allows components to be considered independently. One such method is based on using so-called contracts, first introduced in the field of computer science. While contract theories have been developed for various system classes, these are generally restricted to systems with discrete variables evolving in discrete time.Motivated by this, in this thesis, we develop a contract theory for a class of dynamical control systems with continuous variables in continuous time. In particular, we introduce notions of implementation, refinement, and conjunction, which allow us to express, compare, and combine specifications using contracts. Furthermore, we introduce two notions of contract composition, which, together with the notion of refinement, enable the independent design of components within interconnected systems.

The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton---Jacobi---Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems.