Control problem of the asynchronous spectrum of linear periodic systems with degenerate right lower diagonal block of averaging of coefficient matrix

We consider a linear control system with an almost periodic matrix of the coefficients. The control has the form of feedback that is linear on the phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency modulus of the coefficient matrix. The following problem is formulated: choose a control from an admissible set for which the system closed by this control has almost periodic solutions with the frequency spectrum (a set of Fourier exponents) containing a predetermined subset, and the intersection of the frequency modules of solution and the coefficient matrix is trivial. The problem is called as the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with the target set of frequencies. At present, this problem has been studied only in a very special case, when the average value of the almost periodic coefficients matrix of the system is zero. In the case of nontrivial averaging, the question remains open. In the paper, a sufficient condition is obtained under which the control problem of the asynchronous spectrum of linear almost periodic systems with diagonal averaging of the coefficient matrix has no solution.

A linear control system with an almost periodic matrix of coefficients and the control in the form of feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a zero average value of the matrix of coefficients. For the described class of systems, the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies is solved. This task is as follows: to construct such a control from an admissible set so that the system closed by this control has almost periodic solutions, the set of Fourier exponents (frequency spectrum) that are contained in a predetermined subset; the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for solvability of the control problem of the asynchronous spectrum are obtained.

We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.

A linear control system with a periodic matrix of coefficients and program control is considered. The matrix under control is constant, rectangular (the number of columns does not exceed the number of rows) and its rank is not maximum, i. e. less than the number of columns. It is assumed that the control is periodic, and the module of its frequencies, i. e. the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. The following task is posed: to construct such a control from an admissible set that switches the system to asynchronous mode, i. e. the system must have periodic solutions such that the intersection of the frequency moduli of the solution and the coefficient matrix is trivial. The problem posed is called the problem of synthesis of asynchronous mode. The solution to the formulated problem significantly depends on the structure of the average value of the coefficient matrix. In particular, its solution is known for systems with zero average. In addition, solvability conditions were obtained in the case when the matrix under control has zero rows, the averaging of the coefficient matrix is reduced to the form with upper left diagonal block and with zero remaining blocks. In this paper we consider a more general case with a nontrivial left lower block. Assuming an incomplete column rank of the matrix function composed from the rows of oscillation path of the coefficient matrix, we construct the control explicitly. This control switches the system to asynchronous mode.

A linear control system with an almost periodic matrix of coefficients and control in the form of the feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a triangular average value of the matrix of coefficients. For the described class of systems, the control problem of the asynchronous spectrum with a target set of frequencies is solved. This task is to construct such a control from an admissible set that the system closed by this control has almost periodic solutions, a set of the Fourier exponents of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for the solvability of this problem are obtained.

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

We analyse the solution spaces of limit periodic homogeneous linear difference systems, where the coefficient matrices of the considered systems are taken from a commutative group which does not need to be bounded. In particular, we study such systems whose fundamental matrices are not asymptotically almost periodic or which have solutions vanishing at infinity. We identify a simple condition on the matrix group which guarantees that the studied systems form a dense subset in the space of all considered systems. The obtained results improve previously known theorems about non-almost periodic and non-asymptotically almost periodic solutions. Note that the elements of the coefficient matrices are taken from an infinite field with an absolute value and that the corresponding almost periodic case is treated as well.

Using the method of averaging, we give sufficient conditions for the existence of an almost periodic solution of an undamped oscillatory system with almost periodic forcing, and show that these can be applied to a Duffing equation with almost periodic forcing provided the nonlinear term is sufficiently small, and the natural frequency of the linear part of the system is included in the set of Fourier exponents of the forcing function. It is well known that the real linear second order scalar differential equation + x = f (t), with f almost periodic (a.p. for short), cannot have almost periodic solutions, or even a solution bounded on the real line, whenever 1 is a Fourier exponent off It is the purpose of this note to obtain a result which will show that for such so-called resonance cases, there exist sufficiently small perturbations of this equation, involving only functions of x which have a.p. solutions. More precisely, iff has Fourier exponent 1 and one of the corresponding coefficients of the sine or cosine term in the Fourier series is zero, then there exist positive numbers v and E0 = E0(v) such that for 0 < E < E0 the equation (1) X +X-EVX+ E3X3f=(t) will have an a.p. solution. The magnitude of v is proportional to the absolute value of the Fourier coefficient corresponding to the Fourier exponent 1 in the series forf This a.p. solution is unstable in the sense of exhibiting the so-called saddle-point property, and the suprema of its absolute value and that of its derivative over all real t become unbounded as E --0. This property of (1) seems interesting for cases wheref has a sequence {2k}, k = 1, 2, * , of Fourier exponents, 2' < 1, and 2'-? 1 as k-xo. Received by the editors February 22, 1971. AMS 1970 subject classifications. Primary 34C25; Secondary 34C30.

The problem of designing cooperative control of linear system with a quadratic performance index is considered. A necessary and sufficient condition for optimal cooperative strategies is given. Then, the decentralized control of the large-scale linear system is studied based on the new viewpoint of cooperative control and the necessary and sufficient condition for optimal decentralized control is obtained. Furthermore, the problem for designing the suboptimal decentralized control of linear quadratic large-scale system is formulated to a concave optimization problem with BMI constrains and solved by an alternative optimization algorithm via LMIs. Finally, an example is given to illustrate the main results of this paper.

Dear SCPE readers, This issue is devoted to Professor Ian Gladwell on the occasion of his retirement from the Mathematics Department of Southern Methodist University (SMU) in Dallas, Texas, USA. In his long career, Ian Gladwell was a member of the faculty of the University of Manchester (England) from 1967 to 1987 and a member of the faculty of SMU from 1987 to date. Whilst at SMU he was Chair of the Mathematics Department for two terms and served multiple terms as Director of Undergraduate Studies and as Director of Graduate Studies. Between 1975 and 2006 he supervised more than 20 Ph.D. students (see [1] ) who are now working in the USA, Europe, and Asia. Professor Gladwell has been very influential as an editor. This activity includes serving as Editor-in-Chief of the ACM Transactions on Mathematical Software from 2005 to date. He was also Associate Editor of the IMA Journal on Numerical Analysis ; Scalable Computing: Practice and Experience ; and the SIAM Journal on Numerical Analysis . He has served as editor for several books and special issues of journals. At present he serves as editor for the chapter in Scholarpedia devoted to Boundary Value Problems [2] . The many publications listed in [3] show that his research activity has focussed on the numerical integration of ordinary differential equations, quadrature, parallel computing, and mathematical software. His recent research has been concerned with the numerical solution of almost block diagonal (ABD) systems and applications. His most recent book is Solving ODEs with MATLAB , which he wrote with L.F. Shampine and S. Thompson. It was published by Cambridge University Press in 2003 [4] . Professor Gladwell was a pioneer in the development of mathematical software, especially software for the numerical solution of ordinary differential equations. Three software packages were published by the ACM and several of his programs were included in the NAG Fortran 77 library [5] . His association with NAG began in 1975 with his numerical ODE programs for the first NAG Library. He is a founder member of both the NAG Ltd. Technical Policy Committee and the NAG Inc. Advisory Panel. He has also been a long-term consultant for Texas Instruments. The Special Issue contains five papers dealing with subjects related to the research activity of Ian Gladwell. Most of the authors had a fruitful collaboration with Ian in the past. We thank all of them to have agreed to our call. The first paper Vectorized Solution of ODEs in Matlab is by L. F. Shampine, from the Southern Methodist University (USA). The author investigates a class of Runge-Kutta methods, able to efficiently exploit vectorization in the popular problem-solving environment Matlab. Local error estimates and continuous extensions that require no additional function evaluations are also derived. As a result, a (7,8) pair is derived and implemented in the program BV78 that well compares with the well-known Matlab ODE solver ode45, based on a (4,5) pair. The second paper Conditioning and Hybrid Mesh Selection Algorithms for Two-Point Boundary Value Problems is by J. R. Cash, from the Imperial College, London (England), and F. Mazzia, from the University of Bari (Italy). The authors deal with the use of conditioning of the problem in the stepsize variation strategy. This allows to obtain very reliable algorithms, which are implemented in state of the art numerical codes for boundary value problems for ordinary differential equations. In particular, they speak about different choices of monitor functions that are used in the BVP codes and analyze the setting of the parameters in order to optimize the stepsize variation strategy. The third paper Preconditioning of Implicit Runge-Kutta Methods is by L. O. Jay, from the University of Iowa, Iowa City (USA). A major problem in obtaining an efficient implementation of fully implicit Runge-Kutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations, usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. In this article the author presents a cheap, and parallelizable, preconditioner for solving the linear systems with the approximate Jacobian matrix. The fourth paper Parallel Numerical Solution of ABD and BABD Linear Systems Arising from BVPs is by P. Amodio, from the University of Bari (Italy), and G. Romanazzi, from the University of Coimbra (Portugal). The authors describe a parallel algorithm (based on the cyclic reduction) for the solution of linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. They also report numerical tests involving parallel OpenMP versions of the Fortran 90 codes BABDCR and GBABDCR and compare them with COLROW . Finally, they discuss about the use of GBABDCR inside paralell version of BVP codes. The fifth paper Parallel Factorizations in Numerical Analysis is by P. Amodio, from the University of Bari (Italy), and L. Brugnano, from the University of Florence (Italy). The authors review a number of parallel solvers for large, sparse, and structured linear systems through the use of the so called parallel factorizations , which provide parallel extensions of usual matrix factorizations. In particular, the paper is focused on the use of parallel factorizations for solving linear systems deriving from the numerical solution of ODEs. Moreover, the so called Parareal algorithm is derived within the framework of parallel factorizations. Pierluigi Amodio, Universita di Bari, Bari, Italy Luigi Brugnano, Universita di Firenze, Firenze, Italy References [1]. Ian Gladwell Graduate Student Supervision [2]. Scholarpedia Boundary value problem [3]. MathSciNet Publication results for Items authored by or related to Gladwell, Ian [4]. L. F. Shampine, I. Gladwell, S. Thompson Solving ODEs with MATLAB Cambridge University Press [5]. Nag Fortran library

For the deterministic case, a linear controlled system is alwayspth order stable as long as we use the control obtained as the solution of the so-called LQ-problem. For the stochastic case, however, a linear controlled system with multiplicative noise is not alwayspth mean stable for largep, even if we use the LQ-optimal control. Hence, it is meaningful to solve the LP-optimal control problem (i.e., linear system,pth order cost functional) for eachp. In this paper, we define the LP-optimal control problem and completely solve it for the scalar case. For the multidimensional case, we get some results, but the general solution of this problem seems to be impossible. So, we consider thepth mean stabilization problem more intensively and give a sufficient condition for the existence of apth mean stabilizing control by using the contraction mapping method in a Hilbert space. Some examples are also given.

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

We study limit periodic and almost periodic homogeneous linear difference systems. The coefficient matrices of the considered systems are taken from a given commutative group. We mention a condition on the group which ensures that, by arbitrarily small changes, the considered systems can be transformed to new systems, which do not possess any almost periodic solution other than the trivial one. The elements of the coefficient matrices are taken from an infinite field with an absolute value.

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.

This paper presents the central finite-dimensional H∞ controller for linear systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the paper reduces the original H∞ controller problem to the corresponding H2 controller problem, using the technique proposed in [1]. The paper yields the central suboptimal H∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H2 controller obtained in [16]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H∞ controller for linear systems with exactly known parameter values.