pole assignment by output feedback using cylindrical algebraic decomposition recently a new polynomial method for pole assignment by static output feedback has been developed in this paper we show how this method can be generalized to partial pole clustering some poles are exactly assigned the others being clustered in a specified region of the complex plane a constructive and systematic algorithm for the numerical resolution of both problems the collins cylindrical algebraic decomposition algorithm is presented and applied to an illustrative example

new results on the separation of matrix eigenvalues and the clustering of matrix elements the problem of separation of matrix eigenvalues is considered in this paper necessary and sufficient conditions are given for a matrix to have all eigenvalues contained in an open set defined on the complex plane by a separable polynomial spl gamma z sub 1 z sub 2 the problem of clustering of matrix elements is also considered in this case a set of matrices is identified having all eigenvalues in a given subset of the complex plane the presented results can be useful in both robust stability analysis and the design of control systems

linear quadratic optimal output feedback control for systems with poles in a specified region the problem of output feedback linear quadratic regulator synthesis with regional pole constraints is studied the synthesis algorithm is presented and its convergence is also proved the effectiveness of the algorithm is illustrated by an example

pseudo bosons arising from standard ladder operators pseudo bosons in the form as sa sa bs sa sa with as bs a a i are considered the s and s being real coefficients which depend on real parameters s1 sn the eigenstates of the two number operators and their norm are explicitly obtained pseudo bosons in bagarello s sense are recovered the states form two sets of biorthogonal bases of the full hilbert space but riesz bases are obtained only in the ordinary bosonic case some examples of this setting are analyzed in detail

pole assignment of linear uncertain systems in a sector via a lyapunov type approach the problem of designing robust control laws in performance and in stability for uncertain linear systems is considered performances are taken into account using root clustering of the closed loop dynamic matrix in a sector of the complex plane a synthesis procedure based on a sufficient condition for quadratic stabilization and root clustering such as stabilizability is given using an auxiliary convex problem the results are illustrated by a significant example from the literature

classification of eight vertex solutions of the coloured yang baxter equation in this paper all eight vertex type solutions of the coloured yang baxter equation dependent on spectral as well as colour parameters are given it is proved that they are composed of three groups of basic solutions three groups of their degenerate forms and two groups of trivial solutions up to five solution transformations moreover all non trivial solutions can be classified into two types called baxter type and free fermion type

controller design with regional pole constraints a design procedure is developed that combines linear quadratic optimal control with regional pole placement specifically a static and dynamic output feedback control problem is addressed in which the poles of the closed loop system are constrained to lie in specified regions of the complex plane these regional pole constraints are embedded within the optimization process by replacing the covariance lyapunov equation by a modified lyapunov equation whose solution in certain cases leads to an upper bound on the quadratic cost functional the results include necessary and sufficient conditions for characterizing static output feedback controllers with bounded performance and regional pole constraints sufficient conditions are also presented for the fixed order i e full and reduced order dynamic output feedback problem with regional pole constraints circular elliptical vertical strip parabolic and section regions are considered

nonlinear n pseudo fermions nonlinear pseudo fermions of degree n n pseudo fermions are introduced as pseudo particles with creation and annihilation operators b and a b a obeying the simple nonlinear anticommutation relation ab bnan 1 the n 1 order nilpotency of these operators follows from the existence of unique up to a bi normalization factor a vacuum supposing appropriate n 1 order nilpotent para grassmann variables and integration rules the sets of n pseudo fermion number states right and left ladder operator bi overcomplete sets of coherent states are constructed explicit examples of n pseudo fermion ladder operators are provided and the relation of pseudo fermions to finite level pseudo hermitian systems is briefly considered this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

root distribution for the ellipse it is shown that the number of characteristic roots of a given matrix or polynomial inside outside and on the boundary of a given ellipse can be determined by a common half plane test e g routh test the test should be performed on a polynomial which is received from the original characteristic polynomial by a simple transformation root exclusion and root clustering criteria are included as special cases

extended pseudo fermions from non commutative bosons we consider some modifications of the two dimensional canonical commutation relations leading to non commutative bosons and we show how biorthogonal bases of the hilbert space of the system can be obtained out of them our construction extends those recently introduced by one of us f b modifying the canonical anticommutation relations we also briefly discuss how bicoherent states producing a resolution of the identity can be defined

an algebraic algorithm for determining the desired gain of multivariable feedback systems in this paper an algebraic method is presented whereby one can obtain the relevant values of the gain k of a multivariable multi input multi output feedback system to be stable or relatively stable the basis of the method involves the algebraic solution of the two equations obtained from the real and imaginary parts of the characteristic equation of the system the proposed method offers an alternative procedure for determining stability relative stability to the graphical methods based on the generalized nyquist criterion and the multivariable root locus method this method is analogous to the analytical form of the root locus combined with the routh hurwitz criterion as compared to the nyquist or graphical root locus methods when applied to single input single output feedback systems several examples are presented illustrating the applications of the method and compared with the graphical methods

loose eigenstructure assignment via rank one lmi approach with application to transient response shaping in h control this article proposes a new regional eigenstructure assignment via rank one lmi approach a gain parameter condition for the regional eigenvalue eigenstructure assignment is newly derived this assignment condition is easily combined with h design by means of enhanced lmi characterisation in the present approach the desired assignment of closed loop eigenvalues i e poles are not previously fixed but constrained into individual assignment regions to bring us more design freedom than classical exact assignment the regional assignment discussed in this article never falls into an ordinary lmi root clustering because the union of assignment regions is disjoint in general each closed loop eigenvector is also constrained into individual alignment cone to show the practical use of the extra design freedom brought by regional assignment a transient response shaping in h design is also discussed a useful design algorithm based on linearisation algorithm is proposed

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

robust root clustering for linear uncertain systems using generalized lyapunov theory in this paper the problem of matrix root clustering in sub regions of complex plane for linear state space models with real parameter uncertainty is considered the nominal matrix root clustering theory of gutman and jury 1981 ieee trans aut control ac 26 403 using generalized lyapunov equation is extended to the perturbed matrix case and bounds are derived on the perturbation to maintain root clustering inside a given region the theory allows us to get an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space the current literature available on perturbation bounds for robust stability becomes a special case of this unified theory

perturbation bounds for root clustering of linear systems in a specified second order subregion sufficient bounds for structured and unstructured uncertainties for root clustering in a specified second order subregion of the complex plane for both continuous time and discrete time systems are given using the generalized lyapunov theory furthermore for unstructured uncertainties a still less conservative result is obtained by shifting the center or focus of the subregion along the real axis to the origin and by applying root clustering to the shifted eigenvalue system matrix which is obtained by shifting the eigenvalues of the system matrix correspondingly

generalized grassmannian coherent states for pseudo hermitian n level systems the purpose of this paper is to generalize fermionic coherent states for two level systems described by pseudo hermitian hamiltonian cherbal et al 2007 j phys a math theor 40 1835 to n level systems central to this task is the expression of the coherent states in terms of generalized grassmann variables these kinds of grassmann coherent states satisfy the bi overcompleteness condition instead of the overcompleteness one as it is reasonably expected because of the biorthonormality of the system choosing an appropriate grassmann weight function the resolution of identity is examined moreover grassmannian coherent and squeezed states of the deformed group suq 2 for the three level pseudo hermitian system are presented

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

interval stability and stabilization of linear stochastic systems this technical note applies the spectrum technique to deal with the interval stability and stabilization of linear stochastic time invariant systems first the notion of beta alpha stability of stochastic systems is defined and some relationships among the beta alpha stability the decay rate of system state response and the second order moment lyapunov exponent are revealed we then address the beta alpha stabilization problem and give a sufficient condition via a linear matrix inequality approach

quantum dynamical r matrices and quantum frobenius group we propose an algebraic scheme for quantizing the rational ruijsenaars schneider model in the r matrix formalism we introduce a special parametrization of the cotangent bundle over in new variables the standard symplectic structure is described by a classical frobenius r matrix and by a new dynamical matrix quantizing both of them we find the quantum l operator algebra and construct its particular representation corresponding to the rational ruijsenaars schneider system using the dual parametrization of the cotangent bundle we also derive the algebra for the l operator of the hyperbolic calogero moser system

d stability of continuous time delay systems subjected to a class of highly structured perturbations the d stability testing problem for continuous time delay systems subjected to a class of highly structured parametric perturbations is addressed in this note by means of the lyapunov stability theorem razumikhin type theorem gersgorin theorem concept of spectral radius and norm and matrix measure techniques the author has developed several new sufficient conditions for guaranteeing that all characteristic roots of the above systems are located inside a specified disk d spl alpha r with center at spl alpha j0 and radius r in the left half complex plane for the present results it is not necessary to solve any lyapunov equation which may be unsolvable though the lyapunov stability theorem is utilized

robust matrix root clustering analysis through extended kyp lemma this paper is dedicated to robust matrix eigenvalue clustering in a subregion mathcal d of the complex plane a norm bounded uncertainty is considered cd is any intersection and or union of half planes discs and outsides of discs this work aims at computing the associated complex cd stability radius this challenge was already tackled in a recent paper but the technique therein may fail or lead to conservative results when the subregions overlap the present paper proposes to circumvent those limits thanks to an extended version of the kalman yakubovich popov kyp lemma which is deduced from the so called generalized s procedure

generalized pole placement via static output feedback a methodology based on projections this paper presents an algorithm for solving static output feedback pole placement problems of the following rather general form given n subsets of the complex plane find a static output feedback that places in each of these subsets a pole of the closed loop system the algorithm presented is iterative in nature and is based on alternating projection ideas each iteration of the algorithm involves a schur matrix decomposition a standard least squares problem and a combinatorial least squares problem while the algorithm is not guaranteed to always find a solution computational results are presented demonstrating the effectiveness of the algorithm

on finding eigenvalue distribution of a matrix in several regions of the complex plane a direct method is proposed for determining eigenvalue distribution of a matrix with respect to several important regions of the complex plane these regions include half planes shifted half planes hyperbolas sectors quadrants imaginary axis region contained within two straight lines that pass through the orgin etc the method neither requires computation of the characteristic polynomial of the given matrix nor solution of any matrix equations the method seems to be more efficient than the eigenvalue and matrix equations methods

the matrix equation xa atx and an associated algorithm for solving the inertia and stability problems abstract we develop a constructive procedure for generating nonsingular solutions of the matrix equation xa a t x and establish an interesting relationship between a given solution x of the above equation and the associated matrix polynomial p a the latter is then used to develop an algorithm for computing the inertia of a matrix the algorithm is more efficient than the other common procedures

relative stability gain in multivariable feedback systems the problem of finding the stability range of a scalar gain or parameter is a simple one for multivariable feedback systems we propose an algebraic procedure for the relative stability range using the so called critical constraint this method is simpler than the frequency domain approach and compared favourably with a recently proposed approach

common lyapunov function approach to matrix root clustering abstract this paper proposes an alternative route to the matrix root clustering problem which enables to use a result on the common lyapunov function for solving the problem efficiently a necessary and sufficient condition is obtained in terms of the existence of a common positive definite solution to a set of lyapunov inequalities for eigenvalues of a matrix to lie in a prescribed subregion of the complex plane applications to root clustering in sector regions are shown for illustration

classification of six vertex type solutions of the colored yang baxter equation in this paper symmetries and spectral dependent six vertex type solutions of the colored yang baxter equation are discussed it is shown that each nondegenerate six vertex type solution of the colored yang baxter equation is equivalent to one of the six basic solutions up to five solution transformations and all solutions can be classified into two types called the baxter type and the free fermion type

regular pseudo bosons versus bosons we discuss in which sense the so called regular pseudo bosons recently introduced by trifonov and analyzed in some detail by the author are related to ordinary bosons we repeat the same analysis for pseudo bosons and analyze the role played by certain intertwining operators which may be bounded or not

eigenvalue clustering in subregions of the complex plane eigenvalue clustering in subregions of the complex plane is investigated for multivariable linear feedback systems critical constraints of all the eigenvalues of a perturbed system causing it to stay within a specified region of the nominal system arc discussed a design procedure is developed for the selection of a parametrized class of state feedback gains for eigenvalue clustering an illustration of the method for the case of a bounded frequency region is included

the gervais neveu felder equation and the quantum calogero moser systems we quantize the spin calogero moser model in ther matrix formalism the quantumr matrix of the model is dynamical thisr matrix has already appeared in gervais neveu s quantization of toda field theory and in felder s quantization of the knizhnik zamolodchikov bernard equation

on stabilization of 2d roesser models this note is devoted to the stabilization of 2d roesser models which are discrete continuous or mixed continuous discrete a recent linear matrix inequalities lmis necessary and sufficient condition for stability of such models is used to derive a quasi non conservative technique for state feedback stabilization

robust control of the characteristic values of systems with possible parameter variations an analytical method to control by multiparameter output feedback the characteristic values of a closed loop multi input multi output system inside or outside general aubregions of the complex plane is presented the method pertains to analogue as well as discrete systems the design method is robust in the sense that the characteristic values are designed to cluster inside the desired subregion even though any number of the specifying parameters are only known to within a given interval the interval may possibly be disjoint and possibly infinite completely unknown examples are provided which illustrate the broadness and effectiveness of the method

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

an lmi approach for designing sliding mode observers this paper presents a method to design sliding mode observers for a class of uncertain systems using linear matrix inequalities the objective is to exploit the degrees of freedom available in the design which have hitherto been ignored because of the lack of a tractable solution framework the relationship between the linear component of the sliding mode observer and a particular sub optimal observer arising from classical linear quadratic gaussian lqg theory is demonstrated this helps motivate how the design weighting matrices inherent in the method may be chosen in practice it will also be shown how the weighting matrices affect the dynamics of the sliding motion furthermore using pole placement and pole clustering methods the poles of the sliding motion can be forced to lie in a specified region in the complex plane so that the performance of the sliding motion can be tuned this paper will also present a more general solution where the eigenvalues of the linear part of the observer are forced

a new approach to optimal regional pole placement abstract the multi constraints optimal regional pole placement problem is studied the constrained region is represented by several algebraic inequalities necessary conditions for which the optimal solution must be satisfied are derived furthermore a solution algorithm is provided finally an analysis of robustness with rsepest to structured uncertainty for continuous system is discussed 1997 elsevier science ltd

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

roots of composite polynomials an application to root clustering abstract in 1900 stephanos showed that given a c n n with spectrum a i the eigenvalues of the composite matrix p q c pq a p a q are the n 2 values p q c pq p i q j this fact has been recently used in the theory of root clustering throug the characteristics polynomial of if a characteristic polynomial is given rather than a we can use the companion matrix to generate the characteristic coefficients of in this paper we develop a direct theory for composite polynomials

stability of a matrix polynomial in continuous systems two sufficient conditions under which the roots of the determinant of a given m times m matrix polynomial of n th order lie in the open left half plane have been obtained the first condition is given in terms of the positive definiteness of an mn times mn symmetric matrix while the second condition is given in terms of the positive definiteness of an m times m matrix that is a function of s re s leq 0 these conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial therefore the explicit computation of the determinant polynomial is not required

root clustering for convex combination of complex polynomials in some important applications such as the edge theorem it is required that a polynomial is alpha stable along a parameter segment using the critical constraint the necessary and sufficient conditions for root clustering alpha stability of convex combinations of complex polynomials are presented the approach is general and requires that a certain real polynomial has no zeros in the open interval

quantum dynamical matrix with spectral parameter from fusion a quantum dynamical matrix with a spectral parameter is constructed by fusion procedure this spin 1 matrix is connected with lie algebra so 3 and does not satisfy the condition of translation invariance