pole placement by parametric output feedback abstract this note presents a new analytical solution to the problem of pole placement via constant output feedback under the condition m p n where n m and p are the number of states inputs and outputs respectively the approach is based upon parametric eigenstructure assignment of linear time invariant multivariable systems in combination with a special explicit formulation of the pole assignment equations thus the resulting analytical solution explicitly offers all remaining m p n degrees of freedom beyond eigenvalue assignment which can be used for additional design goals such as response shaping minimizing the norm of the feedback matrix and robust control respectively

survey paper static output feedback a survey this paper reviews the static output feedback problem in the control of linear time invariant lti systems it includes analytical and computational methods and presents in a unified fashion the knowledge gained in the decades of research into this important open problem the paper shows that although many approaches and techniques exist to approach different versions of the problem no efficient algorithmic solutions are available

parametric output feedback controller design abstract a parameterization of fixed gain output feedback controllers which will assign a complete set of distinct eigenvalues to a given state space system is presented the main result is a compact parametric expression for the output feedback controller gain matrix explicitly characterized by two sets of free parameters the set of n closed loop system eigenvalues and a set of mr n design parameters for n state r input m output systems the free design parameters are determined by a sequential design procedure which automatically satisfies the basic orthogonality constraints on the left and right closed loop eigenvectors under the condition m r n the principal benefits of the explicit characterization of a parametric class of output feedback controllers lie in the ability to directly accommodate various different design criteria an illustration of the parametric output feedback design procedure is given by way of approximating the optimal full state regulator response of a fifth order pilot plant evaporator model

eigenstructure assignment for componentwise ultimate bound minimization in discrete time linear systems this technical note presents a feedback design approach to minimise state ultimate bounds ubs in linear time invariant discrete time systems subject to non vanishing bounded disturbances a target closed loop eigenstructure is first constructed to achieve the lowest possible ubs for one or more state components of the system structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback when the required structural conditions are satisfied an eigenvector assignment procedure can be applied to simultaneously minimise multiple ubs in systems with multiple inputs the results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter based microgrids

eigenvalue generalized eigenvector assignment by output feedback this note generalizes the previous results of the closed loop eigenstructure assignment via output feedback in linear multivariable systems necessary and sufficient conditions for the closed loop eigenstructure assignment by output feedback are presented some known results on entire eigenstructure assignment are deduced from this result

dynamic least squares kernel density modeling of fokker planck equations with application to neural population the dynamic least squares kernel density lsqkd model c pantano and b shotorban phys rev e 76 066705 2007 is used to solve the fokker planck equations in this model the probability density function pdf is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least squares approximation of the pdf in the phase space in this work basis functions are set to be gaussian for which the mean variance and covariances are governed by a set of partial differential equations pdes or ordinary differential equations odes depending on what phase space variables are approximated by gaussian functions three sample problems of univariate double well potential bivariate bistable neurodynamical system g deco and d mart i phys rev e 75 031913 2007 and bivariate brownian particles in a nonuniform gas are studied the lsqkd is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases for the double well potential problem it is observed that for low to moderate diffusivity the dynamic lsqkd well predicts the stationary pdf for which there is an exact solution a similar observation is made for the bistable neurodynamical system in both these problems least squares approximation is made on all phase space variables resulting in a set of odes with time as the independent variable for the gaussian function parameters in the problem of brownian particles in a nonuniform gas this approximation is made only for the particle velocity variable leading to a set of pdes with time and particle position as independent variables solving these pdes a very good performance by lsqkd is observed for a wide range of diffusivities

sufficient lmi conditions for output feedback control problems the authors present linear matrix inequality lmi conditions for output feedback control problems the results are based on sufficient conditions because they are dependent on the particular state space representation used for describing the system nevertheless the conditions are not sensitive to a certain class of state space transformations and if the control problem is feasible then there exists some state space transformation leading the conditions to be necessary and sufficient for the problem the authors approach can be used for designing decentralized controllers and is easily extended to h sub 2 h sub spl infin and mixed h sub 2 h sub spl infin problems via standard lmi techniques the continuous and discrete time cases are considered and numerical examples are given to illustrate the results

parametric eigenstructure assignment by output feedback control the case of multiple eigenvalues this paper deals with closed loop eigenstructure assignment by output feedback control in linear multivariable systems for the general case of multiple eigenvalues the approach is algebraic in nature and operates directly on the closed loop characteristic equation using fundamental properties of determinants and their derivatives a compact form for the controller gain matrix is derived and expressed in terms of a set of parameter vectors that specify the non uniqueness of the solution of the eigenvalue assignment problem some of these parameter vectors assume restricted forms while others are free a computational scheme is given in which the free parameter vectors are determined in such a way as to assign the corresponding eigenvectors and generalized eigenvectors in a weighted least square error sense an illustrative numerical example is included

band limited functions lp convergence we consider the set bp q functions of lp tl whose fourier spectrum lies in i2 i2 we prove that the prolate spheroidal wave func tions constitute a basis of this space if and only if 4 3 p 4 the result is obtained as a consequence of the analogous problem for the spherical bessel functions the proof rely on a weighted inequality for the hubert transform

static output feedback controllers stability and convexity the main objective of this paper is to solve the following stabilizing output feedback control problem given matrices a b sub 2 c sub 2 with appropriate dimensions find if one exists a static output feedback gain l such that the closed loop matrix a b sub 2 lc sub 2 is asymptotically stable it is known that the existence of l is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs hence this paper provides a new sufficient but not necessary condition for the solvability of the above stabilizing output feedback control problem as a natural extension we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index some examples borrowed from the literature are solved to illustrate the theoretical results

transmission zero assignment using semistate descriptions the problem of finite infinite transmission zero assignment is examined by squaring a system from the outputs to the inputs in particular this problem is studied in two cases state accessible systems and partially state accessible systems it is shown that the problem of transmission zero assignment for state accessible state variable systems is equivalent to a pole placement problem with state feedback for a generalized system which always has a solution in the case of partially state accessible systems it is shown that the transmission zero assignment problem is equivalent to a pole placement problem with output feedback for a generalized system in both cases the block hessenberg form of the system and the extended lower triangular hessenberg form are exploited to formulate the problem

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

fisher equation with turbulence in one dimension as an example of life at high reynolds number we investigate the dynamics of the fisher equation for the spreading of micro organisms in one dimension subject to both turbulent convection and diffusion we show that for strong enough turbulence bacteria for example track in a quasilocalized fashion with remarkably long persistence times sinks in the turbulent field an important consequence is a large reduction in the carrying capacity of the fluid medium we analytically determine the regimes where this quasi localized behavior occurs and test our predictions by numerical simulations

convex analysis of output feedback control problems robust stability and performance this paper addresses the problem of optimal h sub 2 control by output feedback necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function the results can be easily extended to deal with linear uncertain systems where uncertainties are supposed to belong to convex bounded domains providing an h sub 2 guaranteed cost output feedback control thanks to the properties of the above mentioned function we show that under certain conditions convex programming tools can be used for numerical purposes examples illustrate the theoretical results

on the solution to the sylvester matrix equation av bw evf a simple direct complete and explicit parametric solution of the generalized sylvester matrix equation av bw evf is proposed for matrix f in the jordan form with arbitrary eigenvalues using the smith canonical form of the matrix a se b the obtained solution is in a clear neat form and may have important applications in descriptor linear system theory

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

output feedback eigenstructure assignment using a new reduced orthogonality condition this paper presents a new method of output feedback eigenstructure assignment a new reduced orthogonality condition is derived which is less restrictive on the design degrees of freedom than others in the literature from this a novel general formulation for the gain matrix is admitted which utilizes a two stage design process in the first stage a subset of desired eigenvectors is assigned either left or right in the second stage a dual set is assigned which adheres to the necessary and sufficient conditions for output feedback eigenstructure assignment proved earlier in the paper a numerical example is presented to illustrate the whole procedure

multistage parametric eigenstructure assignment by output feedback control this paper develops an efficient multistage parametric approach for eigenstructure assignment in linear multivariable systems by output feedback control the sets of closed loop eigenvalues and associated eigenvectors are suitably divided into subsets and the entire eigenstructure is constructed by parts in two or more consecutive stages this allows the subsets of right and left eigenvectors to be assigned in separate stages and thus the computational algorithm is relaxed from the orthogonality conditions the number of effective free parameters beyond eigenvalue assignment is determined and the notion of redistributing these parameters among the assignable right and left eigenvectors is introduced numerical examples are included to demonstrate the design procedure

some necessary and sufficient conditions for eigenstructure assignment abstract some necessary and sufficient conditions for closed loop eigenstructure assignment by output feedback in time invariant linear multivariable control systems are presented a simple condition on a square matrix necessary and sufficient for it to be the closed loop plant matrix of a given system with some output feedback is the basis of the paper some known results on entire eigenstructure assignment are deduced from this the concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors

output feedback eigenstructure assignment using two sylvester equations the eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p m n the main tool used is the concept of c a b invariance and two coupled sylvester equations the solution of which leads to the computation of an output stabilizing feedback a computationally efficient algorithm for the solution of these two coupled equations which leads to the computation of a desired output feedback is presented

quadratic characterization and use of output stabilizable subspaces this note treats the problem of stabilization of linear systems by static output feedback using the concept of c a b invariant subspaces the work provides a new characterization of output stabilizable c a b invariant subspaces through two coupled quadratic stabilization conditions an equivalence is shown between the existence of a solution to this set of conditions and the possibility to stabilize the system by static output feedback an algorithm is provided and numerical examples are reported to illustrate the approach

eigenstructure assignment for state constrained linear continuous time systems abstract linear state constraints define a polyhedron in the state space of a dynamical system if this polyhedron can be made positively invariant by state feedback the constraints are satisfied all along the trajectory for any initial point in this polyhedron in the context of continuous time linear systems some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra the existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input output system in particular the number of zeros of this system should be appropriate and they should be stable if these conditions are satisfied the problem can easily be solved by eigenvalue and eigenvector assignment

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

eigenvalue assignment robustness the analysis for autonomous system matrices abstract new techniques are presented for the analysis of eigenvalue assignment robustness the basic idea is to use a root locus approach to ensure that the eigenvalues of a dynamic system under parameter perturbations are restricted in the distinct desired regions no restriction is imposed on the shapes of the specified regions both the norm bound and the element bound for the allowable perturbations are obtained scaling techniques are proposed to increase the allowable element bound the proposed methods are applicable to the continuous time case as well as to the discrete time case when the eigenvalues are just required to locate in the stable region the proposed criteria will become the stability robustness criteria

robust eigenstructure assignment for the extended medium range air to air missile abstract a new sufficient condition for the robust stability of a linear time invariant system subject to linear time varying structured state space uncertainty is presented this sufficient condition is a sum of terms each of which involves the ith right eigenvector the ith left eigenvector and the real part of the ith eigenvalue this new robustness result is less conservative than an earlier result which involves the modal matrix and the real part of the eigenvalue which is closest to the imaginary axis in the complex plane a robust design is computed for the extended medium range air to air missile and the new design is compared with an earlier orthogonal projection eigenstructure assignment design the new design method uses the matlabtm optimization toolbox to minimize the integral of the deviations of the integrated roll rate which is approximately the bank angle from zero the integral is approximated by a sum of discrete samples of the integrated roll rate while normalizing the integration width to unity constraints are placed on the real part of the dutch roll and roll mode eigenvalues the damping ratios of the dutch roll and roll mode the aileron and rudder deflection rates and the sufficient condition for robust stability the new design satisfies the robustness condition while also yielding an improved transient response

perturbation theory of output feedback pole assignment perturbation theory for output feedback eigenstructure assignment is considered the analysis is based on general conditions for eigenstructure assignment together with the well known perturbation theory for the pseudo inverse of a matrix this enables us to present a variety of sensitivity analyses for this problem and to indicate some precautions when choosing eigenvectors in the pole assignment problem

a new approach to eigenstructure assignment by output feedback the eigenstructure assignment problem for a linear time invariant multi input multi output system using output feedback is considered a new approach is developed which identifies the eigenspaces for the desired set of all the closed loop eigenvalues for the assignment of this desired set necessary and sufficient conditions are established these conditions contain two coupled sylvester matrix equations one of which is proven to be a reduced order square sylvester matrix equation this results in an efficient analytical procedure numerically superior to known techniques for the determination of the output feedback gain matrix

solutions of the equation av bw vf and their application to eigenstructure assignment in linear systems two new simple complete analytical and restriction free solutions with complete and explicit freedom of the matrix equation av bw vf are proposed here ab is known and is controllable and f is in the jordan form with arbitrary given eigenvalues based on the proposed solutions of this matrix equation a complete parametric approach for eigenstructure assignment in linear systems via state feedback is proposed and two new algorithms are presented the proposed solutions of the matrix equation and the eigenstructure assignment result are generalizations of some previous results and are simpler and more effective

exact pole assignment with regularity by output feedback in descriptor systems part 2 in this paper the first of two we examine the problem of modal control of linear time invariant multivariable descriptor systems in particular for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix possibly complex which exactly assigns an arbitrary set of r the number of independent semi state variables of the system distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition furthermore it is shown that if the set of proposed poles contains only real elements the assignment can be carried out using real feedback next we present conditions that are necessary and sufficient for pole assignability by real feedback of a self conjugate set of complex numbers provided that the system satisfies a further inequality

a bilinear formulation for the output feedback problem in linear systems this note ties together a geometric theory and previous algebraic results for the closed loop eigenstructure assignment by output feedback in multivariable systems necessary and sufficient conditions for the closed loop stabilization eigenstructure assignment are presented in terms of a bilinear sylvester equation the solutions of which completely characterize the closed loop eigenstructure problem with output feedback some preliminary results for the solution of this bilinear equation are presented