quadratic stabilizability of uncertain linear systems existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control this note is concerned with the problem of stabilizing an uncertain linear system via state feedback control an uncertain system which admits a stabilizing state feedback control and some associated quadratic lyapunov function is said to be quadratically stabilizable in a number of recent papers conditions are given under which quadratic stabilizability via nonlinear control implies quadratic stabilizability via linear control these papers restrict the manner in which the uncertain parameters are permitted to enter structurally into the state equation in order to establish this result this note presents an example which shows that this implication is not true for more general uncertain linear systems to this end we describe an uncertain linear system which is quadratically stabilizable via nonlinear control but not quadratically stabilizable via linear control

robust stabilization of discrete time linear systems with norm bounded time varying uncertainty abstract this paper presents an algorithm for the stabilization of a class of discrete time uncertain systems suffering from uncertainty of the norm bounded time varying type the stabilizing control gain matrix is obtained by solving a certain discrete riccati equation the solution of this riccati equation is also a lyapunov matrix which is used to establish the stability of closed loop system the results presented in this paper generalize those obtained in previous works for continuous time uncertain systems

designing stabilizing controllers for uncertain systems using the riccati equation approach a useful technique for determining a linear feedback control law stabilizes an uncertain system is the riccati equation approach of i r petersen and c v hollot 1986 they consider systems with time varying uncertainty in the system matrix and obtain the constant feedback gains for the linear stabilizing controller in terms of the solutions of a riccati equation the technique is extended to include problems with time varying uncertainty in the input connection matrix several examples are included to demonstrate the efficacy of this result

linear quadratic differential games with cheap control abstract this paper considers a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero it is shown that if a certain minimum phase condition is satisfied then the value of the game will approach zero as the weighting on the minimizing control approaches zero

balanced model reduction of symmetric composite systems in this paper we investigate the structural properties of balanced realizations and their truncation of a class of systems with symmetric interconnected and identical structure it is established that the balanced realizations of such symmetric composite systems can be constructed via the balanced realizations of two auxiliary systems of comparatively low dimensions the corresponding truncated realizations which also have a certain symmetric structure are explicitly constructed the extension of the results to generalized symmetric composite systems is also studied

an optimal control approach to robust tracking of linear systems in our early work we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem if the system is linear then the optimal control problem becomes a linear quadratic regulator lqr problem which can be solved by solving an algebraic riccati equation in this article we extend the optimal control approach to robust tracking of linear systems we assume that the control objective is not simply to drive the state to zero but rather to track a non zero reference signal we assume that the reference signal to be tracked is a polynomial function of time we first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic riccati equation because the state feedback is not always available in practice we also investigated the output feedback we show that if we place the poles of the observer

a stabilization algorithm for a class of uncertain linear systems abstract this paper presents an algorithm for the stabilization of a class of uncertain linear systems the uncertain systems under consideration are described by state equations which depend on time varying unknown but bounded uncertain parameters the construction of the stabilizing controller involves solving a certain algebraic riccati equation furthermore the solution to this riccati equation defines a quadratic lyapunov function which is used to establish the stability of the closed loop system this leads to a notion of quadratic stabilizability it is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable the paper also deals with a notion of overbounding for uncertain linear systems this procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process

on the perfect regulation of optimal regulators an alternative proof is provided for an earlier result of francis 1 on the perfect regulation of optimal regulators our proof shown in the present paper 1 is an interesting application of anderson s result on the spectral factorization and may be simpler and more transparent than the one in 1

stability and performance in the presence of magnitude bounded real uncertainty riccati equation based state space approaches abstract this paper is concerned with riccati equation based analysis methods for determining stability and performance of perturbed feedback systems the elements of the state space representation of the systems are assumed to be linearly perturbed with real magnitude bounded uncertaintites performance is defined as the value of the h norm of the transfer function of interest an analysis method based on checking the existence of a common solution to a finite set of riccati inequalities is proposed it represents a nonconvex optimization problem this problem is then transformed into a convex optimization problem by the use of properties of certain matrix inequalities

the gpl stability of rosenbrock methods for delay differential equation this paper deals with the stability analysis of the rosenbrock method for the numerical solution of delay differential equation the stability behavior of rosenbrock method is analyzed for the solutions of linear test equation we will give that the rosenbrock method is gpl stable if and only if it is l stable

a linear matrix inequality approach for robust control of systems with delayed states abstract this paper addresses the robust stabilization problem for uncertain systems with delayed states the parameter uncertainties are unknown but norm bounded and the delay is time varying in this study a linear matrix inequality lmi approach for robust stability analysis for the nominal unforced system and a method for robust stabilization for a class of uncertain delay systems via linear memoryless state feedback control are presented the results depend on the size of the delay and are given in terms of linear matrix inequalities

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

successive approximation procedure of optimal tracking control for nonlinear similar composite systems abstract the optimal tracking control otc problem for a class of affine nonlinear composite systems with similar structure is considered by using a modeling technique the nonlinear similar composite system is first transformed into some quasi decoupled subsystems then the high order strongly coupled nonlinear two point boundary value tpbv problem is transformed into a sequence of linear decoupled tpbv problems through a successive approximation procedure the obtained otc law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence a suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term

pole assignment for uncertain systems in a specified disk by state feedback this paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system the uncertainty being norm bounded for this the quadratic d stabilizability concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived this condition expressed as a parameter dependent discrete riccati equation enables one to design the control gain matrix by solving iteratively a discrete riccati equation

h guaranteed cost control for uncertain continuous time linear systems abstract this paper deals with the h guaranteed cost control problems for continuous time uncertain systems it consist of the determination of a stabilizing state feedback gain which imposes on all possible closed loop models an h norm upper bound 0 assuming that the uncertain domain is convex bounded and the uncertain system is quadratic stabilizable with disturbance attenuation it is shown how to determine by means of a convex programming problem the global minimum of as a particular and important case for precisely known linear systems the last problem reduces to the classical h optimal control problem the results follow from the definition of a special parameter space on which the above mentioned problems are convex

symmetric failures in symmetric control systems abstract this paper discusses the fault tolerance of symmetric systems with respect to controllability which is a fundamental characteristic of control systems in particular we reveal the underlying mathematical mechanism of the loss of controllability for symmetric systems induced by failures based on the decomposition of the symmetric systems into subsystems under the symmetry the controllability of the entire system can be discussed by checking that of each subsystem the analysis of the fault tolerance in this paper is an extension of this idea with the aid of the chain adapted transformation matrix for the decomposition the result is shown as a necessary condition for symmetric systems to retain the controllability despite some symmetric failures we also discuss sufficient conditions

fast linear algebra is stable in demmel et al numer math 106 2 199 224 2007 we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense and that in fact if multiplication of n by n matrices can be done by any algorithm in o n operations for any 0 then it can be done stably in o n operations for any 0 here we extend this result to show that essentially all standard linear algebra operations including lu decomposition qr decomposition linear equation solving matrix inversion solving least squares problems generalized eigenvalue problems and the singular value decomposition can also be done stably in a normwise sense in o n operations

stability of discrete delay systems boundary criteria this paper deals with the stability of linear discrete delay systems two stability criteria that are described by evaluating a corresponding harmonic function on the boundary of a torus region are presented testing these criteria can be proved to be much easier than those reported in the literature

asymptotic stability for the system of neutral delay differential equations abstract this paper deals with the asymptotic stability of the system of neutral delay differential equations in an angle which has fewer reports stability regions are presented geometrically as well as two criteria which are obtained through the evaluation of a harmonic function on the boundary of a certain region numerical experiments on various circumstances are shown to locate the specific region which can make the system unstable so as to exclude it from the whole complex plane

reduction and non linear controllability of symmetric distributed systems this paper considers the reduction problem for distributed control systems in particular we consider controllability of systems containing multiple instances of diffeomorphic components where the overall system dynamics is invariant with respect to a discrete group action a subclass of such systems are systems with a set of identical components where the overall system dynamics are invariant with respect to physically interchanging these components the main result is a proposition which shows that for an equivalence class of symmetric systems of this type controllability of the entire class of systems can be determined by analysing the member of the equivalence class with the smallest state space the reduction methods developed are illustrated by considering the controllability of a team of mobile robots and a platoon of underwater vehicles

technical communique delay dependent output feedback guaranteed cost control for uncertain time delay systems this paper considers the problem of output feedback guaranteed cost controller design for uncertain time delay systems by representing the time delay system in the descriptor form and using a recent result on bounding of cross products of vectors we obtain new delay dependent sufficient conditions for the existence of the guaranteed cost output feedback controller in terms of matrix inequalities a numerical algorithm is proposed to construct a full order output feedback controller achieving a suboptimal guaranteed cost such that the system can be stabilized for all admissible uncertainties an example is presented which shows that the proposed method can produce a lower guaranteed cost than the delay independent method

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

delay dependent robust stabilization of uncertain systems with multiple state delays this paper deals with the problem of robust stabilization for uncertain systems with multiple state delays the parameter uncertainties are time varying and unknown but are norm bounded and the delays are time varying a new method for achieving robust stabilization is presented for a class of uncertain time delay systems via linear memoryless state feedback control the results depend on the size of the delays and are given in terms of several linear matrix inequalities

design of robust state feedback laws existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices the uncertainties are modelled as sector bounded non linearities and feedback laws ensuring global asymptotic closed loop stability are obtained using lyapunov e direct method the algorithms essentially amount to repeatedly solving a parameter dependent riccati equation until the maximal solution becomes positive definite

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions

matrix multiplication a little faster strassen s algorithm 1969 was the first sub cubic matrix multiplication algorithm winograd 1971 improved the leading coefficient of its complexity from 6 to 7 there have been many subsequent asymptotic improvements unfortunately most of these have the disadvantage of very large often gigantic hidden constants consequently strassen winograd s o nlog27 algorithm often outperforms other fast matrix multiplication algorithms for all feasible matrix dimensions the leading coefficient of strassen winograd s algorithm has been generally believed to be optimal for matrix multiplication algorithms with a 2 2 base case due to the lower bounds by probert 1976 and bshouty 1995 surprisingly we obtain a faster matrix multiplication algorithm with the same base case size and asymptotic complexity as strassen winograd s algorithm but with the leading coefficient reduced from 6 to 5 to this end we extend bodrato s 2010 method for matrix squaring and transform matrices to an alternative basis we also prove a generalization of probert s and bshouty s lower bounds that holds under change of basis showing that for matrix multiplication algorithms with a 2 2 base case the leading coefficient of our algorithm cannot be further reduced and is therefore optimal we apply our method to other fast matrix multiplication algorithms improving their arithmetic and communication costs by significant constant factors

robust reduced order nonstrictly proper state estimation via the optimal projection equations with petersen hollot bounds a state estimation design problem involving parametric plant uncertainties is considered an error bound suggested by recent work of petersen and hollot is utilized for guaranteeing robust estimation necessary conditions which generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator which minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators an additional feature is the presence of a static estimation gain in conjunction with the dynamic kalman estimator i e a nonstrictly proper estimator

stabilizing controllers for uncertain symmetric composite systems abstract this paper discusses the problem of designing stabilizing controllers for uncertain symmetric composite systems a procedure of designing a linear decentralized feedback control law which will stabilize a given symmetric composite system with uncertainty is presented by using the riccati equation approach a feature of the procedure is that the stabilizing control law can be conducted by solving two lower order riccati equations or inequalities the uncertainties considered in the systems may be time varying however the values of the uncertainties are constrained to lie within some known admissible bounds

stabilizing controllers for uncertain linear systems with additive disturbances this paper is concerned with the problem of designing stabilizing controllers for linear systems containing uncertainty the uncertainty is time varying with values lying in a specified set and enters through the system matrix through the input connection matrix and additively a method is presented for obtaining a linear state feedback controller which guarantees that the system response ultimately lies in a prespecified neighbourhood of the origin in some situations this neighbourhood can be made arbitrarily small and practical stability is achieved the solution of an algebraic riccati equation is used to generate the feedback gains for the linear controller several examples are included to illustrate the application of the method

graph expansion and communication costs of fast matrix multiplication the communication cost of algorithms also known as i o complexity is shown to be closely related to the expansion properties of the corresponding computation graphs we demonstrate this on strassen s and other fast matrix multiplication algorithms and obtain the first lower bounds on their communication costs in the sequential case where the processor has a fast memory of size m too small to store three n by n matrices the lower bound on the number of words moved between fast and slow memory is for a large class of matrix multiplication algorithms n m 0 m where 0 is the exponent in the arithmetic count e g 0 lg 7 for strassen and 0 3 for conventional matrix multiplication with p parallel processors each with fast memory of size m the lower bound is asymptotically lower by a factor of p these bounds are attainable both for sequential and for parallel algorithms and hence optimal

explicit solution and eigenvalue bounds in the lyapunov matrix equation an explicit solution to the algebraic lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices this enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix based on this explicit formula upper and lower bounds for each eigenvalue of the solution are derived which always give nontrivial estimates

quadratic stability and stabilization of dynamic interval systems this note presents necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems the results are obtained in terms of linear matrix inequalities lmis and extended to the quadratic stability and stabilization of linear systems with uncertain parameters with the powerful lmi toolbox it is very convenient to solve these problems the illustrative examples show that this method is effective and less conservative to check the robust stability and to design the stabilizing controller for dynamic interval systems

nonlinear versus linear control in the robust stabilizability of linear uncertain systems via fixed order output feedback this paper considers the problem of robustly stabilizing an uncertain system via fixed order output feedback control this problem is considered by defining a notion of robust stabilizability with a quadratic storage function this notion is closely related to the notion of quadratic stabilizability the main result of the paper shows that if an uncertain system is robustly stabilizable with a quadratic storage function via a nonlinear time varying controller then it will also be robustly stabilizable with a quadratic storage function via a linear time invariant controller of the same order

a note on observer design for one sided lipschitz nonlinear systems in this note observer design of a class of nonlinear systems is considered based on the one sided lipschitz condition sufficient conditions for the existence of observers of the class of nonlinear systems are presented a design method of the observer that is dependent on the solution of the linear matrix inequality is then presented furthermore a gain matrix of the observer is given the method given in this note makes the applicable class larger than that given in the literature it should be noticed that the proposed method is applicable not only to the monotone nonlinearities and lipschitz nonlinearities but also to the one sided lipschitz nonlinearities which are neither monotone nonlinearities nor usual lipschitz nonlinearities finally we use the proposed method to design observers for three simulative examples and the effect of each state trajectory tracking is very satisfactory

generalized riccati equations associated with guaranteed cost control an overview of solutions and features robust control of dynamic linear systems with parametric uncertainties is based on the solution of generalized riccati equations i e riccati equations with additional terms depending on the uncertainties in particular guaranteed cost controller design requires the existence and computation of positive definite solutions of such equations in the present paper an overview of the riccati equations arising in guaranteed cost control approaches is presented it is shown that different uncertainty descriptions and the corresponding upper bounding functions lead to various forms of generalized riccati equations that may admit either analytical or computational positive definite solutions the associated closed loop systems robustness features are also discussed

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

optimal guaranteed cost control of uncertain discrete time delay systems this paper considers the problem of robust guaranteed cost control of linear discrete time delay systems with parametric uncertainties by matrix inequality approach the robust quadratic stability of the system is studied a control design method is developed such that the closed loop system with a cost function has a upper bound irrespective of all admissible parameter uncertainties and unknown time delays furthermore the upper bound cost can be optimized by incorporating with a minimization problem a numerical example is given to show the potential of the proposed techniques

robustness bounds for lq regulators a robustification procedure for lq state feedback design is presented such a procedure consists of choosing the state and input weighting matrices according to the kind of uncertainties on the system both structured and norm bounded additive uncertainties are addressed and upper bounds for the uncertainties that do not destabilize the closed loop system are presented connections with the quadratic stabilizability problem are established

an implicit small gain condition and an upper bound for the real structured singular value in this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem the implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and unlike prior bounds does not require frequency dependent scales or multipliers numerical results show that the implicit small gain bound compares favorably with real bounds that involve frequency dependent scales and multipliers

robust exponential stabilization for large scale uncertain impulsive systems with coupling time delays abstract in this paper we aim to study robust exponential stabilization for a large scale uncertain impulsive system with coupling time delays furthermore we also provide an estimation of the rate of convergence of exponential stabilization by utilizing the lyapunov method and razumikhin technique we shall design the feedback hybrid controllers in terms of linear matrix inequalities under which the robust exponential stability is achieved for a closed loop large scale uncertain impulsive system with coupling time delays moreover we shall also use the results obtained to design impulsive controllers for a large scale uncertain continuous system under which the closed loop continuous system achieves robust and exponential stability to illustrate our results one example is solved

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

memoryless stabilization of uncertain linear systems including time varying state delays the problem of stabilizing a class of uncertain time delay systems via memoryless linear feedback is examined the systems under consideration are linear systems with time varying state delays they also contain uncertain parameters possibly time varying whose values are known only to within a prescribed compact bounding set the main contribution given is to enlarge the class of time delay systems for which one can construct a stabilizing memoryless linear feedback controller within this framework a novel notion of robust memoryless stabilizability is first introduced via the method of lyapunov functionals then a sufficient condition for the stabilizability is proposed it is shown that solvability of a parameterized riccati equation can be used to determine whether the time delay system satisfies the sufficient condition if there exists a positive definite symmetric solution satisfying the riccati equation a suitable memoryless linear feedback law can be derived

qualitative analysis and decentralized controller synthesis for a class of large scale systems with symmetrically interconnected subsystems abstract a number of large scale interconnected systems often encountered in practice are composed of subsystems with similar dynamics interconnected in a symmetrical fashion and the synthesis of controllers for such systems must exploit the special structural properties in order to avoid overly conservative designs and to take advantage of the possible beneficial effects of the interconnections an analysis of some important qualitative properties of such symmetrically interconnected systems focussing on the spectrum characterization controllability and observability and the solutions of the algebraic riccati equation and the matrix lyapunov equation is conducted in this paper and procedures for constructing the solutions to the analysis problems at the overall system level from the computationally simple subsystem level solutons are developed a decentralized controller design procedure is presented as an illustration of the utilization of the available structural information in addressing synthesis problems numerical examples are included to demonstrate the superiority of the presented designs over the use of existing approaches which do not take full advantage of the structural knowledge in these large scale systems

reliable control using redundant controllers this paper presents a methodology for the design of reliable control systems by using multiple identical controllers to a given plant the resulting closed loop control system is reliable in the sense that it provides guaranteed internal stability and h sub spl infin performance in terms of disturbance attenuation not only when all controllers are operational but also when some controller outages sensor and or actuator occur a numerical example is given to illustrate the proposed design procedures

some new results on algebraic riccati equations arising in linear quadratic differential games and the stabilization of uncertain linear systems abstract this paper presents some new results on algebraic riccati equations arising in linear quadratic differential games the first result is a uniqueness result for solutions of the riccati equations under consideration the second result is concerned with riccati equations which arise in differential games in which the weighting on the minimizing control is allowed to approach zero it is shown that if a certain minimum phase condition is satisfied then the corresponding solution to the riccati equation will also approach zero

brief paper structured semidefinite programs for the control of symmetric systems in this paper we show how the symmetry present in many linear systems can be exploited to significantly reduce the computational effort required for controller synthesis this approach may be applied when controller design specifications are expressible via semidefinite programming in particular when the overall system description is invariant under unitary coordinate transformations of the state space matrices synthesis semidefinite programs can be decomposed into a collection of smaller semidefinite programs

effects of using observers on stabilization of uncertain linear systems the effects of observers on robust linear feedback controllers are studied sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback the effects of the resulting high gain observers on the disturbance rejection bounds are also studied it is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection however if there is only additive plant disturbance and no measurement disturbance this degradation can be prevented

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

quadratic games and h type problems for time varying systems abstract h type control problems with state feedback are considered for time varying systems first general quadratic differential games are studied and then the relationship between h problems and differential games is clarified a necessary and sufficient condition for the existence of a stabilizing controller such that the norm of the operator from the disturbance to the output is less than a prescribed number is given in terms of a riccati equation earlier results on time invariant systems are found to be special cases of our main results the finite horizon problem is also studied

the majorant lyapunov equation a nonnegative matrix equation for robust stability and performance of large scale systems a new robust stability and performance analysis technique is developed the approach involves replacing the state covariance by its block norm matrix i e the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics a bound i e majorant for the block norm matrix is given by the majorant lyapunov equation a lyapunov type nonnegative matrix equation existence uniqueness and computational tractability of solutions to the majorant lyapunov equation are shown to be completely characterized in terms of m matrices two examples are considered for a damped simple harmonic oscillator with uncertain but constant natural frequency the majorant lyapunov equation predicts unconditional stability and for a pair of nominally uncoupled oscillators with uncertain coupling the majorant lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators a result not predictable from quadratic or vector lyapunov functions

a new simple approach to the derivation of exact analytical formulae for the zeros of analytic functions a very simple method for the construction of the polynomial whose zeros coincide with the zeros of an analytic function inside and along a simple closed contour in the complex plane based on an appropriate application of the cauchy theorem in complex analysis is proposed the present method was motivated by the classical burniston siewert method based on the theory of the riemann hilbert boundary value problem for the construction of the aforementioned polynomial but although essentially equivalent to the burniston siewert method it is much simpler

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

robust nonfragile kalman filtering for uncertain linear systems with estimator gain uncertainty the note is concerned with the problem of a robust nonfragile kalman filter design for a class of uncertain linear systems with norm bounded uncertainties the designed state estimator can tolerate multiplicative uncertainties in the state estimator gain matrix the robust nonfragile state estimator designs are given in terms of solutions to algebraic riccati equations the designs guarantee known upper bounds on the steady state error covariance a numerical example is given to illustrate the results

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

synthesis of controllers for continuous time delay systems with saturating controls via lmis the stabilization of linear continuous time systems with time delay in the state and subject to saturating controls is addressed sufficient conditions obtained via a linear matrix inequality lmi formulation are stated to guarantee both the local stabilization and the satisfaction of some performance requirements the method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed loop system is guaranteed when control saturations effectively occur

a connection between h control and the absolute stabilizability of uncertain systems abstract this paper considers a stabilization problem for a class of uncertain systems in which the uncertainty satisfies a certain integral quadratic constraint such uncertain systems have been previously studied in the theory of absolute stability with most publications in this area being in the russian literature the paper introduces a notion of absolute stabilizability and shows that an uncertain system is absolutely stabilizable if and only if there exists a solution to a corresponding h control problem the paper also shows that if an uncertain system can be absolutely stabilized via nonlinear output feedback control then it can be absolutely stabilized via linear output feedback control

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

asymptotic h sup infinity disturbance attenuation based on perfect observation a design method of controllers which ensure internal stability and attain asymptotically h sup infinity disturbance attenuation is presented the design procedure consists of two steps 1 to design an h sup infinity state feedback control via an algebraic riccati equation approach and 2 under a certain minimum phase condition to recover the achievable performance asymptotically by applying high gain observers it makes use of the perfect observation for the design of high gain observers it is shown that the asymptotic recovery can be attained by using reduced order observers provided there is no direct feedthrough of controls and disturbances in observations

structural stabilization of uncertain systems necessity of the matching condition this paper investigates one aspect of the problem of stabilizing an uncertain linear system that is the systems under consideration contain uncertain parameters which are unknown but bounded the question arises as to whether such a system can be stabilized via feedback control in some of the previous papers in this area the system is assumed to satisfy a so called matching condition this type of assumption is used to assure that the uncertain system can be stabilized it is known however that the matching condition is not a necessary condition for stabilizability this paper introduces a strengthened notion of stabilizability referred to as structural stabilizability via a nominally determined quadratic lyapunov function it is then shown that if a system is to have this stronger property the matching condition must necessarily be satisfied

an approach to h control of a class of nonlinear systems this paper considers robust h control of a class of nonlinear systems that can be represented by a heterogeneous model a new controller design method using both state feedback is proposed based on quadratic stability theory a number of necessary and sufficient conditions are derived for quadratic stabilizability with disturbance attenuation of the control system concerned an example is also given to demonstrate the performance of the proposed controller design methods

absolute stability of systems under block diagonal memoryless uncertainties abstract absolute stability of systems under block diagonal uncertainties is studied a quadratic lyapunov function is used it is found that the quadratic stability condition of such a system is equivalent to that of systems with uncertainties restricted to be rank one matrices the main technique used is exact penalization in nonsmooth analysis

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

fast matrix multiplication is stable we perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of bini and lotti numer math 36 63 72 1980 as a consequence of our analysis we show that the exponent of matrix multiplication the optimal running time can be achieved by numerically stable algorithms we also show that new group theoretic algorithms proposed in cohn and umans foundations of computer science 44th annual ieee symposium pp 438 449 2003 and cohn et al foundations of computer science 46th annual ieee symposium pp 379 388 2005 are all included in the class of algorithms to which our analysis applies and are therefore numerically stable we perform detailed error analysis for three specific fast group theoretic algorithms

on the complexity of matrix product our main result is a lower bound of omega m 2 log m for the size of any arithmetic circuit for the product of two matrices over the real or complex numbers as long as the circuit does not use products with field elements of absolute value larger than 1 where m m is the size of each matrix that is our lower bound is superlinear in the number of inputs and is applied for circuits that use addition gates product gates and products with field elements of absolute value up to 1 we also prove size depth tradeoffs for such circuits we show that if a circuit as above is of depth d then its size is omega m 2 1 o d