reversible christoffel factorizations we define a family of natural decompositions of sturmian words in christoffel words called reversible christoffel rc factorizations they arise from the observation that two sturmian words with the same language have almost always arbitrarily long abelian equivalent prefixes using the three gap theorem we prove that in each rc factorization only 2 or 3 distinct christoffel words may occur we begin the study of such factorizations considered as infinite words over 2 or 3 letters and show that in the general case they are either sturmian words or obtained by a three interval exchange transformation

extended optimality properties of the linear quadratic regulator and stationary kalman filter it is shown that the constant gain state feedback solution to the infinite time linear quadratic regulator problem is optimal not only for arbitrarily initial conditions or white noise disturbances but also for worst case l sup 1 disturbances using a similar technique it is shown that in the stationary kalman filter the white disturbance and measurement noise can be replaced by unknown bounded energy signals and that optimality still holds if the performance criterion is a time domain l sub infinity normal of the state estimation errors in the presence of worst case energy signals

relationship between the trace and maximum eigenvalue norms for linear quadratic control design the use of a sum of squares h sub 2 norm is considered for optimal control system design the relative advantages of the linear quadratic gaussian lqg trace norm and the h sub infinity sup norm cost functions are discussed an argument is presented to show that the sum of squares norm ssn problem might be considered alongside the lqg and h sub infinity cost problems when exploring the relationship between the lqg and ssn problems it is found that an lqg controller could be found which correspond to an ssn problem with larger disturbance inputs this suggests that the lqg controller can be computed with the knowledge that it minimizes a well defined ssn problem with worst case uncertainty or disturbances the range of a certain set of eigenvalues is found to provide a measure of the difference between the lqg and ssn problem designs

minimization of the l sup infinity induced norm for sampled data systems it is shown that given any degree of accuracy there exists a standard discrete time l sup 1 problem that can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l sup infinity induced norm this is accomplished by first converting the hybrid system into an equivalent infinite dimensional discrete time system using the lifting technique in continuous time and then approximating the infinite dimensional parts of the system which model the intersample dynamics a thorough analysis of the approximation procedure is presented and it is shown that it is convergent at the rate of 1 n explicit bounds that are independent of the controller are obtained to characterize the approximation it is also shown that the geometry of the induced norm for the sampled data problem is different from that of the standard l sup 1 norm and hence there might not exist a linear isometry that maps the sampled data problem exactly to a standard discrete time problem

the mean square error of a random fuzzy vector based on the support function and the steiner point metrics between fuzzy values are a topic with interest for different purposes among them statistics with fuzzy data is growing in modelling and techniques largely through the use of suitable distances between such data this paper introduces a generalized actually parameterized l 2 metric between fuzzy vectors which is based on their representation in terms of their support function and steiner points consequently the metric takes into account the deviation in central location represented by the steiner points and the deviation in shape represented by a deviation defined in terms of the support function and steiner points then sufficient conditions can be given for this representation to characterize fuzzy vectors which is valuable for different aims like optimization studies properties of the metric are analyzed and its application to quantify the mean square error of a fuzzy value in estimating the value of a fuzzy vector valued random element is examined some immediate implications from this mean error are finally described

fr operator approach to the h sub 2 analysis and synthesis of sampled data systems recently a frequency domain operator called frequency response fr operator was defined and shown to represent the transfer characteristics of a stable sampled data system using this novel frequency domain notion and introducing its extended notion called hybrid fr operator we define an h sub 2 norm for sampled data systems in this paper then sampled data h sub 2 control problems are formulated and solved whereby the usefulness of these frequency domain notions is demonstrated both in the analysis and synthesis aspects of sampled data systems for the case of sampled data systems with hybrid i e both continuous time and discrete time input and output signals the h sub 2 norm defined by a hybrid fr operator turns out to be slightly different from that defined in previous studies the source of the discrepancy is also identified

the generalized h 2 control problem abstract in this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary but bounded energy exogenous inputs more specifically we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from l 2 0 to l 0 is below a specified level we consider both state feedback and output feedback problems in the state feedback case it is shown that if this synthesis problem is solvable then a solution can be chosen to be a constant state feedback gain necessary and sufficient conditions for the existence of solutions as well as a formula for a state feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program after showing the separation properties of this synthesis problem the output feedback case is reduced to a state feedback problem it is shown that in the output feedback case generalized h 2 controllers can be chosen to be observer based controllers the theory is demonstrated with a numerical example

a lifting technique for linear periodic systems with applications to sampled data control abstract a lifting technique is developed for periodic linear systems and applied to the h and h 2 sampled data control problems

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

extensive theoretical numerical comparative studies on h2 and generalised h2 norms in sampled data systems abstractthis paper is concerned with linear time invariant lti sampled data systems by which we mean sampled data systems with lti generalised plants and lti controllers and studies their h2 norms from the viewpoint of impulse responses and generalised h2 norms from the viewpoint of the induced norms from l2 to l a new definition of the h2 norm of lti sampled data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions we then establish unified treatment of the three definitions of the h2 norm through a matrix function g defined on the sampling interval 0 h this paper next considers the generalised h2 norms in which two types of the l norm of the output are considered as the temporal supremum magnitude under the spatial 2 norm and norm of a vector valued function we further give unified treatment of the generalised h2 norms through another matrix function f which is also defined on 0 h through a close connection between g

sturmian words lyndon words and trees abstract we prove some new combinatorial properties of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 4 3 we show that aperb a b st lynd where st is the set of the finite factors of all infinite sturmian words and lynd is the set of the lyndon words on the alphabet a b it is also shown that aperb a b cp where cp is the set of christoffel primitive words such words can be defined in terms of the slope of the words and of their prefixes 1 from this result one can derive in a different way by using a theorem of borel and laubie that the elements of the set aperb are lyndon words we prove the following correspondence between the ratio p q of the periods p q p q of w per a a b and the slope w b 1 w a 1 of the corresponding christoffel primitive word awb if p q has the development in continued fractions 0 h 1 h n 1 h n 1 then has the development in continued fractions 0 h n h 2 h 1 1 this and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the stern brocot and the raney numbers of a node in a complete binary tree however one needs some further results more precisely we label the binary tree with standard pairs standard tree christoffel pairs christoffel tree and the elements of per farey tree the main theorem is the following if the node w is labeled by the standard pair u v by the christoffel pair x y and by w per then uv wab xy awb the stern brocot number sb w is equal to the slope of the standard word uv and of the christoffel word xy while the raney number ra w is equal to the ratio of the minimal period of wa and the minimal period of wb some further auxiliary results are also derived

extended h 2 and h norm characterizations and controller parametrizations for discrete time systems this paper presents new synthesis procedures for discrete time linear systems it is based on a recently developed stability condition which contains as particular cases both the celebrated lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters these new synthesis conditions have some nice properties a they can be expressed in terms of lmi linear matrix inequalities and b the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic lyapunov function used to test stability this second feature is important for several reasons first structural constraints as those appearing in the decentralized and static output feedback control design can be addressed less conservatively second parameter dependent lyapunov function can be considered with a very positive impact on the design of robust h 2 and h x control problems third the design of controller with mixed

an lqr weight selection approach to the discrete generalized h2 control problem it is known that a generalized h2 gh2 controller for a continuous time system takes the form of a conventional kalman filter together with a state feedback control law and that the feedback gains can be determined using convex programming in this paper the discrete time version of the gh2 control problem and the related convex programmes involving weight selection in a full information lqr problem are considered

minimization of the l induced norm for sampled data systems the following problem is addressed given a continuous time plant with continuous time performance objectives expressed in terms of the l induced norm design a digital controller that delivers or optimizes this performance this problem differs from the standard discrete time methods in that it takes into consideration the inter sample behavior of the closed loop system the resulting closed loop system dynamics consist of both continuous time and discrete time dynamics and thus such systems are known as hybrid systems it is shown that given any degree of accuracy there exists a standard discrete time l 1 problem which can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l induced norm

l sub 1 analysis and design of sampled data systems the focus of this work is l sub 1 optimal control of sampled data systems a converging approximation procedure is derived to compute the l sub infinity induced norm of closed loop finite dimensional linear time invariant lti sampled data systems an approximation method is developed to synthesize l sub 1 optimal sampled data regulators finally an example is provided that illustrates the l sub 1 analysis and design techniques presented

l2 to l gains for sampled data systems four kinds of 2 to gains for sampled data systems are defined and derived in this paper it turns out that these gains can be expressed by analytical formulae which are easy to compute only a discrete lyapunov equation is involved

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

the h 2 problem for sampled data systems m for sampled data systems in this paper we pose and give a complete solution to an analog of the h2 problem for sampled data system we motivate and develop a natural generalization of the h2 cost to periodic systems which is then applied to the continuous time closed loop mapping in a sampled data control system it is shown that the cost criterion developed is actually a norm in an space of hilbert schmidt operator valued functions we give state space solutions to the optimal and suboptimal controllers synthesis problems in this new norm by establishing an equivalent standard h2 problem

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

h sub 2 optimal sampled data control the h sub 2 optimal control of continuous time linear time invariant systems by sampled data controllers is discussed two different solutions state space and operator theoretic are given in both cases the h sub 2 sampled data problem is shown to be equivalent to a certain discrete time h sub 2 problem other topics discussed include input output stability of sampled data systems performance recovery in digital implementation of analog controllers and sampled data control of systems with the possibility of multiple time delays