feedback control for linear time invariant systems with state and control bounds in the presence of disturbances the problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered the only information required about the disturbances is a finite convex polyhedral bound discrete and continuous time systems are considered the property of positive d invariance of a region is introduced and it is proved that a solution of the problem is achieved by the selection of a polyhedral set s and the computation of a feedback matrix k such that s is positively d invariant for the closed loop system it is shown that if polyhedral sets are considered the solution involves simple linear programming algorithms however the procedure suggested requires a great amount of computational work offline if the state space dimension is large because the feedback matrix k is obtained as a solution of a large set of linear inequalities all of the vertices of s are required

rational spl lscr sup 1 suboptimal compensators for continuous time systems the persistent disturbance rejection problem spl lscr sup 1 optimal control for continuous time systems leads to nonrational compensators even for single input single output systems as noted in dahleh and pearson 1987 the difficulty of physically implementing these controllers suggests that the most significant application of the continuous time spl lscr sup 1 theory is to furnish achievable performance bounds for rational controllers in this paper the authors use the theory of positively invariant sets to provide a design procedure based upon the use of the discrete euler approximating system for suboptimal rational spl lscr sup 1 controllers with a guaranteed cost the main results of the paper show that i the spl lscr sup 1 norm of a continuous time system is bounded above by the l sup 1 norm of an auxiliary discrete time system obtained by using the transformation z 1 rs and ii the proposed rational compensators yield spl lscr sup 1 cost arbitrarily close to the optimum even in cases where the design procedure proposed in the above mentioned paper fails due to the existence of plant zeros on the stability boundary

minimization of a closed loop response to a fixed input for siso systems it is shown that the problem of minimizing a regulated response of a single input single output system due to a fixed bounded input can be converted via polynomial techniques to a linear infinite dimensional chebyshev data fitting problem approximating feasible solutions within any specified degree of accuracy can be obtained by converting the original problem into a sequence of increasingly large finite dimensional chebyshev approximation problems for which solution stable and efficient numerical methods exist a direct formula for calculating tight upper bounds to the approximation error is provided the link between the present algebraic approach and the dahleh and pearson functional analytic one 1988 is also discussed

minimization of a regulated response to a fixed input the problem of minimizing a regulated output due to a specific bounded input is solved in the siso single input single output discrete time case it is shown that in most situations the optimal solution will not exist but given any epsilon 0 it is possible to find a suboptimal solution with performance within epsilon of the optimal performance the proposed approach is extended to the problem of shaping the error transient response it is shown how time domain templates can be used in this problem in the same way that frequency weighting functions are used in the h sup infinity and h sup 2 problems

minimization of the maximum peak to peak gain the general multiblock problem a comprehensive study of the general l sub 1 optimal multiblock problem and a new linear programming algorithm for computing suboptimal controllers are presented by formulating the interpolation conditions in a concise and natural way the general theory is developed in simpler terms and with a minimum number of assumptions in addition further insight is gained into the structure of the optimal solution and different classes of multiblock problems are distinguished this leads to a conceptually attractive iterative method for finding approximate solutions

on the stabilization of nonlinear systems we extend the applicability of the global q parametrization method of controller design to a large class of unstable nonlinear plants the main result is a two step compensation theorem analogous to that of zames for unstable linear plants if p l e2 rightarrow l e1 is a nonlinear possibly unstable plant and f 0 is any incrementally stable controller such that p 1 p i f 0 p 1 is incrementally stable then the class of controllers f which yields an f g stable closed loop system in the unity feedback configuration for p is globally parametrized by finite gain stable maps q l e1 rightarrow l e2 with f f 0 q i p 1 q 1

a heuristic approach to solving a class of bilinear matrix inequality problems abstract many canonical and modern control problems can be recast into the problem of a group of matrix inequalities some of them are in the form of linear matrix inequalities lmis which can be solved very efficiently by the powerful lmi toolbox in matlab but some others are in the form of bilinear matrix inequalities the characteristic of this latter class of problems is that when the so called communicating variables are fixed the overall problem will be reduced to the problem in lmis thus how to find the communicating variables is the key to solve the whole problem in this paper an optimal estimate for the communicating variables is presented we will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems numerical examples are provided to show the effectiveness of the proposed method

design of spl lscr sub 1 optimal controllers with flexible disturbance rejection level this note presents a new design methodology that allows for flexible management of the tradeoff between the ability of a system to attenuate disturbance signals versus its expected worst peak to peak amplification the proposed strategy applies to linear time invariant systems which are subject to persistent disturbance signals and combines a recently developed quasi robust linear programming concept with a well known spl lscr sub 1 optimal controller synthesis approach the benefit of the resulting technique is demonstrated using an example

hierarchies of nonlinear integrable q difference equations from series of lax pairs we present two hierarchies of nonlinear integrable q difference equations one of which includes a q difference form of each of the second and fifth painleve equations and the other includes all the equations have multiple free parameters a method to calculate a 2 2 lax pair for each equation in the hierarchy is also given

optimal pole placement design for siso discrete time systems in this paper we incorporate the placement of one real closed loop pole into a compensator design framework based upon the youla parameterization duality theory and linear programming this framework has been used to design discrete time compensators to solve the l sub 1 controller design problem as well as other related time domain optimization problems previous work on these problems has focused on deadbeat systems it is known that these can require high order controllers part of the motivation for this work is to improve the tradeoff between controller order and performance over that using deadbeat control

perturbation induced radiative losses in collision of nse solitons the total energy and its spectral density emitted during the collision of two solitons described by the conservatively perturbed non linear schrodinger equation nse is calculated by means of perturbation theory based on the inverse scattering transform it is shown that the total emitted energy at large relative velocity does not depend on the relative internal phase of the colliding solitons the same results are obtained for solitons in a system of two weakly coupled non linear schrodinger equations

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

block wise discretization accounting for structural constraints this paper presents an approximate discretization method named mixed euler zoh me zoh which can be applied to any continuous time linear system this method has been explicitly developed to preserve the system sparsity a property that is particularly important when dealing with the analysis and design of distributed controllers for large scale systems in terms of stability preservation as a function of the sampling interval we show that me zoh outperforms the classical forward euler fe approach which is the only known discretization method guaranteeing the preservation of sparsity for all possible sampling times it is then shown that this new discretization method is capable of preserving stability for all sampling times for a wide classes of dynamical systems including the important class of positive systems besides stability also positivity of the resulting discrete time system is preserved contrarily to what happens for the fe approach a couple of examples are reported to illustrate the main theoretical results of the paper

dual isomonodromic deformations and moment maps to loop algebras the hamiltonian structure of the monodromy preserving deformation equations of jimboet al jmms is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras the nonautonomous hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps this construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations as illustrative examples involving discrete and continuous reductions a higher rank generalization of the hamiltonian equations governing the correlation functions for an impenetrable bose gas is obtained as well as dual pairs of isomonodromy representations for the equations of the painleve transcendentspv andvi

suboptimal receding horizon control for continuous time systems solving a continuous time optimal control problem under state and control constraints is known to be a very hard task in this note we propose a suboptimal solution based on the euler auxiliary system eas we show that we can determine a continuous time stabilizing control whose cost not only converges to the optimal as the eas time parameter vanishes but it is also upper bounded by the discrete time cost no matter how such a parameter is chosen in particular continuous time linear problems with convex cost can be solved by considering a fictitious receding horizon scheme both stability and constraints satisfaction are guaranteed for the continuous time system this scheme turns out to be very useful when due to unstable or poorly damped dynamics the digital implementation of the control requires a very small virtually zero sampling time since the time parameter of the eas can be much greater than the sampling time without compromising stability with a strongly reduced computational burden

effect of quintic nonlinearity on soliton collisions in optical fibers abstract we investigate the effect of quintic nonlinearity on collisions between solitons in different frequency channels in optical fibers we calculate the intensity and dynamics of the collision induced emitted radiation and changes in the soliton parameters due to the collision this is done by means of a perturbation theory with two small parameters the quintic nonlinearity coefficient q and the reciprocal of the inter channel frequency difference 1 we find that the amplitude of the collision induced emitted radiation is proportional to q this radiation can be described as originating from a variation in the kerr nonlinearity coefficient with respect to distance along the fiber we also find that the only other effect of quintic nonlinearity on the collision up to second order in the perturbation theory is an additional phase shift proportional to q the results obtained by our numerical simulations of the perturbed nonlinear schrodinger equation are in very good agreement with theory

correspondence a comment on l optimal control of siso continuous time systems the paper l optimal control of siso continuous time systems by wang and sznaier wang sznaier 1997 automatica 33 1 85 90 studies the problem of designing a controller that optimally minimizes the peak absolute value of system output due to a fixed input signal with a newly defined function space a it was claimed that the set of all l bounded outputs could be parameterized and that the problem could be transformed to a minimal distance problem on l space we believe however their formulation has essential flaws

quantum deformation of kdv hierarchies and their exact solutions q deformed solitons a complete procedure is given by which one can construct q deformed kdv hierarchies and obtain their exact solutions with arbitrarily many q deformed solitons as an elementary example we obtain the exact solution of the q deformed single soliton of the simplest 3 2 system i e the q deformed kdv equations and investigate this solution in the case of small deformation it is interesting that this method can supply many exact solutions for systems of new conventional nonlinear partial differential equations

on finding the minimum possible peak value of multi input infinite horizon steering controls a computational method for the infinite dimensional optimization problem of finding the minimum possible peak value of a multi input infinite horizon steering control is presented it is shown that the problem is equivalent to a convex program in a finite dimensional euclidean space the convex program is approximated by a sequence of linear programs the objective values of which converge to the objective of the convex program a bound above and below are given for the error between the objective value of the convex program and the objective of any linear program in the approximating sequence the bounds show convergence and are geometric with ratio given by the minimum modulus over all unstable plant poles

dual periodicity in l 1 norm minimisation problems the topic of this paper is the discrete time l1 norm minimisation problem with convolution constraints we find primal initial conditions for which the dual optimal solution is periodic periodicity of the dual optimal solution implies satisfaction of a simple linear recurrence relation by the primal optimal solution

minimum overshoot design for siso discrete time systems the problem of designing a controller to minimize the maximum negative error due to a specific bounded input is studied it is found that the solution can be obtained by solving an infinite linear programming problem in the special case of a step input a simple analytical formula for minimum overshoot is derived for a plant with one unstable pole and one nonminimum phase zero the solution technique has potential application to other control problems which seek to minimize quantities not directly representable as a norm

analytic expressions of transfer function responses and choice of numerator coefficients zeros general analytic expressions of transfer function responses are derived in this paper the analytic forms include the numerator coefficients of the transfer function a vandermonde like matrix and a vector containing the transfer function eigenresponses the latter two only depending on the eigenvalues several characteristics of basic transient responses are derived e g their exact number of zeros an upper bound on the number of extrema of the step response is obtained an algorithm is suggested for the selection of the numerator coefficients assuming fixed eigenvalues using the derived characteristics of the basic transient responses effectively leading to a suboptimal choice furthermore the analytic formulas are used to calculate the numerator coefficients optimally by minimizing the step transient the ramp transient the parabolic transient etc simulations are presented indicating the properties of the basic transient responses and the transfer function response properties based on optimal and suboptimal choices of the numerator coefficients

on nonlinear state feedback controllers for persistent disturbance rejection abstract an l 1 optimal linear time invariant lti compensator may have an order significantly higher than that of the plant in the state feedback case there has been recent work exploring the use of nonlinear static feedback controllers which provide near optimal performance in the case of d 12 square and invertible we show that it is enough to have the nonlinearity on a particular substate which can be used to significantly simplify the design problem

technical communique optimization based design of fixed order controllers for command following for discrete time scalar systems we propose an approach for designing feedback controllers of fixed order to minimize an upper bound on the peak magnitude of the tracking error to a given command input the work makes use of linear programming to design over a class of closed loop systems recently proposed for the rejection of non zero initial conditions and bounded disturbances we incorporate performance robustness in the form of a guaranteed upper bound on the peak magnitude of the tracking error under plant coprime factor uncertainty

q painleve vi equation arising from q uc hierarchy we study the q difference analogue of the sixth painleve equation q pvi by means of tau functions associated with the affine weyl group of type d5 we prove that a solution of q pvi coincides with a self similar solution of the q uc hierarchy as a consequence we obtain in particular algebraic solutions of q pvi in terms of the universal character which is a generalization of the schur polynomial attached to a pair of partitions

some results on minimum magnitude regulated response analytical results are obtained for the minimum peak tracking error magnitude achievable by some finite settling time control systems in response to a step reference input the limits of these results as the settling time approaches infinity are also obtained these represent performance bounds which apply for any finite order linear time invariant lti controller of a given configuration i e one parameter or two parameter the systems considered are one parameter discrete time single input single output siso where the plant has one unstable pole and one non minimum phase zero the result for a two parameter compensator for plants with one non minimum phase zero is presented

rational suboptimal continuous time controller design we obtain rational suboptimal continuous time solutions to some optimal control problems specified via time domain performance criteria these include spl lscr sub 1 and spl lscr sub spl infin norms and peak overshoot the approach uses linear semi infinite programming to compute weights for a given finite set of rational basis functions and makes the best possible use of the basis the closed loop transfer functions obtained satisfy appropriate interpolation constraints for internal stability and the formulation allows additional linear constraints to be incorporated

on an integrable system of q difference equations satisfied by the universal characters its lax formalism and an application to q painleve equations the universal character is a generalization of the schur function attached to a pair of partitions we study an integrable system of q difference equations satisfied by the universal characters which is an extension of the q kp hierarchy and is called the lattice q uc hierarchy we describe the lattice q uc hierarchy as a compatibility condition of its associated linear system lax formalism and explore an application to the q painleve equations via similarity reduction in particular a higher order analogue of the q painleve vi equation is presented

optimal and robust controllers for periodic and multirate systems the problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete time periodic systems the solution consists of solving an equivalent time invariant standard l sup 1 optimization problem subject to an additional constraint this constraint assures the causality of the resulting periodic controller by the duality theory the problem is shown to be equivalent to a linear programming problem which is no harder than the standard l sup 1 problem also it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems finally the results are applied to the problem of robust stabilization of periodic and multirate systems

a construction of rational wavelets and frames in hardy sobolev spaces with applications to system modeling using the daubechies wavelet theory we establish rational wavelet decompositions of the hardy sobolev classes on the half plane the decay of wavelet coefficients is analyzed and error bounds for approximation are given we give applications to the modeling of linear systems and to the model reduction of infinite dimensional systems

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

minimization of the worst case peak to peak gain via dynamic programming state feedback case considers the problem of designing a controller that minimizes the worst case peak to peak gain of a closed loop system in particular we concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude we apply the principle of optimality and derive a dynamic programming formulation of the optimization problem under mild assumptions we show that at each step of the dynamic program the cost to go has the form of a gauge function and can be recursively determined through simple transformations we study both the finite horizon and the infinite horizon case under different information structures the proposed approach allows us to encompass and improve earlier results based on viability theory in particular we present a computational scheme alternative to the standard bisection algorithm or gamma iteration that allows us to compute the exact value of the worst case peak to peak gain for any finite horizon we show that the sequence of finite horizon optimal costs converges as the length of the horizon goes to infinity to the infinite horizon optimal cost the sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem we also show the existence of an optimal state feedback strategy that is globally exponentially stabilizing and derive suboptimal globally exponentially stabilizing strategies from the solutions of finite horizon problems

nonovershooting and monotone nondecreasing step responses of a third order siso linear system this paper presents the necessary and sufficient conditions for a third order single input single output siso linear system to have a nonovershooting or monotone nondecreasing step response if the transfer function of an overall system has real poles a necessary and sufficient condition is found for the nonovershooting or monotone nondecreasing step response in the case of complex poles one sufficient condition and two necessary conditions are obtained the resulting conditions are all in terms of the coefficients of the numerator of the transfer function simple calculations can be used to check a system for the nonovershooting or monotone nondecreasing step response another feature is that the conditions in terms of pole zero configurations can be easily derived from the present results

rejection of persistent spl lscr sub spl infin bounded disturbances for nonlinear systems a nonlinear spl lscr sup 1 control problem which deals with the rejection of persistent bounded disturbances for nonlinear systems is investigated the concept of invariance plays an important role in the spl lscr sup 1 performance analysis and synthesis the main idea is to construct an invariant subset in the state space such that achieving disturbance rejection is equivalent to restricting the state dynamics to the subset the relation between the spl lscr sup 1 control of a continuous time system and the l sup 1 control of its euler approximated discrete time system is established

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

mimo l 1 optimal control problems via the polynomial equations approach the general mimo multi block l 1 optimal control problem is considered a novel solution is derived by resorting to polynomial techniques and simple algebraic conditions on the free parameter of the yjbk parameterization as a result we arrive at unconstrained lp formulations less affected by redundancy and solvable by standard lp solvers as usual because the optimal controller is in general infinite dimensional a solution scheme based on solving sequences of increasing larger finite dimensional sub super optimal optimization problem is proposed viz sequences of finite dimensional optimization problems whose solutions provide lower and respectively upper bounds to the optimum monotonically converging to it finally an example is presented in order to exemplify the theory and show the effectiveness of the method

fundamental limitations on the time domain shaping of response to a fixed input in this paper we examine time domain limitation of performance problems in feedback control system design the main result is a theorem which gives a dual formulation to the problem of determining absolute limits on the time domain shaping of the response to a fixed input the duals that arise are particularly illuminating they have a simple interpretation as optimizing the impulse response of a transfer function whose poles are readily constructed from the data much insight is obtainable from the dual for example in helping to classify classes of plants with the same quantitative behavior new results on classical performance measures such as overshoot and undershoot are presented we also examine minimization of the difference between the maximum and minimum values of the error response a quantity we term fluctuation

a reduction of the resonant three wave interaction to the generic sixth painleve equation among the reductions of the resonant three wave interaction system to six dimensional differential systems one of them has been specifically mentioned as being linked to the generic sixth painleve equation p6 we derive this link explicitly and we establish the connection to a three degree of freedom hamiltonian previously considered for p6

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

further results on rational approximations of spl lscr sup 1 optimal controllers the continuous time persistent disturbance rejection problem spl lscr sup 1 optimal control leads to nonrational compensators even for siso systems the difficulty of physically implementing these controllers suggests that the most significant applications of the continuous time spl lscr sup 1 theory is to furnish bounds for the achievable performance of the plant previously two different rational approximations of the optimal spl lscr sup 1 controller were developed by ohta et al 1992 and by blanchini and sznaier 1994 in this paper the authors explore the connections between these two approximations the main result of the paper shows that both approximations belong to the same subset spl omega sub t of the set of rational approximations and that the method proposed by blanchini and sznaier gives the best approximation in the sense of providing the tightest upper bound of the approximation error among the elements of this subset additionally the authors exploit the structure of the dual to the spl lscr sup 1 optimal control problem to obtain rational approximations with approximation error smaller than a prespecified bound

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

controller design for plants with structured uncertainty this paper addresses the problem of designing feedback controllers to achieve good performance in the presence of structured plant uncertainty and bounded but unknown disturbances a general formulation for the performance robustness problem is presented and exact computable conditions are furnished these conditions are then utilized for synthesizing robust controllers which involves solving l1 optimization problems these solutions are computed using the duality theory of lagrange multipliers approximations and computational issues are discussed

simultaneous performance achievement via compensator blending in this paper we consider the problem of designing a state feedback controller that simultaneously achieves different optimality criteria defined on different input output pairs precisely if r optimal target transfer functions are given as the result of local optimal controllers it is shown that under mild assumptions there exists a unique controller capable of replicating these transfer functions in the closed loop system so simultaneously achieving the performances inherited by the chosen local transfer functions an explicit and constructive procedure we refer to such procedure as compensator blending is provided the possibility of designing a stable blending compensator or the generalization to dynamic local controllers or time varying systems are also discussed we finally consider the dual version of the problem precisely we show how to achieve simultaneous optimality by filter blending

approximation and identification of continuous time systems an approximation approach to the identification of stable continuous time possibly infinite dimensional systems is studied for this purpose concrete exponential leg endre series methods are proposed and analysed here for the l1 and l2 approximations of continuous time systems rate of approximation results are given for the proposed methods for an important class of delay systems the proposed exponential legendre series method for the l2 approximation has a certain optimal rate of approximation property for this class of delay systems these results show that exponential legendre series techniques have promising properties in connection with the l1 and l2 identifications of stable continuous time systems the proposed l1 identification method has a direct application to the robust design of control systems

control system design via infinite linear programming in this paper we deal with the problem of designing non overshooting feedback control systems when the input is a step we also consider the more general problem of designing controllers to track a step optimally with some predetermined amount of allowable overshoot these problems are cast as infinite linear programming problems we show that both these problems always have a solution under some standard assumptions it is then shown that the counterpart of this problem that is the problem of designing non undershooting feedback control systems need not have a solution in general but it is proved that one can always design a feedback controller to achieve as little percentage undershoot as desired finally it is shown that the problem of designing controllers to simultaneously satisfy bounds on the overshoot and undershoot always has a solution if some undershoot is allowed for the duals to the infinite linear programming problems are formulated and the absense of a duality gap is established fo

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

i 1 optimal control of multivariable systems with output norm constraints in this paper we consider the l1 optimal control problem for general rational plants it is shown that for plants with no poles or zeros on the unit circle an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs exactly or approximately optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed loop transfer function no assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators it is shown that constrained problems in which a set of outputs is regulated subject to l norm constraints on another set of outputs can be solved using a slight modification of the same algorithm

rational l sub spl infin suboptimal controllers for siso continuous time systems we study the problem of minimizing the weighted amplitude of the time response due to a given fixed input signal for single input single output siso continuous time systems and focus on obtaining rational suboptimal solutions an eas euler approximating system based method is proposed for designing a rational l sub spl infin suboptimal controller for siso systems it is shown that this rational approximation is the best one among a set of rational approximations in the sense of providing the tightest upper bound and that it can approximate the optimal cost arbitrarily close

mixed l sub 1 h sub spl infin control of mimo systems via convex optimization we present a methodology for designing mixed l sub 1 h sub spl infin controllers for mimo systems these controllers allow for minimizing the worst case peak output due to persistent disturbances while at the same time satisfying an h sub spl infin norm constraint upon a given closed loop transfer function therefore they are of particular interest for applications dealing with multiple performance specifications given in terms of the worst case peak values both in the time and frequency domains the main results of the paper show that 1 contrary to the h sub 2 h sub spl infin case the l sub 1 h sub spl infin problem admits a solution in l sub 1 and 2 rational suboptimal controllers can be obtained by solving a sequence of problems each one consisting of a finite dimensional convex optimization and a four block h sub spl infin problem moreover this sequence of controllers converges in the l sub 1 topology to an optimum

relatively optimal control and its linear implementation motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality roughly optimality for a specific nominal plant initial condition we consider a generic discrete time finite horizon constrained optimal control problem for linear systems and we seek for a state feedback possibly dynamic controller as a fundamental requirement we do not admit preactions or controller state initialization based on the plant initial state and we assume our controller to be time invariant in particular we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory a relatively optimal control is a stabilizing controller such that if initialized at its zero state produces the optimal constrained trajectory for the nominal initial condition of the plant we show that one of such controllers is linear dead beat and its order is equal to the length of the horizon minus the plant order thus of complexity which is known a priori some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed we show that by means of the proposed approach we can face several problems such as optimal point to point operations optimal impulse response and optimal tracking

persistent disturbance rejection via static state feedback in contrast with spl hscr sub spl infin and spl hscr sub 2 control theories the problem of persistent disturbance rejection l sup 1 optimal control leads to dynamic controllers even when the states of the plant are available for feedback using viability theory shamma showed 1993 in a nonconstructive way that in the state feedback case the same performance achieved by any dynamic linear time invariant controller can be achieved using memoryless nonlinear state feedback in this paper we give an alternative constructive proof of these results for discrete and continuous time systems the main result of the paper shows that in both cases the l sup 1 norm achieved by any stabilizing state feedback linear dynamic controller can be also achieved using a memoryless variable structure controller

controller design with multiple objectives in this paper we study multi objective control problems that give rise to equivalent convex optimization problems we develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone we present some specialized results on duality theory and we apply them to the study of three multi objective control problems the optimal l sub 1 control with time domain constraints on the response to some fixed input the mixed h sub 2 l sub 1 control problem and the l sub 1 control with magnitude constraint on the frequency response what makes these problems complicated is that they are often equivalent to infinite dimensional optimization problems the characterization of the duality relationship between the primal and dual problem allows us to derive several results these results establish connections with special convex problems linear programming or linear matrix inequality problems uncover finite dimensional structures in the optimal solution when possible and provide finite dimensional approximations to any degree of accuracy when the problem does not appear to have a finite dimensional structure to illustrate the theory and highlight its potential several numerical examples are presented

rational approximation of l sub 1 optimal controllers for siso systems the l sub 1 optimal control problem with rational controllers for continuous time systems is considered in which it is shown that the optimal l sub 1 performance index with rational controllers is equal to that of irrational controllers a sequence of rational controllers that approximates the optimal index is constructed convergence properties of such a sequence are studied that the corresponding sequence of objective transfer functions is shown to converge in weak topology in bv r sub in the time domain and uniformly in a wider sense in the frequency domain

l optimal control of siso continuous time systems abstract in this paper we study the problem of designing a controller that minimizes the weighted amplitude of the time response due to a given fixed input signal for siso continuous time systems the main result of the paper shows that this problem admits a minimizing solution in l and that the optimal closed loop system has a special structure a sum of delayed step functions all having the same amplitude thus the optimal controller has a non rational transfer function although in the general case finding this controller entails solving an infinite dimensional linear programming problem we show that in some special cases the optimal solutions have closed form expressions and can be found by solving a set of algebraic equations finally we address the issue of selecting time and frequency domain weighting functions this paper together with our paper rational l suboptimal controller for siso continuous time systems ieee trans autom control ac 41 1358 1363 1996 which deals with the design of rational l suboptimal controllers for general systems give a complete solution of the l control problem

elements of q harmonic analysis abstract measures that act as q analogs of lebesgue measure are introduced some q analogs of fourier and laplace transforms are studied last the connection between certain spaces of q exponential type and discrete q fourier transforms is discussed

a technique for choosing zero locations for minimal overshoot an approach is presented to the specification of optimal overshoot controllers when the controller order is fixed and the closed loop system poles are at a fixed location the essential effect of the technique is to pick the best possible locations of the free zeros of the system in terms of minimizing the overshoot the solution is obtained by solving an affine minimax optimization problem an example illustrates how this technique can be used as a design aid when it is necessary to trade off conflicting design requirements such as overshoot and settling time specifications it is shown that the method is not restricted to the deadbeat case but can also be used for any prescribed set of closed loop poles

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

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

further results on the optimal rejection of persistent bounded disturbances the problem of designing controllers that optimally reject persistent disturbances is studied the focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary i e the unit circle in the discrete time case and the extended j omega axis in the continuous time case for the discrete time case the problem of minimizing a cost functional of the form f rg sub 1 where the transform g of g has some unit circle zeros is studied a previously published dual problem formulation is extended and it is shown that an optimal controller need not exist the construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied it is shown that two results which hold in the case of h sub infinity optimization do not hold in the presented situation specifically the introduction of unit circle zeros can increase the value of the infimum even when every unit circle zero of g is also a zero of f and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence similar results are obtained for the continuous time case

optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization the problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case the minimum value of the objective function can be obtained by solving a semi infinite linear programming problem and an iterative procedure for obtaining approximate solutions is introduced application of the l sup 1 optimal problem to robustness is discussed a mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations

model reference adaptive control for nonminimum phase systems in this paper we define a new model reference adaptive control problem which allows a large amount of plant uncertainty instead of the usual goal of asymptotic error regulation here we require that the limit superior of the magnitude of the error be less than a predefined constant times the limit superior of the magnitude of the external reference input this provides a natural way to incorporate uncertainty which includes nonminimum phase systems

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

a q analogue of hierarchy and q painleve vi a q analogue of the drinfel d sokolov hierarchy is proposed as a reduction of the q kp hierarchy applying a similarity reduction and a q laplace transformation to the hierarchy one can obtain the q painleve vi equation proposed by jimbo and sakai

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

bezout factors and l sup 1 optimal controllers for delay systems using a two parameter compensator scheme the authors consider the simultaneous problem of optimal robust stabilization and optimal tracking for single input single output siso systems in an l sup spl infin setting using a two parameter compensator scheme optimal robustness is linked to the work done by georgiou and smith 1992 in the l sup 2 setting optimal tracking involves the resolution of l sup 1 optimization problems the authors consider in particular the robust control of delay systems they determine explicit expressions of the bezout factors for general delay systems which are in the callier desoer class spl bscr spl circ 0 finally they solve several general l sup 1 optimization problems and give an algorithm to solve the optimal robust control problem for a large class of delay systems

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

on optimal e to e filtering abstract in this paper we consider the problem of finding a filter that minimizes the worst case magnitude l of the estimation error in the case of linear time invariant systems subjected to unknown but magnitude bounded l inputs these inputs consist of process and observation noise as well as initial conditions also the optimization problem is considered over an infinite time horizon taking a model matching approach suboptimal solutions are presented that stem from the resulting l induced norm minimization problem

input output stability of sampled data systems the authors study the l sub p input output stability of a continuous time controller using the usual arrangement of periodic sampling and zero order hold it is noted that even if the hybrid system is exponentially stable this arrangement does not yield l sub p 1

sign weighted peak minimization problem for feedback systems this note studies the problem of minimizing a parameterized convex combination of the overshoot and undershoot of a single input single output siso discrete time system to a fixed input we provide a simple condition for the solution existence and characterize the structure of optimal solutions it is shown that a known l sub spl infin suboptimal scheme also works in our problem in addition we make a comparative analysis with the l sub spl infin problem

h sub 2 control with time domain constraints theory and an application in this paper we study the problem of minimizing the h sub 2 norm of a given transfer function subject to time domain constraints on the time response of a different transfer function to a given test signal the main result of this paper shows that this problem admits a minimizing solution in r h sub 2 moreover rational solutions with performance arbitrarily close to optimal can be found by constructing families of approximating problems each one of these problems entails solving a finite dimensional quadratic programming problem whose dimension can be determined before hand these results are illustrated and experimentally validated by designing a controller for an active vision application

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm

l 1 optimal compensators for continuous time systems previous work has been concerned with minimizing the l 1 norm of an error pulse response for discrete time siso 1 and mimo 2 systems in this paper we study the problem of minimizing the l1 norm of the error impulse response for siso continuous time systems this problem is quite different from the discrete time problem in that irrational solutions are obtained even when the problem data are rational two methods are suggested for the solution of the continuous time problem an exact method which leads to a finite dimensional nonlinear programming problem and an approximate method which leads to a linear programming problem

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations

diverging probability density functions for flat top solitary waves we investigate the statistics of flat top solitary wave parameters in the presence of weak multiplicative dissipative disorder we consider first propagation of solitary waves of the cubic quintic nonlinear schrodinger equation cqnlse in the presence of disorder in the cubic nonlinear gain we show by a perturbative analytic calculation and by monte carlo simulations that the probability density function pdf of the amplitude eta exhibits loglognormal divergence near the maximum possible amplitude eta m a behavior that is similar to the one observed earlier for disorder in the linear gain a peleg phys rev e 72 027203 2005 we relate the loglognormal divergence of the amplitude pdf to the superexponential approach of eta to eta m in the corresponding deterministic model with linear nonlinear gain furthermore for solitary waves of the derivative cqnlse with weak disorder in the linear gain both the amplitude and the group velocity beta become random we therefore study analytically and by monte carlo simulations the pdf of the parameter p where p eta 1 epsilon s beta 2 and epsilon s is the self steepening coefficient our analytic calculations and numerical simulations show that the pdf of p is loglognormally divergent near the maximum p value