discrete time nonlinear h control with measurement feedback abstract this paper deals with the problem of h control via measurement feedback with internal stability for discrete time nonlinear systems using the theory of discrete time dissipative systems and differential game we first present a parameterization of a family of static state feedback h nonlinear controllers then we provide sufficient conditions for the existence of a dynamic output compensator which solves the problem of disturbance attenuation with internal stability for discrete time nonlinear systems the theory developed in this paper incorporates the well known results for discrete time linear systems and generalizes some recent advances in h control of continuous time nonlinear systems to their discrete time counterparts

a family of nonlinear h sup spl infin output feedback controllers state space formulas are derived for a family of controllers solving the nonlinear h sup spl infin output feedback control problem the formulas given are expressed in terms of the solutions to two hamilton jacobi inequalities in n independent variables these controllers are obtained by interconnecting the central controller with an asymptotically stable free system having l sup 2 gain spl les spl gamma all proofs given are simple and clear and provide deeper insight in the synthesis of the corresponding linear h sup spl infin controllers

which logic is the real fuzzy logic this paper is a contribution to the discussion of the problem whether there is a fuzzy logic that can be considered as the real fuzzy logic we give reasons for taking imtl bl email protected and ev l fuzzy logic with evaluated syntax as those fuzzy logics that should be indeed taken as the real fuzzy logics

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

what is mathematical fuzzy logic the paper comments on the development and present state of fuzzy logic as a kind branch of mathematical logic it is meant just as a contribution to the discussion on what fuzzy logic is not as a systematic presentation of mathematical fuzzy logic

keeping options open an optimal control model with trajectories that reach a dnss point in positive time the so called dnss points of indifference are of interest because they give decision makers in optimal control problems a choice between following either of two or more trajectories while still achieving optimality usually they are described in terms of initial conditions so that if the system starts at a dnss point the decision maker can proceed in either of two or more directions here we present a model that has an entire curve of indifference points away from which the decision maker can move in only one direction but does so by choosing either of two trajectories that initially coincide in the state space but later diverge approaching different long run steady states

generalized riccati equations for the full and reduced order mixed norm h 2 h abstract this paper considers the mixed norm h 2 h standard problem specifically an lqg control design problem involving a constraint on h disturbance attenuation is addressed it is shown that the h 2 h dynamic compensator gains are completely characterized via coupled riccati lyapunov equations the principal result involves sufficient conditions for characterizing full and reduced order controlleers that satisfy bounds on both h 2 and h performance costs as a special case of this unified result we obtain the full order h solution to the standard control problem the pure reduced order h solution with no h 2 contribution further extensions include nonstrictly proper dynamics a direct transmission term from disturbances to h performance variables cross weighting and sensor noise plant disturbance correlation and a treatment of the pure reduced order h control problem

h sup infinity control for nonlinear systems with output feedback the basic question of nonlinear h sup infinity control theory is to decide for a given two port system when feedback that makes the full system dissipative and internally stable exists this problem can also be viewed as a question about circuits and after translation also has a game theoretic statement several necessary conditions for solutions to exist are presented and sufficient conditions for a certain construction to lead to a solution are given

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

brief paper disturbance attenuation and trajectory tracking via a reduced order output feedback controller for robot manipulators this paper presents a solution to the problem of trajectory tracking with external disturbance attenuation in robotics systems via a reduced order output feedback controller without velocity measurement the proposed control law ensures both semi global asymptotic stability of the closed loop system and external disturbance attenuation the approach is based on the notion of the l 2 gain and requires to solve two algebraic riccati inequalities the proposed controller design is illustrated by a simulation example

spl hscr sub spl infin control of nonlinear systems a convex characterization the nonlinear spl hscr sub spl infin control problem is considered with an emphasis on developing machinery with promising computational properties the solutions to spl hscr sub spl infin control problems for a class of nonlinear systems are characterized in terms of nonlinear matrix inequalities which result in convex problems the computational implications for the characterization are discussed

a j spectral factorization approach to control necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with mathcal h infty control j control are derived using j spectral factorization theory the existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled j spectral factorization problems with the second factor providing a parametrization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of nonnegative definite stabilizing solutions to two indefinite algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given a virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework and no additional auxiliary riccati equations are required

brief paper a controller design based on a functional h filter for descriptor systems the time and frequency domain cases in this paper we investigate the time and frequency domain designs of a controller based on a functional filter for both continuous time and discrete time multivariable descriptor systems the order of this controller is equal to the dimension of the state to be estimated and the control gain is designed using h techniques the time procedure is based on the unbiasedness of the estimation error using a sylvester equation then a new method for avoiding the derivative of the disturbance vector in the observation error dynamics is proposed this problem is solved by means of linear matrix inequalities lmis the frequency domain procedure design is derived from time domain results where we propose some suitable matrix fraction descriptions mfds and most importantly the factorization approach permits us to give a polynomial parameterization of the functional h filter based controller

some characterizations of optimal trajectories in control theory several characterizations of optimal trajectories for the classical mayer problem in optimal control are provided for this purpose the regularity of directional derivatives of the value function is studied for instance it is shown that for smooth control systems the value function v is continuously differentiable along an optimal trajectory x t 0 1 to bf r n provided v is differentiable at the initial point t 0 x t 0 then the upper semicontinuity of the optimal feedback map is deduced the problem of optimal design is addressed obtaining sufficient conditions for optimality finally it is shown that the optimal control problem may be reduced to a viability one

eq algebras we introduce a new class of algebras called eq algebras an eq algebra has three basic binary operations meet multiplication and a fuzzy equality and a top element these algebras are intended to become algebras of truth values for a higher order fuzzy logic a fuzzy type theory ftt the motivation stems from the fact that until now the truth values in ftt were assumed to form either an imtl bl or mv algebra all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation multiplication and its residuum the latter is a natural interpretation of implication in fuzzy logic the equivalence is then interpreted by the biresiduum a derived operation the basic connective in ftt however is a fuzzy equality and therefore it is not natural to interpret it by a derived operation this defect is expected to be removed by the class of eq algebras introduced and studied in this paper from the algebraic point of view the class of eq algebras generalizes in a certain sense the class of residuated lattices and so they may become an interesting class of algebraic structures as such

spl hscr sub spl infin control of nonlinear systems via output feedback controller parameterization the standard state space solutions to the spl hscr sub spl infin control problem for linear time invariant systems are generalized to nonlinear time invariant systems a class of local nonlinear output feedback spl hscr sub spl infin controllers are parameterized as nonlinear fractional transformations on contractive stable nonlinear parameters as in the linear case the spl hscr sub spl infin control problem is solved by its reduction to state feedback and output estimation problems together with a separation argument sufficient conditions for spl hscr sub spl infin control problem to be locally solved are also derived with this machinery

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

h sub spl infin compensator design with minimal order observers a method based on a bounded real lemma is presented in the form of sufficient conditions for designing observer based proper reduced order h sub spl infin compensators the proposed compensators are shown to include the strictly proper full order central h sub spl infin compensators as a special class a compensator design algorithm is included to illustrate the use of the proposed compensator design equations

h sub spl infin control of discrete time nonlinear systems this paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback state feedback and dynamic output feedback respectively for discrete time nonlinear systems the h sub spl infin control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality differential game and lasalle s invariance principle in discrete time a substantial difficulty that v a x b x u e x w respectively v f x u w is no longer quadratic in sub w sup u arising in the case of discrete time nonlinear systems has been surmounted in the paper in the case of a linear system we show how the results reduce to the well known ones recently proposed in the literature

parameterization of nonlinear h state feedback controllers state space formulas are derived for a family of controllers solving the nonlinear h state feedback control problem these controllers are obtained by interconnecting the central controller with an asymptotically stable free system that satisfies one additional cascade condition all proofs given are simple and clear and also provide a deeper insight into the synthesis of the corresponding linear h controllers

h sup spl infin control for more general nonlinear systems the authors consider the standard h sup spl infin control problem for more general nonlinear systems modeled by equations in which the penalty output and the measured output are in general functions of the state the exogenous input and the control input in particular we characterize a family of h sup spl infin controllers via output feedback as well as state feedback solving the problem the results obtained generalize some recent results in the literature

conjugate points and shocks in nonlinear optimal control in this paper the authors use the method of characteristics to extend the jacobi conjugate points theory to the bolza problem arising in nonlinear optimal control this yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix riccati differential equation the same approach allows to investigate as well smoothness of the value function

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

technical communique reduced order h controller design an algebraic riccati equation approach an algebraic riccati equation are approach to the reduced order h controller design problem is proposed it is shown that the order of h controller can be reduced to r where r is the rank of the stabilizing solution w to an are in w developed in petersen anderson and jonckheere 1991 international journal of robust nonlinear control 1 171 185 state space formulas for the reduced order h controller design are also given in terms of the stabilizing solutions to the two standard h ares the development uses only elementarily algebraic ideas mainly the bounded real lemma thus the proofs given are simple and clear

observer based compensators for fixed order h control this paper present a two riccati inequality approach for the design of fixed order h suboptimal controllers with an observer based structure the two riccati inequalities are only one way coupled and can be solved consecutively the fixed order controller has the structure of a fixed order estimator of the state feedback control law in the presence of the worst case disturbances it retains the observed based compensator structure of the full order central controller of the standard h design but adds a fixed order dynamic and mappings between the full order and the fixed order state spaces in the full order special case this controller recovers to the minimal entropy central controller of the standard h design

simplifying the h theory via loop shifting matrix pencil and descriptor concepts the 2 riccati h controller formulae and derivations are simplified via various loop shifting transformations that are naturally expressed in terms of a degree one polynomial system matrix psm closely related to the luenberger descriptor form of a system the technique enables one without loss of generality to restrict attention to the simple case in which d 11 0 d 22 0 d t 12 0 l d 21 0 l d l 12 c1 0 and b 1 d t 21 0 matrix fraction descriptions mfds for the algebraic riccati equation solutions afford another change of variables which brings the 2 riccati h controller formulae into a cleaner more symmetric descriptor form having the important practical advantage that it eliminates the numerical difficulties that can occur in cases where one or both of the riccati solutions pi and q blow up and in cases where l qp is nearly singular numerical difficulties previously associated with verifying the existence conditions p 0 q 0 and maxlpar qp 1 are largely eliminated by e

reduced order controllers for continuous and discrete time singular h control problems based on lmi this paper describes a method for constructing a petri net controller for a discrete event system modeled by a petri net the controller consists only of places and arcs and is computed based on the concept of petri net place invariants the size of the controller is proportional to the number of constraints that must be satisfied this method is computationally efficient and can accommodate constraints written as boolean logic formulas in the conjunctive normal form or algebraic inequalities that contain elements of the marking and or the firing vectors

liapunov and covariance controllers the early work of liapunov produced some of the most powerful tools for stability analysis that remain to this day to capture the class of all stabilizing controllers one would be well served by posing the problem in terms of the existence of a liapunov function since a liapunov function is known to exist for stable systems for linear systems this paper derives the set of all quadratic liapunov functions for output feedback control problems and in this way parametrizes the set of all stabilizing controllers of fixed order this is a unifying framework from which all other controllers can be produced by special choices of the free parameters in these controllers we will show how to choose the free parameters to produce all covariance controllers and all h controllers of fixed order these results also apply to robustness analysis and provide a closed form expression for the set of all stabilizing real structured perturbations due to the assignment of a matrix property to the system e g covaria

brief h controller reduction for nonlinear systems sufficient conditions are proposed for the existence of reduced order fixed order controllers solving the standard nonlinear h output feedback control problem state space formulas for such reduced order h controllers are also derived in terms of the solutions of two hamilton jacobi inequalities the development uses only elementary concepts of dissipativity and differential game thus the proofs given are simple and clear

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

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

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

robust h sub spl infin compensator design for time delay systems with norm bounded time varying uncertainties considers robust h sub spl infin compensator design for linear systems that involve unknown time delays both in states and controls and time varying norm bounded parameter uncertainties sufficient conditions for designing robust compensators are obtained in terms of two riccati inequalities which are not mutually coupled through which a full order robust h sub spl infin compensator can be obtained finally a small example is given to show the applicability of the proposed approach

luenberger observer based h sub spl infin compensator design for nonstandard problems this paper presents luenberger observer based h sub spl infin compensator design methods for different nonstandard cases these design methods are developed based on the bounded real lemma and the resultant design algorithms are two riccati inequalities which are not mutually coupled

h sub spl infin control via measurement feedback for general nonlinear systems this paper shows how the problem of local disturbance attenuation via measurement feedback with internal stability can be solved for a nonlinear system of rather general structure the solution of the problem is shown to be related to the existence of solutions of a pair of hamilton jacobi inequalities in n independent variables which are associated with state feedback and respectively output injection design