two dimensional alternating turing machines abstract this paper introduces a two dimensional alternating turing machine 2 atm which can be considered as a natural extension of a one dimensional alternating turing machine to two dimensions this paper also introduces a three way two dimensional alternating turing machine tr2 atm which is an alternating version of a three way two dimensional turing machine we first investigate a relationship between the accepting powers of space bounded 2 atm s or tr2 atm s and ordinary space bounded two dimensional turing machines or three way two dimensional turing machines we then introduce a simple natural new complexity measure for 2 atm s or tr2 atm s called leaf size and provide a spectrum of complexity classes based on leaf size bounded computations we finally investigate recognizability of connected patterns by 2 atm s or tr2 atm s

a note on three way two dimensional alternating turing machines abstract we investigate the properties of three way two dimensional alternating turing machines tr2 atms which are restricted versions of ordinary four way two dimensional alternating turing machines 2 atms we first show that 2 atms are more powerful than tr2 atms with spaces below log m then we investigate the closure properties of tr2 atms on square and general rectangular input tapes finally we show that o log m space bounded tr2 atms cannot accept the set of all square connected pictures

alternating multihead finite automata abstract we define alternating multihead finite automata a generalization of nondeterministic multihead finite automata based on the alternating turing machine model introduced by chandra kozen and stockmeyer 1981 we study the relationships between the classes of languages accepted by alternating multihead finite automata and the classes accepted by deterministic and nondeterministic multihead finite automata and pushdown automata we also examine basic questions about alternating multihead finite automata for example are k 1 heads better than k we conclude by placing upper bounds on the deterministic time and space complexity of the classes of languages accepted by alternating multihead finite automata as corollaries to our results about alternating multihead finite automata we obtain several facts about multihead pushdown automata indicating that the study of alternating multihead finite automata may lead to useful results about nonalternating automata

a framework for robustness analysis of constrained finite receding horizon control a framework for robustness analysis of input constrained finite receding horizon control is presented under the assumption of quadratic upper bounds on the finite horizon costs we derive sufficient conditions for robust stability of the standard discrete time linear quadratic receding horizon control formulation this is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms lending itself to s procedure tools which are used to convert robustness questions to tractable convex conditions robustness with respect to plant model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities simple examples demonstrate the approach

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

global predictive regulation of null controllable input saturated linear systems a predictive regulator is described for stabilizing linear plants subject to polyhedral input constraints while the asymptotic stability of general linear plants is ensured subject to a set membership condition on the initial state global asymptotic and semiglobal exponential regulation is achieved for input saturated plants which are asymptotically null controllable with bounded inputs the design knobs of the regulator can be chosen off line so as to compromise between nonconservative regulation performance and computational complexity and an online optimization problem of prequantifiable numerical burden results

a new approach to stability analysis for constrained finite receding horizon control without end constraints we present a new approach to the stability analysis of finite receding horizon control applied to constrained linear systems by relating the final predicted state to the current state through a bound on the terminal cost it is shown that knowledge of upper and lower bounds for the finite horizon costs is sufficient to determine the stability of a receding horizon controller this analysis is valid for receding horizon schemes with arbitrary positive definite terminal weights and does not rely on the use of stabilizing constraints the result is a computable test for stability and two simple examples are used to illustrate its application

semiglobal stabilization of linear discrete time systems subject to input saturation via linear feedback an are based approach we revisit the problem of semiglobal stabilization of linear discrete time systems subject to input saturation and give an algebraic riccati equation are based approach to the proof of a fact established earlier lin and saberi 1995 i e a linear discrete time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open loop poles are located inside or on the unit circle moreover we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work

properties of generalized predictive control control design using long range prediction based on a dynamic model of the plant has become an important contender for high performance applications the model can be derived analytically but more typically it is provided by simple experiments or in the adaptive context by recursive estimation many methods have been proposed in the literature depending on the assumed model structure and the choice of cost function here the approach called generalized predictive control or gpc is reviewed of special interest is the derivation of prediction equations using an observer polynomial and a demonstration that the selection of particular horizons the costing horizons and the control horizon leads to well understood basic techniques such as dead beat pole placement and lq simulated examples show that gpc is suitable for controlling a complex plant such as unstable inverse unstable systems and how the observer polynomial independently tailors the response to the disturbances preprogrammed set points as with robot trajectory control and actuator nonlinearities can also be catered for a discussion of adaptive applications of gpc to several industrial process concludes that the method is easy to use and effective

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

computational complexity certification for real time mpc with input constraints based on the fast gradient method this paper proposes to use nesterov s fast gradient method for the solution of linear quadratic model predictive control mpc problems with input constraints the main focus is on the method s a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre specified suboptimality is obtained for any possible state of the system we investigate cold and warm starting strategies and provide an easily computable lower iteration bound for cold starting and an asymptotic characterization of the bounds for warm starting moreover we characterize the set of mpc problems for which small iteration bounds and thus short solution times are expected the theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal dual interior point method

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

necessary and sufficient conditions of stability a multiloop generalization of the circle criterion a linear time invariant system with a vector output and a vector input is described this system is closed by an uncertain nonlinear time varying feedback the only information on this feedback is given by several integral quadratic inequalities i e the uncertainties under consideration generalize the so called conic nonlinearities necessary and sufficient frequency domain conditions of stability are obtained an advanced version of the s procedure losslessness theorem and some other tools of the absolute stability theory are used

moving horizon control of linear systems with input saturation and plant uncertainty part 2 disturbance rejection and tracking we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem in part 1 we showed that the proposed moving horizon control system is robustly stable in this paper we show that it is capable of asymptotically suppressing a class of l bounded disturbances that are restricted to a subspace and of asymptotically following a class of reference inputs that are continuously differentiable bounded in magnitude and slope and restricted to a subspace

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

surfaces in three dimensional digital images this is one of a series of reports on the digital geometry of three dimensional images such as those produced by computed tomography in this report we define simple surface points and simple closed surfaces and show that any connected collection of simple surface points form a simple closed surface thus proving a three dimensional analog of the two dimensional jordan curve theorem we also show that the converse is not a theorem in contrast to the two dimensional case and discuss more complex surface types finally we show that the two dimensional analog of our definition of simple closed surface characterizes simple closed curves but that several other characterizations of 2 d curves when extended to 3 d are not adequate to characterize surfaces

a note on three dimensional finite automata abstract this paper investigates the relationships between the accepting powers of three dimensional six way finite automata 3 fa s and three dimensional five way turing machines 5wtm s where the input tapes of these automata are restricted to cubic ones a 3 fa 5wtm can be considered as a natural extension of the two dimensional four way finite automaton two dimensional three way turing machine to three dimensions the main results are 1 n2log n n3 space is necessary and sufficient for deterministic 5wtm s to simulate deterministic nondeterministic 3 fa s 2 n2 n2 space is necessary and sufficient for nondeterministic 5wtm s to simulate deterministic nondeterministic 3 fa s

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

discrete time stability with perturbations application to model predictive control exponential stability and asymptotic perturbed stability results are derived for nonlinear discrete time systems some related results are available in the literature but proofs for the discrete time case are difficult to find our motivation for obtaining these results was to determine conditions under which discrete time nonlinear model predictive control is stabilizing in the face of perturbations this problem has particular relevance when output rather than state feedback is used a nonlinear state estimator is then required to reconstruct the state and it is necessary to identify the class of reconstruction errors for which the cascaded estimator controller assembly is stabilizing

on the trade off between feasibility and performance in bilinear and state affine model based predictive control closed loop stabilizing model based predictive control mpc algorithms for discrete time bilinear and state affine models are presented stability of the closed loop is obtained through the use of an appropriate end point weighting and end point inequality constraint in this way the infinite horizon performance index is bounded from above by the objective function that is minimized in the mpc algorithm this paper presents an algorithm that aims at obtaining a large feasibility region by maximizing off line the region that is defined by the end point inequality constraint in order to improve the performance of the mpc algorithm the conservatism of the upper bound on the infinite horizon performance index is reduced in the on line computations

robust constrained predictive control of uncertain norm bounded linear systems a novel robust predictive control algorithm is presented for uncertain discrete time input saturated linear systems described by structured norm bounded model uncertainties the solution is based on the minimization at each time instant of a semi definite convex optimization problem subject to a number of lmi feasibility constraints which grows up only linearly with the control horizon length n the general case of arbitrary n is considered closed loop stability and feasibility retention over the time are proved and comparisons with robust multi model polytopic mpc algorithms are reported

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

semi global stabilizability of linear null controllable systems with input nonlinearities an h sub spl infin based lyapunov proof is provided for a result established by lin and saberi 1993 if a linear system is asymptotically null controllable with bounded controls then when subject to input saturation it is semi globally stabilizable by linear state feedback a new result is that if the system is also detectable then it is semi global stabilizable by completely linear output feedback further an extension which relaxes the requirements on the input characteristic is obtained

model predictive control for linear parameter varying constrained systems using ellipsoidal set prediction this paper proposes a new model predictive control mpc method for linear parameter varying systems with bounded parameter variation subject to input constraints the method adopts closed loop prediction and constructs ellipsoidal sets to predict the future states with reasonable computational effort then the information on the parameter variation rate is exploited to improve the accuracy of the prediction furthermore a relaxed terminal condition which guarantees the stability for infinite horizon is introduced to enlarge the stabilizable region it is shown that the feasibility of the mpc problem at the initial step ensures the stability of the closed loop system finally a simulation result illustrates the effectiveness of the proposed method

brief nonlinear model predictive control with polytopic invariant sets ellipsoidal invariant sets have been widely used as target sets in model predictive control mpc these sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope the choice of inclusion polytope has a significant effect on the size of the target ellipsoid but the optimal inclusion polytope cannot in general be computed systematically this paper shows that use of polytopic invariant sets overcomes this difficulty allowing larger stabilizable sets without loss of performance in the interests of online efficiency consideration is focused on interpolation based mpc

random context structure grammars and automata a formal approach abstract in an earlier paper results of a research project to define grammars and automata of two and three dimensions were reported in that paper many of the relevant theorems were only discussed in a cursory way and were not formally proved in this paper we formalize the major concepts introduced because of current interest and the fact that the two dimensional grammars and automata defined are but a special case of the three dimensional grammars and automata we concentrate on three dimensional concepts a brief overview of current three dimensional grammars and automata is given formal definitions for random context structure grammars and random context structure automata are given a formal proof is given for the equivalence of the class of languages generated by random context structure grammars and the class of languages accepted by random context structure grammars

the braided structures for ω smash coproduct hopf algebras abstract braided bialgebras were introduced by larson and towber by considering a bilinear form with certain properties here we study the braided structures of smash coproduct hopf algebras b h as constructed by caenepeel ion militaru and zhu necessary and sufficient conditions for a class of smash coproduct hopf algebras to be braided hopf algebras are given in terms of properties of their components as applications of our results some special cases are discussed and explicit examples are given

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

brief an efficient off line formulation of robust model predictive control using linear matrix inequalities the practicality of model predictive control mpc is partially limited by its ability to solve optimization problems in real time moreover on line computational demand for synthesizing a robust mpc algorithm will likely grow significantly with the problem size in this paper we use the concept of an asymptotically stable invariant ellipsoid to develop a robust constrained mpc algorithm which gives a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off line one within another in state space this off line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed loop system and substantial reduction of the on line mpc computation the controller design is illustrated with two examples

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

asymptotic properties of receding horizon optimal control problems the asymptotic behavior of receding horizon optimal control problems with terminal cost chosen as a control liapunov function is analyzed for regulator as well as disturbance attenuation problems both the continuous as well as the discrete time cases are treated further the approximation of the continuous time optimal control problem by the discrete time receding horizon problems is studied

three dimensional alternating turing machines with only universal states abstract we investigate several properties of three dimensional alternating turing machines whose input tapes are restricted to cubic ones we first investigate the relationship between the classes of sets accepted by space bounded and finitely leaf size bounded five way three dimensional alternating turing machines and the classes of sets which are finite intersections of sets accepted by space bounded five way three dimensional nondeterministic turing machines then we investigate the accepting powers of three dimensional alternating turing machines with only universal states

nonconvex optimization problem the infinite horizon linear quadratic control problem with quadratic constraints abstract we present a general method for the solution of the problem of minimization of a quadratic functional in a hilbert space over the intersection of some subspace and a set defined by special quadratic constraints many optimal control problems can be formulated in this abstract form as an example a stationary infinite horizon linear quadratic optimization problem with quadratic constraints is posed and solved

robust optimal feedback for terminal linear quadratic control problems under disturbances we consider a linear dynamic system in the presence of an unknown but bounded perturbation and study how to control the system in order to get into a prescribed neighborhood of a zero at a given final moment the quality of a control is estimated by the quadratic functional we define optimal guaranteed program controls as controls that are allowed to be corrected at one intermediate time moment we show that an infinite dimensional problem of constructing such controls is equivalent to a special bilevel problem of mathematical programming which can be solved explicitely an easy implementable algorithm for solving the bilevel optimization problem is derived based on this algorithm we propose an algorithm of constructing a guaranteed feedback control with one correction moment we describe the rules of computing feedback which can be implemented in real time mode the results of illustrative tests are given

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

on linear programming and robust modelpredictive control using impulse responses abstract campo and morari have derived a linear programming problem with a potentially large number of constraints which is equivalent to a min max formulation for robust model predictive control of linear systems that formulation involves minimization with respect to the controls of the maximum with respect to the system s impulse response from a set of possible impulse responses of the infinity norm of the error between the predicted and required system output sequences here an alternative linear programming problem is derived which has a smaller number of constraints and is therefore potentially more convenient for on line control

the group of automorphisms of a semisimple hopf algebra over a field of characteristic 0 is finite introduction suppose that a is a semisimple hopf algebra over a field of characteristic 0 or that a is a semisimple cosemisimple involutory hopf algebra over a field k of characteristic p dim a in this paper we prove that the group authopf a of hopf algebra automorphisms of a is finite we show that the semigroup endhopf a of hopf algebra endomorphisms of a is also finite as a corollary generally the group of hopf algebra automorphisms of a finite dimensional hopf algebra need not be finite for any positive integer n and any field k we construct a family of finite dimensional hopf algebras over k with automorphism group gln k the fact that authopf a is finite is a consequence of a theorem of

robust receding horizon state estimation for uncertain discrete time linear systems an approach to robust receding horizon state estimation for discrete time linear systems is presented estimates of the state variables can be obtained by minimizing a worst case quadratic cost function according to a sliding window strategy this leads to state the estimation problem in the form of a regularized least squares one with uncertain data the optimal solution involving on line scalar minimization together with a suitable closed form approximation are given the stability properties of the estimation error for both the optimal filter and the approximate one have been studied and conditions to select the design parameters are proposed simulation results are reported to show the effectiveness of the proposed approach

on stability of constrained receding horizon control with finite terminal weighting matrix in this paper a new stabilizing receding horizon control based on a finite input and state horizon cost with a finite terminal weighting matrix is proposed for time varying discrete linear systems with constraints we propose matrix inequality conditions on the terminal weighting matrix under which closed loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints we show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality lmi in the case of constrained time invariant systems an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints using the invariant ellipsoid constraints a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control

on the robustness of receding horizon control with terminal constraints robustness properties of nonlinear receding horizon controllers with zero terminal state constraints are investigated with respect to gain and additive perturbations some robustness margins are derived by extending to the receding horizon case the analysis originally proposed by geromel and da cruz for infinite horizon controllers in the linear case it is shown that the zero terminal state receding horizon controller exhibits worse robustness margins compared to standard infinite horizon lq control

an interpolation strategy for discrete time bilinear mpc problems input output i o feedback linearization suffers from a number of restrictions which have limited its use in model based predictive control some of these restrictions do not apply to the case of bilinear systems but problems with input constraints and unstable zero dynamics persist this paper overcomes these difficulties by means of an interpolation strategy involved in this interpolation is a feasible and stabilizing trajectory which is computed through the use of invariant feasible sets and a more aggressive trajectory which can be chosen to be either the unconstrained optimal trajectory or any alternative one

on the quasitriangular structures of a semisimple hopf algebra in this paper we show that the theorem of 4 on the order of the group of hopf algebra automorphisms aut a of certain semisimple hopf algebras over a field k has an interesting implication for quantum groups if a is a finite dimensional hopf algebra over the field k there may be infinitely many re a a such that a r is a quasitriangular hopf algebra however if a is semisimple and the characteristic of k is 0 or if a is semisimple cosemisimple and involutory and the characteristic of k is p dim a we show in the theorem of this paper that there are at most finitely r ea a such that a r is quasitriangular our proof is based on the fact that the group aut a is finite 4 theorem 33 for such a hopf algebra a we will assume that the reader is familiar with the elementary aspects of the theory of hopf algebras a good general reference is s

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

quasi min max fuzzy model predictive control of direct methanol fuel cells abstract direct methanol fuel cells dmfcs are known as a promising power source in future in this paper we consider steering a dmfc plant to a desired operating point while optimizing the transient performance according to a quadratic cost function quasi min max fuzzy model predictive control fmpc with input constraints is proposed for the dmfc in order to reduce the computational burden for real time implementation a partial off line quasi min max fmpc is also proposed in this case a bank of invariant sets together with the corresponding feedback control laws are obtained by solving some linear matrix inequalities lmis off line leaving the online part a bisection search and a much simplified constrained optimization problem online computation complexity for both the quasi min max fmpc and the partial off line one is also analyzed simulation results are given to show the effectiveness of the proposed controllers

semi global exponential stabilization of linear discrete time systems subject to input saturation via linear feedbacks abstract in this paper we show that a linear discrete time system subject to input saturation is semi globally exponentially stabilizable via linear state and or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable and has all its poles located inside or on the unit circle furthermore the semi globally stabilizing feedback laws are explicitly constructed the results presented here are parallel to our earlier results on the continuous time counterpart lin and saberi 1993

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method