optimal controllers for finite wordlength implementation when a controller is implemented by a digital computer with a d and d a conversion numerical errors can severely affect the performance of the control system there exist realizations of a given controller transfer function exhibiting arbitrarily large effects from computational errors assuming sufficient excitation of the system the problem of designing an optimal controller in the presence of both external disturbances and internal roundoff errors is solved the results reduce to the standard lqg controller when infinite precision computation is used for finite precision however the separation principle does not hold a penalty is also added to the cost function to penalize the sum of the wordlengths used to compute the fractional part of each state variable of the controller this sum can be used to represent the lower bound on computer memory needed for controller synthesis it measures controller complexity and is minimized penalized here

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

nonparametric estimation for a class of levy processes we consider estimation for a class of levy processes modelled as a sum of a drift a symmetric stable process and a compound poisson process we propose a nonparametric approach to estimating unknown parameters of our model including the drift the scale and index parameters in the stable law the mean of the poisson process and the underlying jump size distribution we show that regression and nonparametric deconvolution methods based on the empirical characteristic function can be used for inference interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution

on finite potential games a linear system called the potential equation pe is presented it is proved that a finite game is potential if and only if its potential equation has solution some properties of the potential equation are obtained based on these properties a closed form solution of the pe is obtained moreover a formula based on the solution of the pe is obtained to calculate the potential function finally it is proved that a networked evolutionary game is potential if and only if its fundamental network game is potential some interesting examples are presented to illustrate the theoretical results

on the periodic trajectories of boolean control networks abstract in this note we first characterize the periodic trajectories or equivalently the limit cycles of a boolean network and their global attractiveness we then investigate under which conditions all the trajectories of a boolean control network may be forced to converge to the same periodic trajectory if every trajectory can be driven to such a periodic trajectory this is possible by means of a feedback control law

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

balanced realizations via gradient flow techniques abstract the task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations by viewing such algebraic tasks in the context of calculus they are amenable to analog computational solutions or parallel processing machines perhaps even neural networks the convergence rates of the differential equations is exponential and consequentially convergence is rapid and numerical stability properties are attractive

optimal realizations of floating point implemented digital controllers with finite word length considerations the closed loop stability issue of finite precision realizations is investigated for digital controllers implemented in floating point arithmetic unlike the existing methods which only address the effect of the mantissa bits in floating point implementation to the sensitivity of closed loop stability the sensitivity of closed loop stability is analyzed with respect to both the mantissa and exponent bits of floating point implementation a computationally tractable fwl closed loop stability measure is then defined and the method of computing the value of this measure is given the optimal controller realization problem is posed as searching for a floating point realization that maximizes the proposed fwl closed loop stability measure and a numerical optimization technique is adopted to solve for the resulting optimization problem simulation results show that the proposed design procedure yields computationally efficient controller realizations with enhanced fwl closed loop stability performance

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

optimizing stability bounds of finite precision pid controller structures this paper investigates a recently derived lower bound stability measure for sampled data controller structures subject to finite wordlength fwl constraints the optimal realization of the digital pid controller with fwl considerations is formulated as a nonlinear optimization problem and an efficient strategy based on adaptive simulated annealing asa is adopted to solve this complex optimization problem a numerical example of optimizing the finite precision pid controller structure for a simulated steel rolling mill system is presented to illustrate the effectiveness of the proposed strategy

brief paper an efficient controller structure with minimum roundoff noise gain based on a polynomial operator approach a new sparse controller structure is derived which is actually an improved version of the recently proposed structure li g 2004 a polynomial operator based dfiit structure for iir filters ieee transactions on circuits and systems ii 51 147 151 the performance of the proposed structure is analyzed by deriving the corresponding expression of roundoff noise gain and the problem of finding optimized sparse structures is solved efficiently with a genetic algorithm ga a numerical example is given which shows that the newly developed structure can achieve much better performance than some well known structures and particularly outperforms the traditional optimal fully parametrized realization greatly in terms of reducing roundoff noise and implementation complexity

generalised modal realisation as a practical and efficient tool for fwl implementation finite word length fwl effects have been a critical issue in digital filter implementation for almost four decades although some optimisations may be attempted to get an optimal realisation with regards to a particular effect for instance the parametric sensitivity or the round off noise gain the purpose of this article is to propose an effective one i e taking into account all the aspects based on the specialised implicit form a new effective and sparse structure named modal realisation is proposed this realisation meets simultaneously accuracy low sensitivity round off noise gain and overflow risk few and flexible computational efforts with a good readability thanks to sparsity and simplicity no tricky optimisation is required to obtain it as well two numerical examples are included to illustrate the modal realisation s interest

matrix approach to model matching of asynchronous sequential machines in this note we propose a matrix based approach for asynchronous sequential machines using semi tensor product of matrices we convert an asynchronous machine into a discrete time bilinear system and study its dynamics by investigating its structure matrices we give simple algorithms for cycle detection and reachability analysis and moreover provide a control design method for the model matching of two input state machines

optimal supervisory control of finite state automata this paper presents optimal supervisory control of dynamical systems that can be represented by deterministic finite state automaton dfsa models the performance index for the optimal policy is obtained by combining a measure of the supervised plant language with possible penalty on disabling of controllable events the signed real measure quantifies the behaviour of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in earlier publications synthesis of the optimal control policy requires at most n iterations where n is the number of states of the dfsa model generated from the unsupervised plant language the computational complexity of the optimal control synthesis is polynomial in n syntheses of the control algorithms are illustrated with two application examples

optimum realizations of sampled data controllers for fwl sensitivity minimization the problem of an optimal finite word length state space realization of a digital controller is investigated the closed loop to be considered consists of a continuous time plant a discrete time controller a sampler a zero order hold and an antialiasing filter an effective algorithm is proposed to find the optimal sampled data controller realization minimizing the sensitivity of the closed loop performance with respect to coefficient errors in the state variable matrices of the controller realization in order to get a tractable problem a two step procedure is to be used very fast sampling at a multiple of the sampling frequency followed by blocking or lifting to obtain a single rate system the procedure allows consideration of the system s intersample behaviour

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

local asymptotic mixed normality for semimartingale experiments we give conditions for local asymptotic mixed normality of experiments when the observed process is a semimartingale and the observation time increases to infinity as a consequence we obtain asymptotic efficiency of various estimators several special models for counting process s diffusion processes and diffusions with jumps are studied

the detection and stabilisation of limit cycle for deterministic finite automata abstractin this paper the topological structure properties of deterministic finite automata dfa under the framework of the semi tensor product of matrices are investigated first the dynamics of dfa are converted into a new algebraic form as a discrete time linear system by means of boolean algebra using this algebraic description the approach of calculating the limit cycles of different lengths is given second we present two fundamental concepts namely domain of attraction of limit cycle and prereachability set based on the prereachability set an explicit solution of calculating domain of attraction of a limit cycle is completely characterised third we define the globally attractive limit cycle and then the necessary and sufficient condition for verifying whether all state trajectories of a dfa enter a given limit cycle in a finite number of transitions is given fourth the problem of whether a dfa can be stabilised to a limit cycle by the state feedback controller is discussed criteri

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

a linear representation of dynamics of boolean networks a new matrix product called semi tensor product of matrices is reviewed using it a matrix expression of logic is proposed where a logical variable is expressed as a vector a logical function is expressed as a multiple linear mapping under this framework a boolean network equation is converted into an equivalent algebraic form as a conventional discrete time linear system analyzing the transition matrix of the linear system formulas are obtained to show a the number of fixed points b the numbers of cycles of different lengths c transient period for all points to enter the set of attractors and d basin of each attractor the corresponding algorithms are developed and used to some examples

a search algorithm for a class of optimal finite precision controller realization problems with saddle points with game theory we review the optimal digital controller realization problems that maximize a finite word length fwl closed loop stability measure for a large class of these optimal fwl controller realization problems which have saddle points a minimax based search algorithm is derived for finding a global optimal solution the algorithm consists of two stages in the first stage the closed form of a transformation set is constructed which contains global optimal solutions in the second stage a subgradient approach searches this transformation set to obtain a global optimal solution this algorithm does not suffer from the usual drawbacks associated with using direct numerical optimization methods to tackle these fwl realization problems furthermore for a small class of optimal fwl controller realization problems which have no saddle point the proposed algorithm also provides useful information to help solve them

limit cycle structure for dynamic bi threshold systems in this paper we determine the limit cycle structures of synchronous and sequential finite dynamical systems governed by dynamic bi threshold functions over non uniform networks so far work in this area has been concerned with static threshold values and our results generalize these in particular this includes the celebrated result by goles and olivos on synchronous neural networks 1981 5 the work by kuhlman et al 2012 12 on static bi threshold systems and the work by chang et al 2014 2 on dynamic standard threshold systems in our work the state of each vertex v includes the usual binary state x v 0 1 the dynamic up thresholds k v and down thresholds k v whose values change only upon a transition of x v we show that the projection of the periodic orbits onto the boolean x dynamics have maximal size 2 under the synchronous update method regardless of how the up and down thresholds evolve the proof is a careful extension of the technique originally developed in the proof by goles and olivos 1981 5 and which was later modified by kuhlman et al 2012 12 to cover bi threshold systems we also derive sufficient conditions on the evolution of the up and down thresholds that ensures that the sequential systems only have fixed points as limit sets the results should be relevant for modeling and analyzing a large range of social and biological systems where agent adaptation occurs and needs to be accounted for

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

observability of boolean control networks a unified approach based on finite automata the problem on how to determine the observability of boolean control networks bcns has been open for five years already in this technical note we propose a unified approach to determine all the four types of observability of bcns in the literature we define the concept of weighted pair graphs for bcns in the sense of each observability we use the so called weighted pair graph to transform a bcn to a finite automaton and then we use the automaton to determine observability in particular the two types of observability that rely on initial states and inputs in the literature are determined finally we show that no pairs of the four types of observability are equivalent which reveals the essence of nonlinearity of bcns