Set Of Possible Initial States Research Articles

Symbolic state estimation in bounded timed labeled Petri nets

This paper addresses the state estimation of timed discrete event systems, where the occurrence of an event is associated with a time delay and the initial state of the system belongs to a set of possible initial states. A discrete event system is modeled with a bounded timed labeled Petri net whose unobservable subnet is acyclic, and the behavior of the system under a global time clock can be observed. The procedure for state estimations of partially observed Petri nets is refined by the introduction of time in discrete event systems. We apply the symbolic techniques to the representation of states and transition relations of bounded timed labeled Petri nets. Then we propose an online algorithm for the computation of state estimations generated by timed observations.

Interval methods in the problems of optimal control

The problem of optimal control of a linear dynamic object in conditions of incomplete information about the initial data is considered. The approaches based on various models of the uncertainty description are analyzed. It is shown that the use of the approach based on a probabilistic model for describing uncertainty is advisable when the uncertainty is associated only with randomness, the description of other sources of uncertainty within this model is rather difficult and the formal application of the regression analysis provides the results that are far from true. Though the fuzzy model is suitable for describing a wide range of the uncertainty sources, application of the model faces methodological difficulties when comparing and ranking fuzzy numbers and smoothing fuzzy data. In this regard, it seems promising to use an approach based on an interval model which allows description of a wide class of uncertain and inaccurate initial data. To unify control algorithms for the systems described by equations of state of different types with interval-given parameters, we developed an algorithm of equivalent transformations that provides the transition to special forms of representing the state matrix while maintaining preservation of all the dynamic properties of the original system. The problems of constructing the range of values of the roots of the matrix of the system and its description by an approximation in the form of an interval vector are solved to ensure the implementation of the algorithm. An approach is proposed to solve the problems of terminal control and maximum performance, when the uncertainty of the initial data is described by the interval model and the apparatus of interval analysis is used to solve the problem. It is shown that in this case with the direct use of the classical formulation of the optimal control problem without taking the uncertainty into account, there is no single optimal control which guarantees the exact transfer of the object to the required final state for any value of the parameters from a given range of their possible values. Therefore, in the presence of the interval uncertainty of the initial data, the solution of the problem can’t be obtained in the sense in which it is understood with precisely known parameters and the approach to the formulation of the control problem itself should be revised in order to determine in the future a solution that ensures guaranteed accuracy of the system translation. In this regard, we propose to formulate the control problem in conditions of the interval uncertainty as a problem of determining the set of control actions that guarantee the solution with the accuracy set up to the interval, on the set of initial data known up to the interval. Using an example of the problem with an inaccurately known initial state, it is shown that if the set of possible initial states of an object belongs to a n-dimensional rectangular parallelepiped, then when implementing a control on an object calculated for any initial state from a given set, the set of final states is convex and represents a n-dimensional parallelepiped. To construct the parallelepiped, it is sufficient to determine the coordinates of the vertices corresponding to the vertices of the abovementioned n-dimensional rectangular parallelepiped. We propose a system of inequalities which determines the condition for the membership of a set of finite states of the object obtained upon implementation of control for any possible value of the initial state from a given set to the required set of finite states. On the basis of this system of inequalities, conditions are formulated that provide a priori determination of the solvability of the problem.

The mini-batch adaptive method of random search (MAMRS) for parameters optimization in the tracking control problem

The tracking control problem for nonlinear dynamic system linear in control and signal tracked is considered. It is proposed to formalize the tracking problem as a problem of controlling a bunch of trajectories emanating from a given set of possible initial states. The quality of a single trajectory control is described by the value of the integral criterion, which depends on the square of the tracking error and the intensity of the control used. The control is sought from the condition of the minimum value of the average criterion defined on the set of the bunch trajectories. When choosing a control structure, the well-known result was used that provides internal stability and the desired quality of the output control. The unknown parameters of the control law within the selected structure are found using the new mini-batch adaptive method of random search (MAMRS). This new optimization algorithm combines the ideas of the adaptive random search and stochastic gradient descent methods. The examples of solving the tracking problem as a result of parametric optimization of controlling a bunch of trajectories are presented.

Reference state for arbitrary U-consistent subspace

The reduced dynamics of the system S, interacting with the environment E, is not given by a linear map, in general. However, if it is given by a linear map, then this map is also Hermitian. In order that the reduced dynamics of the system is given by a linear Hermitian map, there must be some restrictions on the set of possible initial states of the system-environment or on the possible unitary evolutions of the whole SE. In this paper, adding an ancillary reference space R, we assign to each convex set of possible initial states of the system-environment , for which the reduced dynamics is Hermitian, a tripartite state , which we call the reference state, such that the set is given as the steered states from the reference state . The set of possible initial states of the system is also given as the steered set from a bipartite reference state . The relation between these two reference states is as , where is the identity map on R and is a Hermitian assignment map, from S to SE. As an important consequence of introducing the reference state , we generalize the result of Buscemi (2014 Phys. Rev. Lett. 113 140502): we show that, for a U-consistent subspace, the reduced dynamics of the system is completely positive, for arbitrary unitary evolution of the whole system-environment U, if and only if the reference state is a Markov state. In addition, we show that the evolution of the set of system-environment (system) states is determined by the evolution of the reference state ().

Решение задачи терминального управления линейным объектом в условиях интервальной неопределённости

An approach to solving an optimal control problem subject to a quadratic criterion under the assumption that the initial data are known imperfectly is considered. The uncertainty in the initial data is described by an interval model based on the assumption that the values of certain parameters of the problem truly belong to intervals with the known lower and upper boundaries. Noteworthy is that no probability measure is defined on the interval as a probability density function (as in a stochastic model) or a membership function (as in a fuzzy model). The problem is solved by means of interval analysis techniques. It is shown that an attempt to solve the optimal control problem in its classical formulation without taking into account the uncertainty in its parameters cannot provide a unique optimal control that would guarantee that the plant will be transferred into the required final state with any values of the parameters from the specified interval of their possible values. This means that an attempt to perform control of the system without taking into account the uncertainty may result in the system having been transferred into inadmissible state. Hence, if there is interval uncertainty in the parameters of the problem, one cannot speak about its solution in the sense in which it is understood with precisely known parameters. Therefore, the very approach to the control problem formulation should be revised with a purpose to determine a solution ensuring that the system will be transferred into the desired state with guaranteed accuracy. In this connection, the control problem involving interval uncertainty is proposed to be formulated as a problem of determining the set of control actions guaranteeing its solution with the accuracy specified to the interval in a set of parameters known to within an interval. It is shown using the example of a problem with an imperfectly known initial state that, if the set of possible initial states of a plant belongs to a set represented by an n-dimensional rectangular parallelepiped in the state space, then, when being implemented in a controlled plant calculated for any parameter value from a given set of possible initial states, the set of final states is a convex one and is represented by an n-dimensional parallelepiped. To construct this parallelepiped it is sufficient to determine only the coordinates of its vertices corresponding to the vertices of the n-dimensional rectangular parallelepiped determined above that defines the domain of possible values of the problem’s parameters determined to within an interval. A system of inequalities depending on the plant’s parameters, on the range of its initial states and on the duration of the control period is obtained that determines the condition under which the set of the plant final states belongs to the desired rectangular set. The conditions subject to which the question whether or not the problem is solvable can a priori be answered are formulated proceeding from this system of inequalities.

Verifying chemical reaction network implementations: A pathway decomposition approach

The emerging fields of genetic engineering, synthetic biology, DNA computing, DNA nanotechnology, and molecular programming herald the birth of a new information technology that acquires information by directly sensing molecules within a chemical environment, stores information in molecules such as DNA, RNA, and proteins, processes that information by means of chemical and biochemical transformations, and uses that information to direct the manipulation of matter at the nanometer scale. To scale up beyond current proof-of-principle demonstrations, new methods for managing the complexity of designed molecular systems will need to be developed. Here we focus on the challenge of verifying the correctness of molecular implementations of abstract chemical reaction networks, where operation in a well-mixed “soup” of molecules is stochastic, asynchronous, concurrent, and often involves multiple intermediate steps in the implementation, parallel pathways, and side reactions. This problem relates to the verification of Petri nets, but existing approaches are not sufficient for providing a single guarantee covering an infinite set of possible initial states (molecule counts) and an infinite state space potentially explored by the system given any initial state. We address these issues by formulating a new theory of pathway decomposition that provides an elegant formal basis for comparing chemical reaction network implementations, and we present an algorithm that computes this basis. Our theory naturally handles certain situations that commonly arise in molecular implementations, such as what we call “delayed choice,” that are not easily accommodated by other approaches. We further show how pathway decomposition can be combined with weak bisimulation to handle a wider class that includes most currently known enzyme-free DNA implementation techniques. We anticipate that our notion of logical equivalence between chemical reaction network implementations will be valuable for other molecular implementations such as biochemical enzyme systems, and perhaps even more broadly in concurrency theory.

Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems

The problem of optimal control of automatic deterministic discrete systems in conditions of parametric uncertainty is considered. The state change (switchover) of a system is described by a recurrent equation in which instant multiple switchovers are admitted. The times and the number of switchovers are not specified in advance--they are found as a result of optimization of a functional. The initial state of the system is not known exactly, but the set of possible initial states is known; therefore, problems of average optimal control and guaranteed optimal control of bunches of trajectories are formulated. For solving these problems, it is suggested to use the principle of separation: namely, to control the bunch of trajectories by applying a control optimal for one specially chosen trajectory of the bunch. The resulting control of the bunch is suboptimal, but it may be satisfactory in practice. On the basis of sufficient conditions for optimality of a full-feedback control, an algorithm for constructing a suboptimal control in conditions of uncertainty is developed. Counterexamples of linear---quadratic problems of control of automaton-type systems in which the principle of separation fails are presented.

Formal and Compositional Analysis of Power Systems Using Reachable Sets

Power system stability analysis becomes more important in the presence of ever increasing variations in operating conditions. Traditionally, the operation of power systems is verified for specific operating conditions. In this work, the stability analysis is performed for a set of operating conditions using reachability analysis, which makes it possible to compute the bounds of all possible system trajectories. Thus, reachability analysis can be used to rigorously check specifications. Contrary to previous work, the presented approach does not require model simplifications when the system is described by semi-explicit, nonlinear, index-1 differential-algebraic equations. The main obstacle in reachability analysis is the scalability towards larger systems, which is addressed by investigating compositional techniques. As a result, transient stability and variable energy production can be analyzed for the IEEE 14-bus and 30-bus benchmark systems, for which the computation times are orders of magnitude faster than the simulation of all cases starting in the corners of the set of possible initial states.

Strong solutions of Tsirel’son’s equation in discrete time taking values in compact spaces with semigroup action

Under the assumption that the infinite product of the evolution process converges almost surely, the set of strong solutions is characterized by a compact space T, which may be regarded as the set of possible initial states. More precisely, any strong solution may be represented as the result of a uniquely specified element of T acted by the infinite product of the evolution process.

Reachability analysis of continuous-time piecewise affine systems

This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.

Robust observability for a class of time-varying discrete-time uncertain systems

This paper introduces a concept of robust observability for a class of uncertain discrete-time systems. This notion is an extension of the standard notion of observability for discrete-time, linear time-varying systems. The uncertainty and noise are modelled deterministically via a sum quadratic constraint. A necessary and sufficient condition for robust observability is presented in terms of existence of a suitable solution to a Riccati difference equation. The set of possible initial states given noisy measurements over a finite number of sampling periods is shown to be an ellipsoid.

Generalized information-lossless automata. I

Huffman and Even introduced classes of abstract automata, which they called respectively information-lossless automata (ILL) and information-lossless automata of finite order (ILLFO). The underlying property of these automata is the ability to reconstruct unknown input sequences from observations of the output response, assuming that the true initial state of the automaton is known. Similar classes of automata introduced in are called essentially information-lossless automata, and they are capable of reconstructing the unknown input word without knowledge of the initial state of the automaton. It is only assumed that the set of possible initial states of the automaton is the set of all automaton states. In this paper we analyze a structural analog of an abstract ILL-automaton whose set of initial states may be of arbitrary cardinality. This class of automata is thus a generalization of the classical ILL-automata, which allows not only for the structure of the input and output alphabets, but also for the configuration of the set of possible initial states.

Finding reachable states of finite-state concurrent systems

We present detailed algorithms for finding, for a given finitestate concurrent software system, all states reachable from a given set of possible initial states. Our systems consist of processes that send and receive messages through links of bounded capacity. Input to the algorithm is a software system, described in a modeling notation that resembles a programming language. Of particular interest are the techniques for modeling the effects of concurrency and nondeterminism, for detecting infinite loops, and for selecting a set of particularly significant states to report to the user. We discuss performance of the algorithms and provide data from implementations.

A minimax theorem for guaranteed terminal error

A control law u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> guaranteeing a prespecified terminal error is found subject to bounded uncertainties in the initial state. In the general case, this u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is described by a finite-dimensional minimax point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y_{0}^{*}</tex> . A refinement of this result is given for the case when the set of possible initial states is polyhedral convex.