State-space concept

The three-dimensional AIM model is a first attempt to concretize and to visualize the state space concept. Because it has three dimensions, it is a space, not a plane, as are traditional representations of waking, sleeping, and dreaming. Furthermore, when realistic values are assigned to the three dimensions of the model—and with time as the fourth dimension—orbital trajectories of conscious state change emerge from the mapping. 

The set of states and the probability distribution together fully define the stochastic variable, but a number of additional concepts are often used. The average or expectation value of any function /(X) defined on the same state space is... 

Since, in process control, input-output linearization techniques are usually preferred to state-space approaches, mostly due to the higher complexity of the latter, in the following, only input-output feedback linearization basic concepts are briefly reviewed. 

I. C., and Green, D. N., "Energy Concepts in the State-Space Theory of Nonlinear N-Ports 2. Losslessness,"... 

The most basic concept is that of a dynamical (or a semidynamical) system. Let 7T M X R- M be a function of two variables, where M is R" and R denotes the real numbers. (We use M for the first variable or state space to suggest that the results are true in greater generality.) The function 7T is said to be a continuous dynamical system if tt is continuous and has the following properties ... 

Fig. 19.3. Molecular transitions and states utilized to break the diffraction barrier. Each nanoscopy modality resorts to a specific pair of bright and dark states. Several concepts share the same states, but differ by the direction in which the molecule is driven optically (say A B or B A) oi hy whether the transition is performed in a targeted way or stochastically. The targeted read-out modality drives the transition with an optical intensity I and hence operates with probabilities of the molecule of being in A or B. This probability depends on the rates k of the transitions between the two states and hence also on the applied intensity I. The probability pA of the molecule to remain in A typically decreases as indicated in the panel, pa -C 1 means that the molecule is bound or switched to the state B. This switching from A to B or vice versa allows the confinement of A to subdiffraction-sized coordinates of extent Ar at a position rt where /(r) is zero. In the stochastic read-out mode, the probability that state A emerges in space is evenly distributed across the sample and kept so low that the molecules in state A are further apart from each other than the diffraction limit. An optically nonlinear aspect of the stochastic concept is the fact that the molecules undergo a switch to A from where they suddenly emit 1 detectable photons in a row...

In order to suppress this oscillatory behaviour, the use of the automatic feedback control has been considered [4]. State space model and nonlinear full-state feedback have been used for stabilization of the system [5]. But, some of these state variables are not measurable, therefore, concept of state estimation from well-head measurements has been considered. A nonlinear observer is used for state estimation [6] which has shown satisfactory result in experiment [7]. As noted in [7], estimation is affected by noise. The standard Kalman filter has been used for state estimation and down-hole soft-... 

Metastability of some subset of the state space is characterized by the property that the Markov process is likely to remain within the subset for a long period of time, until it exits and a transition to some other region of the state space occurs. There are in fact several related but different definitions of metastability in the literature (see, e.g., [16-20]) we will focus on the so-called ensemble concept introduced in (6), for a comparison with, e.g., the exit time concept, see [2]. 

The basic concepts of Markov chains are system, the state space, i.e., the set of all possible states a system can occupy and the state transition, namely, the transfer of the system from one state to the other. Alternative synonyms are event as well as observation of an event. It should be emphasized that the concepts system and state are of a wide meaning and must be specified for each case under consideration. This will be elaborated in the numerous examples demonstrated in the following. 

The basic elements of Markov-chain theory are the state space, the one-step transition probability matrix or the policy-making matrix and the initial state vector termed also the initial probability function In order to develop in the following a portion of the theory of Markov chains, some definitions are made and basic probability concepts are mentioned. 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

Subspace state-space models are developed by using techniques that determine the largest directions of variation in the data to build models. Two subspace methods, PCA and PLS have already been introduced in Sections 4.2 and 4.3. Usually, they are used with steady-state data, but they could also be used to develop models for dynamic relations by augmenting the appropriate data matrices with lagged values of the variables. In recent years, dynamic model development techniques that rely on subspace concepts have been proposed [158, 159, 307, 313]. Subspace methods are introduced in this section to develop state-space models for process monitoring and closed-loop control. 

Momentum-space concepts are not, in general, familiar to the chemist and so we outline first the calculation of momentum-space electron densities, p p), from ab initio wavefunctions. The form of pip) for different molecules is discussed, using as examples (i) the ground state of H2, (ii) bond formation in BH", and (iii) the n-orbitals in large conjugated polyenes. 

A necessary and sufficient condition for identifia-bility is the concept that with p-estimable parameters at least p-solvable relations can be generated. Traditionally, for linear compartment models this involves using Laplace transforms. For example, going back to the 1-compartment model after first-order absorption with complete bioavailability, the model can be written in state-space notation as... 

When the state space spanned by /(0) includes more than 2 eigenstates, the dynamics in state space can begin to look very complicated. However, the concepts of bright state, dark state, state-selective detection, and... 

A fundamental concept in DP is that of a state, denoted by s. The set S of all possible states is called the state space. The decision problem is often describe as a controlled stochastic process that... 

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