Dynamical systems state vector

The principle of least action, 8A = 0 for a particular choice of system state vector li/r), yields the equations that describe the system dynamics. In the case of a state vector that can explore the entire Hilbert space, the stationarity of the action yields the time-dependent Schrodinger equation. For any approximate family of state vectors this procedure yields an equation that approximates the time-dependent Schrodinger equation in a manner that is variationally optimal for the particular choice of state vector form. 

When the system state vector v(t) is difficult to determine or totally unavailable, then the system behaviour must be described on the basis of the system input-output relation, i.e. the relations between controlled input signals and the system response. A frequently used approach is the description of the object dynamics by an equation which combines the previous outputs and inputs directly with the fijture outputs from the object... 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Since the concept of observability was primarily defined for dynamic systems, observability as a property of steady-state systems will be defined in this chapter. Instead of a measurement trajectory, only a measurement vector is available for steady-state systems. Estimability of the state process variables is the concept associated with the analysis of a steady-state situation. 

The mapping Xss = tt to, p) represents the steady state zero output submanifold and Uss = 7 eo,p) is the steady state input which makes invariant this steady state zero output submanifold. Condition (48) expresses the fact that this steady state input can be generated, independently of the values of the parameter vector p, by the linear dynamic system... 

The vertex A4 is the fixed point for the discrete dynamical system. There is no reaction A4 For convenience, we include the eigenvectors f" and for zero eigenvalue, K4 = 0. These vectors correspond to the steady state is the steady-state vector, and the functional f is the conservation law. 

Let the phase portrait of the system be characterized by some set of co-limit points. The concepts of an "co-limit point and an "co-limit set have been extensively used in the theory of dynamic systems. The thing is that the trajectory does not necessarily enter into a steady state. In the general case (as well as in the case of chemical kinetic equations), the existence of limit cycles is possible. The letter co is a symbol for the region of the phase space into which at t—>co the trajectory tends ("from a to co ). Let x0 be a vector... 

Figure 4.29 shows a block diagram of a reactor with manipulated inputs U. other measured inputs W, and unknowm or unmeasured inputs N. We may assume that this reactor is more complicated than a simple plug-flow reactor or a CSTR. It may be more along the lines of the fluidized catalytic cracker that we showed in Fig. 4.4. The reactor can be described by a set of nonlinear differential equations as we have previously demonstrated. This results in a set of dynamic state variables X The state vector is often of high dimension and we normally only measure a subset of all the states. Y is the vector of all measurements made on the system.

The letters A and B denote quantities characterising the vectors. They refer to dynamical aspects of the state of the system. The vectors are called state vectors, a term which we abbreviate to states. The length of a state vector, considered apart from related state vectors, has no physical significance. We are free to choose it, a process known as normalisation. 

A particularly important class of solutions are the constant ones, which are called steady states, rest points, or equilibrium points. In terms of (3.1), such a solution is a zero of f(y), that is, a vector >> 6 K" such that f(y ) = 0. In the terminology of dynamical systems, a rest point is an element peM such that Tr p,t) = p for all telR. Similarly, a periodic orbit is one that satisfies -K(p,t + T) = -ir(p, t) for all t and for some fixed number T. The corresponding solution of (3.1) will be a periodic function. 

Quantum mechanics involves two distinct sets of hypotheses—the general mathematical scheme of linear operators and state vectors with its associated probability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substituting a single quantum dynamical principle for the conventional array of assumptions based on classical Hamiltonian dynamics and the correspondence principle. 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

Where the initial state x(t) corresponds to the vector of parameters in the multi-parametric programming framework. Balanced truncation is then applied to Eq. (1). We work with the dynamic system (xt+k+i t = Axt+k t + But+kl yt+k t = Cxt+k t) and seek to find a transformation T such that the transformed system is balanced. Following the procedure as described in [5], we describe the dynamic system in an equivalent balanced form ... 

Apparently as an independent development, A.R. Johnson (1988a) proposed the idea of using a multivariate approach to the analysis of multispecies toxicity tests. This state space analysis is based upon the common representation of complex and dynamic systems as an n-dimensional vector. In other words, the... 

The system dynamics equation defines the model for the propagation of the state vector X(k) (which comprises the best estimates of the n parameters describing the system state)... 

In classical molecular dynamics, a molecular system with a fixed number of N atoms is given by a state vector q,p) X = x where q denotes the position vector and p R the momentum vector. The dynamical behavior, given a specified potential energy function V, a mass matrix M and initial conditions qo,Po), is described by the Hamilton s equations... 

The phenomenon known as the quantum Zeno effect takes place in a system which is subject to frequent measurements projecting it onto its (necessarily known) initial state if the time interval between two projections is small enough the evolution of the system is nearly "frozen". This effect, and its inverse (the anti-Zeno effect), have been widely investigated theoretically [Khalhn 1957-58 Winter 1961 Misra 1977 Fonda 1978 Kofman 1996 Kof-man 2000 Lewenstein 2000 Kofman 2001 (a) Schmidt 2003 / 2004] as well as experimentally [Cook 1988 Itano 1990 Wilkinson 1997 Fischer 2001], Generalizations have been proposed which employ incomplete measurements [Facchi 2002] in this setting, the Hilbert space is split into "Zeno subspaces" (degenerate multidimensional eigenspaces of the measured observable), and the state vector of the system is compelled by frequent measurements of the physical observable to remain in its initial Zeno subspace. The dynamics of the system in the Zeno subspaces has also been studied in different specific situations [Facchi 2001 (b)]. 

Various choices of families of approximate state vectors are characterized by sets of time-dependent parameters, which serve as dynamical variables as the system of electrons and atomic nuclei evolves in time. Such parameters are, for example, molecular orbital coefficients, the coefficients of the various configurations in a multi-configurational electronic state vector, average nuclear positions and momenta, etc. Minimal END is characterized by the state vector... 

One of the most powerful techniques for identifying the presence of chaos is the phase- ace reconstruction of tune-series data. The phase space of a dynamic system is defined as an -dimensional mathematical space, with ordiogonal coordinates representing the n variables needed to specify the instantaneous state of the system (48, 49). The trajectories of the system s vector in the n-dimensional phase-space evolve in time from initial conditions onto the geometrical object called an attractor. The attractor is a set of points in a phase space towards which nearly all trajectories converge, and the attractor describes an ensemble of states of die system. Different variables of die phase space can be used as coordinates to graphically construct the attractor. Exaiqiles of die variables are ... 

The projection-based model order reduction algorithm begins with a spatial discretization of the governing PDEs to attain the dynamic system equations as Eq. 11. Specifically, here, X(t) is the state vector of unknowns (a function of time) on the discrete nodes, n is the total number of nodes A is formulated by the numerical discretization Z defines the functions of boundary conditions and source terms and B relates the input function to each state X. Equation 11 can be recast into the frequency domain in terms of transfer function T(s). T(s) then is expanded as a Taylor series at s = 0 yielding... 

Consider a n-components dynamic system described by an irreducible homogenous Markov process = Xj, t > 0 (initial state /) with finite state space E and the transition rate matrix M. This Markov process is ergodic and a single stationary distribution exists (Ross 1996). Let a row vector jt = (tti, 7T2,. ..) be the vector of steady state probabilities (stationary distribution vector). Chapman-Kolmogorov equations at steady state can be written as ... 

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