Subspace state-space models

Canonical variates will be used in the formulation of subspace state-space models in Section 4.5. 

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. 

The data used in subspace state-space model development consists of the time series data of output and input variables. For illustration, assume a case with only output data and the objective is to build a model of the form Eq. 4.62. Since the whole data set is already known, it can be partitioned as past and future with respect to any sampling time. Defining a past data window of length K and a future data window of length J that are shifted from the beginning to the end of the data set, stacked vectors of data are formed. The Hankel matrix (Eq. 4.64) is used to develop subspace... 

The subspace state-space model that includes external inputs is of the form ... 

To include the information about process d3mamics in the models, the data matrix can be augmented with lagged values of data vectors, or model identification techniques such as subspace state-space modeling can be used (Section 4.5). Negiz and Cinar [209] have proposed the use of state variables developed with canonical variates based realization to implement SPM to multivariable continuous processes. Another approach is based on the use of Kalman filter residuals [326]. MSPM with dynamic process models is discussed in Section 5.3. The last section (Section 5.4) of the chapter gives a brief survey of other approaches proposed for MSPM. 

MSPM techniques rely on the model of the process. If the process has significant dynamic variations, state-space and subspace state-space models... 

To include the information about process dynamics in the models, the data matrix can be augmented with lagged values of data vectors, or model identification techniques such as subspace state-space modeling can be used (Section 5.3). Other approaches proposed for MSPM are summarized in Section 5.4). 

The states in Eqn (25.2) are now being formed as linear combinations of the -step ahead predicted outputs k= 1, 2,. ..). The literature on state space identification has shown how the states can be estimated directly from the process data by certain projections. (Verhaegen, 1994 van Overschee and de Moor, 1996 Ljung and McKelvey, 1996). The MATLAB function n4sid (Numerical Algorithms for Subspace State Space System Identification) uses subspace methods to identify state space models (Matlab 2000, van Overschee and de Moor, 1996) via singular value decomposition and estimates the state x directly from the data. 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

Consider a complex scalar product space V that models the states of a quantum system. Suppose G is the symmetry group and (G, V, p) is the natural representation. By the argument in Section 5.1, the only physically natural subspaces are invariant subspaces. Suppose there are invariant subspaces Gi, U2, W c V such that W = U U2. Now consider a state w of the quantum system such that w e W, but w Uy and w U2. Then there is a nonzero mi e Gi and a nonzero M2 e U2 such that w = ui + U2. This means that the state w is a superposition of states ui and U2. It follows that w is not an elementary state of the system — by the principle of superposition, anything we want to know about w we can deduce by studying mi and M2. 

We know from numerous experiments that every quantum system has elementary states. An elementary state of a quantum system should be observer-independent. In other words, any observer should be able (in theory) to recognize that state experimentally, and the observations should all agree. Second, an elementary state should be indivisible. That is. one should not be able to think of the elementary state as a superposition of two or more more elementary states. If we accept the model that every recognizable state corresponds to a vector subspace of the state space of the system, then we can conclude that elementary states correspond to irreducible representations. The independence of the choice of observer compels the subspace to be invariant under the representahon. The indivisible nature of the subspace requires the subspace to be irreducible. So elementary states correspond to irreducible representations. More specifically, if a vector w represents an elementary state, then w should lie in an irreducible invariant subspace W, that is, a subspace whose only invariant subspaces are itself and 0. In fact, every vector in W represents a state indistinguishable from w, as a consequence of Exercise 6.6. 

Thus the decoherence model of Fig. 8 exactly reproduces the anticipated behavior. All decoherence processes resulting from individual and uncorrelated reservoir interactions of the atoms are either exponentially suppressed by the energy gap (33) or are proportional to l/N. The latter is due to the large effective distance of the collective states in state space. In this way a quasi decoherence free subspace of dimension two is generated which allows to protect a stored photonic qubit from decoherence much more efficiendy than possible in quantum memories based on single particles. 

In this section, classical state-space models are discussed first. They provide a versatile modeling framework that can be linear or nonlinear, continuous- or discrete-time, to describe a wide variety of processes. State variables can be defined based on physical variables, mathematical solution convenience or ordered importance of describing the process. Subspace models are discussed in the second part of this section. They order state variables according to the magnitude of their contributions in explaining the variation in data. State-space models also provide the structure for... 

Subspace modeling can be cast as a reduced rank regression (RRR) of collections of future outputs on past inputs and outputs after removing the effects of future inputs. CVA performs this regression. In the case of a linear system, an approximate Kalman filter sequence is recovered from this regression. The state-space coefficient matrices are recovered from the state sequence. The nonlinear approach extends this regression to allow for possible nonlinear transformations of the past inputs and outputs, and future inputs and outputs before RRR is performed. The model structure consists of two sub models. The first model is a multivariable dynamic model for a set of latent variables, the second relates these latent variables to outputs. The latent variables are linear combinations of nonlinear transformations of past inputs and outputs. These nonlinear transformations or functions are... 

M Verhaegen and P Dewilde. Subspace model identification. Part I The output error state space model identification class of algorithms. Int. J. Control 56 1187-1210, 1992. 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

In a quasi-equilibrium state, a -exponential function [20] whose tail is of power type is reported in the HMF model with a certain type of initial condition [15]. The initial condition known as waterbag seems special because it lies on a one-dimensional line on two-dimensional p-space. To check generality, we must compute distribution functions for another type of initial condition that spreads on two-dimensional subspace of the p-space. 

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