Linear state-space framework

In addition to specifying operating policies for the supply chain, one may find the linear state-space framework useful in studying important issues in supply chain management, such as the assessment of the potential benefits of information sharing between the retailer and the supplier (see more details in 4.3). In fact, it has been shown that in the elementary AR(1) and ARIMA(0,1,1) cases, there is no value for the retailer sharing information about the actual demand realizations with the supplier. This is because the supplier can observe the values of A-2, from the history of retailer s... 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

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... 

The solution of the electronic Schrodinger equation, Eq. (8.76), requires the integration of a complicated partial differential equation depending on 3N electronic coordinates as variables. The complexity of this problem increases with the number of electrons in the molecule. In order to establish a general solution strategy it is mandatory to first study the underlying formal framework of many-particle quantum mechanics, namely that of a linear vector space — the Hilbert space (cf. section 4.1). The N-electron Hilbert space, which hosts the total quantum mechanical state vector, is then constructed by direct multiplication of the one-electron Hilbert spaces. 

Most of the current MFC research is based on state-space models, because they provide an important theoretical advantage, namely, a unified framework for both linear and nonlinear control problems. State-space models are also more convenient for theoretical analysis and facilitate a wider range of output feedback strategies (Rawlings, 2000, Maciejowski, 2002 Qin and Badgwell, 2003). 

The theory of deterministic linear systems plays a fundamental role in the dynamic analysis of structures subjected to stochastic excitations. For this reason in this section the fundamental of deterministic analysis of SDoF subjected to deterministic excitation is synthetically reviewed. Particular care has been devoted to the state-space approach. This approach is the best suited for the development of formulations in the framework of random vibrations. In fact, its adaptability to numerical method of solution of differential equations and its extension to multi-degree-of-freedom (MDoF) systems are very straightforward. 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

As we shall see, linear algebraic constraints arising from steady state mass balance form the basis of metabolic flux analysis (MFA) and flux balance analysis (FBA). Thermodynamic laws, while introducing inherent non-linearities into the mathematical description of the feasible flux space, allow determination of feasible reaction directions and facilitate the introduction of reactant concentrations to the constraint-based framework. 

The framework has two axes. One axis tells us the number of variables needed to characterize the state of the system. Equivalently, this number is the dimension of the phase space. The other axis tells us whether the system is linear or nonlinear. 

---