Classification of Linear Systems

The examples in the last section had the special feature that two of the entries in the matrix A were zero. Now we want to study the general case of an arbitrary 2x2 matrix, with the aim of classifying all the possible phase portraits that can occur. 

Example 5.1.2 provides a clue about how to proceed. Recall that the x and y axes played a crucial geometric role. They determined the direction of the trajectories as z — 1. They also contained special straight-line trajectories a trajectory starting on one of the coordinate axes stayed on that axis forever, and exhibited simple exponential growth or decay along it. 

For the general case, we would like to find the analog of these straight-line trajectories. That is, we seek trajectories of the form 

To find the conditions on v and A, we substitute x(r) = e v into x = Ax, and obtain Ae v = e Av. Canceling the nonzero scalar factor yields 

Let s recall how to find eigenvalues and eigenvectors. (If your memory needs more refreshing, see any text on linear algebra.) In general, the eigenvalues of a matrix A are given by the characteristic equation det(A - Af) = 0, where 1 is the identity matrix. For a 2 x 2 matrix 

Models of a diffusionless chemical reaction are described by systems of autonomous ordinary differential equations of first order  

At such points several integral curves of equation (5.3) may intersect. The singular points (x, y) also have a physical interpretation. It follows from equations (5.2a), (5.2b) that dx/dt = 0, dy/dt = 0, i.e. the points (x, j ) are stationary points of the system (5.2). When equations (5.2) or (5.1) describe chemical systems, stationary points (sometimes also called rest points) must satisfy the additional condition 

Properties of stationary points, for example their number and stability (the stationary point (x, y) is stable if the points x(f), y(t close to the point x, y, approach it when t - oo) may be examined by linearizing the system (5.2) in the neighbourhood of its stationary points. To investigate the stability of the stationary point (x,y) we make a substitution in (5.2) 

A stationary pojnt of the linearized system (5.6) is the origin of the coordinate system ( , rj) = (0, 0). The matrix Ftj = dFi/dxj will be called the Jacobian matrix whereas the matrix ai = FiJ(x1,x2) will be called the stability matrix of the system (5.2). 

Apparently, the condition for existence of non-zero solutions to the system of equations (5.8) is vanishing of the determinant 

The position and velocity coefficient matrices (i.e., Ap and Ay) in (3.8) are, in general, not symmetric and represent both conservative and nonconservative forces. We consider the following types of autonomous general forces [48]  

In the general case where all of the above forces are present, the linearized equations of motion of the n-DOF autonomous mechanical system takes the form 

Based on (3.21), the following general classification of linear systems can be made  

Specific to the case of dynamic systems with frictional constraints, two other subcategories of circulatory systems can be included in the above list where the inertia matrix, M, is asymmetric due to friction  

Remark 3.5 Systems given by (3.22) and (3.23) can be converted to the regular form of circulatory systems with a symmetric positive definite inertia matrix and asynunetric stiffness and damping matrices. First notice that in the absence of 

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To arouse your interest in the classification of linear systems, we now discuss a simple model for the dynamics of love affairs (Strogatz 1988). The following story illustrates the idea. 

Another classification of systems which is very important for deciding the algorithm for model solution, is that of linear and non-linear systems. The equations of linear systems can usually be solved analytically, while the equations of non-linear systems are almost always solved numerically. In this respect, it is important to recognize the important fact that physical systems are almost always non-linear and linear systems are either an approximation that should be justified, or the equations are intentionally linearized in the neighbourhood of a certain state of the system and are strictly valid only in this neighbourhood. 

Fig. 7-3. Canonical format of matrix of linear system (7.2.3) (The verbal classification is explained below)...

A universal criterion is that petroleum systems are mostly multiphase and heterogeneous with highly developed interfaces. The degree of dispersity is inversely proportional to a characteristic linear scale of inclusions. The degree of dispersity is a kernel of classification of disperse systems and should be accoimted for as an additional variable in all equations describing the thermodynamic state of a system At nano-scale ranges, this fact becomes especially important (Anisimov, 2004). 

Neural networks have been applied to IR spectrum interpreting systems in many variations and applications. Anand [108] introduced a neural network approach to analyze the presence of amino acids in protein molecules with a reliability of nearly 90%. Robb and Munk [109] used a linear neural network model for interpreting IR spectra for routine analysis purposes, with a similar performance. Ehrentreich et al. [110] used a counterpropagation network based on a strategy of Novic and Zupan [111] to model the correlation of structures and IR spectra. Penchev and co-workers [112] compared three types of spectral features derived from IR peak tables for their ability to be used in automatic classification of IR spectra. 

The concepts of structural observability are the basic tools for developing variable classification strategies. Some approaches presented in Chapter 3 are based on the fact that the classification of process variables results from the topology of the system and the placement of instruments and has nothing to do with the functional form of the balance equations. Thus, the linearity restriction will be removed and efficient reduction of the large-scale problem will be accomplished. 

For linear plant models Crowe et al. (1983) used a projection matrix to obtain a reduced system of equations that allows the classification of measured variables. They identified the unmeasured variables by column reduction of the submatrix corresponding to these variables. 

Crowe et al. (1983) proposed an elegant strategy for decoupling measured variables from the linear constraint equations. This procedure allows both the reduction of the data reconciliation problem and the classification of process variables. It is based on the use of a projection matrix to eliminate unmeasured variables. Crowe later extended this methodology (Crowe, 1986, 1989) to bilinear systems. 

The first case study consists of a section of an olefin plant located at the Orica Botany Site in Sydney, Australia. In this example, all the theoretical results discussed in Chapters 4,5,6, and 7 for linear systems are fully exploited for variable classification, system decomposition, and data reconciliation, as well as gross error detection and identification. 

In the present work we will deal with all the above problems and provide a unified framework to deal with the error correction for static or dynamic systems using multicomponent mass and energy balances. The topological character of the complex process is exploited for an easy classification of the measured and unmeasured variable independently of the linearity or nonlinearity of the balance equations. 

From the set E of equations the construction of the system matrices follow directly. The approach outlined before, making use of the classification strategy allows the general reduction of the initial balances into a set of equations smaller in size than that suggested by Vaclavek. The reduced set of balance equations given by eq. (15), or (18) define now the following weighted least squares problem for the reconciliation of the measurement errors. In the linear case... 

Now definitions or frameworks of modem thermodynamics in a broad sense, of classical thermodynamics, and of modem thermodynamics in a narrow sense are very clear. Modern thermodynamics in a broad sense includes all fields of thermodynamics (both classical thermodynamics and modem thermodynamics in a narrow sense) for any macroscopic system, but modem thermodynamics in a narrow sense includes only three fields of thermodynamics, i.e., nonequilibrium nondissipative thermodynamics, linear dissipative thermodynamics and nonlinear dissipative thermodynamics. The modem thermodynamics in a narrow sense should not be called nonequilibrium thermodynamics, because the classical nonequilibrium thermodynamics is not included. Meanwhile, the classical thermodynamics should only be applied to simpler systems without reaction coupling. That is, the application of classical thermodynamics to some modem inorganic syntheses and to the life science may be not suitable. Without the self-consistent classification of modem thermodynamics it was very difficult to really accept the term of modem thermodynamics even only for teaching courses. 

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