Conditionally Linear Systems

In engineering we often encounter conditionally linear systems. These were defined in Chapter 2 and it was indicated that special algorithms can be used which exploit their conditional linearity (see Bates and Watts, 1988). In general, we need to provide initial guesses only for the nonlinear parameters since the conditionally linear parameters can be obtained through linear least squares estimation. 

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In algebraic equation models we also have the special situation of conditionally linear systems which arise quite often in engineering (e.g., chemical kinetic models, biological systems, etc.). In these models some of the parameters enter in a linear fashion, namely, the model is of the form,... 

Solving linear systems via orthogonal factorization of the coefficient matrix is a stable method, particularly effective for singular and iU-conditioned linear systems. 

Substitution of the known values of (x, f x,)) in Eq. (3.112) yields a set of ( + 1) simultaneous linear algebraic equations whose unknowns are the coefficients, ..., a of the polynomial equation. The. solution of this set of linear algebraic equations may be obtained using one of the algorithms discussed in Chap. 2. However, this solution results in an ill-conditioned linear system therefore, other methods have been favored in the development of interpolating polynomials. 

Minimizing the square of the gradient vector under the condition c/ = I yields the following linear system of equations... 

Bioprocess Control An industrial fermenter is a fairly sophisticated device with control of temperature, aeration rate, and perhaps pH, concentration of dissolved oxygen, or some nutrient concentration. There has been a strong trend to automated data collection and analysis. Analog control is stiU very common, but when a computer is available for on-line data collec tion, it makes sense to use it for control as well. More elaborate measurements are performed with research bioreactors, but each new electrode or assay adds more work, additional costs, and potential headaches. Most of the functional relationships in biotechnology are nonlinear, but this may not hinder control when bioprocess operate over a narrow range of conditions. Furthermore, process control is far advanced beyond the days when the main tools for designing control systems were intended for linear systems. 

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... 

The matrix A is known as the preconditioner and has to be chosen such that the condition number of the transformed linear system is smaller than that of the original system. 

The condition number of a matrix A is intimately connected with the sensitivity of the solution of the linear system of equations A x = b. When solving this equation, the error in the solution can be magnified by an amount as large as cortd A) times the norm of the error in A and b due to the presence of the error in the data. 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

Caglayan, A. E. (1980). Necessary and sufficient conditions for detectability of jumps in linear systems. IEEE Trans. Autom. Control AC-25, 833-834. 

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

The preceding approach applies to all linear systems that is, those involving mechanisms in which only first-order or pseudo-first-order homogeneous reactions are coupled with the heterogeneous electron transfer steps. As seen, for example, in Section 2.2.5, it also applies to higher-order systems, involving second-order reactions, when they obey pure kinetic conditions (i.e., when the kinetic dimensionless parameters are large). If this is not the case, nonlinear partial derivative equations of the type... 

As in the case of continuous linear systems, the exponential holder will then ensure the fulfilment of the regulation conditions for a continuous linear system with a discrete controller. This result is summarized in the following theorem. 

Theorem 3. Assume that for the linear system (1) the following conditions hold ... 

Nonlinearity In addition, it is well known that the process kinetics shows a highly nonlinear behavior. This a serious drawback in instrumentation and automatic control because, in contrast to linear systems where the observability can be established independently of the process inputs, the nonlinear systems must accomplish with the detectability condition depending on the available on-line measurements, including process inputs in the case of non autonomous systems [23]. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

Encouraged by the previous results, more recently, SOAGP has been used as the reference state in PT [13, 14]. The full Hamiltonian is split into two parts, of which one, Hq, can be solved for a state (Eqs. (8) and (9)). The resulting conditions for first order (Eqs. (9)-(12)) describe an easily solved linear system of equations ... 

While the point - area method is very convenient in terms of computational efforts, it has a serious drawback. The matrix of the linear system (5.6B) is inherently ill - conditioned (ref. 25), and the result is very sensitive to the errors in the observations. 

Jacobian becomes increasingly ill-conditioned. With sufficiently great scale disparity, the Jacobian can become effectively singular and the linear system can not be solved numerically. 

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