Functionality matrix

It is important to note that the two surface calculations will be carried out in the diabatic representation. One can get the initial diabatic wave function matrix for the two surface calculations using the above adiabatic initial wave function by the following orthogonal transformation,... 

From the partitioned matrix in equation (9.161), the closed-loop transfer function matrix relating yi and uj is... 

One method used to summarize the required devices and show the function performed by each device is with a function matrix. Figure 14-4 is a completed function matrix chart for the simple process flow diagram shown in Figure 14-5. The function matrix is from RP I4C and is called a SAFE chart. Each component is listed in the left hand column with an identification number and description. Under Device I.D., each of the devices listed in the SAC is listed. If the device is not present, the appropriate SAC reference number is listed. If the SAC rationale requires that another device be present on another component, that device is listed under Alternate Device, if applicable. 

A function matrix can also form the basis for the design of the logic necessary to carry out the functions that are to be performed when a sig-... 

This prescription transforms the effective Hamiltonian to a tridiagonal form and thus leads directly to a continued fraction representation for the configuration averaged Green function matrix element = [G i]at,. This algorithm is usually continued... 

We conclude this section by giving the equation for the alloy Green s function matrix, which is relevant for electronic structure calculations. Elaborating Eq. (23) one finds... 

In addition to the insoluble polymers described above, soluble polymers, such as non-cross-linked PS and PEG have proven useful for synthetic applications. However, since synthesis on soluble supports is more difficult to automate, these polymers are not used as extensively as insoluble beads. Soluble polymers offer most of the advantages of both homogeneous-phase chemistry (lack of diffusion phenomena and easy monitoring) and solid-phase techniques (use of excess reagents and ease of isolation and purification of products). Separation of the functionalized matrix is achieved by either precipitation (solvent or heat), membrane filtration, or size-exclusion chromatography [98,99]. 

The returned vector p is obviously the characteristic polynomial. The matrix ql is really the first column of the transfer function matrix in Eq. (E4-30), denoting the two terms describing the effects of changes in C0 on Ci and Cj Similarly, the second column of the transfer function matrix in (E4-30) is associated with changes in the second input Q, and can be obtained with ... 

Thus, in this problem, the process transfer function matrix Eq. (10-27) can be written in terms of the steady state gain matrix ... 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, 4,ad(, t) to Tatime step to use in Eq. (17) and as the two surface calculations are performed in the diabatic representation, the wave function matrix is back transformed to the adiabatic representation in each time step as... 

Of course, the matrices must have the same order, i.e., they must have the same number of rows and columns. Most of the matrices used in this book will be square (same number of rows and columns), and they wilt usually be the same order as system (the number of inputs and outputs). If there are JV manipulated, variables and N controlled variables, the process is an th order system and the process transfer function matrix will be an x JV square matrix. 

Comparing this with Eq. (15.47), we can see how the transfer function matrix and transfer function vector are related to the A and C matrices and to the D vector. 

Equation (15.64) gives the effects of setpoint and load changes on tbe controlled variables in the closedloop multivariable environment. The matrix (of order N X N) multiplying the vector of setpoints is the closedloop servo transfer function matrix. The matrix (N x 1) multiplying the load disturbance is the closed-loop regulator transfer function vector. 

Exarngde 15.15. Determine the dosedloop characteristic equation for the system whose openloop transfer function matrix was derived in Example 15.14. Use a diagonal controller structure (two SI SO.controllers) that are proportional only. 

A fairly useful stability analysis method is the Niederlinski index. It can be used to eliminate unworkable pairings of variables at an early stage in the design. The settings of the controllers do not have to be known, but it applies only when integral action is used in all loops. It uses only the steadystate gains of the process transfer function matrix. 

The MRl is the minimum singular value of the process openloop transfer function matrix It can be evaluated over a range of frequencies (o or just... 

Wardle and Wood (I. Ghent. E. Symp. Series, 1969, No. 32, p. 1) give the following transfer function matrix for an industrial distillation column ... 

A distillation column has the following transfer function matrix ... 

Derive the openloop plant transfer function matrix relating controlled variables Xi and manipulated variables rtlj. 

C EVALUATE PROCESS TRANSFER FUNCTION MATRIX C GAIN KP(I), DEADTIME D(I), LEAD TAU(1,I,J)... 

In practice, however, this extension is not as straightforward as in DMC. In multivariable DMC, there is a definite design procedure to follow. In multi-variable IMC, there are steps in the design procedure that are not quantitative but involve some art. The problem is in the selection of the invertible part of the process transfer function matrix. Since there are many possible choices, the design procedure becomes cloudy. 

The basic idea in multivariable IMC is the same as in single-loop IMC. The ideal controller would be the inverse of the plant transfer function matrix. This would give perfect control. However, the inverse of the plant transfer function matrix is not physically realizable because of deadtimes, higher-order denominators than numerators, and RHP zeros (which would give an openloop unstable controller). 

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