Transfer function matrix

OPENLOOP SYSTEM. Let us first consider an openloop process with N controlled variables, N manipulated variables, and one load disturbance. The system can be described in the Laplace domain by N equations that show how all of the manipulated variables and the load disturbance affect each of the controlled variables through their appropriate transfer functions. 

All the variables are in the Laplace domain, as are all of the transfer functions. This set of N equations is very conveniently represented by one matrix equation. 

Qm = N X N matrix of process openloop transfer functions relating the controlled variables and the manipulated variables m = vector of N manipulated variables 

These relationships are shown pictorially in Fig. 15.1. We have chosen to use only one load variable in this development in order to keep things as simple as possible. Clearly, there could be several load disturbances. We would just add additional terms to Eqs. (15.46). Then the L, in Eq. (15.47) becomes a vector and becomes a matrix with N rows and as many columns as there are load disturbances. Since the effects each of the load disturbances can be considered one at a time, we will do it. way to simplify the mathematics. Note that the effects of each of the manipu.ti ted variables can also be considered one at a time if we were looking only at the open loop, system or if we were considering controlling only one variable. However, when we go to a multivariable closedloop system, the effects of all manipulated variables must be considered simultaneously. 

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

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 ... 

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). 

A distillation column has the following openloop transfer function matrix relating controlled variables (x, and Xg) to manipulated variables (reflux ratio RR and... 

From Eq.(23) it is possible to deduce the (2x2) transfer function matrix, G(s), of the linearized MIMO system. 

The application of the SVD technique provides a measure of the controllability properties of a given d mamic system. More than a quantitative measure, SVD should provide a suitable basis for the comparison of the theoretical control properties among the thermally coupled sequences under consideration. To prepare the information needed for such test, each of the product streams of each of the thermally coupled systems was disturbed with a step change in product composition and the corresponding d3mamic responses were obtained. A transfer function matrix relating the product compositions to the intended manipulated variables was then constructed for each case. The transfer function matrix can be subjected to SVD ... 

Known results from control theory give the transfer function matrix for the companion model in terms of the model s matrices, Ji, Jx and C, as ... 

The matrix of the transfer functions is called the transfer function matrix. 

For a system with two inputs and two outputs, such as the one discussed above, we have 2x2 = 4 transfer functions to relate all outputs to all inputs. For a general process with M inputs and N outputs, we will have N x M transfer functions or a transfer function matrix with N rows (number of outputs) and M columns (number of inputs). 

For a process with four inputs (disturbances and manipulated variables) and three measured outputs, how many transfer functions should you formulate, and why What is the corresponding transfer function matrix ... 

For a discrete system with multiple inputs and outputs, we define the discrete transfer function matrix D(z) as follows ... 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Transfer Function Matrix of a Process with Multiple Outputs 163... 

In Example 9.2 we developed the transfer function matrix for a continuous stirred tank reactor. Determine ... 

The disturbance transfer function (matrix), G/s), can be defined in a similar manner. Then, the output of the liniar system... 

Again, poles and zeros are important for evaluating stability and controllability properties of the physical system. To find the poles of an open-loop MIMO system one can use the transfer function matrix or the state-space description. They are related by ... 

The poles of the transfer function and the eigenvalues of the equivalent system matrix in the state-space form are the same. The knowledge of poles is linked with stability. A multivariable system is stable if all the poles of the transfer function matrix lie in the left-half (LHP) plane otherwise it is unstable. 

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