Steady-state gain matrix

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

Before we design the MIMO system, we need to check the paring of variables. The steady state gain matrix is... 

If the pairing had been reversed, the steady-state gain matrix would be... 

Select manipulated variables. Find the set of manipulated variables that gives the largest minimum singular value of the steady-state gain matrix. 

The entries in this table are the elements in the steady-state gain matrix of the column... 

For systems of higher rank, a sufficient condition for the stabilizability of a decentralized control system is the selection of pairings such that k, > 0, and hence the RGA provides a useful screening tool. The decentralized integral controllability (DIG) conditions (see Morari and Zafiriou, 1989, pp. 359-367) provide additional sufficient conditions for the stability of higher-order systems, which depend only on the steady-state gain matrix, P 0. ... 

The output vector, = [02,04], is recomputed for small positive and negative perturbations of magnitude Am, to each manipulated variable, m one at a time, with the results stored in the vectors ypj andy respectively. Then, column i of the steady-state gain matrix, F 0, is computed Py, 0 = Am (y ij - y ij)IAUj,J = 1,. 3. Note that a factor of Am scales the input variables such that m, s 1. 

The SVD results are similar to the sensitivity results. They suggest that Stage 8 can be controlled by reflux and Stage 29 by heat input. The singular values of the steady-state gain matrix are ai = 0.479 and 02 = 0.166, which gives a condition number CN = 0 102 = 2.88. This indicates that the two temperatures are fairly independent, so a dual-temperature control scheme should be feasible, at least from a steady-state point of view. 

While both the RGA and the RGA number are computable as functions of frequency, the commonly-used procedure involves calculating A based on the steady-state gain matrix of the process, thus taking into consideration only steady-state interactions. 

It is a relatively straightforward task to obtain the steady state gain matrix K for a multi-variable system from process data (e.g., see Ljung [11]), from which the degree of interaction... 

Methods for Obtaining the Steady-State Gain Matrix... 

A 2 X 2 process has the following steady-state gain matrix ... 

The SVA and RGA methods can be used as a way to screen subsets of the possible manipulated variables (MVs) and controlled variables (CVs) for a MIMO control system. Because these analyses are based on the steady-state gain matrix, it is recommended that promising combinations of MVs and CVs be identified and then investigated in more detail using simulation and dynamic analysis. The two steps shown below can be used to identify promising subsets of MVs and CVs, recognizing that for multiloop control the number of MVs should equal the number of CVs (a square system). 

Determine the preferred muitiloop control strategy for a process with the following steady-state gain matrix, which has been scaled by dividing the process variables by their maximum values ... 

These preliminary decisions leave four controlled variables and four manipulated variables that can be analyzed using a 4 X 4 relative gain array. For the core plant model (mass balance equations only) and values of the operating parameters given in Appendix I, the steady-state gain matrix is... 

The steady-state gain matrix can be calculated if one assumes that the steady-state condition is linear around each of the manipulated variables. It is a calculation that shows how each of the manipulated variables contributes to the overall effect on the controlled variables at steady state. It is also referred to as the open-loop gain matrix . 

The steady-state gain matrix G of the process can be derived based on the process model. The steady-state gain matrix is defined as follows ... 

Therefore, if n = 3, the steady-state gain matrix would be derived by first holding m2 and m3 constant while taking the partial derivative of yi with respect to mi to calculate gii, the partial derivative of y2 with mi to calculate g2i, and the partial derivative of ys with nil to calculate gsf, then for m2 one would hold mi and m3 constant while taking the partial derivative of yi with respect to m2 resulting in g2i, and so on. 

If there are no process model equations to be differentiated to obtain gij, then one can use experimental results to calculate the steady-state gain matrix. Refer to experiment 1 of the RGA calculation (Section 9.2.1), where, as in Equation 9.3 ... 

Once the steady-state gain matrix is known it can then be manipulated to generate the RGA as described previously. 

SVD can be applied to the steady-state gain matrix. The gain matrix is first decomposed into the product of three matrices, where two are eigenvectors and one is an eigenvalue [2,3,6] matrix ... 

To calculate the steady-state gain matrix, open-loop gains can be found by differentiating the model with respect to m, while holding mj (j i ) constant. 

---