Minimum singular value

The maximum singular value rr of A and the minimum singular value o of A are defined by... 

A matrix that is JV x iV has JV singular values. We use the symbol a for singular values. The a, that is the biggest in magnitude is called the maximum singular value and the notation is used. The that is the smallest in magnitude is called the minimum singular value and the notation <7 ° is used. 

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... 

B. MATRIX MULTIVARIABLE SYSTEMS. For multivariable systems, the Doyle-Stein criterion for robustness is very similar to the reciprocal plot discussed above. The minimum singular value of the matrix given in Eq. (16.43) is plotted as a function of frequency (a. This gives a measure of the robustness of a closed-... 

Doyle-Stein criterion minimum singular value of Q + (i[Pg.586]

The procedure is summarized in the program given in Table 16.4 which calculates the minimum singular value of the [ + g(7m)] matrix for the Wood and Berry column with the empirical controller settings. Figure 16.10 gives plots of the singular values as a function of frequency. The lowest dip occurs at 0.23 radians per minute and is about — 10.4 dB. 

The minimum singular value is a measure of the invertibility of the system and therefore represents a measure of the potential problems of the system under feedback control. The condition number reflects the sensitivity of the system under uncertainties in process parameters and modelling errors. These parameters provide a qualitative assessment of the dynamic properties of a... 

Control Performance of Thermally Coupled Columns 63 Table 2. Minimum singular value and condition number for each arrangement... 

In the case where the SVD is used for the study of the theoretical control properties, two parameters are of interest the minimum singular value (o,),.the maximum singular value (o ), and its ratio known as condition number (y). The systems with higher minimum singular values and lower condition numbers are expected to show the best dynamic performance under feedback control (Klema and Laub, 1980). Also, it is important to note that a full SVD analysis should cover a wide range of frequencies. 

Table 4. Minimum singular value and eondition number for each structure (Ml).

The minimum singular value of the plant, a(G(ja))) is a useful measure for evaluating the feasibility to achieve acceptable control without input saturation. For scaled variables, we can achieve an output magnitude of at least ct(g) in any output direction, with a manipulated input of unit magnitude. 

The requirement is that ct (G) > 1 and preferably as large as possible. Minimum singular value can be used for the selection of the controlled outputs from a list of candidates paired with given inputs. It is desirable to have as outputs those that correspond to the highest minimum singular value. 

The condition number y of a C matrix is defined as the ratio between the maximum and minimum singular values ... 

Minimum singular value is designated in some works by Morarl Resiliency Index. ... 

Calculate the RGA, Niederlinski index, minimum singular value, and condition number of the following matrix of steady-state gains. 

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

This selection of control structure is independent of variable pairing and controller tuning. The minimum singular value is a measure of the inherent ability of the process (with the specified choice of manipulated variables) to handle disturbances, changes in operating conditions, etc. 

Vinson and Georgakis [7, 8] developed a direct measure, called the Operability Index, which effectively captures the inherent steady-state operability of linear and non-linear continuous processes. Its geometrical interpretation makes it easy to understand and it also addresses multi-variable interactions. The index was demonstrated to more accurately reflect the true operability than other indices, such as minimum singular value, CN, and RGA, on representative linear examples. In addition it has been proven to be independent of inventory control structure [39]... 

Two broad approaches to dynamic operability analysis are the use of so-called open-loop indicators, and the solution of a suitably formulated optimization problem. Characteristics of the former are that they are based on steady-state or linear dynamic models, are relatively easy to compute and seek to provide indications of potential plant-inherent control problems independent of the choice of control system. Examples are the minimum singular value and the plant condition number which reflect sensitivity to input constraints and model uncertainty respectively. More detail may be found in [1] with a good overview of these methods given in [2]. 

A characteristic of A for ill-posed problems is that it has a very large condition number. In other words, the ill-conditioned matrix A is very near to being singular. Briefly, the condition number is defined as k A) = A IIA II or the ratio of maximum to minimum singular values measured in the l2 norm. The ideal problem conditioning occurs for orthogonal matrices which have k (A) 1, while an ill-conditioned... 

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