Linear system identifiability

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

Fig. 1.55.6. Collapse plot in log (TOF) versus UK. The extrapolated intersection of the two linear portions identifies the collapse temperature of the system. (Fig. 10 from [1.126]).

Verotta, D., Concepts, properties, and applications of linear systems to describe distribution, identify input, and control endogenous substances and drugs in biological systems, Crit. Rev. Biomed. Eng., 24, 73-139, 1996. 

Mineralization was represented by quartz-chlorite-carbonate-sulfide veins with visible chalcopyrite, galena, sphalerite, pyrrhotite, cobaltite, limonite and malachite. Fractures filled by veins are identified on aerial photos as one submeridional zone up to 20 m thick and 500 m long. The northeastern linear system represented by a shear zone containing veinlets with arsenopyrite, bismuthine, gold and silver sulfosalts. Twelve veins that differ in extent and intensity of mineralization were discovered, with some veins yielding 3 wt. % Sn, 3 wt. % Cu, and up to 250 g/t Ag. 

Consider now the problem of identifying a linear system in the form of its weighting function h(t), using the relationship (5.66). This problem is called deconvolution. Discrete Fourier transformation offers a standard technique performing numerical deconvolution as mentioned in Section 4.3.3. It... 

The fibres of pr, are linear systems of quadrics containing a line, we can therefore identify EH with P(p (2)), where... 

Here p is the set of characteristic values of the parameters i.e. p(x) = p co(jt) where w(x) has values centered on 1. Often we can set p = / p(x) g(x) dx. The proof is really a statement of what linearity means, for if g(x)djt is the input concentration, g(x)dx.A(p(x)) is the output when the parameter values are p(x). Here x serves merely as an identifying mark, being truly an index variable and the integration in equation (14) follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the out-ports to a unit input at one of the in-ports, the input at all the others being zero. The detail of this case will not be elaborated here, but rather shall we pass to an application. 

Morris et al. [1.126] proposed the use of dielectric analysis (DEA) to predict the collapse temperature of two-component systems. The background of DEA is explained and the >take-off frequency< (TOF) is chosen as the best analytical method to identify the collapse temperature. Figure 1.55.5 shows the dielectric loss factor as a function of the frequency. The frequency at the minimum of this curve is called TOF by the authors. TOF varies with the temperature as shown in Figure 1.55.6. The extrapolated intersection of the two linear portions identifies the collapse temperature. The predicted Tc by TOF for 10% sucrose, 10% trehalose, 10% sorbitol and 11% Azactam solution deviates from observations with a freeze-drying microscope (Table 1 in [1.126]) to slightly lower temperatures, the differences being -3, -1.4, 2.2 and 0.7 °C. 

If the regression expression is a polynomial, then, by applying the method of least squares to identify the coefficients and compute the values of the coefficients, we obtain a simple linear system. If we particularize the case for a regression expression given by a polynomial of second order, the general relation (5.3) is reduced to ... 

If we assume that the function <1> from the general FIj and 112 relationship is a power expression, then the relationship = allj will be obtained. If we apply the logarithm to this relationship, we can identify a and b using a normalized linear system Eq. (5.15). 

Soderstrom, T., Gustavsson, I., and Ljung, L., Identifiability conditions for linear systems operating in closed loop, Int. J. Control 21, 243 (1975). 

Although there is a large literature on identifiability for linear systems with constant coefficients, less has been done on nonlinear systems. Two general properties should be remembered. Whereas for linear systems one can substitute impulsive inputs for the experimental inputs for die analysis of identifiability, one cannot do that for nonlinear systems. One must analyze the input-output experiment for the actual inputs used. That is a drawback. On the other hand, experience shows dial frequently the introduction of nonlinearities makes a formerly nonidentifiable model identifiable for a given input-output experiment. Two methods are available. 

An important finding is that if one has initial estimates of the basic parameters one can determine local identifiability numerically at the initial estimates directly without having to generate the observational parameters as explicit functions of the basic parameters. That is the approach used in the IDENT programs which use the method of least squares (Jacquez and Perry, 19W Perry, 1991). It is important to realize that the method works for linear and nonlinear systems, compartmental or noncompartmental. Furthermore, for linear systems it gives structural local identifiability. 

If the determinant of g g is nonzero, except possibly on a subspace of the parameter space of lower dimensions, the model is locally identifiable if the system is a linear system, it is structurally locally identifiable. 

Cobelli, C., and DiStefano, J. J., Ill (1980). Parameter and structural identifiability concepts and ambiguities A critical review and analysis. Am. J. Physiol. 239, R7-R24. DiStefano, J. J., Ill (1983). Complete parameter bounds and quasiidentifiability conditions for a class of unidentifiable linear systems. Math. Biosci. 65,51-68. 

The representative household determines its composition of total consumption to maximise a given utility function. In the top nest, the consumer system determines the composition of a number of aggregate goods by a Stone-Geaiy linear expenditure system. The expenditure system identifies four... 

The modal frequencies of the building are also identified. They correspond to the equivalent linear system of the building since this structure may exhibit nonlinear behavior under this level of excitation. Figure 3.29 shows the variation of the modal frequencies with the associated plus and minus three standard derivations ( 3a) confidence intervals during the typhoon Kammuri. This interval includes a probability of 99.7% for the equivalent modal frequency falling in this range since the posterior PDF is approximately Gaussian. When the intensity of the excitation... 

Generally, the efforts to identify fully the structure of a system [Eq. (1)] are generously awarded at the solution stage. Special cases and special algorithms are so numerous, however, that we shall first study the algorithms that work for general linear systems [Eq. (1)]. We shall follow the customary pattern of subdividing the methods for... 

Vajda, S. (1979). Comments on structural identifiability in linear time-invariant systems. IEEE Trans, on Automatic Control, AC-24, 495-Vajda, S. (1981). Structural equivalence of linear systems and compartmental models. 

X serves merely as an identifying mark, being truly an index variable, and the integration in Equation 14 follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the outports to a unit input at one of the in-ports, the input at all the others being zero. 

ABSTRACT This contribution discusses some of the lessons learned in computational stochastic dynamics. Among them is the role of the design point for calculating failure probabilities of systems involving a large number of random variables and structural non linearities. Several selected numerical examples are carried out. The results obtained allow to identify the range of applicability of the reliability techniques based on the design point. Finally, the properties of the failure domains for non linear systems are discussed. 

We have thus completed the classification. The group I of columns represents the observable variables, the group II the unobservable ones. In this manner, the canonical format of the extended matrix enables one to identify uniquely the following invariants of the given linear system (independent of the admissible equivalent transformations). 

The neural network approach is an alternative way of solving the problem. Unlike multiple linear or nonlinear regression techniques, which require a predefined empirical form, the neural network can identify and learn the correlative patterns between the input and the corresponding output values once a training set is provided. This approach avoids some of the shortcomings encountered in more traditional correlative methods, and with modem software it can provide useful models in a relatively short time for both linear and non-linear systems. 

---