Nonlinear 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 subject of multiplicative fluctuations (in linear and especially nonlinear systems) is still deeply fraught with ambiguity. The authors of Chapter X set up an experiment that simulates the corresponding nonlinear stochastic equations by means of electric circuits. This allows them to shed light on several aspects of external multiplicative fluctuation. The results of Chapter X clearly illustrate the advantages resulting from the introduction of auxiliary variables, as recommended by the reduced model theory. It is shown that external multiplicative fluctuations keep the system in a stationary state distinct from canonical equilibrium, thereby opening new perspectives for the interpretation of phenomena that can be identified as due to the influence of multiplicative fluctuations. 

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. 

Neural network approaches have been used as an alternative to other nonhnear techniques for modeling physiological systems [Chon et al., 1998]. Several neural network control systems have utilized model-based approaches in which the neural network is used to identify a forward nonlinear system... 

It is not easy to identify the sequence of variable groupings that minimizes the number of calls to the nonlinear system. Ciutk, Powell, and Reid (1974) proposed a heuristic algorithm that is often optimal. It is easy to describe We start with the first variable and identify the functions that depend on it Next we check if the second variable does not interfere with the functions with which the first variable interacts. If so, we go on with the third variable. Any new variable introduced into the sequence ako increments the number of functions involved. When no additional variables can be added to the list, thk means that the first group has been identified and we can go on with the next group until all the N variables of the system have been collected. Clearly, the matrix structure of the Jacobian must be known for this procedure to be applied. Thk means that the user must identify the Boolean of the Jacobian, that k, the matrix that contains the dependencies of each function from the system variables (see Figure 2.11). 

The first necessary (but not sufficient) condition to identify the achievement of the solution is based on the maximum value of the residuals at the support points after the solution of the nonlinear system. If such a value is too large (meaning that no solution is found for the nonlinear system), the number of elements is increased and the order of the polynomial is often increased as well. The selection of the new points does not use the solution achieved since it is unreliable. 

On the other hand, when more than one fault can influence the system at the same time, advanced diagnostic methods are used. These methods are based on parameter estimation. Sensitivity bond graph formulation [12] allows real-time parameter estimation and thus it is possible not only to isolate multiple faults but also to quantify the fault severities. Parameter estimation in single fault [2] or multiple fault scenarios [12] are essential steps to be performed before fault accommodation. The parameter estimation scheme also gives the temporal evolution of system parameters. Thus, it is possible to identify and quantify different kinds of fault occurrences. A progressive fault shows gradual drift in estimated parameter values and intermittent fault shows spikes in the estimated parameter values. The advances made in the field of control theory have made it possible to develop state and parameter estimators for various classes of nonlinear systems. Analytical redundancy relations may also be used in optimization loop for parameter estimation because it avoids the need for state estimation. Interested readers may see Ref. [3] for further details and some solved examples. 

The reduction technique works best for linear (such as unimolecular) systems, or weakly coupled nonlinear systems. Coupled nonlinear reactions will cause degeneracies and erroneously identify fast species in the above procedure. In these cases, condition C.l will not be satisfied, even after the described transformation of variables. 

Saccomani, M.P., Audoly, S., and D Angio , L. 2003. Parameter identifiability of nonlinear systems the role of initial conditions. Automatica 39 619 32. 

Chapman, M.J., Godfrey, K.R., Chappell, M.J., and Evans, N.D. 2003. Structural identifiability of nonlinear systems using linear/non-linear sphtting. Int.. Control. 76 209-216. 

Jacquez, J.A. 1996. Compartmental Analysis in Biology and Medicine. 3rd ed., Biomedware, Ann Arbor, MI. Jacquez, J.A. and Simon, C.P. 1993. Qualitative theory of compartmental systems. Siam. Rev., 35 43-718. Joly-Blanchard, G. and Denis-Vidal, L. 1998. Some remarks about identifiability of controlled and uncontrolled nonlinear systems. Automatica 34 1151-1152. 

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

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