Linearization of nonlinear systems

Chap. 6 Computer Simulation and the Linearization of Nonlinear Systems... 

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). 

When eompiling the material for the book, deeisions had to be made as to what should be ineluded, and what should not. It was deeided to plaee the emphasis on the eontrol of eontinuous and diserete-time linear systems. Treatment of nonlinear systems (other than linearization) has therefore not been ineluded and it is suggested that other works (sueh as Feedbaek Control Systems, Phillips and Harbor (2000)) be eonsulted as neeessary. 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

W.T. Baumaim and W.J. Rugh. Feedback control of nonlinear systems byex-tended linearization. IEEE Trans. Automat. Contr., 31(1) -, 1986. 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

J. E. Harrar and C. L. Pomernacki, Linear and Nonlinear System Characteristics of Controlled-Potential Electrolysis Cells, Anal. Chem. 45 51 (1973). 

This is a positive situation. It is also positive that for the same endpoint, there are, even now, different models. This offers a more robust series of tools to the user, mainly if these tools are based on different assumptions and adopt different techniques, such as fragments of chemical descriptors, linear or nonlinear systems, etc. It is not ideal to search for the best model. It is preferable to have a battery of tools, which, together, are in general more robust toward errors. 

Whether considering linear or nonlinear systems, and regardless of the controller design method employed, the fast component of a composite control system is typically designed to stabilize the fast dynamics (if it is unstable), while the... 

A linear system is referred to as minimum phase if all the zeros of its transfer function lie in the left-hand plane else, the system is non-minimum phase. Naturally, the inverse of the transfer function of a non-minimum phase system will be unstable. In the case of nonlinear systems, the concept of transfer function zeros is replaced by the zero dynamics (Isidori 1995) a nonlinear system is minimum phase if its zero dynamics are stable. 

In the case of nonlinear lattice topology, electron hops mix different spin configurations and the corresponding eigenvalue problem becomes much more complicated. A complete analytical solution of this problem is known only for some special cases (e.g. for the linear chain with periodic boundary conditions [22]). In the absence of exact results a reliable way to describe the properties of nonlinear systems is to perform a numerical study of the electron structure for finite lattice clusters. 

To illustrate how to proceed using the cumulant generating functions, the well-known two-compartment model and the enzymatic reaction will be presented as examples of linear and nonlinear systems, respectively. In these examples, there are two interacting populations (m = 2) and the cumulant generating function is... 

Veng-Pedersen, P., Linear and nonlinear system approaches in pharmacokinetics How much do they offer. I. General considerations, Journal of Pharmacokinetics and Biopharmaceutics, Vol. 16, No. 4, 1988, pp. 413-472. 

The Piecewise Linear Reasoner (PLR) [5] takes parameterized ordinary differential equations and produces maps with the global description of dynamic systems. Despite its present limitations, PLR is a typical example of a new approach that attempts to endow the computer with large amounts of analytical knowledge (dynamics of nonlinear systems, differential topology, asymptotic analysis, etc.) so that it can complement and expand the capabilities of numerical simulations. 

This principle holds as an absolute principle only for the quasi-linear case treated in the preceding, that is, for constant rate coefficients As a local principle it covers also a larger group of nonlinear systems involving concentration- or time-dependent rate parameters. 

Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. 

As we ve seen, in one-dimensional phase spaces the flow is extremely confined— all trajectories are forced to move monotonically or remain constant. In higherdimensional phase spaces, trajectories have much more room to maneuver, and so a wider range of dynamical behavior becomes possible. Rather than attack all this complexity at once, we begin with the simplest class of higher-dimensional systems, namely linear systems in two dimensions. These systems are interesting in their own right, and, as we ll see later, they also play an important role in the classification of fixed points of nonlinear systems. We begin with some definitions and examples. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

The linear model structures discussed in this section can handle mild nonlinearities. They can also result from linearization around an operating point. Simple alternatives can be considered for developing linear models with better predictive capabilities than a traditional ARMAX model for nonlinear processes. If the nature of nonlinearity is known, a transformation of the variable can be utilized to improve the linear model. A typical example is the knowledge of the exponential relationship of temperature in reaction rate expressions. Hence, the log of temperature with the rate constant can be utilized instead of the actual temperature as a regressor. The second method is to build a recursive linear model. By updating model parameters frequently, mild nonlinearities can be accounted for. The rate of change of the process and the severity of the nonlinearities are critical factors for the success of this approach. Another approach is based on the estimation of nonlinear systems by using multiple linear models [11, 82, 83]. 

Linearity. A system is linear when its response to a sum of individual input signals is equal to the sum of the individual responses. This also implies that the system is described by a system of linear differential equations [see e.g., Eqs. (2) and (7)]. Electrochemical systems are usually highly nonlinear and the impedance is obtained by the linearization of equations [see e.g., Eqs. (42) and (130)] for small amplitudes. For linear systems, the response is independent of the amplitude. It is easy to verify the linearity of the system if the impedance obtained is the same when the amplitude of the applied ac signal is halved, then the system is... 

The notion and the characteristics of the Taylor series expansion as well as the linear approximation of nonlinear systems can be found in all the standard texts on calculus. 

The methods differ for linear and nonlinear systems I present them in that order. First, let us make clear the distinction between linear and nonlinear systems and linear and nonlinear parameters. For a linear system, the rates of change of the state variables are given by linear di erential equations. Such systems have the superposition or input linearity property. By that I mean, the response to a sum of two inputs equals the sum of the responses to the individual inputs. In contrast, the rates of change of the state variables of nonlinear systems are given by nonlinear differential equations, and superposition does not hold. 

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. 

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