Linear system analysis

The correlation function corresponds to the memory function, which indicates to which degree values of one function at time t are comparable to values of another function at time t — a before. For statistical signals, the similarity usually decreases rapidly with increasing shift a. For white noise, all values are independent of the others, and the auto-correlation function is proportional to a delta function. The proportionality factor is the second moment (12 of the noise signal. 

If x(f) is white noise with a zero mean value, and yi(t) is the linear system response (4.2.6), then the cross-correlation function is proportional to the memory function k (a) (Fig. 4.3.1), 

Therefore, for measurements with noise excitation, the linear transfer function K (co) (cf. Fig. 4,1.1 (a)) is obtained after cross-correlation of excitation and response and subsequent Fourier transformation of the cross-correlation function Ci (tr) (cf. Fig. 4.1.1 (c)). 

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A modification of the forcing function approach makes use of linear systems analysis for individual tissue compartments [59], Parametric or nonparamet-ric functions are fitted to observed blood drug concentration-time data and are then combined with tissue drug concentration-time measurements deconvolved... 

R.E. Skelton. Dynamic Systems Control. Linear Systems Analysis and Synthesis. John Wiley Sons, New York, 1988. 

J. A. Aseltine, Transform Methods in Linear System Analysis, McGraw-Hill, New York, 1958. 

Chapter 16, detailing linear systems analysis, demonstrates the potential for utilizing the characteristic response and the input function to analyze thoroughly the distribution characteristics of the drug. 

The descriptive pharmacokinetic terms illustrated above only begin to touch the surface of the analysis that can be done. When the rate of change of the response function is evaluated, the derivative of C(t), then other clearance terms, peripheral bioavailabilities, and mean time parameters can be computed. These will be examined in Chapter 16 describing linear systems analysis. In addition, it is possible to use intravenous results and extravascular results to deconvolve the input function from the characteristic response function to assess various mean time parameters that examine the arrival of the drug into the sampling compartment. 

Veng-Pedersen, R, Stochastic interpretation of linear pharmacokinetics a linear system analysis approach, J. Pharm. ScL, 80(7) 621-631, 1991. 

Cutler, D. J., Linear systems analysis in pharmacokinetics, J. Pharmacokinet. Biopharm., 6 265-282, 1978. 

The relative advantages and disadvantages of linear system analysis (LSA) and noncompartmentally based pharmacokinetic (PK) modeling to other modeling... 

It is important to note that we are using the formalism of linear systems analysis that is, Eq. (2) is considered to hold independently of the magnitude of the input perturbation. Electrochemical systems do not, in general, have linear current-voltage characteristics. However, since any continuous, differentiable function can be considered linear for limitingly small input perturbation amplitudes (Taylor expansion), this presents more of a practical problem than a theoretical one. 

Again, in the formalism of linear systems analysis, the transfer function is the mathematical description of the relationship between any two signals. In the special case where the signals of interest are the input (current excitation) and output (voltage response) of a linear electrical system, the transfer function is equivalent to the system impedance. 

Mathematics drives all aspects of chemical engineering. Calculations of material and energy balances are needed to deal with any operation in which chemical reactions are carried out. Kinetics, the study dealing with reaction rates, involves calculus, differential equations, and matrix algebra, which is needed to determine how chemical reactions proceed and what products are made and in what ratios. Control system design additionally requires the understanding of statistics and vector and non-linear system analysis. Computer mathematics including numerical analysis is also needed for control and other applications. 

Virtually every real process is not an ideally linear system. Nevertheless, linear system analysis and linear control has proven to be adequate in many applications. Obviously, there are nonlinear systems that can be described very well by linear models, whereas other nonlinear systems have a behaviour that is very different from the behaviour of any linear system. Recognizing a system as being nonlinear does therefore not suffice, but the extent and severity of a system s inherent nonlinearity is the cracial characteristic in order to decide whether linear system analysis and controller synthesis methods are adequate. 

For the stochastic equation given by (70), using the standard linear system analysis, it follows that... 

In general, the equations of motion are nonlinear. Making special assumptions on the MBS under consideration some multibody formalisms result in partially linear equations. For example the kinematics can be assumed to be linear in some applications of vehicle dynamics If the motion of a railway vehicle on a track is considered, the deviation of the vehicle from the nominal motion defined by the track can be assumed to be so small that one can linearize around that nominal motion. If in addition also the forces are linear, the resulting equations of motion are linear. For a straight line track and linear, time independent force laws the equations are linear with constant coefficients. For details on multibody formalisms which establish linear or partially linear equations, the reader is referred to [Wal89]. In this section we will assume that the equations of motion are given in a nonlinear form and we will show how the linearized equations look like. In subsequent chapters we will refer then to the linear equations when discussing linear system analysis methods. 

Many methods for linear system analysis are based on explicit linear ODEs. Constrained linear systems have therefore to be reduced first to an explicit linear ODE. In Sec. 1.4 such a reduction was obtained for tree structured systems by formulating the system in relative coordinates. For general systems this reduction has to be performed numerically. The reduction to this so-called state space form will be the topic of the first part of this chapter. Then, the exact solution of linear ODEs and DAEs is discussed. 

For more details on linear system analysis in multibody dynamics we refer to [KL94]. 

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