Linear systems theory

Van Overschee, P. and De Moor, B, Subspace Identification for Linear Systems Theory. Implementation, Applications. Kluwer Academic Publishers Dordrecht, 1996. 

The idea behind the use of Eqs. (4-1) and (4-2) is that we can make use of linear system theories, and complex systems can be analyzed much more effectively. There is no unique way to define the state variables. What we will show is just one of many possibilities. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

This completes our "feel good" examples. It may not be too obvious, but the hint is that linear system theory can help us analysis complex problems. We should recognize that state space representation can do everything in classical control and more, and feel at ease with the language of... 

Thirdly, the binary-tree tabulation algorithm used in ISAT is very different from the grid-based method described above for pre-computed lookup tables. We will look at each of these aspects in detail below. However, we will begin by briefly reviewing a few points from non-linear systems theory that will be needed to understand ISAT. 

C. T. Chen, Introduction to Linear System Theory, Holt, Rinehart and Winston, New York, 1970. 

Coin, M. C. Senturia, S. D. "The Application of Linear System Theory to Parametric Micro sensors", Proc. Third Int l Conf on Solid-State Sensors and Actuators, Philadelphia, June 1985 (in press). 

To know whether or not a system is linear, we may use linear systems theory (LST) [37], LST is a good time-saving theory for testing whether a system is linear. The two basic tests are homogeneity and additivity. 

Heeger, D (1999) Linear systems theory handout, http //www.cim.mcgill.ca/ siddiqi/ linear-systems.pdf. Accessed 19 April 1999... 

T.F. Bogard, Basic Concepts in Linear Systems Theory and Experiments, Wiley, New York, 1984. 

Some definitions from linear system theory are required. The weighting function h(t) is the response (output) of a linear system applied to an impulse signal, theoretically a Dirac-delta function 6(t), or more precisely, a 6-distribution with the properties ... 

In order to prove the theorem, we need some concepts and results from linear system theory. For convenience, we state the results here (see Kailath, 1980, for details). 

Equation 122-9 is a special case of the general convolution result from linear systems theory, based on the fact that E 0) is the impulse response of the flow system ... 

It is assumed that the reader is familiar with the basic concepts of linear systems theory. This material is covered typically in an introductory course on process dynamics and control at the undergraduate level. The subjects in that course that are prerequisite to understanding the concepts in this chapter are... 

Chen, C.-T., 1995. Linear System Theory and Design. Oxford University Press, Inc., Oxford. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

Another related problem that seems worth investigation is, how the present result changes in the case of compartmental systems if one considers, instead of the core defined here, the controllable and observable part as defined in linear system theory, see, for example, Brockett (1970). 

Zadeh, L.A., Desoer, C. A. Linear systems theory The state space approach. In W. Linvill, L. A. Zadeh, and G. Dantzia, editors. Series in Systems Science, page 628. McGraw-Hill, New York, NY, 1963. 

EIS changed the ways electrochemists interpret the electrode-solution interface. With impedance analysis, a complete description of an electrochemical system can be achieved using equivalent circuits as the data contains aU necessary electrochemical information. The technique offers the most powerful analysis on the status of electrodes, monitors, and probes in many different processes that occur during electrochemical experiments, such as adsorption, charge and mass transport, and homogeneous reactions. EIS offers huge experimental efficiency, and the results that can be interpreted in terms of Linear Systems Theory, modeled as equivalent circuits, and checked for discrepancies by the Kramers-Kronig transformations [1]. 

Linear behaviour of processes is desirable because linear system theory is highly developed and there exist many powerful and mature tools for the purpose of process control. When dealing with nonlinear systems, it is crucial to know whether linear modeling, analysis and design tools can be applied. Nonlinearity measures do deliver important insight about the degree of nonlinearity of a system. 

From the above results it is clear that we need to modify the process design in order to remove the non-minimum phase behavior and simultaneously increase the effect of the control input on the product composition. However, since the disturbance sensitivity requires a relatively high bandwidth of the control system, i.e., as > 0.2, it may be relevant to also modify the design with the aim of reducing the disturbance sensitivity at higher frequencies. In order to achieve these goals it is necessary to understand the source of the relevant behaviors, and for this purpose we shall in the next section consider decomposition of models for integrated process systems by means of tools from linear systems theory. 

This is a well known result from linear systems theory. The function Sp is called the sensitivity function as it gives the relative change in the input-output sensitivity due the presence of feedback. The index p is here used to denote that it is the sensitivity function of the process itself For the case with multivariable recycle, we get identical results, except that the sensitivity function in this case is a matrix Sp = I — G22 s)Gr s)). ... 

With the above decomposition we are in a position to determine the part of the dynamics that can be uniquely attributed to individual process units, and the part which is caused by interactions involving several units. Furthermore, the effect of interactions can by the use of linear systems theory easily be related to the properties of the individual units, i.e., to G s) and sr s). 

We stress that design for controllability can either aim at reducing control bandwidth limitations, imposed by fundamental process properties, or at reducing the control requirements imposed by disturbance sensitivities. Based on results from linear systems theory we have presented simple model based tools, based on the decomposed models above, which can be used to improve stability, non-minimum phase behavior and disturbance sensitivities in plants with recycle. One important conclusion of the presented results is that the phase-lag properties of the individual process units play a crucial role for the disturbance sensitivity of an integrated plant. In particular, by a careful design of the recycle loop phase lag, it is possible to tailor the effect of process interactions such that they serve to effectively dampen the effect of disturbances in the most critical frequency region, that is, around the bandwidth of the control system. 

Analyzing the dynamic behavior of a corrosion system requires special techniques, which differ essentially from conventional dc techniques, such as measurements of the open circuit potential, polarization curves, weight loss, or other physicochemical parameters. Based on dynamic system analysis and linear system theory (LST), electrochemical impedance spectroscopy (EIS) is one of the most powerful nonconventional techniques. 

The procedure shown in Fig. 7-1 describes the perturbation of a system by a signal X (0 superimposed to the steady state, which causes the system to respond by a signal of the conjugated variable y(t) (Jiittner et al., 1985). Regardless of the shape of the X (t) perturbation, linear system theory predicts that the dynamic behavior of the system is fully determined by its transient response y(t) in the time domain or by its transfer function H(s) in the frequency domain. In the time domain, the correlation between system perturbation x (t) and response y(t) is given by the convolution of both functions, jc (t)=y (t)xh (r), defined by the integral... 

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